1 Introduction

C. Anantharaman-Delaroche introduced in [1] the class of factorizable completely positive maps between von Neumann algebras equipped with a normal faithful state, while studying non-commutative analogues of classical ergodic theory results, including, e.g., G.-C. Rota’s “Alternierende Verfahren” theorem.

It was shown in [5] that not all unital completely positive, trace-preserving maps on \(M_n({\mathbb C})\) (also referred to as unital quantum channels in dimension n) are factorizable when \(n\ge 3\), which also led to a negative answer to the so-called asymptotic quantum Birkoff conjecture, [15]. The tool was the characterization established in [5] that a unital, completely positive, trace-preserving map \(T :M_n({\mathbb C}) \rightarrow M_n({\mathbb C})\) is factorizable if and only if there exist a finite von Neumann algebra N equipped with a (faithful, normal) tracial state \(\tau \) and a unitary u in \(M_n({\mathbb C}) \otimes N\) such that \(T(x) = (\mathrm {id_n} \otimes \tau )(u(x \otimes 1_N)u^*)\), for all \(x \in M_n({\mathbb C})\). Following the terminology introduced in [6], we say in this case that T admits an exact factorization through\(M_n({\mathbb C}) \otimes N\), and N is called the ancilla. In all previously studied cases of factorizable maps (see [5, 6], and the recent paper [11] for the case \(n=2\)), the ancilla could be taken to be finite dimensional, and even a full matrix algebra. It was, however, first remarked in [12] that one cannot always take the ancilla to be a full matrix algebra. In this paper we show that for each \(n \ge 11\), there are factorizable maps on \(M_n({\mathbb C})\) that do not admit a finite dimensional ancilla, nor an ancilla of type I, and we give concrete examples of such maps, see Example 3.5 and Theorem 4.1. Therefore, one needs to employ ancillas of type \(\hbox {II}_1\) to describe a general factorizable channel in dimension \(n\ge 11\). Observe that all factorizable quantum channels do admit an exact factorization through a type \(\hbox {II}_1\) von Neumann algebra (even a type \(\hbox {II}_1\) factor), by the well-known fact that each finite von Neumann algebra equipped with a fixed faithful normal tracial state embeds in a trace-preserving way into a type \(\hbox {II}_1\) factor.

The proof of our result uses very recent developments of analysis in quantum information theory concerning the non-closure of certain sets of correlation matrices. The first such result was due to Slofstra, who proved the failure of what is referred to as the strong Tsirelson conjecture. This was recently refined by Dykema, Paulsen and Prakash, [3], to show that the set of synchronous correlation matrices \(C_q^s(n,k)\) is non-closed, when \(n \ge 5\) and \(k \ge 2\).

We consider the set \({\mathcal {D}}(n)\) of \(n \times n\) matrices arising from second-order moments of n-tuples of projections in finite von Neumann algebras with a (normal, faithful) tracial state, and the subset \({\mathcal {D}}_\mathrm {fin}(n)\) consisting of those matrices that arise likewise from n-tuples of projections in finite-dimensional von Neumann algebras (or \(C^*\)-algebras). We show that the set \({\mathcal {D}}_\mathrm {fin}(n)\) is not closed, when \(n \ge 5\). Our proof uses a theorem of Kruglyak, Rabanovich and Samoilenko, [10], also employed in [3], which describes which scalar multiples of the identity operator on a (finite dimensional) Hilbert space can arise as the sum of n projections. We use this to give a shorter and more direct proof of the main result from [3] that the set of synchronous quantum correlation matrices \(C_q^s(n,2)\) is non-closed, when \(n \ge 5\).

Kirchberg, [9], reformulated the Connes Embedding Problem in terms of the set \({\mathcal {G}}(n)\) of \(n \times n\) matrices of correlations arising from unitaries in finite von Neumann algebras with a (normal, faithful) tracial state. This result was further refined by Dykema and Juschenko, [2], and in their formulation, the Connes Embedding Problem is equivalent to the statement that \({\mathcal {F}}(n) = {\mathcal {G}}(n)\), for all \(n \ge 3\), where \({\mathcal {F}}(n)\) is the closure of the set of \(n \times n\) matrices of correlations arising from unitaries in full matrix algebras. A trick originating in (as of yet unpublished) work of Regev, Slofstra and Vidick (communicated to us by W. Slofstra in May 2018), which we carry out in the setting of finite von Neumann algebras in Sect. 3, allows us to conclude further that the set \({\mathcal {F}}_{\mathrm {fin}}(2n+1)\) of matrices of correlations arising from unitaries in finite dimensional von Neumann algebras is non-closed, whenever \({\mathcal {D}}_\mathrm {fin}(n)\) is non-closed, i.e., for all \(n \ge 5\).

A connection between the set \({\mathcal {G}}(n)\) and the set of factorizable Schur multipliers on \(M_n({\mathbb C})\) was established in [6]. This connection gives the final link between the established non-closure of the sets \({\mathcal {F}}_{\mathrm {fin}}(2n+1)\), for \(n \ge 5\), and existence of factorizable Schur multipliers with no finite dimensional ancilla (or, even stronger, non type I), in each dimension \(\ge 11\).

In the Appendix, written by N. Ozawa, it is shown that the construction by Kruglyak, Rabanovich and Samoilenko in [10] of an n-tuple of projections with sum equal to a multiple \(\alpha \) of the identity can be realized in the hyperfinite \(\hbox {II}_1\) factor \({\mathcal {R}}\), for all admissible values of \(\alpha \), except, possibly, for two extremal ones. This, in turn, implies that the factorizable Schur multipliers with no finite dimensional ancilla found in this article do admit \({\mathcal {R}}\) as an ancilla (except, possibly, for the cases corresponding to the above mentioned extremal values of \(\alpha \)).

Different sets of matrices of correlations arising from the generators of L. Brown’s universal \({C}^*\)-algebra were shown to be non-closed by Harris and Paulsen [7]. This was used by Gao, Harris and Junge [4] to obtain further non-closure results in a matrix-valued setting.

2 Non-closure of Sets of Matrices of Quantum Correlations

For \(n\ge 2\), let \({\mathcal {D}}(n)\) and \({\mathcal {D}}_{\mathrm {fin}}(n)\) be the set of \(n \times n\) matrices \(\big [\tau (p_jp_i)\big ]_{i,j=1}^n\), where \(p_1,\dots , p_n\) are projections in some arbitrary finite von Neumann algebra, respectively, in some finite dimensional von Neumann algebra, equipped with a (normal) faithful tracial state \(\tau \). We show in this section that \({\mathcal {D}}_{\mathrm {fin}}(n)\) is non-closed, when \(n \ge 5\), and we use this to give a more direct proof, avoiding graph correlation functions, of the very recent result of Dykema, Paulsen and Prakash, [3], that the set of synchronous quantum correlations \(C_q^s(5,2)\) is non-closed.

Standard arguments involving ultralimits, as in the proof of Proposition 2.4 (v) below, respectively, direct sums of finite von Neumann algebras, show that the set \({\mathcal {D}}(n)\) is compact and convex. One can likewise show that the set \({\mathcal {D}}_{\mathrm {fin}}(n)\) is convex. The subset \({\mathcal {D}}_{\mathrm {matrix}}(n)\) of \({\mathcal {D}}_{\mathrm {fin}}(n)\) consisting of \(n \times n\) matrices \(\big [\mathrm {tr}_k(p_jp_i)\big ]_{i,j=1}^n\), where \(p_1,\dots , p_n\) are projections in a matrix algebra \(M_k({\mathbb C})\), for \(k \ge 1\), is not convex (and not closed), for any \(n \ge 1\), since each diagonal entry of such a matrix is the (normalized) trace of a projection in a matrix algebra, which is a rational number. It is not hard to see that \({\mathcal {D}}_{\mathrm {matrix}}(n)\) is a dense subset of \({\mathcal {D}}_{\mathrm {fin}}(n)\).

Note that the closure of \({\mathcal {D}}_{\mathrm {matrix}}(n)\) is equal to \({\mathcal {D}}(n)\), for all \(n \ge 3\), if and only if the Connes Embedding Problem has an affirmative answer (see Sect. 3 for a sketch of a proof of this fact). Let us also observe that \({\mathcal {D}}_{\mathrm {fin}}(2)= {\mathcal {D}}(2)\) is closed.

