Trace inequalities

In mathematics, there are many kinds of inequalities involving matrices and linear operators on Hilbert spaces. This article covers some important operator inequalities connected with traces of matrices.^[1]^[2]^[3]^[4]

Basic definitions

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 $A \in H n$ with eigenvalues $λ$ in $I$ by defining it on the eigenvalues and corresponding projectors $P$ as

f(A)\equiv \sum_j f(\lambda_j)P_j ~,

given the spectral decomposition

A=\sum_j\lambda_j P_j.

Operator monotone

A function $f : I \to ℝ$ defined on an interval $I$ ⊂ ℝ is said to be operator monotone if ∀ $n$ , and all $A,B \in H n$ with eigenvalues in $I$ , the following holds,

A \geq B \Rightarrow f(A) \geq f(B),

where the inequality $A \geq B$ means that the operator $A - B \geq 0$ is positive semi-definite. One may check that $f(A)=A 2$ is, in fact, not monotone!

Operator convex

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

f(\lambda A + (1-\lambda)B) \leq \lambda f(A) + (1 -\lambda)f(B) .

Note that the operator $\lambda A + (1-\lambda)B$ has eigenvalues in $I$ , since $A$ and $B$ have eigenvalues in $I$ .

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

Joint convexity

A function $g: I\times J \rightarrow \mathbb{R}$ , defined on intervals $I,J\subset \mathbb{R}$ is said to be jointly convex if for all $n$ and all $A_1, A_2\in \mathbf{H}_n$ with eigenvalues in $I$ and all $B_1,B_2\in \mathbf{H}_n$ with eigenvalues in $J$ , and any $0\leq \lambda\leq 1$ the following holds

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 $g$ is jointly concave if − $g$ is jointly convex, i.e. the inequality above for $g$ is reversed.

Trace function

Given a function $f$ : ℝ → ℝ, the associated trace function on H_n is given by

A\mapsto {\rm Tr} f(A)=\sum_j f(\lambda_j),

where $A$ has eigenvalues $λ$ and Tr stands for a trace of the operator.

Convexity and monotonicity of the trace function

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

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

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

Löwner–Heinz theorem

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

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

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

f(t) = \log(t)

is operator concave and operator monotone, while

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]

Klein's inequality

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,

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

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.

Proof

Let $C = A - B$ so that, for 0 < $t$ < 1,

B + tC = (1 -t)B + tA

Define

\phi(t) = {\rm Tr}[f(B + tC)]~.

By convexity and monotonicity of trace functions, $φ$ is convex, and so for all 0 < $t$ < 1,

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

and, in fact, the right hand side is monotone decreasing in $t$ . Taking the limit $t$ →0 yields Klein's inequality.

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

Golden–Thompson inequality

Main article: Golden–Thompson inequality

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

For any matrices $A, B\in\mathbf{H}_n$ ,

{\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 $A, B, C\in\mathbf{H}_n^+$ ,

{\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

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

{\rm Tr}\, e^{F}e^R \geq {\rm Tr}\, e^{F+R}\geq e^g.

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

Gibbs variational principle

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

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

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

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.

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

F(A,B,K) = {\rm Tr}(K^*A^qKB^r)

is jointly concave in $(A,B)$
is convex in $K$ .

Here $K^*$ stands for the adjoint operator of $K.$

Lieb's theorem

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

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

is concave on $\mathbf{H}_n^{++}$ .

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.

Ando's convexity theorem

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

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

(A,B) \mapsto {\rm Tr}(K^*A^qKB^{-r})

is convex.

Joint convexity of relative entropy

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

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

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

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

Statement

The map $R(A\|B): \mathbf{H}^{++}_n \times \mathbf{H}^{++}_n \rightarrow \mathbf{R}$ is jointly convex.

Proof

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

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

is convex. But

\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 $f$ on an interval $I$ satisfies Jensen's Operator Inequality if the following holds

f\left(\sum_kA_k^*X_kA_k\right)\leq\sum_k A_k^*f(X_k)A_k,

for operators $\{A_k\}_k$ with $\sum_k A^*_kA_k=1$ and for self-adjoint operators $\{X_k\}_k$ with spectrum on $I$ .

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

Jensen's trace inequality

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

{\rm Tr}\Bigl(f\Bigl(\sum_{k=1}^nA_k^*X_kA_k\Bigr)\Bigr)\leq {\rm Tr}\Bigl(\sum_{k=1}^n A_k^*f(X_k)A_k\Bigr),

for all ( $X$ ₁, ... , $X$ _n) self-adjoint $m$ × $m$ matrices with spectra contained in $I$ and all ( $A$ ₁, ... , $A$ _n) of $m$ × $m$ matrices with

\sum_{k=1}^nA_k^*A_k=1

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

Jensen's operator inequality

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

$f$ is operator convex.
For each natural number $n$ we have the inequality

f\Bigl(\sum_{k=1}^nA_k^*X_kA_k\Bigr)\leq\sum_{k=1}^n A_k^*f(X_k)A_k,

for all $(X_1, \ldots , X_n)$ bounded, self-adjoint operators on an arbitrary Hilbert space ${\mathcal {H}}$ with spectra contained in $I$ and all $(A_1, \ldots , A_n)$ on ${\mathcal {H}}$ with $\sum_{k=1}^n A^*_kA_k=1$ .

$f(V^*XV) \leq V^*f(X)V$ for each isometry $V$ on an infinite-dimensional Hilbert space ${\mathcal {H}}$ and

every self-adjoint operator $X$ with spectrum in $I$ .

$Pf(PXP + \lambda(1 -P))P \leq Pf(X)P$ for each projection $P$ on an infinite-dimensional Hilbert space ${\mathcal {H}}$ , every self-adjoint operator $X$ with spectrum in $I$ and every $\lambda$ in $I$ .

