Three-mode Gaussian States Research Articles

Monogamy inequality and entanglement sharing in optomechanics

In this paper, we quantitatively study the dynamics of the tripartite entanglement of the three-mode Gaussian state in a cavity-free optomechanical system. After deriving the explicit form of the evolved state matrix covariance, we elucidate in detail the evidence of entanglement monogamy in different mode partitions, based on Coofman–Kundu–Wotters inequalities, using the Gaussian Contangle quantifying bipartite entanglement. Next, we focus on the genuine tripartite entanglement quantified by means of the Residual Gaussian Contangle.

Quantum illumination with three-mode Gaussian state

The quantum illumination is examined by making use of the three-mode maximally entangled Gaussian state, which involves one signal and two idler beams. It is shown that the quantum Bhattacharyya bound between rho (state for target absence) and sigma (state for target presence) is less than the previous result derived by two-mode Gaussian state when N_S, average photon number per signal, is less than 0.295. This indicates that the quantum illumination with three-mode Gaussian state gives less error probability compared to that with two-mode Gaussian state when N_S < 0.295.

Bipartite and tripartite entanglement in a Bose-Einstein acoustic black hole

We investigate quantum entanglement in an analog black hole realized in the flow of a Bose-Einstein condensate. The system is described by a three-mode Gaussian state and we construct the corresponding covariance matrix at zero and finite temperature. We study associated bipartite and tripartite entanglement measures and discuss their experimental observation. We identify a simple optical setup equivalent to the analog Bose-Einstein black hole which suggests a different way of determining the Hawking temperature and gray-body factor of the system.

Tripartite Einstein-Podolsky-Rosen steering with linear and nonlinear beamsplitters in four-wave mixing of Rubidium atoms.

Multipartite Einstein-Podolsky-Rosen (EPR) steering is an essential resource for secure one-sided device-independent quantum secret sharing. Here, we analyze the EPR steering properties exhibited in three-mode Gaussian states created by four-wave mixing (FWM) in Rubidium atoms combined with a linear beamsplitter and a nonlinear beamsplitter (second FWM), respectively. By quantifying Gaussian steerability based on a measure determined by the covariance matrix of the produced states, we compare the performance of two schemes to achieve one-way, collective, and genuine tripartite steering, as well as the monogamy constraints for distributing steering among three parties. We show that the scheme with nonlinear beamsplitter is feasible to create stronger bipartite steering and genuine tripartite steering and has more flexibility to manipulate the monogamy relation through the cooperation of the two cascaded FWM processes.

Multipartite Gaussian steering: Monogamy constraints and quantum cryptography applications

We derive laws for the distribution of quantum steering among different parties in multipartite Gaussian states under Gaussian measurements. We prove that a monogamy relation akin to the generalized Coffman-Kundu-Wootters inequality holds quantitatively for a recently introduced measure of Gaussian steering. We then define the residual Gaussian steering, stemming from the monogamy inequality, as an indicator of collective steering-type correlations. For pure three-mode Gaussian states, the residual acts a quantifier of genuine multipartite steering, and is interpreted operationally in terms of the guaranteed key rate in the task of secure quantum secret sharing. Optimal resource states for the latter protocol are identified, and their possible experimental implementation discussed. Our results pin down the role of multipartite steering for quantum communication.

Nonclassicality Invariant of General Two-Mode Gaussian States.

We introduce a new quantity for describing nonclassicality of an arbitrary optical two-mode Gaussian state which remains invariant under any global photon-number preserving unitary transformation of the covariance matrix of the state. The invariant naturally splits into an entanglement monotone and local-nonclassicality quantifiers applied to the reduced states. This shows how entanglement can be converted into local squeezing and vice versa. Twin beams and their transformations at a beam splitter are analyzed as an example providing squeezed light. An extension of this approach to pure three-mode Gaussian states is given.

