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

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 $[A,B]=0$.