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arXiv:1804.05116

Unconditional separation of finite and infinite-dimensional quantum correlations

Andrea Coladangelo, Jalex Stark

Published Apr 13, 2018• 39 citations
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Abstract

Determining the relationship between quantum correlation sets is a long-standing open problem. The most well-studied part of the hierarchy is captured by the chain of inclusions $\mathcal C_q \subseteq \mathcal C_{qs} \subsetneq \mathcal C_{qa} \subseteq \mathcal C_{qc}$. The separation $\mathcal C_{qs} \neq \mathcal C_{qa}$, showing that the set of quantum spatial correlations is not closed, was proven in breakthrough work by Slofstra [arXiv:1606.03140 (2016), arXiv:1703.08618 (2017)]. Resolving the question of $\mathcal C_{qa} = \mathcal C_{qc}$ would resolve the Connes Embedding Conjecture and would represent major progress in the mathematical field of operator algebras. In this work, we resolve the ambiguity in the first inclusion, showing that $\mathcal{C}_q \neq \mathcal{C}_{qs}$. We provide an explicit construction of a correlation that can be attained on a tensor product of infinite-dimensional Hilbert spaces but not finite-dimensional ones. This property is also conjectured to be possessed by any correlation which maximally violates the $I_{3322}$ inequality.

DOI
10.48550/arxiv.1804.05116

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Determining the relationship between quantum correlation sets is a long-standing open problem. The most well-studied part of the hierarchy is captured by the chain of inclusions Cq⊆Cqs⊊Cqa⊆Cqc\mathcal C_q \subseteq \mathcal C_{qs} \subsetneq \mathcal C_{qa} \subseteq \mathcal C_{qc}Cq​⊆Cqs​⊊Cqa​⊆Cqc​. The separation Cqs≠Cqa\mathcal C_{qs} \neq \mathcal C_{qa}Cqs​=Cqa​, showing that the set of quantum spatial correlations is not closed, was proven in breakthrough work by Slofstra [arXiv:1606.03140 (2016), arXiv:1703.08618 (2017)]. Resolving the question of Cqa=Cqc\mathcal C_{qa} = \mathcal C_{qc}Cqa​=Cqc​ would resolve the Connes Embedding Conjecture and would represent major progress in the mathematical field of operator algebras. In this work, we resolve the ambiguity in the first inclusion, showing that Cq≠Cqs\mathcal{C}_q \neq \mathcal{C}_{qs}Cq​=Cqs​. We provide an explicit construction of a correlation that can be attained on a tensor product of infinite-dimensional Hilbert spaces but not finite-dimensional ones. This property is also conjectured to be possessed by any correlation which maximally violates the I3322I_{3322}I3322​ inequality.
Completely determining the relationship between quantum correlation sets is a long-standing open problem, known as Tsirelson's problem. Following recent progress by Slofstra [arXiv:1606.03140 (2016), arXiv:1703.08618 (2017)] only two instances of the problem remain open. One of them is the question of whether the set of finite-dimensional quantum correlations is strictly contained in the set of infinite-dimensional ones (i.e. whether Cq≠Cqs\mathcal C_{q} \neq \mathcal C_{qs}Cq​=Cqs​). The usual formulation of the question assumes finite question and answer sets. In this work, we show that, when one allows for either infinite answer sets (and finite question sets) or infinite question sets (and finite answer sets), there exist correlations that are achievable using an infinite-dimensional quantum strategy, but not a finite-dimensional one. For the former case, our proof exploits a recent result [Nat. Comm. 8, 15485 (2017)], which shows self-testing of any pure bipartite entangled state of arbitrary local dimension ddd, using question sets of size 3 and 4 and answer sets of size ddd. For the latter case, a key step in our proof is to show a novel self-test, inspired by [Nat. Comm. 8, 15485 (2017)], of all bipartite entangled states of any local dimension d, using question sets of size O(d)O(d)O(d), and answer sets of size 4 and 3 respectively.
We relate Connes' embedding problem in operator algebras to the Brown algebra Unc(n), which is defined as the universal unital C*-algebra generated by the entries of an n x n unitary matrix. In particular, we show that the embedding problem is equivalent to determining whether or not the algebra Unc(n) has the weak expectation property for any (equivalently, all) n at least 2. From this perspective, we develop a theory of what we call unitary correlation sets. These sets are analogous to the usual sets of bipartite probabilistic correlations arising from the typical models in quantum information theory. We show that the analogue of the weak Tsirelson problem for unitary correlation sets is again equivalent to Connes' embedding problem. Moreover, we show that as long as Alice and Bob's unitaries are of size at least 2, the set of spatial unitary correlations is never a closed set. This result is analogous to a recent theorem of Slofstra, which states that the set of quantum probabilistic correlations is not closed, so long as the input and output sets are large enough. \n \nWe also discuss several applications of the theory of unitary correlation sets. First, we show that the class of extended non-local games known as quantum XOR games is a rich enough class to detect the validity of Connes' embedding problem. That is, the embedding problem has a positive answer if and only if the value of every quantum XOR game in the commuting model agrees with the value of the game in the approximate finite-dimensional model. Second, we use a C*-algebraic analogue of the quantum teleportation and super-dense coding maps from quantum information theory to obtain separations between the tensor product model (or "quantum spatial" model) and the approximate finite-dimensional model (or "quantum approximate" model), for matrix-valued generalizations of the usual Tsirelson corrleation sets. We use some of the intermediate results to also obtain separations between the matrix versions of the finite-dimensional model (or "quantum model") and the tensor product model.
