Golden–Thompson inequality

If a and b are two real numbers, then the exponential of a+b is the product of the exponential of a with the exponential of b:

${\displaystyle e^{a+b}=e^{a}e^{b}.}$

This relationship is not true if we replace a and b with symmetric/hermitian square matrices A and B. Golden and Thompson proved that, while the matrix given by ${\displaystyle e^{A+B}}$ is not always equal to the matrix given by ${\displaystyle e^{A}e^{B}}$, their traces are related by the following inequality:

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

The inequality is well defined as the expression on right hand side of the inequality is a positive real number, as can be seen by rewriting it as ${\displaystyle \operatorname {tr} \left(e^{\frac {A}{2}}e^{B}e^{\frac {A}{2}}\right)}$ (using the cyclic property of the trace).

If A and B commute, then the equality ${\displaystyle e^{A+B}=e^{A}e^{B}}$ holds, just like in the case of real number. In this situation the Golden-Thompson inequality is actually an equality. Petz (1994) harvtxt error: no target: CITEREFPetz1994 (help) proved that this is the only situation in which this happens: if A and B are two Hermitian matrices for which the Golden-Thomposon inequality is verified as an equality, then the two matrices commute.

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). For three matrices, it takes the following formulation:

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

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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