Proposition 2.1

We have

$$\begin{aligned} {\mathcal {D}}(2) = \Big \{ \begin{pmatrix}s &{} \quad u \\ u &{}\quad t \end{pmatrix}: \, 0 \le s,t \le 1, \; \max \{0,s+t-1\} \le u \le \min \{s,t\}\Big \}. \end{aligned}$$
(2.1)

Moreover, each matrix in \({\mathcal {D}}(2)\) arises from a pair of projections in the commutative finite-dimensional \(C^*\)-algebra \({\mathbb C}\oplus {\mathbb C}\oplus {\mathbb C}\oplus {\mathbb C}\) (with respect to a suitable trace).

In particular, it follows that \({\mathcal {D}}_{\mathrm {fin}}(2) = {\mathcal {D}}(2)\) and \({\mathcal {D}}_{\mathrm {fin}}(2)\) is closed.

Proof

Let \((M,\tau )\) be a finite von Neumann algebra equipped with a tracial state, and let \(p,q \in M\) be projections. The associated matrix in \({\mathcal {D}}(2)\) is

$$\begin{aligned} \begin{pmatrix} \tau (p) &{}\quad \tau (pq) \\ \tau (qp) &{}\quad \tau (q) \end{pmatrix} = \begin{pmatrix} s &{} \quad u \\ u &{}\quad t \end{pmatrix}, \end{aligned}$$

where \(s = \tau (p)\), \(t = \tau (q)\) and \(u = \tau (pq)=\tau (qp) = \tau (pqp) = \tau (qpq)\). It is clear that \(0 \le s,t \le 1\). Since \(0 \le pqp \le p\) and \(0 \le qpq \le q\), we further conclude that \(0 \le u \le \min \{s,t\}\).

By Kaplansky’s formula (see, e.g., [8]), we get \(\tau (p \wedge q) = \tau (p)+\tau (q)-\tau (p \vee q) \ge \tau (p)+\tau (q)-1\). As \(p \wedge q = q(p \wedge q)q \le qpq\), we infer that \(u = \tau (qpq) \ge \tau (p \wedge q) \ge s+t-1\).

To complete the proof, we show that each matrix in the right-hand side of (2.1) arises from projections p and q in \({\mathbb C}\oplus {\mathbb C}\oplus {\mathbb C}\oplus {\mathbb C}\), with respect to the trace with weight \((\alpha _1,\alpha _2,\alpha _3,\alpha _4)\), satisfying \(\alpha _j \ge 0\) and \(\sum _{j=1}^4 \alpha _j=1\). Set \(p = (1,1,0,0)\) and \(q = (1,0,1,0)\). Then \(pq = (1,0,0,0)\) and

$$\begin{aligned} \begin{pmatrix} \tau (p) &{} \quad \tau (pq) \\ \tau (qp) &{} \quad \tau (q) \end{pmatrix} = \begin{pmatrix} \alpha _1+\alpha _2 &{}\quad \alpha _1 \\ \alpha _1 &{} \quad \alpha _1+\alpha _3 \end{pmatrix}. \end{aligned}$$

We must therefore choose \(\alpha _1 = u \ge 0\), \(\alpha _2 = s-u \ge 0\), \(\alpha _3 = t-u \ge 0\), and \(\alpha _4 = 1-\alpha _1-\alpha _2-\alpha _3\). The inequality \(u \ge s+t-1\) ensures that \(\alpha _1+\alpha _2+\alpha _3 \le 1\), whence \(\alpha _4 \ge 0\). \(\square \)

We do not know if the sets \({\mathcal {D}}_{\mathrm {fin}}(n)\) are closed for \(n=3,4\).

We proceed to prove that the sets \({\mathcal {D}}_\mathrm {fin}(n)\) are non-closed, for \(n \ge 5\), following the idea of Dykema, Paulsen and Prakash to use Theorem 2.2 below from [10]. As in [10], let \(\Sigma _n\) be the set of all \(\alpha \ge 0\) for which there exist projections \(p_1,\dots , p_n\) on a Hilbert space H such that \(\sum _{j=1}^n p_j = \alpha \cdot I_H\). The sets \(\Sigma _n\) are completely described in [10] and have the following properties: They are symmetric, i.e., if \(\alpha \in \Sigma _n\), then \(n-\alpha \in \Sigma _n\). Moreover, \(\Sigma _2 = \{0,1,2\}\) and \(\Sigma _3 = \{0,1,\frac{3}{2},2,3\}\). The set \(\Sigma _4\) is a countably infinite subset of the rational numbers, \({\mathbb Q}\), with one accumulation point, namely 2. For \(n \ge 5\), \(\Sigma _n\) is the union of the interval \(\big [\frac{1}{2} (n-\sqrt{n^2-4n}), \frac{1}{2} (n+\sqrt{n^2-4n})\big ]\) and a countably infinite discrete subset of rational numbers, containing \(\{0,1,\frac{n}{n-1}, n-\frac{n}{n-1}, n-1,n\}\). Observe that \(\frac{n}{n-1} < \frac{1}{2} (n-\sqrt{n^2-4n})\). Set

$$\begin{aligned} \Pi _n = \big [{\textstyle {\frac{1}{2}}}(1-\sqrt{1-4/n}), {\textstyle {\frac{1}{2}}}(1+\sqrt{1-4/n})\big ] \subset \Big (\textstyle {\frac{2n-1}{n(n-1)}}, 1-\textstyle {\frac{2n-1}{n(n-1)}}\Big ). \end{aligned}$$
(2.2)

Then \(\Pi _n\) is an interval with non-empty interior, and \(n^{-1}\Sigma _n {\setminus } \Pi _n\) is contained in \({\mathbb Q}\).

Theorem 2.2

(Kruglyak, Rabanovich and Samoilenko, [10, Theorem 6]). Let \(n \ge 2\) be an integer. Then there exist projections \(p_1, \dots , p_n\) on some finite dimensional Hilbert space H such that \(\sum _{j=1}^n p_j = \alpha \cdot I_H\) if and only if \(\alpha \in \Sigma _n \cap {\mathbb Q}\).

Note that the “only if” part of the theorem above is trivial: Apply the standard trace on B(H) to both sides of the equation \(\sum _{j=1}^n p_j = \alpha \cdot I_H\), to obtain \(\sum _{j=1}^n \mathrm {dim}(p_j)= \alpha \, \mathrm {dim}(H)\). This argument also gives the following quantitative result for rational values of \(\alpha \): If \(\alpha = a/b\) is irreducible, with a, b positive integers, and if there exist projections \(p_1, \dots , p_n\) on a Hilbert space H such that \(\sum _{j=1}^n p_j = \alpha \cdot I_H\), then \(\mathrm {dim}(H) \ge b\).

For each \(n \ge 2\) and each \(t \in [1/n,1]\), define the \(n \times n\) matrix \(A^{(n)}_t = \big [A^{(n)}_t(i,j)\big ]_{i,j=1}^n\) by

$$\begin{aligned} A^{(n)}_t(i,j) = {\left\{ \begin{array}{ll} t, &{} i=j,\\ \displaystyle {\frac{t(nt-1)}{n-1}}, &{} i \ne j. \end{array}\right. } \end{aligned}$$
(2.3)

Proposition 2.3

Let \(n \ge 2\) be an integer, let \(1/n \le t \le 1\), and let \(\alpha = nt\).

  1. (i)

    Let \(\mathcal{A}\) be a unital \(C^*\)-algebra with a faithful tracial state \(\tau \) and let \(p_1, \dots , p_n\) be projections in \(\mathcal{A}\) satisfying \(\tau (p_jp_i) = A^{(n)}_t(i,j)\), for all \(1\le i, j\le n\). Then \(\sum _{j=1}^n p_j = \alpha \cdot 1_\mathcal{A}\). Furthermore, if t is irrational, then \(\mathcal{A}\) is necessarily infinite dimensional. Respectively, if \(\alpha = a/b\) is rational with a, b positive integers and a / b is irreducible, then \(\mathcal{A}\) has no representation on a Hilbert space of dimension less than b.

  2. (ii)

    Conversely, let \(\mathcal{A}\) be a unital \(C^*\)-algebra with a tracial state \(\tau \), and let \(p_1, \dots , p_n\) be projections in \(\mathcal{A}\) satisfying \(\sum _{j=1}^n p_j = \alpha \cdot 1_\mathcal{A}\). Then there exist projections \({\widetilde{p}}_1, \dots , {\widetilde{p}}_n\) in some matrix algebra \(M_m(\mathcal{A})\) over \(\mathcal{A}\) satisfying

    $$\begin{aligned} \sum _{j=1}^n {\widetilde{p}}_j = \alpha \cdot 1_{M_m(\mathcal{A})}, \qquad {\widetilde{\tau }}({\widetilde{p}}_j{\widetilde{p}}_i) = A^{(n)}_t(i,j), \quad 1\le i, j\le n, \end{aligned}$$

    where \({\widetilde{\tau }}\) is the normalized trace on \(M_m(\mathcal{A})\) induced by \(\tau \).

