Bipartite entangled Gaussian states provide an important resource for quantum information based on continuous variables. Perfect complementarity between one-particle visibility and two-particle visibility in discrete systems, has been previously generalized to an inequality, rather than equality, for Gaussian states characterized by varying amounts of squeezing and entanglement. The latter quantum inequality, however, appears to be due to assessment of two-particle visibility using an indirect method that first corrects the two-particle probability distribution by adding and subtracting distributions with varying degree of entanglement. In this work, we develop a new direct method for quantifying two-particle visibility based on measurement of a pair of two-particle observables that commute with the measured pair of single-particle observables. This direct method treats the two pairs of quantum observables on equal footing by formally identifying all four observable distributions as Radon transforms of the original two-particle probability distribution. The quantum complementarity relation between one-particle visibility and two-particle visibility obtained through the direct method deviates from unity by an exponentially small quantity bounded by the amount of squeezing, which is shown to vanish in the limit of infinite squeezing when the entangled Gaussian state approaches an ideal Einstein-Podolsky-Rosen state. The presented results demonstrate the theoretical utility of Radon transforms for elucidating the nature of two-particle visibility and provide new tools for the development of quantum applications employing continuous variables.
Bipartite entangled Gaussian states provide an important resource for quantum information based on continuous variables. Perfect complementarity between one-particle visibility and two-particle visibility in discrete systems, has been previously generalized to an inequality, rather than equality, for Gaussian states characterized by varying amounts of squeezing and entanglement. The latter quantum inequality, however, appears to be due to assessment of two-particle visibility using an indirect method that first corrects the two-particle probability distribution by adding and subtracting distributions with varying degree of entanglement. In this work, we develop a new direct method for quantifying two-particle visibility based on measurement of a pair of two-particle observables that commute with the measured pair of single-particle observables. This direct method treats the two pairs of quantum observables on equal footing by formally identifying all four observable distributions as Radon transforms of the original two-particle probability distribution. The quantum complementarity relation between one-particle visibility and two-particle visibility obtained through the direct method deviates from unity by an exponentially small quantity bounded by the amount of squeezing, which is shown to vanish in the limit of infinite squeezing when the entangled Gaussian state approaches an ideal Einstein-Podolsky-Rosen state. The presented results demonstrate the theoretical utility of Radon transforms for elucidating the nature of two-particle visibility and provide new tools for the development of quantum applications employing continuous variables.
Complementarity between one-particle visibility and two-particle visibility in discrete systems can be extended to bipartite quantum-entangled Gaussian states implemented with continuous-variable quantum optics. The meaning of the two-particle visibility originally defined by Jaeger, Horne, Shimony, and Vaidman with the use of an indirect method that first corrects the two-particle probability distribution by adding and subtracting other distributions with varying degree of entanglement, however, deserves further analysis. Furthermore, the origin of complementarity between one-particle visibility and two-particle visibility is somewhat elusive and it is not entirely clear what is the best way to associate particular two-particle quantum observables with the two-particle visibility. Here, we develop a direct method for quantifying the two-particle visibility based on measurement of a pair of two-particle observables that are compatible with the measured pair of single-particle observables. For each of the two-particle observables from the pair is computed corresponding visibility, after which the absolute difference of the latter pair of visibilities is considered as a redefinition of the two-particle visibility. Our approach reveals an underlying mathematical symmetry as it treats the two pairs of one-particle or two-particle observables on equal footing by formally identifying all four observable distributions as rotated marginal distributions of the original two-particle probability distribution. The complementarity relation between one-particle visibility and two-particle visibility obtained with the direct method is exact in the limit of infinite Gaussian precision where the entangled Gaussian state approaches an ideal Einstein-Podolsky-Rosen state. The presented results demonstrate the theoretical utility of rotated marginal distributions for elucidating the nature of two-particle visibility and provide tools for the development of quantum applications employing continuous variables.
In this somewhat pedagogical paper we revisit complementarity relations in bipartite quantum systems. Focusing on continuous-variable systems, we examine the influential class of EPR-like states through a generalization to Gaussian states and present some new quantitative relations between entanglement and local interference within symmetric and asymmetric double-double-slit scenarios. This approach is then related to ancilla-based quantum measurements, and weak measurements in particular. Finally, we tie up the notions of distinguishability, predictability, coherence and visibility while drawing some specific connections between them.
