Macroscopic Reality from Quantum Complexity

Abstract

Beginning with the Everett–DeWitt many-worlds interpretation of quantum mechanics, there have been a series of proposals for how the state vector of a quantum system might split at any instant into orthogonal branches, each of which exhibits approximately classical behavior. Here we propose a decomposition of a state vector into branches by finding the minimum of a measure of the mean squared quantum complexity of the branches in the branch decomposition. In a non-relativistic formulation of this proposal, branching occurs repeatedly over time, with each branch splitting successively into further sub-branches among which the branch followed by the real world is chosen randomly according to the Born rule. In a Lorentz covariant version, the real world is a single random draw from the set of branches at asymptotically late time, restored to finite time by sequentially retracing the set of branching events implied by the late time choice. The complexity measure depends on a parameter b with units of volume which sets the boundary between quantum and classical behavior. The value of b is, in principle, accessible to experiment.

Appendices

Appendix 1: Truncated Hermitian Operator Hilbert Space

Let ${\mathcal {H}}_x^n$ be the subspace of ${\mathcal {H}}_x$ with less than n bosons. The dimension $d_n$ of each ${\mathcal {H}}_x^n$ is finite. Let ${\mathcal {H}}^n$ be the product over x of all ${\mathcal {H}}^n_x$

$$\begin{aligned} {\mathcal {H}}^n = \bigotimes _x {\mathcal {H}}_x^n. \end{aligned}$$

(A1)

For any site x, let ${\mathcal {F}}^n_x$ consist of all Hermitian $f_x$ on ${\mathcal {H}}^n_x$ with finite

$$\begin{aligned} \parallel f_x \parallel ^ 2 = \mathrm {Tr}_x( f_x)^2, \end{aligned}$$

(A2)

and vanishing trace

$$\begin{aligned} \mathrm {Tr}_x f_x = 0. \end{aligned}$$

(A3)

For any pair of nearest neighbor sites $\{x, y\}$, let ${\mathcal {F}}^n_{xy}$ consist of all Hermitian $f_{xy}$ on ${\mathcal {H}}^n_x \otimes {\mathcal {H}}^n_y$ with finite

$$\begin{aligned} \parallel f_{xy} \parallel ^ 2 = \mathrm {Tr}_{xy}( f_{xy})^2, \end{aligned}$$

(A4)

and vanishing traces

$$\begin{aligned} \mathrm {Tr}_x f_{xy}= & {} 0, \end{aligned}$$

(A5a)

$$\begin{aligned} \mathrm {Tr}_y f_{xy}= & {} 0. \end{aligned}$$

(A5b)

Inner products on ${\mathcal {F}}^n_x$ and ${\mathcal {F}}^n_{xy}$ are

$$\begin{aligned} \langle f_x, f'_x \rangle= & {} \mathrm {Tr}_x( f_x f'_x), \end{aligned}$$

(A6a)

$$\begin{aligned} \langle f_{xy}, f'_{xy} \rangle= & {} \mathrm {Tr}_{xy}( f_{xy} f'_{xy}). \end{aligned}$$

(A6b)

Operators $f_x \in {\mathcal {F}}^n_x$ and $f_{xy} \in {\mathcal {F}}^n_{xy}$ can be made into operators on ${\mathcal {H}}^n$ by

$$\begin{aligned} {\hat{f}}_x= & {} f_x \bigotimes _{q \ne x} I_q, \end{aligned}$$

(A7a)

$$\begin{aligned} {\hat{f}}_{xy}= & {} f_{xy} \bigotimes _{q \ne x,y} I_q, \end{aligned}$$

(A7b)

where $I_q$ is the identity operator on ${\mathcal {H}}^n_q$. As usual, we now drop the hat and use the same symbol for operators on ${\mathcal {H}}^n_x$, ${\mathcal {H}}^n_x \otimes {\mathcal {H}}^n_y$, and the corresponding operators on ${\mathcal {H}}^n$.

Let $K^n$ be the vector space over the reals of linear operators k on ${\mathcal {H}}^n$ given by sums of the form

$$\begin{aligned} k = \sum _{x y} f_{x y} + \frac{1}{\sqrt{d_n}} \sum _x f_x \end{aligned}$$

(A8)

for any collection of $f_{x y} \in {\mathcal {F}}^n_{x y}$ for a set of nearest neighbor pairs $\{x, y\}$ and any collection of $f_x \in {\mathcal {F}}^n_x$ in a set of sites x. The inner product on K is

$$\begin{aligned} \langle k, k' \rangle = \sum _{xy} \langle f_{xy}, f'_{xy} \rangle + \sum _x \langle f_x, f'_x \rangle . \end{aligned}$$

(A9)

An equivalent inner product on $K^n$, which is a version of the inner product on operator Hilbert space in [10], is

$$\begin{aligned} \langle k, k' \rangle = \frac{ \mathrm {Tr}( k k')}{ d_n^{n_L - 2}}, \end{aligned}$$

(A10)

where $\mathrm {Tr}$ is the trace on all of ${\mathcal {H}}^n$ and $n_L$ is the number of sites in the lattice L. As a result of the factor of $\frac{1}{\sqrt{d_n}}$ in Eq. (A8), if $d_n$ is made large, matrix elements of k given by Eq. (A8) will approach those of k given by Eq. (15) and $K^n$ will become equivalent to the operator space K of Sect. 3.2.

Appendix 2: Lower Bound on the Complexity of Entangled States

The proof of Eq. (37) proceeds as follows. The trajectories $k(\nu ) \in K$ and $U_k(\nu )$ which determine any $C( |\psi \rangle , |\omega \rangle )$, according to Eqs. (17a)–(20), we characterize by a corresponding set of trajectories of Schmidt spectrum vectors. We then find the rotation matrices which govern the motion of these vectors as $\nu$ changes. A bound on the time integral of the angles which occur in these matrices by a time integral of $\parallel k(\nu ) \parallel$ yields Eq. (37).

1.1 2.1 Schmidt Spectra

Consider some entangled n-fermion $|\psi \rangle$ of form Eq. (36). For a trajectory $k(\nu ) \in K$, let $U_k(\nu )$ be the solution to Eqs. (17a) and (17b). Define $|\omega (\nu ) \rangle$ to be

$$\begin{aligned} |\omega ( \nu ) \rangle = U_k(\nu )|\omega \rangle , \end{aligned}$$

(B1)

for some product state $|\omega \rangle$ and assume that $k(\nu )$ has been chosen to give

$$\begin{aligned} |\omega (1) \rangle = \xi |\psi \rangle , \end{aligned}$$

(B2)

for a phase factor $\xi$. Since all $k(\nu )$ conserve fermion number, $|\omega \rangle$ according to Eq. (9) must have the form

$$\begin{aligned}&|\omega \rangle = d_f^\dagger ( p_{n-1}) \ldots d_f^\dagger ( p_0) \times d_b^\dagger ( q_{m-1}) \ldots d_b^\dagger ( q_0) |\Omega \rangle , \end{aligned}$$

(B3)

for some number of bosons m.

We now divide the lattice L into a collection of disjoint regions and define a corresponding collection of Schmidt decompositions of the trajectory of states which determine any $C( |\psi \rangle , |\omega \rangle )$. Divide L into subsets $L^e, L^o$, with, respectively, even or odd values of the sums of components ${\hat{x}}_i$. The sites in each subset have nearest neighbors only in the other. Let $D^e_{ij}, D^o_{ij}, D^e, D^o$ be

$$\begin{aligned} D^e_{ij}= & {} L^e \cap D_{ij} \end{aligned}$$

(B4a)

$$\begin{aligned} D^o_{ij}= & {} L^o \cap D_{ij} \end{aligned}$$

(B4b)

$$\begin{aligned} D^e= & {} \cup _{ij} D^e_{ij}. \end{aligned}$$

(B4c)

$$\begin{aligned} D^o= & {} \cup _{ij} D^o_{ij}. \end{aligned}$$

(B4d)

Between $D^e$ and $D^o$ choose the larger, or either if they are equal. Assume the set chosen is $D^e$. Among the nm spins $s_{ij}$, at least $\frac{nm}{2}$ will have the same value and therefore correspond to $D_{ij}$ which do not intersect. The corresponding collection of $D^e_{ij}$ will then include at least $\frac{nmV}{4}$ points.

From this set of $D^e_{ij}$ construct a set of subsets $E_\ell$ each consisting of 2n distinct points chosen from 2n distinct $D^e_{ij}$. The total number of $E_\ell$ will then be at least $\frac{m V}{8}$. We will consider only the first $\frac{m V}{8}$ of these.

The Hilbert space ${\mathcal {H}}$ is given by a tensor product

$$\begin{aligned} {\mathcal {H}} = {\mathcal {H}}^f \otimes {\mathcal {H}}^b, \end{aligned}$$

(B5)

of a fermion space ${\mathcal {H}}^f$ and a boson space ${\mathcal {H}}^b$. Similarly the space ${\mathcal {H}}_x$ at each x is given by a tensor product

$$\begin{aligned} {\mathcal {H}}_x = {\mathcal {H}}_x^f \otimes {\mathcal {H}}_x^b, \end{aligned}$$

(B6)

of a fermion space ${\mathcal {H}}_x^f$ and a boson space ${\mathcal {H}}_x^b$. The dimensions of ${\mathcal {H}}_x^f$ and ${\mathcal {H}}_x^b$ are, respectively, 4 and $\infty$.

For each set $E_\ell$ form the tensor product spaces

$$\begin{aligned} {\mathcal {Q}}_\ell= & {} \bigotimes _{x \in E_\ell } {\mathcal {H}}_x^f, \end{aligned}$$

(B7a)

$$\begin{aligned} {\mathcal {R}}_\ell= & {} {\mathcal {H}}^b\bigotimes _{q \ne E_\ell } {\mathcal {H}}_q^f. \end{aligned}$$

(B7b)

It follows that ${\mathcal {Q}}_\ell$ has dimension $4^{2n}$ and

$$\begin{aligned} {\mathcal {H}} = {\mathcal {Q}}_\ell \otimes {\mathcal {R}}_\ell . \end{aligned}$$

(B8)

A Schmidt decomposition of $|\omega (\nu ) \rangle$ according to Eq. (B8) then becomes

$$\begin{aligned} |\omega (\nu ) \rangle = \sum _j \lambda _{j\ell }(\nu ) |\phi _{j\ell }(\nu ) \rangle |\chi _{j\ell }(\nu ) \rangle , \end{aligned}$$

(B9)

where

$$\begin{aligned} |\phi _{j\ell }(\nu ) \rangle\in & {} {\mathcal {Q}}_\ell \end{aligned}$$

(B10a)

$$\begin{aligned} |\chi _{j\ell }( \nu ) \rangle\in & {} {\mathcal {R}}_\ell , \end{aligned}$$

(B10b)

for $0 \le j < 4^{2n}$ and real non-negative $\lambda _{j\ell }( \nu )$ which fulfill the normalization condition

$$\begin{aligned} \sum _j [ \lambda _{j\ell }( \nu )]^2 = 1. \end{aligned}$$

(B11)

Each $|\phi _{j\ell }(\nu ) \rangle$ is a pure fermion state while the $|\chi _{j\ell }(\nu ) \rangle$ can include both fermions and bosons.

The fermion number operators $N[{\mathcal {Q}}_\ell ]$ and $N[{\mathcal {R}}_\ell ]$ commute and $|\omega (\nu ) \rangle$ is an eigenvector of the sum with eigenvalue n. It follows that the decomposition of Eq. (B9) can be done with $|\phi _{j\ell }( \nu ) \rangle$ and $|\chi _{j\ell }(\nu ) \rangle$ eigenvectors of $N[{\mathcal {Q}}_\ell ]$ and $N[{\mathcal {R}}_\ell ]$, respectively, with eigenvalues summing to n. Let $|\phi _{0\ell } \rangle$ be $|\Omega _\ell \rangle$, the vacuum state of ${\mathcal {Q}}_\ell$, and let $|\phi _{i\ell } (\nu ) \rangle , 1 \le i \le 4n$, span the 4n-dimensional subspace of ${\mathcal {Q}}_\ell$ with $N[{\mathcal {Q}}_\ell ]$ of 1. We assume the corresponding $\lambda _{i\ell }( \nu ), 1 \le i \le 4n$, are in nonincreasing order. Consider Eq. (B9) for $\nu = 1$. By Eq. (B2), for any choice of $\ell$ there will be a set of 2n nonzero orthogonal $|\phi _{1\ell }( 1) \rangle , \ldots |\phi _{2n\ell }( 1) \rangle$ with

$$\begin{aligned} \lambda _{j\ell }( 1) = \sqrt{\frac{1}{mV}}, \end{aligned}$$

(B12)

for $1 \le j \le 2n$.

On the other hand, for $\nu = 0$, Eq. (B9) becomes a decomposition of the product state $|\omega \rangle$. The boson part of $|\omega (0) \rangle$ will occur as the same overall tensor factor in each $|\chi _{1\ell }(0) \rangle ,\ldots |\chi _{n\ell }(0) \rangle$. The fermion part of $|\omega (0) \rangle$ is a product of n independent single fermion states, the space spanned by the projection of these into some ${\mathcal {Q}}_\ell$ is at most n dimensional, and as a result at most n orthogonal $|\phi _{1\ell }(0) \rangle ,\ldots |\phi _{n\ell }(0) \rangle$ can occur. Therefore at $\nu = 0$, there will be at most n nonzero $\lambda _{1\ell }(0), \ldots \lambda _{n\ell }(0)$. For $n < j \le 2n$, we have

$$\begin{aligned} \lambda _{j\ell }( 0) = 0. \end{aligned}$$

(B13)

But according to Eq. (B11), for each fixed value of $\ell$ the set of components $\{\lambda _{j\ell }( \nu )\}$ indexed by j is a unit vector. Eqs. (B13) and (B12) then imply that as $\nu$ goes from 0 to 1, $\{\lambda _{j\ell }( \nu )\}$ must rotate through a total angle of at least $\arcsin (\sqrt{\frac{n}{mV}})$.

For the small interval from $\nu$ to $\nu + \delta \nu$ let $\mu _{j\ell }(\nu )$ and $\theta _{\ell }(\nu )$ be

$$\begin{aligned} \lambda _{j\ell }(\nu + \delta \nu )= & {} \lambda _{j\ell }( \nu ) + \delta \nu \mu _{j\ell }(\nu ), \end{aligned}$$

(B14a)

$$\begin{aligned} \theta _{\ell }( \nu )^2= & {} \sum _j [ \mu _{j\ell }(\nu )]^2. \end{aligned}$$

(B14b)

We then have

$$\begin{aligned} \int _0^1 | \theta _{\ell }(\nu )| d \nu \ge \arcsin \left( \sqrt{\frac{n}{mV}} \right) . \end{aligned}$$

(B15)

Summed over the $\frac{mV}{8}$ values of $\ell$, Eq. (B15) becomes

$$\begin{aligned} \sum _{\ell } \int _0^1 | \theta _{\ell }(\nu )| d \nu \ge \frac{ m V}{8} \arcsin \left( \sqrt{\frac{n}{mV}} \right) , \end{aligned}$$

(B16)

and therefore

$$\begin{aligned} \sum _{\ell } \int _0^1 | \theta _{\ell }(\nu )| d \nu \ge \frac{1}{4\pi } \sqrt{mnV}. \end{aligned}$$

(B17)

1.2 2.2 More Schmidt Spectra

Replacing the subsets $E_\ell$ defined in Appendix B1, with subsets of L obtained from the $S_\ell$ of Sect. 4 leads to an additional bound similar to Eq. (B17).

For each $0 \le \ell < q$, of the two subsets of L defined by $S_\ell$, let $T_\ell$ be the subset which, for each $0 \le i < m$, holds $n_0$ of the sets $D_{ij}, 0 \le j < n$. Redefine ${\mathcal {Q}}_\ell , {\mathcal {R}}_\ell$ of Eqs. (B7a) and (B7b), to be

$$\begin{aligned} {\mathcal {Q}}^T_\ell= & {} \bigotimes _{x \in T_\ell } {\mathcal {H}}_x^f, \end{aligned}$$

(B18a)

$$\begin{aligned} {\mathcal {R}}^T_\ell= & {} {\mathcal {H}}^b\bigotimes _{q \ne T_\ell } {\mathcal {H}}_q^f. \end{aligned}$$

(B18b)

For each $0 \le \ell < q$ there is again a corresponding Schmidt decomposition of $|\omega (\nu ) \rangle$ of Eqs. (B1) and (B2)

$$\begin{aligned} |\omega (\nu ) \rangle = \sum _j \lambda ^T_{j\ell }(\nu ) |\phi ^T_{j\ell }(\nu ) \rangle |\chi ^T_{j\ell }(\nu ) \rangle , \end{aligned}$$

(B19)

where

$$\begin{aligned} |\phi ^T_{j\ell }(\nu ) \rangle\in & {} {\mathcal {Q}}^T_\ell , \end{aligned}$$

(B20a)

$$\begin{aligned} |\chi ^T_{j\ell }( \nu ) \rangle\in & {} {\mathcal {R}}^T_\ell . \end{aligned}$$

(B20b)

Each $|\phi ^T_{j\ell }(\nu ) \rangle$ is a pure fermion state while the $|\chi ^T_{j\ell }(\nu ) \rangle$ can include both fermions and bosons. For $\nu = 1$, for every $0 \le \ell < q$, the sum over j in Eq. (B19) has m nonzero entries each with

$$\begin{aligned} \lambda ^T_{j \ell }(1) = \frac{1}{\sqrt{m}}, \end{aligned}$$

(B21)

with $|\phi ^T_{j\ell }(1) \rangle$ carrying fermion number $n_0$ and $|\chi ^T_{j\ell }(1) \rangle$ carrying fermion number $n_1$.

Duplicating the discussion of Appendix B1, a trajectory of angles $\theta ^T_\ell (\nu )$ can be defined which rotates the unit vector $[ \lambda ^T_{j\ell }(0) ]$ arising from the product state $|\omega (0) \rangle$ into the unit vector $[\lambda ^T_{j\ell }(1) ]$ of Eq. (B21). For each $0 \le \ell < q$, a version of the lower bound of Eq. (B15) can be obtained by finding the product state $|\omega (0) \rangle$ which gives $[ \lambda ^T_{j\ell }(0) ]$ closest to $[ \lambda ^T_{j\ell }(1) ]$ for the set of $0 \le j < m$ corresponding to $|\phi ^T_{j\ell }(0) \rangle$ and $|\chi ^T_{j\ell }(0) \rangle$ with fermion numbers $n_0$ and $n_1$, respectively.

