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International Journal of Theoretical Physics

State Determination and Sufficiency of Observables

Katarzyna Lubnauer, Hanna Podsȩdkowska

Published May 18, 2013• 1 citation
Publisher

Abstract

Informational completeness and the possibility of state distinction and determination are among the more important issues of quantum statistics. We use spectral and semispectral (POV) measures to analyse these questions. For a given W ∗-algebra and a family of normal states on it we investigate the relation between sufficiency in Petz’s sense of a W ∗-subalgebra generated by a spectral measure, and the possibility of determination of these states by means of an observable.

DOI
10.1007/s10773-013-1635-y

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Informational completeness and the possibility of state distinction and determination are among the more important issues of quantum statistics. We use spectral and semispectral (POV) measures to analyse these questions. For a given W ∗-algebra and a family of normal states on it we investigate the relation between sufficiency in Petz’s sense of a W ∗-subalgebra generated by a spectral measure, and the possibility of determination of these states by means of an observable.
The paper is devoted to the investigation of the notion of sufficiency in quantum statistics. Three kinds of this notion are considered: plain sufficiency (called simply: sufficiency), Petz’s sufficiency, and Umegaki’s sufficiency. The problem of the existence and structure of the minimal sufficient subalgebra is analyzed in some detail, conditions yielding equivalence of the three modes of sufficiency are considered, and quantum Basu’s theorem is obtained. Moreover, it is shown that an interesting “factorization theorem” of Jenčová and Petz needs some corrections to hold true.
Abstract Some features of the notion of sufficiency in quantum statistics are investigated. Three kinds of this notion are considered: plain sufficiency (called simply: sufficiency), strong sufficiency and Umegaki’s sufficiency. It is shown that for a finite von Neumann algebra with a faithful family of normal states the minimal sufficient von Neumann subalgebra is sufficient in Umegaki’s sense. Moreover, a proper version of the factorization theorem of Jenčová and Petz is obtained. The structure of the minimal sufficient subalgebra is described in the case of pure states on the full algebra of all bounded linear operators on a Hilbert space.
For convex optimization problems Bregman divergences appear as regret functions. Such regret functions can be defined on any convex set, but if a sufficiency condition is added the regret function must be proportional to information divergence and the convex set must be spectral. Spectral sets are sets where different orthogonal decompositions of a state into pure states have unique mixing coefficients. Only on such spectral sets it is possible to define well behaved information theoretic quantities like entropy and divergence. It is only possible to perform measurements in a reversible way if the state space is spectral. The most important spectral sets can be represented as positive elements of Jordan algebras with trace 1. This means that Jordan algebras provide a natural framework for studying quantum information.
This paper attempts to give an overview about sufficiency in the setting of quantum statistics. The basic concepts are treated in parallel to the measure theoretic case. It turns out that several classical examples and results have a noncommutative analogue. Some of the results are presented without proof (but with exact references) and the presentation is intended to be self-contained. The main examples discussed in the paper are related to the Weyl algebra and to the exponential family of states. The characterization of sufficiency in terms of quantum Fisher information is a new result.
Various forms of optimality for quantum observables described as normalized positive-operator-valued measures (POVMs) are studied in this paper. We give characterizations for observables that determine the values of the measured quantity with probabilistic certainty or a state of the system before or after the measurement. We investigate observables that are free from noise caused by classical post-processing, mixing, or pre-processing of quantum nature. Especially, a complete characterization of pre-processing and post-processing clean observables is given, and necessary and sufficient conditions are imposed on informationally complete POVMs within the set of pure states. We also discuss joint and sequential measurements of optimal quantum observables.