Golden–Thompson inequality

In physics and mathematics, the Golden–Thompson inequality is a trace inequality between exponentials of symmetric and Hermitian 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.

Statement

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

[math]\displaystyle{ \operatorname{tr}\, e^{A+B} \le \operatorname{tr} \left(e^A e^B\right). }[/math]

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 [math]\displaystyle{ \operatorname{tr}(e^{A/2}e^B e^{A/2}) }[/math] using the cyclic property of the trace.

Motivation

The Golden–Thompson inequality can be viewed as a generalization of a stronger statement for real numbers. 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:

[math]\displaystyle{ e^{a+b} = e^a e^b . }[/math]

If we replace a and b with commuting matrices A and B, then the same inequality [math]\displaystyle{ e^{A+B} = e^A e^B }[/math] holds.

This relationship is not true if A and B do not commute. In fact, (Petz 1994) proved that if A and B are two Hermitian matrices for which the Golden–Thompson inequality is verified as an equality, then the two matrices commute. The Golden–Thompson inequality shows that, even though [math]\displaystyle{ e^{A+B} }[/math] and [math]\displaystyle{ e^Ae^B }[/math] are not equal, they are still related by an inequality.

Generalizations

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

[math]\displaystyle{ \|e^{A+B}\| \leq \|e^{A/2}e^Be^{A/2}\| . }[/math]

The standard Golden–Thompson inequality is a special case of the above inequality, where the norm is the Schatten norm with [math]\displaystyle{ p=1 }[/math]. Since [math]\displaystyle{ e^{A+B} }[/math] and [math]\displaystyle{ e^{A/2}e^Be^{A/2} }[/math] are both positive semidefinite matrices, [math]\displaystyle{ \operatorname{tr}(e^{A+B}) = \|e^{A+B}\|_1 }[/math] and [math]\displaystyle{ \operatorname{tr}(e^{A/2}e^Be^{A/2}) = \|e^{A/2}e^Be^{A/2}\|_1 }[/math].

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

[math]\displaystyle{ \operatorname{tr}(e^{A+B+C}) \leq |\operatorname{tr}(e^Ae^Be^C)| }[/math]

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

[math]\displaystyle{ \operatorname{tr}\, e^{A+B+C} \le \operatorname{tr} \left(e^A \mathcal{T}_{e^{-B}} e^C\right), }[/math]

where the operator [math]\displaystyle{ \mathcal{T}_f }[/math] is the derivative of the matrix logarithm given by [math]\displaystyle{ \mathcal{T}_f(g) = \int_0^\infty \operatorname{d}t \, (f+t)^{-1} g (f+t)^{-1} }[/math]. Note that, if [math]\displaystyle{ f }[/math] and [math]\displaystyle{ g }[/math] commute, then [math]\displaystyle{ \mathcal{T}_f(g) = gf^{-1} }[/math], 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 
• Cohen, J.E.; Friedland, S.; Kato, T.; Kelly, F. (1982), "Eigenvalue inequalities for products of matrix exponentials", Linear Algebra and Its Applications 45: 55–95, doi:10.1016/0024-3795(82)90211-7 
• Golden, Sidney (1965), "Lower bounds for the Helmholtz function", Phys. Rev., Series II 137 (4B): B1127–B1128, doi:10.1103/PhysRev.137.B1127, Bibcode: 1965PhRv..137.1127G 
• 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, http://www.numdam.org/item?id=ASENS_1973_4_6_4_413_0 
• 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, http://www.numdam.org/item/RCP25_1973__19__A4_0/ 
• Petz, D. (1994), A survey of trace inequalities, in Functional Analysis and Operator Theory, 30, Warszawa: Banach Center Publications, pp. 287–298, http://www.renyi.hu/%7Epetz/pdf/64.pdf 
• Sutter, David; Berta, Mario; Tomamichel, Marco (2016), "Multivariate Trace Inequalities", Communications in Mathematical Physics 352 (1): 37–58, doi:10.1007/s00220-016-2778-5, Bibcode: 2017CMaPh.352...37S 
• Thompson, Colin J. (1965), "Inequality with applications in statistical mechanics", Journal of Mathematical Physics 6 (11): 1812–1813, doi:10.1063/1.1704727, ISSN 0022-2488, Bibcode: 1965JMP.....6.1812T 

External links

• Tao, T. (2010), The Golden–Thompson inequality, http://terrytao.wordpress.com/2010/07/15/the-golden-thompson-inequality/ 
• Forrester, Peter J; Thompson, Colin J (2014). "The Golden-Thompson inequality --- historical aspects and random matrix applications". Journal of Mathematical Physics 55 (2): 023503. doi:10.1063/1.4863477. Bibcode: 2014JMP....55b3503F.