According to Eq. (9), the product state $|\omega (0) \rangle$ includes n fermion creation operators $d^\dagger _f( p_i)$ given by Eq. (8a). Since $|\omega (1) \rangle$ and therefore $|\omega (0) \rangle$ are normalized to 1, we can require the $p_i( x, s)$ to be orthonormal. The simplest way to insure $n_0$ and $n_1$, respectively, for $|\phi _{0\ell }(0) \rangle$ and $|\chi _{0\ell }(0) \rangle$ is for the support of $p_i( x, s)$ to be entirely within $T_\ell$ for $0 \le i < n_0$ and entirely outside $T_\ell$ for $n_0 \le i < n$. The Schmidt decomposition of Eq. (B19) then yields a vector $[ \lambda ^T_{j\ell }(0) ]$ with only a single nonzero entry and therefore

$$\begin{aligned} \sum _j \lambda ^T_{j \ell }(0) \lambda ^T_{j \ell }(1) = \frac{1}{\sqrt{m}}. \end{aligned}$$

(B22)

A larger value of the sum in Eq. (B22) is possible only if an even number of $p_i(x,s)$ have support both within $T_\ell$ and outside $T_\ell$. For some $r \le n_0, n_1,$, define z to be the set

$$\begin{aligned} z = \{ i | 0 \le i< r \} \cup \{ i | n_0 \le i < n_0 + r \}. \end{aligned}$$

(B23)

Then for $i \in z$, suppose

$$\begin{aligned} p_i( x, s) = p^0_i( x, s) + p^1_i(x, s), \end{aligned}$$

(B24)

where the $p^0_i( x, s)$ have support entirely within $T_\ell$ and the $p^1_i( x, s)$ have support entirely outside $T_\ell$. Since $p_i(x,s)$ is normalized and the support of $p^0_i(x,s)$ is disjoint from the support of $p^1_i(x,s)$ we have

$$\begin{aligned} \parallel | p^0_i \rangle \parallel ^2 + \parallel | p^1_i \rangle \parallel ^2 = 1. \end{aligned}$$

(B25)

The piece $|{\hat{\omega }}( 0) \rangle$ of $|\omega ( 0) \rangle$ with fermion number $n_0$ on $T_\ell$ and $n_1$ outside $T_\ell$ is given by

$$\begin{aligned}&|{\hat{\omega }}(0) \rangle \nonumber \\&\quad =\sum _u [\bigotimes _{i \in u} |p^0_i \rangle \bigotimes _{ j \in z - u } |p^1_j \rangle ]\bigotimes _{i \notin z} |p_i \rangle , \end{aligned}$$

(B26)

where the sum is over all r element subsets $u \subset z$.

The vector $[ \lambda ^T_{j\ell }(0) ]$ corresponding to $|{\hat{\omega }}(0) \rangle$ will have at most $\frac{(2r)!}{(r!)^2}$ nonzero entries, one for each of the sets u in the sum in Eq. (B26). The Cauchy-Schwartz inequality then yields

$$\begin{aligned} \sum _j \lambda ^T_{j \ell }(0) \lambda ^T_{j \ell }(1) \le \sqrt{ \frac{ (2r)!}{m (r!)^2}} \parallel |{\hat{\omega }}(0) \rangle \parallel \end{aligned}$$

(B27)

Let $|{\hat{p}}^0_i \rangle$ be the projection of $|p^0_i \rangle$ orthogonal to all $|{\hat{p}}^0_j \rangle , j < i$, and let $|{\hat{p}}^1_i \rangle$ be the projection of $|p^1_i \rangle$ orthogonal to all $|{\hat{p}}^1_j \rangle , j < i$. Substituting $\{ |{\hat{p}}^0_i \rangle \}$ and $\{ |{\hat{p}}^1_i \rangle \}$ for $\{|p^0_i \rangle \}$ and $\{ |p^1_i \rangle \}$, respectively, in Eq. (B26) leaves $|{\hat{\omega }}(0) \rangle$ unchanged. The value of $\parallel |{\hat{\omega }}(0) \rangle \parallel$ will then be maximized if the resulting $\parallel | {\hat{p}}^0_i \rangle \parallel$ and $\parallel | {\hat{p}}^1_i \rangle \parallel$ are increased as needed to satisfy Eq. (B25).

Suppose now that $\parallel |{\hat{\omega }}(0) \rangle \parallel$ has been maximized with respect to $\parallel | {\hat{p}}^0_i \rangle \parallel$ and $\parallel | {\hat{p}}^1_i \rangle \parallel$ for all $0 \le i < r$, expect some pair of values j, k. The remaining dependence on $\parallel |{\hat{p}}^0_i \rangle \parallel$ and $\parallel |{\hat{p}}^1_i \parallel$ for $i = j, k,$ is maximized at

$$\begin{aligned} \parallel |{\hat{p}}^0_j \rangle \parallel= & {} \parallel |{\hat{p}}^0_k \rangle \parallel , \end{aligned}$$

(B28a)

$$\begin{aligned} \parallel |{\hat{p}}^1_j \rangle \parallel= & {} \parallel | {\hat{p}}^1_k \rangle \parallel . \end{aligned}$$

(B28b)

If $\parallel |{\hat{\omega }}(0) \rangle \parallel$ is then maximized with respect to the remaining i independent $\parallel |{\hat{p}}^0_i \rangle \parallel$ and $\parallel | {\hat{p}}^1_i \rangle \parallel$, Eq. (B27) becomes

$$\begin{aligned} \sum _j \lambda ^T_{j \ell }(0) \lambda ^T_{j \ell }(1) \le \sqrt{ \frac{ (2r)!}{m 2^r (r!)^2}} \end{aligned}$$

(B29)

Suppose m has the form $\frac{(2r)!}{(r!)^2}$. For any $r' \le r$ Eq. (B29) becomes

$$\begin{aligned} \sum _j \lambda ^T_{j \ell }(0) \lambda ^T_{j \ell }(1) \le \frac{ (2 r')!}{\sqrt{m}2^{r'} (r'!)^2}. \end{aligned}$$

(B30)

An induction argument then shows that Eq. (B30) is an increasing function of $r'$. For $r' > r$ on the other hand, the sum in Eq. (B30) becomes

$$\begin{aligned} \sum _j \lambda ^T_{j \ell }(0) \lambda ^T_{j \ell }(1) \le \frac{ \sqrt{m}}{ 2^{r'}}, \end{aligned}$$

(B31)

which is a decreasing function of $r'$. The maximum of Eq. (B30) will therefore be at $r' = r$.

Now suppose m lies between $\frac{(2r)!}{(r!)^2}$ and $\frac{(2r + 2)!}{[(r+1)!]^2}$. For $r' \le r$ the maximum Eq. (B30) will still be at $r' = r$ and given by

$$\begin{aligned} \sum _j \lambda ^T_{j \ell }(0) \lambda ^T_{j \ell }(1)\le & {} \frac{ (2 r)!}{\sqrt{m}2^{r} (r!)^2}, \end{aligned}$$

(B32a)

$$\begin{aligned}< & {} \sqrt{ \frac{ (2 r)!}{2^{2r} (r!)^2}}. \end{aligned}$$

(B32b)

For $r' = r + 1$, Eq. (B30) becomes

$$\begin{aligned} \sum _j \lambda ^T_{j \ell }(0) \lambda ^T_{j \ell }(1)\le & {} \frac{ \sqrt{m}}{ 2^{r+1}}, \end{aligned}$$

(B33a)

$$\begin{aligned}< & {} \sqrt{\frac{ (2 r+2)!}{2^{2r+2} [(r+1)!]^2}}. \end{aligned}$$

(B33b)

A further induction argument shows that Eqs. (B32b) and (B33b) are decreasing functions of r. Thus for $m \ge 2$, we have

$$\begin{aligned} \sum _j \lambda ^T_{j \ell }(0) \lambda ^T_{j \ell }(1) \le \frac{1}{\sqrt{2}}. \end{aligned}$$

(B34)

A duplicate of the argument leading to Eq. (B17) then yields

$$\begin{aligned} \sum _{\ell } \int _0^1 | \theta ^T_{\ell }(\nu )| d \nu \ge \frac{\pi q}{4}. \end{aligned}$$

(B35)

1.3 2.3 Schmidt Rotation Matrix

A lower bound on $C(|\psi \rangle )$ follows from Eqs. (B17) and (B35). Appendices 2.3 and 2.4 derive the consequence of Eq. (B17). A derivation of the additional terms in the bound on $C(|\psi \rangle )$ which follow from Eq. (B35) is briefly summarized in Appendix 2.5.

The rotation of $\lambda _{j\ell } (\nu )$ during the interval from $\nu$ to $\nu + \delta \nu$ will be determined by $k(\nu )$. For each $f_{xy}$ in Eq. (15) for $k(\nu )$ which can contribute to a nonzero value of $\theta _{\ell }(\nu )$, the nearest neighbor pair $\{x, y\}$ has one point, say x in $E_{\ell }$. Since $E_{\ell } \subset D^e$ and the nearest neighbors of all points in $D^e$ are in $D^o$, y can not be in $E_\ell$. Let $g_{\ell }(\nu )$ be the sum of all such $f_{xy}$. The effect of all other terms in Eq. (15) on the Schmidt decomposition of Eq. (B9) will be a unitary transformation on ${\mathcal {R}}_\ell$ and identity on ${\mathcal {Q}}_\ell$. All other terms will therefore leave $\lambda _{j\ell }( \nu )$ unchanged.

The effect of $g_{\ell }(\nu )$ on $\lambda _{j\ell }(\nu )$ over the interval from $\nu$ to $\nu + \delta \nu$ can be determined from the simplification

$$\begin{aligned} |\omega (\nu + \delta \nu ) \rangle = \exp [ i \delta \nu g_{\ell }(\nu )] |\omega (\nu ) \rangle . \end{aligned}$$

(B36)

From $|\omega (\nu + \delta \nu ) \rangle \langle \omega (\nu + \delta \nu )|$ of Eq. (B36), construct the density matrix $\rho (\nu + \delta \nu )$ by a partial trace over ${\mathcal {R}}_{\ell }$, using the basis for ${\mathcal {R}}_{\ell }$ from the Schmidt decomposition in Eq. (B9) of $|\omega (\nu ) \rangle$ at t

$$\begin{aligned}&\rho (\nu + \delta \nu ) = \sum _j [ \langle \chi _{j\ell }(\nu )|\omega (\nu + \delta \nu ) \rangle \times \langle \omega (\nu + \delta \nu )|\chi _{j\ell }(\nu ) \rangle ]. \end{aligned}$$

(B37)

An eigenvector decomposition of $\rho (\nu + \delta \nu )$ exposes the $\lambda _{j\ell }(\nu + \delta \nu )$

$$\begin{aligned}&\rho (\nu + \delta \nu ) = \sum _j [\lambda _{j\ell }( \nu + \delta \nu )^2 \times |\phi _{j\ell }(\nu + \delta \nu ) \rangle \langle \phi _{j\ell }( \nu + \delta \nu )|]. \end{aligned}$$

(B38)

A power series expansion through first order in $\delta \nu$ applied to Eqs. (B36), (B37) and (B38) then gives for $\mu _{j\ell }(\nu )$ of Eq. (B14a)

$$\begin{aligned} \mu _{j\ell }(\nu ) = \sum _k r_{jk\ell }(\nu ) \lambda _{k\ell }(\nu ), \end{aligned}$$

(B39)

for the rotation matrix $r_{jk\ell }(\nu )$

$$\begin{aligned}&r_{jk\ell }(\nu ) = -{\text {Im}}[ \langle \phi _{j\ell }(\nu )| \langle \chi _{j\ell }(\nu )| g_{\ell }(\nu )|\phi _{k\ell }(\nu ) \rangle |\chi _{k\ell }(\nu ) \rangle ]. \end{aligned}$$

(B40)

1.4 2.4 Rotation Angle Bounds

Since the $f_{xy}$ contributing to $g_\ell (\nu )$ conserve total fermion number N, $g_\ell (\nu )$ can be expanded as

$$\begin{aligned} g_{\ell }(\nu )= & {} \sum _{xy} g_{\ell }( x, y, \nu ), \end{aligned}$$

(B41a)

$$\begin{aligned} g_{\ell }(x,y,\nu )= & {} \sum _{i = 0,1} a^i(x, y, \nu ) z^i(x, y, \nu ) \end{aligned}$$

(B41b)

where $z^0( x, y, \nu )$ acts only on states with $N( {\mathcal {H}}_x \otimes {\mathcal {H}}_y)$ of 0, $z^1( x, y, \nu )$ acts only on states with $N( {\mathcal {H}}_x \otimes {\mathcal {H}}_y)$ strictly greater than 0, and the $z^i(x,y,\nu )$ are normalized by

$$\begin{aligned} \parallel z^i(x, y, \nu ) \parallel = 1. \end{aligned}$$

(B42)

The operator $z^0(x, y, \nu )$ will be

$$\begin{aligned} z^0(x,y,\nu )= & {} z^{0f}(x,y) \otimes g^b(x,y,\nu ), \end{aligned}$$

(B43a)

$$\begin{aligned} z^{0f}(x,y,\nu )= & {} P^f(x,y) \bigotimes _{q \ne x,y} I_q, \end{aligned}$$

(B43b)

where $P^f(x,y)$ projects onto the vacuum state of ${\mathcal {H}}^f_x \otimes {\mathcal {H}}^f_y$ and $g^b(x,y,\nu )$ is a normalized Hermitian operator acting on ${\mathcal {H}}^b_x \otimes {\mathcal {H}}^b_y$

Combining Eqs. (B14b),(B39) - (B41b) gives

$$\begin{aligned} |\theta _\ell (\nu )|\le & {} \sum _{xyi}|\theta ^i_{\ell }(x,y,\nu )| \end{aligned}$$

(B44a)

$$\begin{aligned}{}[\theta ^i_{\ell }( x,y,\nu )]^2= & {} \sum _j [ \mu ^i_{j\ell }(x,y,\nu )]^2, \end{aligned}$$

(B44b)

with the definitions

$$\begin{aligned} \mu ^i_{j\ell }(x,y,\nu )= & {} -a^i(x,y,\nu ) \sum _k {\text {Im}}\{ \langle \phi _{j\ell }(\nu )| \langle \chi _{j\ell }(\nu )| \nonumber \\&\quad \times z^i(x,y,\nu )|\phi _{k\ell }(\nu ) \rangle |\chi _{k\ell }(\nu ) \rangle \lambda _{k\ell }(\nu )\}. \end{aligned}$$

(B45)

Since the $|\phi _{j\ell }(\nu ) \rangle$ are orthonormal, $g^b(x,y,\nu )$ is Hermitian and the $\lambda _{k\ell }(\nu )$ are real, we have

$$\begin{aligned}&{\text {Im}}\{ \langle \phi _{j\ell }(\nu )|\phi _{k\ell }(\nu ) \rangle \langle \chi _{j\ell }(\nu )|g^b(x,y,\nu )|\chi _{k\ell }(\nu ) \rangle \lambda _{k\ell }(\nu )\} = 0. \end{aligned}$$

(B46)

Equation (B45) for $i = 0$ can then be turned into

$$\begin{aligned} \mu ^0_{j\ell }(x,y,\nu )= & {} a^0(x,y,\nu ) \sum _k {\text {Im}}\{ \langle \phi _{j\ell }(\nu )| \langle \chi _{j\ell }(\nu )| [I - z^{0f}(x,y)] \nonumber \\&\quad \times g^b(x,y,\nu ) |\phi _{k\ell }(\nu ) \rangle |\chi _{k\ell }(\nu ) \rangle \lambda _{k\ell }(\nu )\}. \end{aligned}$$

(B47)

But in addition

$$\begin{aligned} |\omega ( \nu ) \rangle = \sum _k |\phi _{k\ell }(\nu ) \rangle |\chi _{k\ell }(\nu ) \rangle \lambda _{k\ell }(\nu ). \end{aligned}$$

(B48)

Also $I - z^{0f}(x,y)$ is a projection operator so that

$$\begin{aligned}{}[I - z^{0f}(x,y)]^2 = I - z^{0f}(x,y). \end{aligned}$$

(B49)

The normalization condition on $z^0(x,y,\nu )$ implies $[g^b(x,y,\nu )]^2$ has trace 1 as an operator on ${\mathcal {H}}^b_x \otimes {\mathcal {H}}^b_y$ and therefore all eigenvalues bounded by 1. Eqs. (B43a), (B44b), (B47), (B48), and (B49) then give

$$\begin{aligned}&[\theta ^0_{\ell }(x,y,\nu )]^2 \le [a^0(x,y,\nu )]^2 \langle \omega (\nu )|[I - z^{0f}(x,y)]|\omega (\nu ) \rangle . \end{aligned}$$

(B50)

For $\mu ^1_{j\ell }(x,y,\nu )$, since $z^1(x,y,\nu )$ is nonzero only on the subspace with $N({\mathcal {H}}_x \otimes {\mathcal {H}}_y)$ nonzero, we have

$$\begin{aligned} \mu ^1_{j\ell }(x,y,\nu )= & {} -a^1(x,y,\nu ) {\text {Im}}\{ \langle \phi _{j\ell }(\nu )| \langle \chi _{j\ell }(\nu )| \nonumber \\&\quad \times z^1(x,y,\nu ) [I - z^{0f}(x,y)]|\omega (\nu ) \rangle \}. \end{aligned}$$

(B51)

Equations (B44b) and (B51) give

$$\begin{aligned}{}[\theta ^1_{\ell }(x,y,\nu )]^2\le & {} [a^1(x,y,\nu )]^2 \langle \omega (\nu )| [I - z^{0f}(x,y)] \nonumber \\&\quad \times [z^1(x,y,\nu )]^2[I - z^{0f}(x,y)]|\omega (\nu ) \rangle . \end{aligned}$$

(B52)

But by Eq. (B42), $[z^1(x,y,\nu )]^2$ as an operator on ${\mathcal {H}}_x \otimes {\mathcal {H}}_y$, has trace 1 and therefore all eigenvalues bounded by 1. Thus Eq. (B52) implies

$$\begin{aligned}{}[\theta ^1_{\ell }(x,y,\nu )]^2 \le [a^1(x,y,\nu )]^2 \langle \omega (\nu )| [I - z^{0f}(x,y)]|\omega (\nu ) \rangle . \end{aligned}$$

(B53)

By construction of $D^e$, each nearest neighbor pair $\{x, y\}$ with $x \in D^e$ must have $y \in D^o$. Also any $x \in D^e$ is contained in at most a single $E_\ell$. As a result Eqs. (B44a), (B50) and (B53) imply

$$\begin{aligned} \sum _{\ell } |\theta _{\ell }(\nu )|\le & {} \sum _{x \in D^e, y \in D^o} \{ [|a^0(x,y,\nu )| + |a^1(x,y,\nu )|] \nonumber \\&\quad \times \sqrt{ \langle \omega (\nu )| [I - z^{0f}(x,y)]|\omega (\nu ) \rangle } \}. \end{aligned}$$

(B54)

The Cauchy–Schwartz inequality then gives

$$\begin{aligned}&[\sum _{\ell } |\theta _{\ell }(\nu )|] ^ 2 \le \sum _{x \in D^e, y \in D^o} [|a^0(x,y,\nu )| + |a^1(x,y,\nu )|]^2\nonumber \\&\quad \times \sum _{x \in D^e, y \in D^o} \langle \omega (\nu )| [I - z^{0f}(x,y)]|\omega (\nu ) \rangle . \end{aligned}$$

(B55)

The state $|\omega (\nu ) \rangle$ can be expanded as a linear combination of orthogonal states each with n fermions each at a single position. A state with fermions at n positions will survive the projection $I - z^{0f}(x,y)$ only if at least one of the fermions is either at x or y. Each $x \in D^e$ can be the member of only a single such pair of nearest neighbor $\{x, y\}$. A $y \in D^o$ can be in 6 x, y pairs for an $x \in D^e$. Thus a term with n fermion positions in the expansion of $|\omega (\nu ) \rangle$ will pass $I - z^{0f}(x,y)$ for at most 6n pairs of x and y. Therefore

$$\begin{aligned} \sum _{x \in D^e, y \in D^o} \langle \omega (\nu )| [I - z^{0f}(x,y)]|\omega (\nu ) \rangle \le 6n. \end{aligned}$$

(B56)

By Eq. (16)

$$\begin{aligned} \parallel k(\nu ) \parallel ^ 2 \ge \sum _{\ell , x \in D^e, y \in D^o} \parallel g_\ell ( x, y, \nu ) \parallel ^2 \end{aligned}$$

