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arXiv:1311.0509 · Physical Review A

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

G. Giedke, Barbara Kraus

Published Jan 30, 2014• 13 citations
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Abstract

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.

DOI
10.1103/physreva.89.012335

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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.
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.
Necessary and sufficient conditions for arbitrary multimode (pure or mixed) Gaussian states to be equivalent under incoherent Gaussian operations are derived. We show that two Gaussian states are incoherent equivalence if and only if they are related by incoherent unitaries. This builds the counterpart of the celebrated result that two pure entangled states are equivalent under local operations and classical communication (LOCC) if and only if they are related by local unitaries. Furthermore, incoherent equivalence of Gaussian states is equivalent to frozen coherence [Phys. Rev. Lett. 114, 210401 (2015)]. Basing this as foundation, we find all measures of coherence are frozen for an initial Gaussian state under strongly incoherent Gaussian operations if and only if the relative entropy measure of coherence is frozen for the state. This gives an entropy-based dynamical condition in which the coherence of an open quantum system is totally unaffected by noise.
We investigate the action of local unitary operations on multimode (pure or mixed) Gaussian states and single out the minimal number of locally invariant parameters which completely characterize the covariance matrix of such states. For pure Gaussian states, central resources for continuous-variable quantum information, we investigate separately the parameter 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 parameters 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. Finally, 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.
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.
Local operations assisted by classical communication (LOCC) play a fundamental role in the resource-theoretic formulation of entanglement theory inducing an operationally meaningful ordering in the set of entangled states. The particular class within LOCC of local unitary (LU) transformations is obviously closed under inversion. Thus, LU-equivalent states are interconvertible by LOCC and have therefore the same entanglement. For this reason LU-equivalence has been thoroughly studied in the context of entanglement theory. However, as I note here with a simple example, the converse is not true: there exist entangled states that are interconvertible by LOCC but which are not LU-equivalent. This motivates studying under which conditions LOCC interconvertibility is characterized by LU-equivalence. I provide a sufficient condition for this to hold. When one considers multipartite states with full entanglement dimensionality (as quantified by the Schmidt number vector) LOCC interconvertibility can only hold under LU-equivalence.