In 2020, Ji et al. [arXiv:2001.04383 and Comm.~ACM 64}, 131 (2021)] provided a proof that the complexity classes MIP∗\text{MIP}^\astMIP∗ and RE\text{RE}RE are equivalent. This result implies a negative resolution of Tsirelson's problem, that is, CqaC_{qa}Cqa​ (the closure of the set of tensor product correlations) and CqcC_{qc}Cqc​ (the set of commuting correlations) can be separated by a hyperplane (that is, a Bell-like inequality). In particular, there are correlations produced by commuting measurements (a finite number of them and with a finite number of outcomes) on an infinite-dimensional quantum system which cannot be approximated by sequences of finite-dimensional tensor product correlations. Here, we point out that there are four logical possibilities of this result. Each possibility is interesting because it fundamentally challenges the nature of spacially separated systems in different ways. We list open problems for making progress for deciding which of the possibilities is correct.
Bell's theorem, a landmark result in the foundations of physics, establishes that quantum mechanics is a non-local theory. It asserts, in particular, that two spatially separated, but entangled, quantum systems can be correlated in a way that cannot be mimicked by classical systems. A direct operational consequence of Bell's theorem is the existence of statistical tests which can detect the presence of entanglement. Remarkably, certain correlations not only witness entanglement, but they give quantitative bounds on the minimum dimension of quantum systems attaining them. In this work, we show that there exists a correlation which is not attainable by quantum systems of any arbitrary finite dimension, but is attained exclusively by infinite-dimensional quantum systems (such as infinite-level systems arising from quantum harmonic oscillators). This answers the long-standing open question about the existence of a finite correlation witnessing infinite entanglement.
We construct a linear system nonlocal game which can be played perfectly using a limit of finite-dimensional quantum strategies, but which cannot be played perfectly on any finite-dimensional Hilbert space, or even with any tensor-product strategy. In particular, this shows that the set of (tensor-product) quantum correlations is not closed. The constructed nonlocal game provides another counterexample to the ‘middle’ Tsirelson problem, with a shorter proof than our previous paper (though at the loss of the universal embedding theorem). We also show that it is undecidable to determine if a linear system game can be played perfectly with a finite-dimensional strategy, or a limit of finite-dimensional quantum strategies.
Abstract For given quantum (non-commutative) spaces P{\mathbb {P}}P and O{\mathbb {O}}O, we study the quantum space of maps MP,O{\mathbb {M}}_{{\mathbb {P}},{\mathbb {O}}}MP,O​ from P{\mathbb {P}}P to O{\mathbb {O}}O. In case of finite quantum spaces, these objects turn out to be behind a large class of maps which generalize the classical qc\textrm {qc}qc-correlations known from quantum information theory to the setting of quantum input and output sets. We prove various operator algebraic properties of the C∗{\textrm C}^\astC∗-algebras C⁡(MP,O){\operatorname {C}}({\mathbb {M}}_{{\mathbb {P}},{\mathbb {O}}})C(MP,O​) such as the lifting property and residual finite dimensionality. Inside C⁡(MP,O){\operatorname {C}}({\mathbb {M}}_{{\mathbb {P}},{\mathbb {O}}})C(MP,O​) we construct a universal operator system SP,O{\mathbb {S}}_{{\mathbb {P}},{\mathbb {O}}}SP,O​ related to P{\mathbb {P}}P and O{\mathbb {O}}O, and show, among other things, that the embedding SP,O⊂C⁡(MP,O){\mathbb {S}}_{{\mathbb {P}},{\mathbb {O}}}\subset {\operatorname {C}}({\mathbb {M}}_{{\mathbb {P}},{\mathbb {O}}})SP,O​⊂C(MP,O​) is hyperrigid and has another interesting property, which we call the strong extension property. Furthermore, C⁡(MP,O){\operatorname {C}}({\mathbb {M}}_{{\mathbb {P}},{\mathbb {O}}})C(MP,O​) is the C∗{\textrm C}^\astC∗-envelope of SP,O{\mathbb {S}}_{{\mathbb {P}},{\mathbb {O}}}SP,O​ and a large class of non-signalling correlations on the quantum sets P{\mathbb {P}}P and O{\mathbb {O}}O arise from states on C⁡(MP,O)⊗maxC⁡(MP,O){\operatorname {C}}({\mathbb {M}}_{{\mathbb {P}},{\mathbb {O}}})\otimes _{\text {\tiny {\textrm {max}}}}{\operatorname {C}}({\mathbb {M}}_{{\mathbb {P}},{\mathbb {O}}})C(MP,O​)⊗max​C(MP,O​) as well as states on the commuting tensor product SP,O⊗cSP,O{\mathbb {S}}_{{\mathbb {P}},{\mathbb {O}}}\otimes _{\text {\tiny {\textrm {c}}}}{\mathbb {S}}_{{\mathbb {P}},{\mathbb {O}}}SP,O​⊗c​SP,O​. Finally, we introduce and study the notion of a synchronous correlation with quantum input and output sets and prove several characterizations of such correlations and their relation to traces on C⁡(MP,O){\operatorname {C}}({\mathbb {M}}_{{\mathbb {P}},{\mathbb {O}}})C(MP,O​).