We provide a class of bound entangled states that have positive distillable secure key rate. The smallest state of this kind is 4 ⊗ 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 by one-way Devetak-Winter protocol. Subsequently the volume of bound entangled keydistillable states in arbitrary dimension 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 7 collective settings is proven to be optimal in number of settings.Quantum cryptography is one of the very interesting practical phenomena within quantum information theory [1,2,3,4]. There were in general two ideas to produce cryptographic key. The first was based on sending nonorthogonal states [5], the second -on specially chosen measurements of maximally entangled pairs [6]. They have been shown to be equivalent in general [7] including most general eavesdropper attack. An important ingredient of the protocol was so called Quantum Privacy Amplification [8] based on distillation of EPR paris [9]. Despite of natural expectations, that distillability of EPR pairs is a precondition of secure key it has been recently shown [10,11] that the class of states which contain ideal key (private states) is much wider than class of maximally entangled states. It has been shown that certain bound entangled (BE) states [12] (from which no EPR pair can be distilled) can be distilled to (approximate) private states. In other words -surprisingly -there are bound entangled states with K D > 0.However the BE states with nonzero distillable key K D > 0 of Ref. [10,11] require Hilbert space of quite large dimension, which makes impression that BE states useful for cryptography are rather exceptional, and far from experimental regime. In this paper we provide a general construction of class of BE states with K D > 0 and give examples of 4 ⊗ 4 states having this property. To this end we consider binary mixture {p 1 , p 2 } of two orthogonal private bits (private states with at least one bit of ideal key). We first show that any such non equal mixture has K D > 0, and can be distilled by -quite remarkablyone-way Devetak-Winter protocol [13]. Next we construct special pairs of private bits, and show that for certain probability their mixture is key distillable and bound entangled. To obtain this we assure that the state remains positive (and even invariant) under partial transposition (PPT) [14] which is sufficient condition for a state to be non distillable [12]. We then provide an example of the smallest state of our construction which resides on 4 qubits. Basing on this family of BE key distillable states we argue, that the volume of BE states with K D > 0 is nonzero in arbitrary dimension. We exploit their properties, and consider their experimental preparation. We then show how to verify that experimentally prepared state has nonzero disti...

We consider the problem of existence of bound entangled states with non-positive partial transpose (NPT). As one knows, existence of such states would in particular imply nonadditivity of distillable entanglement. Moreover it would rule out a simple mathematical description of the set of distillable states. Distillability is equivalent to so called n-copy distillability for some n. We consider a particular state, known to be 1-copy nondistillable, which is supposed to be bound entangled. We study the problem of its two-copy distillability, which boils down to show that maximal overlap of some projector Q with Schmidt rank two states does not exceed 1/2. Such property we call the half-property. We first show that the maximum overlap can be attained on vectors that are not of the simple product form with respect to cut between two copies. We then attack the problem in twofold way: a) prove the half-property for some classes of Schmidt rank two states b) bound the required overlap from above for all Schmidt rank two states. We have succeeded to prove the half-property for wide classes of states, and to bound the overlap from above by c < 3/4. Moreover, we translate the problem into the following matrix analysis problem: bound the sum of the squares of the two largest singular values of matrix A ⊗ I + I ⊗ B with A, B traceless 4 × 4 matrices, and TrA † A + TrB † B = 1 4 .

We conjecture new uncertainty relations which restrict correlations between results of measurements performed by two separated parties on a shared quantum state. The first uncertainty relation bounds the sum of two mutual informations when one party measures a single observable and the other party measures one of two observables. The uncertainty relation does not follow from MaassenUffink uncertainty relation and is much stronger than Hall uncertainty relation derived from the latter. The second uncertainty relation bounds the sum of two mutual informations when each party measures one of two observables. We provide numerical evidence for validity of conjectured uncertainty relations and prove them for large classes of states and observables.

We present a constructive example of violation of additivity of minimum output Rényi entropy for each p > 2. The example is provided by antisymmetric subspace of a suitable dimension. We discuss possibility of extension of the result to go beyond p > 2 and obtain additivity for p = 0 for a class of entanglement breaking channels.

The problem of sharing entanglement over large distances is crucial for implementations of quantum cryptography. A possible scheme for long-distance entanglement sharing and quantum communication exploits networks whose nodes share Einstein-Podolsky-Rosen (EPR) pairs. In Perseguers et al. [Phys. Rev. A 78, 062324 (2008)] the authors put forward an important isomorphism between storing quantum information in a dimension D and transmission of quantum information in a D + 1-dimensional network. We show that it is possible to obtain long-distance entanglement in a noisy two-dimensional (2D) network, even when taking into account that encoding and decoding of a state is exposed to an error. For 3D networks we propose a simple encoding and decoding scheme based solely on syndrome measurements on 2D Kitaev topological quantum memory. Our procedure constitutes an alternative scheme of state injection that can be used for universal quantum computation on 2D Kitaev code. It is shown that the encoding scheme is equivalent to teleporting the state, from a specific node into a whole two-dimensional network, through some virtual EPR pair existing within the rest of network qubits. We present an analytic lower bound on fidelity of the encoding and decoding procedure, using as our main tool a modified metric on space-time lattice, deviating from a taxicab metric at the first and the last time slices.Suppose we have a network of laboratories with some fixed distance between neighboring ones, and one of them wants to establish quantum communication with another one. We assume that neighboring labs can directly exchange quantum communication with some small, fixed error. This can be used e.g. to share some noisy Einstein-Podolsky-Rosen (EPR) pairs between the neighboring labs. We also assume that all operations performed within each lab may be faulty with some fixed, small probability. If two distant labs can achieve quantum communication with the help of all the labs in the network, then they can exploit it to share cryptographic key that will be known only to these two labs. This scenario was put forward in [1], and is an alternative to quantum repeaters [2,3]. It is also closely related to entanglement percolation [4]. In [1] the question was posed whether for a 2-dimensional network, in principle, one can perform quantum communication over an arbitrary distance, provided that one can execute gates between the adjacent nodes (i.e. local gates), and the size of the system in each node is constant (i.e. it does not depend on the distance), so that the nodes do not need quantum memory. The answer was affirmative. Namely, the authors represented nearest neighbour quantum computation on a line as a teleportation process on quantum 2-dimensional square networks, where entangled pairs are shared between adjacent nodes (i.e. between those that are separated by a size of an elementary cell a of the network). 1-dimensional system for quantum computation is formed by all nodes belonging to a chosen line forming a diagonal of elementary ce...