Fock expansion of multimode pure Gaussian states

The Fock expansion of multimode pure Gaussian states is derived starting from their representation as displaced and squeezed multimode vacuum states. The approach is new and appears to be simpler and more general than previous ones starting from the phase-space representation given by the characteristic or Wigner function. Fock expansion is performed in terms of easily evaluable two-variable Hermite–Kampé de Fériet polynomials. A relatively simple and compact expression for the joint statistical distribution of the photon numbers in the different modes is obtained. In particular, this result enables one to give a simple characterization of separable and entangled states, as shown for two-mode and three-mode Gaussian states.

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

We derive necessary and sufficient conditions for arbitrary multi--mode (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 iff their standard form coincides. 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 generated all three-mode Gaussian states by local Gaussian operations.

Theory of Genuine Tripartite Nonlocality of Gaussian States

We investigate the genuine multipartite nonlocality of three-mode Gaussian states of continuous variable systems. For pure states, we present a simplified procedure to obtain the maximum violation of the Svetlichny inequality based on displaced parity measurements, and we analyze its interplay with genuine tripartite entanglement measured via Rényi-2 entropy. The maximum Svetlichny violation admits tight upper and lower bounds at fixed tripartite entanglement. For mixed states, no violation is possible when the purity falls below 0.86. We also explore a set of recently derived weaker inequalities for three-way nonlocality, finding violations for all tested pure states. Our results provide a strong signature for the nonclassical and nonlocal nature of Gaussian states despite their positive Wigner function, and lead to precise recipes for its experimental verification.

Exact solution and short-time dynamics of multimode Gaussian states embedded in a common non-Markovian environment

We investigate a system of N mutually coupled harmonic oscillators interacting with the same environment and derive the corresponding exact non-Markovian master equation using our introduced approach. We obtain an explicit formula for the covariance matrix of the evolved state by taking a Gaussian state as initial one. With this, we analyze the short-time non-Markovian dynamics of three-mode Gaussian state in high-temperature limit. Our results show that the short-time evolution behavior of bipartite entanglement for 1 × 2 bipartition is similar to that of genuine tripartite entanglement, while the 1 × 1 bipartition entanglement does not. It is also shown that there exists a threshold that makes the initial tri- and bipartite entanglement increase or decrease. For the squeezing degree of the initial state than this critical value, the entanglement is increased with the evolution time; otherwise it is decreased. Finally, we present a physical positivity criterion for the covariance matrix of the evolved state.

Genuine Tripartite Entanglement and Nonlocality in Bose-Einstein Condensates by Collective Atomic Recoil

We study a system represented by a Bose-Einstein condensate interacting with a cavity field in presence of a strong off-resonant pumping laser. This system can be described by a three-mode Gaussian state, where two are the atomic modes corresponding to atoms populating upper and lower momentum sidebands and the third mode describes the scattered cavity field light. We show that, as a consequence of the collective atomic recoil instability, these modes possess a genuine tripartite entanglement that increases unboundedly with the evolution time and is larger than the bipartite entanglement in any reduced two-mode bipartition. We further show that the state of the system exhibits genuine tripartite nonlocality, which can be revealed by a robust violation of the Svetlichny inequality when performing displaced parity measurements. Our exact results are obtained by exploiting the powerful machinery of phase-space informational measures for Gaussian states, which we briefly review in the opening sections of the paper.

Three-step approach to Fokker-Planck equation for an interacting open waveguide system and localizable entanglement dynamics

We consider a system of three coupled single-mode waveguides each locally interacting with its own Gaussian environment and present a general solution for this coupled system initially in any Gaussian state using the symplectic operations. We investigate the dynamics of two-mode localizable entanglement contained in the evolved state when the system is initially in three-mode bisymmetric Gaussian state in contact with the independent decoherence. We show that such an entanglement exhibits a damped oscillation in a regime of weak waveguide-waveguide coupling and small mean photon numbers of the bath. Remarkably, we find that the entanglement can reappear after the long-time death and arrives at a steady-state oscillation, whose maximum depends strongly on both the squeezing of the bath and the coupling strength between these waveguides. Finally, we generalize the approach to a common squeezed environment case.