Proof

(i). Set \(q = (nt)^{-1} \sum _{j=1}^n p_j\). To show that \(q=1_\mathcal{A}\), it suffices to check that \(1 = \tau (q) = \tau (q^2)\), since this will entail that \(\tau ((1-q)^*(1-q)) = \tau ((1-q)^2) = 0\). These are straightforward calculations: Indeed, \(\tau (q) = (nt)^{-1} \sum _{j=1}^n \tau (p_j) = 1\), and

$$\begin{aligned} \tau (q^2) = (nt)^{-2}\Big (\sum _{j=1}^n \tau (p_j) + \sum _{i \ne j} \tau (p_ip_j)\Big ) = (nt)^{-2} \left( nt + n(n-1) \cdot \frac{t(nt-1)}{n-1} \right) = 1. \end{aligned}$$

If t is irrational, then it follows from (the easy part of) Theorem 2.2 that \(\mathcal{A}\) does not have any finite dimensional representation on a Hilbert space, so \(\mathcal{A}\) must be infinite dimensional. Respectively, if \(\alpha =a/b\) is rational with a, b positive integers and a / b irreducible, then \(\mathcal{A}\) has no representation on a Hilbert space of dimension less than b, by the (explicit) comment after the statement of Theorem 2.2.

(ii). We follow the same strategy as in the proof of [3, Theorem 4.2]. Let \(m = n!\) and let \(S_n\) denote the collection of all permutations of the set \(\{1,2,\dots , n\}\). For \(1\le j\le n\), set

$$\begin{aligned} {\widetilde{p}}_j = \bigoplus _{\sigma \in S_n} p_{\sigma (j)} \in \bigoplus _{\sigma \in S_n} \mathcal{A}\subset M_m(\mathcal{A}), \end{aligned}$$

where the inclusion above is given by identifying the diagonal of \(M_m(\mathcal{A})\) with the direct sum \(\bigoplus _{\sigma \in S_n} \mathcal{A}\) (for some enumeration of \(S_n\)). It is easy to check that \(\sum _{j=1}^n {\widetilde{p}}_j = \alpha \cdot 1_{M_m(\mathcal{A})}\). Moreover, for \(1\le i\le n\),

$$\begin{aligned} {\widetilde{\tau }}({\widetilde{p}}_i) = \frac{1}{n!} \sum _{\sigma \in S_n} \tau (p_{\sigma (i)}) = \frac{1}{n} \sum _{k=1}^n \tau (p_k) :\!\!= t_0, \end{aligned}$$

and for \(1\le i \ne j\le n\),

$$\begin{aligned} {\widetilde{\tau }}({\widetilde{p}}_i{\widetilde{p}}_j) = \frac{1}{n!} \sum _{\sigma \in S_n} \tau (p_{\sigma (i)}p_{\sigma (j)}) = \frac{1}{n(n-1)} \sum _{k \ne \ell } \tau (p_k p_\ell ) :\!\!= s_0. \end{aligned}$$

Since \(\sum _{j=1}^n p_j = \alpha \cdot 1_\mathcal{A}\), we deduce that \(t_0 = \alpha /n = t\), and that

$$\begin{aligned} n^2t^2 = \alpha nt = \sum _{k=1}^n \tau (\alpha \cdot p_k) = \sum _{k=1}^n \tau \big (p_k \sum _{\ell =1}^n p_\ell \big ) = \sum _{k, \ell =1}^n \tau (p_kp_\ell ) = n t + n(n-1)s_0, \end{aligned}$$

from which we conclude that \(s_0 =(n-1)^{-1}t(nt-1)\), as desired. \(\square \)

Combining Theorem 2.2 and Proposition 2.3, we obtain the following:

Proposition 2.4

For \(n \ge 2\) and \(1/n \le t \le 1\) we have:

  1. (i)

    \(A_t^{(n)} \in {\mathcal {D}}(n)\) if and only if \(A_t^{(n)} \in \overline{{\mathcal {D}}_{\mathrm {fin}}(n)}\) if and only if \(t \in n^{-1} \Sigma _n\),

  2. (ii)

    \(A_t^{(n)} \in {\mathcal {D}}_{\mathrm {fin}}(n)\) if and only if \(A_t^{(n)} \in {\mathcal {D}}_{\mathrm {matrix}}(n)\) if and only if \(t \in n^{-1} \Sigma _n \cap {\mathbb Q}\).

Moreover, for \(n \ge 5\),

  1. (iii)

    \(A_t^{(n)}\) belongs to \(\overline{{\mathcal {D}}_{\mathrm {fin}}(n)} {\setminus } {\mathcal {D}}_{\mathrm {fin}}(n)\), for \(t \in n^{-1}\Sigma _n {\setminus } {\mathbb Q}= \Pi _n {\setminus } {\mathbb Q}\).

  2. (iv)

    If \(t \in n^{-1}\Sigma _n {\setminus } {\mathbb Q}\) and \(p_1, \dots , p_n\) are projections in a von Neumann algebra \((N,\tau )\) with faithful tracial state satisfying \(A_t^{(n)}(i,j) = \tau (p_jp_i)\), for all \(1\le i,j\le n\), then N must be of type \(\hbox {II}_1\).

  3. (v)

    For each \(A \in \overline{{\mathcal {D}}_{\mathrm {fin}}(n)}\), there exist projections \(p_1, \dots , p_n\) in the ultrapower \({\mathcal {R}}^\omega \) of the hyperfinite type \(\hbox {II}_1\) factor \({\mathcal {R}}\) such that \(A(i,j) = \tau _{{\mathcal {R}}^\omega }(p_jp_i)\), for all \(1\le i,j\le n\).

Proof

Suppose that \(n \ge 2\) and \(1/n \le t \le 1\). From the first part of Proposition 2.3 (i) we see that \(A_t^{(n)} \in {\mathcal {D}}(n)\) implies \(t \in n^{-1} \Sigma _n\), while its second part shows that \(A_t^{(n)} \in {\mathcal {D}}_{\mathrm {fin}}(n)\) implies \(t \in n^{-1} \Sigma _n \cap {\mathbb Q}\). Proposition 2.3 (ii) gives that \(t \in n^{-1} \Sigma _n\) implies \(A_t^{(n)} \in {\mathcal {D}}(n)\).

Let \(t \in n^{-1} \Sigma _n \cap {\mathbb Q}\). Then by Theorem 2.2, there exist projections \(p_1,\dots , p_n\) on a finite dimensional Hilbert space H satisfying \(\sum _{j=1}^n p_j = \alpha \cdot I_H\), where \(\alpha = nt\). Identifying the bounded operators on H with a full matrix algebra, we may assume that the projections \(p_j\) belong to \(M_k({\mathbb C})\), for some \(k \ge 2\). By Proposition 2.3, we can find projections \({\widetilde{p}}_1, \dots , {\widetilde{p}}_n\) in some larger matrix algebra \(M_{m}({\mathbb C})\) with normalized trace \(\mathrm {tr}\), satisfying \(A^{(n)}_t(i,j) = \mathrm {tr}({\widetilde{p}}_i{\widetilde{p}}_j)\), for all \(1\le i,j\le n\). This shows that \(A^{(n)}_t\) belongs to \({\mathcal {D}}_{\mathrm {matrix}}(n)\). As \({\mathcal {D}}_{\mathrm {matrix}}(n) \subset {\mathcal {D}}_{\mathrm {fin}}(n)\), this completes the proof of (ii).

To complete the proof of (i) we must show that \(A_t^{(n)} \in \overline{{\mathcal {D}}_{\mathrm {fin}}(n)}\) when \(t \in n^{-1} \Sigma _n\). This follows directly from (ii) when t is rational. Suppose that t is irrational. As remarked below (2.2), \(n^{-1}\Sigma _n {\setminus } \Pi _n \subset {\mathbb Q}\), for all \(n \ge 2\). Hence \(t \in \Pi _n {\setminus } {\mathbb Q}\). We can therefore find a sequence \(\{t_k\}_{k=1}^\infty \) of rational numbers in the interval \(\Pi _n\) converging to t. Then \(A^{(n)}_{t_k} \rightarrow A^{(n)}_t\), as \(k\rightarrow \infty \), and \(A^{(n)}_{t_k}\) belongs to \({\mathcal {D}}_{\mathrm {fin}}(n)\), for each \(k\ge 1\), by (i). This shows that \(A^{(n)}_t\) belongs to the closure of \({\mathcal {D}}_{\mathrm {fin}}(n)\).

(iii) follows from (i) and (ii).

