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arXiv:1808.10153 · Journal of Physics A Mathematical and Theoretical

Typical Gaussian quantum information

Philipp Sohr, Valentin Link, Kimmo Luoma, Walter T Strunz

Published Nov 23, 2018• 2 citations
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Abstract

Abstract We investigate different geometries and invariant measures on the space of mixed Gaussian quantum states. We show that when the global purity of the state is held fixed, these measures coincide and it is possible, within this constraint, to define a unique notion of volume on the space of mixed Gaussian states. We then use the so defined measure to study typical non-classical correlations of two mode mixed Gaussian quantum states, in particular entanglement and steerability. We show that under the purity constraint alone, typical values for symplectic invariants can be computed very elegantly, irrespectively of the non-compactness of the underlying state space. Then we consider finite volumes by constraining the purity and energy of the Gaussian state and compute typical values of quantum correlations numerically.

DOI
10.1088/1751-8121/aaf365

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Abstract We investigate different geometries and invariant measures on the space of mixed Gaussian quantum states. We show that when the global purity of the state is held fixed, these measures coincide and it is possible, within this constraint, to define a unique notion of volume on the space of mixed Gaussian states. We then use the so defined measure to study typical non-classical correlations of two mode mixed Gaussian quantum states, in particular entanglement and steerability. We show that under the purity constraint alone, typical values for symplectic invariants can be computed very elegantly, irrespectively of the non-compactness of the underlying state space. Then we consider finite volumes by constraining the purity and energy of the Gaussian state and compute typical values of quantum correlations numerically.
We study the Hilbert-Schmidt measure on the manifold of mixed Gaussian states in multi mode continuous variable quantum systems. An analytical expression for the Hilbert-Schmidt volume element is derived. Its corresponding probability measure can be used to study typical properties of Gaussian states. It turns out that although the manifold of Gaussian states is unbounded, an ensemble of Gaussian states distributed according to this measure still has a normalizable distribution of symplectic eigenvalues, from which unitarily invariant properties can be obtained. By contrast, we find that for an ensemble of one-mode Gaussian states based on the Bures measure the corresponding distribution cannot be normalized. As important applications, we determine the distribution and the mean value of von Neumann entropy and purity for the Hilbert-Schmidt measure.
We study the Hilbert–Schmidt measure on the manifold of mixed Gaussian states in multi-mode continuous variable quantum systems. An analytical expression for the Hilbert–Schmidt volume element is derived. Its corresponding probability measure can be used to study typical properties of Gaussian states. It turns out that although the manifold of Gaussian states is unbounded, an ensemble of Gaussian states distributed according to this measure still has a normalizable distribution of symplectic eigenvalues, from which unitarily invariant properties can be obtained. By contrast, we find that for an ensemble of one-mode Gaussian states based on the Bures measure the corresponding distribution cannot be normalized. As important applications, we determine the distribution and the mean value of von Neumann entropy and purity for the Hilbert–Schmidt measure.
Abstract. Quantum entanglement of pure states of a bipartite system is defined as the amount of local or marginal (i.e. referring to the subsystems) entropy. For mixed states this identification vanishes, since the global loss of information about the state makes it impossible to distinguish between quantum and classical correlations. Here we show how the joint knowledge of the global and marginal degrees of information of a quantum state, quantified by the purities or in general by information entropies, provides an accurate characterization of its entanglement. In particular, for Gaussian states of continuous variable systems, we classify the entanglement of two–mode states according to their degree of total and partial mixedness, comparing the different roles played by the purity and the generalized p−entropies in quantifying the mixedness and bounding the entanglement. We prove the existence of strict upper and lower bounds on the entanglement and the existence of extremally (maximally and minimally) entangled states at fixed global and marginal degrees of information. This results allow for a powerful, operative method to measure mixedstate entanglement without the full tomographic reconstruction of the state. Finally, we briefly discuss the ongoing extension of our analysis to the quantification of multipartite entanglement in highly symmetric Gaussian states of arbitrary 1 × N-mode partitions. 1.
Quantum entanglement of pure states of a bipartite system is defined as the amount of local or marginal (i.e. referring to the subsystems) entropy. For mixed states this identification vanishes, since the global loss of information about the state makes it impossible to distinguish between quantum and classical correlations. Here we show how the joint knowledge of the global and marginal degrees of information of a quantum state, quantified by the purities or, in general, by information entropies, provides an accurate characterization of its entanglement. In particular, for Gaussian states of continuous variable systems, we classify the entanglement of two-mode states according to their degree of total and partial mixedness, comparing the different roles played by the purity and the generalized p-entropies in quantifying the mixedness and bounding the entanglement. We prove the existence of strict upper and lower bounds on the entanglement and the existence of extremally (maximally and minimally) entangled states at fixed global and marginal degrees of information. This results allow for a powerful, operative method to measure mixed-state entanglement without the full tomographic reconstruction of the state. Finally, we briefly discuss the ongoing extension of our analysis to the quantification of multipartite entanglement in highly symmetric Gaussian states of arbitrary 1 × N-mode partitions.
We present a derivation of the Von Neumann entropy and mutual information of arbitrary two-mode Gaussian states, based on the explicit determination of the symplectic eigenvalues of a generic covariance matrix. The key role of the symplectic invariants in such a determination is pointed out. We show that the Von Neumann entropy depends on two symplectic invariants, while the purity (or the linear entropy) is determined by only one invariant, so that the two quantities provide two different hierarchies of mixed Gaussian states. A comparison between mutual information and entanglement of formation for symmetric states is considered, taking note of the crucial role of the symplectic eigenvalues in qualifying and quantifying the correlations present in a generic state.
The relation between the symplectic and Lorentz groups is explored to investigate entanglement features in a two-mode bipartite Gaussian state. We verify that the correlation matrix of arbitrary Gaussian states can be associated with a hyperbolic space with a Minkowski metric, which is divided in two regions: separabilitylike and entanglementlike, in equivalence to timelike and spacelike in special relativity. This correspondence naturally allows the definition of two insightful invariant squared distance measures: one related to the purity and another related to amount of entanglement. The second distance allows us to define a measure for entanglement in terms of the invariant interval between the given state and its closest separable state, given in a natural manner without the requirement of a minimization procedure.