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arXiv:1507.04609

A class of permutation-invariant measurements and their relation to quantum relative entropies

Janis Nötzel

Published Jul 16, 2015• 0 citations
arXivHTMLPDF

Abstract

We characterize the asymptotic performance of a class of positive operator valued measurements (POVMs) where the only task is to make measurements on independent and identically distributed quantum states on finite-dimensional systems. The POVMs we utilize here can be efficiently described in terms of a reasonably small set of parameters. Their analysis furthers the development of a quantum method of types. They deliver provably optimal performance in asymmetric hypothesis testing and in the transmission of classical messages over quantum channels. We now relate them to the recently developed $α-z$ divergences $D_{α,z}$ by giving an operational interpretation for the limiting case $\lim_{α\to1}D_{α,1-α}$ in terms of probabilities for certain measurement outcomes. This explains one of the more surprising findings of [1] in terms of the theory of group representations. In addition, we provide a Cauchy-Binet type formula for unitary matrices which connects the underlying representation theoretic objects to partial sums of the entries of unitary matrices. At last, we concentrate on the special case of qubits. We are able to give a complete description of the asymptotic detection probabilities for all POVM elements described here. We take the opportunity to define a family of functions on pairs of semi-definite matrices which obeys the quantum generalizations of Rényi's axioms except from the generalized mean value axiom. This family is described by limiting values of $α-z$ divergences for the extremal values of the parameter.

DOI
10.48550/arxiv.1507.04609

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Abstract A quantum measurement, often referred to as positive operator-valued measurement (POVM), is a set of positive operators summing to identity. This can be seen as a generalization of a probability distribution of positive real numbers summing to unity, whose evolution is given by a stochastic matrix. Discrete transformations in the set of quantum measurements can be described by blockwise stochastic matrices, composed of positive blocks that sum columnwise to identity, and the notion of sequential product of matrices. We show that such transformations correspond to a sequence of quantum measurements. Imposing additionally the dual condition that the sum of blocks in each row is equal to identity we arrive at blockwise bistochastic matrices (also called quantum magic squares) and study their properties. Analyzing the dynamics they induce we formulate a quantum analog of the Ostrowski description of the classical Birkhoff polytope and introduce the notion of majorization between quantum measurements. Our framework provides a dynamical characterization of the set of blockwise bistochastic matrices and establishes a resource theory in this set.
We propose a simple method for constructing positive operator-valued measures (POVMs) using any set of matrices which form an orthonormal basis for the space of complex matrices. Considering the orthonormal set of irreducible spherical tensors, we examine the properties of the construction on the [Formula: see text]-dimensional subspace of the [Formula: see text]-dimensional Hilbert space of [Formula: see text] qubits comprising the permutationally symmetric states. Using the notion of vectorization, the constructed POVMs are interpretable as projection operators in a higher-dimensional space. We then describe a method to physically realize the constructed POVMs for symmetric states using the Clebsch–Gordan decomposition of the tensor product of irreducible representations of the rotation group. We illustrate the proposed construction on a spin-1 system, and show that it is possible to generate entangled states from separable ones.