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Journal of Mathematical Physics

Some Inequalities Involving Traces of Operators

C. L. Mehta

Published May 1, 1968• 11 citations
Publisher

Abstract

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.

DOI
10.1063/1.1664630

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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.
It is known that for a completely positive and trace preserving (cptp) map N{\cal N}N, Tr\text{Tr}Tr exp⁡\expexp{log⁡σ\{ \log σ{logσ +++ N†[log⁡N(ρ){\cal N}^\dagger [\log {\cal N}(ρ)N†[logN(ρ) −log⁡N(σ)]}-\log {\cal N}(σ)] \}−logN(σ)]} ⩽\leqslant⩽ Tr\text{Tr}Tr ρρρ when ρρρ, σσσ, N(ρ){\cal N}(ρ)N(ρ), and N(σ){\cal N}(σ)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.
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.
The trace of a positive bounded operator in a Hilbert space is defined and shown to be independent of the basis used in the definition. The trace of the product WA is defined for bounded A and positive bounded W with unit trace and is also shown to be basis-invariant. An integral representation is derived and used to define Tr(WA) for unbounded A. Linearity, isotony, and continuity with respect to uniform covergence are proved for Tr(WA) as a function of (bounded) A. Several formulas used in statistical mechanics are derived, and it is proved that if A and B are commuting positive operators, then Tr(W(A + B)) = Tr(WA) + Tr(WB). A counter example is given which shows that the commonly used definition of Tr(WA) in terms of an orthonormal basis is not invariant under permutation of the basis vectors even in the case of a very simple unbounded operator A. An expectation-value function M which associates the expectation value M(A) to the operator A is assumed given and subject to certain restrictions. For bounded operators, these include positivity, normalization, additivity, and a condition which may be considered as a requirement of regularity for the probability function induced by M. Modifications of these requirements are imposed for unbounded operators, and von Neumann's statistical formula is proved: there is a unique bounded positive operator W with unit trace for which M(A) = Tr(WA). The requirements placed on M are weaker than those of von Neumann, and are in fact satisfied by Tr(WA) as a function of A.
The Golden–Thompson trace inequality, which states that Tr eH+K ≤ Tr eHeK, has proved to be very useful in quantum statistical mechanics. Golden used it to show that the classical free energy is less than the quantum one. Here, we make this G–T inequality more explicit by proving that for some operators, notably the operators of interest in quantum mechanics, H = Δ or H=−−Δ+m and K = potential, Tr eH+(1−u)KeuK is a monotone increasing function of the parameter u for 0 ≤ u ≤ 1. Our proof utilizes an inequality of Ando, Hiai, and Okubo (AHO): Tr XsYtX1−sY1−t ≤ Tr XY for positive operators X, Y and for 12≤s,t≤1, and s+t≤32. The obvious conjecture that this inequality should hold up to s + t ≤ 1 was proved false by Plevnik [Indian J. Pure Appl. Math. 47, 491–500 (2016)]. We give a different proof of AHO and also give more counterexamples in the 32,1 range. More importantly, we show that the inequality conjectured in AHO does indeed hold in the full range if X, Y have a certain positivity property—one that does hold for quantum mechanical operators, thus enabling us to prove our G–T monotonicity theorem.