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arXiv:1405.0262 · Physical Review Letters

Steering Bound Entangled States: A Counterexample to the Stronger Peres Conjecture

Tobias Moroder, Oleg Gittsovich, Marcus Huber, Otfried Gühne

Published Aug 1, 2014• 114 citations
arXivHTMLPDF

Abstract

Quantum correlations are at the heart of many applications in quantum information science and, at the same time, they form the basis for discussions about genuine quantum effects and their difference to classical physics. On one hand, entanglement theory provides the tools to quantify correlations in information processing and many results have been obtained to discriminate useful entanglement, which can be distilled to a pure form, from bound entanglement, being of limited use in many applications. On the other hand, for discriminating quantum phenomena from their classical counterparts, Schrödinger and Bell introduced the notions of steering and local hidden variable models. We provide a method to generate systematically bound entangled quantum states which can still be used for steering and, therefore, to rule out local hidden state models. This sheds light on the relations between the various views on quantum correlations and disproves a widespread conjecture known as the stronger Peres conjecture. For practical applications, it implies that even the weakest form of entanglement can be certified in a semidevice independent way.

DOI
10.1103/physrevlett.113.050404

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Quantum correlations are at the heart of many applications in quantum information science and, at the same time, they form the basis for discussions about genuine quantum effects and their difference to classical physics. On one hand, entanglement theory provides the tools to quantify correlations in information processing and many results have been obtained to discriminate useful entanglement, which can be distilled to a pure form, from bound entanglement, being of limited use in many applications. On the other hand, for discriminating quantum phenomena from their classical counterparts, Schrödinger and Bell introduced the notions of steering and local hidden variable models. We provide a method to generate systematically bound entangled quantum states which can still be used for steering and, therefore, to rule out local hidden state models. This sheds light on the relations between the various views on quantum correlations and disproves a widespread conjecture known as the stronger Peres conjecture. For practical applications, it implies that even the weakest form of entanglement can be certified in a semidevice independent way.
Einstein-Podolsky-Rosen steering incarnates a useful nonclassical correlation which sits between entanglement and Bell nonlocality. While a number of qualitative steering criteria exist, very little has been achieved for what concerns quantifying steerability. We introduce a computable measure of steering for arbitrary bipartite Gaussian states of continuous variable systems. For two-mode Gaussian states, the measure reduces to a form of coherent information, which is proven never to exceed entanglement, and to reduce to it on pure states. We provide an operational connection between our measure and the key rate in one-sided device-independent quantum key distribution. We further prove that Peres' conjecture holds in its stronger form within the fully Gaussian regime: namely, steering bound entangled Gaussian states by Gaussian measurements is impossible.
Entanglement and nonlocality are both fundamental aspects of quantum theory, and play a prominent role in quantum information science. The exact relation between entanglement and nonlocality is however still poorly understood. Here we make progress in this direction by showing that, contrary to what previous work suggested, quantum nonlocality does not imply entanglement distillability. Specifically, we present analytically a 3-qubit entangled state that is separable along any bipartition. This implies that no bipartite entanglement can be distilled from this state, which is thus fully bound entangled. Then we show that this state nevertheless violates a Bell inequality. Our result also disproves the multipartite version of a longstanding conjecture made by Asher Peres.
Quantum steering, loosely speaking the distribution of entanglement from an untrusted party, is a form of quantum nonlocality which is intermediate between entanglement and Bell nonlocality. Determining which states can be steered is important from a conceptual point of view, but also for applications, e.g. in quantum cryptography. Here we show that bound entanglement, although it represents the weakest form of entanglement, can nevertheless lead to quantum steering. This is done by noticing that steering inequalities can be derived from entropic uncertainty relations. Our result has implications on the connection between entanglement distillability and nonlocality, and shows that bound entangled states can be useful for information-theoretic tasks featuring an untrusted party.
High-dimensional quantum entanglement can give rise to stronger forms of nonlocal correlations compared to qubit systems, offering significant advantages for quantum information processing. Certifying these stronger correlations, however, remains an important challenge, in particular in an experimental setting. Here we theoretically formalize and experimentally demonstrate a notion of genuine high-dimensional quantum steering. We show that high-dimensional entanglement, as quantified by the Schmidt number, can lead to a stronger form of steering, provably impossible to obtain via entanglement in lower dimensions. Exploiting the connection between steering and incompatibility of quantum measurements, we derive simple two-setting steering inequalities, the violation of which guarantees the presence of genuine high-dimensional steering, and hence certifies a lower bound on the Schmidt number in a one-sided device-independent setting. We report the experimental violation of these inequalities using macropixel photon-pair entanglement certifying genuine high-dimensional steering. In particular, using an entangled state in dimension d=31, our data certifies a minimum Schmidt number of n=15.
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.
High-dimensional entanglement can give rise to stronger forms of nonlocal correlations compared to qubit systems. Beyond being of fundamental interest, this offers significant advantages for quantum information processing. The problem of certifying these stronger correlations, however, remains an important challenge. Here we theoretically formalise and experimentally demonstrate a notion of genuine high-dimensional quantum steering. We show that high-dimensional entanglement combined with judiciously chosen local measurements can lead to a stronger form of steering, provably impossible to obtain via entanglement in lower dimensions. Exploiting the connection between steering and incompatibility of quantum measurements, we derive two-setting inequalities for certifying the presence of genuine high-dimensional steering. We report the experimental violation of these inequalities using macro-pixel photon-pair entanglement certifying genuine high-dimensional steering in dimensions up to 15.