Quantum squeezing, a major resource for quantum information processing and quantum metrology, is best analyzed in terms of the field quadratures - the quantum optical analogues of position and momentum, which form the continuous-variable formalism of quantum light. Degenerate squeezing admits a very helpful and simple description in terms of the single-mode quadrature operators, but the non-degenerate case, i.e. when the squeezing involves pairs of modes, requires a more complicated treatment involving correlations between the quadratures of the different modes. We introduce a generalized set of complex quadrature operators that treats degenerate and non-degenerate squeezing on equal footing. We describe the mode-pairs (and photon-pairs) as a single entity, generalizing the concept of single-mode quadrature operators to two-mode fields of any bandwidth. These complex operators completely describe the SU(1,1) algebra of two-photon devices and directly relate to observable physical quantities, like power and visibility. Based on this formalism, we discuss the measurement of optically-broad squeezed signals with direct detection, and present a compact set of phase-dependent observables that completely and intuitively determine the two-mode squeezed state, and quantify the degree of inseparability and entanglement between the modes.
In this chapter we review the characterization of entanglement in Gaussian states of continuous variable systems. For two-mode Gaussian states, we discuss how their bipartite entanglement can be accurately quantified in terms of the global and local amounts of mixedness, and efficiently estimated by direct measurements of the associated purities. For multimode Gaussian states endowed with local symmetry with respect to a given bipartition, we show how the multimode block entanglement can be completely and reversibly localized onto a single pair of modes by local, unitary operations. We then analyze the distribution of entanglement among multiple parties in multimode Gaussian states. We introduce the continuous-variable tangle to quantify entanglement sharing in Gaussian states and we prove that it satisfies the Coffman-Kundu-Wootters monogamy inequality. Nevertheless, we show that pure, symmetric three-mode Gaussian states, at variance with their discrete-variable counterparts, allow a promiscuous sharing of quantum correlations, exhibiting both maximum tripartite residual entanglement and maximum couplewise entanglement between any pair of modes. Finally, we investigate the connection between multipartite entanglement and the optimal fidelity in a continuous-variable quantum teleportation network. We show how the fidelity can be maximized in terms of the best preparation of the shared entangled resources and, viceversa, that this optimal fidelity provides a clearcut operational interpretation of several measures of bipartite and multipartite entanglement, including the entanglement of formation, the localizable entanglement, and the continuous-variable tangle.
Based on total variance of a pair of Einstein–Podolsky–Rosen (EPR) type operators, the generalized EPR entangled states in continuous variable systems are defined. We show that such entangled states must correspond to two-mode squeezing states whether these states are Gaussian or not and whether they are pure or not. With help of the relation between the total variance and the entanglement, the degree of such entanglement is also defined. Through analysing some specific cases, we see that this method is very convenient and easy in practical applications. In addition, an entangled state with no squeezing is studied, which reveals that there certainly exists something unknown about entanglement in continuous variable systems.
Quantum entanglement of pure states of a bipartite system is defined as the amount of local or marginal (i.e. referring to the subsystems) entropy. For mixed states this identification vanishes, since the global loss of information about the state makes it impossible to distinguish between quantum and classical correlations. Here we show how the joint knowledge of the global and marginal degrees of information of a quantum state, quantified by the purities or, in general, by information entropies, provides an accurate characterization of its entanglement. In particular, for Gaussian states of continuous variable systems, we classify the entanglement of two-mode states according to their degree of total and partial mixedness, comparing the different roles played by the purity and the generalized p-entropies in quantifying the mixedness and bounding the entanglement. We prove the existence of strict upper and lower bounds on the entanglement and the existence of extremally (maximally and minimally) entangled states at fixed global and marginal degrees of information. This results allow for a powerful, operative method to measure mixed-state entanglement without the full tomographic reconstruction of the state. Finally, we briefly discuss the ongoing extension of our analysis to the quantification of multipartite entanglement in highly symmetric Gaussian states of arbitrary 1 × N-mode partitions.