Derivatives of symplectic spectral functions
Abstract
A symplectic spectral function of a 2n × 2n real positive definite matrix A is a function of its symplectic eigenvalues 0< d1(A) ≤ ⋯ ≤ dn(A), which is given by a composition f ∘ d of some symmetric function f on the set of n-vectors with positive entries and the symplectic eigenvalue vector map d(A)=(d1(A),…, dn(A)). In this work, we rigorously study various types of differentiability and Clarke generalized gradient of symplectic spectral functions, and also provide several applications of our findings. We show that the symplectic spectral function f ∘ d is Fréchet differentiable at A if and only if f is Fréchet differentiable at d(A), and we compute the derivative expression explicitly. We also show that f ∘ d is strictly Fréchet differentiable (respectively, continuously Gâteaux differentiable) at A if and only if f is strictly Fréchet differentiable (respectively, continuously Gâteaux differentiable) at d(A). We determine the Clarke generalized gradient of a symplectic spectral function f ∘ d at A, which is given in terms of the Clarke generalized gradient of f at d(A). As an application of our work, we show that the purity, von Neumann entropy, and Rényi entropy of a bosonic faithful Gaussian state are Fréchet differentiable functions of the covariance matrix of the Gaussian state. We also provide explicit expressions of their Fréchet derivatives.
Related
- Lidskii theorem for directional derivatives of symplectic eigenvalues
- Möbius functions for pseudo-Levi subgroups in finite general linear and symplectic groups
- Integral representations of high-order derivatives of Bessel functions with respect to the order
- Extreme values of derivatives of Dedekind zeta functions on the $1$-line
- Derivatives of Theta Functions and a Problem of Lyubarskii and Nes
BibTeX
@misc{mishra2026derivatives,
title = {{Derivatives of symplectic spectral functions}},
author = {Hemant K. Mishra and Niloy Paul and Temjensangba},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.40175},
eprint = {2609.40175},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.40175},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Derivatives of symplectic spectral functions
AU - Hemant K. Mishra
AU - Niloy Paul
AU - Temjensangba
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2609.40175
UR - https://arxiv.org/abs/2609.40175
AB - A symplectic spectral function of a 2n × 2n real positive definite matrix A is a function of its symplectic eigenvalues 0< d_1(A) ≤ ⋯ ≤ d_n(A), which is given by a composition f ∘ d of some symmetric function f on the set of n-vectors with positive entries and the symplectic eigenvalue vector map d(A)=(d_1(A),…, d_n(A)). In this work, we rigorously study various types of differentiability and Clarke generalized gradient of symplectic spectral functions, and also provide several applications of our findings. We show that the symplectic spectral function f ∘ d is Fréchet differentiable at A if and only if f is Fréchet differentiable at d(A), and we compute the derivative expression explicitly. We also show that f ∘ d is strictly Fréchet differentiable (respectively, continuously Gâteaux differentiable) at A if and only if f is strictly Fréchet differentiable (respectively, continuously Gâteaux differentiable) at d(A). We determine the Clarke generalized gradient of a symplectic spectral function f ∘ d at A, which is given in terms of the Clarke generalized gradient of f at d(A). As an application of our work, we show that the purity, von Neumann entropy, and Rényi entropy of a bosonic faithful Gaussian state are Fréchet differentiable functions of the covariance matrix of the Gaussian state. We also provide explicit expressions of their Fréchet derivatives.
ER -