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arXiv:quant-ph/9903001 · Physical Review A

Method of areas for manipulating the entanglement properties of one copy of a two-particle pure entangled state

Lucién Hardy

Published Sep 1, 1999• 65 citations
arXivHTMLPDF

Abstract

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 certain coloring problems. We consider two problems. First, we find the most general way of manipulating the state to obtain maximally entangled states. After such a manipulation, the entangled states $|11〉+|22〉+\ensuremath{\cdot}\ensuremath{\cdot}\ensuremath{\cdot}+|\mathrm{mm}〉$ are obtained with probability ${p}_{m}.$ We obtain an expression for the optimal average entanglement obtainable. Also, some results of Lo and Popescu (e-print quant-ph/9707038) pertaining to this problem are given simple geometric proofs. Second, we consider how to manipulate one two-particle entangled state $|\ensuremath{\psi}〉$ to another $|{\ensuremath{\psi}}^{\ensuremath{'}}〉$ with certainty. We derive Nielsen's theorem (which states a necessary and sufficient condition for this to be possible) using the method of areas.

DOI
10.1103/physreva.60.1912

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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 certain coloring problems. We consider two problems. First, we find the most general way of manipulating the state to obtain maximally entangled states. After such a manipulation, the entangled states ∣11〉+∣22〉+\ensuremath⋅\ensuremath⋅\ensuremath⋅+∣mm〉|11〉+|22〉+\ensuremath{\cdot}\ensuremath{\cdot}\ensuremath{\cdot}+|\mathrm{mm}〉∣11〉+∣22〉+\ensuremath⋅\ensuremath⋅\ensuremath⋅+∣mm〉 are obtained with probability pm.{p}_{m}.pm​. We obtain an expression for the optimal average entanglement obtainable. Also, some results of Lo and Popescu (e-print quant-ph/9707038) pertaining to this problem are given simple geometric proofs. Second, we consider how to manipulate one two-particle entangled state ∣\ensuremathψ〉|\ensuremath{\psi}〉∣\ensuremathψ〉 to another ∣\ensuremathψ\ensuremath′〉|{\ensuremath{\psi}}^{\ensuremath{'}}〉∣\ensuremathψ\ensuremath′〉 with certainty. We derive Nielsen's theorem (which states a necessary and sufficient condition for this to be possible) using the method of areas.
Suppose two distant observers Alice and Bob share a pure bipartite quantum state. By applying local operations and communicating with each other using a classical channel, Alice and Bob can manipulate it into some other states. Previous investigations of entanglement manipulations have been largely limited to a small number of strategies and their average outcomes. Here we consider a general entanglement manipulation strategy, and go beyond the average property. For a pure entangled state shared between two separated persons Alice and Bob, we show that the mathematical interchange symmetry of the Schmidt decomposition can be promoted into a physical symmetry between the actions of Alice and Bob. Consequently, the most general (multistep two-way-communications) strategy of entanglement manipulation of a pure state is, in fact, equivalent to a strategy involving only a single (generalized) measurement by Alice followed by one-way communications of its result to Bob. We also prove that strategies with one-way communications are generally more powerful than those without communications. In summary, one-way communications is necessary and sufficient for entanglement manipulations of a pure bipartite state. The supremum probability of obtaining a maximally entangled state (of any dimension) from an arbitrary state is determined, and a strategy for achieving this probability is constructed explicitly. One important question is whether collective manipulations in quantum mechanics can greatly enhance the probability of large deviations from the average behavior. We answer this question in the negative by showing that, given n pairs of identical partly entangled pure states $(|\ensuremath{\Psi}〉)$ with entropy of entanglement $E(|\ensuremath{\Psi}〉),$ the probability of getting nK\mathrm{nK}nK $[K>E(|\ensuremath{\Psi}〉)]$ singlets out of entanglement concentration tends to zero as n tends to infinity.