Entanglement Swapping with Local Certification: Application to Remote Micromechanical Resonators

We propose a protocol for entanglement swapping which involves tripartite systems. The generation of remote entanglement induced by the Bell measurement can be easily certified by additional local measurements. We illustrate the protocol in the case of continuous variable systems where the certification is effective for an appropriate class of three-mode gaussian states. We then apply the protocol to optomechanical systems, showing how mechanical entanglement between two remote micromechanical resonators can be generated and certified via local optical measurements.

Dynamical Gaussian state transfer with quantum-error-correcting architecture

Transferring a quantum state of a light field to a memory is of particular importance. However, this transfer is usually hampered because the memory system is subjected to some noise and this can limit the performance of the state transfer greatly. In this paper, we consider the transfer of a Gaussian state of light to a linear medium memory such as an optomechanical oscillator and propose a dynamical feedback controller that suppresses the noise in the memory system. To protect an unknown state, the feedback scheme employs the specific configuration of the quantum error correction; that is, a three-mode Gaussian state having appropriate syndromes is taken as the input. Correspondingly, the memory consists of three independent linear systems. The syndrome errors are estimated continuously in time through the measurement of the output field, and the results are then fed back to control the system. Because the input is Gaussian and the systems are all linear, it is possible to formulate the problem using the framework of the celebrated classical Kalman filtering and linear quadratic Gaussian control. A numerical simulation demonstrates the effectiveness of the control scheme.

Entanglement distribution using three-mode separable state with imperfect equipments

We show a scheme to distribute the entanglement by using three-mode separable Gaussian state prepared with imperfect equipments. The scheme achieves the aim that the entanglement is distributed between two distant parties with only Gaussian operations and linear optics elements. Moreover, we analyse the logarithmic negativity of the entanglement shared between the two parties when the systems are imperfect and arrive at the conclusion that the logarithmic negativity is asymptotically stable with fluctuations within a certain space range.

Quantum versus Classical Correlations in Gaussian States

Quantum discord, a measure of genuinely quantum correlations, is generalized to continuous variable systems. For all two-mode Gaussian states, we calculate analytically the quantum discord and a related measure of classical correlations, solving an optimization over all Gaussian measurements. Almost all two-mode Gaussian states are shown to have quantum correlations, while for separable states, the discord is smaller than unity. For a given amount of entanglement, it admits tight upper and lower bounds. Via a duality between entanglement and classical correlations, we derive a closed formula for the Gaussian entanglement of formation of a family of three-mode Gaussian states.

Calculation of Covariance Matrix for Multi-mode Gaussian States in Decoherence Processes

We investigate the dynamics of n single-mode continuous variable systems in a generic Gaussian state under the influence of the independent and correlated noises making use of the characteristic function method. In two models the bath is assumed to be a squeezed thermal one. We derive an explicit input-output expression between the initial and final covariance matrices. As an example, we study the evolution of entanglement of three-mode Gaussian state embedded in two noisy models.

Enhancement of three-mode Gaussian state light field nonlocality

Here we propose a scheme for enhancing the nonlocality of three-mode Gaussian state light fields. It is found that three-mode Gaussian State can become non-Gaussian by the non-Gaussian operation of an IPS process, It is vertified that nonlocality of Gaussian state can be enhanced by Bell inquality.

Mixed-state localizable entanglement for continuous variables

We investigate localization of entanglement of multipartite mixed Gaussian states into a pair of modes by local Gaussian measurements on the remaining modes and classical communication. We provide a detailed proof that for arbitrary symmetric Gaussian state maximum entanglement can be localized by homodyne detection of either amplitude or phase quadrature on each mode. We then consider arbitrary mixed three-mode Gaussian states and show that the optimal Gaussian measurement on one mode yielding maximum entanglement among the other two modes can be determined by calculating roots of a high-order polynomial. Finally, we discuss localization of entanglement with single-photon detection.

The separability of tripartite Gaussian states with amplification and amplitude damping

The most general evolution solution of multi-mode continuous variable states is given when it undergoes amplification, amplitude damping and thermal noise. In the case of some kind of initial Gaussian state totally symmetrically interacting with the environment, the conditions for full separability and full entanglement of the final tripartite three-mode Gaussian state are worked out.