A variant of Horn’s problem and the derivative principle. (English) Zbl 1430.15009
A conjectural eigenvalues characterization of a Hermitian matrix \(C=A+B\) in terms of linear inequalities in eigenvalues of Hermitian matrices \(A\) and \(B\) was given by A. Horn [Pac. J. Math. 12, 225–241 (1962; Zbl 0112.01501)]. The conjecture
was answered affirmative by A. Knutson and T. Tao [J. Am. Math. Soc. 12, No. 4, 1055–1090 (1999; Zbl 0944.05097)] in 1999.
In this paper, the authors study a variant of Horn’s problem using a derivative principle. More precisely, they obtain the joint probability density function of the diagonal entries of \(C=A+B\) for random matrices \(A,B\). The formula has been simplified by applying a determinant equality. The derivative principle offers a connection between the joint distribution of a random matrix and the joint distribution of its diagonal elements. This connection is applied to re-derive the joint probability density function of \(C=A+B\) for random Hermitian matrices \(A,B\).
The authors then restrict the attention to random Hermitian matrices from GUE ensemble which is the class of matrices generated by the real part of standard complex Gaussian random matrices. The probability density function of the sum of \(k\) random matrices is given. Connections of these results with a trace inequality of exponentials of Hermitian matrices, known as the Golden-Thompson inequality, are studied. An asymptotic form on the ratio between expectations of trace of exponentials on Hermitian matrices is derived. This provides evidence for an open question asked by P. J. Forrester and C. J. Thompson [J. Math. Phys. 55, No. 2, 023503, 12 p. (2014; Zbl 1308.15018)].
Examples of \(2\times 2\) random Hermitian matrices are presented to illustrate the results. Moreover, the authors suggest two applications of the results in quantum information theory. In particular, the results are applied to obtain an explicit expression on the uniform average quantum Jensen-Shannon divergence and average coherence of uniform mixture of two unitary orbits in a two-dimensional space.
In this paper, the authors study a variant of Horn’s problem using a derivative principle. More precisely, they obtain the joint probability density function of the diagonal entries of \(C=A+B\) for random matrices \(A,B\). The formula has been simplified by applying a determinant equality. The derivative principle offers a connection between the joint distribution of a random matrix and the joint distribution of its diagonal elements. This connection is applied to re-derive the joint probability density function of \(C=A+B\) for random Hermitian matrices \(A,B\).
The authors then restrict the attention to random Hermitian matrices from GUE ensemble which is the class of matrices generated by the real part of standard complex Gaussian random matrices. The probability density function of the sum of \(k\) random matrices is given. Connections of these results with a trace inequality of exponentials of Hermitian matrices, known as the Golden-Thompson inequality, are studied. An asymptotic form on the ratio between expectations of trace of exponentials on Hermitian matrices is derived. This provides evidence for an open question asked by P. J. Forrester and C. J. Thompson [J. Math. Phys. 55, No. 2, 023503, 12 p. (2014; Zbl 1308.15018)].
Examples of \(2\times 2\) random Hermitian matrices are presented to illustrate the results. Moreover, the authors suggest two applications of the results in quantum information theory. In particular, the results are applied to obtain an explicit expression on the uniform average quantum Jensen-Shannon divergence and average coherence of uniform mixture of two unitary orbits in a two-dimensional space.
Reviewer: Tin Yau Tam (Reno)
MSC:
| 15A18 | Eigenvalues, singular values, and eigenvectors |
| 15A42 | Inequalities involving eigenvalues and eigenvectors |
| 15B52 | Random matrices (algebraic aspects) |
| 15B57 | Hermitian, skew-Hermitian, and related matrices |
| 22E70 | Applications of Lie groups to the sciences; explicit representations |
| 81Q10 | Selfadjoint operator theory in quantum theory, including spectral analysis |
| 46L30 | States of selfadjoint operator algebras |
| 81R05 | Finite-dimensional groups and algebras motivated by physics and their representations |
Keywords:
Horn’s problem; derivative principle; probability density function; quantum Jensen-Shannon divergence; quantum coherenceSoftware:
DLMFReferences:
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