Golden-Thompson inequality

Trace inequality bounding Tr exp(A+B) by Tr(exp(A)exp(B)) for Hermitian matrices.

Golden-Thompson inequality

Let $A$ and $B$ be Hermitian (self-adjoint) matrices (equivalently, finite-dimensional self-adjoint operators; see self-adjoint-operator-observable ). The Golden-Thompson inequality states that

\operatorname{Tr}\!\big(e^{A+B}\big)\ \le\ \operatorname{Tr}\!\big(e^{A}e^{B}\big),

where $e^{X}$ denotes the matrix exponential and $\operatorname{Tr}$ is the trace (see trace-operator ).

Using cyclicity of the trace, the right-hand side can also be written as

\operatorname{Tr}\!\big(e^{A}e^{B}\big)=\operatorname{Tr}\!\big(e^{A/2}e^{B}e^{A/2}\big).

Equality condition

If $A$ and $B$ commute (so they are simultaneously diagonalizable), then

e^{A+B}=e^Ae^B

and equality holds. In general, noncommutativity forces $\operatorname{Tr}(e^{A+B})$ to be no larger than $\operatorname{Tr}(e^{A}e^{B})$ .

Uses and context

Golden-Thompson is a foundational trace inequality in matrix analysis and quantum theory. It is often used to control partition functions in quantum statistical mechanics and appears in proofs of entropy and monotonicity statements involving quantum-relative-entropy .