Asymmetric steering is an effect whereby an inseparable bipartite system can be found to be described by either quantum mechanics or local hidden variable theories depending on which one of Alice or Bob makes the required measurements. We show that, even with an inseparable bipartite system, situations can arise where Gaussian measurements on one half are not sufficient to answer the fundamental question of which theory gives an adequate description and the whole system must be considered. This phenomenon is possible because of an asymmetry in the definition of the original Einstein-Podolsky-Rosen paradox and in this article we show theoretically that it may be demonstrated, at least in the case where Alice and Bob can only make Gaussian measurements, using the intracavity nonlinear coupler.
Asymmetric steering is an effect whereby an inseparable bipartite system can be found to be described by either quantum mechanics or local hidden variable theories depending on which one of Alice or Bob makes the required measurements. We show that, even with an inseparable bipartite system, situations can arise where Gaussian measurements on one half are not sufficient to answer the fundamental question of which theory gives an adequate description and the whole system must be considered. This phenomenon is possible because of an asymmetry in the definition of the original Einstein-Podolsky-Rosen paradox and in this article we show theoretically that it may be demonstrated, at least in the case where Alice and Bob can only make Gaussian measurements, using the intracavity nonlinear coupler.
Einstein-Podolsky-Rosen steering is a form of quantum nonlocality exhibiting an inherent asymmetry between the observers, Alice and Bob. A natural question is then whether there exist entangled states which are one-way steerable, that is, Alice can steer Bob's state, but it is impossible for Bob to steer the state of Alice. So far, such a phenomenon has been demonstrated for continuous variable systems, but with a strong restriction on allowed measurements, namely, considering only Gaussian measurements. Here we present a simple class of entangled two-qubit states which are one-way steerable, considering arbitrary projective measurements. This shows that the nonlocal properties of entangled states can be fundamentally asymmetrical.
Entanglement is a property of composite physical systems distinct from any classical description and therefore has fundamental significance in quantum theory. Although a great deal of progress has been made in understanding entanglement, there is still much unknown about this phenomenon and interest in the nature of entanglement alone continues to motivate research in this area. Moreover, a better understanding of entanglement inevitably leads to a deeper knowledge of physical processes in quantum systems. This thesis explores continuous-variable entanglement in many-body nonlinear bosonic systems with the overall aim of broadening our knowledge of entanglement, and in turn, the quantum optical systems under consideration. The secondary aim of this thesis is to further the development of entanglement as a resource for use in quantum information, communication or computing. After introducing some of the terminology and basics of entanglement theory (Chapter 1), we then provide a brief review of the main theoretical tools employed throughout this thesis (Chapter 2). We begin our investigations of entanglement by considering an evanescently coupled pair of nonlinear crystals contained in a pumped optical cavity (Chapter 3). This two-mode system is known as the intracavity Kerr nonlinear coupler and it is the output beams emerging from the cavity that provide a source of continuous-variable entangled states. Within this framework, we examine the concept of steering, a term introduced by Schr{\o}dinger to generalize the effect described by the Einstein-Podolsky-Rosen paradox. In particular, we investigate bipartite asymmetric steering. This is an effect whereby an inseparable bipartite system can be found to be described by either quantum mechanics or local hidden variable theories, depending on which one of Alice or Bob makes the required measurements. We show that, even with an inseparable bipartite system, situations can arise where Gaussian measurements on one half are not sufficient to answer the fundamental question of which theory gives an adequate description and the whole system must be considered. We predict that asymmetric steering should arise in experiments using the intracavity nonlinear coupler, at least in the case where Alice and Bob can only make Gaussian measurements. We continue our investigations of nonlinear quantum optical systems and entanglement by studying an experimentally feasible intra-cavity coupled down-conversion scheme that makes use of concurrent nonlinearities (Chapter 4). From a theoretical perspective, we demonstrate that this scheme gives rise to a set of continuous-variable multipartite entangled output beams. Specifically, we verify that genuine quadripartite entanglement is present in this system by calculating the output fluctuation spectra and then considering violations of multipartite entanglement inequalities. The entanglement characteristics both above and below the oscillation threshold of the cavity are considered. Finally, we build on these studies of quadripartite entanglement and examine a modified intra-cavity concurrent down-conversion scheme (Chapter 5). We analyze the feasibility and potential of this scheme for generating continuous-variable cluster states. Cluster states have been proposed as a resource state for continuous-variable one-way quantum computing. We demonstrate genuine quadripartite entanglement and investigate the degree of entanglement present. We find that above the oscillation threshold, the basic cluster state geometry under consideration suffers from phase diffusion. We alleviate this problem by incorporating a small injected signal into our analysis. We also investigate squeezed joint operators, which must approach zero in the limit of large squeezing if the cluster state defining relation is to be satisfied. While the squeezed joint operators approach zero in the undepleted regime, we find that this is not the case when we consider the full interaction Hamiltonian and the presence of a cavity. In fact, we find that the decay of these operators is minimal in a cavity and even depletion alone inhibits cluster state formation.