Araki-Lieb-Thirring inequality

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

{\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 $A\geq 0$ , $B\geq 0$ and $q\geq 0,$

{\rm Tr}(B^{1/2}AB^{1/2})^{rq}\leq {\rm Tr}(B^{r/2}A^rB^{r/2})^q,

for

r\geq 1,

and

{\rm Tr}(B^{r/2}A^rB^{r/2})^q\leq {\rm Tr}(B^{1/2}AB^{1/2})^{rq},

for

0\leq r\leq 1.

Lieb-Thirring inequality also enjoys the following generalization:^[21] for any $A\geq 0$ , $B\geq 0$ and $\alpha \in [0,1],$

{\rm {Tr}}(B^{1/2}A^{\alpha }B^{1/2}B^{1/2}A^{1-\alpha }B^{1/2})\leq {\rm {Tr}}(BAB).

Effros's theorem and its extension

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

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

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

is jointly convex, i.e. if $L=\lambda L_1+(1-\lambda)L_2$ and $R=\lambda R_1+(1-\lambda)R_2$ with $[L_i, R_i]=0$ (i=1,2), $0\leq\lambda\leq 1$ ,

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 $L$ and $R$ do not commute . ^[23]

Von Neumann's trace inequality

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 $\alpha_1 \ge \alpha_2 \ge \cdots \ge \alpha_n$ and $\beta_1 \ge \beta_2 \ge \cdots \ge \beta_n$ respectively,^[24]

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

The equality is achieved when $A$ and $B$ are simultaneously unitarily diagonalizable (see trace).

References

1 2 3 E. Carlen, Trace Inequalities and Quantum Entropy: An Introductory Course, Contemp. Math. 529 (2009).
↑ R. Bhatia, Matrix Analysis, Springer, (1997).
1 2 B. Simon, Trace Ideals and their Applications, Cambridge Univ. Press, (1979); Second edition. Amer. Math. Soc., Providence, RI, (2005).
↑ M. Ohya, D. Petz, Quantum Entropy and Its Use, Springer, (1993).
↑ K. Löwner, "Uber monotone Matrix funktionen", Math. Z. 38, 177–216, (1934).
↑ W.F. Donoghue, Jr., Monotone Matrix Functions and Analytic Continuation, Springer, (1974).
↑ S. Golden, Lower Bounds for Helmholtz Functions, Phys. Rev. 137, B 1127–1128 (1965)
↑ C.J. Thompson, Inequality with Applications in Statistical Mechanics, J. Math. Phys. 6, 1812–1813, (1965).
1 2 3 E. H. Lieb, Convex Trace Functions and the Wigner–Yanase–Dyson Conjecture, Advances in Math. 11, 267–288 (1973).
↑ D. Ruelle, Statistical Mechanics: Rigorous Results, World Scient. (1969).
↑ E. P. Wigner, M. M. Yanase, On the Positive Semi-Definite Nature of a Certain Matrix Expression, Can. J. Math. 16, 397–406, (1964).
1 2 . Ando, Convexity of Certain Maps on Positive Definite Matrices and Applications to Hadamard Products, Lin. Alg. Appl. 26, 203–241 (1979).
↑ H. Epstein, Remarks on Two Theorems of E. Lieb, Comm. Math. Phys., 31:317–325, (1973).
↑ M. B. Ruskai, Inequalities for Quantum Entropy: A Review With Conditions for Equality, J. Math. Phys., 43(9):4358–4375, (2002).
↑ M. B. Ruskai, Another Short and Elementary Proof of Strong Subadditivity of Quantum Entropy, Reports Math. Phys. 60, 1–12 (2007).
↑ G. Lindblad, Expectations and Entropy Inequalities, Commun. Math. Phys. 39, 111–119 (1974).
1 2 C. Davis, A Schwarz inequality for convex operator functions, Proc. Amer. Math. Soc. 8, 42–44, (1957).
↑ F. Hansen, G. K. Pedersen, Jensen's Operator Inequality, Bull. London Math. Soc. 35 (4): 553–564, (2003).
↑ 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).
↑ H. Araki, On an Inequality of Lieb and Thirring, Lett. Math. Phys. 19, 167-170 (1990).
↑ Z. Allen-Zhu, Y. Lee, L. Orecchia, Using Optimization to Obtain a Width-Independent, Parallel, Simpler, and Faster Positive SDP Solver, in ACM-SIAM Symposium on Discrete Algorithms, 1824-1831 (2016).
↑ E. Effros, A Matrix Convexity Approach to Some Celebrated Quantum Inequalities, Proc. Natl. Acad. Sci. USA, 106, n.4, 1006–1008 (2009).
↑ A. Ebadian, I. Nikoufar, and M. Gordjic, "Perspectives of matrix convex functions," Proc. Natl Acad. Sci. USA, 108(18), 7313--7314 (2011)
↑ Mirsky, L. (December 1975). "A trace inequality of John von Neumann". Monatshefte für Mathematik. 79 (4): 303–306. doi:10.1007/BF01647331.

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Trace inequalities

Basic definitions

Operator monotone

Operator convex

Joint convexity

Trace function

Convexity and monotonicity of the trace function

Löwner–Heinz theorem

Klein's inequality

Proof

Golden–Thompson inequality

Peierls–Bogoliubov inequality

Gibbs variational principle

Lieb's concavity theorem

Lieb's theorem

Ando's convexity theorem

Joint convexity of relative entropy

Statement

Proof

Jensen's operator and trace inequalities

Jensen's trace inequality

Jensen's operator inequality

Araki-Lieb-Thirring inequality

Effros's theorem and its extension

Von Neumann's trace inequality

See also

References