A Result About the Classification of Quantum Covariance Matrices Based
on Their Eigenspectra
- URL: http://arxiv.org/abs/2308.03439v2
- Date: Mon, 5 Feb 2024 13:50:02 GMT
- Title: A Result About the Classification of Quantum Covariance Matrices Based
on Their Eigenspectra
- Authors: Arik Avagyan
- Abstract summary: We find a non-trivial class of eigenspectra with the property that the set of quantum covariance matrices corresponding to any eigenspectrum in this class are related by symplectic transformations.
We show that all non-degenerate eigenspectra with this property must belong to this class, and that the set of such eigenspectra coincides with the class of non-degenerate eigenspectra.
- Score: 0.0
- License: http://creativecommons.org/licenses/by/4.0/
- Abstract: The set of covariance matrices of a continuous-variable quantum system with a
finite number of degrees of freedom is a strict subset of the set of real
positive-definite matrices due to Heisenberg's uncertainty principle. This has
the implication that, in general, not every orthogonal transform of a quantum
covariance matrix produces a positive-definite matrix that obeys the
uncertainty principle. A natural question thus arises, to find the set of
quantum covariance matrices consistent with a given eigenspectrum. For the
special class of pure Gaussian states the set of quantum covariance matrices
with a given eigenspectrum consists of a single orbit of the action of the
orthogonal symplectic group. The eigenspectrum of a covariance matrix of a
state in this class is composed of pairs that each multiply to one. Our main
contribution is finding a non-trivial class of eigenspectra with the property
that the set of quantum covariance matrices corresponding to any eigenspectrum
in this class are related by orthogonal symplectic transformations. We show
that all non-degenerate eigenspectra with this property must belong to this
class, and that the set of such eigenspectra coincides with the class of
non-degenerate eigenspectra that identify the physically relevant thermal and
squeezing parameters of a Gaussian state.
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