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arXiv:1304.0270

An optimal problem for relative entropy

Fan Wang, Jun Tao Zhu, Lin Zhang

Published Apr 1, 2013• 0 citations
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Abstract

Relative entropy is an essential tool in quantum information theory. There are so many problems which are related to relative entropy. In this article, the optimal values which are defined by $\displaystyle\max_{U\in{U(\cX_{d})}} S(Uρ{U^{\ast}}\parallelσ)$ and $\displaystyle\min_{U\in{U(\cX_{d})}} S(Uρ{U^{\ast}}\parallelσ)$ for two positive definite operators $ρ,σ\in{\textmd{Pd}(\cX)}$ are obtained. And the set of $S(Uρ{U^{\ast}}\parallelσ)$ for every unitary operator $U$ is full of the interval $[\displaystyle\min_{U\in{U(\cX_{d})}} S(Uρ{U^{\ast}}\parallelσ),\displaystyle\max_{U\in{U(\cX_{d})}} S(Uρ{U^{\ast}}\parallelσ)]$

DOI
10.48550/arxiv.1304.0270

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Relative entropy is an essential tool in quantum information theory. There are so many problems which are related to relative entropy. In this article, the optimal values which are defined by max⁡U∈U(\cXd)S(UρU∗∥σ)\displaystyle\max_{U\in{U(\cX_{d})}} S(Uρ{U^{\ast}}\parallelσ)U∈U(\cXd​)max​S(UρU∗∥σ) and min⁡U∈U(\cXd)S(UρU∗∥σ)\displaystyle\min_{U\in{U(\cX_{d})}} S(Uρ{U^{\ast}}\parallelσ)U∈U(\cXd​)min​S(UρU∗∥σ) for two positive definite operators ρ,σ∈Pd(\cX)ρ,σ\in{\textmd{Pd}(\cX)}ρ,σ∈Pd(\cX) are obtained. And the set of S(UρU∗∥σ)S(Uρ{U^{\ast}}\parallelσ)S(UρU∗∥σ) for every unitary operator UUU is full of the interval [min⁡U∈U(\cXd)S(UρU∗∥σ),max⁡U∈U(\cXd)S(UρU∗∥σ)][\displaystyle\min_{U\in{U(\cX_{d})}} S(Uρ{U^{\ast}}\parallelσ),\displaystyle\max_{U\in{U(\cX_{d})}} S(Uρ{U^{\ast}}\parallelσ)][U∈U(\cXd​)min​S(UρU∗∥σ),U∈U(\cXd​)max​S(UρU∗∥σ)]
Numerical range of a Hermitian operator X is defined as the set of all possible expectation values of this observable among a normalized quantum state. We analyze a modification of this definition in which the expectation value is taken among a certain subset of the set of all quantum states. One considers, for instance, the set of real states, the set of product states, separable states, or the set of maximally entangled states. We show exemplary applications of these algebraic tools in the theory of quantum information: analysis of k-positive maps and entanglement witnesses, as well as study of the minimal output entropy of a quantum channel. Product numerical range of a unitary operator is used to solve the problem of local distinguishability of a family of two unitary gates.
Recently, an interesting quantity called the quantum Renyi divergence (or "sandwiched" Renyi relative entropy) was defined for pairs of positive semi-definite operators ρ\rhoρ and σ\sigmaσ. It depends on a parameter α\alphaα and acts as a parent quantity for other relative entropies which have important operational significances in quantum information theory: the quantum relative entropy and the min- and max-relative entropies. There is, however, another relative entropy, called the 0-relative Renyi entropy, which plays a key role in the analysis of various quantum information-processing tasks in the one-shot setting. We prove that the 0-relative Renyi entropy is obtainable from the quantum Renyi divergence only if ρ\rhoρ and σ\sigmaσ have equal supports. This, along with existing results in the literature, suggests that it suffices to consider two essential parent quantities from which operationally relevant entropic quantities can be derived - the quantum Renyi divergence with parameter α≥1/2\alpha \ge 1/2α≥1/2, and the α\alphaα-relative R\'enyi entropy with α∈[0,1)\alpha\in [0,1)α∈[0,1).
The measured relative entropy and measured Rényi relative entropy are quantifiers of the distinguishability of two quantum states ρρρ and σσσ. They are defined as the maximum classical relative entropy or Rényi relative entropy realizable by performing a measurement on ρρρ and σσσ, and they have interpretations in terms of asymptotic quantum hypothesis testing. Crucially, they can be rewritten in terms of variational formulas involving the optimization of a concave or convex objective function over the set of positive definite operators. In this paper, we establish foundational properties of these objective functions by analyzing their matrix gradients and Hessian superoperators; namely, we prove that these objective functions are βββ-smooth and γγγ-strongly convex / concave, where βββ and γγγ depend on the max-relative entropies of ρρρ and σσσ. A practical consequence of these properties is that we can conduct Nesterov accelerated projected gradient descent / ascent, a well known classical optimization technique, to calculate the measured relative entropy and measured Rényi relative entropy to arbitrary precision. These algorithms are generally more memory efficient than our previous algorithms based on semi-definite optimization [Huang and Wilde, arXiv:2406.19060], and for well conditioned states ρρρ and σσσ, these algorithms are notably faster.