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Quantum correlations: Schrodinger's steering in lossy conditions; Heisenberg's limit to laser coherence.

Travis J. Baker

Published Jun 16, 2021• 0 citations
Publisher

Abstract

Quantum correlations are a fundamental resource for technologies arising out of quantum information science. This thesis contains a body of work consisting of published and unpublished papers, which document theoretical developments in two active topics within this field - Einstein-Podolsky-Rosen (EPR) steering, and optical laser coherence. While these two topics might seem unrelated at first glance, all results contained within this work fundamentally arise from exploring the correlations between nonlocal quantum systems of low dimension. As such, this thesis is composed of two parts. The first is dedicated to studies of EPR steering, where one observer can appear to remotely steer (in the terminology of Schrodinger) the state of a distant party into different ensembles of quantum states. A number of novel results are presented which address the fundamental problem of determining both steerability and non-steerability of bipartite quantum states. Motivated by experiments aiming to demonstrate one-way steering, we begin by deriving a condition that can be applied to any two-qubit state, which is sufficient to determine if it is non-steerable when passed through a channel with a given amount of loss. This result is further expanded in the context of a new experiment, and applied to tomographically reconstructed two-qubit states which demonstrate one-way EPR steering in this rigorous way for the first time. Next, we develop a different idea for proving non-steerability of two-qubit entangled states, based on the idea of allowing the steering party to perform deterministic local quantum operations prior to measuring. Finally, we turn to the general issue of certifying steerability while closing the detection loophole, which has been previously closed in a number of experiments by violating specific loss-tolerant steering inequalities. Introducing a semi-definite programming formulation of the problem, we exploit the symmetry of a class of quantum states which naturally arise in quantum networks (the so-called inept states), to efficiently calculate the amount of loss they can tolerate while maintaining steerability. The collection of results in this first part are expected to be useful, especially as the field moves toward applications in networks where qubit loss becomes a limiting factor. The second part of this thesis concerns a defining property of a laser beam-optical coherence. This can be quantified by a dimensionless number, C, which can be roughly understood as the mean number of photons emitted into the beam with the same phase. While the fundamental theory of practical laser systems was developed in the 1950's and 60's, the ultimate quantum limits to the coherence of the beams a laser can produce has not been previously studied. In terms of the mean number of excitations stored inside the laser cavity, μ, it was previously thought that C = ø(μ2). Here, under some natural assumptions we show analytically that the ultimate limit permitted by quantum theory, or the Heisenberg limit, is quadratically better; C = O(μ4). Moreover, by treating the beam produced by a laser cavity as an infinite chain of entangled qubits, we use state-of-the-art matrix product state methods to numerically find a model of laser dynamics which attains this new ultimate limit. That is, C = ø(μ4). Finally, we present a system constructed with technology already existing in the field of circuit quantum electrodynamics, and use techniques from quantum control to prove that a coherent train of pulses attaining this limit can be created. The results presented in this part are a prescription for engineering a laser system operating at the best possible limit.

