Gaussian local unitary equivalence of $n$-mode Gaussian states and Gaussian transformations by local operations with classical communication

Abstract

We derive necessary and sufficient conditions for arbitrary multi--mode (pure or mixed) Gaussian states to be equivalent under Gaussian local unitary operations. To do so, we introduce a standard form for Gaussian states, which has the properties that (i) every state can be transformed into its standard form via Gaussian local unitaries and (ii) it is unique and (iii) it can be easily computed. Thus, two states are equivalent under Gaussian local unitaries iff their standard form coincides. We explicitly derive the standard form for two-- and three--mode Gaussian pure states. We then investigate transformations between these classes by means of Gaussian local operations assisted by classical communication. For three-mode pure states, we identify a global property that cannot be created but only destroyed by local operations. This implies that the highly entangled family of symmetric three--mode Gaussian states is not sufficient to generated all three-mode Gaussian states by local Gaussian operations.

16+pages; published version

References in corpus (9)

• Ultra-Large-Scale Continuous-Variable Cluster States Multiplexed in the Time Domain
• Experimental generation of four-mode continuous-variable cluster states
• Continuous variable tangle, monogamy inequality, and entanglement sharing in Gaussian states of continuous variable systems
• The maximally entangled set of multipartite quantum states
• Gaussian localizable entanglement
• Generic Entanglement and Standard Form for N-mode Pure Gaussian States
• Optical state engineering, quantum communication, and robustness of entanglement promiscuity in three-mode Gaussian states
• Not-so-normal mode decomposition
• Standard forms and entanglement engineering of multimode Gaussian states under local operations

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• Are all Gaussian states also cluster states? Essential diagnostic tools for continuous-variable one-way quantum computing
• Interconversion of pure Gaussian states using non-Gaussian operations
• Einstein-Podolsky-Rosen uncertainty limits for bipartite multimode states
• Incoherent Gaussian equivalence of $m-$mode Gaussian states
• Graphical Framework for Non-Gaussian Quantum States
• Transformations of symmetric multipartite Gaussian states by Gaussian LOCC