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arXiv:1701.06688 · IEEE International Symposium on Information Theory (ISIT)

Quantum information on spectral sets

Peter Harremöes D

Published Jun 1, 2017• 3 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1109/isit.2017.8006789

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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.
The notion of Bregman divergence and sufficiency will be defined on general convex state spaces. It is demonstrated that only spectral sets can have a Bregman divergence that satisfies a sufficiency condition. Positive elements with trace 1 in a Jordan algebra are examples of spectral sets, and the most important example is the set of density matrices with complex entries. It is conjectured that information theoretic considerations lead directly to the notion of Jordan algebra under some regularity conditions.
We show that the strongly symmetric spectral convex compact sets are precisely the normalized state spaces of finite-dimensional simple Euclidean Jordan algebras and the simplices. Spectrality is the property that every state has a convex decomposition into perfectly distinguishable pure states; strong symmetry is transitivity, for each integer N, of the affine automorphism group of the state space on lists of N perfectly distinguishable pure states. Additional assumptions combine with this theorem to give simple characterizations of finite-dimensional complex quantum state space. Important aspects of quantum and classical thermodynamics and of query complexity have been generalized to classes of general probabilistic theories (GPTs) satisfying natural postulates including or implying spectrality and strong symmetry; our result shows that these apply to a narrower class of theories than might have been hoped. Sorkin's notion of irreducibly k-th order interference has been studied in the GPT framework and looked for in experiments. Our result shows that the assumption of no higher-order (k > 2) interference, used along with spectrality and strong symmetry to characterize the same class of Jordan-algebraic convex sets by Barnum, Mueller, and Ududec in arXiv:1403.4147, was superfluous. It also implies that Lee and Selby's extension, on the assumption that interference has fixed maximal degree k, of the important order square root of N lower bound on the quantum black-box query complexity of searching N possibilities for one having a desired property (which is achieved by Grover's quantum algorithm), to a class of theories satisfying certain postulates allowing the formulation of a generalized notion of query algorithm, actually applies in the Jordan-algebraic setting where higher-order interference is not possible.
The spectral geometric mean of Koufany and Lim (2026), already known to encode the eigenvalue geometry of a pair of positive elements in a Euclidean Jordan algebra, sits at a largely unexplored intersection of quantum information theory and post-quantum cryptography. On the cone of positive-definite Hermitian matrices the spectral mean recovers the Fiedler–Pták construction whose spectrum consists of the geometric means of the eigenvalues of the product; the same construction extends verbatim to any simple Euclidean Jordan algebra, including the exceptional Albert algebra h3(O)\mathfrak{h}_3(\mathbb{O})h3​(O). This extension examines three interlocking questions: (i) the relation between the spectral geometric mean and quantum relative entropies (Umegaki, Belavkin–Staszewski, and their Jordan-algebraic analogues); (ii) the possible use of Mostow product metrics dσd_\sigmadσ​ as alternative geometries for quantum state spaces and for divergence-based learning; and (iii) the prospect that the exceptional Jordan structure of the Albert algebra, together with its Mostow decompositions, may furnish new hardness assumptions or algebraic primitives for post-quantum cryptography. Concrete open problems and a stratified ontological reading of the exceptional cone are offered. C S Also available on Academia.edu: https://www.academia.edu/attachments/133701404/download_file?s=portfolio
In standard quantum theory, the ideas of information-entropy and of pure states are closely linked. States are represented by density matrices ρρρ on a Hilbert space and the information-entropy −tr(ρlog⁡ρ)-tr(ρ\logρ)−tr(ρlogρ) is minimised on pure states (pure states are the vertices of the boundary of the convex set of states). The space of decoherence functions in the consistent histories approach to generalised quantum theory is also a convex set. However, by showing that every decoherence function can be written as a convex combination of two other decoherence functions we demonstrate that there are no `pure' decoherence functions. The main content of the paper is a new notion of information-entropy in generalised quantum mechanics which is applicable in contexts in which there is no a priori notion of time. Information-entropy is defined first on consistent sets and then we show that it decreases upon refinement of the consistent set. This information-entropy suggests an intrinsic way of giving a consistent set selection criterion.
In standard quantum theory, the ideas of information entropy and of pure states are closely linked. States are represented by density matrices \ensuremath{\rho} on a Hilbert space and the information entropy - tr(\ensuremath{\rho} ln \ensuremath{\rho}) is minimized on pure states (pure states are the vertices of the boundary of the convex set of states). The space of decoherence functions in the consistent histories approach to generalized quantum theory is also a convex set. However, by showing that every decoherence function can be written as a convex combination of two other decoherence functions, we demonstrate that there are no ``pure'' decoherence functions. The main content of the paper is a notion of information entropy in generalized quantum mechanics which is applicable in contexts in which there is no a priori notion of time. Information entropy is defined first on consistent sets and then we show that it decreases upon refinement of the consistent set. This information entropy suggests an intrinsic way of giving a consistent set selection criterion.
Quantum f-divergences are a quantum generalization of the classical notion of f-divergences, and are a special case of Petz' quasi-entropies. Many well-known distinguishability measures of quantum states are given by, or derived from, f-divergences. Special examples include the quantum relative entropy, the Rényi relative entropies, and the Chernoff and Hoeffding measures. Here we show that the quantum f-divergences are monotonic under substochastic maps whenever the defining function is operator convex. This extends and unifies all previously known monotonicity results for this class of distinguishability measures. We also analyze the case where the monotonicity inequality holds with equality, and extend Petz' reversibility theorem for a large class of f-divergences and other distinguishability measures. We apply our findings to the problem of quantum error correction, and show that if a stochastic map preserves the pairwise distinguishability on a set of states, as measured by a suitable f-divergence, then its action can be reversed on that set by another stochastic map that can be constructed from the original one in a canonical way. We also provide an integral representation for operator convex functions on the positive half-line, which is the main ingredient in extending previously known results on the monotonicity inequality and the case of equality. We also consider some special cases where the convexity of f is sufficient for the monotonicity, and obtain the inverse Hölder inequality for operators as an application. The presentation is completely self-contained and requires only standard knowledge of matrix analysis.