Trace inequality


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.

Basic definitions

Let Hn denote the space of Hermitian × matrices, Hn+ denote the set consisting of positive semi-definite × Hermitian matrices and Hn++ 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 on an interval ⊂ ℝ, one may define a matrix function for any operator with eigenvalues in by defining it on the eigenvalues and corresponding projectors as

Operator monotone

A function defined on an interval ⊂ ℝ is said to be operator monotone if ∀, and all with eigenvalues in, the following holds,
where the inequality means that the operator is positive semi-definite. One may check that is, in fact, not operator monotone!

Operator convex

A function is said to be operator convex if for all and all with eigenvalues in, and, the following holds
Note that the operator has eigenvalues in, since and have eigenvalues in.
A function is operator concave if is operator convex, i.e. the inequality above for is reversed.

Joint convexity

A function, defined on intervals is said to be jointly convex if for all and all
with eigenvalues in and all with eigenvalues in, and any the following holds
A function is jointly concave if − is jointly convex, i.e. the inequality above for is reversed.

Trace function

Given a function : ℝ → ℝ, the associated trace function on Hn is given by
where has eigenvalues and Tr stands for a trace of the operator.

Convexity and monotonicity of the trace function

Let : ℝ → ℝ be continuous, and let be any integer. Then, if is monotone increasing, so
is on Hn.
Likewise, if is convex, so is on Hn, and
it is strictly convex if is strictly convex.
See proof and discussion in, for example.

Löwner–Heinz theorem

For, the function is operator monotone and operator concave.
For, the function is operator monotone and operator concave.
For, the function is operator convex. Furthermore,
The original proof of this theorem is due to K. Löwner who gave a necessary and sufficient condition for to be operator monotone. An elementary proof of the theorem is discussed in and a more general version of it in.

Klein's inequality

For all Hermitian × matrices and and all differentiable convex functions
convex functions : → ℝ, the following inequality holds,
In either case, if is strictly convex, equality holds if and only if =.
A popular choice in applications is, see below.

Proof

Let so that, for,
varies from to.
Define
By convexity and monotonicity of trace functions, is convex, and so for all,
which is,
and, in fact, the right hand side is monotone decreasing in.
Taking the limit yields,
which with rearrangement and substitution is Klein's inequality:
Note that if is strictly convex and, then is strictly convex. The final assertion follows from this and the fact that is monotone decreasing in.

Golden–Thompson inequality

In 1965, S. Golden and C.J. Thompson independently discovered that
For any matrices,
This inequality can be generalized for three operators: for non-negative operators,

Peierls–Bogoliubov inequality

Let be such that Tr eR = 1.
Defining, we have
The proof of this inequality follows from the above combined with [|Klein's inequality]. Take.

Gibbs variational principle

Let be a self-adjoint operator such that is trace class. Then for any with
with equality if and only if

[|Lieb's concavity theorem]

The following theorem was proved by E. H. Lieb in. It proves and generalizes a conjecture of E. P. Wigner, M. M. Yanase and F. J. Dyson. Six years later other proofs were given by T. Ando and B. Simon, and several more have been given since then.
For all matrices, and all and such that and, with the real valued map on given by
Here stands for the adjoint operator of

Lieb's theorem

For a fixed Hermitian matrix, the function
is concave on.
The theorem and proof are due to E. H. Lieb, Thm 6, where he obtains this theorem as a corollary of Lieb's concavity Theorem.
The most direct proof is due to H. Epstein; see M.B. Ruskai papers, for a review of this argument.

Ando's convexity theorem

T. Ando's proof of Lieb's concavity theorem led to the following significant complement to it:
For all matrices, and all and with, the real valued map on given by
is convex.

Joint convexity of relative entropy

For two operators define the following map
For density matrices and, the map is the Umegaki's quantum relative entropy.
Note that the non-negativity of follows from Klein's inequality with.

Statement

The map is jointly convex.

Proof

For all, is jointly concave, by Lieb's concavity theorem, and thus
is convex. But
and convexity is preserved in the limit.
The proof is due to G. Lindblad.

Jensen's operator and trace inequalities

The operator version of Jensen's inequality is due to C. Davis.
A continuous, real function on an interval satisfies Jensen's Operator Inequality if the following holds
for operators with and for self-adjoint operators with spectrum on.
See, for the proof of the following two theorems.

Jensen's trace inequality

Let be a continuous function defined on an interval and let and be natural numbers. If is convex, we then have the inequality
for all self-adjoint × matrices with spectra contained in and
all of × matrices with
Conversely, if the above inequality is satisfied for some and , where > 1, then is convex.

Jensen's operator inequality

For a continuous function defined on an interval the following conditions are equivalent:
for all bounded, self-adjoint operators on an arbitrary Hilbert space with
spectra contained in and all on with
every self-adjoint operator with spectrum in.
E. H. Lieb and W. E. Thirring proved the following inequality in in 1976: For any, and
In 1990 H. Araki generalized the above inequality to the following one: For any, and
and
The Lieb–Thirring inequality also enjoys the following generalization: for any, and

Effros's theorem and its extension

E. Effros in proved the following theorem.
If is an operator convex function, and and are commuting bounded linear operators, i.e. the commutator, the perspective
is jointly convex, i.e. if and with ,,
Ebadian et al. later extended the inequality to the case where and do not commute.

Von Neumann's trace inequality and related results

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 and respectively,
A simple corollary to this is the following result: For hermitian n × n positive semidefinite complex matrices A, B where now the eigenvalues are sorted decreasingly,