Quantum steering has recently been formalized in the framework of a resource theory of steering, and several quantifiers have already been introduced. Here, we propose an information-theoretic quantifier for steering called intrinsic steerability, which uses conditional mutual information to measure the deviation of a given assemblage from one having a local-hidden-state model. We thus relate conditional mutual information to quantum steering and introduce monotones that satisfy certain desirable properties. The idea behind the quantifier is to suppress the correlations that can be explained by an inaccessible quantum system and then quantify the remaining intrinsic correlations. A variant of the intrinsic steerability finds operational meaning as the classical communication cost of sending the measurement choice and outcome to an eavesdropper who possesses a purifying system of the underlying bipartite quantum state that is being measured.
Quantum steering has recently been formalized in the framework of a resource theory of steering, and several quantifiers have already been introduced. Here, we propose an information-theoretic quantifier for steering called intrinsic steerability, which uses conditional mutual information to measure the deviation of a given assemblage from one having a local-hidden-state model. We thus relate conditional mutual information to quantum steering and introduce monotones that satisfy certain desirable properties. The idea behind the quantifier is to suppress the correlations that can be explained by an inaccessible quantum system and then quantify the remaining intrinsic correlations. A variant of the intrinsic steerability finds operational meaning as the classical communication cost of sending the measurement choice and outcome to an eavesdropper who possesses a purifying system of the underlying bipartite quantum state that is being measured.
The aim of this thesis is to understand the fundamental limitations on secret key distillation in various settings of quantum key distribution. We first consider quantum steering, which is a resource for one-sided device-independent quantum key distribution. We introduce a conditional mutual information based quantifier for quantum steering, which we call intrinsic steerability. Next, we consider quantum non-locality, which is a resource for device-independent quantum key distribution. In this context, we introduce a quantifier, intrinsic non-locality, which is a monotone in the resource theory of Bell non-locality. Both these quantities are inspired by intrinsic information and squashed entanglement and are based on conditional mutual information. The idea behind these quantifiers is to suppress the correlations that can be explained by a local hidden variable or by an inaccessible quantum system, thus quantifying the remaining intrinsic correlations. We then prove various properties of these two monotones, which includes the following: monotonicity under free operations, additivity under tensor product of objects, convexity, and faithfulness, among others. Next, we prove that intrinsic steerability is an upper bound on the secret-key-agreement capacity of an assemblage, and intrinsic non-locality is an upper bound on the secret-key-agreement capacity of a quantum probability distribution. Thus we prove that these quantities are upper bounds on the achievable key rates in one-sided device-independent and device-independent quantum key distribution protocols. We also calculate these bounds for certain honest devices. The study of these upper bounds is instrumental in understanding the limitations of protocols that can be designed for various settings. These upper bounds inform us that, even if one considers the best possible protocol, there is no possibility of exceeding the upper bounds on key rates without a quantum repeater. The upper bounds introduced in this thesis are an important step for initiating this line of research in one-sided device-independent and in device-independent quantum key distribution.
The bipartite state is said to be steerable iff it does not have single system description, i.e., the bipartite state can not be explained by local hidden state model [Phys. Rev. Lett. {\bf 98}, 140402 (2007)]. Several steering inequalities have been derived using different local uncertainty relations to verify the control on the state of one subsystem by other. Here, we derive complementarity kind of relations between coherences measured on mutually unbiased bases using various coherence measures like the l1-norm, relative entropy and skew information. Using these relations, we formulate steering inequalities to check whether one subsystem has control to the coherence of the other entangled subsystem. According to our results, all pure entangled states are maximally steerable but for mixed entangled states, coherence steerability needs stronger non-local correlation than the existing state steering inequalities. We also show that our steering criteria is monogamous.
