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arXiv:2006.03139

A Topos Theoretic Notion of Entropy

Carmen-Maria Constantin, Andreas Doering

Published Jun 4, 2020• 1 citation
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Abstract

In the topos approach to quantum theory, the spectral presheaf plays the role of the state space of a quantum system. We show how a notion of entropy can be defined within the topos formalism using the equivalence between states and measures on the spectral presheaf. We show how this construction unifies Shannon and von Neumann entropy as well as classical and quantum Renyi entropies. The main result is that from the knowledge of the contextual entropy of a quantum state of a finite-dimensional system, one can (mathematically) reconstruct the quantum state, i.e., the density matrix, if the Hilbert space is of dimension $3$ or greater. We present an explicit algorithm for this state reconstruction and relate our result to Gleason's theorem.

DOI
10.48550/arxiv.2006.03139

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•Jan 1, 2013•0 citations
In the topos approach to quantum theory, the spectral presheaf plays the role of the state space of a quantum system. We show how a notion of entropy can be defined within the topos formalism using the equivalence between states and measures on the spectral presheaf. We show how this construction unifies Shannon and von Neumann entropy as well as classical and quantum Renyi entropies. The main result is that from the knowledge of the contextual entropy of a quantum state of a finite-dimensional system, one can (mathematically) reconstruct the quantum state, i.e., the density matrix, if the Hilbert space is of dimension 333 or greater. We present an explicit algorithm for this state reconstruction and relate our result to Gleason's theorem.
In this paper, we construct a sheaf-based topos quantum theory. It is well known that a topos quantum theory can be constructed on the topos of presheaves on the category of commutative von Neumann algebras of bounded operators on a Hilbert space. Also, it is already known that quantization naturally induces a Lawvere-Tierney topology on the presheaf topos. We show that a topos quantum theory akin to the presheaf-based one can be constructed on sheaves defined by the quantization-induced Lawvere-Tierney topology. That is, starting from the spectral sheaf as a state space of a given quantum system, we construct sheaf-based expressions of physical propositions and truth objects, and thereby give a method of truth-value assignment to the propositions. Furthermore, we clarify the relationship to the presheaf-based quantum theory. We give translation rules between the sheaf-based ingredients and the corresponding presheaf-based ones. The translation rules have “coarse-graining” effects on the spaces of the presheaf-based ingredients; a lot of different proposition presheaves, truth presheaves, and presheaf-based truth-values are translated to a proposition sheaf, a truth sheaf, and a sheaf-based truth-value, respectively. We examine the extent of the coarse-graining made by translation.
Given the algebra of observables of a quantum system subject to selection rules, a state can be represented by different density matrices. As a result, different von Neumann entropies can be associated with the same state. Motivated by a minimality property of the von Neumann entropy of a density matrix with respect to its possible decompositions into pure states, we give a purely algebraic definition of entropy for states of an algebra of observables, thus solving the above ambiguity. The entropy so-defined satisfies all the desirable thermodynamic properties and reduces to the von Neumann entropy in the quantum mechanical case. Moreover, it can be shown to be equal to the von Neumann entropy of the unique representative density matrix belonging to the operator algebra of a multiplicity-free Hilbert-space representation.
In this thesis we use the language of sheaf theory in order to develop a deeper understanding of some of the fundamental differences - such as entanglement, contextuality and non-locality - between quantum and classical physics. We first present, based on the work of Abramsky and Brandenburger, how sheaves, defined over certain posets of physically meaningful contexts, give a natural setting for capturing and analysing important quantum mechanical phenomena, such as quantum non-locality and contextuality. We also describe how this setting naturally leads to a three level hierarchy of quantum contextuality: weak contextuality, logical non-locality and strong contextuality. We use these insights in order to classify certain multipartite entangled states. Almost all of these turn out to be at least logically non-local, and a number of them even turn out to be strongly contextual. We further extend this result by showing that all n-qubit entangled states, with the exception of tensor products of single-qubit and bipartite maximally-entangled states, are logically non-local. Our proof is constructive: given any n-qubit state, we present an algorithm which produces n+2 local observables witnessing its logical non-locality. In the second half of the thesis we use the same basic principle of sheaves defined over physically meaningful contexts, in order to present an elegant mathematical language, known under the name of the Topos Approach, in which many quantum mechanical concepts, such as states, observables, and propositions about these, can be expressed. We then show that the language of the Topos Approach is as least as expressive, in logical terms, as traditional quantum logic. Finally, starting from a topos-theoretic perspective, we develop the construction of contextual entropy in order to give a unified treatment of classical and quantum notions of information theoretic entropy.
Baudot and Bennequin introduced a cohomological construction adapted to information theory, called \emph{information cohomology}; it characterizes Shannon entropy as the only nontrivial cohomology class in degree one, for suitable coefficients. This text develops the relation between information cohomology and topos theory. To this end, we redefine information structures, as categories of random variables related by a notion of extension or refinement; classical and quantum probability spaces appear as models (or representations) of them. These generalized information structures form a category with finite products and coproducts. Information cohomology is a derived functor in the abelian category of modules over an algebra of random variables, a subcategory of the presheaf topos associated with the information structure. The cohomology is invariant under isomorphisms of generalized structures. We prove that the relatively free bar construction gives a projective object for the computation of this invariant. We provide detailed computations of H0H^0H0 and H1H^1H1 when the coefficients are functionals of classical probabilities, including the degenerate cases. In the process, we also establish the homological nature of Tsallis entropies, as the only cocycles in degree one for certain one-parameter family of modules.
The theory of majorization is a powerful mathematical tool which naturally arises in the quantum theory as a consequence of its fundamental connection with unitarity. In this work, we look for majorization relations based on entropic inequalities. We begin by introducing the theory of majorization. We then give some notions of quantum mechanics, like the density matrix and some algebra, before presenting the concept of entropy which is closely related to the theory of majorization. We define in the process the Shannon, von Neumann and Renyi entropies, along with some of their properties and applications. We then define quantum entanglement and give some criteria for separability, which turn out to be applications of the concepts of majorization and entropy in the quantum theory. We present some new majorization relations similar to already existing entropic inequalities, and exhibit their advantages and limitations. The rest of the report is dedicated to the study of the field of Gaussian quantum information. We present the theory, before introducing other majorization relations, which turn out to be useful because they allow one to know if it is possible to transform one quantum state into another when there exists majorization relations between them.