A Method of Areas for Manipulating the Entanglement Properties of One Copy of a Two-Particle Pure State

We consider the problem of how to manipulate the entanglement properties of a general two-particle pure state, shared between Alice and Bob, by using only local operations at each end and classical communication between Alice and Bob. A method is developed in which this type of problem is found to be equivalent to a problem involving the cutting and pasting of certain shapes along with a certain colouring problem. We consider two problems. Firstly we find the most general way of manipulating the state to obtain maximally entangled states. After such a manipulation the entangled state |11>+|22>+....|mm> is obtained with probability p_m. We obtain an expression for the optimal average entanglement. Also, some results of Lo and Popescu pertaining to this problem are given simple geometric proofs. Secondly, we consider how to manipulate one two particle entangled pure state to another with certainty. We derive Nielsen's theorem (which states the necessary and sufficient condition for this to be possible) using the method of areas.Comment: 29 pages, 9 figures. Section 2.4 clarified. Error in second colouring theorem (section 3.2) corrected. Some other minor change

Eavesdropping without quantum memory

In quantum cryptography the optimal eavesdropping strategy requires that the eavesdropper uses quantum memories in order to optimize her information. What happens if the eavesdropper has no quantum memory? It is shown that the best strategy is actually to adopt the simple intercept/resend strategy.Comment: 9 pages LaTeX, 3 figure

Fault tolerant quantum key distribution protocol with collective random unitary noise

We propose an easy implementable prepare-and-measure protocol for robust quantum key distribution with photon polarization. The protocol is fault tolerant against collective random unitary channel noise. The protocol does not need any collective quantum measurement or quantum memory. A security proof and a specific linear optical realization using spontaneous parametric down conversion are given.Comment: Accepted by PRA as a Rapid Communicatio

Broadcasting of three qubit entanglement via local copying and entanglement swapping

In this work,We investigate the problem of secretly broadcasting of three-qubit entangled state between two distant partners. The interesting feature of this problem is that starting from two particle entangled state shared between two distant partners we find that the action of local cloner on the qubits and the measurement on the machine state vector generates three-qubit entanglement between them. The broadcasting of entanglement is made secret by sending the measurement result secretly using cryptographic scheme based on orthogonal states. Further we show that this idea can be extended to generate three particle entangled state between three distant partners.Comment: 18 pages, 4 figures, Accepted in Physical Review

When only two thirds of the entanglement can be distilled

We provide an example of distillable bipartite mixed state such that, even in the asymptotic limit, more pure-state entanglement is required to create it than can be distilled from it. Thus, we show that the irreversibility in the processes of formation and distillation of bipartite states, recently proved in [G. Vidal, J.I. Cirac, Phys. Rev. Lett. 86, (2001) 5803-5806], is not limited to bound-entangled states.Comment: 4 pages, revtex, 1 figur

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

A classical analogue of entanglement

We show that quantum entanglement has a very close classical analogue, namely secret classical correlations. The fundamental analogy stems from the behavior of quantum entanglement under local operations and classical communication and the behavior of secret correlations under local operations and public communication. A large number of derived analogies follow. In particular teleportation is analogous to the one-time-pad, the concept of ``pure state'' exists in the classical domain, entanglement concentration and dilution are essentially classical secrecy protocols, and single copy entanglement manipulations have such a close classical analog that the majorization results are reproduced in the classical setting. This analogy allows one to import questions from the quantum domain into the classical one, and vice-versa, helping to get a better understanding of both. Also, by identifying classical aspects of quantum entanglement it allows one to identify those aspects of entanglement which are uniquely quantum mechanical.Comment: 13 pages, references update

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

Entanglement Swapping Chains for General Pure States

We consider entanglement swapping schemes with general (rather than maximally) entangled bipartite states of arbitary dimension shared pairwise between three or more parties in a chain. The intermediate parties perform generalised Bell measurements with the result that the two end parties end up sharing a entangled state which can be converted into maximally entangled states. We obtain an expression for the average amount of maximal entanglement concentrated in such a scheme and show that in a certain reasonably broad class of cases this scheme is provably optimal and that, in these cases, the amount of entanglement concentrated between the two ends is equal to that which could be concentrated from the weakest link in the chain.Comment: 18 pages, 5 figure

Thermodynamics and the Measure of Entanglement

We point out formal correspondences between thermodynamics and entanglement. By applying them to previous work, we show that entropy of entanglement is the unique measure of entanglement for pure states.Comment: 8 pages, RevTeX; edited for clarity, additional references, to appear as a Rapid Communication in Phys. Rev.