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Quantum correlation matrices and Tsirelson’s problem: Previous work and three-player considerations

Gerrit Vos

Published Jan 1, 2019• 0 citations
OpenAlex

Abstract

Tsirelson once claimed that the set of quantum correlations, defined by strategies of non-local two-player games, does not depend on which of two possible models is chosen: the tensor product model or the commuting operator model. He later came back from this claim, and the resulting conjecture is now known as Tsirelson’s problem. The problem has since been proven equivalent to notoriously hard problems in operator theory, such as the Connes’ Embedding Problem and the QWEP conjecture. In this master thesis, we look at the finite dimensional case of Tsirelson’s problem, working out all the details of an existing proof and giving a new, shorter proof which also extends to the nuclear case. Moreover, we give an overview of the equivalence of Tsirelson’s problem and two of Kirchberg’s conjectures, including the QWEP conjecture. Finally, we give some results and considerations for the three-player case of Tsirelson’s problem. The appendix contains proofs of many related results used throughout the thesis, and also a beginner’s introduction to quantum mechanics.

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Tsirelson’s problem and an embedding theorem for groups arising from non-local games

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Tsirelson’s problem asks whether the commuting operator model for two-party quantum correlations is equivalent to the tensor-product model. We give a negative answer to this question by showing that there are non-local games which have perfect commuting-operator strategies, but do not have perfect tensor-product strategies. The weak Tsirelson problem, which is known to be equivalent to the Connes embedding problem, remains open. The examples we construct are instances of (binary) linear system games. For such games, previous results state that the existence of perfect strategies is controlled by the solution group of the linear system. Our main result is that every finitely-presented group embeds in some solution group. As an additional consequence, we show that the problem of determining whether a linear system game has a perfect commuting-operator strategy is undecidable.

arXiv:1606.03140•Journal of the American Mathematical Society•Jun 28, 2019•122 citations

Connes implies Tsirelson: a simple proof

Alexander Frei

More precisely, we give a simple and very short proof of "the Connes embedding problem implies the synchronous Tsirelson conjecture" that relies on only two elementary ingredients: 1) the well-known description of synchronous correlations as traces on the algebra per player $C^*(\mathrm{player})=C^*(\mathrm{inputs}|\mathrm{outputs})$ and 2) an elementary lifting result by Kim, Paulsen and Schafhauser. Moreover, this bypasses every of the deep results by Kirchberg as well as any other implicit reformulation as the microstates conjecture and thelike. Meanwhile, we also give a different construction of Connes' algebra $\mathcal{R}^ω$ appearing in the Connes embedding problem, which is more suitable for the purposes of quantum information theory and much easier to comprehend for the reader without any prior knowledge in operator algebras. Most importantly, however, we present this proof for the following reason: Since the recent refutation of the synchronous Tsirelson conjecture by MIP*=RE, there exists a nonlocal game which violates the synchronous Tsirelson conjecture, and by the proof of MIP*=RE even a synchronous such game. The approach however is based on contradiction with the undecidability of the Halting problem, and so remains implicit. As such the quest now has started to give an explicit example of a synchronous game violating the synchronous Tsirelson conjecture together with a direct argument for the failure, and the current article serves as a direct translation to the corresponding operator algebra and its tracial state violating the Connes embedding problem.

arXiv:2209.07940•Sep 16, 2022•0 citations

Unitary Correlation Sets and their Applications

S.J. Harris

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.

