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Chinese Physics Letters

Methods for Derivation of Density Matrix of Arbitrary Multi-Mode Gaussian States from Its Phase Space Representation*

ShengLi Zhang, Song Yang

Published Sep 1, 2019• 1 citation
Publisher

Abstract

Abstract We present a method for derivation of the density matrix of an arbitrary multi-mode continuous variable Gaussian entangled state from its phase space representation. An explicit computer algorithm is given to reconstruct the density matrix from Gaussian covariance matrix and quadrature average values. As an example, we apply our method to the derivation of three-mode symmetric continuous variable entangled state. Our method can be used to analyze the entanglement and correlation in continuous variable quantum network with multi-mode quantum entanglement states.

DOI
10.1088/0256-307x/36/9/090301

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Abstract We present a method for derivation of the density matrix of an arbitrary multi-mode continuous variable Gaussian entangled state from its phase space representation. An explicit computer algorithm is given to reconstruct the density matrix from Gaussian covariance matrix and quadrature average values. As an example, we apply our method to the derivation of three-mode symmetric continuous variable entangled state. Our method can be used to analyze the entanglement and correlation in continuous variable quantum network with multi-mode quantum entanglement states.
We present a complete analysis of the multipartite entanglement of three-mode Gaussian states of continuous-variable systems. We derive standard forms which characterize the covariance matrix of pure and mixed three-mode Gaussian states up to local unitary operations, showing that the local entropies of pure Gaussian states are bound to fulfill a relationship which is stricter than the general Araki-Lieb inequality. Quantum correlations can be quantified by a proper convex roof extension of the squared logarithmic negativity, the continuous-variable tangle, or contangle. We review and elucidate in detail the proof that in multimode Gaussian states the contangle satisfies a monogamy inequality constraint [G. Adesso and F. Illuminati, New J. Phys8, 15 (2006)]. The residual contangle, emerging from the monogamy inequality, is an entanglement monotone under Gaussian local operations and classical communications and defines a measure of genuine tripartite entanglements. We determine the analytical expression of the residual contangle for arbitrary pure three-mode Gaussian states and study in detail the distribution of quantum correlations in such states. This analysis yields that pure, symmetric states allow for a promiscuous entanglement sharing, having both maximum tripartite entanglement and maximum couplewise entanglement between any pair of modes. We thus name these states GHZ∕W\mathrm{GHZ}∕WGHZ∕W states of continuous-variable systems because they are simultaneous continuous-variable counterparts of both the GHZ and the WWW states of three qubits. We finally consider the effect of decoherence on three-mode Gaussian states, studying the decay of the residual contangle. The GHZ∕W\mathrm{GHZ}∕WGHZ∕W states are shown to be maximally robust against losses and thermal noise.
Gaussian states are the backbone of quantum information protocols with continuous-variable systems whose power relies fundamentally on the entanglement between the different modes. In the case of global pure states, knowledge of the reduced states in a given bipartition of a multipartite quantum system bears information on the entanglement in such bipartition. For Gaussian states, the reduced states are also Gaussian, so their determination requires essentially the experimental determination of their covariance matrix. Here we develop strategies to determine the covariance matrix of an arbitrary nnn-mode bosonic Gaussian state through measurement of the total phase acquired when appropriate metaplectic evolutions, associated with quadratic Hamiltonians, are applied. Simply one-mode metaplectic evolutions, such rotations, squeezing, and shear transformations, in addition to a single two-mode rotation, allows us to determine all the covariance matrix elements of an nnn-mode bosonic system. All the single-mode metaplectic evolutions are applied conditionally to a state in which an ancilla qubit is entangled with the nnn-mode system. The ancillary system provides, after measurement, the value of the total phase of each evolution. The proposed method is experimentally suited to implement in the most currently used continuous-variable systems.
The entanglement of multi-mode Gaussian state in continuous variable system is investigated. In recent studies, the determination in entanglement of a general two-mode Gaussian state depend on purity measurement, and some studies have already extended it to the case of quasi-symmetry multi-mode Gaussian state. In this paper, entanglement is quantified by logarithmic negativity. The study focuses on the entanglement of quasi-symmetric eight-mode Gaussian state. Our aim is to find the analytic description of the correlation between the quasi-symmetric Gaussian state and two-mode Gaussian state. The results show that: the correlation is not unique. A two-mode Gaussian state may correspond to many different multi-mode Gaussian states, and these multi-mode Gaussian states have the same entanglement information.
Abstract We derive several expressions for the quantum Fisher information matrix (QFIM) for the multi-parameter estimation of multi-mode Gaussian quantum states, the corresponding symmetric logarithmic derivatives, and conditions for saturability of the quantum Cramér–Rao bound. This bound determines the ultimate precision with which parameters encoded into quantum states can be estimated. We include expressions for mixed states, for the case when the Williamson decomposition of the covariance matrix is known, expressions in terms of infinite series, and expressions for pure states. We also discuss problematic behavior when some modes are pure, and present a method that allows the use of expressions that are defined only for mixed states, to compute QFIM for states with any number of pure modes.