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.
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 being d2 is equivalent to the existence of a SIC POVM in dimension d. This opens up another approach to tackle this existence problem.
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.
This document is an extended abstract of [Fri10b]. See [JNP + 10] for closely related results obtained by different methods. Quantum correlations in composite quantum systems. In the study of quantum entanglement and quantum correlations, one usually assumes that the state space of a composite quantum system is a tensor product HA ⊗ HB, so that the correlations take on the form P (a, b | x, y) = 〈ψ, (A a x ⊗ B b y)ψ〉. (1) with POVM observables Ax a and B y b. However, how is this justified from physical principles? Can we really be sure that this tensor product assumption is appropriate? One possible alternative assumption might be to say that a composite system is defined in terms of a joint Hilbert space H together with, for each site, a set of local observables on H, such that each observable located on the first site commutes with each observable located on the second site; in physical terms, this means that the observables located at different sites are compatible, and can in particular be measured jointly. This is the “commutativity assumption”. In the case of finite-dimensional case, this is effectively equivalent to the
The situation of two independent observers conducting measurements on a joint quantum system is usually modelled using a Hilbert space of tensor product form, each factor associated to one observer. Correspondingly, the operators describing the observables are then acting non-trivially only on one of the tensor factors. However, the same situation can also be modelled by just using one joint Hilbert space, and requiring that all operators associated to different observers commute, i.e. are jointly measurable without causing disturbance. The problem of Tsirelson is now to decide the question whether all quantum correlation functions between two independent observers derived from commuting observables can also be expressed using observables defined on a Hilbert space of tensor product form. Tsirelson showed already that the distinction is irrelevant in the case that the ambient Hilbert space is of finite dimension. We show here that the problem is equivalent to the question whether all quantum correlation functions can be approximated by correlation function derived from finite-dimensional systems. We also discuss some physical examples which fulfill this requirement.
In the past few decades, the connections between non-local games arising in quantum information theory and the theory of operator algebras have undergone a phase of significant development. Operator algebras provide a particularly fruitful framework for approaching questions of non-locality in quantum systems, as the input-output behavior of measurements on bipartite quantum systems can be encoded through noncommutative operator algebras and their state spaces. This thesis is the compilation of a series of papers, written by the author in collaboration with Ivan G. Todorov, utilizing this framework by using the theory of operator algebras and completely positive maps to study questions involving non-local games. ☐ Using the simulation paradigm from information theory, we define a quantized version of homomorphisms and isomorphisms between classical hypergraphs, generalizing quantum homomorphisms and isomorphisms of graphs from the literature. We show that these quantum homomorphisms and isomorphisms constitute pre-orders and equivalence relations, respectively. We use quantum homomorphisms of hypergraphs to provide multiple examples of strict separation between correlation classes of varying types. Specializing to the case when our underlying hypergraphs arise from classical non-local games, we define quantum non-local game homomorphisms and isomorphisms. We also show how the existence of a homomorphism or isomorphism of some fixed type between games is reflected in a comparison of optimal and asymptotic game values with respect to this type. These non-local game homomorphisms are defined via a new class of non-signalling correlations which we introduce here. We develop the multivariate tensor product theory in the category of operator systems, which we then use to provide a characterization of game homomorphisms in terms of state spaces for various tensor products of canonically associated operator systems. We then apply these results to the study of synchronous non-local games, defining jointly synchronous correlations and showing how they correspond to tracial states of tensor products of C*-algebras canonically associated to each game party. ☐ We then move to a second level of quantization by proposing a "quantized" notion of hypergraphs (where "quantum" here is in the sense of a non-commutative analogue for a discrete combinatorial object), and introduce quantum homomorphisms between these quantum objects. We provide an initial foray into the properties quantum morphisms between quantum hypergraphs display, showing they satisfy analogous properties to quantum homomorphisms between classical hypergraphs. We also indicate initial connections these quantum homomorphisms between quantum hypergraphs have to the study of quantum input-output games. We end the thesis showing that homomorphisms of a local type between quantum hypergraphs is closely related, and in some cases identical, to the TRO equivalence of finite-dimensionally acting operator spaces canonically associated with each hypergraph. This suggests a quantum information theoretic approach to Morita equivalence in the category of operator spaces.
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.