(B57)

In addition, $z^0(x,y,\nu )$ is orthogonal to $z^1(x, y, \nu )$. It follows that

$$\begin{aligned} \parallel k(\nu ) \parallel ^2 \ge \sum _{x \in D^e, y \in D^o} [|a^0(x,y,\nu )|^2 + |a^1(x,y,\nu )|^2]. \end{aligned}$$

(B58)

Assembling Eqs. (B55), (B56) and (B58) gives

$$\begin{aligned} \parallel k(\nu ) \parallel ^2\ge & {} \frac{1}{2} \sum _{x \in D^e, y \in D^o} [|a^0(x,y,\nu )| + |a^1(x,y,\nu )|]^2 \nonumber \\\ge & {} \frac{1}{12 n} [\sum _{\ell } |\theta _{\ell }(\nu )|] ^ 2 \end{aligned}$$

(B59)

Eq. (B17) then implies

$$\begin{aligned} \int _0^1 \parallel k(\nu ) \parallel \ge \frac{1}{\pi } \sqrt{ \frac{ mV}{192}}, \end{aligned}$$

(B60)

and therefore

$$\begin{aligned} C( |\psi \rangle , |\omega \rangle ) \ge \frac{1}{\pi }\sqrt{ \frac{ mV}{192}}. \end{aligned}$$

(B61)

Since Eq. (B61) holds for all product $|\omega \rangle$ we obtain

$$\begin{aligned} C( |\psi \rangle ) \ge \frac{1}{\pi } \sqrt{ \frac{ mV}{192}}. \end{aligned}$$

(B62)

1.5 2.5 Additional Terms

The nearest neighbor $\{x,y\}$ which contribute to each $\theta ^T_{\ell }(\nu )$ in Eq. (B35) are all distinct from the pairs which contribute to $\theta _{\ell }(\nu )$ in Eq. (B17). A repeat of the steps leading to Eq. (B59) yields

$$\begin{aligned} \parallel k(\nu ) \parallel ^2 \ge \frac{1}{12 n} [\sum _{\ell } |\theta _{\ell }(\nu )| + \sum _{\ell } |\theta ^T_{\ell }(\nu )| ] ^ 2. \end{aligned}$$

(B63)

Eq. (B62) becomes

$$\begin{aligned} C( |\psi \rangle ) \ge \frac{1}{\pi } \sqrt{ \frac{ mV}{192}} + \frac{\pi q}{ \sqrt{192}}. \end{aligned}$$

(B64)

Appendix 3: Upper Bound on the Complexity of Entangled States

An upper bound on $C( |\psi \rangle )$ of the n-particle entangled state of Eq. (36) is given by $C( |\psi \rangle , |\omega \rangle )$ for any n-particle product state $|\omega \rangle$, for which in turn an upper bound is given by

$$\begin{aligned} C( |\psi \rangle , |\omega \rangle ) \le \int _0^1 d t \parallel k( \nu ) \parallel , \end{aligned}$$

(C1)

for any trajectory $k(\nu ) \in K$ fulfilling Eqs. (B1) and (B2). Beginning with an $|\omega \rangle$ consisting of n particles each at one of a corresponding set of n single points, we construct a sufficient $k(\nu )$ in three stages. First,$|\omega \rangle$ is split into a sum of m orthogonal product states, each again consisting of n particles one at each of a corresponding set of n single points. Then the position of each of the particles in the product states is moved to the center of the wave function of one of the single particle states of Eq. (35). Finally, by approximately $\ln ( V) / \ln ( 8)$ iterations of a fan-out tree, the mn wave functions concentrated at points are spread over the mn cubes $D_{ij}$.

1.1 3.1 Product State to Entangled State

Define the set of positions $x_{ij}$ to be

$$\begin{aligned} (x_{ij})^1= & {} i + (x_{00})^1 , \end{aligned}$$

(C2a)

$$\begin{aligned} (x_{ij})^2= & {} j + (x_{00})^2, \end{aligned}$$

(C2b)

$$\begin{aligned} (x_{ij})^3= & {} (x_{00})^3, \end{aligned}$$

(C2c)

for $0 \le i< m, 0 \le j < n$ and arbitrary base point $x_{00}$. Let the set of n-particle product states $|\omega _i \rangle$ be

$$\begin{aligned} |\omega _i \rangle = \prod _{0 \le j < n} \Psi ^\dagger ( x_{ij}, 1) |\Omega \rangle . \end{aligned}$$

(C3)

The entangle n-particle state $|\chi \rangle$

$$\begin{aligned} |\chi \rangle = \sqrt{\frac{1}{m}} \sum _i |\omega _i \rangle \end{aligned}$$

(C4)

we generate from a sequence of unitary transforms acting on $|\omega \rangle = |\omega _0 \rangle$.

Let $k_{0}$ acting on ${\mathcal {H}}_{x_{00}} \otimes {\mathcal {H}}_{x_{01}}$ have matrix elements

$$\begin{aligned}&\langle \Omega | \Psi ( x_{00}, -1) \Psi ( x_{01}, -1) k_0 \nonumber \\&\quad \Psi ^{\dagger }(x_{00},1) \Psi ^{\dagger }( x_{01},1)|\Omega \rangle = -i, \end{aligned}$$

(C5)

$$\begin{aligned}&\langle \Omega | \Psi (x_{00},1) \Psi ( x_{01}, 1) k_0 \nonumber \\&\quad \Psi ^{\dagger }(x_{00},-1) \Psi ^{\dagger }( x_{01},-1)|\Omega \rangle = i, \end{aligned}$$

(C6)

and extend $k_0$ to ${\mathcal {H}}$ by Eq. (14). We then have

$$\begin{aligned}&\exp ( i \theta _0 k_0) |\omega _0 \rangle = \sqrt{\frac{1}{m}} |\omega _0 \rangle + \sqrt{\frac{m - 1}{m}} \prod _{0 \le j < n} \Psi ^{\dagger }( x_{0j}, s_{1j}) |\Omega \rangle , \end{aligned}$$

(C7)

where

$$\begin{aligned} \theta _0 = \arcsin ( \sqrt{\frac{m - 1}{m}}), \end{aligned}$$

(C8)

and the set of spin indices $s_{ij}, 0 \le i,j < n$ is

$$\begin{aligned} s_{ij}= & {} -1, j \le i, \end{aligned}$$

(C9a)

$$\begin{aligned} s_{ij}= & {} 1, j > i. \end{aligned}$$

(C9b)

Now let $k_{1}$ acting on ${\mathcal {H}}_{x_{01}} \otimes {\mathcal {H}}_{x_{02}}$ have matrix elements

$$\begin{aligned}&\langle \Omega | \Psi ( x_{01}, -1) \Psi ( x_{02}, -1) k_1 \nonumber \\&\quad \Psi ^{\dagger }(x_{01},-1) \Psi ^{\dagger }( x_{02}, 1)|\Omega \rangle = -i, \end{aligned}$$

(C10)

$$\begin{aligned}&\langle \Omega | \Psi (x_{01},-1) \Psi ( x_{02}, 1) k_1 \nonumber \\&\quad \Psi ^{\dagger }(x_{01},-1) \Psi ^{\dagger }( x_{02},-1)|\Omega \rangle = i, \end{aligned}$$

(C11)

and extend $k_1$ to ${\mathcal {H}}$ by Eq. (14). We then have

$$\begin{aligned}&\exp ( i \theta _1 k_1) \exp ( i \theta _0 k_0)|\omega _0 \rangle =\sqrt{\frac{1}{m}} |\omega _0 \rangle + \sqrt{\frac{m - 1}{m}} \prod _{0 \le j < n} \Psi ^{\dagger }( x_{0j}, s_{2j}) |\Omega \rangle , \end{aligned}$$

(C12)

for $\theta _1$ given by $\frac{\pi }{2}$.

Continuing in analogy to Eqs. (C5)–(C12), for a sequence of operators $k_j$, $0 \le j < n-1$, acting on ${\mathcal {H}}_{x_{0j}} \otimes {\mathcal {H}}_{x_{0j+1}}$, and corresponding $\theta _j$ we obtain

$$\begin{aligned}&\exp ( i \theta _{n-2} k_{n-2}) \ldots \exp ( i \theta _0 k_0) |\omega _0 \rangle \nonumber \\&\quad =\sqrt{\frac{1}{m}} |\omega _0 \rangle + \sqrt{\frac{m - 1}{m}} \prod _{0 \le j < n} \Psi ^{\dagger }( x_{0j}, -1) |\Omega \rangle , \end{aligned}$$

(C13)

Let $k_{n-1}$ acting on ${\mathcal {H}}_{x_{00}} \otimes {\mathcal {H}}_{x_{10}}$ have matrix elements

$$\begin{aligned} \langle \Omega | \Psi ( x_{10}, 1) k_{n-1} \Psi ^{\dagger }(x_{00},-1)|\Omega \rangle= & {} -i, \end{aligned}$$

(C14a)

$$\begin{aligned} \langle \Omega | \Psi (x_{00},-1) k_{n-1} \Psi ^{\dagger }( x_{10},1)|\Omega \rangle= & {} i, \end{aligned}$$

(C14b)

extend $k_{n-1}$ to ${\mathcal {H}}$ by Eq. (14), and let $\theta _{n-1}$ be $\frac{\pi }{2}$. Applying $\exp (i \theta _{n-1} k_{n-1})$ to Eq. (C13), followed by a similar sequence of $\exp (i \theta _j k_j), n \le j < 2n -1$ acting on ${\mathcal {H}}_{x_{0(j-n+1)}} \otimes {\mathcal {H}}_{x_{1(j-n+1)}}$ gives

$$\begin{aligned}&\exp ( i \theta _{2n-2} k_{2n-2}) \ldots \exp ( i \theta _0 k_0) |\omega _0 \rangle = \sqrt{\frac{1}{m}} |\omega _0 \rangle + \sqrt{\frac{m - 1}{m}} |\omega _1 \rangle \end{aligned}$$

(C15)

Multiplying Eq. (C15) by $\exp (i\theta _jk_j), 2n-1 \le j < 3n-2$ on ${\mathcal {H}}_{x_{1(j-2n+1)}} \otimes {\mathcal {H}}_{x_{1(j-2n +2)}}$, and then $\exp (i\theta _jk_j), 3n -2 \le j < 4n -2$ on ${\mathcal {H}}_{x_{1(j-3n+2)}} \otimes {\mathcal {H}}_{x_{2(j-3n+2)}}$ gives

$$\begin{aligned}&\exp ( i \theta _{4n-3} k_{4n-3}) \ldots \exp ( i \theta _0 k_0) |\omega _0 \rangle = \sqrt{\frac{1}{m}} |\omega _0 \rangle + \sqrt{\frac{1}{m}} |\omega _1 \rangle + \sqrt{\frac{m - 2}{m}} |\omega _2 \rangle . \end{aligned}$$

(C16)

The end result of a sequence of $2mn - m$ such steps is $|\chi \rangle$ of Eq. (C4)

$$\begin{aligned}&\exp ( i \theta _{2mn-m-1} k_{2mn-m-1}) \ldots \exp ( i \theta _0 k_0) |\omega _0 \rangle \nonumber \\&\quad =\sqrt{\frac{1}{m}} \sum _i |\omega _i \rangle . \end{aligned}$$

(C17)

The $k_i$ and $\theta _i$ of Eq. (C17) have

$$\begin{aligned} \parallel k_i \parallel= & {} \sqrt{2}, \end{aligned}$$

(C18a)

$$\begin{aligned} | \theta _i |\le & {} \frac{\pi }{2}. \end{aligned}$$

(C18b)

Thus Eq. (C17) implies

$$\begin{aligned} C( |\chi \rangle , |\omega \rangle ) \le \sqrt{2} \pi m \bigg (n - \frac{1}{2}\bigg ). \end{aligned}$$

(C19)

1.2 3.2 Entangled State Repositioned

Let $y_{ij}$ be the center of cube $D_{ij}$ of Eq. (35), $s_{ij}$ the spins of Eq. (35) and $\zeta _i$ the phases of Eq. (36). Define the entangled n-particle state $|\phi \rangle$ be

$$\begin{aligned} |\phi \rangle = \sum _{i} \zeta _i \prod _j \Psi ^{\dagger }( y_{ij}, s_{ij}) |\Omega \rangle . \end{aligned}$$

(C20)

For each $0 \le i< m, 0 \le j < n$, let $z^0_{ij}, z^1_{ij} \ldots z^{r_{ij}}_{ij}$ be the shortest sequence of nearest neighbor sites such that

$$\begin{aligned} z^0_{ij}= & {} x_{ij}, \end{aligned}$$

(C21a)

$$\begin{aligned} z^{r_{ij}}_{ij}= & {} y_{ij}, \end{aligned}$$

(C21b)

for the $x_{ij}$ in Eqs. (C2a)–(C4) and such that all $z^\ell _{ij}$ for distinct $\ell , i, j,$ are themselves distinct. For each $0 \le \ell < r_{ij} -1$, for nearest neighbor pair $z^\ell _{ij}, z^{\ell +1}_{ij}$, let $k^\ell _{ij}$ acting on ${\mathcal {H}}_{z^\ell _{ij}} \otimes {\mathcal {H}}_{z^{\ell +1}_{ij}}$ have matrix elements

$$\begin{aligned} \langle \Omega | \Psi (z^{\ell +1}_{ij} , 1) k^\ell _{ij} \Psi ^{\dagger }(z^\ell _{ij},1)|\Omega \rangle= & {} -i, \end{aligned}$$

(C22a)

$$\begin{aligned} \langle \Omega | \Psi (z^\ell _{ij},1) k^\ell _{ij} \Psi ^{\dagger }( z^{\ell +1}_{ij},1)|\Omega \rangle= & {} i, \end{aligned}$$

(C22b)

and extend $k^\ell _{ij}$ to ${\mathcal {H}}$ by Eq. (14). For each i, j pair with $j<n-1$, for the final nearest neighbor step $\exp ( i k^\ell _{ij}), \ell = r_{ij} - 1,$ Eqs. (C22a) and (C22b) are modified to produce spin orientation $s_{ij}$ at $y_{ij}$

$$\begin{aligned} \langle \Omega | \Psi (z^{\ell +1}_{ij} , s_{ij}) k^\ell _{ij} \Psi ^{\dagger }(z^\ell _{ij},1)|\Omega \rangle= & {} -i, \end{aligned}$$

(C23a)

$$\begin{aligned} \langle \Omega | \Psi (z^\ell _{ij},1) k^\ell _{ij} \Psi ^{\dagger }( z^{\ell +1}_{ij},s_{ij})|\Omega \rangle= & {} i, \end{aligned}$$

(C23b)

and for $j = n-1$ for the final $\exp ( i k^\ell _{in-1}), \ell = r_{in-1} - 1,$ Eqs. (C22a) and (C22b) are modified in addition to generate the phase $\zeta _i$

$$\begin{aligned} \langle \Omega | \Psi (z^{\ell +1}_{ij} , s_{ij}) k^\ell _{ij} \Psi ^{\dagger }(z^\ell _{ij},1)|\Omega \rangle= & {} -i \zeta _i, \end{aligned}$$

(C24a)

$$\begin{aligned} \langle \Omega | \Psi (z^\ell _{ij},1) k^\ell _{ij} \Psi ^{\dagger }( z^{\ell +1}_{ij},s_{ij})|\Omega \rangle= & {} i \zeta ^*_i. \end{aligned}$$

(C24b)

Define r to be

$$\begin{aligned} r = \max _{ij} r_{ij}, \end{aligned}$$

(C25)

and for each i, j pair define

$$\begin{aligned} k ^\ell _{ij} = 0, r_{ij} \le \ell < r. \end{aligned}$$

(C26)

Let $k^\ell$ be

$$\begin{aligned} k^\ell = \sum _{ij} k^\ell _{ij}. \end{aligned}$$

(C27)

Then we have

$$\begin{aligned} \prod _{ij}[ \exp ( i\frac{\pi }{2} k^{s-1}) \ldots \exp (i\frac{\pi }{2} k^0)] |\chi \rangle = |\phi \rangle , \end{aligned}$$

(C28)

for $|\chi \rangle$ of Eq. (C4).

The $k^\ell$ of Eqs. (C27), (C22a) - (C24b) have

$$\begin{aligned} \parallel k^\ell _{ij} \parallel \le \sqrt{2 mn}. \end{aligned}$$

(C29)

Thus Eq. (C28) implies

$$\begin{aligned} C( |\phi \rangle , |\chi \rangle ) \le \frac{ \pi \sqrt{mn} r}{\sqrt{2}} . \end{aligned}$$

(C30)

We now minimize r over the base point $x_{00}$

$$\begin{aligned} {\hat{r}} = \min _{x_{00}} r, \end{aligned}$$

(C31)

with the result

$$\begin{aligned} C( |\phi \rangle , |\chi \rangle ) \le \frac{ \pi \sqrt{mn} r}{\sqrt{2}}, \end{aligned}$$

(C32)

where we have dropped the hat on r.

1.3 3.3 Fan-Out

The state $|\phi \rangle$ with particles at the centers of the cubes $D_{ij}$ we now fan-out to the state $|\psi \rangle$ with particle wave functions spread uniformly over the cubes $D_{ij}$. For sufficiently small lattice spacing a nearly all of the complexity of the bound on $C(|\psi \rangle )$ is generated in this step.

Let d be the length of the edge of the $D_{ij}$. Each edge of $D_{ij}$ then consists of $d+1$ sites. The volume V is then $d^3$. We begin with case

$$\begin{aligned} d = 2^p, \end{aligned}$$

(C33)

for some integer p. For simplicity we present the fan-out applied to a prototype single particle state $|\upsilon _0 \rangle$ on prototype cube G with edge length d, and center at some point y

$$\begin{aligned} |\upsilon _0 \rangle = \Psi ^{\dagger }( y, 1) |\Omega \rangle . \end{aligned}$$

(C34)

The first stage of the fan-out process consists of splitting $|\upsilon _0 \rangle$ into a pair of components displaced from each other in lattice direction 1. For integer $-2^{p-2} \le i \le 2^{p-2}$ define y(i) to be y incremented by i nearest neighbor steps in lattice direction 1. For $1 \le j \le 2^{p-2}$ define $k_j$ on ${\mathcal {H}}_{y( j -1)} \otimes {\mathcal {H}}_{y(j)}$ to have matrix elements

$$\begin{aligned} \langle \Omega | \Psi [y(j),1] k_j \Psi ^{\dagger }[y(j-1),1]|\Omega \rangle= & {} -i, \end{aligned}$$

(C35a)

$$\begin{aligned} \langle \Omega | \Psi [y(j-1), 1] k_j \Psi ^{\dagger }[ y(j),1]|\Omega \rangle= & {} i. \end{aligned}$$

(C35b)

For $-2^{p-2} \le j \le -1$ define $k_j$ by Eqs. (C35a) and (C35b) but with $j+1$ in place of $j-1$. Then define ${\bar{k}}_j$ by

$$\begin{aligned} {\bar{k}}_1= & {} \frac{1}{\sqrt{2}}( k_1 + k_{-1}), \end{aligned}$$

(C36a)

$$\begin{aligned} {\bar{k}}_j= & {} k_j + k_{-j}, 2 \le j \le 2^{p-2}. \end{aligned}$$

(C36b)

With these definitions it then follows that

$$\begin{aligned} |\upsilon _1 \rangle = \exp (i \frac{\pi }{2}\bar{ k}_m) \ldots \exp (i \frac{\pi }{2} {\bar{k}}_1) |\upsilon _0 \rangle , \end{aligned}$$

(C37)

for $m = 2^{p-2}$, is given by

$$\begin{aligned} |\upsilon _1 \rangle = \frac{1}{\sqrt{2}}\sum _{i = -2^{p-2},2^{p-2}} \Psi ^{\dagger }[ y(i), 1] |\Omega \rangle . \end{aligned}$$

(C38)

Equations (C36a) and (C36b) imply

$$\begin{aligned} \parallel {\bar{k}}_1 \parallel= & {} \sqrt{2}, \end{aligned}$$

(C39a)

$$\begin{aligned} \parallel {\bar{k}}_j\parallel= & {} 2, 2 \le j \le 2^{p-2}. \end{aligned}$$

(C39b)

It then follows that

$$\begin{aligned} C( |\upsilon _1 \rangle , |\upsilon _0 \rangle ) < 2^{p-2} \pi , \end{aligned}$$

(C40)

where for simplicity we have used an overestimate for $\parallel {\bar{k}}_1 \parallel$.

