Bell inequalities and operator algebras

Cite this problem as Problem 33.

Problem

Quantum Bell-type inequalities are defined in terms of two (or more) subsystems of a quantum system. The subsystems may be treated either via (local) Hilbert spaces, — tensor factors of the given (global) Hilbert space, or via commuting (local) operator algebras. The latter approach is less restrictive, it just requires that the given operators commute whenever they belong to different subsystems.

Are these two approaches equivalent?

Background

Three convex sets are considered in [1] and Problem 26, denoted $X_{\text{HDB}} \subset X_{\text{QB}} \subset X_{\text{B}}$ and $C \subset Q \subset P$ respectively. Their elements are called `behaviors’ and `correlations’, respectively. However, in both cases one should consider four sets, say, $C \subset Q' \subset Q'' \subset P$ , where $Q', Q''$ correspond to the two approaches (more restrictive and less restrictive, respectively).

The question is, whether $Q' = Q''$ , or not.

If $Q' \ne Q''$ then another (even more important) question arises naturally: is $Q'$ dense in $Q''$ , or not?

If $Q'$ is essentially different from $Q''$ (that is, not dense) then we should decide, which one is more relevant to physics. I believe that $Q'$ is. Here is why. In physics we deal not just with commuting operators, but with operators belonging to local algebras corresponding to disjoint domains (on a positive distance). True, it is usually believed that local algebras are type III factors. But it is also usually believed that they are separated by type I factors, provided that the domains are disjoint. (This is called “F property” or “funnel property” in the local quantum field theory.) And type I factors mean a tensor product Hilbert space.

Still another question appears if $Q' \ne Q''$ : which properties of operator algebras are discerned by $Q$ ? One may introduce $Q_{\text{I}}$ , $Q_{\text{II}}$ , $Q_{\text{III}}$ for type I, II, III factors, then $Q_{\text{I}} = Q'$ , but what about $Q_{\text{II}}, Q_{\text{III}}$ ? Are they equal?

Partial Results

Finite dimensions

If the given (global) Hilbert space $H$ is finite-dimensional and only two subsystems are dealt with, then the two approaches are equivalent. The given operators of the first subsystem generate an

operator algebra $\mathcal A_1$ . Its center decomposes $H$ into the direct sum $H_1 \oplus \dots \oplus H_n$ of subspaces (sectors) $H_k$ . On each sector, $\mathcal A_1$ boils down to a factor. Thus, $H_k = H'_k \otimes H''_k$ and

$H = H'_1 \otimes H''_1 \oplus \dots \oplus H'_n \otimes H''_n \, .$

It remains to embed $H$ into $H' \otimes H''$ where

$H' = H'_1 \oplus \dots \oplus H'_n \, , \quad H'' = H''_1 \oplus \dots \oplus H''_n \, .$

Infinite Dimensions

This problem is connected to questions of finite approximability of C*-algebras [2]. In [3], [4], the problem is shown to be related to one major open question in operator algebra theory, the embedding problem of Connes’, asking whether any type $II_1$ factor can be embedded into an ultrapower of the hyperfinite type $II_1$ factor. More specifically, if Connes’ embedding conjecture is true, then the closure of $Q'$ must equal $Q''$ . The converse relation, namely, that $\mbox{closure}(Q')=Q''$ implies that Connes’ embedding conjecture is true, is proven in [5].

In [6], Slofstra solved part of the problem, by demonstrating that $Q'\not= Q''$ .

Solution

The question of whether $Q'$ is dense in $Q''$ was solved in the negative by Ji et al. [7]. In this pre-print, the authors consider the problem of deciding whether the supreme value of a given non-local game within the set $Q'$ is either 1 or lower than 1/2. They show that there is a one-to-one correspondence between this question and the halting problem. Since the latter is known to be undecidable, so is the problem of characterizing the set $Q'$ up to an approximation.

Now, given any nonlocal game, one can obtain, via variational methods, a converging sequence of lower bounds on its supreme value in $Q'$ . Furthermore, via the Navascués-Pironio-Acín (NPA) hierarchy of semidefinite programs [8], one can provide a converging sequence of upper bounds on its maximum value within $Q''$ . From Ji et al.’s result, it follows that those two limits cannot coincide for all nonlocal games, as otherwise one could solve the halting problem by applying these two methods in parallel. Invoking a fixed-point theorem, Ji et al. show how to construct an explicit example of such “separating” games.

References

[1] B.S. Tsirelson, Hadronic Journal Supplement 8, pp 329, (1993)

[2] V.B. Scholz, R.F. Werner, arXiv:0812.4305

[3] M. Junge, M. Navascues, C. Palazuelos, D. Perez-Garcia, V. B. Scholz, R. F. Werner, J. Math. Phys. 52, 012102 (2011).

[4] T. Fritz, Rev. Math. Phys. 24(5), 1250012 (2012).

[5] N. Ozawa, Jpn. J. Math. 8, no. 1, 147–183 (2013).

[6] W. Slofstra, arXiv:1606.03140.

[7] Z. Ji, A. Natarajan, T. Vidick, J. Wright and H. Yuen, MIP*=RE, arXiv:2001.04383.

[8] M. Navascués, S. Pironio and A. Acín, Phys. Rev. Lett. 98, 010401 (2007).