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arXiv:2105.07820 · International Mathematics Research Notices

Quantum Correlations on Quantum Spaces

Arkadiusz Bochniak, Paweł Kasprzak, Piotr Mikołaj Sołtan

Published May 9, 2022• 1 citation
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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.
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​).
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 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.
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.
The project concerned the study of quantum correlations (QC) in compound systems, i.e. statistical correlations more general than entanglement which are predicted by quantum mechanics but not described in any classical scenario. I aimed to understand the technical and operational properties of the measures of QC, their interplay with entanglement quantifiers and the experimental accessibility.\nIn the first part of my research path, after having acquired the conceptual and technical rudiments of the project, I provided solutions for some computational issues: I developed analytical and numerical algorithms for calculating bipartite QC in finite dimensional systems. Then, I tackled the problem of the experimental detection of QC. There is no Hermitian operator associated with entanglement measures, nor with QC ones. However, the information encoded in a density matrix is redundant to quantify them, thus the full knowledge of the state is not required to accomplish the task. I reported the first protocol to measure the QC of an unknown state by means of a limited number of measurements, without performing the tomography of the state. My proposal has been implemented experimentally in a NMR (Nuclear Magnetic Resonance) setting.\nIn the final stage of the project, I explored the foundational and operational merits of QC. I showed that the QC shared by two subsystems yield a genuinely quantum kind of uncertainty on single local observables. The result is a promising evidence of the potential exploitability of separable (unentangled) states for quantum metrology in noisy conditions.