Steering is a manifestation of quantum correlations that embodies the Einstein–Podolsky–Rosen (EPR) paradox. While there have been recent attempts to quantify steering, continuous variable systems remained elusive. We introduce a steering measure for two-mode continuous variable systems that is valid for arbitrary states. The measure is based on the violation of an optimized variance test for the EPR paradox by quadrature measurements and admits a computable and experimentally friendly lower bound only depending on the second moments of the state, which reduces to a recently proposed quantifier of steerability by Gaussian measurements. We further show that Gaussian states are extremal with respect to our measure, minimizing it among all continuous variable states with fixed second moments. As a byproduct of our analysis, we generalize and relate well-known EPR-steering criteria. Finally an operational interpretation is provided, as the proposed measure is shown to quantify a guaranteed key rate in semi-device-independent quantum key distribution.
Steering is a manifestation of quantum correlations that embodies the Einstein–Podolsky–Rosen (EPR) paradox. While there have been recent attempts to quantify steering, continuous variable systems remained elusive. We introduce a steering measure for two-mode continuous variable systems that is valid for arbitrary states. The measure is based on the violation of an optimized variance test for the EPR paradox by quadrature measurements and admits a computable and experimentally friendly lower bound only depending on the second moments of the state, which reduces to a recently proposed quantifier of steerability by Gaussian measurements. We further show that Gaussian states are extremal with respect to our measure, minimizing it among all continuous variable states with fixed second moments. As a byproduct of our analysis, we generalize and relate well-known EPR-steering criteria. Finally an operational interpretation is provided, as the proposed measure is shown to quantify a guaranteed key rate in semi-device-independent quantum key distribution.
Steering is a manifestation of quantum correlations that embodies the Einstein-Podolsky-Rosen (EPR) paradox. While there have been recent attempts to quantify steering, continuous variable systems remained elusive. We in-troduce a steering measure for two-mode continuous variable systems that is valid for arbitrary states. The measure is based on the violation of an optimized variance test for the EPR paradox, and admits a computable and exper-imentally friendly lower bound only depending on the second moments of the state, which reduces to a recently proposed quantifier of steerability by Gaussian measurements. We further show that Gaussian states are extremal with respect to our measure, minimizing it among all continuous variable states with fixed second moments. As a byproduct of our analysis, we generalize and relate well-known EPR-steering criteria. Finally an operational inter-pretation is provided, as the proposed measure is shown to quantify the optimal guaranteed key rate in semi-device independent quantum key distribution. 1.
Einstein-Podolsky-Rosen steering incarnates a useful nonclassical correlation which sits between entanglement and Bell nonlocality. While a number of qualitative steering criteria exist, very little has been achieved for what concerns quantifying steerability. We introduce a computable measure of steering for arbitrary bipartite Gaussian states of continuous variable systems. For two-mode Gaussian states, the measure reduces to a form of coherent information, which is proven never to exceed entanglement, and to reduce to it on pure states. We provide an operational connection between our measure and the key rate in one-sided device-independent quantum key distribution. We further prove that Peres' conjecture holds in its stronger form within the fully Gaussian regime: namely, steering bound entangled Gaussian states by Gaussian measurements is impossible.
Einstein-Podolsky-Rosen (EPR) steering is an asymmetric form of correlations which is intermediate between quantum entanglement and Bell nonlocality, and can be exploited as a resource for quantum communication with one untrusted party. In particular, steering of continuous-variable Gaussian states has been extensively studied theoretically and experimentally, as a fundamental manifestation of the EPR paradox. While most of these studies focused on quadrature measurements for steering detection, two recent works revealed that there exist Gaussian states which are only steerable by suitable non-Gaussian measurements. In this paper we perform a systematic investigation of EPR steering of bipartite Gaussian states by pseudospin measurements, complementing and extending previous findings. We first derive the density-matrix elements of two-mode squeezed thermal Gaussian states in the Fock basis, which may be of independent interest. We then use such a representation to investigate steering of these states as detected by a simple nonlinear criterion, based on second moments of the correlation matrix constructed from pseudospin operators. This analysis reveals previously unexplored regimes where non-Gaussian measurements are shown to be more effective than Gaussian ones to witness steering of Gaussian states in the presence of local noise. We further consider an alternative set of pseudospin observables, whose expectation value can be expressed more compactly in terms of Wigner functions for all two-mode Gaussian states. However, according to the adopted criterion, these observables are found to be always less sensitive than conventional Gaussian observables for steering detection. Finally, we investigate continuous-variable Werner states, which are non-Gaussian mixtures of Gaussian states, and find that pseudospin measurements are always more effective than Gaussian ones to reveal their steerability. Our results provide useful insights on the role of non-Gaussian measurements in characterizing quantum correlations of Gaussian and non-Gaussian states of continuous-variable quantum systems.
Steering is a form of quantum nonlocality that is intimately related to the famous Einstein-Podolsky-Rosen (EPR) paradox that ignited the ongoing discussion of quantum correlations. Within the hierarchy of nonlocal correlations appearing in nature, EPR steering occupies an intermediate position between Bell nonlocality and entanglement. In continuous variable systems, EPR steering correlations have been observed by violation of Reid's EPR inequality, which is based on inferred variances of complementary observables. Here we propose and experimentally test a new criterion based on entropy functions, and show that it is more powerful than the variance inequality for identifying EPR steering. Using the entropic criterion our experimental results show EPR steering, while the variance criterion does not. Our results open up the possibility of observing this type of nonlocality in a wider variety of quantum states.
Abstract Einstein–Podolsky–Rosen (EPR) steering is an important resource for one-sided device-independent quantum information processing. This steering property can be destroyed by the interaction between a quantum system and its environment in practical applications. In this paper, we employ the characteristic function representation of probability distributions to investigate the quantum steering of two-mode continuous-variable states in a laser channel, where both the gain factor and the loss effect are taken into account. Firstly, we analyse the steering time of the two-mode squeezed vacuum state under one-mode and two-mode laser channels, respectively. We find that the gain process introduces additional noise into the two-mode squeezed vacuum state, thereby reducing the steerable time. Secondly, by quantifying EPR steering, we show that two-side loss exhibits smaller steerability than one-side loss, although they share the same two-way steerable time. In addition, we find that the more-gained party can steer the other party’s state, whereas the other party cannot steer the gained party beyond a certain threshold value. In this sense, the gain effect in one party appears to be equivalent to the loss effect in the other party. Our results pave the way for the distillation of EPR steering and quantum information processing in practical quantum channels.
Using Venn diagrams, we classify the different types of two-mode Gaussian continuous variable quantum correlation including directional entanglement and Einstein-Podolsky-Rosen (EPR) steering. We establish unified signatures for one- and two-way quantum steering, entanglement, and discord beyond entanglement in terms of an EPR-type variance. By focusing on Gaussian states, we link an optimized condition for entanglement based on an EPR variance to the Simon-Peres condition. This allows us to quantify the asymmetry of the Gaussian entanglement, and to relate the asymmetry to a directional quantum teleportation protocol where Alice and Bob possess asymmetrically noisy channels. Our analysis enables a determination of the type and direction of quantum correlation in a way that is easily measured in experiment. We also find that for symmetric states, when discord exceeds a certain threshold, the states are necessarily steerable.