Golden–Thompson inequality

The Golden–Thompson inequality states that for (real) symmetric or (complex) Hermitian matrices A and B, the following trace inequality holds:

${\displaystyle \operatorname {tr} \,e^{A+B}\leq \operatorname {tr} \left(e^{A}e^{B}\right).}$

This inequality is well defined, since the quantities on either side are real numbers. For the expression on right hand side of the inequality, this can be seen by rewriting it as ${\displaystyle \operatorname {tr} (e^{A/2}e^{B}e^{A/2})}$ using the cyclic property of the trace.

The Golden–Thompson inequality generalizes to any unitarily invariant norm. If A and B are Hermitian matrices and ${\displaystyle \|\cdot \|}$ is a unitarily invariant norm, then

${\displaystyle \|e^{A+B}\|\leq \|e^{A/2}e^{B}e^{A/2}\|.}$

The standard Golden–Thompson inequality is a special case of the above inequality, where the norm is the Schatten norm with ${\displaystyle p=1}$. Since ${\displaystyle e^{A+B}}$ and ${\displaystyle e^{A/2}e^{B}e^{A/2}}$ are both positive semidefinite matrices, ${\displaystyle \operatorname {tr} (e^{A+B})=\|e^{A+B}\|_{1}}$ and ${\displaystyle \operatorname {tr} (e^{A/2}e^{B}e^{A/2})=\|e^{A/2}e^{B}e^{A/2}\|_{1}}$.

The inequality has been generalized to three matrices by Lieb (1973) and furthermore to any arbitrary number of Hermitian matrices by Sutter, Berta & Tomamichel (2016). A naive attempt at generalization does not work: the inequality

${\displaystyle \operatorname {tr} (e^{A+B+C})\leq |\operatorname {tr} (e^{A}e^{B}e^{C})|}$

is false. For three matrices, the correct generalization takes the following form:

${\displaystyle \operatorname {tr} \,e^{A+B+C}\leq \operatorname {tr} \left(e^{A}{\mathcal {T}}_{e^{-B}}e^{C}\right),}$

where the operator ${\displaystyle {\mathcal {T}}_{f}}$ is the derivative of the matrix logarithm given by ${\displaystyle {\mathcal {T}}_{f}(g)=\int _{0}^{\infty }\operatorname {d} t\,(f+t)^{-1}g(f+t)^{-1}}$. Note that, if ${\displaystyle f}$ and ${\displaystyle g}$ commute, then ${\displaystyle {\mathcal {T}}_{f}(g)=gf^{-1}}$, and the inequality for three matrices reduces to the original from Golden and Thompson.

Bertram Kostant (1973) used the Kostant convexity theorem to generalize the Golden–Thompson inequality to all compact Lie groups.

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