# Trace inequalities

In mathematics, there are many kinds of inequalities connected with matrices and linear operators on Hilbert spaces. This article reviews some of the most important operator inequalities connected with traces of matrices.

Useful references here are,.[1][2][3][4]

## Basic definitions

Let ${\displaystyle \mathbf {H} _{n}}$ denote the space of Hermitian ${\displaystyle n\times n}$ matrices and ${\displaystyle \mathbf {H} _{n}^{+}}$ denote the set consisting of positive semi-definite ${\displaystyle n\times n}$ 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.

### Operator monotone

A function ${\displaystyle f:I\rightarrow \mathbb {R} }$ defined on an interval ${\displaystyle I\subset \mathbb {R} }$ is said to be operator monotone if for all ${\displaystyle n}$, and all ${\displaystyle A,B\in \mathbf {H} _{n}}$ with eigenvalues in ${\displaystyle I}$, the following holds:

${\displaystyle A\geq B\Rightarrow f(A)\geq f(B),}$

where the inequality ${\displaystyle A\geq B}$ means that the operator ${\displaystyle A-B\geq 0}$ is positive semi-definite.

### Operator convex

A function ${\displaystyle f:I\rightarrow \mathbb {R} }$ is said to be operator convex if for all ${\displaystyle n}$ and all ${\displaystyle A,B\in \mathbf {H} _{n}}$ with eigenvalues in ${\displaystyle I}$, and ${\displaystyle 0<\lambda <1}$, the following holds

${\displaystyle f(\lambda A+(1-\lambda )B)\leq \lambda f(A)+(1-\lambda )f(B).}$

A function ${\displaystyle f}$ is operator concave if ${\displaystyle -f}$ is operator convex, i.e. the inequality above for ${\displaystyle f}$ is reversed.

### Joint convexity

A function ${\displaystyle g:I\times J\rightarrow \mathbb {R} }$, defined on intervals ${\displaystyle I,J\subset \mathbb {R} }$ is said to be jointly convex if for all ${\displaystyle n}$ and all ${\displaystyle A_{1},A_{2}\in \mathbf {H} _{n}}$ with eigenvalues in ${\displaystyle I}$ and all ${\displaystyle B_{1},B_{2}\in \mathbf {H} _{n}}$ with eigenvalues in ${\displaystyle J}$, and any ${\displaystyle 0\leq \lambda \leq 1}$ the following holds

${\displaystyle g(\lambda A_{1}+(1-\lambda )A_{2},\lambda B_{1}+(1-\lambda )B_{2})\leq \lambda g(A_{1},B_{1})+(1-\lambda )g(A_{2},B_{2}).}$

A function ${\displaystyle g}$ is jointly concave if ${\displaystyle -g}$ is jointly convex, i.e. the inequality above for ${\displaystyle g}$ is reversed.

### Trace function

Given a function ${\displaystyle f:\mathbb {R} \rightarrow \mathbb {R} }$, the associated trace function on ${\displaystyle \mathbf {H} _{n}}$ is given by

${\displaystyle A\mapsto {\rm {Tr}}f(A)=\sum _{j}f(\lambda _{j}),}$

where ${\displaystyle A}$ has eigenvalues ${\displaystyle \lambda }$ and ${\displaystyle {\rm {Tr}}}$ stands for a trace of the operator.

## Convexity and monotonicity of the trace function

Let ${\displaystyle f:\mathbb {R} \rightarrow \mathbb {R} }$ be continuous, and let ${\displaystyle n}$ be any integer.

it is strictly convex if ${\displaystyle f}$ is strictly convex.