The next stage of the fan-out consists of splitting each of the 2 components of $|\upsilon _1 \rangle$ but now in lattice direction 2. For ${\bar{k}}_j, 2^{p-2} < j \le 2^{p-1}$, defined by adapting of Eqs. (C35a)–(C36b), we have

$$\begin{aligned} |\upsilon _2 \rangle = \exp (i \frac{\pi }{2}\bar{ k}_m) \ldots \exp (i \frac{\pi }{2} {\bar{k}}_1) |\upsilon _0 \rangle , \end{aligned}$$

(C41)

with $m = 2^{p-1}$, given by

$$\begin{aligned}&|\upsilon _2 \rangle \frac{1}{2}\sum _{i = -2^{p-2},2^{p-1}} \sum _{j = -2^{p-2},2^{p-2}} \Psi ^{\dagger }[ y(i,j), 1] |\Omega \rangle , \end{aligned}$$

(C42)

for y(i, j) defined to be y(i) displaced j steps in lattice direction 2. Eqs. (C36a) and (C36b) adapted to the fan-out in direction 2 give ${\bar{k}}_j, 2^{p-2} < j \le 2^{p-1}$ each acting on twice as many sites as was the case for the direction 1 fan-out and therefore

$$\begin{aligned} \parallel {\bar{k}}_{2^{p-2} + 1} \parallel= & {} 2, \end{aligned}$$

(C43a)

$$\begin{aligned} \parallel {\bar{k}}_j\parallel= & {} 2\sqrt{2}, 2^{p-2} + 2 \le j \le 2^{p-1}. \end{aligned}$$

(C43b)

It then follows that

$$\begin{aligned} C( |\upsilon _2 \rangle , |\upsilon _1 \rangle ) < 2^{p-2} \sqrt{2}\pi . \end{aligned}$$

(C44)

Splitting yet again, now in lattice direction 3, yields

$$\begin{aligned} |\upsilon _3 \rangle = \exp (i \frac{\pi }{2}{\bar{k}}_m) \ldots \exp (i \frac{\pi }{2}{\bar{k}}_1) |\upsilon _0 \rangle , \end{aligned}$$

(C45)

for $m =2^{p-1} + 2^{p-2}$, given by

$$\begin{aligned}&|\upsilon _3 \rangle = \frac{1}{\sqrt{8}}\sum _{i = -2^{p-2},2^{p-1}} \sum _{j = -2^{p-2},2^{p-2}} \sum _{\ell = -2^{p-2},2^{p-2}} \Psi ^{\dagger }[ y(i,j, \ell ), 1] |\Omega \rangle , \end{aligned}$$

(C46)

for $y(i,j, \ell )$ defined to be y(i, j) displaced $\ell$ steps in lattice direction 3.

Eqs. (C36a) and (C36b) adapted to the fan-out in direction 3 give ${\bar{k}}_j, 2^{p-1} < j \le 2^{p-1} + 2^{p-2}$, each acting on twice as many sites as was the case for the direction 2 fan-out and therefore

$$\begin{aligned} \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \parallel {\bar{k}}_ {2^{p-1} + 1} \parallel= & {} 2\sqrt{2}, \end{aligned}$$

(C47a)

$$\begin{aligned} \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \parallel {\bar{k}}_j\parallel= & {} 4, 2^{p-1}+ 2 \le j \le 2^{p-1} + 2^{p-2}. \end{aligned}$$

(C47b)

It then follows that

$$\begin{aligned} C( |\upsilon _3 \rangle , |\upsilon _2 \rangle ) <2^{p-1} \pi . \end{aligned}$$

(C48)

The weight originally concentrated in $|\upsilon _0 \rangle$ at the center point y of G, with edge length d, in $|\upsilon _3 \rangle$ is distributed equally over the center points of 8 sub-cubes of G each with edge length $\frac{d}{2}$. Combining Eqs. (C40), (C44) and (C48) gives

$$\begin{aligned} C( |\upsilon _3 \rangle , |\upsilon _0 \rangle ) < (3+ \sqrt{2}) 2^{p-2} \pi . \end{aligned}$$

(C49)

The fan-out process of Eqs. (C37)–(C49) we now repeat a total of $p-1$ iterations arriving at a state $|\upsilon _{3 p - 3} \rangle$ with weight equally distributed over the center points of $2^{3 p - 3}$ cubes each with edge length 2. Eqs. (C49) rescaled for iteration $\ell$ give

$$\begin{aligned} C( |\upsilon _{3 \ell } \rangle , |\upsilon _{3 \ell - 3} \rangle ) < (3+\sqrt{2}) 2^{p-\ell -1} 2^{\frac{3\ell - 3}{2}} \pi . \end{aligned}$$

(C50)

The term $2^{p-\ell - 1}$ counts the decreasing number of lattice steps between cube centers as the fan-out process is iterated, while the term $2^{\frac{3\ell - 3}{2}}$ counts the growing number of cubes and therefore of sites which each subsequent operator ${\bar{k}}(i)$ acts on simultaneously.

To complete the fan-out process, the weight at the center of each of the cubes with edge length 2 needs to be distributed to the 26 points forming its boundary. This process can be carried out in 3 additional steps thereby defining $|\upsilon _{3p -2} \rangle , |\upsilon _{3p-1} \rangle$ and $|\upsilon _{3p} \rangle$.

To obtain $|\upsilon _{3p -2} \rangle$ from $|\upsilon _{3p -3} \rangle$, the weight at the center of each edge length 2 cube is distributed simultaneously and equally to the points at the centers of the 6 edge length 2 squares forming the cube’s boundary. This process itself is done simultaneously across all $2^{3 p - 3}$ cubes. The result is

$$\begin{aligned} C( |\upsilon _{3p -2} \rangle , |\upsilon _{3 p - 3} \rangle ) \le \frac{\pi }{2} 2^{\frac{3\ell - 3}{2}}. \end{aligned}$$

(C51)

To obtain $|\upsilon _{3p -1} \rangle$ from $|\upsilon _{3p -2} \rangle$, the weight at the center of each edge length 2 square is distributed simultaneously and equally to the center point of the 4 length 2 lines forming the boundary of that square. This process itself is done simultaneously across all faces of all $2^{3 p - 3}$ cubes. The result is

$$\begin{aligned} C( |\upsilon _{3p -1} \rangle , |\upsilon _{3 p - 2} \rangle ) \le \frac{\sqrt{3}\pi }{2\sqrt{2}} 2^{\frac{3\ell - 3}{2}}. \end{aligned}$$

(C52)

To obtain $|\upsilon _{3p} \rangle$ from $|\upsilon _{3p -1} \rangle$, the weight at the center of each length 2 line is distributed simultaneously and equally to that line’s pair of end points. This process itself is done simultaneously across all lines forming the boundaries of the faces of all $2^{3 p - 3}$ cubes. The result is

$$\begin{aligned} C( |\upsilon _{3p} \rangle , |\upsilon _{3 p - 1} \rangle ) \le \frac{\pi }{2} 2^{\frac{3\ell - 3}{2}}. \end{aligned}$$

(C53)

The bound on $C( |\upsilon _{3p} \rangle , |\upsilon _{3 p - 3} \rangle )$ obtained by summing Eqs. (C51)–(C53) turns out to be less than the bound in Eq. (C50) for $\ell = p$. We therefore sum Eq. (C50) from $\ell$ of 1 to p and obtain

$$\begin{aligned} C(|\upsilon _{3p} \rangle , |\upsilon _0 \rangle ) < \frac{(3 + \sqrt{2})(2+\sqrt{2})}{4\sqrt{2}}\pi 2^{\frac{3 p}{2}}. \end{aligned}$$

(C54)

Substituting V for $2^{3p}$, we then have

$$\begin{aligned} C(|\upsilon _{3p} \rangle , |\upsilon _0 \rangle ) < \frac{(3 + \sqrt{2})(2 + \sqrt{2})}{4\sqrt{2}}\pi \sqrt{V}. \end{aligned}$$

(C55)

The bound of Eq. (C54) is derived assuming Eq. (C33) giving the edge d of cube G as an even power of 2. Consider now the case

$$\begin{aligned} 2^{p-1}< d < 2^p. \end{aligned}$$

(C56)

Assume again that at each iteration $\ell$ of the fan-out process, each edge length of each parent cube is split as evenly as possible into halves to produce 8 child cubes with all edges nearly equal. Suppose d is $2^p - 1$. After iteration $\ell$ has been completed, the total number of cubes will still be $2^{3 \ell }$. Orthogonal to each direction, the cubes can be grouped into $2^\ell$ planes, each holding $2^{2 \ell }$ cubes. But for each direction one of these orthogonal planes will have an edge in that direction which is one lattice unit shorter than the corresponding edge of the other $2^\ell$ planes. It follows that the update process in each direction can proceed with $2^{p - \ell - 1} - 1$ steps occuring simultaneously across all cubes, and one final update skipped for the cubes with a single edge in that direction one lattice unit shorter. The bound of Eq. (C50) will hold without modification. For d given by $2^p - 2$, after iteration $\ell$, for each direction, there will be two planes of $2^{2 \ell }$ cubes each with the edge in that direction one lattice unit shorter. The bound of Eq. (C50) will continue to hold. Similarly for d given by $2^p - q$ for any $q < 2^{p-1}$.

For d of Eq. (C56), when $\ell$ reaches $p - 1$ the resulting cubes (no longer exactly cubes) will have a mix of edges of length 2 and of length 1. The argument leading to Eqs. (C51)–(C53) can be adapted to show they continue to hold for the final pass with $\ell$ of p. The bound of Eq. (C54) remains in place for the net result of the entire fan-out process. By assumption, according to Eq. (C56) we have

$$\begin{aligned} 2 d > 2^p. \end{aligned}$$

(C57)

Then since V is $d^3$, Eq. (C54) gives

$$\begin{aligned} C(|\upsilon _{3p-1} \rangle , |\upsilon _0 \rangle ) < \frac{(3 + \sqrt{2})(2 + \sqrt{2})}{2} \pi \sqrt{V}, \end{aligned}$$

(C58)

which is weaker than Eq. (C55) and therefore holds whether or not d is an even power of 2.

The bound of Eq. (C55) applies to a fan-out process on a single prototype state on cube G. Assume the process repeated in parallel on the mn cubes $D_{ij}$, thereby generating $|\psi \rangle$ of Eq. (36). For $|\phi \rangle$ of Eq. (C20) we then have

$$\begin{aligned} C( |\psi \rangle , |\phi \rangle ) \le \frac{(3 + \sqrt{2})(2+\sqrt{2})}{2} \pi \sqrt{mnV}. \end{aligned}$$

(C59)

From Eqs. (C19) and (C32), it follows that for a product state $|\omega \rangle$ we have

$$\begin{aligned} C(|\psi \rangle ,|\omega \rangle ) \le c_1 \sqrt{ mnV} + c_2 m n + c_3 \sqrt{mn} r, \end{aligned}$$

(C60)

where

$$\begin{aligned} c_1= & {} \frac{(3 + \sqrt{2})(2 + \sqrt{2})}{2} \pi , \end{aligned}$$

(C61a)

$$\begin{aligned} c_2= & {} \sqrt{2} \pi , \end{aligned}$$

(C61b)

$$\begin{aligned} c_3= & {} \frac{\pi }{\sqrt{2}}, \end{aligned}$$

(C61c)

for r of Eq. (C32). Eq. (38) then follows.

Appendix 4: Complexity Group

We now show that the topological closure of the group G of all $U_k( 1)$ realizable as solutions to Eqs. (17a) and (17b) has as a subgroup the direct product

$$\begin{aligned} {\hat{G}} = \times _n SU(d_n), \end{aligned}$$

(D1)

where $SU(d_n)$ acts on the subspace of ${\mathcal {H}}$ with eigenvalue n of the fermion number operator N, $d_n$ is the dimension of this subspace, and the product is over the range $0 \le n \le 16 B^3$.

1.1 4.1 Lie Algebras

The $8 B^3$ sites of the lattice L we reorder as a 1-dimensional array of distinct sites, successive pairs of which are nearest neighbors with respect to the original lattice L. The new array of sites we label with an integer valued index z ranging from 0 to $8 B^3 -1$.

For any pair of nearest neighbor $\{z, z'\}$, let ${\mathcal {F}}_{z z'}$ be the set of operators of the form

$$\begin{aligned} f_{zz'} = g_{zz'} \bigotimes _{q \ne z,z'} I_q, \end{aligned}$$

(D2)

where $I_q$ is the identity operator on ${\mathcal {H}}_q$ and $g_{zz'}$ is a traceless Hermitian operator acting on ${\mathcal {H}}_z \otimes {\mathcal {H}}_{z'}$ which commutes with $N_{zz'}$, the fermion number operator on ${\mathcal {H}}_z \otimes {\mathcal {H}}_{z'}$. Let $K_p$ be the vector space over the reals of operators of the form

$$\begin{aligned} k = \sum _{zz'} f_{zz'}, \end{aligned}$$

(D3)

for any collection of $f_{z z'} \in {\mathcal {F}}_{zz'}$ for $z, z' \le p$.

Let $G_p$ be the group on ${\mathcal {H}}$ of all $U_k(1)$ realizable as solutions to Eq. (17a) for $k(\nu ) \in K_p$. The topological closure of the group $G_p$ consists of all operators of the form $\exp ( i h)$ for $h \in L_p$, where $L_p$ is the Lie algebra generated by $K_p$ [21]. Said differently, $L_p$ is the smallest set of operators such that $K_p \subseteq L_p$ and, in addition, for any $h_0, h_1 \in L_p$, and any real $r_0, r_1$, there are $h_2, h_3 \in L_p$ given by

$$\begin{aligned} h_2= & {} r_0 h_0 + r_1 h_1, \end{aligned}$$

(D4a)

$$\begin{aligned} h_3= & {} i [ h_0, h_1]. \end{aligned}$$

(D4b)

The requirement that $L_p$ be closed under sums in Eq. (D4a) follows from the Trotter product formula applied to the large t limit

$$\begin{aligned}&\exp ( i r_0 h_0 + i r_1 h_1) = \lim _{t \rightarrow \infty }[ \exp ( i t^{-1}r_0 h_0) \exp ( i t^{-1} r_1 h_1)]^t. \end{aligned}$$

(D5)

The requirement that $L_p$ be closed under commutation in Eq. (D4b) follows from the Baker–Campbell–Hausdorff formula applied to the large t limit

$$\begin{aligned}&\exp ( [ h_0, h_1]) = \lim _{t \rightarrow \infty } [ \exp ( i t^{-1/2} h_0) \exp ( -i t^{-1/2} h_1) \nonumber \\&\quad \times \exp ( -i t^{-1/2} h_0) \exp ( i t^{-1/2} h_1)]^t. \end{aligned}$$

(D6)

The requirement of taking a topological closure of the group generated by $U_k(1)$ in order to generate $L_p$ is a consequence of the appearance of limits in Eqs. (D5) and (D6).

1.2 4.2 Induction

For any integer $0 < p \le 8 B^3$ - 1, divide ${\mathcal {H}}$ into the product

$$\begin{aligned} {\mathcal {Q}}_p= & {} \bigotimes _{q \le p} {\mathcal {H}}_q, \end{aligned}$$

(D7a)

$$\begin{aligned} {\mathcal {R}}_p= & {} \bigotimes _{q > p} {\mathcal {H}}_q, \end{aligned}$$

(D7b)

$$\begin{aligned} {\mathcal {H}}= & {} {\mathcal {Q}}_p \otimes {\mathcal {R}}_p. \end{aligned}$$

(D7c)

By induction on p, we will show that the closure of $G_p$ includes the subgroup ${\hat{G}}_p$

$$\begin{aligned} {\hat{G}}_p= & {} \times _n {\hat{G}}_{p n}, \end{aligned}$$

(D8a)

$$\begin{aligned} {\hat{G}}_{p n}= & {} SU(d_{p n}) \bigotimes _{z > p} I_z, \end{aligned}$$

(D8b)

where $SU(d_{p n})$ acts on the subspace ${\mathcal {Q}}_{p n}$ of ${\mathcal {Q}}_p$ with eigenvalue n of the total number operator N, and $d_{p n}$ is the dimension of ${\mathcal {Q}}_{p n}$. The product in Eq. (D8a) is over $0 \le n \le 2 p + 2$. Equations (D8a) and (D8b) for the case $p = 8B^3-1$ become Eq. (D1).

The set of $g_{zz'}$ in Eq. (D2) is a subset of the set of $f_{xy}$ in Eq. (14) of Sect. 3. Thus ${\hat{G}}_p$ for $p = 8B^3-1$ is a subgroup of the group G of Sect. 3. Proof of Eq. (D8a) therefore implies Eq. (18) of Sect. 3.

For $p = 1$, Eqs. (D8a) and (D8b) follow immediately from the definition of $K_p$. Assuming Eqs. (D8a) and (D8b) for some $p - 1$, we will prove them for p.

Let $S_{p n}$ be an orthonormal basis for ${\mathcal {Q}}_{p n}$ consisting of all n-fermion, m-boson, $m \le b_{max}( p + 1)$, vectors of the form

$$\begin{aligned} | \psi \rangle= & {} \prod _{i \le n} \Psi ^{\dagger }( z^f_i, s_i) \prod _{ j \le m}\Phi ^{\dagger }(z^b_j) |\Omega \rangle \end{aligned}$$

(D9a)

$$\begin{aligned} s_i\in & {} \{ -1, 1 \}, \end{aligned}$$

(D9b)

$$\begin{aligned} z^f_i, z^b_j\le & {} p, \end{aligned}$$

(D9c)

for any list of n distinct pairs of $(z^f_i, s_i)$ and any list of m integers $z^b_j$ such that each $z^b_j$ coincides with at most $b_{max} -1$ other $z^b_{j'}$. For any pair of distinct $|\psi _0 \rangle , |\psi _1 \rangle \in S_{p n}$, and 2 $\times$ 2 traceless Hermitian h, define

$$\begin{aligned}&H( |\psi _0 \rangle , |\psi _1 \rangle , h) = \sum _{ij} |\psi _i \rangle \langle \psi _j| h_{ij}, \end{aligned}$$

(D10a)

$$\begin{aligned}&H_p( |\psi _0 \rangle , |\psi _1 \rangle , h) \nonumber \\&\quad =H( |\psi _0 \rangle , |\psi _1 \rangle , h) \bigotimes _{z>p} I_z. \end{aligned}$$

(D10b)

The set of all such $H_p( |\psi _0 \rangle , |\psi _1 \rangle , h)$ is a linear basis for the Lie algebra $L_{p n}$ of the group ${\hat{G}}_{p n}$ of Eq. (D8b).

