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arXiv:quant-ph/0410046 · Physical Review A

Efficiency of deterministic entanglement transformation

Runyao Duan, Yuan Feng, Zhengfeng Ji, Mingsheng Ying

Published Feb 10, 2005• 5 citations
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Abstract

We prove that sufficiently many copies of a bipartite entangled pure state can always be transformed into some copies of another one with certainty by local quantum operations and classical communication. The efficiency of such a transformation is characterized by deterministic entanglement exchange rate, and it is proved to be always positive and bounded from top by the infimum of the ratios of Renyi's entropies of source state and target state. A careful analysis shows that the deterministic entanglement exchange rate cannot be increased even in the presence of catalysts. As an application, we show that there can be two incomparable states with deterministic entanglement exchange rate strictly exceeding 1.

DOI
10.1103/physreva.71.022305

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We prove that sufficiently many copies of a bipartite entangled pure state can always be transformed into some copies of another one with certainty by local quantum operations and classical communication. The efficiency of such a transformation is characterized by deterministic entanglement exchange rate, and it is proved to be always positive and bounded from top by the infimum of the ratios of Renyi's entropies of source state and target state. A careful analysis shows that the deterministic entanglement exchange rate cannot be increased even in the presence of catalysts. As an application, we show that there can be two incomparable states with deterministic entanglement exchange rate strictly exceeding 1.
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>→∣ϕ>∣χ′>| ψ> | χ> \to| ϕ> | χ^{\prime}>∣ψ>∣χ>→∣ϕ>∣χ′>. We also design an efficient algorithm to detect whether a k×kk\times kk×k general catalyst exists for a given entanglement transformation. This algorithm can as well be exploited to witness the existence of standard catalysts.
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 deterministic and conclusive transformation properties of bipartite entanglement in the nonasymptotic scenario (when many but a finite number of copies of a source state are collectively manipulated) are fundamentally different from those in both the single-copy and asymptotic limits. For instance, by generalizing the notion of local comparability of entanglement in the single-copy case, we provide a complete classification of bipartite entanglement transformations in the nonasymptotic scenario. We also show that, unlike the asymptotic case, collective operations need not always be advantageous for the many-copy case. In particular, we show that (1) there exists a class of states for which the optimal conclusive transformation probability decreases exponentially with increasing number of copies, even if the source state has more entropy of entanglement, and (2) optimal conclusive transformation probability need not be a monotonic function of the number of copies.
Quantum entanglement of pure states is usually quantified via the entanglement entropy, the von Neumann entropy of the reduced state. Entanglement entropy is closely related to entanglement distillation, a process for converting quantum states into singlets, which can then be used for various quantum technological tasks. The relation between entanglement entropy and entanglement distillation has been known only for the asymptotic setting, and the meaning of entanglement entropy in the single-copy regime has so far remained open. Here we close this gap by considering entanglement catalysis. We prove that entanglement entropy completely characterizes state transformations in the presence of entangled catalysts. Our results imply that entanglement entropy quantifies the amount of entanglement available in a bipartite pure state to be used for quantum information processing, giving asymptotic results an operational meaning also in the single-copy setup.
Quantum resource theories provide a mathematically rigorous way of understanding the nature of various quantum resources. An important problem in any quantum resource theory is to determine how quantum states can be converted into each other within the physical constraints of the theory. The standard approach to this problem is to study approximate or probabilistic transformations. Here, we investigate the intermediate regime, providing limits on both, the fidelity and the probability of state transformations. We derive limitations on the transformations, which are valid in all quantum resource theories, by providing bounds on the maximal transformation fidelity for a given transformation probability. As an application, we show that these bounds imply an upper bound on the asymptotic rates for various classes of states under probabilistic transformations. We also show that the deterministic version of the single copy bounds can be applied for drawing limitations on the manipulation of quantum channels, which goes beyond the previously known bounds of channel manipulations. Furthermore, we completely solve the question of stochastic-approximate state conversion via local operations and classical communication in the following two cases: (i) Both initial and target states are pure bipartite entangled states of arbitrary dimensions. (ii) The target state is a two-qubit entangled state and the initial state is a pure bipartite state.
We present an optimal scheme to realize the transformations between single copies of two bipartite entangled states without classical communication between the sharing parties. The scheme achieves the upper bound for the success probabilities [Phys. Rev. A 63, 022301 (2001); Phys. Rev. Lett. 83, 1455 (1999)] of generating maximally entangled states if applied to entanglement concentration. Such a strategy also dispenses with the interaction with an ancilla system in the implementation. We also show that classical communications are indispensable in realizing the deterministic transformations of a single bipartite entangled state. With a finite number of identical pairs of two entangled bosons, on the other hand, we can realize the deterministic transformation to any target entangled state of equal or less Schmidt rank through an extension of the scheme.