Invariant measures on multimode quantum Gaussian states

Abstract

We derive the invariant measure on the manifold of multimode quantum Gaussian states, induced by the Haar measure on the group of Gaussian unitary transformations. To this end, by introducing a bipartition of the system in two disjoint subsystems, we use a parameterization highlighting the role of nonlocal degrees of freedom -- the symplectic eigenvalues -- which characterize quantum entanglement across the given bipartition. A finite measure is then obtained by imposing a physically motivated energy constraint. By averaging over the local degrees of freedom we finally derive the invariant distribution of the symplectic eigenvalues in some cases of particular interest for applications in quantum optics and quantum information.

17 pages, comments are welcome. v2: presentation improved and typos corrected. Close to the published version

References in corpus (24)

• Quantum entanglement
• A computable measure of entanglement
• Quantum information with continuous variables
• Gaussian Quantum Information
• Average Entropy of a Subsystem
• The logarithmic negativity: A full entanglement monotone that is not convex
• Entanglement monotones
• Quantum data processing and error correction
• Aspects of generic entanglement
• Efficient Generation of Generic Entanglement
• Phase Transitions in the Distribution of Bipartite Entanglement of a Random Pure State
• Optimised generation of heralded Fock states using parametric down conversion
• Mode-Wise Entanglement of Gaussian States
• Phase transitions of bipartite entanglement
• Emergence of typical entanglement in two-party random processes
• Phase transitions and metastability in the distribution of the bipartite entanglement of a large quantum system
• Generic Entanglement and Standard Form for N-mode Pure Gaussian States
• Canonical and micro-canonical typical entanglement of continuous variable systems
• Statistical distribution of the local purity in a large quantum system
• Classical Tensors and Quantum Entanglement I: Pure States
• Teleportation fidelities of squeezed states from thermodynamical state space measures
• Gaussian maximally multipartite entangled states
• Standard forms and entanglement engineering of multimode Gaussian states under local operations
• Entanglement frustration in multimode Gaussian states

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• Entanglement Typicality
• Typical entanglement for Gaussian states
• Polarized ensembles of random pure states
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• Typical Gaussian Quantum Information
• Gaussian work extraction from random Gaussian states is nearly impossible
• Universal quantum uncertainty relations: minimum-uncertainty wave packet depends on measure of spread