Thus to prove Eqs. (D8b) and (D8a) for p it is sufficient to show that any $H_p( |\psi _0 \rangle , |\psi _1 \rangle , h)$ for some $|\psi _0 \rangle , |\psi _1 \rangle \in S_{p n}$ and 2 $\times$ 2 traceless Hermitian h, given the induction hypothesis, is contained in the Lie algebra generated by $L_{p-1 n'}$ for some $n'$ and ${\mathcal {F}}_{p-1 p}$.

1.3 4.3 Without Bosons

We consider first $|\psi _0 \rangle$ and $|\psi _1 \rangle$ both with m of 0 in Eqs. (D9a)–(D9c). We will work backwards starting from some $H_p( |\psi _0 \rangle , |\psi _1 \rangle , h)$ for $|\psi _0 \rangle , |\psi _1 \rangle \in S_{p n}$. Since $|\psi _0 \rangle$ and $|\psi _1 \rangle$ have the same value of total N on the region $z \le p$, it follows that a $U_0$ can be found in ${\hat{G}}_{p-1}$ such that

$$\begin{aligned} |\psi _2 \rangle= & {} U_0 |\psi _0 \rangle , \end{aligned}$$

(D11a)

$$\begin{aligned} |\psi _3 \rangle= & {} U_0 |\psi _1 \rangle \end{aligned}$$

(D11b)

are orthogonal vectors in $S_{p n}$, their restrictions to the region $p-1 \le z \le p$ are also orthogonal but have equal total particle counts on $p-1 \le z \le p$. The particle count difference between $|\psi _0 \rangle$ and $|\psi _1 \rangle$ at point p is at most 2, and equal and opposite to the difference between the corresponding totals on the region $z \le p-1$. This compensating difference can be moved by $U_0$ to the point $p -1$.

A k in ${\mathcal {F}}_{p-1 p}$ can then be found such that

$$\begin{aligned} |\psi _4 \rangle= & {} \exp ( i k) |\psi _2 \rangle , \end{aligned}$$

(D12a)

$$\begin{aligned} |\psi _5 \rangle= & {} \exp ( i k)|\psi _3 \rangle , \end{aligned}$$

(D12b)

$$\begin{aligned} |\psi _4 \rangle= & {} |\psi _6 \rangle \otimes |\upsilon \rangle \end{aligned}$$

(D12c)

$$\begin{aligned} |\psi _5 \rangle= & {} |\psi _7 \rangle \otimes |\upsilon \rangle , \end{aligned}$$

(D12d)

for some $|\upsilon \rangle \in {\mathcal {H}}_p$, with particle number $n_{\upsilon }$ and $|\psi _6 \rangle$ and $|\psi _7 \rangle$ orthogonal vectors in $S_{(p-1) m}$ with $m = n - n_{\upsilon }$.

It is then possible to find a $U_2$ in ${\hat{G}}_{p-1}$ such that

$$\begin{aligned} |\psi _8 \rangle= & {} U_2 |\psi _4 \rangle , \end{aligned}$$

(D13a)

$$\begin{aligned} |\psi _9 \rangle= & {} U_2 |\psi _5 \rangle , \end{aligned}$$

(D13b)

$$\begin{aligned} |\psi _8 \rangle= & {} | \chi \rangle \otimes |\phi _0 \rangle \otimes |\upsilon \rangle , \end{aligned}$$

(D13c)

$$\begin{aligned} |\psi _9 \rangle= & {} | \chi \rangle \otimes |\phi _1 \rangle \otimes |\upsilon \rangle , \end{aligned}$$

(D13d)

$$\begin{aligned} |\phi _0 \rangle= & {} \Psi ^{\dagger }( p-1, -1) |\Omega \rangle , \end{aligned}$$

(D13e)

$$\begin{aligned} |\phi _1 \rangle= & {} \Psi ^{\dagger }( p-1, 1) |\Omega \rangle , \end{aligned}$$

(D13f)

for a some $|\chi \rangle$ in $S_{(p-2) (m-1)}$.

Combining Eqs. (D11a)–(D13f), the induction hypothesis implies the existence of $U_0, U_2 \in {\hat{G}}_{p-1}$ and $k \in {\mathcal {F}}_{(p-1) p}$ such that

$$\begin{aligned}&U_2 \exp ( i k) U_0 H_p( |\psi _0 \rangle , |\psi _1 \rangle , h) U_0^\dagger \exp ( -i k) U_2^\dagger \nonumber \\&\quad = |\chi \rangle \langle \chi | \otimes \sum _{ij} |\phi _i \rangle \langle \phi _j| h_{ij} \otimes |\upsilon \rangle \langle \upsilon |. \end{aligned}$$

(D14)

The expression on the right-hand side of Eq. (D14) can then be obtained from a commutator between an operator $k \in {\mathcal {F}}_{(p-1) p}$ and an operator $g \in L_{(p-1) m}$ for $m = n - n_{\upsilon }$. For 2 $\times$ 2 traceless Hermitian $k_{ij}$, define

$$\begin{aligned} k = \sum _{ij} |\phi _i \rangle \langle \phi _j| k_{ij} \otimes |\upsilon \rangle \langle \upsilon | \bigotimes _{q \ne p-1, p} I_q, \end{aligned}$$

(D15)

and for a 2 $\times$ 2 traceless Hermitian $g_{ij}$, define

$$\begin{aligned} g = |\chi \rangle \langle \chi | \otimes \sum _{ij} |\phi _i \rangle \langle \phi _j| g_{ij} \bigotimes _{q > p-1} I_q. \end{aligned}$$

(D16)

For any traceless, Hermitian 2 $\times$ 2 $h_{ij}$, there are $k_{ij}$ and $g_{ij}$ such that

$$\begin{aligned} h = i [ k, g]. \end{aligned}$$

(D17)

Combining Eqs. (D14), (D15), (D16) and (D17) then gives

$$\begin{aligned}&H_p( |\psi _0 \rangle , |\psi _1 \rangle , h) = U_0^\dagger \exp (-ik) U_2^\dagger i[k, g] U_2 \exp (ik) U_0, \end{aligned}$$

(D18)

which completes the induction step and for $|\psi _0 \rangle$ and $|\psi _1 \rangle$ with m of 0 in Eqs. (D9a) - (D9c).

1.4 4.4 With Bosons

We consider next $|\psi _0 \rangle$ and $|\psi _1 \rangle$ both with nonzero n and m in Eqs. (D9a)–(D9c).

Suppose $0< n < 2 p + 2$.

If the boson factors in $|\psi _0 \rangle$ and $|\psi _1 \rangle$ are identical, then by a combination of a rotation by a $U_0$ in ${\hat{G}}_{p-1}$ and by a $U_1$ in the group generated by $k \in {\mathcal {F}}_{(p-1) p}$ the boson factors can both be turned into the case m of 0, already covered in Appendix 4.3.

Suppose the boson factors in $|\psi _0 \rangle$ and $|\psi _1 \rangle$ are not identical but the fermion factors in $|\psi _0 \rangle$ and $|\psi _1 \rangle$ are identical. Then again, but a combination of a rotation by a $U_0$ in ${\hat{G}}_{p-1}$ and by a $U_1$ in the group generated by $k \in {\mathcal {F}}_{(p-1) p}$ the boson factors can both be turned into the case m of 0 but with orthogonal fermion factors in $|\psi _0 \rangle$ and $|\psi _1 \rangle$. Thus back to the case covered in Appendix 4.3.

Suppose both the fermion factors and the boson factors in $|\psi _0 \rangle$ and $|\psi _1 \rangle$ are not identical. The induction step of Appendix 4.3 shows that the action of ${\hat{G}}_p$ is available at least on the fermion factors in $|\psi _0 \rangle$ and $|\psi _1 \rangle$. A $U_0$ in ${\hat{G}}_p$ can therefore be found which makes the fermion factors in $|\psi _0 \rangle$ and $|\psi _1 \rangle$ distinct both on the region $p-1 \le z \le p$ and on the region $0 \le z \le p - 1$. It follows that a $U_1$ in ${\hat{G}}_{p-1}$ and a $U_2$ in the group generated by $k \in {\mathcal {F}}_{(p-1) p}$ can then be found which take $|\psi _0 \rangle$ and $|\psi _1 \rangle$ back to m of 0.

Suppose finally either n is 0 and $|\psi _0 \rangle$ and $|\psi _1 \rangle$ have only fermions or n is $2 p + 2$ and all sites are filled with fermions. In either case, ${\hat{G}}_{p-1}$ and ${\hat{G}}_p$ act purely on boson states. The induction step to show that the Lie algebra of ${\hat{G}}_p$ is generated by the Lie algebra of $L_{p-1 n'}$, either for $n'$ of 0 or $n'$ of 2p, and ${\mathcal {F}}_{p-1 p}$ becomes nearly a direct translation of the induction step in Appendix 4.3 from fermion states to boson states. We omit the details.

Appendix 5: Auxiliary Field Algebra

We will construct a Hilbert space ${\mathcal {H}}^B$ generated by the algebra B of polynomials in the $\Sigma _i( x, s)$ and $\Upsilon _i( x)$ acting purely as creation operators on $|\Omega ^B \rangle$ and satisfying Eqs. (153a)–(153d).

Let $B^\Sigma$ be the algebra generated by the set of all $\Sigma _i( x, s)$, for any x, s and i, and let $B^\Upsilon$ be the algebra generated by the set of all $\Upsilon _i(x)$, for any x and i. Since every $a \in B^\Sigma$ commutes with every $b \in B^\Upsilon$, the algebra B is the tensor product

$$\begin{aligned} B = B^\Sigma \otimes B^\Upsilon . \end{aligned}$$

(E1)

For every x, let $B^\Sigma _x$ be the algebra generated by the set of $\Sigma _i( x, s)$, for any s and i, and let $B^\Upsilon _x$ be the algebra generated by the set of $\Upsilon _i( x)$ for any i. Then for every $x \ne y$, every $a_x \in B^\Sigma _x$ commutes or anticommutes with every $a_y \in B^\Sigma _y$, and every $a_x \in B^\Upsilon _x$ commutes with every $a_y \in B^\Upsilon _y$. Therefore the algebras $B^\Sigma$ and $B^\Upsilon$ are the products

$$\begin{aligned} B^\Sigma= & {} \bigotimes _x B^\Sigma _x, \end{aligned}$$

(E2a)

$$\begin{aligned} B^\Upsilon= & {} \bigotimes _x B^\Upsilon _x. \end{aligned}$$

(E2b)

Now let $\eta _x$ be a boost that takes the point x to the point $(\tau , 0, 0, 0)$. For Eqs. (155a) and (155b) to be covariant, $\Sigma _0(x,s)$ has to transform under boosts like $\Psi ( x, s)$ and $\Sigma _1(x,s)$ has to transform under boosts like $\Psi ^\dagger ( x, s)$. Let $S^x_{ss'}$ and ${\bar{S}}^x_{ss'}$ be the spin transformation matrices corresponding to $\eta _x$ and define ${\hat{\Sigma }}_0(x,s)$ and ${\hat{\Sigma }}_1(x,s)$ to be

$$\begin{aligned} {\hat{\Sigma }}_0(x,s) = \sum _{s'} S^x_{ss'} \Sigma _0( x, s'), \end{aligned}$$

(E3a)

$$\begin{aligned} {\hat{\Sigma }}_1(x,s) = \sum _{s'} {\bar{S}}^x_{ss'} \Sigma _1( x, s'). \end{aligned}$$

(E3b)

For each x and s, let $B^\Sigma _{xs}$ be the algebra generated by ${\hat{\Sigma }}_0(x,s)$ and ${\hat{\Sigma }}_1(x,s)$. Then for $s \ne s'$, every $a_{xs} \in B^\Sigma _{xs}$ either commutes or anticommutes with every $a_{xs'} \in B^\Sigma _{xs'}$. Therefore the algebra $B^\Sigma _x$ is the product

$$\begin{aligned} B^\Sigma _x = \bigotimes _s B^\Sigma _{xs}. \end{aligned}$$

(E4)

Equation (E1) implies ${\mathcal {H}}^B$ is a tensor product

$$\begin{aligned} {\mathcal {H}}^B = {\mathcal {H}}^\Sigma \otimes {\mathcal {H}}^\Upsilon , \end{aligned}$$

(E5)

of a space generated by $B^\Sigma$ acting on $|\Omega ^B \rangle$ and a space generated by $B^\Upsilon$ acting on $|\Omega ^B \rangle$ and Eqs. (E2a) and (E2b) imply ${\mathcal {H}}^\Sigma$ and ${\mathcal {H}}^\Upsilon$ are themselves products of spaces ${\mathcal {H}}^\Sigma _x$ and ${\mathcal {H}}^\Upsilon _y$ generated, respectively, by $B^\Sigma _x$ and $B^\Upsilon _x$ acting on $|\Omega ^B \rangle$

$$\begin{aligned} {\mathcal {H}}^\Sigma= & {} \bigotimes _x {\mathcal {H}}^\Sigma _x, \end{aligned}$$

(E6a)

$$\begin{aligned} {\mathcal {H}}^\Upsilon= & {} \bigotimes _x {\mathcal {H}}^\Upsilon _x. \end{aligned}$$

(E6b)

Similarly, Eq. (E4) implies ${\mathcal {H}}^\Sigma _x$ is a product of ${\mathcal {H}}^\Sigma _{xs}$ generated by $B^\Sigma _{xs}$ acting on $|\Omega ^B \rangle$

$$\begin{aligned} {\mathcal {H}}^\Sigma _x = \bigotimes _s {\mathcal {H}}^\Sigma _{xs}. \end{aligned}$$

(E7)

For the pair of operators ${\hat{\Sigma }}_0(x,s)$ and ${\hat{\Sigma }}_1(x,s)$ which generate $B^\Sigma _{xs}$, Eqs. (153a) and (153c) become

$$\begin{aligned}{}[{\hat{\Sigma }}_0( x, s)] ^ 2= & {} 0, \end{aligned}$$

(E8a)

$$\begin{aligned}{}[{\hat{\Sigma }}_1( x, s)] ^ 2= & {} 0, \end{aligned}$$

(E8b)

$$\begin{aligned} \{{\hat{\Sigma }}_0( x, s), {\hat{\Sigma }}_0( x, s)\}= & {} \gamma ^0_{ss}. \end{aligned}$$

(E8c)

Equations (E8a)–(E8c) combined with approximate Lorentz and charge conjugation invariance of the complexity of states in ${\mathcal {H}}$ imply that for the field polynomials $P_i[ {\hat{\Sigma }}_0( x, s), {\hat{\Sigma }}_1( x, s)]$

$$\begin{aligned} P_0[ {\hat{\Sigma }}_0( x, s), {\hat{\Sigma }}_1( x, s)]= & {} 1, \end{aligned}$$

(E9a)

$$\begin{aligned} P_1[ {\hat{\Sigma }}_0( x, s), {\hat{\Sigma }}_1( x, s)]= & {} u {\hat{\Sigma }}_0( x, s), \end{aligned}$$

(E9b)

$$\begin{aligned} P_2[ {\hat{\Sigma }}_0( x, s), {\hat{\Sigma }}_1( x, s)]= & {} u {\hat{\Sigma }}_1( x, s), \end{aligned}$$

(E9c)

$$\begin{aligned} P_3[ {\hat{\Sigma }}_0( x, s), {\hat{\Sigma }}_1( x, s)]= & {} v [{\hat{\Sigma }}_0( x, s), {\hat{\Sigma }}_1( x, s)], \end{aligned}$$

(E9d)

where u and v are normalization constants independant of x and s, an orthonormal basis for ${\mathcal {H}}^\Sigma _{xs}$ must have the form

$$\begin{aligned} | x, s, i \rangle = P_i[ {\hat{\Sigma }}_0( x, s), {\hat{\Sigma }}_1( x, s)] |\Omega ^B \rangle , \end{aligned}$$

(E10)

up to an overall unitary rotation of the basis. Equations (E8a)–(E8c) imply the result of any other polynomial in ${\hat{\Sigma }}_0( x, s)$ and ${\hat{\Sigma }}_1( x, s)$ acting on $|\Omega ^B \rangle$ is equal to some corresponding linear combination of the $|x, s, i \rangle$ of Eq. (E10). The complexity of a state in ${\mathcal {H}}^B$ is independent of overall normalization, however, so u can be arbitrarily set to 1. The remaining constant v determines the contribution to complexity arising from sites occupied by more than a single fermion. In the continuum limit of complexity, if a continuum limit exists, the weight of multiply occupied sites in any state will go to 0. The continuum limit should therefore be independent of v.

For the pair of operators $\Upsilon _0(x)$ and $\Upsilon _1(x)$ which generate $B^\Upsilon _x$, Eq. (153d) becomes

$$\begin{aligned}{}[\Upsilon _0( x), \Upsilon _1( x)] = i. \end{aligned}$$

(E11)

Equation (E11) combined with approximate Lorentz and charge conjugation invariance of the complexity of states in ${\mathcal {H}}$ imply that, up to an overall unitary rotation of the basis, an orthonormal basis for ${\mathcal {H}}^\Upsilon _x$ will consist of a family of states $\{|x, n_0, n_1 \rangle ^\Upsilon \}$ labeled by a pair of nonnegative integers $n_0, n_1$. For each $n_0, n_1$ pair $P_{n_0 n_1}[\Upsilon _0(x),\Upsilon _1(x)]$ is an ordered product, independent of x, of $n_0$ copies of $\Upsilon _0(x)$ and $n_1$ copies of $\Upsilon _1(x)$ subject to the requirement

$$\begin{aligned} P_{n_0 n_1}[\Upsilon _0(x),\Upsilon _1(x)] = P_{n_1 n_0}[\Upsilon _1(x),\Upsilon _0(x)]. \end{aligned}$$

(E12)

The $\{|x, n_0, n_1 \rangle ^\Upsilon \}$ are given by

$$\begin{aligned} | x, n_0, n_1 \rangle ^\Upsilon = u_{n_0 n_1} P_{n_0 n_1}[\Upsilon _0(x),\Upsilon _1(x)] |\Omega \rangle ^B, \end{aligned}$$

(E13)

where the $u_{n_0 n_1}$ are normalization constants independant of x and symmetric in the indices $n_0, n_1$. Equation (E11) implies the result of any other polynomial in $\Upsilon _0( x)$ and $\Upsilon _1( x)$ acting on $|\Omega ^B \rangle$ is equal to some corresponding linear combination of the $|x, n_0, n_1 \rangle$ of Eq. (E13). To be consistent with the normalization choice for fermions, $u_{0 0}$, $u_{0 1}$ and $u_{1 0}$ will be set to 1. The remaining $u_{n_0 n_1}$ determine the contribution to complexity arising from sites occupied by more than a single boson and should have no effect on the continuum limit of complexity, if a continuum limit exists.

Equation (154) implies the $P_{n_0 n_1}[\Upsilon _0(x),\Upsilon _1(x)]$ identically vanish for $n_0 \ge n$ or $n_1 \ge n$.

The end result of Eqs. (E5)–(E7) is an ${\mathcal {H}}^B$ generated by the algebra B acting on $|\Omega ^B \rangle$ which is an ordered tensor product

$$\begin{aligned} {\mathcal {H}}^B = \bigotimes _x {\mathcal {H}}^B_x, \end{aligned}$$

(E14)

on which, according to Eqs. (E8a)–(E13), the $\Sigma _i( x, s), \Upsilon _i( x),$ satisfy Eqs. (153a)–(153d).

