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arXiv:2004.14204

On the geometric phase for Gaussian states

Angel García-Chung

Published Apr 28, 2020• 0 citations
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Abstract

We show the explicit expression of the geometric phase for $n$-partite Gaussian states. In our analysis, the covariance matrix can be obtained as a boundary term of the geometric phase.

DOI
10.48550/arxiv.2004.14204

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We show the explicit expression of the geometric phase for nnn-partite Gaussian states. In our analysis, the covariance matrix can be obtained as a boundary term of the geometric phase.
We present a rigorous analysis of the algebraic and geometric structure of the quantum complexity resource of a system of bosonic modes in Gaussian boson sampling. This resource underlies the quantum advantage of the system: its photon-counting statistics require the evaluation of a hafnian of the resource covariance matrix, and that computation is #P-hard. The resource covariance matrix is the solution of a semidefinite program that extracts the minimum-trace physical quantum part of the total covariance matrix; the complementary part is positive semidefinite and can therefore be simulated classically. Earlier work characterized this resource only through the trace of the quantum part, equal to its photon number. We characterize the optimizer itself, as a quantum state and as a geometric object, beyond the scalar given by its trace. We prove that it is a unique pure Gaussian state and construct an explicit oracle map, obeying an algebraic Riccati identity, that reconstructs the resource. We prove that the full problem compresses exactly onto the active symplectic sector that the dual program support generates. The passive-diagonalizable states are solved in closed form, the first explicit solvable class, and the whole program is shown to be equivalent to a minimization over the symplectic group, that is, over the Siegel upper half-space. Together these results establish that the program determines a canonical localized pure Gaussian component of the resource, and they provide the structural foundation for its detailed analysis.
We establish a direct connection between quantum thermodynamics and information geometry by introducing an ergotropic decomposition of the Wigner-Fisher information for Gaussian quantum states. We derive an analytical expression for the Wigner-Fisher information in terms of the phase-space covariance matrix and mean vector, and show that it naturally separates into passive and ergotropic contributions. The passive term is shown to be entirely determined by the Wigner entropy rate of the associated passive state. The ergotropic contribution, in turn, quantifies how displacement and squeezing resources modify the statistical velocity and length of the system trajectory in phase space. As an application, we analyze the recently proposed ergotropic Mpemba effect and demonstrate that it can be traced to the anomalous geometric evolution of the passive state associated with squeezed thermal states. Our results reveal how the extractable work stored in a quantum state constrains its information-geometric structure, establishing a framework that links ergotropy, entropy production rate, and statistical geometry in continuous-variable quantum systems.