We study the entanglement properties of two-mode Gaussian light emerging from a generic SU (1,1) interferometer. Our tool is the two-mode characteristic function which is determined by the 4 × 4 covariance matrix. For an initial product of two mixed single-mode Gaussian states we investigate the output two-mode covariance matrix. Its structure displays the noise properties of the reduced states as well as the correlations between modes. Classicality of the output two-mode state is characterized by the existence of the Glauber-Sudarshan P representation. For testing separability we apply the Peres-Simon criterion requiring preservation of the positivity of the density matrix under partial transposition. Since inseparability entails nonclassicality, the threshold gain above which nonclassicality of the output state becomes manifest is lower than that allowing for its inseparability. We find that only for a thermal input do nonclassicality and inseparability of the output have the same threshold gain.
We study the entanglement properties of two-mode Gaussian light emerging from a generic SU (1,1) interferometer. Our tool is the two-mode characteristic function which is determined by the 4 × 4 covariance matrix. For an initial product of two mixed single-mode Gaussian states we investigate the output two-mode covariance matrix. Its structure displays the noise properties of the reduced states as well as the correlations between modes. Classicality of the output two-mode state is characterized by the existence of the Glauber-Sudarshan P representation. For testing separability we apply the Peres-Simon criterion requiring preservation of the positivity of the density matrix under partial transposition. Since inseparability entails nonclassicality, the threshold gain above which nonclassicality of the output state becomes manifest is lower than that allowing for its inseparability. We find that only for a thermal input do nonclassicality and inseparability of the output have the same threshold gain.
Nonclassicality of a single-mode field is equivalent to the rank of two-mode entanglement this field generates at the output of a beam-splitter [Phys. Rev. A 89, 052302 (2014)]. We derive a measure for the nonclassicality of a single-mode field using the two-mode inseparability measure of Simon-Peres-Horodecki. Degree of the single-mode nonclassicality requires the knowledge of only ⟨a^2⟩ and ⟨a^†a^⟩. This condition/measure is both a necessary and sufficient condition for Gaussian single-mode states. We show that this measure increases with squeezing strength and display a jump at the critical coupling for a superradiant phase transition. Additionally, we derive a simple analytical condition, ∣⟨a^2⟩∣>⟨a^†a^⟩, for nonclassicality from Duan-Giedke-Cirac-Zoller criterion. In a forthcoming study, we use the derived measure to show the following. Non-causal linear optical response of an optomechanical cavity emerges at the same critical cavity-mechanical coupling where output field becomes nonclassical.
Entanglement detection and quantification usually require the knowledge of correlations and thus is a global issue. For both theoretical and practical reasons, it is desirable to characterize entanglement via local quantities. Of course, this can only be achieved in special setups with some underlying structure encoding the correlations. For the often encountered bilinear interactions such as beamsplitter experiments, we propose an operational and friendly entanglement criterion for the two-mode Gaussian output states. Specifically, we show that in order to detect and evaluate the entanglement of the output states of a bilinear interaction involving uncorrelated Gaussian input states, we do not need to know the joint covariance matrix of the whole two-mode output states. Instead, we only need the respective covariances of the reduced states of the two input modes and that of the two output modes, which are all local quantities pertaining to a single mode and can be easily measured locally without explicitly considering the correlations between the modes. This is achieved by exploiting the input-output characteristics of bilinear interactions. The result recasts the celebrated entanglement criteria of Simon ( Phys. Rev. Lett. , 84 (2000) 2726) and Duan et al . ( Phys. Rev. Lett. , 84 (2000) 2722) into a more convenient and simple form.
Abstract Generating and controlling continuous-variable entanglement is a central requirement for quantum information processing protocols. In the present contribution, we investigate the bipartite Gaussian entanglement of the two-mode Gaussian state produced by a parametric converter, with the two cavity modes initially prepared as arbitrary, non-classical single-mode Gaussian states of tunable purity and non-classicality. The entanglement is witnessed using the Simon's separability criterion, which is necessary and sufficient for Gaussian states, along with the Duan's inseparability criterion. We solve its dynamics as functions of the coherent pump-field phase, the cavity decay rates, and the non-classicality and purity of the initial modes. The entanglement evolves as a damped oscillation and emerges only once there is a difference between the non-classicalities of the two input states, becoming stronger as this difference grows. Likewise, the phase of the pump-field controls both the strength and time duration of entanglement. Higher input purity moderately strengthens and prolongs the correlations, whereas the cavity decay rates are the main source of decoherence. Finally, we compare the two criteria and find that the Simon's condition shows nearly continuous entanglement, whereas Duan's criterion responds only where the correlations are strong, leaving transient windows in which the state is separable under Duan yet still inseparable under Simon. These theoretical results suggest that, the non-classicality of the input field and the phase of the pump field can serve as effective control parameters for achieving tunable bipartite Gaussian entanglement, offering guidance for future practical realizations in cavity-based systems.
Quantum universal invariants of a Gaussian state's covariance matrix, which can be derived from intensity correlation moments, have been adopted to characterize the entanglement properties of Gaussian states via the positive partial transpose criterion, also known as the Peres-Horodecki separability criterion. Such intensity correlation moments enable the extraction of information about the covariance matrix without the need for a coherent local oscillator. Here, we experimentally detect the entanglement properties of multimode Gaussian states using high-order\,(up to sixth-order) intensity correlation moments. These multimode Gaussian states are prepared via spontaneous and cascaded parametric down-conversion pumped by a high-peak-energy pulsed laser. Their intensity correlation moments are measured using a pseudo-photon-number-resolving detector constructed through spatial multiplexing of 32 threshold superconducting nanowire single photon detectors. This method is successfully demonstrated for two-mode and three-mode Gaussian states and can be extended to N-mode Gaussian states with N>3.
In the framework of the theory of open systems based on completely positive quantum dynamical semigroups, we present a description of continuous-variable entanglement for a system consisting of two non-interacting modes embedded in a thermal environment. By using the Peres–Simon necessary and sufficient criterion for the separability of two-mode Gaussian states, we describe the evolution of entanglement in terms of the covariance matrix for Gaussian input states. For all values of the temperature of the thermal reservoir, an initial separable Gaussian state remains separable for all times. In the case of an entangled initial Gaussian state, entanglement suppression (entanglement sudden death) takes place for non-zero temperatures of the environment. Only for zero temperature of the thermal bath, the initial entangled state remains entangled for finite times. We also show that, independent of its type, namely separable or entangled, the initial state evolves asymptotically to an equilibrium state that is always separable.
In Quantum Optics, the widely accepted definition of nonclassicality is based on the P function of Glauber and Sudarshan. When it fails to be interpreted as a classical probability density, the corresponding quantum state is said to be a nonclassical one. Here we present the first reconstruction of a nonclassical P function of a single-photon added thermal state. We also consider the nonclassical properties of general spacegtime dependent correlation functions of radiation fields. For the detection of these correlation functions, a balanced homodyne correlation technique was proposed. It is shown that the measurable correlation functions also allow one to completely characterize bipartite entangled quantum states with a negative partial transposition. Finally, we present a method for identifying general bipartite entanglement for continuous variables.