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arXiv:0712.3999

On quantum cryptography with bipartite bound entangled states

Paweł Horodecki, Remigiusz Augusiak

Published Dec 24, 2007• 2 citations
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Abstract

Recently the explicit applicability of bound entanglement in quantum cryptography has been shown. In this paper some of recent results respecting this topic are reviewed. In particular relevant notions and definitions are reminded. The new construction of bound entangled states containing secure correlations is presented. It provides low dimensional 6\otimes6 bound entangled states with nonzero distillable key.

DOI
10.48550/arxiv.0712.3999

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Recently the explicit applicability of bound entanglement in quantum cryptography has been shown. In this paper some of recent results respecting this topic are reviewed. In particular relevant notions and definitions are reminded. The new construction of bound entangled states containing secure correlations is presented. It provides low dimensional 6\otimes6 bound entangled states with nonzero distillable key.
Abstract An entangled state is bound entangled if one cannot combine any number of copies of the state to a maximally entangled state by using only local operations and classical communication. If one formalizes this notion of bound entanglement, one arrives immediately at four different definitions. In addition, at least three more definitions are commonly used in the literature, in particular in the very first paper on bound entanglement. Here we review critical distillation protocols and we examine how different results from quantum information theory interact in order to prove that all seven definitions are eventually equivalent. Our self-contained analysis unifies and extends previous results scattered in the literature and reveals details of the structure of bound entanglement.
Recently it has been shown that quantum cryptography beyond pure entanglement distillation is possible and a paradigm for the associated protocols has been established. Here we systematically generalize the whole paradigm to the multipartite scenario. We provide constructions of new classes of multipartite bound entangled states, i.e., those with underlying twisted Greenberger-Horne-Zeilinger (GHZ) structure and nonzero distillable cryptographic key. We quantitatively estimate the key from below with the help of the privacy squeezing technique.
Bound entanglement is a special form of quantum entanglement that cannot be used for distillation, i.e. the local transformation of copies of arbitrarily entangled states into a smaller number of approximately maximally entangled states. Implying an inherent irreversibility of quantum resources, this phenomenon highlights the gaps in our current theory of entanglement. This review provides a comprehensive exploration of the key findings on bipartite bound entanglement. We focus on systems of finite dimensions, an area of high relevance for many quantum information processing tasks. We elucidate the properties of bound entanglement and its interconnections with various facets of quantum information theory and quantum information processing. The paper highlights areas where our understanding of bound entangled states, particularly in their detection and characterization, remains underdeveloped. By highlighting the need for further research into this phenomenon and underscoring relevant open questions, this paper invites researchers to unravel its relevance for our understanding of entanglement in Nature and how this resource can most effectively be used for applications in quantum technology.
It is proven that recently introduced states with perfectly secure bits of cryptographic key (private states representing secure bit) [K. Horodecki et al., Phys. Rev. Lett. 94, 160502 (2005)] as well as its multipartite and higher dimension generalizations always represent distillable entanglement. The corresponding lower bounds on distillable entanglement are provided. We also present a simple alternative proof that for any bipartite quantum state entanglement cost is an upper bound on a distillable cryptographic key in a bipartite scenario.
We construct a class of entangled states in ℋ = ℋA ⊗ ℋB ⊗ ℋC quantum systems with dim ℋA = dim ℋB = dim ℋC = 2 and classify those states with respect to their distillability properties. The states are bound entanglement for the bipartite split (AB) - C. The states are non-positive partial transpose (NPT) entanglement and 1-copy undistillable for the bipartite splits A - (BC) and B - (AC). Moreover, we generalize the results of 2 ⊗ 2 ⊗ 2 systems to the case of 2n ⊗ 2n ⊗ 2n systems.
We characterize the set of shared quantum states which contain a cryptographically private key. This allows us to recast the theory of privacy as a paradigm closely related to that used in entanglement manipulation. It is shown that one can distill an arbitrarily secure key from bound entangled states. There are also states that have less distillable private keys than the entanglement cost of the state. In general, the amount of distillable key is bounded from above by the relative entropy of entanglement. Relationships between distillability and distinguishability are found for a class of states which have Bell states correlated to separable hiding states. We also describe a technique for finding states exhibiting irreversibility in entanglement distillation.