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arXiv:1208.2046

Contextual Entropy and Reconstruction of Quantum States

Carmen Constantin, Andreas Doering

Published Aug 9, 2012• 2 citations
arXivHTMLPDF

Abstract

We introduce a new notion of entropy for quantum states, called contextual entropy, and show how it unifies Shannon and von Neumann entropy. The main result is that from the knowledge of the contextual entropy of a quantum state of a finite-dimensional system, one can 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.1208.2046

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We introduce a new notion of entropy for quantum states, called contextual entropy, and show how it unifies Shannon and von Neumann entropy. The main result is that from the knowledge of the contextual entropy of a quantum state of a finite-dimensional system, one can 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.
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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.
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Infinite-dimensional systems play an important role in the continuous-variable quantum computation model, which can compete with a more standard approach based on qubit and quantum circuit computation models. But, in many cases, the value of entropy unfortunately cannot be easily computed for states originating from an infinite-dimensional Hilbert space. Therefore, in this article, the unified quantum entropy (which extends the standard von Neumann entropy) notion is extended to the case of infinite-dimensional systems by using the Fredholm determinant theory. Some of the known (in the finite-dimensional case) basic properties of the introduced unified entropies were extended to this case study. Certain numerical examples for computing the proposed finite- and infinite-dimensional entropies are outlined as well, which allowed us to calculate the entropy values for infinite Hilbert spaces.
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Quantum-state reconstruction on a finite number of copies of a quantum system with informationally incomplete measurements, as a rule, does not yield a unique result. We derive a reconstruction scheme where both the likelihood and the von Neumann entropy functionals are maximized in order to systematically select the most-likely estimator with the largest entropy, that is, the least-bias estimator, consistent with a given set of measurement data. This is equivalent to the joint consideration of our partial knowledge and ignorance about the ensemble to reconstruct its identity. An interesting structure of such estimators will also be explored.