(iv). The finite von Neumann algebra N can have no representation on a finite dimensional Hilbert space, by the second part of Proposition 2.3 (i), whence N must be of type \(\hbox {II}_1\).

(v). Suppose that A is the limit of a sequence \(\{A_k\}_{k=1}^\infty \) of matrices in \({\mathcal {D}}_{\mathrm {fin}}(n)\). Since each finite dimensional \(C^*\)-algebra with a distinguished trace can be embedded in a trace preserving way into the hyperfinite type \(\hbox {II}_1\) factor \({\mathcal {R}}\), we can find projections \(p^{(k)}_1, \dots , p^{(k)}_n\) in \({\mathcal {R}}\) satisfying \(\tau _{\mathcal {R}}(p^{(k)}_jp^{(k)}_i) = A_k(i,j)\), for all \(1\le i,j\le n\). Let \(p_i\) be the image in \(\mathcal {R^\omega }\) of the sequence \(\{p^{(k)}_i\}_{k=1}^\infty \in \ell ^\infty ({\mathcal {R}})\). Then

$$\begin{aligned} \tau _{{\mathcal {R}}^\omega }(p_jp_i) = \lim _\omega \tau _{\mathcal {R}}(p^{(k)}_jp^{(k)}_i) = \lim _\omega A_k(i,j) = A(i,j), \end{aligned}$$

for all \(1\le i,j\le n\), as wanted. \(\square \)

By Proposition 2.4 (iii) and the fact that \(\Pi _n\) is an interval with non-empty interior, when \(n \ge 5\), cf. (2.2), we obtain the following theorem:

Theorem 2.5

The set \({\mathcal {D}}_{\mathrm {fin}}(n)\) is non-compact, when \(n \ge 5\).

Remark 2.6

It is shown in Proposition 2.4 above that one can realize the \(n \times n\) matrix \(A_t^{(n)}\) using projections in \({\mathcal {R}}^\omega \), for all \(n \ge 2\) and all \(t \in n^{-1}\Sigma _n\). If moreover t is rational, then \(A_t^{(n)}\) can be realized using projections in some matrix algebra, by Proposition 2.3 (ii).

Using Proposition 2.3, this also shows that for each \(n \ge 2\) and for each \(\alpha \) in \(\Sigma _n\), one can find an n-tuple of projections summing up to \(\alpha \cdot 1_N\) in some type \(\hbox {II}_1\) factor N, e.g., \(N= {\mathcal {R}}^\omega \). One can also reach this conclusion directly from Theorem 2.2, using an ultraproduct argument as in the proof of Proposition 2.4 (v).

In Theorem A.1 in the Appendix by N. Ozawa it is shown that for each \(n \ge 5\) and each \(\alpha \in \big ({\textstyle {\frac{1}{2}}}(n-\sqrt{n^2-4n}), {\textstyle {\frac{1}{2}}}(n+\sqrt{n^2-4n})\big )\), one can find an n-tuple of projections in \({\mathcal {R}}\) summing up to \(\alpha \cdot 1_{{\mathcal {R}}}\). It follows that the matrix \(A^{(n)}_t\) can be realized using projections in \({\mathcal {R}}\) for all t in \(n^{-1}\Sigma _n\), except, possibly, for the endpoints of the interval \(\big [{\textstyle {\frac{1}{2}}}(1-\sqrt{1-4/n}), {\textstyle {\frac{1}{2}}}(1+\sqrt{1-4/n})\big ]\).

Recall, e.g., from [3, Section 2], that for \(n, k\ge 2\), the set \(C_{{qc}}(n,k)\) consists of \(nk \times nk\) quantum correlation matrices \(\big [(p(i,j | v,w)\big ]_{i,j,v,w}\) with entries given by

$$\begin{aligned} p(i,j | v,w) = \big \langle P_{v,i}Q_{w,j} \psi , \psi \big \rangle , \quad 1\le i,j\le k, \; \; 1\le v,w\le n, \end{aligned}$$

where, for each v and w, \(\{P_{v,i}\}_{i=1}^k\) and \(\{Q_{w,j}\}_{j=1}^k\) are projection-valued measures on some Hilbert space H, satisfying \(P_{v,i}Q_{w,j} =Q_{w,j}P_{v,i}\), for all i, j, v, w, and where \(\psi \) is a unit vector in H. Let \(C_{{q}}(n,k)\) be the same set of quantum correlation matrices, with the additional assumption that \(H=H_A \otimes H_B\), for some finite dimensional Hilbert spaces \(H_A\) and \(H_B\), and \(P_{v,i}\) belongs to \(B(H_A) \otimes I_{H_B}\), while \(Q_{w,j}\) belongs to \(I_{H_A} \otimes B(H_B)\). The closure of the set \(C_{{q}}(n,k)\) is denoted by \(C_{{qa}}(n,k)\). We have the following inclusions: \(C_q(n,k) \subseteq C_{qa}(n,k) \subseteq C_{qc}(n,k)\). Furthemore, the sets of synchronous correlation matrices, denoted by \(C^s_{{q}}(n,k)\), \(C^s_{{qa}}(n,k)\), and \(C^s_{{qc}}(n,k)\), respectively, consist of those quantum correlation matrices \(\big [(p(i,j | v,w)\big ]_{i,j,v,w}\) in \(C_{{q}}(n,k)\), \(C_{{qs}}(n,k)\), and \(C_{{qc}}(n,k)\), respectively, where \(p(i,j | v,v) = 0\), whenever \(i \ne j\).

We use Theorem 2.5 and Proposition 2.7 below from [13] to give a shorter proof of [3, Theorem 4.2] of Dykema, Paulsen and Prakash, which, again, was refining Slofstra’s result (in the general, non-synchronous case) from [14]. For this, using the notation from [13], for \(n,k \ge 2\) let \({\mathcal {D}}_q^s(n,k)\) be the set of matrices \(\big [\tau (e_{v,i}e_{w,j})\big ]_{i,j,v,w}\), where \(e_{v,i}\) are projections in a finite dimensional von Neumann algebra with a (faithful) tracial state \(\tau \), satisfying \(\sum _{i=1}^k e_{v,i} = 1\), for all \(1\le v\le n\). Note that \({\mathcal {D}}_q^s(n,k)\) is a subset of \({\mathcal {D}}(nk)\).

Proposition 2.7

(Paulsen, Severini, Stahlke, Todorov and Winter, [13]). For all \(n,k \ge 2\), one has \(C_q^s(n,k) ={\mathcal {D}}_q^s(n,k)\).

Theorem 2.8

(Dykema, Paulsen and Prakash, [3]). The set \(C_q^s(n,2)\) of synchronous quantum correlation matrices is non-compact, for all \(n \ge 5\).

Proof

Use Theorem 2.5 to find a matrix A in \({\mathcal {D}}(n) {\setminus } {\mathcal {D}}_{\mathrm {fin}}(n)\) and a sequence \(\{A_k\}_{k=1}^\infty \) of matrices in \( {\mathcal {D}}_{\mathrm {fin}}(n)\) converging to A. Let \(p_1, \dots , p_n\) be projections in \({\mathcal {R}}^\omega \) such that \(\tau _{{\mathcal {R}}^\omega }(p_jp_i) = A(i,j)\), for all i, j, cf. Proposition 2.4 (v), and let further \(p^{(k)}_1, \dots , p^{(k)}_n\) be projections in some matrix algebra \(M_{m_k}({\mathbb C})\) with normalized trace \(\mathrm {tr}_{m_k}\) such that \(\mathrm {tr}_{m_k}(p^{(k)}_jp^{(k)}_i) = A_k(i,j)\), \(1\le i, j\le n\).

Set \(e_{0,v} = p_v\) and \(e_{1,v} = 1-p_v\) in \({\mathcal {R}}^\omega \), and set \(e^{(k)}_{0,v} = p_v^{(k)}\) and \(e^{(k)}_{1,v} = 1- p_v^{(k)}\) in \(M_{m_k}({\mathbb C})\), for all \(k \ge 1\) and \(1\le v \le n\). It then follows that

$$\begin{aligned} \lim _{k \rightarrow \infty } \big [\mathrm {tr}_{m_k}(e^{(k)}_{v,i}e^{(k)}_{w,j})\big ]_{i, j, v, w} = \big [\tau _{{\mathcal {R}}^\omega }(e_{v,i}e_{w,j})\big ]_{i, j, v, w}, \end{aligned}$$

and each of the matrices \(\big [\mathrm {tr}_{m_k}(e^{(k)}_{v,i}e^{(k)}_{w,j})\big ]_{i, j, v, w}\) belongs to \({\mathcal {D}}_q^s(n,2) = C_q^s(n,2)\), cf. Proposition 2.7. However, the matrix \(\big [\tau (e_{v,i}e_{w,j})\big ]_{i, j, v, w}\) itself does not belong to \(C_q^s(n,2)\), since the matrix \(A = \big [\tau (e_{v,0}e_{w,0})\big ]_{v,w}\) does not belong to \({\mathcal {D}}_{\mathrm {fin}}(n)\). \(\square \)

Note that the set \(C_q(n,2)\) of (non-synchronous) quantum correlation matrices contains \(C_q^s(n,2)\) as a relatively closed subset, which shows that \(C_q(n,2)\) also is non-closed, when \(n \ge 5\).