It is convenient to define at this point an orthonormal basis P for B. In particular, no linear combination of elements of P is 0 as a result of the anticommutation and commutation relations of Eqs. (153a)–(153d). Each $p \in P$ consists of a product of a $p^\Sigma \in P^\Sigma$ and a $p^\Upsilon \in P^\Upsilon$, where $P^\Sigma$ and $P^\Upsilon$ are orthonormal bases for the fermion field algebra $B^\Sigma$ and the boson field algebra $B^\Upsilon$, respectively. Each $p^\Sigma$ is defined to be a product over all distinct x and s of one of the fermion field combinations in Eqs. (E9a)–(E9d). Each $p^\Upsilon$ is defined to be a product over all distinct x of one of the normalized boson field combinations $u_{n_0 n_1} P_{n_0 n_1}[\Upsilon _0(x),\Upsilon _1(x)]$.

Appendix 6: Lower Bound on the Complexity of Entangled Relativistic States

The proof of Eq. (180) bounding from below the complexity of the entangled relativistic state $|\psi ^B \rangle$ of Eq. (178b) is a version of the proof in Appendix 2 of a lower bound on the complexity of the entangled non-relativistic state of Eq. (36), but with the regular lattice of Sect. 3.1 replaced by the random lattice of Sect. 13 and with the inclusion in ${\mathcal {H}}^B$ of anti-fermion states. The proof in Appendix 2 can be adapted to the presence of anti-fermion states in ${\mathcal {H}}^B$ by treating fermion-anti-fermion pairs in ${\mathcal {H}}^B$ following the treatment of bosons in Appendix 2. To do this we convert the complexity calculation in ${\mathcal {H}}^B$ into an equivalent complexity calculation in yet another auxiliary Hilbert space.

1.1 6.1 More Auxiliary Hilbert Spaces

For a trajectory $k^B(\nu ) \in K^B$, let $U_{k^B}(\nu )$ be the solution to

$$\begin{aligned} \frac{dU_{k^B}(\nu )}{d \nu }= & {} -i k^B( \nu ) U_{k^B}( \nu ), \end{aligned}$$

(F1a)

$$\begin{aligned} U_{k^B}( 0)= & {} I. \end{aligned}$$

(F1b)

Define $|\omega (\nu )^B \rangle$ to be

$$\begin{aligned} |\omega ^B( \nu ) \rangle = U_{k^B}(\nu )|\omega ^B \rangle . \end{aligned}$$

(F2)

for a product state $|\omega ^B(0) \rangle \in {\mathcal {H}}^B$

$$\begin{aligned} |\omega ^B(0) \rangle= & {} d_f( p_{j - 1}) \ldots d_f( p_0) d_{{\bar{f}}}( q_{k - 1}) \ldots d_{{\bar{f}}}( q_0) \nonumber \\&\times \quad d_b( r_{\ell -1}) \ldots d_b( r_0) |\Omega ^B \rangle , \end{aligned}$$

(F3)

with j fermions, k anti-fermions, and $\ell$ bosons, Assume that $|\omega ^B(0) \rangle$ and $k^B(\nu )$ have been chosen to give

$$\begin{aligned} |\omega (1)^B \rangle = \xi |\psi ^B \rangle , \end{aligned}$$

(F4)

for a phase factor $\xi$. Fermion number conservation by $k^B(\nu )$ implies $j - k$ must equal the fermion number n of $|\psi ^B \rangle$.

To deal with the presence of anti-fermions in ${\mathcal {H}}^B$, we will make use of yet one more auxiliary Hilbert space, ${\mathcal {H}}^C$, which consists purely of fermion states generated by all polynomials in an auxiliary field $\Sigma ^C_1(x,s)$ acting on an auxiliary vacuum $|\Omega ^C \rangle$. The tensor product ${\mathcal {H}}^C \otimes {\mathcal {H}}^B$ we name ${\mathcal {H}}^D$.

There is a natural map M from ${\mathcal {H}}^D$ to ${\mathcal {H}}^B$ defined by

$$\begin{aligned} M [ P( \Sigma _1^C) |\Omega ^{\mathcal {C}} \rangle \otimes |\psi ^B \rangle ] = P( \Sigma _1) |\psi ^B \rangle , \end{aligned}$$

(F5)

where $P( \Sigma _1^C)$ is a polynomial in the field $\Sigma _1^C(x, s)$, $P( \Sigma _1)$ is the corresponding polynomial but in the field $\Sigma _1(x, s)$ and $|\psi ^B \rangle$ is any state in ${\mathcal {H}}^B$. The map M takes a subspace of ${\mathcal {H}}^D$ to the null vector in ${\mathcal {H}}^B$ and thus does not have an inverse.

Corresponding to the decomposition of ${\mathcal {H}}^D$ and ${\mathcal {H}}^B$ as tensor products over all sites

$$\begin{aligned} {\mathcal {H}}^D= & {} \bigotimes _x {\mathcal {H}}^D_x, \end{aligned}$$

(F6a)

$$\begin{aligned} {\mathcal {H}}^B= & {} \bigotimes _x {\mathcal {H}}^B_x, \end{aligned}$$

(F6b)

the map M is given by the product

$$\begin{aligned} M = \prod _x M_x, \end{aligned}$$

(F7)

where each $M_x$ maps ${\mathcal {H}}^D_x$ to ${\mathcal {H}}^B_x$. The maps $M_x$ and $M_y$ for distinct x and y commute.

Let $K^D$ be the Hilbert space of Hermitian operators of Sect. 15 for ${\mathcal {H}}^D$ in place of ${\mathcal {H}}^B$ and with the additional requirement that $k^D \in K^D$ separately conserve both the fermion number $N^B$ of ${\mathcal {H}}^B$ and the fermion number $N^C$ of ${\mathcal {H}}^C$.

We now convert $k^B(\nu ) \in K^B, |\omega ^B(\nu ) \rangle \in {\mathcal {H}}^B$ connecting

$$\begin{aligned} |\omega ^B(1) \rangle = \xi |\psi ^B \rangle , \end{aligned}$$

(F8)

for a phase factor $\xi$, to the product state $|\omega ^B(0) \rangle$ into corresponding $k^D(\nu ) \in K^D, |\omega ^D(\nu ) \rangle \in {\mathcal {H}}^D$ connecting some $|\omega ^D(1) \rangle$ to a product state $|\omega ^D(0) \rangle$ along a path such that for $0 \le \nu \le 1$

$$\begin{aligned} M |\omega ^D(\nu ) \rangle= & {} |\omega ^B(\nu ) \rangle , \end{aligned}$$

(F9a)

$$\begin{aligned} \parallel k^D(\nu ) \parallel\le & {} 9 \parallel k^B(\nu ) \parallel . \end{aligned}$$

(F9b)

In addition, while $|\omega ^B(\nu ) \rangle$ is an eigenvector of $N^B$ with eigenvalue n, $|\omega ^D(\nu ) \rangle$ is an eigenvector of $N^B$ with eigenvalue 0 and of $N^C$ with eigenvalue n. Equation (F9b) implies

$$\begin{aligned}&C^D[ |\omega ^D(1) \rangle , |\omega ^D( 0) \rangle ] \le \nonumber \\&\quad 9 C^B[ |\omega ^B(1) \rangle , |\omega ^B( 0) \rangle ]. \end{aligned}$$

(F10)

Thus a lower bound on $C^D[ |\omega ^D(1) \rangle , |\omega ^D( 0) \rangle ]$ will give a lower bound on $C^B[ |\omega ^B(1) \rangle , |\omega ^B( 0) \rangle ]$.

Let the product state $|\omega ^D(0) \rangle$ be $|\omega ^C \rangle \otimes |\omega ^B \rangle$ where

$$\begin{aligned}&|\omega ^C \rangle = d^C_f( p_{n+m-1}) \ldots d^C_f( p_m) |\Omega ^C \rangle \end{aligned}$$

(F11a)

$$\begin{aligned} |\omega ^B \rangle= & {} d_f( p_{m-1}) \ldots d_f( p_0) d_{{\bar{f}}}( q_{m - 1}) \ldots d_{{\bar{f}}}( q_0) \nonumber \\&\quad \times d_b( r_{\ell -1}) \ldots d_b( r_0) |\Omega ^B \rangle . \end{aligned}$$

(F11b)

for $d_f( p_i)$, $d_{{\bar{f}}}( q_i)$ and $d_b( r_i)$ from Eq. (F3), and $d_f^C( p_i)$ constructed from $d_f( p_i)$ of Eq. (F3) by substituting $\Sigma _1^C(x, s)$ for $\Sigma _1(x, s)$.

Equations (F8) and (F9a) imply the state $|\omega ^D(1) \rangle$ will satisfy

$$\begin{aligned} M |\omega ^D(1) \rangle = \xi |\psi ^B \rangle . \end{aligned}$$

(F12)

In addition, since the trajectory $k^D(\nu )$ conserves $N^B$ and $N^C$ and $|\omega ^D(0) \rangle$, by Eqs. (F11a) and (F11b), has $N^B$ of 0 and $N^C$ of n, $|\omega ^D(1) \rangle$ must have these same eigenvalues. Also, since M acts only on the $\Sigma _1^C(x,s)$ fermion content of $|\omega ^D(1) \rangle$ and $|\psi \rangle ^B$, by Eqs. (178a) and (178b), has no boson content and no $\Sigma _0^B(x,s)$ anti-fermion content, $|\omega ^D(1) \rangle$ must have no boson content, no $\Sigma _0^B(x,s)$ and $\Sigma _1^B(x,s)$ content and be given instead by

$$\begin{aligned} |\omega ^D(1) \rangle = \xi |\psi ^C \rangle \otimes |\Omega ^B \rangle , \end{aligned}$$

(F13)

where $|\psi ^C \rangle$ is

$$\begin{aligned} q^C= & {} z^{-1} m^{-\frac{1}{2}}\sum _{0 \le i < m} \zeta _i p^C_i, \end{aligned}$$

(F14a)

$$\begin{aligned} |\psi ^C \rangle= & {} q^C|\Omega ^C \rangle , \end{aligned}$$

(F14b)

for $p^C_i$ given by

$$\begin{aligned} p^C_i = V^{-\frac{n}{2}}\prod _{0 \le j <n} \left[ \sum _{x \in D_{ij},k} u^k(x) \Sigma ^C_1( x,k )\right] , \end{aligned}$$

(F15)

for the same $\zeta _i, u^k(x)$ and $D_{ij}$ in Eqs. (175)–(178b) for $|\psi ^B \rangle$.

For both the nonrelativistic version of complexity in Sect. 3.3 and the relativistic version in Sect. 16, $C( |\psi \rangle )$ is actually independent of the normalization of $|\psi \rangle$. We can therefore safely set z to 1 in Eq. (F14a). The result is that $|\omega ^D(1) \rangle$ in Eq. (F13) is normalized to 1, which for consistency we now assume also for $|\omega ^D(0) \rangle$.

Now approximate Eqs. (17a), (17b), (B1) and (B2) for $|\omega ^B(\nu ) \rangle$ by a series of discrete steps

$$\begin{aligned} |\omega ^B( \nu + \delta ) \rangle = [1 -i\delta k^B( \nu )] |\omega ^B( \nu ) \rangle . \end{aligned}$$

(F16)

We will prove Eqs. (F9a) and (F9b) by induction in $\nu$. Eqs. (F5), (F3), (F11a) and (F11b) give Eq. (F9a) and (F9b) for $\nu = 0$

Now assume $k^D( \nu )$ satisfying Eq. (F9b) has been found for $\nu < \nu '$ such that $|\omega ^D(\nu ) \rangle$ given by

$$\begin{aligned} |\omega ^D( \nu + \delta ) \rangle = [1 -i\delta k^D( \nu )] |\omega ^D( \nu ) \rangle , \end{aligned}$$

(F17)

satisfies Eq. (F9a) for $\nu \le \nu '$. We will show that a $k^D( \nu ')$ exists satisfying Eq. (F9b) and extending Eq. (F9a) to $\nu ' + \delta$.

According to Eq. (A8), $k^B$ in Eq. (F16) consists of a sum of operators of the form

$$\begin{aligned} {\hat{f}}^B_{xy} = f^B_{xy} \bigotimes _{q \ne x,y} I_q, \end{aligned}$$

(F18)

where $f^B_{xy}$ is a Hermitian operator on ${\mathcal {H}}^B_x \otimes {\mathcal {H}}^B_y$ for a pair of nearest neighbor sites $\{x,y\}$ which conserves $N^B$ and has vanishing partial traces for both ${\mathcal {H}}^B_x$ and ${\mathcal {H}}^B_y$. We assume the dimension $d_{\mathcal {H}}$ of ${\mathcal {H}}^D_x$, and the corresponding slightly smaller dimension of ${\mathcal {H}}^B_x$, are large enough that the contribution to $k^B$ of single site operators of the form given in Eq. (A7a) can be neglected.

Then the required $k^D( \nu ')$ can be found if for every allowed ${\hat{f}}^B_{xy}$ there is a ${\hat{f}}^D_{xy}$ of the form

$$\begin{aligned} {\hat{f}}^D_{xy} = f^D_{xy} \bigotimes _{q \ne x,y} I^D_q, \end{aligned}$$

(F19)

where $f^D_{xy}$ is a Hermitian operator on ${\mathcal {H}}^D_x \otimes {\mathcal {H}}^D_y$ which has vanishing partial traces for both ${\mathcal {H}}^D_x$ and ${\mathcal {H}}^D_y$, conserves $N^B$ and $N^C$ and for which

$$\begin{aligned} M {\hat{f}}^D_{xy}|\omega ^D(\nu ') \rangle= & {} {\hat{f}}^B_{xy}|\omega ^B(\nu ') \rangle , \end{aligned}$$

(F20a)

$$\begin{aligned} \parallel f^D_{xy} \parallel\le & {} 9 \parallel f^B_{xy} \parallel . \end{aligned}$$

(F20b)

To find the required $f^D_{xy}$, decompose ${\mathcal {H}}^D_x \otimes {\mathcal {H}}^D_y$ into a direct sum of subspaces

$$\begin{aligned} {\mathcal {H}}^D_x \otimes {\mathcal {H}}^D_y = \oplus _{mn} {\mathcal {H}}^D_{mn}, \end{aligned}$$

(F21)

with eigenvalues m and n of $N^B$ and $N^C$, respectively. Similarly, decompose ${\mathcal {H}}^B_x \otimes {\mathcal {H}}^B_y$ into a direct sum of subspaces

$$\begin{aligned} {\mathcal {H}}^B_x \otimes {\mathcal {H}}^B_y = \oplus _m {\mathcal {H}}^B_m, \end{aligned}$$

(F22)

with eigenvalue m of $N^B$.

Let $P^D_{mn}$ be the projection operator onto ${\mathcal {H}}^D_{mn}$. Define $M_{mn}$ to be

$$\begin{aligned} M_{mn} = M_x M_y P^D_{mn}, \end{aligned}$$

(F23)

for $M_x$ and $M_y$ from Eq. (F7). Then $M_{mn}$ maps ${\mathcal {H}}^D_{mn}$ onto ${\mathcal {H}}^B_{m+n}$. Let ${\mathcal {H}}^{D\perp }_{mn}$ be the orthogonal complement of the subspace of ${\mathcal {H}}^D_{mn}$ which is mapped to 0 by $M_{mn}$. For each $|\psi ^B \rangle \in {\mathcal {H}}^B_{m+n}$ there is a unique $|\psi ^D \rangle \in {\mathcal {H}}^{D\perp }_{mn}$ such that

$$\begin{aligned} M_x M_y |\psi ^D \rangle = |\psi ^B \rangle . \end{aligned}$$

(F24)

For each such $|\psi ^B \rangle$, define $M_{mn}^{-1}$

$$\begin{aligned} M_{mn}^{-1}|\psi ^B \rangle = |\psi ^D \rangle , \end{aligned}$$

(F25)

and for any $|\psi ^B \rangle \in {\mathcal {H}}^B_\ell$ with $\ell$ other than $m + n$

$$\begin{aligned} M_{mn}^{-1}|\psi ^B \rangle = 0. \end{aligned}$$

(F26)

Equations (F24)–(F26) imply

$$\begin{aligned} M_x M_y M_{mn}^{-1} = P^B_{m+n}, \end{aligned}$$

(F27)

where $P^B_{m+n}$ is the projection operator onto ${\mathcal {H}}^B_{m+n}$. Define $g^D_{xy}$ to be

$$\begin{aligned} g^D_{xy} = \sum _{mn} M_{mn}^{-1} f^B_{xy} M_{mn}. \end{aligned}$$

(F28)

By Eq. (F23), $g^D_{xy}$ maps each ${\mathcal {H}}^D_{mn}$ into itself and therefore conserves both $N^C$ and $N^B$.

We then have

$$\begin{aligned} M_x M_y g^D_{xy}= & {} \sum _{mn} P^B_{m+n} f^B_{xy} M_{mn}, \nonumber \\ \quad= & {} f^B_{xy} \sum _{mn} M_{mn}, \nonumber \\ \quad= & {} f^B_{xy} M_x M_y, \end{aligned}$$

(F29)

where the first line follows from by Eq. (F27) and the second follows because $M_{mn}$ maps onto ${\mathcal {H}}^B_{m+n}$ and $f^B_{xy}$ conserves $N_B$. Equations (F7), (F29) and the induction hypothesis, Eq. (F9a) for $\nu '$, give

$$\begin{aligned}&M {\hat{g}}^D_{xy} |\omega ^D(\nu ') \rangle = {\hat{f}}^B_{xy} M|\omega ^D(\nu ') \rangle \nonumber \\&\quad = {\hat{f}}^B_{xy} |\omega ^B(\nu ') \rangle , \end{aligned}$$

(F30)

which is Eq. (F20a).

In addition, since $M_{mn}^{-1}$ maps into ${\mathcal {H}}^D_{mn}$, Eqs. (F23) and (F27) imply

$$\begin{aligned} M_{m'n'} M_{mn}^{-1} = \delta _{m'm} \delta _{n'n} P^B_{m + n}. \end{aligned}$$

(F31)

We then have

$$\begin{aligned} \mathrm {Tr}^D_{xy} ( g^D_{xy})^2= & {} \sum _{mn} \mathrm {Tr}^B_{xy}[ P^B_{m +n} f^B_{xy}P^B_{m+n} f^B_{xy}] \nonumber \\ \quad= & {} \sum _{mn} \mathrm {Tr}^B_{xy}[ P^B_{m +n} (f^B_{xy})^2] \end{aligned}$$

(F32)

where the first line follows from Eqs. (F28) and (F31) and the second holds because $f^B_{xy}$ conserves $N^B$. Since the index s of $\Sigma _1(x,s)$ is in the range $0 \le s < 4$, the maximum possible value of $N^B$ for x and y together is 8. As a result there are at most 9 different combinations of m and n giving any value of $m + n$. Equation (F32) then implies

$$\begin{aligned} \mathrm {Tr}^D_{xy} ( g^D_{xy})^2 \le 9 \mathrm {Tr}^B_{xy}(f^B_{xy})^2, \end{aligned}$$

(F33)

which is Eq. (F20b).

