Elsevier

Physics Letters A

Volume 300, Issue 6, 12 August 2002, Pages 573-580
Physics Letters A

The fully entangled fraction as an inclusive measure of entanglement applications

https://doi.org/10.1016/S0375-9601(02)00884-8Get rights and content

Abstract

Characterizing entanglement in all but the simplest case of a two qubit pure state is a hard problem, even understanding the relevant experimental quantities that are related to entanglement is difficult. It may not be necessary, however, to quantify the entanglement of a state in order to quantify the quantum information processing significance of a state. It is known that the fully entangled fraction has a direct relationship to the fidelity of teleportation maximized under the actions of local unitary operations. In the case of two qubits we point out that the fully entangled fraction can also be related to the fidelities, maximized under the actions of local unitary operations, of other important quantum information tasks such as dense coding, entanglement swapping and quantum cryptography in such a way as to provide an inclusive measure of these entanglement applications. For two qubit systems the fully entangled fraction has a simple known closed-form expression and we establish lower and upper bounds of this quantity with the concurrence. This approach is readily extendable to more complicated systems.

Introduction

A pure quantum state is entangled if it is impossible to factorize into a tensor product of states for the separate systems (e.g., the singlet state of two spin-1/2 particles, (1/2)(|01〉−|10〉), is entangled). This property, originally introduced to sharpen discussions of foundational issues in quantum theory [1], has been studied extensively with regard to nonlocal quantum correlations [2] indicated by the observed violation of Bell's inequality [3]. In the past decade, the focus of entanglement studies has shifted toward applications which use the nonclassical features of quantum systems to surpass classical limitations on communications and computation. Such applications are part of the emerging field of quantum information [4] and include quantum cryptography [5], dense coding [6], teleportation [7], entanglement swapping [8], and quantum computation [9].

Due to this recent interest in quantum entanglement applications, the characterization of entanglement in a mixed bipartite system has become an intensely studied problem. In general, mixed states are entangled if it is impossible to represent the density operator as an incoherent sum of factorizable pure states [10]. There are a number of measures of entanglement for a bipartite system. Three closely related measures are the entanglement of formation, the entanglement of distillation, and the concurrence. The entanglement of formation is defined as the least number of maximally entangled states required to asymptotically prepare a mixed state ρ with local operations and classical communications [11] and the entanglement of distillation is defined as the asymptotic yield of maximally entangled states that can be extracted from ρ with local operations and classical communications [11]. The concurrence [12] is monotonically related to the entanglement of formation, and therefore an equally valid measure of entanglement, but is the only measure described here that provides a closed expression for the simplest case of a two qubit bipartite system [13]. Relative entropy [14] measures entanglement by considering the ability to distinguish ρ from all separable states and negativity [15] quantifies the degree to which the eigenvalues of the partial transpose fail to satisfy the partial transpose separability condition [16]. To be sure all of these entanglement measures can be computed, like any physical quantity in quantum mechanics, from knowledge of the density matrix which can be found experimentally with tomography [17], but their relation to experimental consequences are indirect at best. For example, a two qubit mixed state described by an ensemble of partially entangled states can always be distilled, in a nonunique fashion, into a smaller ensemble of maximally entangled states which can in turn be used for useful quantum information processing [18].

