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arXiv:0807.1362 · Physical Review A

Complete state reconstruction of a two-mode Gaussian state via local operations and classical communication

Gustavo Rigolin, M. C. de Oliveira

Published Mar 11, 2009• 3 citations
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Abstract

We propose a strictly local protocol that is completely equivalent to global quantum state reconstruction for a bipartite system. We show that the joint density matrix of an arbitrary two-mode Gaussian state, entangled or not, is obtained via local operations and classical communication only. In contrast to previous proposals, simultaneous homodyne measurements on both modes are replaced with local homodyne detections and a set of local projective measurements.

DOI
10.1103/physreva.79.030302

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We propose a strictly local protocol that is completely equivalent to global quantum state reconstruction for a bipartite system. We show that the joint density matrix of an arbitrary two-mode Gaussian state, entangled or not, is obtained via local operations and classical communication only. In contrast to previous proposals, simultaneous homodyne measurements on both modes are replaced with local homodyne detections and a set of local projective measurements.
We develop the minimal requirements for the complete entanglement quantification of an arbitrary two-mode bipartite Gaussian state via local measurements and a classical communication channel. The minimal set of measurements is presented as a reconstruction protocol of local covariance matrices and no previous knowledge of the state is required but its Gaussian character. The protocol becomes very simple mostly when dealing with Gaussian states transformed to its standard form, since photocounting or intensity measurements define the whole set of entangled states. In addition, conditional on some prior information, the protocol is also useful for a complete global state reconstruction.
We show that no distillation protocol for Gaussian quantum states exists that relies on (i) arbitrary local unitary operations that preserve the Gaussian character of the state and (ii) homodyne detection together with classical communication and postprocessing by means of local Gaussian unitary operations on two symmetric identically prepared copies. This is in contrast to the finite-dimensional case, where entanglement can be distilled in an iterative protocol using two copies at a time. The ramifications for the distribution of Gaussian states over large distances will be outlined. We also comment on the generality of the approach and sketch the most general form of a Gaussian local operation with classical communication in a bipartite setting.
We present a theory of entanglement transformations of Gaussian pure states with local Gaussian operations and classical communication. This is the experimentally accessible set of operations that can be realized with optical elements such as beam splitters, phase shifts and squeezers, together with homodyne measurements. We provide a simple necessary and sufficient condition for the possibility to transform a pure bipartite Gaussian state into another one. We contrast our criterion with what is possible if general local operations are available.
We will show that no distillation protocol for Gaussian quantum states exists that relies on (i) arbitrary local unitary operations that preserve the Gaussian character of the state and (ii) homodyne detection together with classical communication and postprocessing by means of local Gaussian unitary operations on two symmetric identically prepared copies. This analysis shows that unlike the finite-dimensional case, where entanglement can be distilled in an iterative protocol using two copies at a time, there is no such procedure in the case of continuous variables for Gaussian initial states and the above Gaussian operations. The ramifications for the distribution of Gaussian states over large distances will be outlined. We will also comment on the generality of the approach and sketch the most general form of a Gaussian local operation with classical communication in a bi-partite setting.
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.