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arXiv:1008.1226 · Journal of Physics A Mathematical and Theoretical

Low-dimensional quite noisy bound entanglement with a cryptographic key

Łukasz Pankowski, Michał Horodecki

Published Dec 13, 2010• 4 citations

Abstract

We provide a class of bound entangled states that have a positive distillable secure key rate. The smallest state of this kind is 4⊗4. Our class is a generalization of the class presented in Horodecki et al (2008 IEEE Trans. Inf. Theory 54 2621–5). It is much wider, containing, in particular, states from the boundary of PPT entangled states (all of the states in the previous class were of this kind) and also states inside the set of PPT entangled states, even approaching the separable states. This generalization comes at a price: for the wider class, a positive key rate requires, in general, apart from the one-way Devetak–Winter protocol (used in the previous case) also the recurrence preprocessing and thus is effectively a two-way protocol. We also analyze the amount of noise that can be admixtured to the states of our class without losing the key distillability property which may be crucial for experimental realization. The wider class contains key-distillable states with higher entropy (up to 3.524, as opposed to 2.564 for the previous class).

DOI: 10.1088/1751-8113/44/3/035301

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In this paper, we provide a class of bound entangled states that have positive distillable secure key rate. The smallest state of this kind is 4 otimes 4, which shows that peculiar security contained in bound entangled states does not need high-dimensional systems. We show that for these states a positive key rate can be obtained byone-wayDevetak-Winter (DW) protocol. Subsequently, the volume of bound entangled key-distillable states formotimesnHilbert space withm,n> 4 is shown to be nonzero. We provide a scheme of verification of cryptographic quality of experimentally prepared state in terms of local observables. Proposed set of seven collective settings is proven to be optimal in number of settings.

We present sufficient conditions for states to have positive distillable key rate. Exploiting the conditions, we show that the bound entangled states given by Horodecki et al. [Phys. Rev. Lett. 94, 160502 (2005)] have nonzero distillable key rate and finally exhibit new classes of bound entangled states with positive distillable key rate, but with negative Devetak-Winter lower bound of the distillable key rate for the classical-classical-quantum (ccq) states of their privacy squeezed versions, which represent the states after the local measurement performed in the key part without considering the shield part.

In this paper, we develop a formalism for distilling a classical key from a quantum state in a systematic way, expanding on our previous work on a secure key from bound entanglement (Horodecki, 2005). More detailed proofs, discussion, and examples are provided of the main results. Namely, we demonstrate that all quantum cryptographic protocols can be recast in a way which looks like entanglement theory, with the only change being that instead of distilling Einstein–Podolsky–Rosen (EPR) pairs, the parties distill private states. The form of these general private states are given, and we show that there are a number of useful ways of expressing them. Some of the private states can be approximated by certain states, which are bound entangled. Thus, distillable entanglement is not a requirement for a private key. We find that such bound entangled states are useful for a cryptographic primitive we call a controlled private quantum channel (PQC). We also find a general class of states, which have negative partial transpose (are NPT), but which appear to be bound entangled. The relative entropy distance is shown to be an upper bound on the rate of a key. This allows us to compute theexactvalue of a distillable key for a certain class of private states.

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.

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 introduce systematically with the help of Weyl operators novel classes of multipartite and multidimensional states which are all bound entangled for arbitrary dimension. We find that the entanglement is bound due to different reasons: unlockable due to the multi--particle nature and some states are in addition bound due the fact being positive under partial transposition (PPT). By a general construction (W simplices) we obtain classes of states which have the same geometry concerning separability and entanglement independent of the number of involved particle pairs. Moreover, we introduce a distillation protocol and demonstrate for d=3 that for a certain set of states the entanglement can be increased only up to a certain amount.