Modern conventional wisdom holds that characterizing entanglement in all but the simplest of cases is a hard problem. Even understanding the relevant experimental quantities that are related to entanglement is difficult. It may not be necessary, however, to quantify the entanglement of a state in order to quantify the quantum information processing significance of a state. For example, Horodecki et al. [19] demonstrated that the maximum teleportation fidelity for a general two qubit system is given by FTmax=13(1+2F), where F is the fully entangled fraction [11] and is defined as the overlap between a mixed state ρ̂ and a maximally entangled state |Φ〉 maximized over all |Φ〉: F=max|Φ〉〈Φ|ρ̂|Φ〉. Unlike entanglement, the fully entangled fraction does have a clear experimental interpretation as the optimal ability of a state to teleport and it is clear that the degree to which fully entangled fraction is greater than 1/2 (FT>2/3) can be used to quantify the teleporting ability of a state over the best “classical teleportation” protocols. This suggests that it may be possible to define a measure of entanglement applications directly. Such a mathematical quantity may be just as useful as a true entanglement measure, but more practical from a theoretical standpoint. It is natural to wonder whether the fully entangled fraction is such a quantity, that is, can it be used to measure the general quantum information significance of a state. To answer this question for the case of two qubits, in Section 2, we examine the relationship between the fully entangled fraction and the fidelities of all two qubit applications which have been experimentally demonstrated to date: dense coding, teleportation, entanglement swapping, and quantum cryptography (Bell inequalities). We consider these applications with a general two qubit mixed state in place of the standard maximally entangled pure state and find that the fully entangled fraction does indeed quantify the quantum processing significance of dense coding, teleportation, entanglement swapping, and quantum cryptography (violating Bell's inequality) in an inclusive sense. We are taking “inclusive” to mean that a nonzero value indicates that a state can perform at least one of these applications better than allowed “classically” and a zero value indicates that a state cannot perform any of these applications better than allowed “classically”. Similar approaches have examined continuous variable teleportation [20] and Bell inequality experiments [21]. The fully entangled fraction has a simple closed-form analytic expression, in the case of two qubits, which we rederive in Section 3 under the present context. In Section 4 we establish upper and lower bounds between the fully entangled fraction and the only measure of entanglement described above with a closed-form expression for a general two qubit state, the concurrence, and in Section 5 we conclude and discuss generalizations of these ideas to more complicated systems.

Section snippets

Dense coding

The relationship between the fully entangled fraction and dense coding [6] (see Fig. 1(a)) is clearest. In this entanglement application Alice and Bob each receive one qubit of a maximally entangled state, 1〉≡1/2(|00〉+|11〉), where the first entry denotes Bob's qubit and the second denotes Alice's qubit. Alice can encode 2 bits of information in four orthogonal states by applying one of four local unitaries solely to her own qubit, 1̂1̂Φ1=Φ1,1̂⊗iX̂Φ1=Φ2≡i|01〉+|10〉2,1̂⊗iŶΦ1=Φ3≡−|01〉−|10〉2,1̂⊗i

A simple closed-form expression for the fully entangled fraction

In Section 2 we deduced that the fully entangled fraction can be physically interpreted as an inclusive measure of entanglement applications. That is, when F is greater than 1/2 a mixed state can at least perform dense coding, teleportation, or entanglement swapping with a fidelity that is better than any separable state using classical protocols. It is clear that this quantity is invariant under local unitary operators, which can be viewed passively as a basis transformation, but not under

The relation between the fully entangled fraction and the concurrence

The fully entangled fraction, once measured, establishes lower and upper bounds for the concurrence. It has been proved that the fully entangled fraction is a lower bound for the entanglement of formation [11], [27] and therefore a lower bound for the concurrence which is monotonically related to the entanglement of formation. The states which form the lower bound are given by a convex sum of a maximally mixed state and an arbitrary pure state, ρ̂1̂4+(1−ϵ)|ψpure〉〈ψpure|, where 0<ϵ<1. If the

Conclusions

In conclusion, we have found that the fully entangled fraction can be used as an inclusive measure of entanglement applications in the case of two qubit states. That is, F>1/2 guarantees that a mixed state can be used to achieve, on average, “classically impossible” results in either dense coding, teleportation, entanglement swapping, or quantum cryptography (Ekert protocol); all two qubit quantum information processing applications which have been experimentally demonstrated to date. This

Acknowledgements

We would like to thank Tanmoy Bhattacharya and Bill Munro for useful discussions. D.M.E. would like to thank the Los Alamos Summer School for support and this work was supported by the Los Alamos National Laboratory LDRD program.

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    Present address: University of Rochester, Rochester, NY 14627, USA.

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