Simple Proof of Security of the BB84 Quantum Key Distribution Protocol

We prove the security of the 1984 protocol of Bennett and Brassard (BB84) for quantum key distribution. We first give a key distribution protocol based on entanglement purification, which can be proven secure using methods from Lo and Chau's proof of security for a similar protocol. We then show that the security of this protocol implies the security of BB84. The entanglement-purification based protocol uses Calderbank-Shor-Steane (CSS) codes, and properties of these codes are used to remove the use of quantum computation from the Lo-Chau protocol.Comment: 5 pages, Latex, minor changes to improve clarity and fix typo

The Power of LOCCq State Transformations

Reversible state transformations under entanglement non-increasing operations give rise to entanglement measures. It is well known that asymptotic local operations and classical communication (LOCC) are required to get a simple operational measure of bipartite pure state entanglement. For bipartite mixed states and multipartite pure states it is likely that a more powerful class of operations will be needed. To this end \cite{BPRST01} have defined more powerful versions of state transformations (or reducibilities), namely LOCCq (asymptotic LOCC with a sublinear amount of quantum communication) and CLOCC (asymptotic LOCC with catalysis). In this paper we show that {\em LOCCq state transformations are only as powerful as asymptotic LOCC state transformations} for multipartite pure states. We first generalize the concept of entanglement gambling from two parties to multiple parties: any pure multipartite entangled state can be transformed to an EPR pair shared by some pair of parties and that any irreducible $m$ $(m\ge 2)$ party pure state can be used to create any other state (pure or mixed), using only local operations and classical communication (LOCC). We then use this tool to prove the result. We mention some applications of multipartite entanglement gambling to multipartite distillability and to characterizations of multipartite minimal entanglement generating sets. Finally we discuss generalizations of this result to mixed states by defining the class of {\em cat distillable states}

Can closed timelike curves or nonlinear quantum mechanics improve quantum state discrimination or help solve hard problems?

We study the power of closed timelike curves (CTCs) and other nonlinear extensions of quantum mechanics for distinguishing nonorthogonal states and speeding up hard computations. If a CTC-assisted computer is presented with a labeled mixture of states to be distinguished--the most natural formulation--we show that the CTC is of no use. The apparent contradiction with recent claims that CTC-assisted computers can perfectly distinguish nonorthogonal states is resolved by noting that CTC-assisted evolution is nonlinear, so the output of such a computer on a mixture of inputs is not a convex combination of its output on the mixture's pure components. Similarly, it is not clear that CTC assistance or nonlinear evolution help solve hard problems if computation is defined as we recommend, as correctly evaluating a function on a labeled mixture of orthogonal inputs.Comment: 4 pages, 3 figures. Final version. Added several references, updated discussion and introduction. Figure 1(b) very much enhance

Can non-private channels transmit quantum information?

We study the power of quantum channels with little or no capacity for private communication. Because privacy is a necessary condition for quantum communication, one might expect that such channels would be of little use for transmitting quantum states. Nevertheless, we find strong evidence that there are pairs of such channels that, when used together, can transmit far more quantum information than the sum of their individual private capacities. Because quantum transmissions are necessarily private, this would imply a large violation of additivity for the private capacity. Specifically, we present channels which display either (1) A large joint quantum capacity but very small individual private capacities or (2) a severe violation of additivity for the Holevo information.Comment: We both think so. 4 pages and 3 figures explain wh

The Parity Bit in Quantum Cryptography

An $n$-bit string is encoded as a sequence of non-orthogonal quantum states. The parity bit of that $n$-bit string is described by one of two density matrices, $\rho_0^{(n)}$ and $\rho_1^{(n)}$, both in a Hilbert space of dimension $2^n$. In order to derive the parity bit the receiver must distinguish between the two density matrices, e.g., in terms of optimal mutual information. In this paper we find the measurement which provides the optimal mutual information about the parity bit and calculate that information. We prove that this information decreases exponentially with the length of the string in the case where the single bit states are almost fully overlapping. We believe this result will be useful in proving the ultimate security of quantum crytography in the presence of noise.Comment: 19 pages, RevTe

Exact and Asymptotic Measures of Multipartite Pure State Entanglement

In an effort to simplify the classification of pure entangled states of multi (m) -partite quantum systems, we study exactly and asymptotically (in n) reversible transformations among n'th tensor powers of such states (ie n copies of the state shared among the same m parties) under local quantum operations and classical communication (LOCC). With regard to exact transformations, we show that two states whose 1-party entropies agree are either locally-unitarily (LU) equivalent or else LOCC-incomparable. In particular we show that two tripartite Greenberger-Horne-Zeilinger (GHZ) states are LOCC-incomparable to three bipartite Einstein-Podolsky-Rosen (EPR) states symmetrically shared among the three parties. Asymptotic transformations result in a simpler classification than exact transformations. We show that m-partite pure states having an m-way Schmidt decomposition are simply parameterizable, with the partial entropy across any nontrivial partition representing the number of standard ``Cat'' states (|0^m>+|1^m>) asymptotically interconvertible to the state in question. For general m-partite states, partial entropies across different partitions need not be equal, and since partial entropies are conserved by asymptotically reversible LOCC operations, a multicomponent entanglement measure is needed, with each scalar component representing a different kind of entanglement, not asymptotically interconvertible to the other kinds. In particular the m=4 Cat state is not isentropic to, and therefore not asymptotically interconvertible to, any combination of bipartite and tripartite states shared among the four parties. Thus, although the m=4 cat state can be prepared from bipartite EPR states, the preparation process is necessarily irreversible, and remains so even asymptotically.Comment: 13 pages including 3 PostScript figures. v3 has updated references and discussion, to appear Phys. Rev.