Low-dimensional quite noisy bound entanglement with a cryptographic key

Abstract

We provide a class of bound entangled states that have positive distillable secure key rate. The smallest state of this kind is 4 ⊗ 4. Our class is a generalization of the class presented in [1]. It is much wider, containing, in particular, states from the boundary of PPT entangled states (all of the states in the class in [1] were of this kind) but also states inside the set of PPT entangled states, even, approaching the separable states. This generalization comes with a price: for the wider class a positive key rate requires, in general, apart from the one-way Devetak-Winter protocol (used in [1]) also the recurrence preprocessing and thus effectively is a two-way protocol. We also analyze the amount of noise that can be admixtured to the states of our class without losing 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 class in [1]).

Universal Limitations on Quantum Key Distribution over a Network

“…Nevertheless distillation of Φ GHZ M states is only an example of key distillation technique [46,[114][115][116][117][118]. More general conference key agreement scenario of our interest incorporates distillation of twisted Φ GHZ M states (see Definition 3) [46,59,105,119,120]. In that case, a different approach for upper bounding conference key rates than estimation of conversion Φ W M to Φ GHZ M rates is required.…”

Hybrid zero-capacity channels

“…Setting k = f (n) , where f (n) is the function of the sublinear growth, the state (11) provides the the family of states that are f (n)-hybrid, with vanishing amount of entanglement on a single copy level in the limit of large n. The tensor product η AA BB f (n) ⊗n is an f (n)hybrid state, which is very far from the set SEP of separable states.…”

“…where γ AA BB min is the 4 ⊗ 4 pbit with the smallest possible dimensions of the shield recently introduced by [11], and σ BB is the erasure flag. The authors in [5] exhibit a 3 ⊗ 3 state which achieves the same goal, although there is no known distillation protocol to obtain a key.…”

Construction and properties of a class of private states in arbitrary dimensions

“…A quantum pdit [10][11][12][13][14][15], Q uite recently, an im portant discovery has been m ade in studies betw een security and correlations. In [16], a clean classical analog o f bound entanglem ent and private bound entanglem ent has been provided, w here the authors have constructed private bound entangled states based on unam biguous classical probability distribution to a quantum state that is not based on a "standard" key or shield scheme, opening a new direction in the study o f private states.…”

“…A quantum p-dit is composed from a d ⊗ d AB part called "key", and A ′ B ′ called "shield", shared between Alice (subsystems AA ′ ) and Bob (subsystems BB ′ ) in such a way that the local von Neumann measurements on the key part in a particular basis will make its results completely statistically uncorrelated from the results of any measurement of an eavesdropper Eve on her subsystem E, which is a part of the purification |Ψ ABA ′ B ′ E of the p-dit state ρ ABA ′ B ′ . Pdits (especially pbits), have been studied extensively for some time [10][11][12][13][14][15].…”