Finally, $g^D_{xy}$ can be split into

$$\begin{aligned}&g^D_{xy} = f^D_{xy} + \frac{1}{\sqrt{d_{\mathcal {H}}}} I_x \otimes f^D_y + \frac{1}{\sqrt{d_{\mathcal {H}}}} f^D_x \otimes I_y + \frac{1}{d_{\mathcal {H}}} f^D I_x \otimes I_y, \end{aligned}$$

(F34)

where

$$\begin{aligned} \mathrm {Tr}^D_x f^D_{xy}= & {} 0, \end{aligned}$$

(F35a)

$$\begin{aligned} \mathrm {Tr}^D_y f^D_{xy}= & {} 0, \end{aligned}$$

(F35b)

$$\begin{aligned} \mathrm {Tr}^D_y f^D_y= & {} 0, \end{aligned}$$

(F35c)

$$\begin{aligned} \mathrm {Tr}^D_x f^D_x= & {} 0, \end{aligned}$$

(F35d)

and $d_{\mathcal {H}}$ is the dimension of each ${\mathcal {H}}^D_x$. Eqs. (F34)–(F35d) imply

$$\begin{aligned}&\mathrm {Tr}^D_{xy}(g^D_{xy})^2 = \mathrm {Tr}^D_{xy}(f^D_{xy})^2 + \mathrm {Tr}^D_y(f^D_y)^2 + \nonumber \\&\quad \mathrm {Tr}^D_x(f^D_x)^2 + (f^D)^2. \end{aligned}$$

(F36)

Equations (F36) and (F33) imply $f^D_{xy}$ satisfies Eq. (F20b). For $d_{\mathcal {H}}$ large enough, Eqs. (F34) and (F30) imply ${\hat{f}}^D_{xy}$ satisfies Eq. (F20a).

Which completes the induction step of the proof of Eqs. (F9a) and (F9b). Equation ( F10) follows. To obtain a lower bound on $C^B[ |\omega ^B(1) \rangle , |\omega ^B( 0) \rangle ]$ we now derive a lower bound on $C^D[ |\omega ^D(1) \rangle , |\omega ^D( 0) \rangle ]$.

1.2 6.2 Schmidt Spectra Again

From each region $D_{ij}$, extract a subset ${\hat{D}}_{ij}$, consisting of the center points x of all cells c(x) reached by starting at $y_{ij}$ and traveling along a geodesic in $L(\tau , \sigma )$ in the positive or negative $x_1$ direction a number $\le \frac{d}{2 \rho }$ of discrete steps each of proper length $2 \rho$, then traveling along a geodesic in the positive or negative $x_2$ direction a number $\le \frac{d}{2 \rho }$ of discrete steps each of proper length $2 \rho$, then traveling along a geodesic in the positive or negative $x_3$ direction a number $\le \frac{d}{2 \rho }$ of discrete steps each of proper length $2 \rho$. Since each c(x) is contained within a sphere of radius $\rho$ around its center point, none of the points in ${\hat{D}}_{ij}$ will be nearest neighbors and for large d, the total number of points in each ${\hat{D}}_{ij}$ will be nearly $\frac{d^3}{\rho ^3}$. Since V is between $\frac{48 d^3}{\pi \rho ^3}$ and $\frac{ 6 d^3}{\pi \rho ^3}$, the number of points in each ${\hat{D}}_{ij}$ is zV for z between $\frac{\pi }{6}$ and $\frac{\pi }{48}$. We will assume V is large enough that we can ignore the statistical uncertainty in the number of points in each ${\hat{D}}_{ij}$.

From this set of ${\hat{D}}_{ij}$ construct a set of subsets $E_\ell$ each consisting of 2n distinct points chosen from 2n distinct ${\hat{D}}_{ij}$. Since there are mn sets ${\hat{D}}_{ij}$, there will be $\frac{z m V}{2}$ sets $E_\ell$.

For each $E_\ell$ form the tensor product spaces

$$\begin{aligned} {\mathcal {Q}}_\ell= & {} \bigotimes _{x \in E_\ell } {\mathcal {H}}_x^C, \end{aligned}$$

(F37a)

$$\begin{aligned} {\mathcal {R}}_\ell= & {} [\bigotimes _{x \ne E_\ell } {\mathcal {H}}_x^C] \otimes {\mathcal {H}}^B. \end{aligned}$$

(F37b)

The space ${\mathcal {Q}}_\ell$ has dimension $16^{2n}$ and ${\mathcal {H}}^D$ becomes

$$\begin{aligned} {\mathcal {H}}^D = {\mathcal {Q}}_\ell \otimes {\mathcal {R}}_\ell . \end{aligned}$$

(F38)

A Schmidt decomposition of $|\omega ^D(\nu ) \rangle$ according to Eq. (F38) then becomes

$$\begin{aligned} |\omega ^D(\nu ) \rangle = \sum _j \lambda _{j\ell }(\nu ) |\phi _{j\ell }(\nu ) \rangle |\chi _{j\ell }(\nu ) \rangle , \end{aligned}$$

(F39)

where

$$\begin{aligned} |\phi _{j\ell }(\nu ) \rangle\in & {} {\mathcal {Q}}_\ell , \end{aligned}$$

(F40a)

$$\begin{aligned} |\chi _{j\ell }( \nu ) \rangle\in & {} {\mathcal {R}}_\ell , \end{aligned}$$

(F40b)

for $0 \le j < 16^{2n}$ and real non-negative $\lambda _{j\ell }( \nu )$ which fulfill the normalization condition

$$\begin{aligned} \sum _j [ \lambda _{j\ell }( \nu )]^2 = 1. \end{aligned}$$

(F41)

Each $|\phi _{j\ell }(\nu ) \rangle$ is a pure fermion state while the $|\chi _{j\ell }(\nu ) \rangle$ can include fermions, antifermions and bosons.

The fermion number operators $N^C[{\mathcal {Q}}_\ell ]$ and $N^C[{\mathcal {R}}_\ell ]$ commute and $|\omega ^D(\nu ) \rangle$ is an eigenvector of the sum with eigenvalue n. It follows that the decomposition of Eq. (F39) can be done with $|\phi _{j\ell }( \nu ) \rangle$ and $|\chi _{j\ell }(\nu ) \rangle$ eigenvectors of $N^C[{\mathcal {Q}}_\ell ]$ and $N^C[{\mathcal {R}}_\ell ]$, respectively, with eigenvalues summing to n. Let $|\phi _{0\ell } \rangle$ be $|\Omega _\ell \rangle$, the vacuum state of ${\mathcal {Q}}_\ell$, and let $|\phi _{i\ell } (\nu ) \rangle , 1 \le i \le 8n$, span the 8n-dimensional subspace of ${\mathcal {Q}}_\ell$ with $N^C[{\mathcal {Q}}_\ell ]$ of 1. We assume the corresponding $\lambda _{i\ell }( \nu ), 1 \le i \le 8n$, are in nonincreasing order. Consider Eqs. (F13)–(F15) for $|\omega ^D(1) \rangle$. For any choice of $\ell$ there will be a set of 2n nonzero orthogonal $|\phi _{1\ell }( 1) \rangle , \ldots |\phi _{2n\ell }( 1) \rangle$ with

$$\begin{aligned} \lambda _{j\ell }( 1) = \sqrt{\frac{1}{mV}}. \end{aligned}$$

(F42)

On the other hand, for the product state $|\omega ^D(0) \rangle$ in Eqs. (F11a) and (F11b), the $|\phi _{j\ell }(\nu ) \rangle$ come entirely from $|\omega ^C \rangle$, which is a product of n independent single fermion states. The space spanned by the projection of these into some ${\mathcal {Q}}_\ell$ is at most n dimensional, and as a result at most n orthogonal $|\phi _{1\ell }(0) \rangle ,\ldots |\phi _{n\ell }(0) \rangle$ can occur. Therefore at $\nu = 0$, there will be at most n nonzero $\lambda _{1\ell }(0), \ldots \lambda _{n\ell }(0)$. For $n < j \le 8n$, we have

$$\begin{aligned} \lambda _{j\ell }( 0) = 0. \end{aligned}$$

(F43)

Since $\{\lambda _{j\ell }( \nu )\}$ is a unit vector, Eqs. (F42) and (F43) imply that as $\nu$ goes from 0 to 1, $\{\lambda _{j\ell }( \nu )\}$ must rotate through a total angle of at least $\arcsin (\sqrt{\frac{n}{mV}})$.

For the small interval from $\nu$ to $\nu + \delta \nu$ let $\mu _{j\ell }(\nu )$ and $\theta _{\ell }(\nu )$ be

$$\begin{aligned} \lambda _{j\ell }(\nu + \delta \nu )= & {} \lambda _{j\ell }( \nu ) + \delta \nu \mu _{j\ell }(\nu ), \end{aligned}$$

(F44a)

$$\begin{aligned} \theta _{\ell }( \nu )^2= & {} \sum _j [ \mu _{j\ell }(\nu )]^2. \end{aligned}$$

(F44b)

We then have

$$\begin{aligned} \int _0^1 | \theta _{\ell }(\nu )| d \nu \ge \arcsin (\sqrt{\frac{n}{mV}}). \end{aligned}$$

(F45)

Summed over the $\frac{zmV}{2}$ values of $\ell$, Eq. (F45) becomes

$$\begin{aligned} \sum _{\ell } \int _0^1 | \theta _{\ell }(\nu )| d \nu \ge \frac{z m V}{2} \arcsin \left( \sqrt{\frac{n}{mV}}\right) , \end{aligned}$$

(F46)

and therefore

$$\begin{aligned} \sum _{\ell } \int _0^1 | \theta _{\ell }(\nu )| d \nu\ge & {} \frac{r}{\pi } \sqrt{mnV}, \end{aligned}$$

(F47a)

$$\begin{aligned}\ge & {} \frac{1}{48} \sqrt{mnV}, \end{aligned}$$

(F47b)

since z is greater than $\frac{\pi }{48}$.

1.3 6.3 Rotation Matrix and Rotation Angle Bounds

The rest of the proof of the lower bound on relativistic complexity, Eq. (180), is a copy of Sections 2.3 and 2.4 of the proof in Appendix 2 of the non-relativistic lower bound, Eq. (37), but with the non-relativistic fermion charge N and Hilbert spaces ${\mathcal {H}}^f$ and ${\mathcal {H}}^b$ replaced, respectively, by $N^C$, ${\mathcal {H}}^C$ and ${\mathcal {H}}^B$.

As in Appendix 2.3, the rotation of $\lambda _{j \ell }(\nu )$ during the interval from $\nu$ to $\nu + \delta \nu$ will be determined by the sum $g^D_{\ell }(\nu )$ of all contributions to $k^D(\nu )$ of Eq. (F17) arising from $f^D_{xy}$ for nearest neighbor pairs $\{x,y\}$ with one point, say x, in $E_\ell$. By construction of the $E_\ell$, if x is in $E_\ell$, y can not be in $E_\ell$ or any distinct $E_{\ell '}$. A repeat of the derivation of Eqs. (B36)–(B38) then leads to

$$\begin{aligned} \mu _{j\ell }(\nu ) = \sum _k r_{jk\ell }(\nu ) \lambda _{k\ell }(\nu ), \end{aligned}$$

(F48)

for the rotation matrix $r_{jk\ell }(\nu )$

$$\begin{aligned}&r_{jk\ell }(\nu ) = -{\text {Im}}[ \langle \phi _{j\ell }(\nu )| \langle \chi _{j\ell }(\nu )| g^D_{\ell }(\nu )|\phi _{k\ell }(\nu ) \rangle |\chi _{k\ell }(\nu ) \rangle ], \end{aligned}$$

(F49)

for $|\phi _{k\ell }(\nu ) \rangle$ and $|\chi _{k\ell }(\nu ) \rangle$ of Eq. (F39) and $\mu _{j\ell }(\nu )$ of Eq. (F44a).

Since the $f^D_{xy}$ contributing to $g^D_\ell (\nu )$ conserve total fermion number $N^C$, $g^D_\ell (\nu )$ can be expanded as

$$\begin{aligned} g^D_{\ell }(\nu )= & {} \sum _{x \in E_\ell , y \notin E_\ell } g^D_{\ell }( x, y, \nu ), \end{aligned}$$

(F50a)

$$\begin{aligned} g^D_{\ell }(x,y,\nu )= & {} \sum _{i = 0,1} a^i(x, y, \nu ) z^i(x, y, \nu ) \end{aligned}$$

(F50b)

where $z^0( x, y, \nu )$ acts only on states with $N^C( {\mathcal {H}}^D_x \otimes {\mathcal {H}}^D_y)$ of 0, $z^1( x, y, \nu )$ acts only on states with $N^C( {\mathcal {H}}^D_x \otimes {\mathcal {H}}^D_y)$ strictly greater than 0, and the $z^i(x,y,\nu )$ are normalized by

$$\begin{aligned} \parallel z^i(x, y, \nu ) \parallel = 1. \end{aligned}$$

(F51)

The operator $z^0(x, y, \nu )$ will be

$$\begin{aligned} z^0(x,y,\nu )= & {} z^{0C}(x,y) \otimes g^B(x,y,\nu ), \end{aligned}$$

(F52a)

$$\begin{aligned} z^{0C}(x,y,\nu )= & {} P^C(x,y) \bigotimes _{q \ne x,y} I_q, \end{aligned}$$

(F52b)

where $P^C(x,y)$ projects onto the vacuum state of ${\mathcal {H}}^C_x \otimes {\mathcal {H}}^C_y$ and $g^B(x,y,\nu )$ is a normalized Hermitian operator acting on ${\mathcal {H}}^B_x \otimes {\mathcal {H}}^B_y$

Combining Eqs. (F44b),(F48) - (F50b) gives

$$\begin{aligned} |\theta _\ell (\nu )|\le & {} \sum _{x \in E, y \notin E, i}|\theta ^i_{\ell }(x,y,\nu )| \end{aligned}$$

(F53a)

$$\begin{aligned}{}[\theta ^i_{\ell }( x,y,\nu )]^2= & {} \sum _j [ \mu ^i_{j\ell }(x,y,\nu )]^2, \end{aligned}$$

(F53b)

with the definitions

$$\begin{aligned} \mu ^i_{j\ell }(x,y,\nu )= & {} -a^i(x,y,\nu ) \sum _k {\text {Im}}\{ \langle \phi _{j\ell }(\nu )| \langle \chi _{j\ell }(\nu )| \nonumber \\&\quad \times z^i(x,y,\nu )|\phi _{k\ell }(\nu ) \rangle |\chi _{k\ell }(\nu ) \rangle \lambda _{k\ell }(\nu )\}. \end{aligned}$$

(F54)

A duplicate of the proof of Eqs. (B46)–(B52) then yields

$$\begin{aligned}&[\theta ^0_{\ell }(x,y,\nu )]^2 \le [a^0(x,y,\nu )]^2 \langle \omega (\nu )|[I - z^{0C}(x,y)]|\omega (\nu ) \rangle , \end{aligned}$$

(F55a)

$$\begin{aligned}&[\theta ^1_{\ell }(x,y,\nu )]^2 \le [a^1(x,y,\nu )]^2 \langle \omega (\nu )|[I - z^{0C}(x,y)]|\omega (\nu ) \rangle , \end{aligned}$$

(F55b)

which combined with Eq. (F53a) imply

$$\begin{aligned} \sum _{\ell } |\theta _{\ell }(\nu )|\le & {} \sum _{x \in E, y \notin E} \{ [|a^0(x,y,\nu )| + |a^1(x,y,\nu )|] \nonumber \\&\quad \times \sqrt{ \langle \omega (\nu )| [I - z^{0C}(x,y)]|\omega (\nu ) \rangle } \}. \end{aligned}$$

(F56)

where

$$\begin{aligned} E = \cup _{\ell } E_\ell \end{aligned}$$

(F57)

The Cauchy–Schwartz inequality then gives

$$\begin{aligned}{}[\sum _{\ell } |\theta _{\ell }(\nu )|] ^ 2\le & {} \sum _{x \in E, y \notin E} [|a^0(x,y,\nu )| + |a^1(x,y,\nu )|]^2\nonumber \\&\quad \times \sum _{x \in E, y \notin E} \langle \omega (\nu )| [I - z^{0C}(x,y)]|\omega (\nu ) \rangle . \end{aligned}$$

(F58)

A repeat of the argument leading to Eq. (B56) implies

$$\begin{aligned} \sum _{x \in E, y \notin E} \langle \omega (\nu )| [I - z^{0C}(x,y)]|\omega (\nu ) \rangle \le Mn, \end{aligned}$$

(F59)

where M is the maximum number of nearest neighbors of any lattice point x. An upper bound on M can be found as follows. Recall any c(x) is contained in a sphere with center x and radius $\rho$ and contains a sphere with center x and radius $\frac{\rho }{2}$. It follows that M is less than or equal to the number of disjoint spheres of radius $\frac{\rho }{2}$ that can placed with centers on a sphere with center x and radius $2 \rho$. For each of the $\frac{\rho }{2}$ spheres, a slice through its center orthogonal to the line from its center to x will be contained in a sphere with center x and radius $\frac{\sqrt{17}}{2}$. The area of each of these slices is $\frac{\pi \rho ^2}{4}$, the area of the radius $\frac{\sqrt{17}}{2}$ sphere is $\frac{68 \pi \rho ^2}{4}$, and therefore

$$\begin{aligned} M \le 68. \end{aligned}$$

(F60)

By Eq. (16)

$$\begin{aligned} \parallel k^D(\nu ) \parallel ^ 2 \ge \sum _{\ell , x \in E, y \notin E} \parallel g^D_\ell ( x, y, \nu ) \parallel ^2 \end{aligned}$$

(F61)

In addition, $z^0(x,y,\nu )$ is orthogonal to $z^1(x, y, \nu )$. It follows that

$$\begin{aligned} \parallel k^D(\nu ) \parallel ^2 \ge \sum _{x \in E, y \notin E} [|a^0(x,y,\nu )|^2 + |a^1(x,y,\nu )|^2]. \end{aligned}$$

(F62)

Assembling Eqs. (F58), (F59), (F60) and (F62) gives

$$\begin{aligned} \parallel k^D(\nu ) \parallel ^2\ge & {} \frac{1}{2} \sum _{x \in E, y \notin E} [|a^0(x,y,\nu )| + |a^1(x,y,\nu )|]^2 \nonumber \\\ge & {} \frac{1}{136 n} [\sum _{\ell } |\theta _{\ell }(\nu )|] ^ 2 \end{aligned}$$

(F63)

Eq. (F47b) then implies

$$\begin{aligned} \int _0^1 \parallel k(\nu ) \parallel \ge \frac{1}{2348} \sqrt{mV}, \end{aligned}$$

(F64)

and therefore

$$\begin{aligned} C^D( |\omega ^D(1) \rangle , |\omega ^D(0) \rangle ) \ge \frac{1}{2348}\sqrt{ mV}, \end{aligned}$$

(F65)

which by Eqs. (F10) and (F8) yields

$$\begin{aligned} C^B( |\psi ^B \rangle , |\omega ^B(0) \rangle ) \ge \frac{1}{21132}\sqrt{ mV}. \end{aligned}$$

(F66)

Since Eq. (F66) holds for all product $|\omega ^B(0) \rangle$ we finally obtain

$$\begin{aligned} C^B( |\psi ^B \rangle ) \ge \frac{1}{21132} \sqrt{ mV}. \end{aligned}$$

(F67)

Appendix 7: Upper Bound on the Complexity of Entangled Relativistic States

The proof of Eq. (181) bounding from above the complexity of the entangled relativistic state $|\psi ^B \rangle$ of Eq. (178b) follows the proof in Appendix 3 of an upper bound on the complexity of the entangled non-relativistic state of Eq. (36), but with the regular lattice of Sect. 3.1 replaced by the random lattice of Sect. 13 and ${\mathcal {H}}$ replaced by ${\mathcal {H}}^B$.