We end this section with a remark on the matrices \(A_t^{(n)}\) defined in (2.3).

Remark 2.9

For each integer \(n \ge 2\) and for all \(s,t \in [0,1]\), consider the \(n \times n\) matrix \(A^{(n)}_{t,s}\), whose diagonal entries all are equal to t and whose off-diagonal entries are all equal to s. Note that \(A^{(n)}_{t} = A^{(n)}_{t,s}\), with \(s := t(nt-1)/(n-1)\), when \(t \in [1/n,1]\). The purpose of this remark is to describe the set of “admissible pairs” (t, s), for which \(A^{(n)}_{t,s}\) belongs to \({\mathcal {D}}(n)\), and to show that \(s = t(nt-1)/(n-1)\) is the smallest number for which (t, s) is such an admissible pair, for each fixed \(t \in n^{-1}\Sigma _n\),

Note first that for each fixed \(t \in [0,1]\), the set \(I^{(n)}_t\) of those \(s \in [0,1]\), for which (t, s) is admissible, is a closed interval. This follows by convexity and compactness of the set \({\mathcal {D}}(n)\).

Fix \(t \in [0,1]\), take a projection p of trace t in some finite von Neumann algebra, and let \(p_1= \cdots = p_n = p\). The matrix \(A^{(n)}_{t,t}\) is then equal to \(\big [\tau (p_jp_i)\big ]_{i,j=1}^n\), and therefore belongs to \({\mathcal {D}}(n)\). Moreover, since \(\tau (pq) \le \tau (p)\), whenever p, q are projections in a von Neumann algebra with tracial state \(\tau \), we conclude that \(t=\max I^{(n)}_t\).

For \(0 \le t \le 1/n\) we can find pairwise orthogonal projections \(p_1,\dots , p_n\) in any type \(\hbox {II}_1\) factor M with \(\tau _M(p_j)=t\), for all j. The corresponding \(n \times n\) matrix \(\big [\tau (p_jp_i)\big ]_{i,j=1}^n\) is equal to \(A^{(n)}_{t,0}\). Hence 0 belongs to \(I^{(n)}_t\). This shows that \(I^{(n)}_t = [0,t]\), when \(0 \le t \le 1/n\).

Let \(s,t \in [0,1]\), and suppose that (t, s) is an admissible pair. Let \(p_1, \dots , p_n\) be projections in a finite von Neumann algebra with tracial state \(\tau \) such that \(\big [\tau (p_jp_i)\big ]_{i,j=1}^n = A^{(n)}_{t,s}\). Put \(q_j = 1-p_j\), \(1 \le j \le n\). Then \(\big [\tau (q_jq_i)\big ]_{i,j=1}^n = A^{(n)}_{1-t,1-2t+s}\), which shows that \((1-t,1-2t+s)\) is an admissible pair. The map given by \((t,s) \longmapsto (1-t,1-2t+s)\) is involutive, and therefore it maps the set of admissible pairs in \([0,1]^2\) onto itself. In particular, this involution maps \(\{t\} \times I_t^{(n)}\) onto \(\{1-t\} \times I_{1-t}^{(n)}\). Combining this fact with the result of the previous paragraph, we obtain that \(I_t^{(n)} = [2t-1,t]\), when \(1-1/n\le t \le 1\).

Consider finally the case where \(t \in [1/n,1-1/n]\). Suppose that \(s \in I_t\), and let \(p_1, \dots , p_n\) be projections in some finite von Neumann algebra with tracial state \(\tau \) such that \(\big [\tau (p_jp_i)\big ]_{i,j=1}^n = A^{(n)}_{t,s}\). Then

$$\begin{aligned} nt + n(n-1)s = \tau \Big ( (\sum _{i=1}^n p_i)^2\Big ) \ge \Big (\tau (\sum _{i=1}^n p_i)\Big )^2 = n^2 t^2, \end{aligned}$$

which implies that \(s \ge t(nt-1)/(n-1)\). In other words, \(I_t^{(n)} \subseteq [ t(nt-1)/(n-1),t]\). It was noted in Remark 2.6 that \(A^{(n)}_t= A^{(n)}_{t,t(nt-1)/(n-1)}\) belongs to \({\mathcal {D}}(n)\) if and only if t belongs to \(n^{-1} \Sigma _n\). For those values of t, we therefore obtain that \(s = t(nt-1)/(n-1)\) is the smallest number for which (t, s) is an admissible pair, which shows that \(I^{(n)}_t = [ t(nt-1)/(n-1),t]\).

It remains a curious open problem to detemine the interval \(I^{(n)}_t\), for t belonging to the (non-empty) set \([1/n, 1-1/n] {\setminus } n^{-1} \, \Sigma _n\). In this case, necessarily, \(\min I^{(n)}_t > t(nt-1)/(n-1)\).

3 Correlation Matrices of Unitary Elements

Recall that a correlation matrix is a positive definite matrix whose diagonal entries are equal to 1. For each integer \(n \ge 2\), let \({\mathcal {G}}(n)\), \({\mathcal {F}}_{\mathrm {matrix}}(n)\), and \({\mathcal {F}}_{\mathrm {fin}}(n)\) be the set of \(n \times n\) correlation matrices \(\big [\tau (u_j^*u_i)\big ]_{i,j=1}^n\), where \(u_1,u_2, \dots , u_n\) are unitaries in some finite von Neumann algebra equipped with a faithful tracial state \(\tau \), respectively, in some full matrix algebra with its canonical tracial state, respectively, in some finite dimensional \(C^*\)-algebra with a faithful tracial state. Then the convex hull \(\mathrm {conv}({\mathcal {F}}_{\mathrm {matrix}}(n))\) of \({\mathcal {F}}_{\mathrm {matrix}}(n)\) is equal to \({\mathcal {F}}_{\mathrm {fin}}(n)\), and the two sets \({\mathcal {F}}_{\mathrm {matrix}}(n)\) and \({\mathcal {F}}_{\mathrm {fin}}(n)\) have the same closure, which is denoted by \({\mathcal {F}}(n)\). The sets \({\mathcal {G}}(n)\) and \({\mathcal {F}}(n)\) are compact and convex, see [2, Proposition 1.4].

As shown by Kirchberg, [9] (cf. Dykema–Juschenko, [2]), the Connes Embedding Problem has an affirmative answer if and only if \({\mathcal {G}}(n) = {\mathcal {F}}(n)\), for all \(n \ge 3\). We can use this to show that \({\mathcal {D}}_{\mathrm {matrix}}(n)\) is dense in \({\mathcal {D}}(n)\), for all \(n \ge 3\), if and only if the Connes Embedding Problem has an affirmative answer. Indeed, the proof of the “if” part follows the same strategy as the proof of the “if” part of Kirchberg’s result, where a given finite subset of \({\mathcal {R}}^\omega \) is approximated in trace-norm with a finite subset of a matrix subalgebra of \({\mathcal {R}}^\omega \). To see the “only if” part, assume that \({\mathcal {D}}_{\mathrm {matrix}}(n)\) is dense in \({\mathcal {D}}(n)\), for all \(n \ge 3\). We show that this implies \({\mathcal {G}}(n) = {\mathcal {F}}(n)\), for all \(n \ge 3\). Take an n-tuple \(u_1, \dots , u_n\) of unitaries in some tracial von Neumann algebra \((N,\tau )\). Fix an integer \(m \ge 1\). Each \(u_j\) can be approximated in norm within \(2\pi /m\) by unitaries \(v_1, \dots , v_n\) in N of the form \(v_j = \sum _{k=1}^m \omega ^k p_{k,j}\), where \(\omega = \exp (2\pi i/m)\) and \(p_{1,j}, \dots , p_{m,j}\) are pairwise orthogonal projections in N summing up to 1, for each j. Approximate the second-order moments of the collections of projections \(\{p_{k,j}\}_{k,j}\) by second-order moments of projections \(\{q_{k,j}\}_{k,j}\) in some matrix algebra \((M_r({\mathbb C}), \mathrm {tr}_r)\). Then \(\mathrm {tr}_r(q_{k,j}q_{\ell ,j})\) are small when \(k \ne \ell \), and \(\mathrm {tr}_r\big (\sum _{k=1}^m q_{k,j}\big )\) is close to 1, for all j. A standard lifting argument allows us to replace the projections \(\{q_{k,j}\}_{k,j}\) with new projections, close to the old ones in trace-norm, satisfying \(\sum _{j=1}^m q_{k,j} = 1\), for all j. Set \(w_j = \sum _{k=1}^m \omega ^k q_{k,j}\). The second-order moments of the unitaries \(w_1, \dots , w_n\) are then close to those of \(u_1, \dots , u_n\).