•Jul 4, 2019•3 citations

Tsirelson's problems and entanglement breaking rank

Jitendra Prakash

There are two main themes of this thesis. In the first part, we study the various sets of correlations arising in the study of non-local games and Tsirelson's problems. In the second part we introduce the notion of entanglement breaking rank and study its connection with the SIC POVM existence problem. \n \nTsirelson's problems have been open for quite some time and continue to be an active area of research. These problems seek to identify relationships between different sets of quantum correlations arising from different quantum models and their relaxations. One of the goals of this thesis is to show that the set of correlations arising from the tensor product model is not topologically closed, and hence is distinct from the set of correlations arising from the commuting tensor model. \n \nWe begin our investigation in Chapter 3 by working through an example of a non-local game with three inputs and binary outputs. While this toy example does not help in showing the distinction between the aforementioned sets of correlations, it still provides valuable insights and motivation for a more abstract approach which we take up in the next chapter and which turns out to be fruitful. \n \nIn Chapter 4, we observe that given a graph we can associate functions (of a real variable) corresponding to different correlation sets. We study the properties of these functions. We show that for the complete graph on five vertices, the assumption that the correlation set obtained through tensor model is closed forces its corresponding function to be piecewise linear on an interval, while the function corresponding to the commuting model is not. This shows that the quantum correlation set is not closed hence solving one of Tsirelson's problem and also improves the results of Slofstra by reducing the number of experiments and outcomes to five and two, respectively. \n \nFinally, for the second part of the thesis, we introduce the notion of entanglement breaking rank of an entanglement breaking channel. While technically this notion is not new, this point of view helps us in showing that the entanglement breaking rank of a particular channel on $\\mathbb{M}_d$ being $d^2$ is equivalent to the existence of a SIC POVM in dimension $d$. This opens up another approach to tackle this existence problem.

•Aug 13, 2018•0 citations

Connes’ embedding problem and winning strategies for quantum XOR games

Samuel J. Harris

We consider quantum XOR games, defined in the work of Regev and Vidick [ACM Trans. Comput. Theory 7, 43 (2015)], from the perspective of unitary correlations defined in the work of Harris and Paulsen [Integr. Equations Oper. Theory 89, 125 (2017)]. We show that the winning bias of a quantum XOR game in the tensor product model (respectively, the commuting model) is equal to the norm of its associated linear functional on the unitary correlation set from the appropriate model. We show that Connes’ embedding problem has a positive answer if and only if every quantum XOR game has entanglement bias equal to the commuting bias. In particular, the embedding problem is equivalent to determining whether every quantum XOR game G with a winning strategy in the commuting model also has a winning strategy in the approximate finite-dimensional model.

arXiv:1706.02349•Journal of Mathematical Physics•Dec 1, 2017•5 citations

Property (T) and residually finite groups

Gilles Pisier

This is a brief excursion into a topic that belongs to quantum information theory. We prove here that the Connes-Kirchberg problem is equivalent to a question raised by Tsirelson in connection with quantum mechanics. The question involves correlation matrices of various kinds that we discuss using operator valued measures in order to highlight the equivalence of the Tsirelson problem with the LLP of certain free product C*-algebras.