An upper bound on $C^B( |\psi ^B \rangle )$ is given by $C^B( |\psi ^B \rangle , |\omega ^B \rangle )$ for any product state $|\omega ^B \rangle$, for which in turn an upper bound is given by

$$\begin{aligned} C^B( |\psi ^B \rangle , |\omega ^B \rangle ) \le \int _0^1 d t \parallel k^B( \nu ) \parallel , \end{aligned}$$

(G1)

for any trajectory $k^B(\nu ) \in K^B$ fulfilling

$$\begin{aligned} \frac{d\omega ^B(\nu )}{d \nu }= & {} -i k^B( \nu ) \omega ^B( \nu ), \end{aligned}$$

(G2a)

$$\begin{aligned} \omega ^B( 0)= & {} |\omega ^B \rangle , \end{aligned}$$

(G2b)

$$\begin{aligned} \omega ^B( 1)= & {} \xi |\psi ^B \rangle , \end{aligned}$$

(G2c)

for a phase factor $\xi$.

As in Appendix 3, construction of a sufficient $k^B(\nu )$ begins with an $|\omega ^B \rangle$ consisting of n fermions each at one of a corresponding set of n single points. Then $|\omega ^B \rangle$ is split into a sum of m orthogonal product states, each again consisting of n fermions one at each of a corresponding set of n single points. Then the position of each of the fermions in the product states is moved to the center of of one of the monomials of Eq. (177). Finally, by approximately $\ln ( V) / \ln ( 8)$ iterations of a fan-out tree, the mn wave functions concentrated at points are spread over the mn cubes $D_{ij}$.

1.1 7.1 Cell Count Bound

The bound on $C^B( |\psi ^B \rangle )$ relies on a bound we will derive first on the number of distinct cells $c(x), x \in L( \tau , \sigma , \rho )$, which intersect a geodesic $g( \lambda ) \in L( \tau , \sigma ), 0 \le \lambda \le \lambda _{max},$ of length $\lambda _{max}$.

Let ${\bar{g}}$ be the set of all points within a proper distance $2 \rho$ of any point on $g(\lambda )$. Since every c(x) is contained in a sphere with center x and radius $\rho$, it follows that ${\bar{g}}$ contains all c(x) which intersect $g(\lambda )$. On the other hand, each c(x) within ${\bar{g}}$ contains a sphere with center x and radius $\frac{\rho }{2}$ which is disjoint from all other $c(x')$ contained in ${\bar{g}}$. The total volume occupied by the collection of disjoint radius $\frac{\rho }{2}$ spheres has to be less than the total volume of ${\bar{g}}$. The number $p(\lambda _{max})$ of c(x) which intersect $g(\lambda )$ is therefore bounded by

$$\begin{aligned} p( \lambda _{max}) \le 24 \frac{ \lambda _{max}}{\rho } + 64. \end{aligned}$$

(G3)

1.2 7.2 Product State to Entangled State

For each value of $0 \le i < m$, let $x_{i0}$ be the center point of the cell found by traveling from an arbitrarily chosen starting point, $x_{00}$, along a geodesic in the $x^1$ direction a proper distance of $4 i \rho$. Then from each $x_{i0}$ travel along a geodesic in the $x^2$ direction. For each $0< j < n$, let $x_{ij}$ be the center point of the cell the geodesic beginning at $x_{i0}$ enters after leaving the cell with center point $x_{ij-1}$. All points on the geodesics beginning at $x_{i0}$ and at $x_{i'0}$ for $i \ne i'$ will be at least a distance of $4 \rho$ apart. As a result each $x_{ij}$ will be both distinct from and not a nearest neighbor of each $x_{i'j'}$ with $i \ne i'$. The gap between $x_{ij}$ and $x_{i'j'}$ accomplishes the goal of making it possible, despite the random placement of cells, to insure that $x^B_{ij}$ and $x^B_{ij+1}$ are nearest neighbors as will turn out to be required.

Let the set of n-particle product states $|\omega ^B_i \rangle$ be

$$\begin{aligned} |\omega ^B_i \rangle = \prod _{0 \le j < n} [ \sum _k u^k(x_{ij}) \Sigma _1( x_{ij}, k)] |\Omega ^B \rangle . \end{aligned}$$

(G4)

The entangle n-particle state $|\chi ^B \rangle$

$$\begin{aligned} |\chi ^B \rangle = \sqrt{\frac{1}{m}} \sum _i |\omega ^B_i \rangle \end{aligned}$$

(G5)

we generate from a sequence of unitary transforms acting on $|\omega ^B \rangle = |\omega ^B_0 \rangle$.

The sequence of $k^B$ which convert the product state $|\omega ^B \rangle$ into the entangled state $|\chi ^B \rangle$ follows the sequence of k mapping the product state $|\omega \rangle$ to the entangled state $|\chi \rangle$ in Appendix 3.1, with the non-relativistic fermion operator $\Psi ^\dagger ( x, s)$ replaced by the relativistic ${\hat{\Sigma }}_1( x, s)$.

From $k^B_0, \ldots k^B_{n-2}$ in place of $k_0, \ldots k_{n-2}$ of Eqs. (C5)–(C13) we obtain

$$\begin{aligned}&\exp ( i \theta ^B_{n-2} k^B_{n-2}) \ldots \exp ( i \theta _0 ^Bk^B_0) |\omega ^B_0 \rangle \nonumber \\&\qquad = \sqrt{\frac{1}{m}} |\omega _0 \rangle + \sqrt{\frac{m - 1}{m}} \prod _{0 \le j < n} [\sum _k v^k(x_{0j}) \Sigma _1( x_{0j}, k)] |\Omega ^B \rangle , \end{aligned}$$

(G6)

with

$$\begin{aligned} \parallel k_i^B \parallel= & {} \sqrt{2}, \end{aligned}$$

(G7a)

$$\begin{aligned} | \theta _i^B |\le & {} \frac{\pi }{2}, \end{aligned}$$

(G7b)

as in Eqs. (C18a) and (C18b) and therefore total cost

$$\begin{aligned} \sum _{0 \le j \le n - 2} |\theta ^B_j| \parallel k^B_j \parallel \le \frac{ \pi (n - 1)}{\sqrt{2}}. \end{aligned}$$

(G8)

The spinor $v^k(x)$ in Eq. (G6) , as defined in Section 17, is orthogonal to $u^k(x)$ of Eq. (G4) and obtained, as is $u^k(x)$, by boosting from the origin of $L( \tau , \sigma )$ to x a spin state of a free fermion at rest at the origin of $L( \tau , \sigma )$.

Then from $k^B_{n-1}, \ldots k^B_{n - 1 +p}, 3n - 2 \ge p < 48 n^2 + 159 n$, in place of $k_{n-1}, \ldots k_{2n-2}$ of Eqs. (C14a)–(C15) we obtain

$$\begin{aligned}&\exp ( i \theta _{n-1 + p} k^B_{n-1 + p}) \ldots \exp ( i \theta _0 ^Bk^B_0) |\omega ^B_0 \rangle = \sqrt{\frac{1}{m}} |\omega ^B_0 \rangle + \sqrt{\frac{m -1}{m}} |\omega ^B_1 \rangle , \end{aligned}$$

(G9)

with $\parallel k^B_i \parallel , |\theta _i|$ satisfying Eqs. (G7a) and (G7b) and incremental cost

$$\begin{aligned} \sum _{n-1\le j \le n - 1 +p} |\theta ^B_j| \parallel k^B_j \parallel < 24 \sqrt{2} \pi n^2 + \frac{159 \pi }{\sqrt{2}} n. \end{aligned}$$

(G10)

The count of additional $k^B_i$ required for Eq. (G9) arises as follows. A geodesic between $x^B_{ij}$ and $x^B_{i+1j}$ has proper length $\lambda$ in the range $2 \rho \le \lambda < (2 n + 4) \rho$ and therefore, according to Eq. (G3), can pass through a total of between 3 and $48 n + 160$ cells, and thus requires between 2 and $48 n + 159$ nearest neighbor steps. The sequence of $k^B_{n-1}, \ldots k^B_{n - 1 +p}$ for the map of Eq. (G9) can be required to complete between 2 and $48 n + 159$ such steps from $x^B_{ij}$ and $x^B_{i+1j}$ for each $0 \ge j < n$, hence $3n - 2 \ge p < 48 n^2 + 159 n$.

Following Eqs. (C16) and (C17), we now apply copies of the maps of Eqs. (G6) and (G9) along the $x^2$ direction geodesics at $x_{10}, \ldots x_{m0}$ with end result

$$\begin{aligned}&\exp ( i \theta ^B_q k^B_q) \ldots \exp ( i \theta ^B_0 k^B_0) |\omega ^B_0 \rangle \nonumber \\&\quad =\sqrt{\frac{1}{m}} \sum _i |\omega ^B_i \rangle . \end{aligned}$$

(G11)

where all $k^B_i$ satisfy Eq. (G7a), $\theta ^B_i$ satisfy Eq. (G7b) and

$$\begin{aligned} q < 48 m n^2 + 160 m n. \end{aligned}$$

(G12)

The cost of the transition from $|\omega ^B \rangle$ to $|\chi ^B \rangle$ is then bounded by

$$\begin{aligned}&C^B( |\chi ^B \rangle , |\omega ^B \rangle ) \le 24 \sqrt{2} \pi m n^2 + 80 \sqrt{2} \pi m n. \end{aligned}$$

(G13)

1.3 7.3 Entangled State Repositioned

Let the entangled n-particle state $|\phi ^B \rangle$ be

$$\begin{aligned} |\phi ^B \rangle = \sum _{i} \zeta _i \prod _j[ \sum _k u^k( y_{ij}) \Sigma _1( y_{ij}, k)] |\Omega \rangle . \end{aligned}$$

(G14)

where, as defined in Section 17, $y_{ij}$ is the center of cube $D_{ij}$ in Eq. (175) and $\zeta _i$ is the phase factor of monomial $p_i$ in Eq. (176a).

Equations (C21a)–(C32) translate directly from the non-relativistic field theory to the relativistic case, with the result

$$\begin{aligned} C^B( |\phi ^B \rangle , |\chi ^B \rangle ) \le \frac{ \pi \sqrt{mn} r}{\sqrt{2}}. \end{aligned}$$

(G15)

The distance r is given by

$$\begin{aligned} r = \min _{x_{00}} \max _{ij} r_{ij} \end{aligned}$$

(G16)

where $r_{ij}$ is the number of nearest neighbor steps in the shortest path between lattice points $x_{ij}$ and $y_{ij}$ such that no pair of paths for distinct $\{i, j\}$ intersect, $y_{ij}$ is the center point of $D_{ij}$ and $x_{ij}$ is the $m \times n$ grid of points of Appendix 7.2.

1.4 7.4 Fan-Out

Following Appendix 3.3 of the proof of the non-relativistic upper bound in Appendix 3, the state $|\phi ^B \rangle$ with particles at the centers of the cubes $D_{ij}$ we now fan-out to the state $|\psi ^B \rangle$ with particle wave functions spread uniformly over the cubes $D_{ij}$. For sufficiently small $\rho$ nearly all of the complexity in the bound on $C^B(|\psi ^B \rangle )$ is generated in this step.

We will construct a fan-out initially for $D_{00}$, which will then be duplicated on the remaining $D_{ij}$. Recall the $x \in D_{00}$ are the centers of all cells crossed by starting at $y_{00}$ and traveling along a geodesic in the positive or negative $x^1$ direction a proper distance of less than d, then in the positive or negative $x^2$ direction a proper distance less than d, then in the positive or negative $x^3$ direction a proper distance less than d.

The set of $x \in D_{00}$ we will arrange as the endpoints of a tree constructed in a sequence of stages most of which increase the number of endpoints of the tree by a factor of 8. Starting at $y_{00}$, travel along a geodesic in the positive or negative $x^1$ directions a proper distance of $\frac{d}{2}$. Define this set of 2 points to be s(1). From each of the points of s(1) , travel along a geodesic in the positive or negative $x^2$ direction a proper distance of $\frac{d}{2}$. Let this set of 4 points be s(2). From each of the points of s(2) , travel along a geodesic in the positive or negative $x^3$ direction a proper distance of $\frac{d}{2}$. The resulting set of 8 points is s(3). Repeating this sequence of 3 steps a total of p times yields a set s(3p) of $8^p$ endpoints, each a distance of $\frac{d}{2^{p-1}}$ from its nearest neighbor. For each $y \in s(3p)$ let ${\hat{s}}(y)$ be the set of 8 $y' \in s(3p +3)$ reached by a sequence of geodesic segments originating at y.

Now choose q such that

$$\begin{aligned} \rho < \frac{d}{2^q} \le 2 \rho . \end{aligned}$$

(G17)

Each pair of distinct points in s(3q) will be separated by a distance of at least $2 \rho$. Since every cell c(x) is contained in a sphere of radius $\rho$ around x, each $y \in s(3q)$ will lie in a distinct cell. Similarly, for all $r < q$, each $y \in s(3r)$ will lie in a distinct cell. For each $y \in s(3r), r \le q$, let x(y) be the center point of the cell containing y

At the outset of Sect. 17 we assumed $\rho$ is much smaller than the proper time $\tau$ of the hyperboloid $L( \tau , \sigma )$. The region in $L( \tau , \sigma )$ occupied by a collection of nearby $y \in s(3q)$ will therefore be nearly flat and can be divided up into disjoint cubes each with edge length $\frac{d}{2^{q-1}}$ centered on a corresponding $y \in s(3q)$. Let the cube for $y \in s(3q)$ be t(y). The union of all t(y) covers $D_{00}$. Let w(y) be

$$\begin{aligned} w(y) = t(y) \cap D_{00}. \end{aligned}$$

(G18)

Define n(y) to be the number of points in w(y). Working backwards iteratively from s(3q), define n(y) for $y \in s(3p), p < q$, by

$$\begin{aligned} n(y) = \sum _{y' \in {\hat{s}}(y)} n(y'). \end{aligned}$$

(G19)

Carried back to $n(y_{00})$ the result is V, the total number of points in $D_{00}$.

For any $r \le q$, define the state $\upsilon ^B_{3r}$ to be

$$\begin{aligned} |\upsilon ^B_{3r} \rangle = \sum _{y \in s(3r), k}\sqrt{ \frac{n(y)}{V}} u^k(y) \Sigma _1[x( y), k] |\Omega ^B \rangle . \end{aligned}$$

(G20)

Equations (C35a)–(C49) of the non-relativistic fan-out process in Appendix 3.3 can then be adapted to generate a sequence of $\exp ( i \frac{\pi }{2} k^B)$ which map $|\upsilon ^B_{3r-3} \rangle$ into $|\upsilon ^B_{3r} \rangle$. For the non-relativistic fan-out process, each step in which a state is split yields a pair of equally weighted pieces. For the splitting process corresponding to the states of Eq. (G20) the resulting pair will not in general be weighted equally, but this difference by itself does not affect the complexity bound. In the course of the map taking $|\upsilon ^B_{3r-3} \rangle$ into $|\upsilon ^B_{3r} \rangle$ , each of the 3 geodesic segments by which any point in s(3r) is reached from its parent point in $s(3r - 3)$ will be of length $\frac{d}{2^r}$. Equation (G3) implies that the number of nearest neighbor steps to traverse a segments of length $\frac{d}{2^r}$ is bounded by $24 \frac{d}{\rho 2^r} + 63$. A repetition of the derivation of Eq. (C50) then yields

$$\begin{aligned}&C^B( |\upsilon ^B_{3 r} \rangle , |\upsilon ^B_{3 r - 3} \rangle ) < \nonumber \\&\quad (3+\sqrt{2})( 24 \frac{d}{\rho 2^r} + 63) 2^{\frac{3r-3}{2}} \pi . \end{aligned}$$

(G21)

The last step in the fan-out process consists of splitting the piece of $|\upsilon ^B_{3q} \rangle$ at each x(y) into n(y) equally weighted components, then distributing these across the cube t(y) to produce the state

$$\begin{aligned} |\upsilon ^B_{3q + 1} \rangle = \sum _{x \in D_{00}, k}\frac{1}{\sqrt{V}} u^k(x) \Sigma _1(x, k) |\Omega ^B \rangle . \end{aligned}$$

(G22)

The complexity of the map taking $|\upsilon ^B_{3q} \rangle$ to $|\upsilon ^B_{3q+1} \rangle$ can be bounded as follows. For each $y \in s(3q)$ the length of the shortest line connecting the cell holding y to the cell holding any $x \in w(y)$ is bounded by $\frac{\sqrt{3} d}{2^q}$, the distance from y to a corner of t(y), which according to Eq. (G17) is bounded by $2 \sqrt{3} \rho$. Equation (G3) implies that the number of nearest neighbor steps to traverse a segment of length $2 \sqrt{3} \rho$ is bounded by $48 \sqrt{3} + 63$. For any $x \in w(y)$, at each $z \in w(y)$ along the path from x to y, the remaining path from z to y is the shortest nearest neighbor route to y. It follows that if the paths from some $x \in w(y)$ to y and from a distinct $x' \in w(y)$ to y coincide at z the remaining segments from z to y will also coincide. The collection of shortest paths from all $x \in w(y)$ to y must therefore form a tree, each branch of which consists of at most $48 \sqrt{3} + 63$ nearest neighbor steps. By adapting the derivation of Eq. (C50) the cost of all such paths executed in parallel for all $x \in D_{00}$, the total count of which is V, can then be bounded to give

$$\begin{aligned} C^B(|\upsilon ^B_{3q+1} \rangle ,|\upsilon ^B_{3q} \rangle ) \le (48 \sqrt{3} + 63)\frac{ \pi }{\sqrt{2}} \sqrt{V}. \end{aligned}$$

(G23)

Summing Eq. (G21) over r from 1 to q, adding Eq. (G23) and using Eq. (G17) gives

$$\begin{aligned} C^B(|\upsilon ^B_{3q+1} \rangle ,|\upsilon ^B_0 \rangle ) < c_1 \sqrt{V}, \end{aligned}$$

(G24)

where

$$\begin{aligned} c_1 = [(3 + \sqrt{2})(33 + 42 \sqrt{2}) + \frac{48 \sqrt{3} + 63}{\sqrt{2}}] \pi . \end{aligned}$$

(G25)

The bound of Eq. (G24) applies to a fan-out process on a single cube $D_{00}$. Assume the process repeated in parallel on the mn cubes $D_{ij}$, thereby generating $|\psi ^B \rangle$ of Eq. (178b). For $|\phi ^B \rangle$ of Eq. (G14) we then have

$$\begin{aligned} C^B( |\psi ^B \rangle , |\phi ^B \rangle ) \le c_1 \sqrt{mnV}. \end{aligned}$$

(G26)

From Eqs. (G13) and (G15), it follows that for a product state $|\omega ^B \rangle$ we have

$$\begin{aligned}&C^B(|\psi ^B \rangle ,|\omega ^B \rangle ) \nonumber \\&\quad \le c_1 \sqrt{ mnV} + c_2 m n^2 + c_3 mn + c_4\sqrt{mn} r, \end{aligned}$$

(G27)

for $c_1$ of Eq. (G25), r of Eq. (G16) and

$$\begin{aligned} c_2= & {} 24 \sqrt{2} \pi , \end{aligned}$$

(G28a)

$$\begin{aligned} c_3= & {} 80 \sqrt{2} \pi , \end{aligned}$$

(G28b)

$$\begin{aligned} c_4= & {} \frac{\pi }{\sqrt{2}}. \end{aligned}$$

(G28c)

Equation (181) then follows.