Target space entanglement in quantum mechanics of fermions and matrices

We consider entanglement of first-quantized identical particles by adopting an algebraic approach. In particular, we investigate fermions whose wave functions are given by the Slater determinants, as for singlet sectors of one-matrix models. We show that the upper bounds of the general Rényi entropies are N log 2 for N particles or an N × N matrix. We compute the target space entanglement entropy and the mutual information in a free one-matrix model. We confirm the area law: the single-interval entropy for the ground state scales as 13\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \frac{1}{3} $$\end{document} log N in the large N model. We obtain an analytical ON0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{O}\left({N}^0\right) $$\end{document} expression of the mutual information for two intervals in the large N expansion.


Introduction
It is widely believed that quantum entanglement is closely related to the structure of spacetime in quantum gravity.In the AdS/CFT correspondence, the Ryu-Takayanagi formula 2 states that entanglement about the base space in holographic CFTs is connected to the area of minimal surface in the bulk.As in this example, we often consider the base space entanglement in quantum field theories or statistical models.However, target space instead of base space sometimes directly connects to our spacetime, for example, perturbative string theories or matrix models.Thus, it is natural to investigate a notion of target space entanglement [3][4][5] .See also recent Refs.1,6-9. a In Ref. 3, the target space entanglement is defined using an algebraic approach.We will review this approach in Sec. 2, and apply it to quantum a A concept of entanglement in string theories (matrix models) is investigated in 10 and revisited in 11.  mechanics of fermions in Sec. 3, Sec. 4 and Sec. 5.

Definition of entanglement entropy based on subalgebras of operators
Let us recall the conventional definition of entanglement entropy (EE).Suppose that a total density matrix ρ is given for a Hilbert space H = H B ⊗ H B .The EE for subsystem H B is defined as the von Neumann entropy of the reduced density matrix ρ B = tr B ρ as This definition relies on the tensor product structure of the Hilbert space, H = H B ⊗ H B .However, total Hilbert spaces sometimes do not have such simple tensor-factorized forms.For example, the Hilbert space of a firstquantized (non-relativistic) particle in a space R d is schematically given by a "direct sum" as Thus, even if we divide the space The algebraic approach enables us to define EE without relying on the tensor product structure (see, e.g., the references 1,[12][13][14] ).The algebraic definition is based on the subalgebra of operators (observables).If a total density matrix ρ is given, and we have a restricted set of operators (subalgebra A), an entropy S A (ρ) associated with the subalgebra A is defined.This concept is natural, if we recall the meaning of entropy in information theory.The entropy is a measure of uncertainty about the whole information when we can only know partial information.If an observer can use only a subset of operators A, the whole information is not obtained.Entropy S A (ρ) quantifies the amount of uncertainty (or unknownness).In this sense, the usual EE, S B = − tr B ρ B log ρ B , for H = H B ⊗ H B represents uncertainty for an observer who can probe only subsystem H B .That is, it is the entropy for the subalgebra L(H B ) ⊗ 1 H B .b The choice of subalgebra A is arbitrary, and we do not need the tensor product structure.
For general subalgebra A, the entropy S A (ρ) is computed as follows.First, the 'reduced density matrix' ρ A is uniquely determined from ρ and A as an operator in A satisfying the following equation: For example, if the total Hilbert space has a tensor product form as H = H B ⊗ H B , and we take the subalgebra Here, L(V ) denotes a set of linear operators on linear space V , and 1 V does the identity operator on V .
is given by ρ The point is that the definition Eq. ( 1) is applicable even when the Hilbert space does not have the tensor product structure.Furthermore, for a given subalgebra, we can decompose the Hilbert space into blocks of tensor products where the subalgebra acts nontrivially only on each tensor component as follows: This decomposition is uniquely fixed by the subalgebra A. We represents the projection onto each block by Π k .We define the density matrix ρ k on the projected space Π k H as where p k is a normalization factor defined as p k := tr(Π k ρΠ k ) and is a probability of being in the sector Π k H for the given ρ.Since the projected space Π k H has a simple tensor-factorized form as , we can consider the reduced density matrix of ρ k on H B k as Then, the 'reduced density matrix' ρ A satisfying Eq. ( 1) is given by We define the reduced density matrix ρ B on space EE S A (ρ) is defined as the von Neumann entropy where The first term in the r.h.s. of Eq. ( 7) is called the classical part, and is the Shannon entropy of the probability distribution {p k }.On the other hand, the second term in the r.h.s. of Eq. ( 7) is called the quantum part S q (ρ, A).The expression in Eq. ( 7) is similar to the symmetry resolved entanglement entropy 15,16 .

Example: Entanglement in a single qubit
As a concrete example of EE in the algebraic approach, we consider a single qubit.The Hilbert space is two-dimensional space, H = span{|0 , |1 }.We usually consider entanglement between two qubits.The algebraic approach enables us to consider "EE" even for a single qubit.The full set of operators L(H) is L(H) = span{I, σ x , σ y , σ z }. c If we take the subalgebra as this full algebra, the decomposition Eq. ( 2) is trivial as with A = L(H) ⊗ 1.In this case, ρ B in Eq. ( 6) is just the original ρ.Thus, the EE associated with the full algebra L(H) is just the von Neumann entropy of ρ, In particular, if state ρ is pure, the entropy vanishes as S L(H) (ρ) = 0.It means that the pure state is not ambiguous and is completely determined by quantum tomography if we can use any operators.
Situation changes when we can use only a subset of operators.Let us suppose that we can probe only z-direction.This corresponds to taking subalgebra A = span{1, σ z }.The decomposition (2) for this choice of the subalgebra is H = span{|0 } ⊕ span{|1 } where A = span{1, σ z } can be represented as A = span 1 0 0 0 ⊕ span 0 0 0 1 .The projection Π k are Π k = |k k| (k = 0, 1).We then have p k = k|ρ|k and ρ B k = Π k .Thus, the EE associated with the subalgebra A is where the quantum part S q (ρ, A) always vanishes, and the entropy is just the classical Shannon entropy of the probability distribution that the qubit is measured in 0 or 1 for the given state ρ.Even pure states in general have non-vanishing entropy (except for the case where states are eigenstates of σ z ).The non-vanishing entropy reflects the fact that pure states are ambiguous for restricted observers who can probe only z-direction.In fact, the observers cannot distinguish pure states with mixed states ρ = p 0 0 0 p 1 .

