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1 Catalyst-assisted Probabilistic Entanglement Transformation

Yuan Feng, Runyao Duan, Mingsheng Ying

Published Jan 1, 2004• 14 citations
OpenAlex

Abstract

Abstract — We examine catalyst-assisted probabilistic entanglement transformations. A sufficient and necessary condition of when some appropriate catalyst exists that can increase the maximal transformation probability is presented. We also design an algorithm which leads to an efficient method to find the most economical partial catalyst with minimal dimension. The mathematical structure of catalyst-assisted probabilistic transformation is carefully examined. Index Terms — Probabilistic entanglement transformation, catalyst-assisted transformation, partial catalyst.

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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.

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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.

The Journal of Chemical Physics•Apr 2, 2021•2 citations
Abstract — We examine catalyst-assisted probabilistic entanglement transformations. A sufficient and necessary condition of when some appropriate catalyst exists that can increase the maximal transformation probability is presented. We also design an algorithm which leads to an efficient method to find the most economical partial catalyst with minimal dimension. The mathematical structure of catalyst-assisted probabilistic transformation is carefully examined. Index Terms — Probabilistic entanglement transformation, catalyst-assisted transformation, partial catalyst.
We are concerned with catalyst-assisted probabilistic entanglement transformations. A necessary and sufficient condition is presented under which there exist partial catalysts that can increase the maximal transforming probability of a given entanglement transformation. We also design an algorithm which leads to an efficient method for finding the most economical partial catalysts with minimal dimension. The mathematical structure of catalyst-assisted probabilistic transformation is carefully investigated.
We examine the powers of entanglement-assisted transformation and multiple-copy entanglement transformation. First, we find a sufficient condition of when a given catalyst is useful in producing another specific target state. As an application of this condition, for any non-maximally entangled bipartite pure state and any integer nnn not less than 4, we are able to explicitly construct a set of n×nn\times nn×n quantum states which can be produced by using the given state as a catalyst. Second, we prove that for any positive integer kkk, entanglement-assisted transformation with k×kk\times kk×k-dimensional catalysts is useful in producing a target state if and only if multiple-copy entanglement transformation with kkk copies of state is useful in producing the same target. Moreover, a necessary and sufficient condition for both of them is obtained in terms of the Schmidt coefficients of the target. This equivalence of entanglement-assisted transformation and multiple-copy entanglement transformation implies many interesting properties of entanglement transformation. Furthermore, these results are generalized to the case of probabilistic entanglement transformations.
We show that in some cases, catalyst-assisted entanglement transformation cannot be implemented by multiple-copy transformation for pure states. This fact, together with the result we obtained in R. Y. Duan, Y. Feng, X. Li, and M. S. Ying, Phys. Rev. A 71, 042319 (2005), namely that the latter can be completely implemented by the former, indicates that catalyst-assisted transformation is strictly more powerful than multiple-copy transformation. For the purely probabilistic setting we find, however, these two kinds of transformations are geometrically equivalent in the sense that the sets of pure states that can be converted into a given pure state with maximal probabilities not less than a given value have the same closure, regardless of whether catalyst-assisted transformation or multiple-copy transformation is used.
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.
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.