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Physical Review A

Conditions for the local manipulation of all bipartite Gaussian states

Liang Wang, Shu‐Shen Li, Fuhua Yang, Zhichuan Niu, Songlin Feng, Houzhi Zheng

Published Aug 28, 2003• 5 citations
Publisher

Abstract

We provide a general, necessary, and sufficient condition for the possibility of transforming a mixed bipartite Gaussian state with arbitrarily many modes to another one under arbitrary local Gaussian channels, which do not include classical communication. Moreover, by means of this condition we present a necessary criterion that can be used to check the possibility of a state transformation between two mixed Gaussian states. At the same time, we prove that our criterion can be reduced to the Eisert-Plenio criterion when the mode number is chosen as 1 per side.

DOI
10.1103/physreva.68.020302

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We provide a general, necessary, and sufficient condition for the possibility of transforming a mixed bipartite Gaussian state with arbitrarily many modes to another one under arbitrary local Gaussian channels, which do not include classical communication. Moreover, by means of this condition we present a necessary criterion that can be used to check the possibility of a state transformation between two mixed Gaussian states. At the same time, we prove that our criterion can be reduced to the Eisert-Plenio criterion when the mode number is chosen as 1 per side.
We analyze the conditions under which local operations and classical communication enable entanglement transformations within the set of bipartite pure Gaussian states. A set of necessary and sufficient conditions had been found in [Quant. Inf. Comp. 3, 211 (2003)] for the interconversion between such states that is restricted to Gaussian local operations and classical communication. Here, we exploit majorization theory in order to derive more general (sufficient) conditions for the interconversion between bipartite pure Gaussian states that goes beyond Gaussian local operations. While our technique is applicable to an arbitrary number of modes for each party, it allows us to exhibit surprisingly simple examples of 2 x 2 Gaussian states that necessarily require non-Gaussian local operations to be transformed into each other.
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We derive necessary and sufficient conditions for arbitrary multimode (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 if and only if their standard forms coincide. 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 generate all three-mode Gaussian states by local Gaussian operations.