Entanglement of fermions with a fixed number
We now consider target space entanglement of first-quantized N fermions by the algebraic approach.The Hilbert space of the single particle is represented by H (1) .It is given by H (1) = span{|x |x ∈ M } where M is the target space of particles.The Hilbert space H (N ) of N fermions is given by the N -th exterior power of H (1) as which is spanned as We take a subregion B in the target space M , and consider the EE of this subregion.Since the Hilbert space H (N ) does not have a tensor-factorized structure with respect to the target space coordinates, we adopt the algebraic approach instead of the conventional definition.The subalgebra we take is the set of operators acting non-trivially only on particles in subregion B, which is represented by A(B).For example, when N = 1, the subalgebra A(B) is given by A(B) = span {|y y ||y, y ∈ B} ⊕ span B dz|z z| where B is the complement region of B. For general N , we can decompose H (N ) into a direct sum of the following subsectors as The subsector H To represent the subalgebra A(B), we introduce the following abbreviated notation: The subalgebra A(B) is then given by where and takes the form Since the subalgebra A(B) is specified, we can compute the entropy S A(B) associated with this subalgebra in the manner described in the previous section.d We call this entropy the target space entanglement entropy S B because the subalgebra A(B) is characterized by the subregion B in the target space of particles.In the second quantized picture, we can define the conventional entanglement entropy for subregion B. The target space EE S B agrees with this base space EE 1,3,4 .

Fermions in the Slater determinant states
To be more specific, we focus on pure states whose N -body wave functions are given by the Slater determinants as where χ i (x) are one-body wave functions normalized as The target space EE for subregion B can be evaluated as Eq. ( 7) by computing p k and S(ρ B k ) for the pure states ψ.After some computations (see Ref. 1 for details), we can find that the entropy S B follows the simple formula: where X is a N × N matrix given by We call X overlap matrix.It is easy to show that the eigenvalues λ i of the overlap matrix are in the range 0 From the formula (20), we can find that the entropy has the upper bound e as The maximum entropy N log 2 is proportional to the number of particles N , and thus follows an extensive property like thermal entropy.However, d In this case, the projection Π k in Eq. ( 3) is the projection to the subsector H (N ) k in Eq. ( 13).e We can also confirm that the classical part S cl is bounded as S cl (ρ; A) O(log N ). 1 this upper bound is too generic.We expect that EE for ground states is not extensive but sub-extensive in local models.In fact, we will see in the next section that the entropy of a ground state of N free fermions behaves as S ∼ O(log N ) in the large N limit.

Entanglement for free fermions in a circle
We now apply the formula (20) to N free fermions in a circle with length L, i.e., the target space M is a circle.The Hamiltonian is given by H = N i=1 p 2 i 2m , and we consider its ground state.The one-body eigenfunctions are given by χ n (x) = 1 √ L e 2πi L nx where n are integers.Supposing that the total number of particles N is odd (N = 2K +1), the N -body wave function for the ground state is given by the Slater determinant as Thus, the target space entanglement for a subregion B can be obtained by the formula (20) with the N × N overlap matrix where n, n runs in −K, . . ., K.

Single interval
In this subsection, we consider the case where the subregion B is a single interval I 1 in the circle.We parameterize the length of the interval as rL (0 ≤ r ≤ 1), i.e., r is the ratio of the interval to the circle.
In the large N limit, the asymptotic behavior of the entropy can be obtained as We show the plot of the entropy with the large N result (25) in Fig. 1.It shows that the entropy is sub-extensive (not proportional to N ).Furthermore, the large N behavior Eq. (25) agrees with the EE for the single interval in c = 1 CFTs on the circle 17 if we regard N as a (dimensionless) cutoff.

Entanglement entropy and mutual information for two intervals
In this subsection, we consider two disjoint intervals I 1 and I 2 in the circle.Suppose that the coordinates of the circle is x moving in − L 2 ≤ x ≤ L 2 with the periodic condition x ∼ x + L. We take the two intervals as L .The EE for the subregion I 1 ∪ I 2 can be analytically computed in the large N limit f as We can also evaluated the target space mutual information; The large N behavior is The mutual information is finite even in the large N limit.In addition, Eq. ( 29) agrees with the result in a c = 1 CFT (free compact boson at the self-dual radius 18 ), although the reason is not understood well.The plot of the target space mutual information (28) is Fig.

Brief conclusion
The algebraic approach is a powerful method of characterizing entanglement.This approach might be useful beyond the target space entanglement.A similar idea to define entropy based on observables is also investigated as the observational entropy (see, e.g., Ref. 19).
We have used the algebraic approach to define the target space entanglement of particles.In particular, we consider non-interacting fermions, which can be regarded as the singlet sectors of one-matrix models.It is more interesting to consider entanglement in multi-matrix models, and its relation to holography.
where k particles in B and N − k ones in B as

Fig. 2 .
Fig.2.Mutual information for two intervals.We take N = 101 and set the parameter r as r = 0.01 (length of each interval is rL).The red dots represent the mutual information for some values of d.The blue curve represents the large N result (29). 2.