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Quantum Correlations on Quantum Spaces

Piotr Mikołaj Sołtan, Paweł Kasprzak, Arkadiusz Bochniak

Published Jan 1, 2022• 6 citations
Publisher

Abstract

Abstract For given quantum (non-commutative) spaces ${\mathbb {P}}$ and ${\mathbb {O}}$, we study the quantum space of maps ${\mathbb {M}}_{{\mathbb {P}},{\mathbb {O}}}$ from ${\mathbb {P}}$ to ${\mathbb {O}}$. In case of finite quantum spaces, these objects turn out to be behind a large class of maps which generalize the classical $\textrm {qc}$-correlations known from quantum information theory to the setting of quantum input and output sets. We prove various operator algebraic properties of the ${\textrm C}^\ast$-algebras ${\operatorname {C}}({\mathbb {M}}_{{\mathbb {P}},{\mathbb {O}}})$ such as the lifting property and residual finite dimensionality. Inside ${\operatorname {C}}({\mathbb {M}}_{{\mathbb {P}},{\mathbb {O}}})$ we construct a universal operator system ${\mathbb {S}}_{{\mathbb {P}},{\mathbb {O}}}$ related to ${\mathbb {P}}$ and ${\mathbb {O}}$, and show, among other things, that the embedding ${\mathbb {S}}_{{\mathbb {P}},{\mathbb {O}}}\subset {\operatorname {C}}({\mathbb {M}}_{{\mathbb {P}},{\mathbb {O}}})$ is hyperrigid and has another interesting property, which we call the strong extension property. Furthermore, ${\operatorname {C}}({\mathbb {M}}_{{\mathbb {P}},{\mathbb {O}}})$ is the ${\textrm C}^\ast$-envelope of ${\mathbb {S}}_{{\mathbb {P}},{\mathbb {O}}}$ and a large class of non-signalling correlations on the quantum sets ${\mathbb {P}}$ and ${\mathbb {O}}$ arise from states on ${\operatorname {C}}({\mathbb {M}}_{{\mathbb {P}},{\mathbb {O}}})\otimes _{\text {\tiny {\textrm {max}}}}{\operatorname {C}}({\mathbb {M}}_{{\mathbb {P}},{\mathbb {O}}})$ as well as states on the commuting tensor product ${\mathbb {S}}_{{\mathbb {P}},{\mathbb {O}}}\otimes _{\text {\tiny {\textrm {c}}}}{\mathbb {S}}_{{\mathbb {P}},{\mathbb {O}}}$. 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 ${\operatorname {C}}({\mathbb {M}}_{{\mathbb {P}},{\mathbb {O}}})$.

DOI
10.1093/imrn/rnac139

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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​).
Quantum correlations, crucial for the advantage and advancement of quantum science and technology, arise from the impossibility of expressing a quantum state as a tensor product over a given set of parties. In this work, a generalized notion of correlations via arbitrary products is formulated. Remarkably, as a universal property, the connection between such general products and tensor products is established, allowing one to relate generic non-product states to the common notion of entangled states. We construct the set of free operations for general types of products by extending the local-operation-and-classical-communication paradigm, familiar from standard entanglement theory, thereby establishing a resource theory of correlations for general products. A generalization is provided beyond two factors that can be universally related to multipartite entanglement. Applications that highlight the usefulness of the approach are discussed, such as the factorization of fermionic states, the non-local factorization of multi-photon states into single-photon states, and the interesting possibility of understanding prime numbers as a form of single-party entanglement.
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.
Abstract We show for any d,m≥2d,m\ge 2d,m≥2 with (d,m)≠(2,2)(d,m)\neq (2,2)(d,m)=(2,2), the matrix-valued generalization of the (tensor product) quantum correlation set of ddd inputs and mmm outputs is not closed. Our argument uses a reformulation of super-dense coding and teleportation in terms of C∗C^*C∗-algebra isomorphisms. Namely, we prove that for certain actions of cyclic group Zd{{\mathbb{Z}}}_dZd​, $$\begin{equation*}M_d(C^*({{\mathbb{F}}}_{d^2}))\cong{{\mathcal{B}}}_d\rtimes{{\mathbb{Z}}}_d\rtimes{{\mathbb{Z}}}_d , M_d({{\mathcal{B}}}_d)\cong C^*({{\mathbb{F}}}_{d^2})\rtimes{{\mathbb{Z}}}_d\rtimes{{\mathbb{Z}}}_d,\end{equation*}$$where Bd{{\mathcal{B}}}_dBd​ is the universal unital C∗C^*C∗-algebra generated by the elements ujk, 0≤i,j≤d−1u_{jk}, \, 0 \le i, j \le d-1ujk​,0≤i,j≤d−1, satisfying the relations that [uj,k][u_{j,k}][uj,k​] is a unitary operator, and C∗(Fd2)C^*({{\mathbb{F}}}_{d^2})C∗(Fd2​) is the universal C∗C^*C∗-algebra of d2d^2d2 unitaries. These isomorphisms provide a nice connection between the embezzlement of entanglement and the non-closedness of quantum correlation sets.
We study no-signalling correlations over Cantor spaces, placing the product of infinitely many copies of a finite non-local game in a unified general setup. We define the subclasses of local, quantum spatial, approximately quantum and quantum commuting Cantor correlations and describe them in terms of states on tensor products of inductive limits of operator systems. We provide a correspondence between no-signalling (resp. approximately quantum, quantum commuting) Cantor correlations and sequences of correlations of the same type over the projections onto increasing number of finitely many coordinates. We introduce Cantor games, and associate canonically such a game to a sequence of finite input/output games, showing that the numerical sequence of the values of the games in the sequence converges to the corresponding value of the compound Cantor game.
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.