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Communications in Mathematics and Applications

On the Measure of Quantum Correlations

W. A. Majewski

Published May 9, 2023• 0 citations
Publisher

Abstract

In this paper, we present novel qualities of the measure of noncommutative (so quantum) correlations for general quantum systems. In other words, the fundamental difference between classical and non-commutative probability will be studied. In particular, we introduce the notion of coefficient of quantum correlations \(d(\omega, A)\). The main theorem says that there are quantum correlations if and only if \(d(\omega, A) > 0\). Our presentation is done within \(C^*\)-algebraic description of Quantum Theory.

DOI
10.26713/cma.v14i1.1829

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In this paper, we present novel qualities of the measure of noncommutative (so quantum) correlations for general quantum systems. In other words, the fundamental difference between classical and non-commutative probability will be studied. In particular, we introduce the notion of coefficient of quantum correlations (d(\omega, A)). The main theorem says that there are quantum correlations if and only if (d(\omega, A) > 0). Our presentation is done within (C^*)-algebraic description of Quantum Theory.
A central feature of quantum mechanics is the non-commutativity of operators used to describe physical observables. In this article, we present a critical analysis on the role of non-commutativity in quantum theory, focusing on its consequences in the probabilistic description. Typically, a random phenomenon is described using the measure-theoretic formulation of probability theory. Such a description can also be done using algebraic methods, which are capable to deal with non-commutative random variables (like in quantum mechanics). Here we propose a method to construct a non-commutative probability theory starting from an ordinary measure-theoretic description of probability. This will be done using the entropic uncertainty relations between random variables, in order to evaluate the presence of non-commutativity in their algebraic description.
Abstract: A suitable unified statistical formulation of quantum and classical mechanics in a ∗-algebraic setting leads us to conclude that information itself is noncommutative in quantum mechanics. Specifically we refer here to an observer’s information regarding a physical system. This is seen as the main difference from classical mechanics, where an observer’s information regarding a physical system obeys classical probability theory. Quantum mechanics is then viewed purely as a mathematical framework for the probabilistic description of noncommutative information, with the projection postulate being a noncommutative generalization of conditional probability. This view clarifies many problems surrounding the interpretation of quantum mechanics, particularly problems relating to the measuring process. 1.
Noncommutative mathematics is a significant new trend of mathematics. Initially motivated by the development of quantum physics, the idea of 'making theory noncommutative' has been extended to many areas of pure and applied mathematics. This book is divided into two parts. The first part provides an introduction to quantum probability, focusing on the notion of independence in quantum probability and on the theory of quantum stochastic processes with independent and stationary increments. The second part provides an introduction to quantum dynamical systems, discussing analogies with fundamental problems studied in classical dynamics. The desire to build an extension of the classical theory provides new, original ways to understand well-known 'commutative' results. On the other hand the richness of the quantum mathematical world presents completely novel phenomena, never encountered in the classical setting. This book will be useful to students and researchers in noncommutative probability, mathematical physics and operator algebras.
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 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​).