Golden–Thompson lemma

Trace exponential inequality: for self-adjoint A,B, Tr e^{A+B} ≤ Tr(e^{A}e^{B}).

Golden–Thompson lemma

Definitions and notation

Density operator (states on a Hilbert space).
Quantum partition function (as a trace of an exponential).
See also Golden–Thompson inequality for a general reference statement.

Statement

Let $A$ and $B$ be self-adjoint operators on a finite-dimensional Hilbert space (or assume $e^{A+B}$ is trace-class so the trace is finite). Then

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

If $A$ and $B$ commute, then $e^{A+B}=e^{A}e^{B}$ and equality holds.

Key hypotheses and conclusions

Hypotheses

$A$ and $B$ are self-adjoint.
The traces involved are finite (automatic in finite dimension).

Conclusions

A fundamental bound for trace exponentials: $\operatorname{Tr}(e^{A+B}) \le \operatorname{Tr}(e^{A}e^{B})$ .
Equality when $[A,B]=0$ .

Proof idea / significance

One classical approach uses the Lie–Trotter product formula:

e^{A+B} = \lim_{n\to\infty}\big(e^{A/n}e^{B/n}\big)^n

and combines it with trace inequalities (e.g. Hölder-type bounds for Schatten norms) to pass to the trace and obtain the stated inequality.

In quantum statistical mechanics, the lemma is a workhorse for bounding partition functions of sums of noncommuting terms and for establishing convexity/subadditivity properties of free energy. It is frequently used as an ingredient in the Peierls–Bogoliubov inequality and related variational estimates.