Cambridge University Press eBooks•Feb 10, 2020•0 citations
Tsirelson once claimed that the set of quantum correlations, defined by strategies of non-local two-player games, does not depend on which of two possible models is chosen: the tensor product model or the commuting operator model. He later came back from this claim, and the resulting conjecture is now known as Tsirelson’s problem. The problem has since been proven equivalent to notoriously hard problems in operator theory, such as the Connes’ Embedding Problem and the QWEP conjecture. In this master thesis, we look at the finite dimensional case of Tsirelson’s problem, working out all the details of an existing proof and giving a new, shorter proof which also extends to the nuclear case. Moreover, we give an overview of the equivalence of Tsirelson’s problem and two of Kirchberg’s conjectures, including the QWEP conjecture. Finally, we give some results and considerations for the three-player case of Tsirelson’s problem. The appendix contains proofs of many related results used throughout the thesis, and also a beginner’s introduction to quantum mechanics.
Tsirelson’s problem asks whether the commuting operator model for two-party quantum correlations is equivalent to the tensor-product model. We give a negative answer to this question by showing that there are non-local games which have perfect commuting-operator strategies, but do not have perfect tensor-product strategies. The weak Tsirelson problem, which is known to be equivalent to the Connes embedding problem, remains open. The examples we construct are instances of (binary) linear system games. For such games, previous results state that the existence of perfect strategies is controlled by the solution group of the linear system. Our main result is that every finitely-presented group embeds in some solution group. As an additional consequence, we show that the problem of determining whether a linear system game has a perfect commuting-operator strategy is undecidable.
More precisely, we give a simple and very short proof of "the Connes embedding problem implies the synchronous Tsirelson conjecture" that relies on only two elementary ingredients: 1) the well-known description of synchronous correlations as traces on the algebra per player C∗(player)=C∗(inputs∣outputs)C^*(\mathrm{player})=C^*(\mathrm{inputs}|\mathrm{outputs})C∗(player)=C∗(inputs∣outputs) and 2) an elementary lifting result by Kim, Paulsen and Schafhauser. Moreover, this bypasses every of the deep results by Kirchberg as well as any other implicit reformulation as the microstates conjecture and thelike. Meanwhile, we also give a different construction of Connes' algebra Rω\mathcal{R}^ωRω appearing in the Connes embedding problem, which is more suitable for the purposes of quantum information theory and much easier to comprehend for the reader without any prior knowledge in operator algebras. Most importantly, however, we present this proof for the following reason: Since the recent refutation of the synchronous Tsirelson conjecture by MIP*=RE, there exists a nonlocal game which violates the synchronous Tsirelson conjecture, and by the proof of MIP*=RE even a synchronous such game. The approach however is based on contradiction with the undecidability of the Halting problem, and so remains implicit. As such the quest now has started to give an explicit example of a synchronous game violating the synchronous Tsirelson conjecture together with a direct argument for the failure, and the current article serves as a direct translation to the corresponding operator algebra and its tracial state violating the Connes embedding problem.
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.
There are two main themes of this thesis. In the first part, we study the various sets of correlations arising in the study of non-local games and Tsirelson's problems. In the second part we introduce the notion of entanglement breaking rank and study its connection with the SIC POVM existence problem. \n \nTsirelson's problems have been open for quite some time and continue to be an active area of research. These problems seek to identify relationships between different sets of quantum correlations arising from different quantum models and their relaxations. One of the goals of this thesis is to show that the set of correlations arising from the tensor product model is not topologically closed, and hence is distinct from the set of correlations arising from the commuting tensor model. \n \nWe begin our investigation in Chapter 3 by working through an example of a non-local game with three inputs and binary outputs. While this toy example does not help in showing the distinction between the aforementioned sets of correlations, it still provides valuable insights and motivation for a more abstract approach which we take up in the next chapter and which turns out to be fruitful. \n \nIn Chapter 4, we observe that given a graph we can associate functions (of a real variable) corresponding to different correlation sets. We study the properties of these functions. We show that for the complete graph on five vertices, the assumption that the correlation set obtained through tensor model is closed forces its corresponding function to be piecewise linear on an interval, while the function corresponding to the commuting model is not. This shows that the quantum correlation set is not closed hence solving one of Tsirelson's problem and also improves the results of Slofstra by reducing the number of experiments and outcomes to five and two, respectively. \n \nFinally, for the second part of the thesis, we introduce the notion of entanglement breaking rank of an entanglement breaking channel. While technically this notion is not new, this point of view helps us in showing that the entanglement breaking rank of a particular channel on mathbbMd\\mathbb{M}_dmathbbMd​ being d2d^2d2 is equivalent to the existence of a SIC POVM in dimension ddd. This opens up another approach to tackle this existence problem.
We consider quantum XOR games, defined in the work of Regev and Vidick [ACM Trans. Comput. Theory 7, 43 (2015)], from the perspective of unitary correlations defined in the work of Harris and Paulsen [Integr. Equations Oper. Theory 89, 125 (2017)]. We show that the winning bias of a quantum XOR game in the tensor product model (respectively, the commuting model) is equal to the norm of its associated linear functional on the unitary correlation set from the appropriate model. We show that Connes’ embedding problem has a positive answer if and only if every quantum XOR game has entanglement bias equal to the commuting bias. In particular, the embedding problem is equivalent to determining whether every quantum XOR game G with a winning strategy in the commuting model also has a winning strategy in the approximate finite-dimensional model.