Golden–Thompson inequality

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.

Introduction

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 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) 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.

Generalizations

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:

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

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.

References

• Bhatia, Rajendra (1997), Matrix analysis, Graduate Texts in Mathematics, 169, Berlin, New York: Springer-Verlag, doi:10.1007/978-1-4612-0653-8, ISBN 978-0-387-94846-1, MR 1477662
• J.E. Cohen, S. Friedland, T. Kato, F. Kelly, Eigenvalue inequalities for products of matrix exponentials, Linear algebra and its applications, Vol. 45, pp. 55–95, 1982. doi:10.1016/0024-3795(82)90211-7
• Lieb, Elliott H (1973), "Convex trace functions and the Wigner-Yanase-Dyson conjecture", Advances in Mathematics, 11 (3): 267–288, doi:10.1016/0001-8708(73)90011-X
• Golden, Sidney (1965), "Lower bounds for the Helmholtz function", Phys. Rev., Series II, 137 (4B): B1127–B1128, Bibcode:1965PhRv..137.1127G, doi:10.1103/PhysRev.137.B1127, MR 0189691
• Kostant, Bertram (1973), "On convexity, the Weyl group and the Iwasawa decomposition", Annales Scientifiques de l'École Normale Supérieure, Série 4, 6 (4): 413–455, doi:10.24033/asens.1254, ISSN 0012-9593, MR 0364552
• D. Petz, A survey of trace inequalities, in Functional Analysis and Operator Theory, 287–298, Banach Center Publications, 30 (Warszawa 1994).
• Sutter, David; Berta, Mario; Tomamichel, Marco (2016), "Multivariate Trace Inequalities", Communications in Mathematical Physics, 352 (1): 37–58, arXiv:1604.03023, Bibcode:2017CMaPh.352...37S, doi:10.1007/s00220-016-2778-5
• Thompson, Colin J. (1965), "Inequality with applications in statistical mechanics", Journal of Mathematical Physics, 6 (11): 1812–1813, Bibcode:1965JMP.....6.1812T, doi:10.1063/1.1704727, ISSN 0022-2488, MR 0189688