Trace inequality

Let H_n denote the space of Hermitian n×n matrices, H_n⁺ denote the set consisting of positive semi-definite n×n Hermitian matrices and H_n⁺⁺ denote the set of positive definite Hermitian matrices. For operators on an infinite dimensional Hilbert space we require that they be trace class and self-adjoint, in which case similar definitions apply, but we discuss only matrices, for simplicity.

For any real-valued function f on an interval I ⊂ ℝ, one may define a matrix function f(A) for any operator A ∈ H_n with eigenvalues λ in I by defining it on the eigenvalues and corresponding projectors P as

${\displaystyle f(A)\equiv \sum _{j}f(\lambda _{j})P_{j}~,}$ given the spectral decomposition ${\displaystyle A=\sum _{j}\lambda _{j}P_{j}.}$

Let f: ℝ → ℝ be continuous, and let n be any integer. Then, if ${\displaystyle t\mapsto f(t)}$ is monotone increasing, so is ${\displaystyle A\mapsto \operatorname {Tr} f(A)}$ on H_n.

Likewise, if ${\displaystyle t\mapsto f(t)}$ is convex, so is ${\displaystyle A\mapsto \operatorname {Tr} f(A)}$ on H_n, and it is strictly convex if f is strictly convex.

See proof and discussion in,[1] for example.

For ${\displaystyle -1\leq p\leq 0}$, the function ${\displaystyle f(t)=-t^{p}}$ is operator monotone and operator concave.

For ${\displaystyle 0\leq p\leq 1}$, the function ${\displaystyle f(t)=t^{p}}$ is operator monotone and operator concave.

For ${\displaystyle 1\leq p\leq 2}$, the function ${\displaystyle f(t)=t^{p}}$ is operator convex. Furthermore,

${\displaystyle f(t)=\log(t)}$ is operator concave and operator monotone, while

${\displaystyle f(t)=t\log(t)}$ is operator convex.

The original proof of this theorem is due to K. Löwner who gave a necessary and sufficient condition for f to be operator monotone.[5] An elementary proof of the theorem is discussed in [1] and a more general version of it in.[6]

For all Hermitian n×n matrices A and B and all differentiable convex functions f: ℝ → ℝ with derivative f ' , or for all positive-definite Hermitian n×n matrices A and B, and all differentiable convex functions f:(0,∞) → ℝ, the following inequality holds,

In either case, if f is strictly convex, equality holds if and only if A = B. A popular choice in applications is f(t) = t log t, see below.

In 1965, S. Golden [7] and C.J. Thompson [8] independently discovered that

For any matrices ${\displaystyle A,B\in \mathbf {H} _{n}}$,

${\displaystyle \operatorname {Tr} e^{A+B}\leq \operatorname {Tr} e^{A}e^{B}.}$

This inequality can be generalized for three operators:[9] for non-negative operators ${\displaystyle A,B,C\in \mathbf {H} _{n}^{+}}$,

${\displaystyle \operatorname {Tr} e^{\ln A-\ln B+\ln C}\leq \int _{0}^{\infty }dt\,\operatorname {Tr} A(B+t)^{-1}C(B+t)^{-1}.}$

Let ${\displaystyle R,F\in \mathbf {H} _{n}}$ be such that Tr e^R = 1. Defining g = Tr Fe^R, we have

${\displaystyle \operatorname {Tr} e^{F}e^{R}\geq \operatorname {Tr} e^{F+R}\geq e^{g}.}$

The proof of this inequality follows from the above combined with Klein's inequality. Take f(x) = exp(x), A=R + F, and B = R + gI.[10]

Let ${\displaystyle H}$ be a self-adjoint operator such that ${\displaystyle e^{-H}}$ is trace class. Then for any ${\displaystyle \gamma \geq 0}$ with ${\displaystyle \operatorname {Tr} \gamma =1,}$

${\displaystyle \operatorname {Tr} \gamma H+\operatorname {Tr} \gamma \ln \gamma \geq -\ln \operatorname {Tr} e^{-H},}$

with equality if and only if ${\displaystyle \gamma =\exp(-H)/\operatorname {Tr} \exp(-H).}$

The following theorem was proved by E. H. Lieb in.[9] It proves and generalizes a conjecture of E. P. Wigner, M. M. Yanase and F. J. Dyson.[11] Six years later other proofs were given by T. Ando [12] and B. Simon,[3] and several more have been given since then.

For all ${\displaystyle m\times n}$ matrices ${\displaystyle K}$, and all ${\displaystyle q}$ and ${\displaystyle r}$ such that ${\displaystyle 0\leq q\leq 1}$ and ${\displaystyle 0\leq r\leq 1}$, with ${\displaystyle q+r\leq 1}$ the real valued map on ${\displaystyle \mathbf {H} _{m}^{+}\times \mathbf {H} _{n}^{+}}$ given by

${\displaystyle F(A,B,K)=\operatorname {Tr} (K^{*}A^{q}KB^{r})}$

• is jointly concave in ${\displaystyle (A,B)}$
• is convex in ${\displaystyle K}$.

Here ${\displaystyle K^{*}}$ stands for the adjoint operator of ${\displaystyle K.}$

For a fixed Hermitian matrix ${\displaystyle L\in \mathbf {H} _{n}}$, the function

${\displaystyle f(A)=\operatorname {Tr} \exp\{L+\ln A\}}$

is concave on ${\displaystyle \mathbf {H} _{n}^{++}}$.

The theorem and proof are due to E. H. Lieb,[9] Thm 6, where he obtains this theorem as a corollary of Lieb's concavity Theorem. The most direct proof is due to H. Epstein;[13] see M.B. Ruskai papers,[14][15] for a review of this argument.