In a recent work [Phys. Rev. Lett. 98, 140402 (2007)] we defined ``steering,'' a type of quantum nonlocality that is logically distinct from both nonseparability and Bell nonlocality. In the bipartite setting, it hinges on the question of whether Alice can affect Bob's state at a distance through her choice of measurement. More precisely and operationally, it hinges on the question of whether Alice, with classical communication, can convince Bob that they share an entangled state under the circumstances that Bob trusts nothing that Alice says. We argue that if she can, then this demonstrates the nonlocal effect first identified in the famous Einstein-Podolsky-Rosen paper [Phys. Rev. 47, 777 (1935)] as a universal effect for pure entangled states. This ability of Alice to remotely prepare Bob's state was subsequently called steering by Schr\"odinger, whose terminology we adopt. The phenomenon of steering has been largely overlooked, and prior to our work had not even been given a rigorous definition that is applicable to mixed states as well as pure states. Armed with our rigorous definition, we proved that steerable states are a strict subset of the entangled states, and a strict superset of the states that can exhibit Bell nonlocality. In this work we expand on these results and provide further examples of steerable states. We also elaborate on the connection with the original EPR paradox.
The concept of steering was introduced by Schrödinger in 1935 as a generalization of the Einstein-Podolsky-Rosen paradox for arbitrary pure bipartite entangled states and arbitrary measurements by one party. Until now, it has never been rigorously defined, so it has not been known (for example) what mixed states are steerable (that is, can be used to exhibit steering). We provide an operational definition, from which we prove (by considering Werner states and isotropic states) that steerable states are a strict subset of the entangled states, and a strict superset of the states that can exhibit Bell nonlocality. For arbitrary bipartite Gaussian states we derive a linear matrix inequality that decides the question of steerability via Gaussian measurements, and we relate this to the original Einstein-Podolsky-Rosen paradox.
Einstein-Podolsky-Rosen (EPR) steering is a form of quantum correlation that exhibits a fundamental asymmetry in the properties of quantum systems. Given two observers, Alice and Bob, there are known to exist bipartite entangled states which are one-way steerable in the sense that Alice can steer Bob's state, but Bob cannot steer Alice's state. Here, we generalize this phenomenon to three parties and find a cyclic property of tripartite EPR steering. In particular, we identify a three-qubit state whose reduced bipartite states are one-way steerable for arbitrary projective measurements. Moreover, the three-qubit state has a cyclic steering property in the sense that by arranging the system in a triangular configuration the neighboring parties can steer each other's states only in the same (e.g., clockwise) direction. That is, Alice can steer Bob's state, Bob can steer Charlie's state, and Charlie can steer Alice's state, but not the other way around.
Within the hierarchy of inseparable quantum correlations, Einstein-Podolsky-Rosen steering is distinguished from both entanglement and Bell nonlocality by its asymmetry-there exist conditions where the steering phenomenon changes from being observable to not observable, simply by exchanging the role of the two measuring parties. While this one-way steering feature has been previously demonstrated for the restricted class of Gaussian measurements, for the general case of positive-operator-valued measures even its theoretical existence has only recently been settled. Here, we prove, and then experimentally observe, the one-way steerability of an experimentally practical class of entangled states in this general setting. As well as its foundational significance, the demonstration of fundamentally asymmetric nonlocality also has practical implications for the distribution of the trust in quantum communication networks.