Non-closure of Quantum Correlation Matrices and Factorizable Channels that Require Infinite Dimensional Ancilla (With an Appendix by Narutaka Ozawa)

Abstract

We show that there exist factorizable quantum channels in each dimension $\ge 11$ which do not admit a factorization through any finite dimensional von Neumann algebra, and do require ancillas of type $\hbox {II}_1$, thus witnessing new infinite-dimensional phenomena in quantum information theory. We show that the set of $n \times n$ matrices of correlations arising as second-order moments of projections in finite dimensional von Neumann algebras with a distinguished trace is non-closed, for all $n \ge 5$, and we use this to give a simplified proof of the recent result of Dykema, Paulsen and Prakash that the set of synchronous quantum correlations $C_q^s(5,2)$ is non-closed. Using a trick originating in work of Regev, Slofstra and Vidick, we further show that the set of correlation matrices arising from second-order moments of unitaries in finite dimensional von Neumann algebras with a distinguished trace is non-closed in each dimension $\ge 11$, from which we derive the first result above.

Appendices

Narutaka Ozawa, RIMS, Kyoto University, Sakyo-ku, Kyoto 606-8502, Japan. Email: narutaka@kurims.kyoto-u.ac.jp

Appendix (by Narutaka Ozawa)

Narutaka Ozawa, RIMS, Kyoto University, Sakyo-ku, Kyoto 606-8502, Japan. Email: narutaka@kurims.kyoto-u.ac.jp

Realizing the Kruglyak–Rabanovich–Samoilenko Projections in the Hyperfinite $\hbox {II}_1$ Factor

Kruglyak, Rabanovich, and Samoilenko proved in [10, Theorem 6], cf. Theorem 2.2, that for any $n\ge 5$ and any $\alpha \in [\frac{1}{2}(n-\sqrt{n^2-4n}),\frac{1}{2}(n+\sqrt{n^2-4n})]$, there are orthogonal projections $p_1,\ldots ,p_n$ such that $\sum _i p_i=\alpha$. In this appendix, we observe that the Kruglyak–Rabanovich–Samoilenko construction shows that these projections are realized in the hyperfinite $\mathrm {II}_1$ factor $\mathcal{R}$, possibly except for the extremities.

Theorem A.1

For any $n \ge 5$ and any $\alpha \in (\frac{1}{2}(n-\sqrt{n^2-4n}),\frac{1}{2}(n+\sqrt{n^2-4n}))$, there are projections $p_1,\ldots ,p_n\in \mathcal{R}$ which satisfy $\sum _i p_i=\alpha$.

Let $(M,\tau )$ be a finite von Neumann algebra. By a matricial approximation (or matricial microstates) of a d-tuple $(a_1,\ldots ,a_d)$ in $M^\mathrm {sa}$, we mean a sequence $(x_1(n),\ldots ,x_d(n))$ in $(M_{k(n)}({\mathbb C})^\mathrm {sa})^d$ such that $\lim _n {{\,\mathrm{tr}\,}}(p(x_1(n),\ldots ,x_d(n))) = \tau (p(a_1,\ldots ,a_d))$ for every polynomial p in d non-commuting variables. A matricial approximation of a generating d-tuple $(a_1,\ldots ,a_d)$ of M gives rise to an embedding of $(M,\tau )$ into the tracial ultraproduct of $(M_{k(n)}({\mathbb C}),{{\,\mathrm{tr}\,}}_{k(n)})$. Recall that M satisfies the Connes Embedding Conjecture, i.e., $(M,\tau )\hookrightarrow (\mathcal{R}^\omega ,\tau ^\omega )$, if and only if every (or some) generating d-tuple $(a_i)_{i=1}^d$ in $M^\mathrm {sa}$ admits a matricial approximation. We will give a sufficient condition for hyperfiniteness of M in terms of a matricial approximation. For $x = (x_{ij}) \in M_k({\mathbb C})$, we define its propagation to be $\max \{ |i-j| : x_{ij}\ne 0\}$.

Lemma A.2

Let $(M,\tau )$ be a finite von Neumann algebra generated by $a_1,\ldots ,a_d \in M^\mathrm {sa}$. Assume that $(a_1,\ldots , a_d)$ admits a matricial approximation $(x_1(n),\ldots ,x_d(n))$ with uniformly bounded propagations. Then M is hyperfinite.

Proof

Let k be a positive integer and consider the shift unitary matrix $z_k\in M_k({\mathbb C})$ given by $(z_k)_{i,j}=\delta _{i+1,j}$ (modulo k). It normalizes the diagonal maximal abelian subalgebra $D_k\subset M_k({\mathbb C})$. Observe that any $y \in M_k({\mathbb C})$ that has propagation at most l can be written as $y=\sum _{m=-l}^l f_m z_k^m$ for some $f_m\in D_k$ with $\Vert f_m\Vert \le \Vert y\Vert$.

