---
tag: Z7R754JC7N
kind: statement
role: theorem
name: Golden–Thompson inequality
msc: [15A45, 15B57]
uses: [0P17NGQQMQ, 020M04FN1H, 0XC1RDBY7A, HK3BRMK912]
about: [020M04FN1H, 0XC1RDBY7A]
sources:
  - url: https://en.wikipedia.org/wiki/Golden%E2%80%93Thompson_inequality
same_as:
  wikidata: Q1058622
  wikipedia: Golden–Thompson inequality
---

## Statement

Let $n \ge 1$ and let $A$ and $B$ be $n \times n$ [[Hermitian matrices|0P17NGQQMQ]].
Then
$$\tr e^{A + B} \le \tr\left( e^A e^B \right),$$
where $e^X$ is the [[matrix exponential|020M04FN1H]] and $\tr$ is the [[trace|0XC1RDBY7A]].

## Remarks

The right side is a positive real number, because $\tr(e^A e^B) = \tr(e^{A/2} e^B e^{A/2})$ and $e^{A/2} e^B e^{A/2}$ is positive definite.
One proof uses the inequality $\tr\big((XY)^{2^k}\big) \le \tr\big(X^{2^k} Y^{2^k}\big)$ for positive definite $X$ and $Y$, which follows from the Cauchy–Schwarz inequality for the trace by induction on $k$.
Put $X = e^{A/2^k}$ and $Y = e^{B/2^k}$ and let $k \to \infty$; the [[Lie product formula|HK3BRMK912]] turns the left side into $\tr e^{A + B}$.
If $AB = BA$, then $e^{A + B} = e^A e^B$ and equality holds.
The naive extension to three Hermitian matrices, with $\abs{\tr(e^A e^B e^C)}$ on the right, is false in general.

## Explanations

### L1 human

For numbers, $e^{a + b} = e^a e^b$.
For Hermitian matrices that do not commute, the two sides differ, but their traces compare in one direction: the trace of $e^{A + B}$ is at most the trace of $e^A e^B$.
This one-sided rule is often enough to bound quantities in physics and probability.

### L2 human

In statistical mechanics, $\tr e^{-\beta H}$ is the partition function of a quantum system with Hamiltonian $H$, so the inequality bounds a partition function by a simpler expression.
In probability, it is used in the approach of Ahlswede and Winter to matrix concentration inequalities.
There one must control the trace of the exponential of a sum of random Hermitian matrices.
Lieb found a version for three matrices, with an integral in place of a simple product.