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arXiv:quant-ph/0603085 · Journal of Physics A Mathematical and Theoretical

General entanglement-assisted transformation for bipartite pure quantum states

Wei Song, Yan Hua Huang, Nai-Le Liu, Zeng‐Bing Chen

Published Jan 9, 2007• 2 citations

Abstract

We introduce the general catalysts for pure entanglement transformations under local operations and classical communications in such a way that we disregard the profit and loss of entanglement of the catalysts per se. As such, the possibilities of pure entanglement transformations are greatly expanded. Remarkably, we find an interesting phenomenon that, in some situations, incomparable pairs ${| ψ> ,| ϕ> \} $ and ${| χ> ,| χ^{\prime}> \} $ can assist each other mutually so as to realize the transformation $| ψ> | χ> \to| ϕ> | χ^{\prime}>$. We also design an efficient algorithm to detect whether a $k\times k$ general catalyst exists for a given entanglement transformation. This algorithm can as well be exploited to witness the existence of standard catalysts.

DOI: 10.1088/1751-8113/40/4/014

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Journal of Physics A Mathematical and General•Nov 11, 2004•1 citation

We introduce the general catalysts for pure entanglement transformations under local operations and classical communications in such a way that we disregard the profit and loss of entanglement of the catalysts per se. As such, the possibilities of pure entanglement transformations are greatly expanded. Remarkably, we find an interesting phenomenon that, in some situations, incomparable pairs

{| ψ> ,| ϕ> \}

and

{| χ> ,| χ^{\prime}> \}

can assist each other mutually so as to realize the transformation

| ψ> | χ> \to| ϕ> | χ^{\prime}>

. We also design an efficient algorithm to detect whether a

k\times k

general catalyst exists for a given entanglement transformation. This algorithm can as well be exploited to witness the existence of standard catalysts.

Without additional resources, it is often impossible to transform one entangled quantum state into another with local quantum operations and classical communication. Jonathan and Plenio (Phys. Rev. Lett., vol. 83, p. 3566, 1999) presented an interesting example showing that the presence of another state, called a catalyst, enables such a transformation without changing the catalyst. They also pointed out that in general it is very hard to find an analytical condition under which a catalyst exists. In this paper, we study the existence of catalysts for two incomparable quantum states. For the simplest case of 2/spl times/2 catalysts for transformations from one 4/spl times/4 state to another, a necessary and sufficient condition for existence is found. For the general case, we give an efficient polynomial time algorithm to decide whether a k/spl times/k catalyst exists for two n/spl times/n incomparable states, where k is treated as a constant.

Given a pure state transformation $\ensuremath{\psi}\ensuremath{\mapsto}\ensuremath{\phi}$ restricted to entanglement-assisted local operations with classical communication, we determine a lower bound for the dimension of a catalyst allowing that transformation. Our bound is stated in terms of the generalized concurrence monotones (the usual concurrence of two qubits is one such monotone). We further provide tools for deriving further conditions upon catalysts of pure state transformations.

Auxiliary pure quantum entangled states shared between two parties can act as catalysts in bipartite entanglement transformations. The participation of a catalyst state in the transformation can enhance its success probability. We consider transformations involving entanglement concentration of a finite number of copies of arbitrary two-qubit pure states into a single copy of a maximally entangled two-qubit pure state using bipartite local quantum operations and classical communication aided by a catalyst state. We show the enhancement of success probabilities of the catalyst-assisted procedure compared to the case where no catalysts are used. The optimal two-qubit catalyst and the maximum probability of the transformation using such catalysts can be obtained analytically. The enhancement of concentration success probability using catalysts of higher dimensions is demonstrated numerically. Finally, catalyst aided strategies for obtaining multiple maximally entangled two-qubit states are discussed.

We analytically obtain the maximum probability of converting a finite number of copies of an arbitrary two-qubit pure state to a single copy of a maximally entangled two-qubit pure state via entanglement assisted local operations and classical communications using a two-qubit catalyst state. We show that the optimal catalyst for this transformation is always more entangled than the initial state but any two-qubit state can act as a (non-optimal) catalyst. Interestingly, the entanglement of the optimal two-qubit catalyst state is shown to decrease with that of the initial state. Entanglement assisted strategies for obtaining multiple Bell states are discussed.

The quantum entanglement is the necessary resource,which could realize some quantum tasks.It is very important to study the algorithm for the entanglement transformation between the quantum states determinately or probability.An algorithm first is presented that transforms one pure bipartite quantum state into a mixed bipartite by Local Operations and Classical Communication(LOCC),this algorithm is much simpler than the original algorithm.The algorithm for the transformation between two pure bipartite states with catalysis by LOCC is then discussed.