Remark 3.1

In the case where \(n=2\), the set \({\mathcal {F}}_{\mathrm {matrix}}(2)\) is closed and convex, and \({\mathcal {F}}_{\mathrm {matrix}}(2) = {\mathcal {F}}(2) = {\mathcal {G}}(2)\), which further is equal to the set of \(2 \times 2\) matrices of the form

$$\begin{aligned} \begin{pmatrix} 1 &{} \quad {\bar{z}} \\ z &{} \quad 1 \end{pmatrix},\end{aligned}$$

for \(z \in {\mathbb C}\) with \(|z| \le 1\). To see that each of these \(2 \times 2\) matrices belongs to \({\mathcal {F}}_{\mathrm {matrix}}(2)\), take \(z \in {\mathbb C}\) with \(|z| \le 1\) and find \(\lambda _1\) and \(\lambda _2\) on the complex unit circle \({\mathbb T}\), with \(z = (\lambda _1+\lambda _2)/2\). The correlation matrix arising from the unitary \(2 \times 2\) matrices \(u_1=1\) and \(u_2 = \mathrm {diag}(\lambda _1,\lambda _2)\) is then as desired.

We show in this section that the set \({\mathcal {F}}_{\mathrm {fin}}(n)\) is not closed (hence, not compact), for all \(n \ge 11\). This result originates in a remark made by T. Vidick during his talk at one of the workshops in the Quantitative Linear Algebra program at IPAM, Spring 2018, that led to subsequent discussions with W. Slofstra, who, in particular, communicated to us a version of the following result (to appear, in an approximate case, in a forthcoming paper by O. Regev, W. Slofstra and T. Vidick):

Proposition 3.2

Let M be a finite von Neumann algebra with a faithful tracial state \(\tau _M\), and let \(p_1,\dots , p_n\) be projections in M. Further, let \(u_0,u_1, \dots , u_n\), \(u_{n+1}, \dots , u_{2n}\) be the unitaries in M given by

$$\begin{aligned} u_0 = 1, \quad u_j = 2p_j - 1, \; \; (1 \le j \le n), \quad u_j = (u_{j-n} + i \cdot 1)/{\sqrt{2}}, \; \; (n+1 \le j \le 2n). \end{aligned}$$

Let N be another finite von Neumann algebra with a faithful tracial state \(\tau _N\). Then there exist \(2n+1\) unitaries \(v_0,v_1, \dots , v_{2n}\) in N satisfying

$$\begin{aligned} \tau _N(v_j^*v_i) = \tau _M(u_j^*u_i), \qquad 0 \le i,j\le 2n, \end{aligned}$$
(3.1)

if and only if there exist n projections \(q_1, \dots , q_n\) in N satisfying

$$\begin{aligned} \tau _N(q_jq_i) = \tau _M(p_jp_i),\qquad 1\le i,j\le n. \end{aligned}$$
(3.2)

Proof

Assume that \(q_1, \dots , q_n\) are projections in N satisfying (3.2). Equip the vector spaces \(\mathrm {span} \{p_1,p_2, \dots , p_n\}\) and \(\mathrm {span}\{q_1,q_2, \dots , q_n\}\) with the Euclidean structure arising from the traces \(\tau _M\) and \(\tau _N\), respectively. Using (3.2), we see that the map \(p_j \longmapsto q_j\), \(1 \le j \le n\), extends to a well-defined linear isometry \(\varphi \) from \(\mathrm {span}\{p_1, \dots , p_m\} \) to \(\mathrm {span}\{q_1, \dots , q_m\}\). Set \(v_0 = 1\), \(v_j = 2q_j-1\), for \(1 \le j \le n\), and \(v_j = (v_{j-n}+i \cdot 1)/\sqrt{2}\), for \(n+1 \le j \le n\), and use the isometric property of \(\varphi \) to check that (3.1) holds.

Conversely, assume that we are given unitaries \(v_0,v_1, \dots , v_{2n}\) in N satisfying (3.1). Upon replacing \(v_j\) by \(v_0^*v_j\), for all \(0\le j\le 2n\), we may assume that \(v_0=1\). As above, equip the vector spaces \(\mathrm {span} \{u_0,u_1, \dots , u_{2n}\}\) and \(\mathrm {span}\{v_0,v_1, \dots , v_{2n}\}\) with the Euclidean structure arising from the traces \(\tau _M\) and \(\tau _N\), respectively. Then, by (3.1), we have a well-defined linear isometry \(\psi :\mathrm {span} \{u_0,u_1, \dots , u_{2n}\} \rightarrow \mathrm {span}\{v_0,v_1, \dots , v_{2n}\}\), mapping \(u_j\) to \(v_j\), for \(0 \le j \le 2n\). In particular, for \(1 \le j \le n\),

$$\begin{aligned} \big \Vert v_{j+n} - (v_j + i \cdot v_0)/\sqrt{2}\big \Vert _2 = \big \Vert u_{j+n} - (u_j + i \cdot u_0)/\sqrt{2}\big \Vert _2 = 0,\end{aligned}$$

so \(v_{j+n} = (v_j + i \cdot 1)/\sqrt{2}\).

Note that if u and \((u+i \cdot 1)/\sqrt{2}\) are unitaries in some unital \(C^*\)-algebra, then u is necessarily a symmetry. Indeed, if \(\lambda \) is a complex number such that \( |\lambda | = |(\lambda + i)/\sqrt{2}| = 1\), then \(\lambda \in {\mathbb R}\). Hence, if u is as stated, then its spectrum is contained in \({\mathbb R}\), which entails that it is a symmetry.

We conclude that \(v_1, \dots , v_n\) are symmetries. For \(1\le j\le n\), set \(q_j = (v_j+1)/2\). Then \(q_j\) is a projection and \(v_j = 2q_j-1\). Use the isometric property of \(\psi \) to check (3.2). \(\square \)

Corollary 3.3

The set \({\mathcal {F}}_{\mathrm {matrix}}(m)\) is not compact and not convex, whenever \(m \ge 3\).

Proof

Let \(0< \alpha < 1\) be irrational. Equip \(M := {\mathbb C}\oplus {\mathbb C}\) with the trace \(\tau \) given by \(\tau (x,y) = \alpha x + (1-\alpha )y\), for \(x, y\in {\mathbb {C}}\).

Consider first the case where \(m=3\). Let \(n=1\) and let \(p=p_1 = (1,0) \in M\). Then \(\tau (p) = \alpha \) is irrational. Let \(u_0,u_1,u_2\) be the unitaries in M arising from this projection as in the proposition above (with \(n=1\)). The matrix \(\big [\tau (u_j^*u_i)\big ]_{i,j=0}^{2}\) belongs to \(\mathrm {conv}({\mathcal {F}}_{\mathrm {matrix}}(3))\), and hence to \({\mathcal {F}}(3)\). However, by Proposition 3.2, it does not belong to \({\mathcal {F}}_{\mathrm {matrix}}(3)\) itself, because no full matrix algebra contains a projection of irrational trace, and therefore contains no projection \(q=q_1\) satisfying (3.2) (with \(n=1\)).

