In quantum mechanics, if one party performs a measurement on one system, different outcomes can lead to different states for another system. This phenomenon is known as quantum steering. The thesis begins with some general results about steering: a classification of which states permit the most powerful type of steering, and the implications of this for quantum correlations. The first main topic is steering in two-qubit systems. It turns out that steering provides an excellent way to visualize two-qubit states, leading to a novel criteria for entanglement and a better understanding of separable states. Oddly, the structure of steering can be more subtle in separable states than it can be in entangled ones. Returning to general quantum systems, I then turn to the EPR paradox, and its generalisation using Local Hidden State models. I show that the lack of such models can be used to quantify the amount of entanglement shared by two parties, even when one of them does not trust their measuring devices. Finally, the desire to understand steering without invoking "spooky action-at- a-distance" leads to the idea that quantum states are states of knowledge. I explore some de Finetti theorems that help to make sense of this idea, but then show a significant roadblock the most natural formalisation of it. The main results in chapters 3 and 4 appear in [JPJR13] and [Pus13] respectively. The result in chapter 6 was improved in collaboration with my supervisors and published in [PBR12].
In quantum mechanics, if one party performs a measurement on one system, different outcomes can lead to different states for another system. This phenomenon is known as quantum steering. The thesis begins with some general results about steering: a classification of which states permit the most powerful type of steering, and the implications of this for quantum correlations. The first main topic is steering in two-qubit systems. It turns out that steering provides an excellent way to visualize two-qubit states, leading to a novel criteria for entanglement and a better understanding of separable states. Oddly, the structure of steering can be more subtle in separable states than it can be in entangled ones. Returning to general quantum systems, I then turn to the EPR paradox, and its generalisation using Local Hidden State models. I show that the lack of such models can be used to quantify the amount of entanglement shared by two parties, even when one of them does not trust their measuring devices. Finally, the desire to understand steering without invoking "spooky action-at- a-distance" leads to the idea that quantum states are states of knowledge. I explore some de Finetti theorems that help to make sense of this idea, but then show a significant roadblock the most natural formalisation of it. The main results in chapters 3 and 4 appear in [JPJR13] and [Pus13] respectively. The result in chapter 6 was improved in collaboration with my supervisors and published in [PBR12].
Einstein-Rosen-Podolsky (EPR) steering or quantum steering describes the "spooky-action-at-a-distance" that one party is able to remotely alter the states of the other if they share a certain entangled state. Generally, it admits an operational interpretation as the task of verifying entanglement without trust in the steering party's devices, making it lying intermediate between Bell nonlocality and entanglement. Together with the asymmetrical nature, quantum steering has attracted a considerable interest from theoretical and experimental sides over past decades. In this Perspective, we present a brief overview of the EPR steering with emphasis on recent progress, discuss current challenges, opportunities and propose various future directions. We look to the future which directs research to a larger-scale level beyond massless and microscopic systems to reveal steering of higher dimensionality, and to build up steered networks composed of multiple parties.
The concept of quantum steering was originally introduced by Schrödinger to describe the "spooky action-at-a-distance" effect noted in the Einstein-Podolsky-Rosen (EPR) paradox, whereby local measurements performed on one party apparently adjust (steer) the state of another distant party. In this talk, I will give an introduction about the advances of the EPR steering and its advantage as quantum resource. Then I will present our efforts on characterizing bipartite and multipartite steering and developing its unique applications in quantum information processing. I will give an overview of our recent developments on quantum steering and its applications in quantum information. I will share our view about the current challenges, opportunities and the future directions for this topic.
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.
Phenomena that slip beyond the grasp of our classical intuition reveal uniquely quantum effects that deepen our understanding of the physical world and enable advances in information processing, particularly in quantum communication and computation. One such phenomenon is quantum steering, whereby measurements performed by one party influence the conditional states of another when the two share an entangled quantum system. If the observed correlations cannot be explained by a local hidden state model, the state is said to be steerable. In this work, we investigate the typicality of this behavior: given a generic two-qubit state and m Haar-random projective measurements, what is the probability of observing steering? We derive analytical expressions for the steering probability PS of Werner states in two- and three-setting scenarios, the latter restricted to coplanar projective measurements on the Bloch sphere. For larger numbers of settings and various random states ensembles, we perform numerical analyses showing that PS increases systematically with the number of measurements and substantially exceeds the corresponding probabilities associated with Bell nonlocality. Our results demonstrate that random states with minimal environmental coupling exhibit a high probability of steering for finite m and approach genuine typicality, PS=100%, as the number of settings increases. We provide a detailed characterization of PS across different state ensembles and specific families, including Bell-diagonal and Werner states, identifying those with the greatest non-classical potential and highlighting their relevance for protocols in which steering serves as a key resource.
The phenomenon of quantum steering in bipartite quantum systems can be reduced to the question of whether or not the first party can perform measurements such that the conditional states on the second party can be explained by a local hidden-state model. Clearly, the answer to this depends on the measurements which the first party is able to perform. We introduce a local hidden-state model explaining the conditional states for all generalized measurements on Bell-diagonal states of two qubits. More precisely, it is known for the restricted case of projective measurements and Bell-diagonal states characterized by the correlation matrix T that a local hidden-state model exists if and only if ${R}_{T}=2\ensuremath{\pi}{N}_{T}\left|det(T)\right|\ensuremath{\ge}1$, where NT is defined by an integral over the Bloch sphere. For generalized measurements described by positive operator-valued measures we construct a model if ${R}_{T}\ensuremath{\ge}6/5$. Our work paves the way for a systematic study of steerability of quantum states with generalized measurements beyond the highly symmetric Werner states.
Einstein-Podolsky-Rosen (EPR) steering is a form of bipartite quantum correlation that is intermediate between entanglement and Bell nonlocality. It allows for entanglement certification when the measurements performed by one of the parties are not characterised (or are untrusted) and has applications in quantum key distribution. Despite its foundational and applied importance, EPR steering lacks a quantitative assessment. Here we propose a way of quantifying this phenomenon and use it to study the steerability of several quantum states. In particular we show that every pure entangled state is maximally steerable, the projector onto the anti-symmetric subspace is maximally steerable for all dimensions, we provide a new example of one-way steering, and give strong support that states with positive-partial-transposition are not steerable.