T. Ando's proof [12] of Lieb's concavity theorem led to the following significant complement to it:

For all ${\displaystyle m\times n}$ matrices ${\displaystyle K}$, and all ${\displaystyle 1\leq q\leq 2}$ and ${\displaystyle 0\leq r\leq 1}$ with ${\displaystyle q-r\geq 1}$, the real valued map on ${\displaystyle \mathbf {H} _{m}^{++}\times \mathbf {H} _{n}^{++}}$ given by

${\displaystyle (A,B)\mapsto \operatorname {Tr} (K^{*}A^{q}KB^{-r})}$

is convex.

For two operators ${\displaystyle A,B\in \mathbf {H} _{n}^{++}}$ define the following map

${\displaystyle R(A\parallel B):=\operatorname {Tr} (A\log A)-\operatorname {Tr} (A\log B).}$

For density matrices ${\displaystyle \rho }$ and ${\displaystyle \sigma }$, the map ${\displaystyle R(\rho \parallel \sigma )=S(\rho \parallel \sigma )}$ is the Umegaki's quantum relative entropy.

Note that the non-negativity of ${\displaystyle R(A\parallel B)}$ follows from Klein's inequality with ${\displaystyle f(t)=t\log t}$.

The operator version of Jensen's inequality is due to C. Davis.[17]

A continuous, real function ${\displaystyle f}$ on an interval ${\displaystyle I}$ satisfies Jensen's Operator Inequality if the following holds

${\displaystyle f\left(\sum _{k}A_{k}^{*}X_{k}A_{k}\right)\leq \sum _{k}A_{k}^{*}f(X_{k})A_{k},}$

for operators ${\displaystyle \{A_{k}\}_{k}}$ with ${\displaystyle \sum _{k}A_{k}^{*}A_{k}=1}$ and for self-adjoint operators ${\displaystyle \{X_{k}\}_{k}}$ with spectrum on ${\displaystyle I}$.

See,[17][18] for the proof of the following two theorems.

E. H. Lieb and W. E. Thirring proved the following inequality in [19] in 1976: For any ${\displaystyle A\geq 0}$, ${\displaystyle B\geq 0}$ and ${\displaystyle r\geq 1,}$

${\displaystyle \operatorname {Tr} ((BAB)^{r})\leq \operatorname {Tr} (B^{r}A^{r}B^{r}).}$

In 1990 [20] H. Araki generalized the above inequality to the following one: For any ${\displaystyle A\geq 0}$, ${\displaystyle B\geq 0}$ and ${\displaystyle q\geq 0,}$

${\displaystyle \operatorname {Tr} ((BAB)^{rq})\leq \operatorname {Tr} ((B^{r}A^{r}B^{r})^{q}),}$ for ${\displaystyle r\geq 1,}$

and

${\displaystyle \operatorname {Tr} ((B^{r}A^{r}B^{r})^{q})\leq \operatorname {Tr} ((BAB)^{rq}),}$ for ${\displaystyle 0\leq r\leq 1.}$

The Lieb–Thirring inequality also enjoys the following generalization:[21] for any ${\displaystyle A\geq 0}$, ${\displaystyle B\geq 0}$ and ${\displaystyle \alpha \in [0,1],}$

${\displaystyle \operatorname {Tr} (BA^{\alpha }BBA^{1-\alpha }B)\leq \operatorname {Tr} (B^{2}AB^{2}).}$

E. Effros in [22] proved the following theorem.

If ${\displaystyle f(x)}$ is an operator convex function, and ${\displaystyle L}$ and ${\displaystyle R}$ are commuting bounded linear operators, i.e. the commutator ${\displaystyle [L,R]=LR-RL=0}$, the perspective

${\displaystyle g(L,R):=f(LR^{-1})R}$

is jointly convex, i.e. if ${\displaystyle L=\lambda L_{1}+(1-\lambda )L_{2}}$ and ${\displaystyle R=\lambda R_{1}+(1-\lambda )R_{2}}$ with ${\displaystyle [L_{i},R_{i}]=0}$ (i=1,2), ${\displaystyle 0\leq \lambda \leq 1}$,

${\displaystyle g(L,R)\leq \lambda g(L_{1},R_{1})+(1-\lambda )g(L_{2},R_{2}).}$

Ebadian et al. later extended the inequality to the case where ${\displaystyle L}$ and ${\displaystyle R}$ do not commute .[23]

Von Neumann's trace inequality, named after its originator John von Neumann, states that for any n × n complex matrices A, B with singular values ${\displaystyle \alpha _{1}\geq \alpha _{2}\geq \cdots \geq \alpha _{n}}$ and ${\displaystyle \beta _{1}\geq \beta _{2}\geq \cdots \geq \beta _{n}}$ respectively,[24]

${\displaystyle \left|\operatorname {Tr} (AB)\right|\leq \sum _{i=1}^{n}\alpha _{i}\beta _{i}\,.}$

A simple corollary to this is the following result:[25] For hermitian n × n positive semidefinite complex matrices A, B where now the eigenvalues are sorted decreasingly (${\displaystyle a_{1}\geq a_{2}\geq \cdots \geq a_{n}}$ and ${\displaystyle b_{1}\geq b_{2}\geq \cdots \geq b_{n}}$, respectively),

${\displaystyle \sum _{i=1}^{n}a_{i}b_{n-i+1}\leq \operatorname {Tr} (AB)\leq \sum _{i=1}^{n}a_{i}b_{i}\,.}$

• von Neumann entropy
• Lieb–Thirring inequality
• Schur–Horn theorem

• Scholarpedia primary source.