Standard forms and entanglement engineering of multimode Gaussian states under local operations

Abstract

We investigate the action of local unitary operations on multimode (pure or mixed) Gaussian states and single out the minimal number of locally invariant parametres which completely characterise the covariance matrix of such states. For pure Gaussian states, central resources for continuous-variable quantum information, we investigate separately the parametre reduction due to the additional constraint of global purity, and the one following by the local-unitary freedom. Counting arguments and insights from the phase-space Schmidt decomposition and in general from the framework of symplectic analysis, accompany our description of the standard form of pure n-mode Gaussian states. In particular we clarify why only in pure states with n<=3 modes all the direct correlations between position and momentum operators can be set to zero by local unitary operations. For any n, the emerging minimal set of parametres contains complete information about all forms of entanglement in the corresponding states. An efficient state engineering scheme (able to encode direct correlations between position and momentum operators as well) is proposed to produce entangled multimode Gaussian resources, its number of optical elements matching the minimal number of locally invariant degrees of freedom of general pure n-mode Gaussian states. We demonstrate that so-called "block-diagonal" Gaussian states, without direct correlations between position and momentum, are systematically less entangled, on average, than arbitrary pure Gaussian states.

14 pages, 2 figures, IOP style. Published in J. Phys. A, Special Issue on Quantum Information, Communication, Computation and Cryptography (the arXiv version has an extra note added)

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• Invariant measures on multimode quantum Gaussian states
• Local softening of information geometric indicators of chaos in statistical modeling in the presence of quantum-like considerations
• Unified framework to determine Gaussian states in continuous variable systems
• Einstein-Podolsky-Rosen uncertainty limits for bipartite multimode states
• Rotating quantum Gaussian packets
• Inseparability Criterion Using Higher-Order Schrödinger-Robertson Uncertainty Relation
• Incoherent Gaussian equivalence of $m-$mode Gaussian states
• Entanglement partners and monogamy in de Sitter universes
• Gaussian work extraction from random Gaussian states is nearly impossible