Assume now that \(m > 3\), let \(u_0,u_1,u_2\) be as above, and let unitaries \(u_3,u_4, \dots , u_{m-1}\) in M be arbitrary. If \(v_0,v_1, \dots , v_{m-1}\) are unitaries in some tracial von Neumann algebra \((N,\tau _N)\) satisfying (3.1), then \(v_0,v_1,v_2\) satisfy (3.1) with respect to the set \(\{u_0,u_1,u_2\}\), so \(v_0,v_1,v_2\), and hence \(v_0,v_1, \dots , v_{m-1}\) cannot be found in a full matrix algebra. These arguments also yield the non-convexity of the set \({\mathcal {F}}_{\mathrm {matrix}}(m)\) in all cases. \(\square \)

Example 3.4

For \(m=3\), the correlation matrix \(B=\big [\tau (u_j^*u_i)\big ]_{i,j=0}^{2}\) from the proof above, with unitaries given by

$$\begin{aligned} u_0 = (1,1), \qquad u_1 = (1,-1), \qquad u_2 = \left( \frac{1+i}{\sqrt{2}}, \frac{-1+i}{\sqrt{2}}\right) , \end{aligned}$$

has the following explicit form in terms of the parameter \(\alpha \in (0,1)\):

$$\begin{aligned} B = \begin{pmatrix} 1 &{} \quad \gamma &{} \quad \frac{\gamma - i}{\sqrt{2}} \\ \gamma &{} \quad 1 &{} \quad \frac{1-\gamma i}{\sqrt{2}}\\ \frac{\gamma + i}{\sqrt{2}}&{} \quad \frac{1+\gamma i}{\sqrt{2}}&{}\quad 1 \end{pmatrix}, \end{aligned}$$

where \(\gamma = 2\alpha -1 \in (-1,1)\). Note that the matrix B belongs to \({\mathcal {F}}_{\mathrm {fin}}(3)\), for all \(\gamma \in (-1,1)\), while it does not belong to \({\mathcal {F}}_{\mathrm {matrix}}(3)\), whenever \(\gamma \) is irrational.

Example 3.5

For each \(n \ge 2\) and each \(t \in [1/n,1]\), consider the self-adjoint \((1+2n) \times (1+2n)\) complex matrix

$$\begin{aligned} B_t^{(n)} = \begin{pmatrix}1 &{}\quad X^* &{}\quad Y^* \\ X &{} \quad D_1 &{}\quad C^* \\ Y &{}\quad C &{}\quad D_2 \end{pmatrix}, \end{aligned}$$

where X, Y and \(C, D_1, D_2\) are the \(n \times 1\), respectively, \(n \times n\) complex matrices given by

$$\begin{aligned} X= & {} s \begin{pmatrix}1 \\ 1 \\ \vdots \\ 1 \end{pmatrix}, \qquad Y = \frac{s+i}{\sqrt{2}} \begin{pmatrix} 1\\ 1 \\ \vdots \\ 1 \\ \end{pmatrix}, \qquad C = \frac{1+is}{\sqrt{2}} \, I_n + \frac{4(r-t)+1+is}{\sqrt{2}} \, E,\\ D_1= & {} I_n + (4(r-t)+1) \, E, \qquad D_2 = I_n + (2(r-t)+1) \, E, \end{aligned}$$

where \(s = 2t-1\) and \(r = (n-1)^{-1}t(nt-1)\), and

$$\begin{aligned} E = \begin{pmatrix} 0 &{} \quad 1 &{} \quad \cdots &{} \quad 1 \\ 1 &{} \quad 0 &{}\quad \cdots &{} \quad 1 \\ \vdots &{}\quad \vdots &{} \quad {\ddots } &{} \quad \vdots \\ 1 &{}\quad 1 &{}\quad \cdots &{} \quad 0 \end{pmatrix}\in M_n({\mathbb {C}}). \end{aligned}$$

With this definition, Proposition 3.2 yields that the \((2n+1) \times (2n+1)\) matrix \(B^{(n)}_t\) is the correlation matrix of an \((2n+1)\)-tuple of unitaries in some finite von Neumann algebra M if and only if the \(n \times n\) matrix \(A_t^{(n)}\) is the correlation matrix of an n-tuple of projections in the same von Neumann algebra M. Indeed, if \(p_1,\dots , p_n\) are projections in M such that \(\big [\tau _M(p_jp_i)\big ]_{i,j=1}^n = A_t^{(n)}\), and if \(u_0,u_1, \dots , u_{2n}\) are the \(2n+1\) unitaries in M constructed from these projections as in Proposition 3.2, then \(B_t^{(n)} = \big [\tau _M(u_j^*u_i)\big ]_{i,j=0}^{2n}\). Conversely, if \(v_0,v_1, \dots , v_{2n}\) are unitaries in M such that \(B_t^{(n)} = \big [\tau _M(v_j^*v_i)\big ]_{i,j=0}^{2n}\), then there are projections \(q_1, \dots , q_n \in M\) such that \(A_t^{(n)} = \big [\tau _M(q_jq_i)\big ]_{i,j=1}^n\).

In particular, \(B^{(n)}_t\) belongs to \({\mathcal {G}}(2n+1)\), respectively, to \({\mathcal {F}}_{\mathrm {fin}}(2n+1)\), if and only if \(A^{(n)}_t\) belongs to \({\mathcal {D}}(n)\), respectively, to \({\mathcal {D}}_{\mathrm {fin}}(n)\).

Theorem 3.6

Let \(n \ge 5\) and let \(t \in \Pi _n\).

  1. (i)

    If t is rational, then \(B_t^{(n)}\) belongs to \({\mathcal {F}}_{\mathrm {fin}}(2n+1)\), and if t is irrational, then \(B_t^{(n)}\) belongs \({\mathcal {F}}(2n+1) {\setminus } {\mathcal {F}}_{\mathrm {fin}}(2n+1)\).

  2. (ii)

    If t is irrational and if \(v_0,v_1, \dots , v_{2n}\) are unitaries in some finite von Neumann algebra N with a faithful tracial state \(\tau _N\) such that \(\tau _N(v_j^*v_i) = B_t^{(n)}(i,j)\), for \(0 \le i,j \le 2n\), then N is necessarily of type \(\hbox {II}_1\).

  3. (iii)

    If \(t \in \big ({\textstyle {\frac{1}{2}}}(1-\sqrt{1-4/n}), {\textstyle {\frac{1}{2}}}(1+\sqrt{1-4/n})\big )\), then there are unitaries \(v_0,v_1, \dots , v_{2n}\) in the hyperfinite \(\hbox {II}_1\) factor \({\mathcal {R}}\) such that \(\tau _N(v_j^*v_i) = B_t^{(n)}(i,j)\), for \(0 \le i,j \le 2n\).

  4. (iv)

    The convex sets \({\mathcal {F}}_{\mathrm {fin}}(k)\) are non-compact, for all \(k \ge 11\).

Proof

(i). The map \(t \mapsto B_t^{(n)}\), \(t \in \Pi _n\), is clearly continuous. It follows from Proposition 2.3 and Example 3.5 that \(B_t^{(n)} \in {\mathcal {G}}(2n+1)\), whenever \(A_t^{(n)} \in {\mathcal {D}}(n)\), and in particular whenever \(t \in \Pi _n\). Moreover, if \(t \in \Pi _n\), then \(B_t^{(n)} \in {\mathcal {F}}_{\mathrm {fin}}(2n+1)\), when t is rational, and \(B_t^{(n)} \notin {\mathcal {F}}_{\mathrm {fin}}(2n+1)\), when t is irrational. This implies that \(B_t^{(n)} \in {\mathcal {F}}(2n+1)\), for all \(t \in \Pi _n\), and that \({\mathcal {F}}_{\mathrm {fin}}(2n+1)\) is non-compact, for each \(n \ge 5\).

(ii). Let \(v_0,v_1, \dots , v_{2n}\) be unitaries in some finite von Neumann algebra N with faithful tracial state \(\tau _N\), satisfying \(\tau _N(v_j^*v_i) = B_t^{(n)}(i,j)\), for \(0 \le i,j \le 2n\). Then, by Example 3.5, there exist projections \(q_1,\dots , q_n\) in N such that \(A_t^{(n)}(i,j) = \tau _N(q_jq_i)\). This entails that N has no finite dimensional representations, by Proposition 2.3, so N must be of type \(\hbox {II}_1\).

(iii). Let \(t \in \big ({\textstyle {\frac{1}{2}}}(1-\sqrt{1-4/n}), {\textstyle {\frac{1}{2}}}(1+\sqrt{1-4/n})\big )\). It follows from Theorem A.1 in the Appendix by Ozawa that \(A_t^{(n)}\) can be realized using projections in \({\mathcal {R}}\), cf. Remark 2.6. The claim now follows from Example 3.5.

