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arXiv:1512.00226

More on a trace inequality in quantum information theory

Naresh Kumar Sharma

Published Dec 1, 2015• 1 citation
arXivHTMLPDF

Abstract

It is known that for a completely positive and trace preserving (cptp) map ${\cal N}$, $\text{Tr}$ $\exp$$\{ \log σ$ $+$ ${\cal N}^\dagger [\log {\cal N}(ρ)$ $-\log {\cal N}(σ)] \}$ $\leqslant$ $\text{Tr}$ $ρ$ when $ρ$, $σ$, ${\cal N}(ρ)$, and ${\cal N}(σ)$ are strictly positive. We state and prove a relevant version of this inequality for the hitherto unaddressed case of these matrices being nonnegative. Our treatment also provides an alternate proof for the strictly positive case.

DOI
10.48550/arxiv.1512.00226

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Consider a function F(X,Y) of pairs of positive matrices with values in the positive matrices such that whenever X and Y commute F(X,Y)=XpYq. Our first main result gives conditions on F such that Tr[X log(F(Z,Y))]≤Tr[X(p log X+q log Y)] for all X,Y,Z such that TrZ=TrX. (Note that Z is absent from the right side of the inequality.) We give several examples of functions F to which the theorem applies. Our theorem allows us to give simple proofs of the well-known logarithmic inequalities of Hiai and Petz and several new generalizations of them which involve three variables X,Y,Z instead of just X,Y alone. The investigation of these logarithmic inequalities is closely connected with three quantum relative entropy functionals: The standard Umegaki quantum relative entropy D(X∥Y)=Tr[X(log X− log Y]), and two others, the Donald relative entropy DD(X∥Y), and the Belavkin–Stasewski relative entropy DBS(X∥Y). They are known to satisfy DD(X∥Y)≤D(X∥Y)≤DBS(X∥Y). We prove that the Donald relative entropy provides the sharp upper bound, independent of Z on Tr[X log(F(Z,Y))] in a number of cases in which F(Z,Y) is homogeneous of degree 1 in Z and −1 in Y. We also investigate the Legendre transforms in X of DD(X∥Y) and DBS(X∥Y), and show how our results for these lead to new refinements of the Golden–Thompson inequality.
Completely positive (CP) matrices are ubiquitous in modern science and technology with applications in optimization, graph theory, and quantum entanglement. Recently, Johnston \emph{et al.} [Linear Algebra and its Applications, 2022] have cast CP matrices into the framework of quantum resource theories, where CP states serve as free states and CP-preserving channels act as free operations. This work addresses several questions raised in their work. Specifically, we provide the necessary and sufficient conditions of CP-preserving channels in small dimensions, which are necessary in higher dimensions, and discuss the resource quantification via the trace distance of non-negativity. By constructing an explicit counterexample, we demonstrate that the trace-distance measure of non-negativity violates strong monotonicity. We also provide an alternative proof that every CPDNN channel $Φ:\MM_n\to \MM_2$ is CPCP. Additionally, we show that any unital CPDNN map $Φ:\MM_2\to \MM_n$ is also CPCP.
We prove that for arbitrary completely continuous operators A1, A2, ⋯, An and for positive numbers p1, p2, ⋯, pn with ∑k=1npk−1=1, the inequalities,|Tr (A1A2⋯An)|≤ ∏ k=1n [Tr (Ak†Ak)12pk]pk−1|Tr exp (A1+A2+⋯+An)|≤ ∏ k=1n [Tr exp {12pk(Ak†+Ak)}]pk−1hold. Further if a1, a2, ⋯, an are the annihilation operators of an N-dimensional harmonic oscillator, m and n are any positive integers, and ρ is a nonnegative definite operator, we prove the inequality |Tr (ρai1†⋯aim†aj1⋯ajn)|≤ ∏ k=1m [Tr ρ(aik†aik)m]1/2m ∏ l=1n [Tr ρ(ajl†ajl)n]1/2n. Some consequences of these inequalities, related results, and some applications to correlation functions of the quantized electromagnetic field are discussed.
Itis well known that in quantum information quantum relative is monotonically decreasing under the completely positive and trace-preserving maps. For a new proposed sandwiched R\'enyi quantum relative, a map that is linear trace-preserving and whose Hilber-Schmidt adjoint map satisfies Schwarz inequality, monotonicity still holds. We give a new proof of monotonicity of sandwiched R\'enyi relative entropy for α∈[12,1)\alpha\in[\frac{1}{2},1)α∈[21​,1). The proof is based on complex interplotation techniques, which already has been used to prove monotonicity under trace-preserving and positive map for α∈(1,∞)\alpha\in(1,\infty)α∈(1,∞).