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

TL;DR

This paper characterizes the minimal locally invariant parameters of multimode Gaussian states, analyzes their entanglement properties, and proposes an efficient scheme for engineering such entangled states in quantum information processing.

Contribution

It identifies the minimal set of parameters for pure multimode Gaussian states and introduces a practical scheme for their entanglement-based state engineering.

Findings

Pure Gaussian states with n<=3 modes can have all position-momentum correlations eliminated by local unitaries.

The minimal parameter set fully describes all entanglement forms in these states.

Block-diagonal Gaussian states are generally less entangled than arbitrary pure Gaussian states.

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.