(iv). We show that if \({\mathcal {F}}_{\mathrm {fin}}(k)\) is non-compact, for some positive integer k, then so is \({\mathcal {F}}_{\mathrm {fin}}(k+1)\). To this end, define a map \(\rho :{\mathcal {F}}(k) \rightarrow {\mathcal {F}}(k+1)\) by

$$\begin{aligned} \rho \Big (\big [\tau _M(u^*_ju_i)\big ]_{i,j=1}^k\Big ) = \big [\tau _M(u^*_ju_i)\big ]_{i,j=1}^{k+1}, \end{aligned}$$

whenever \(u_1, \dots , u_k\) are unitaries in some von Neumann algebra M with a faithful tracial state \(\tau \), and where \(u_{k+1}\) is chosen to be equal to \(u_1\). Then the last row and the last column of \(\big [\tau _M(u^*_ju_i)\big ]_{i,j=1}^{k+1}\) is equal to the first row and the first column of the matrix \(\big [\tau _M(u^*_ju_i)\big ]_{i,j=1}^k\), which shows that \(\rho \) is well-defined and continuous. Moreover, \(\rho ^{-1}({\mathcal {F}}_{\mathrm {fin}}(k+1)) = {\mathcal {F}}_{\mathrm {fin}}(k)\). Hence, if \({\mathcal {F}}_{ \mathrm {fin}}(k+1)\) is compact, and thus closed, then \({\mathcal {F}}_{\mathrm {fin}}(k)\) is closed, and thus compact. \(\square \)

4 Factorizable Maps that Require Infinite Dimensional Ancilla

We prove here our claimed result about existence of factorizable quantum channels in all dimensions \(\ge 11\), requiring infinite dimensional ancilla. We first recall necessary prerequisites.

Let \(\mathrm {UCPT}(n)\) denote the convex and compact set of all unital completely positive trace preserving linear maps \(T :M_n({\mathbb C}) \rightarrow M_n({\mathbb C})\), \(n\ge 2\). Maps in \(\mathrm {UCPT}(n)\) are also called unital quantum channels in dimension n. Anantharaman-Delaroche defined in [1] a channel to be factorizable if it admits a factorization (in a suitable way) through a finite von Neumann algebra with a faithful tracial state. This notion was studied extensively in [5], and the following characterization (which we will take to be our definition of factorizable maps) was established therein (cf., [5, Theorem 2.2]): A unital quantum channel T in dimension n is factorizable if and only if there exist a finite von Neumann algebra N, equipped with a normal faithful tracial state \(\tau _N\), and a unitary \(u \in M_n({\mathbb C}) \otimes N\) such that

$$\begin{aligned} T(x) = (\mathrm {id}_n \otimes \tau _N)(u(x \otimes 1_N)u^*), \qquad x \in M_n({\mathbb C}). \end{aligned}$$
(4.1)

The von Neumann algebra N above is also called the ancilla, and, following Definition 3.1 in [6], we say that T has an exact factorization through \(M_n({\mathbb C}) \otimes N\). The set of all factorizable unital channels in dimension n is denoted by \(\mathcal {FM}(n)\). This set is convex and compact, as shown by standard arguments.

Further, let \(\mathcal {FM}_{\mathrm {matrix}}(n)\) and \(\mathcal {FM}_{\mathrm {fin}}(n)\) denote the set of factorizable maps in UCPT(n) that exactly factor through a full matrix algebra, respectively, through a finite dimensional \(C^*\)-algebra, equipped with a faithful tracial state. It was shown in [6, Theorem 3.7] that a positive answer to the Connes Embedding Problem is equivalent to \(\mathcal {FM}_{\mathrm {matrix}}(n)\) being dense in \(\mathcal {FM}(n)\), for all \(n \ge 3\). Moreover, if a unital quantum channel \(T:M_n({\mathbb {C}})\rightarrow M_n({\mathbb {C}})\) belongs to the closure of \(\mathcal {FM}_{\mathrm {matrix}}(n)\), then T admits an exact factorization through an ultrapower \({\mathcal {R}}^\omega \) of the hyperfinite type \(\hbox {II}_1\) factor \(\mathcal {R}\).

It is shown in the upcoming manuscript [12] that \(\mathcal {FM}_{\mathrm {fin}}(n) = \mathrm {conv}(\mathcal {FM}_{\mathrm {matrix}}(n))\), and that \(\mathcal {FM}_{\mathrm {matrix}}(n)\) is non-compact and non-convex, when \(n \ge 3\).

Let \({\mathcal {S}}(n)\) be the set of all Schur multipliers\(T_B :M_n({\mathbb C}) \rightarrow M_n({\mathbb C})\), where \(B \in M_n({\mathbb C})\), i.e., \(T_B(x)\) is the Schur product of B and x, for \(x \in M_n({\mathbb C})\). Let \(\mathcal {FMS}(n) = \mathcal {FM}(n) \cap {\mathcal {S}}(n)\) be the set of all factorizable Schur multipliers, and write \(\mathcal {FMS}_{\mathrm {fin}}(n) = \mathcal {FM}_{\mathrm {fin}}(n) \cap {\mathcal {S}}(n)\).

For the theorem below, recall the definition (2.2) of the set \(\Pi _n\), and the \((2n+1) \times (2n+1)\) matrix \(B_t^{(n)}\) constructed in Example 3.5.

Theorem 4.1

The set \(\mathcal {FM}_{\mathrm {fin}}(k)\) is not compact, for all \(k \ge 11\). Moreover, for each \(n \ge 5\) and each irrational number \(t \in \Pi _n\), the Schur multiplier \(T_B\), where \(B = B_t^{(n)}\), is a factorizable map belonging to the closure of \(\mathcal {FM}_{\mathrm {fin}}(2n+1)\), but not to \(\mathcal {FM}_{\mathrm {fin}}(2n+1)\), and it requires an ancilla of type \(\hbox {II}_1\). This ancilla can be taken to be the hyperfinite \(\hbox {II}_1\) factor \({\mathcal {R}}\), when \({\textstyle {\frac{1}{2}}}(1-\sqrt{1-4/n})< t < {\textstyle {\frac{1}{2}}}(1+\sqrt{1-4/n})\).

Proof

It was shown in [5, Proposition 2.8] that if \(B \in M_k({\mathbb C})\) is a correlation matrix, then the associated Schur multiplier \(T_B\) admits an exact factorization through \(M_k({\mathbb C}) \otimes N\), where N is a finite von Neumann algebra with normal faithful tracial state \(\tau _N\), if and only if \(B = \big [\tau _N(u_j^*u_i)\big ]_{i,j=1}^k\), for some unitaries \(u_1,\dots , u_k \in N\). In particular, \(T_B \) belongs to \(\mathcal {FMS}(k)\), \(\mathcal {FMS}_{\mathrm {fin}}(k)\), and the closure of \(\mathcal {FMS}_{\mathrm {fin}}(k)\), respectively, if and only if B belongs to \({\mathcal {G}}(k)\), \({\mathcal {F}}_{\mathrm {fin}}(k)\), and \({\mathcal {F}}(k)\), respectively.

Since the map from \({\mathcal {G}}(k)\) to \(\mathcal {FM}(k)\) given by \(B \mapsto T_B\) is continuous, it follows from Theorem 3.6 that \(\mathcal {FMS}_{\mathrm {fin}}(k)\) is non-closed in \(\mathcal {FM}(k)\). As the set \({\mathcal {S}}(k)\) is closed, we conclude that \(\mathcal {FM}_{\mathrm {fin}}(k)\) is non-compact.

To prove the second part of the theorem, let \(n \ge 5\), let \(t \in \Pi _n\) be irrational, and let \(B=B_t^{(n)}\). Then B belongs to \({\mathcal {F}}(2n+1){\setminus } {\mathcal {F}}_{\mathrm {fin}}(2n+1)\) by Theorem 3.6. By the argument in the first paragraph, we conclude that \(T_B\) belongs to the closure of \(\mathcal {FMS}_{\mathrm {fin}}(2n+1)\), but not to \(\mathcal {FMS}_{\mathrm {fin}}(2n+1)\). Moreover, if \(T_B\) admits an exact factorization through \(M_n({\mathbb C}) \otimes N\), with ancilla \((N,\tau _N)\) as in the first paragraph, then B is the matrix of correlations of unitaries \(u_0,u_1, \dots , u_{2n}\) in N, which by Theorem 3.6 implies that N must be of type \(\hbox {II}_1\).

The remaining part of the theorem follows from Theorem 3.6 (iii) and the argument in the first paragraph. \(\square \)

We also obtain the following quantitative version of the theorem above, in the case where \(t \in \Pi _n\) is rational and \(n \ge 5\): Let \(B = B_t^{(n)}\). If \(nt = a/b\), with a, b positive integers and a / b irreducible, then the Schur channel \(T_B\) admits an exact factorization through a finite dimensional von Neumann algebra \(M= M_n({\mathbb C}) \otimes N\), where the von Neumann algebra N can have no representation on a Hilbert space of dimension smaller than b. This follows as in the proof of the theorem above and by appealing to Proposition 2.3 (i).

We conclude that for every fixed integer \(n \ge 11\), there is a sequence of factorizable unital quantum channels in dimension n, each admitting finite-dimensional ancillas, but where the size of any such ancillas must tend to infinity.