Suppose that several parties jointly possess a pure multipartite state, |ψ⟩. Using local operations on their respective systems and classical communication (i.e. LOCC), it may be possible for the parties to transform deterministically |ψ⟩ into another joint state |ϕ⟩. In the bipartite case, the Nielsen majorization theorem gives the necessary and sufficient conditions for this process of entanglement transformation to be possible. In the multipartite case, such a deterministic local transformation is possible only if both the states are in the same stochastic LOCC (SLOCC) class. Here, we generalize the Nielsen majorization theorem to the multipartite case, and find necessary and sufficient conditions for the existence of a local separable transformation between two multipartite states in the same SLOCC class. When such a deterministic conversion is not possible, we find an expression for the maximum probability to convert one state to another by local separable operations. In addition, we find necessary and sufficient conditions for the existence of a separable transformation that converts a multipartite pure state into one of a set of possible final states all in the same SLOCC class. Our results are expressed in terms of (i) the stabilizer group of the state representing the SLOCC orbit and (ii) the associate density matrices (ADMs) of the two multipartite states. The ADMs play a similar role to that of the reduced density matrices when considering local transformations that involve pure bipartite states. We show, in particular, that the requirement that one ADM majorizes another is a necessary condition but is, in general, far from also being sufficient as it happens in the bipartite case. In most of the results the twirling operation with respect to the stabilizer group (of the representative state in the SLOCC orbit) plays an important role that provides a deep link between entanglement theory and the resource theory of reference frames.
Suppose two distant observers Alice and Bob share a pure biparticle entangled state secretly chosen from a set, it is shown that Alice (Bob) can probabilistic concentrate the state to a maximally entangled state by applying local operations and classical communication (LQCC) if and only if the states in the set share the same marginal density operator for her (his) subsystem. Applying this result, we present probabilistic superdense coding and show that perfect purification of mixed state is impossible using only LQCC on individual particles.
We derive a sufficient condition for a set of pure states, each entangled in two remote NNN-dimensional systems, to be transformable to kkk-dimensional-subspace equivalent entangled states (k≤Nk\leq Nk≤N) by same local operations and classical communication. If k=Nk=Nk=N, the condition is also necessary. This condition reveals the function of the relative marginal density operators of the entangled states in the entanglement manipulation without sufficient information of the initial states.
In classical information theory one can, in principle, produce a perfect copy of any input state. In quantum information theory, the no cloning theorem prohibits exact copying of non-orthogonal states. Moreover, if we wish to copy multiparticle entangled states and can perform only local operations and classical communication (LOCC), then further restrictions apply. We investigate the problem of copying orthogonal, entangled quantum states with an entangled blank state under the restriction to LOCC. Throughout, the subsystems have finite dimension D . We show that if all of the states to be copied are non-maximally entangled, then novel LOCC copying procedures based on entanglement catalysis are possible. We then study in detail the LOCC copying problem where both the blank state and at least one of the states to be copied are maximally entangled. For this to be possible, we find that all the states to be copied must be maximally entangled. We obtain a necessary and sufficient condition for LOCC copying under these conditions. For two orthogonal, maximally entangled states, we provide the general solution to this condition. We use it to show that for D = 2, 3, any pair of orthogonal, maximally entangled states can be locally copied using a maximally entangled blank state. However, we also show that for any D which is not prime, one can construct pairs of such states for which this is impossible.
Local pure states are an important resource for quantum computing. The problem of distilling local pure states from mixed ones can be cast in an information theoretic paradigm. The bipartite version of this problem where local purity must be distilled from an arbitrary quantum state shared between two parties, Alice and Bob, is closely related to the problem of separating quantum and classical correlations in the state and in particular, to a measure of classical correlations called the one-way distillable common randomness. In Phys. Rev. A 71, 062303 (2005), the optimal rate of local purity distillation is derived when many copies of a bipartite quantum state are shared between Alice and Bob, and the parties are allowed unlimited use of a unidirectional dephasing channel. In the present paper, we extend this result to the setting in which the use of the channel is bounded. We demonstrate that in the case of a classical-quantum system, the expression for the local purity distilled is efficiently computable and provide examples with their tradeoff curves.