DOI
10.25904/1912/4240

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Quantum correlations are a fundamental resource for technologies arising out of quantum information science. This thesis contains a body of work consisting of published and unpublished papers, which document theoretical developments in two active topics within this field - Einstein-Podolsky-Rosen (EPR) steering, and optical laser coherence. While these two topics might seem unrelated at first glance, all results contained within this work fundamentally arise from exploring the correlations between nonlocal quantum systems of low dimension. As such, this thesis is composed of two parts. The first is dedicated to studies of EPR steering, where one observer can appear to remotely steer (in the terminology of Schrodinger) the state of a distant party into different ensembles of quantum states. A number of novel results are presented which address the fundamental problem of determining both steerability and non-steerability of bipartite quantum states. Motivated by experiments aiming to demonstrate one-way steering, we begin by deriving a condition that can be applied to any two-qubit state, which is sufficient to determine if it is non-steerable when passed through a channel with a given amount of loss. This result is further expanded in the context of a new experiment, and applied to tomographically reconstructed two-qubit states which demonstrate one-way EPR steering in this rigorous way for the first time. Next, we develop a different idea for proving non-steerability of two-qubit entangled states, based on the idea of allowing the steering party to perform deterministic local quantum operations prior to measuring. Finally, we turn to the general issue of certifying steerability while closing the detection loophole, which has been previously closed in a number of experiments by violating specific loss-tolerant steering inequalities. Introducing a semi-definite programming formulation of the problem, we exploit the symmetry of a class of quantum states which naturally arise in quantum networks (the so-called inept states), to efficiently calculate the amount of loss they can tolerate while maintaining steerability. The collection of results in this first part are expected to be useful, especially as the field moves toward applications in networks where qubit loss becomes a limiting factor. The second part of this thesis concerns a defining property of a laser beam-optical coherence. This can be quantified by a dimensionless number, C, which can be roughly understood as the mean number of photons emitted into the beam with the same phase. While the fundamental theory of practical laser systems was developed in the 1950's and 60's, the ultimate quantum limits to the coherence of the beams a laser can produce has not been previously studied. In terms of the mean number of excitations stored inside the laser cavity, μ, it was previously thought that C = ø(μ2). Here, under some natural assumptions we show analytically that the ultimate limit permitted by quantum theory, or the Heisenberg limit, is quadratically better; C = O(μ4). Moreover, by treating the beam produced by a laser cavity as an infinite chain of entangled qubits, we use state-of-the-art matrix product state methods to numerically find a model of laser dynamics which attains this new ultimate limit. That is, C = ø(μ4). Finally, we present a system constructed with technology already existing in the field of circuit quantum electrodynamics, and use techniques from quantum control to prove that a coherent train of pulses attaining this limit can be created. The results presented in this part are a prescription for engineering a laser system operating at the best possible limit.
The first description of quantum steering goes back to Schrödinger.In 1935 he named it a necessary and indispensable feature of quantum mechanics in response to Einstein's concerns about the completeness of quantum theory.In recent years investigations of the effect have experienced a revival, as it turned out that steering is a distinct subclass of entanglement.It is strictly stronger than genuine entanglement, meaning that all steerable states are entangled but not vice versa, and strictly weaker than the violation of Bell inequalities.Additionally, the violation of a steering inequality has an intrinsic asymmetry due to the directional construction.One party seemingly steers the other and their roles are in general not interchangeable.The first direct observation of this asymmetry is presented in this thesis.It was demonstrated that certain Gaussian quantum optical states show steering from one party to the other but not in the opposite direction.This is a profound result, as the experiment proves that the class of steerable states itself divides into distinct subclasses.Furthermore, steering found major applications in quantum information.For example in the field of quantum key distribution (QKD) the non-classical correlations can be exploited to establish a secret, cryptographic key.Thereby, steering allows one-sided device independent security, since it violates a description with certain classical models.This thesis reports on the final results of the QKD project at the Leibniz University Hannover that aimed at the generation of a secure quantum key based on mutually entangled conjugate pairs of continuous variables.The protocol of a recent security proof was implemented in a table-top setup and nearly 100 MBit of usable key with composable and one-sided device independent security against coherent attacks were generated.The security analysis included finite size effects, and a newly developed hybrid reconciliation algorithm was used for efficient error correction.Furthermore, a loss study on the entanglement distribution showed that separation of the two detectors by a 5 km optical fibre would be possible.Therefore, the results of this work are a major step towards continuous variable QKD with state-of-the-art security in local area networks.
EPR steering is a form of bipartite quantum correlation that is intermediate between entanglement and Bell nonlocality. It is detected via the violation of steering inequalities, which provide sufficient conditions for the steerability of quantum states. Furthermore, there is a sufficient criterion for guaranteeing that a two-qubit state is unsteerable. But there are some two qubit states that can not determine the steerability of the steering inequality and the sufficient criteria. Here we introduce a way of quantifying this phenomenon and use it to study the steerability of two qubit states. This article recommend a platonic solid measurement schemes. And then discuss the problem, namely expanding the scope of steerable, which guarantees that the state is steerable.
