An unextendible product basis (UPB) for a multipartite quantum system is an incomplete orthogonal product basis whose complementary subspace contains no product state. We give examples of UPBs, and show that the uniform mixed state over the subspace complementary to any UPB is a bound entangled state. We exhibit a tripartite 2x2x2 UPB whose complementary mixed state has tripartite entanglement but no bipartite entanglement, i.e. all three corresponding 2x4 bipartite mixed states are unentangled. We show that members of a UPB are not perfectly distinguishable by local POVMs and classical communication.Comment: 4 pages RevTex + 1 eps fig. This is version 2 with many changes and simplifications, and an additional autho

Active quantum error correction using qubit stabilizer codes has emerged as a promising, but experimentally challenging, engineering program for building a universal quantum computer. In this review we consider the formalism of qubit stabilizer and subsystem stabilizer codes and their possible use in protecting quantum information in a quantum memory. We review the theory of fault-tolerance and quantum error-correction, discuss examples of various codes and code constructions, the general quantum error correction conditions, the noise threshold, the special role played by Clifford gates and the route towards fault-tolerant universal quantum computation. The second part of the review is focused on providing an overview of quantum error correction using two-dimensional (topological) codes, in particular the surface code architecture. We discuss the complexity of decoding and the notion of passive or self-correcting quantum memories. The review does not focus on a particular technology but discusses topics that will be relevant for various quantum technologies.

We show that a class of quantum computations that was recently shown to be efficiently simulatable on a classical computer by Valiant corresponds to a physical model of noninteracting fermions in one dimension. We give an alternative proof of his result using the language of fermions and extend the result to noninteracting fermions with arbitrary pairwise interactions, where gates can be conditioned on outcomes of complete von Neumann measurements in the computational basis on other fermionic modes in the circuit. This last result is in remarkable contrast with the case of noninteracting bosons where universal quantum computation can be achieved by allowing gates to be conditioned on classical bits (quant-ph/0006088).Comment: 26 pages, 1 figure, uses wick.sty; references added to recent results by E. Knil

We report new results and generalizations of our work on unextendible product bases (UPB), uncompletable product bases and bound entanglement. We present a new construction for bound entangled states based on product bases which are only completable in a locally extended Hilbert space. We introduce a very useful representation of a product basis, an orthogonality graph. Using this representation we give a complete characterization of unextendible product bases for two qutrits. We present several generalizations of UPBs to arbitrary high dimensions and multipartite systems. We present a sufficient condition for sets of orthogonal product states to be distinguishable by separable superoperators. We prove that bound entangled states cannot help increase the distillable entanglement of a state beyond its regularized entanglement of formation assisted by bound entanglement.

We introduce the notion of a Schmidt number of a bipartite density matrix, characterizing the minimum Schmidt rank of the pure states that are needed to construct the density matrix. We prove that Schmidt number is nonincreasing under local quantum operations and classical communication. We show that k-positive maps witness Schmidt number, in the same way that positive maps witness entanglement. We show that the family of states which is made from mixing the completely mixed state and a maximally entangled state have increasing Schmidt number depending on the amount of maximally entangled state that is mixed in. We show that Schmidt number does not necessarily increase when taking tensor copies of a density matrix ρ; we give an example of a density matrix for which the Schmidt numbers of ρ and ρ ⊗ ρ are both 2.In quantum information theory the study of bipartite entanglement is of great importance. The usual scenario is one in which two parties, Alice and Bob, share a supply of n pure or mixed states ρ ⊗n which they would like to convert by Local Operations and Classical Communication (denoted as LO + CC) to a supply of k other mixed or pure states σ ⊗k , where k can either be smaller or larger than n. The simple question that underlies many studies in bipartite entanglement is the question: what properties of these two sets of states make it possible or impossible to carry out such a protocol? Much work has been devoted to developing the necessary and sufficient conditions for this LO + CC convertability. In the case of pure state convertability, it has been found that some aspects of this problem can be understood with the mathematics of majorization [1]. In the case of mixed state entanglement the theory of positive maps has been shown to play an important role [2]. The power of positive maps is best illustrated by the Peres separability condition [3] which says that a bipartite density matrix which is unentangled (aka separable) must be positive under the application of the partial transposition map. For low dimensional spin systems this condition is not only necessary but also sufficient to ensure separability [2]. It has been shown [4] that density matrices which are positive under partial transposition are undistillable, that is, nonconvertible by LO + CC to sets of entangled pure states. Many examples of these bound entangled states have been found [5][6][7][8]. Evidence has been found as well for the nondistillability of certain classes of entangled states which are not positive under partial transposition [9,10], and it was shown that this feature relates to the 2-positivity of certain maps [9].In this paper, we extend the LO + CC classification of bipartite mixed states with the use of positive maps. In particular, we extend the notion of the Schmidt rank of a pure bipartite state to the domain of bipartite density matrices. We will show that this new quantity, which we will call Schmidt number, is witnessed by k-positive maps.For a bipartite pure state which we write in its Schmidt decomposition (see ...

We introduce a measure of both quantum as well as classical correlations in a quantum state, the entanglement of purification. We show that the ͑regularized͒ entanglement of purification is equal to the entanglement cost of creating a state asymptotically from maximally entangled states, with negligible communication. We prove that the classical mutual information and the quantum mutual information divided by two are lower bounds for the regularized entanglement of purification. We present numerical results of the entanglement of purification for Werner states in H 2 H 2 .

Abstract-We expand on our work on Quantum Data Hiding [1] -hiding classical data among parties who are restricted to performing only local quantum operations and classical communication (LOCC). We review our scheme that hides one bit between two parties using Bell states, and we derive upper and lower bounds on the secrecy of the hiding scheme. We provide an explicit bound showing that multiple bits can be hidden bitwise with our scheme. We give a preparation of the hiding states as an efficient quantum computation that uses at most one ebit of entanglement. A candidate data hiding scheme that does not use entanglement is presented. We show how our scheme for quantum data hiding can be used in a conditionally secure quantum bit commitment scheme.

We analyze and compare the mathematical formulations of the criterion for separability for bipartite density matrices and the Bell inequalities. We show that a violation of a Bell inequality can formally be expressed as a witness for entanglement. We also show how the criterion for separability and the existence of a description of the state by a local hidden variable theory, become equivalent when we restrict the set of local hidden variable theories to the domain of quantum mechanics. This analysis sheds light on the essential difference between the two criteria and may help us in understanding whether there exist entangled states for which the statistics of the outcomes of all possible local measurements can be described by a local hidden variable theory.