In physics and mathematics, the Golden–Thompson inequality is a trace inequality between exponentials of matrices proved independently by Golden (1965) and Thompson (1965). It has been developed in the context of statistical mechanics, where it has come to have a particular significance.
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:
This relationship is not true if we replace a and b with square matrices A and B. Golden and Thompson proved that, while the matrix given by is not always equal to the matrix given by , their traces are related by the following inequality:
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 (using the cyclic property of the trace).
If A and B commute, then the equality holds, just like in the case of real number. In this situation the Golden-Thompson inequality is actually an equality. Petz (1994) 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 matrices by Sutter, Berta & Tomamichel (2016). For three matrices, it takes the following formulation:
where the operator is the derivative of the matrix logarithm given by . Note that, if and commute, then , and the inequality for three matrices reduces to the original from Golden and Thompson.
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