Shared quantum correlations find application in a wide range of modern quantum information protocols. This thesis presents several optical experiments towards the improved certification of shared quantum correlations, using photonic qubits. The first experiment demonstrates the Einstein–Podolsky–Rosen (EPR) steering effect with no detection loophole. Polarisation–entangled photon pairs generated in a Sagnac interferometer were used to test new loss–tolerant EPR steering inequalities. To test the robustness of the loss–tolerant EPR steering nequalities, a transmission loss of −4.3dB was introduced using a 1km optical fibre loop, finding inequality violations for n = 10 and n = 16 measurement settings with no detection loophole (equating to a total loss of 87%). The findings demonstrate how the new loss–tolerant protocols can be used to certify the EPR steering effect for arbitrarily high losses, and may find application in one–sided device–independent quantum key distribution protocols. The second experiment introduces EPR steering in the context of an EPR steering game. The rules of the game allow for certification of shared quantum correlations via EPR steering inequalites without requiring trust in devices or parties external to a quantum referee. In particular, the experiment successfully demonstrates EPR steering in a measurement–device independent framework; a framework which has typically been accessible only through Bell inequality violations. The findings suggest the possibility for quantum–refereed one–sided device–independent quantum key distribution protocols.
This PhD Dissertation collects results of my own work on the topic of continuous variable (CV) quantum teleportation, which is one of the most important applications of quantum entanglement, as well as on the understanding, quantification, detection, and applications of a type of quantum correlations known as Einstein-Podolsky-Rosen (EPR) steering, for both bipartite and multipartite systems and with a main focus on CV systems. For the first results, we examine and compare two fundamentally different teleportation schemes; the well-known continuous variable scheme of Vaidman, Braunstein and Kimble, and a recently proposed hybrid scheme by Andersen and Ralph. We analyse the teleportation of ensembles of arbitrary pure single-mode Gaussian states using these schemes and compare their performance against classical strategies that utilize no entanglement (benchmarks). Our analysis brings into question any advantage due to non-Gaussianity for quantum teleportation of Gaussian states. For the second part of the results, we study bipartite EPR-steering. We propose a novel powerful method to detect steering in quantum systems of any dimension in a systematic and hierarchical way. Our method includes previous results of the literature as special cases on one hand, and goes beyond them on the other. We proceed to the quantification of steering-type correlations, and introduce a measure of steering for arbitrary bipartite Gaussian states, prove many useful properties, and provide with an operational interpretation of the proposed measure in terms of the key rate in one-sided device independent quantum key distribution. Finally, we show how the Gaussian steering measure gives a lower bound to a more general quantifier of which Gaussian states are proven to be extremal. We proceed to the study of multipartite steering, and derive laws for the distribution of Gaussian steering among different parties in multipartite Gaussian states. We define an indicator of collective steering-type correlations, which is interpreted operationally in terms of the guaranteed secret key rate in the multi-party cryptographic task of quantum secret sharing. The final results look at the cryptographical task of quantum secret sharing, whose security has remained unproven almost two decades after its original conception. By utilizing intuition and ideas from steering, we manage to establish for the first time an unconditional security proof for CV entanglement-based quantum secret sharing schemes, and demonstrate their practical feasibility. Our results establish quantum secret sharing as a viable and practically relevant primitive for quantum communication technologies.
Einstein-Podolsky-Rosen (EPR) steering describes the ability of one observer to nonlocally "steer" the other observer's state through local measurements. EPR steering exhibits a unique asymmetric property; i.e., the steerability can differ between observers, which can lead to one-way EPR steering in which only one observer obtains steerability in the steering process. This property is inherently different from the symmetric concepts of entanglement and Bell nonlocality, and it has attracted increasing interest. Here, we experimentally demonstrate asymmetric EPR steering for a class of two-qubit states in the case of two measurement settings. We propose a practical method to quantify the steerability. We then provide a necessary and sufficient condition for EPR steering and clearly demonstrate one-way EPR steering. Our work provides new insight into the fundamental asymmetry of quantum nonlocality and has potential applications in asymmetric quantum information processing.
Abstract Einstein–Podolsky–Rosen (EPR) steering is an intermediate form of nonlocality which interpolates between entanglement and Bell nonlocality. It has been shown that EPR steerable states are fundamental resources for one-sided device-independent quantum information tasks. Nevertheless, distinguishing quantum steerable states from the unsteerable ones in general cases remains a big challenge under active studies. The generalized Bloch representation of density matrices in terms of the Heisenberg observables is a useful tool for analyzing characteristics of quantum systems and plays an important role in quantum information. In this work, the role of this representation in detection of EPR steering is studied. More specifically, based on the correlation matrices of the Heisenberg–Weyl observables, a family of steerability criteria for arbitrary dimensional bipartite systems are proposed. A large class of Hermitian operators, which can conveniently witness steering in some scenarios, are further provided. Finally, these criteria are illustrated through several examples, are compared with some existing ones, and their power and advantages are exhibited in certain cases.