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

## Löwner–Heinz theorem

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 ${\displaystyle f(t)=t^{p}}$ and 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,[5] where he gave a necessary and sufficient condition for ${\displaystyle f}$ to be operator monotone. An elementary proof of the theorem is discussed in [1] and a more general version of it in [6]

## Klein's inequality

For all Hermitian ${\displaystyle n\times n}$ matrices ${\displaystyle A}$ and ${\displaystyle B}$ and all differentiable convex functions ${\displaystyle f:\mathbb {R} \rightarrow \mathbb {R} }$ with derivative ${\displaystyle f'}$, or for all posotive-definite Hermitian ${\displaystyle n\times n}$ matrices ${\displaystyle A}$ and ${\displaystyle B}$, and all differentiable convex functions ${\displaystyle f:(0,\infty )\rightarrow \mathbb {R} }$ the following inequality holds

${\displaystyle {\rm {Tr}}[f(A)-f(B)-(A-B)f'(B)]\geq 0.}$

In either case, if ${\displaystyle f}$ is strictly convex, there is equality if and only if ${\displaystyle A=B}$.

### Proof

Let ${\displaystyle C=A-B}$ so that for ${\displaystyle 0, ${\displaystyle B+tC=(1-t)B+tA}$. Define ${\displaystyle \phi (t)={\rm {Tr}}[f(B+tC)]}$. By convexity and monotonicity of trace functions, ${\displaystyle \phi }$ is convex, and so for all ${\displaystyle 0,

${\displaystyle \phi (1)=\phi (0)\geq {\frac {\phi (t)-\phi (0)}{t}},}$

and in fact the right hand side is monotone decreasing in ${\displaystyle t}$. Taking the limit ${\displaystyle t\rightarrow 0}$ yields Klein's inequality.

Note that if ${\displaystyle f}$ is strictly convex and ${\displaystyle C\neq 0}$, then ${\displaystyle \phi }$ is strictly convex. The final assertion follows from this and the fact that ${\displaystyle {\frac {\phi (t)-\phi (0)}{t}}}$ is monotone decreasing in ${\displaystyle t}$.

## Golden–Thompson inequality

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

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

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

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

## Peierls–Bogoliubov inequality

${\displaystyle {\rm {Tr}}\,e^{F}e^{R}\geq {\rm {Tr}}\,e^{F+R}\geq e^{f}.}$

The proof of this inequality follows from Klein's inequality. Take ${\displaystyle f(x)=e^{x}}$, ${\displaystyle A=R+F}$ and ${\displaystyle B=R+fI}$.[10]

## Gibbs variational principle

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

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

## Lieb's concavity theorem

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.

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

## Lieb's theorem

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

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

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.

## Ando's convexity theorem

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

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

is convex.

## Joint convexity of relative entropy

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

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

Note that the non-negativity of ${\displaystyle R(A\|B)}$ follows from Klein's inequality with ${\displaystyle f(x)=x\log x}$.

### Proof

For all ${\displaystyle 0, ${\displaystyle (A,B)\mapsto Tr(B^{1-p}A^{p})}$ is jointly concave, by Lieb's concavity theorem, and thus

${\displaystyle (A,B)\mapsto {\frac {1}{p-1}}({\rm {Tr}}(B^{1-p}A^{p})-{\rm {Tr}}\,A)}$

is convex. But

${\displaystyle \lim _{p\rightarrow 1}{\frac {1}{p-1}}({\rm {Tr}}(B^{1-p}A^{p})-{\rm {Tr}}\,A)=R(A\|B),}$

and convexity is preserved in the limit.

The proof is due to G. Lindblad.[16]

## Jensen's operator and trace inequalities

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

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

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

### Jensen's trace inequality

Let ${\displaystyle f}$ be a continuous function defined on an interval ${\displaystyle I}$ and let ${\displaystyle m}$ and ${\displaystyle n}$ be natural numbers. If ${\displaystyle f}$ is convex we then have the inequality

${\displaystyle {\rm {Tr}}{\Bigl (}f{\Bigl (}\sum _{k=1}^{n}A_{k}^{*}X_{k}A_{k}{\Bigr )}{\Bigr )}\leq {\rm {Tr}}{\Bigl (}\sum _{k=1}^{n}A_{k}^{*}f(X_{k})A_{k}{\Bigr )},}$

Conversely, if the above inequality is satisfied for some ${\displaystyle n}$ and ${\displaystyle m}$, where ${\displaystyle n>1}$, then ${\displaystyle f}$ is convex.

