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.
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.
The study of stronger-than-quantum phenomena (i.e., postquantum) has enabled a deeper understanding of the scope of quantum theory. Much is known about the case of correlations in Bell scenarios, where the device-independent framework allowed us to explore its possibilities independently of the formalism of quantum theory. However, less is known about the phenomenon of Einstein-Podolsky-Rosen steering. Here, the `characterised parties' are assumed to describe their systems locally through the quantum formalism, which inconveniences a theory-independent description. In addition, a theorem by Gisin and Hughston, Josza and Wootters further hindered the discovery of the phenomenon. The study of postquantum steering, initiated about a decade ago, has been quite fruitful, including: the development of mathematical formalisms that frame the effect, resource theories that quantify it as a resource, and activation protocols that relate it to Bell correlations. However, all these results have a limitation in common: they apply to scenarios with only one quantum party.
Here we articulate the concept of postquantum steering for scenarios with multiple quantum parties, bringing in the missing piece to the puzzle. We provide an algorithm to certify postquantumness, which in some cases also certifies quantumness. We also define a hierarchy of semidefinite programs that bounds the set of quantum assemblages from the outside. Moreover, we show that the study of postquantum steering is fundamentally relevant since it is not just a mere mathematical curiosity allowed by the no-signalling principle, but it may arise within compositional theories beyond quantum theory. Our work further discovers a peculiarity of steering: its theory-independent description fundamentally prevents a direct connection with Bell nonlocality -- e.g., nonclassical Bell correlations do not imply nonclassical steering.
Non-locality and steering are both non-classical phenomena witnessed in nature as a result of quantum entanglement. It is now well-established that one can study non-locality independently of the formalism of quantum mechanics, in the so-called device-independent framework. With regards to steering, although one cannot study it completely independently of the quantum formalism, ‘post-quantum steering’ has been described, which is steering that cannot be reproduced by measurements on entangled states but does not lead to superluminal signalling. In this work we present a framework based on the study of quantum channels in which one can study steering (and non-locality) in quantum theory and beyond. In this framework, we show that kinds of steering, whether quantum or post-quantum, are directly related to particular families of quantum channels that have been previously introduced by Beckman et al (2001 Phys. Rev. A 64 052309). Utilizing this connection we also demonstrate new analytical examples of post-quantum steering, give a quantum channel interpretation of almost quantum non-locality and steering, easily recover and generalize the celebrated Gisin–Hughston–Jozsa–Wootters theorem, and initiate the study of post-quantum Buscemi non-locality and non-classical teleportation. In this way, we see post-quantum non-locality and steering as just two aspects of a more general phenomenon.
Non-locality and steering are both non-classical phenomena witnessed in nature as a result of quantum entanglement. It is now well-established that one can study non-locality independently of the formalism of quantum mechanics, in the so-called device-independent framework. With regards to steering, although one cannot study it completely independently of the quantum formalism, 'post-quantum steering' has been described, which is steering that cannot be reproduced by measurements on entangled states but does not lead to superluminal signalling. In this work we present a framework based on the study of quantum channels in which one can study steering (and non-locality) in quantum theory and beyond. In this framework, we show that kinds of steering, whether quantum or post-quantum, are directly related to particular families of quantum channels that have been previously introduced by Beckman et al (2001 Phys. Rev. A 64 052309). Utilizing this connection we also demonstrate new analytical examples of post-quantum steering, give a quantum channel interpretation of almost quantum non-locality and steering, easily recover and generalize the celebrated Gisin–Hughston–Jozsa–Wootters theorem, and initiate the study of post-quantum Buscemi non-locality and non-classical teleportation. In this way, we see post-quantum non-locality and steering as just two aspects of a more general phenomenon.
Following recent advancements, we consider a scenario of multipartite postquantum steering and general no-signaling assemblages. We introduce the notion of the edge of the set of no-signaling assemblages and we present its characterization. Next, we use this concept to construct witnesses for no-signaling assemblages without a local hidden state model. Finally, in the simplest nontrivial case of steering with two untrusted subsystems, we discuss the possibility of quantum realization of assemblages on the edge. In particular, for three-qubit states, we obtain a no-go-type result, which states that it is impossible to produce an assemblage on the edge using measurements described by positive operator valued measures as long as the rank of a given state is greater than or equal to 3.
Abstract Characterising non-classical quantum phenomena is crucial not only from a fundamental perspective, but also to better understand its capabilities for information processing and communication tasks. In this work, we focus on exploring the characterisation of Einstein–Podolsky–Rosen inference (a.k.a. steering): a signature of non-classicality manifested when one or more parties in a Bell scenario have their systems and measurements described by quantum theory, rather than being treated as black boxes. We propose a way of characterising common-cause assemblages from the correlations that arise when the trusted party performs tomographically-complete measurements on their share of the experiment, and discuss the advantages and challenges of this approach. Within this framework, we show that so-called almost quantum assemblages satisfy the principle of macroscopic noncontextuality, and demonstrate that a subset of almost quantum correlations recover almost quantum assemblages in this approach.
Quantum steering refers to the non-classical correlations that can be observed between the outcomes of measurements applied on half of an entangled state and the resulting post-measured states that are left with the other party. From an operational point of view, a steering test can be seen as an entanglement test where one of the parties performs uncharacterised measurements. Thus, quantum steering is a form of quantum inseparability that lies in between the well-known notions of Bell nonlocality and entanglement. Moreover, quantum steering is also related to several asymmetric quantum information protocols where some of the parties are considered untrusted. Because of these facts, quantum steering has received a lot of attention both theoretically and experimentally. The main goal of this review is to give an overview of how to characterise quantum steering through semidefinite programming. This characterisation provides efficient numerical methods to address a number of problems, including steering detection, quantification, and applications. We also give a brief overview of some important results that are not directly related to semidefinite programming. Finally, we make available a collection of semidefinite programming codes that can be used to study the topics discussed in this article.