Quantum steering is an asymmetric correlation which occupies a place between entanglement and Bell nonlocality. In the paradigmatic scenario involving the protagonists Alice and Bob, the entangled state shared between them is said to be steerable from Alice to Bob if the steering assemblage on Bob's side does not admit a local hidden state (LHS) description. Quantum conditional entropies, on the other hand, provide for another characterization of quantum correlations. Contrary to our common intuition, conditional entropies for some entangled states can be negative, marking a significant departure from the classical realm. Quantum steering and quantum nonlocality, in general, share an intricate relation with quantum conditional entropies. In the present contribution, we investigate this relationship. For a significant class, namely the two-qubit Weyl states, we show that negativity of conditional R\'enyi 2-entropy and conditional Tsallis 2-entropy is a necessary and sufficient condition for the violation of a suitably chosen three settings steering inequality. With respect to the same inequality we find an upper bound for the conditional R\'enyi 2-entropy, such that the general two-qubit state is steerable. Moving from a particular steering inequality to local hidden-state descriptions, we show that some two-qubit Weyl states which admit a LHS model possess nonnegative conditional R\'enyi 2-entropy. However, the same does not hold true for some non-Weyl states. Our study further investigates the relation between nonnegativity of conditional entropy and LHS models in two-qudits for the isotropic and Werner states. There we find that, whenever these states admit a LHS model, they possess a nonnegative conditional R\'enyi 2-entropy. We then observe that the same holds true for a noisy variant of the two-qudit Werner state.
We present an operational framework for Einstein-Podolsky-Rosen steering as a physical resource. To begin with, we characterize the set of steering non-increasing operations (SNIOs) – i.e., those that do not create steering– on arbitrary-dimensional bipartite systems composed of a quantum subsystem and a black-box device. Next, we introduce the notion of convex steering monotones as the fundamental axiomatic quantifiers of steering. As a convenient example thereof, we present the relative entropy of steering. In addition, we prove that two previously proposed quantifiers, the steerable weight and the robustness of steering, are also convex steering monotones. To end up with, for minimal-dimensional systems, we establish, on the one hand, necessary and sufficient conditions for pure-state steering conversions under stochastic SNIOs and prove, on the other hand, the non-existence of steering bits, i.e., measure-independent maximally steerable states from which all states can be obtained by means of the free operations. Our findings reveal unexpected aspects of steering and lay foundations for further resource-theory approaches, with potential implications in Bell non-locality.
Quantum steering captures the ability of one party, through local measurements on a shared entangled state, to affect the conditional states held by distant parties in a way that admits no local explanation. Its device-independent (DI) quantification -- requiring no characterization of any state or measurement -- has been developed in the bipartite setting, notably through the framework of assemblage moment matrices (AMMs) [Phys. Rev. Lett. 116, 240401 (2016)], but has remained largely unexplored beyond it. Here, we extend the AMM framework to tripartite steering, covering both the scenario in which two parties jointly steer a third one (2-steer-1) and that in which a single party steers the remaining two (1-steer-2). For each scheme, we construct a semidefinite program that, from the observed correlations -- or merely from the violation of a Bell inequality -- yields lower bounds on the steering robustness of the underlying assemblage. As a by-product, our bounds also certify, in a DI manner, the incompatibility of the measurements performed by the steering parties.
We develop a unified approach to classical, quantum and post-quantum steering. The framework is based on uncharacterised (black-box) parties performing quantum measurements on their share of a (possibly unphysical) quantum state, and its starting point is the characterisation of general no-signalling assemblages via non-positive local hidden-state models, which will be defined in this work. By developing a connection to entanglement witnesses, this formalism allows for new definitions of families of assemblages, in particular via (i) non-decomposable positive maps and (ii) unextendible product bases. The former proves to be useful for constructing post-quantum assemblages with the built-in feature of yielding only quantum correlations in Bell experiments, while the latter always gives certifiably post-quantum assemblages. Finally, our framework is equipped with an inherent quantifier of post-quantum steering, which we call the negativity of post-quantum steering. We postulate that post-quantum steering should not increase under one-way quantum operations from the steered parties to the steering parties, and we show that, in this sense, the negativity of post-quantum steering is a convex post-quantum-steering monotone.