### Jensen's operator inequality

For a continuous function ${\displaystyle f}$ defined on an interval ${\displaystyle I}$ the following conditions are equivalent:

${\displaystyle f{\Bigl (}\sum _{k=1}^{n}A_{k}^{*}X_{k}A_{k}{\Bigr )}\leq \sum _{k=1}^{n}A_{k}^{*}f(X_{k})A_{k},}$

every self-adjoint operator ${\displaystyle X}$ with spectrum in ${\displaystyle I}$.

## Araki-Lieb-Thirring inequality

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 ${\displaystyle r\geq 1,}$

${\displaystyle {\rm {Tr}}(B^{1/2}A^{1/2}B^{1/2})^{r}\leq {\rm {Tr}}B^{r/2}A^{r/2}B^{r/2}.}$

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

${\displaystyle {\rm {Tr}}(B^{1/2}AB^{1/2})^{rq}\leq {\rm {Tr}}(B^{r/2}A^{r}B^{r/2})^{q},}$ for ${\displaystyle r\geq 1,}$

and

${\displaystyle {\rm {Tr}}(B^{r/2}A^{r}B^{r/2})^{q}\leq {\rm {Tr}}(B^{1/2}AB^{1/2})^{rq},}$ for ${\displaystyle 0\leq r\leq 1.}$

## Effros's theorem

E. Effros in [21] 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

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

## References

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2. R. Bhatia, Matrix Analysis, Springer, (1997).
3. B. Simon, Trace Ideals and their Applications, Cambridge Univ. Press, (1979); Second edition. Amer. Math. Soc., Providence, RI, (2005).
4. M. Ohya, D. Petz, Quantum Entropy and Its Use, Springer, (1993).
5. K. Löwner, "Uber monotone Matrix funktionen", Math. Z. 38, 177–216, (1934).
6. W.F. Donoghue, Jr., Monotone Matrix Functions and Analytic Continuation, Springer, (1974).
7. S. Golden, Lower Bounds for Helmholtz Functions, Phys. Rev. 137, B 1127–1128 (1965)
8. C.J. Thompson, Inequality with Applications in Statistical Mechanics, J. Math. Phys. 6, 1812–1813, (1965).
9. E. H. Lieb, Convex Trace Functions and the Wigner–Yanase–Dyson Conjecture, Advances in Math. 11, 267–288 (1973).
10. D. Ruelle, Statistical Mechanics: Rigorous Results, World Scient. (1969).
11. E. P. Wigner, M. M. Yanase, On the Positive Semi-Definite Nature of a Certain Matrix Expression, Can. J. Math. 16, 397–406, (1964).
12. . Ando, Convexity of Certain Maps on Positive Definite Matrices and Applications to Hadamard Products, Lin. Alg. Appl. 26, 203–241 (1979).
13. H. Epstein, Remarks on Two Theorems of E. Lieb, Comm. Math. Phys., 31:317–325, (1973).
14. M. B. Ruskai, Inequalities for Quantum Entropy: A Review With Conditions for Equality, J. Math. Phys., 43(9):4358–4375, (2002).
15. M. B. Ruskai, Another Short and Elementary Proof of Strong Subadditivity of Quantum Entropy, Reports Math. Phys. 60, 1–12 (2007).
16. G. Lindblad, Expectations and Entropyy Inequalities, Commun. Math. Phys. 39, 111–119 (1974).
17. C. Davis, A Schwarz inequality for convex operator functions, Proc. Amer. Math. Soc. 8, 42–44, (1957).
18. F. Hansen, G. K. Pedersen, Jensen's Operator Inequality, Bull. London Math. Soc. 35 (4): 553–564, (2003).
19. E. H. Lieb, W. E. Thirring, Inequalities for the Moments of the Eigenvalues of the Schrödinger Hamiltonian and Their Relation to Sobolev Inequalities, in Studies in Mathematical Physics, edited E. Lieb, B. Simon, and A. Wightman, Princeton University Press, 269-303 (1976).
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