Now, let a matricial approximation $(x_1(n),\ldots ,x_d(n))$ be given as in the statement. We denote by $M_\omega$ the tracial ultraproduct of $(M_{k(n)}({\mathbb C}),\mathrm {tr}_{k(n)})$ and by $D_\omega$ the subalgebra arising from the diagonal maximal abelian subalgebras $D_{k(n)}$. From the above discussion, one sees that the element $x_i\in M_\omega$ that corresponds to $(x_i(n))_n$ belongs to the von Neumann subalgebra generated by $D_\omega$ and z, where z is the unitary element corresponding to $(z_{k(n)})_n$. The von Neumann subalgebra generated by $x_1,\ldots ,x_d$ is isomorphic to M and the von Neumann subalgebra generated by $D_\omega$ and z is hyperfinite (as it is isomorphic to $D_\omega \rtimes {\mathbb Z}$, assuming $k(n)\rightarrow \infty$). $\square$

Proof of Theorem A.1

Firstly, note that every separable finite von Neumann algebra $(N,\tau )$ with a faithful normal tracial state is embeddable in a trace-preserving way into a separable $\mathrm {II}_1$ factor M, which can be taken to be the hyperfinite $\hbox {II}_1$ factor $\mathcal{R}$ if M is hyperfinite. Indeed, as observed by U. Haagerup, we may take M to be $(\bigotimes _{n=1}^\infty N) \rtimes S_\infty$, where the infinite tensor product is with respect to the standard representation of N on $L^2(N,\tau )$, and where $S_\infty$ is the (locally finite) group of permutations on the natural numbers with finite support. It therefore suffices to find the projections $p_1, \dots , p_n$ in any hyperfinite finite von Neumann algebra N.

For each $\alpha \in {\mathbb Q}\cap [3/2,2]$, the projections $P_1(\alpha ),\ldots ,P_5(\alpha )$ in $M_{k(\alpha )}({\mathbb C})$ that satisfy $\sum _i P_i(\alpha )=\alpha$ are constructed in [10, Theorem 6] as $R_i$. The proof of Theorem 6 (and Lemma 7) in [10] reveals that the projections $R_i$ are obtained by sewing (see [10, Definition 1]) the projections $P^{(k)}_i \in M_{k_i+2}({\mathbb C})$, $k_i\in \{1,2,3\}$. Since $P^{(k)}_i$’s have propagation at most 4, the projections $R_i$ have propagation at most 8, regardless of $\alpha$.

Let $\alpha \in (3/2,2)$ be given and take a rational sequence $(\alpha _n)_n$ which converges to $\alpha$. Then after passing to a convergent subsequence, $(P_1(\alpha _n),\ldots ,P_5(\alpha _n))$ is a matricial approximation of $(P_1,\ldots , P_5)$ in the tracial ultraproduct $M_\omega$ of $(M_{k(n)}({\mathbb C}),{{\,\mathrm{tr}\,}}_{k(n)})$ and $(P_1,\ldots , P_5)$ satisfies $\sum _i P_i=\alpha$. By Lemma A.2, the projections $P_1,\ldots , P_5$ generate a hyperfinite von Neumann subalgebra. This proves Theorem A.1 for $n=5$ and $\alpha \in [3/2,2]$. By [10, Lemma 5], this implies Theorem A.1 for every $n\ge 5$ and $\alpha \in [2,n-2]$.

Finally, note that all values in $(\frac{1}{2}(n-\sqrt{n^2-4n}),\frac{1}{2}(n+\sqrt{n^2-4n}))$ are obtained by iterating the numerical mappings $\Phi ^+$ and $\Phi ^-$ (see [10, Section 1.2]) starting at $\alpha \in [2,n-2]$ (see [10, Lemma 6]). Thus it suffices to show the functors S and T constructed in Section 1.2 in [10] preserve hyperfiniteness. This is clear for the linear reflection T. For the reader’s convenience, we replicate here the construction of the hyperbolic reflection S, adapted to our setting. Let $P_1,\ldots ,P_n\in N$ be projections such that $\sum _{i=1}^n P_i=\alpha$. We will construct projections $Q_1,\ldots ,Q_n$ such that $\sum _{i=1}^n Q_i=\frac{\alpha }{\alpha -1}$. Put

$$\begin{aligned} V_i:=(\alpha ^2-\alpha )^{-1/2} \, P_i \left[ \begin{matrix} -P_1&\cdots&\alpha -P_i&\cdots&-P_n\end{matrix}\right] \in M_{1,n}(N). \end{aligned}$$

Then, $V_iV_i^*=(\alpha ^2-\alpha )^{-1}P_i(\alpha ^2-2\alpha P_i + \sum _{k=1}^n P_k)P_i=P_i$ and $V_i$ is a partial isometry. Hence $Q_i:=V_i^*V_i \in M_n(N)$ is a projection. A calculation shows

$$\begin{aligned} \sum _k Q_k=(\alpha ^2-\alpha )^{-1}(\alpha ^2 \, \mathrm {diag}(P_1,\ldots ,P_n) - \alpha \, [P_iP_j]_{i.j} )\in M_n(N). \end{aligned}$$

Note that $Q:=\mathrm {diag}(P_1,\ldots ,P_n) - \alpha ^{-1} [P_iP_j]_{i.j}$ is a projection and one has $\sum _k Q_k=\frac{\alpha }{\alpha -1}Q$. Thus, viewing $Q_k$ as projections in $QM_n(N)Q$, we are done. When N is hyperfinite, so is the amplification $QM_n(N)Q$. $\square$