Article
All quantum states useful for teleportation are nonlocal resources
CAVALCANTI, Daniel, et al.
Abstract
Understanding the relation between the different forms of inseparability in quantum mechanics is a longstanding problem in the foundations of quantum theory and has implications for quantum information processing. Here we make progress in this direction by establishing a direct link between quantum teleportation and Bell nonlocality. In particular, we show that all entangled states which are useful for teleportation are nonlocal resources, i.e., lead to deterministic violation of Bell's inequality. Our result also extends the phenomenon of superactivation of quantum nonlocality, recently proven by C. Palazuelos [Phys. Rev. Lett. 109, 190401 (2012)], and suggests that the latter might in fact be more general than initially thought.
Reference
CAVALCANTI, Daniel, et al. All quantum states useful for teleportation are nonlocal resources. Physical Review. A, 2013, vol. 87, no. 4
DOI : 10.1103/PhysRevA.87.042104
Available at: http://archive-ouverte.unige.ch/unige:36420
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1 / 1 PHYSICAL REVIEW A 87, 042104 (2013)
All quantum states useful for teleportation are nonlocal resources
Daniel Cavalcanti,1 Antonio Ac´ın,2,3 Nicolas Brunner,4,5 and Tamas´ Vertesi´ 6 1Centre for Quantum Technologies, University of Singapore, 117542, Singapore 2Institut de Ciencies Fotoniques, 08860 Castelldefels (Barcelona), Spain 3Institucio´ Catalana de Recerca i Estudis Avanc¸ats, Lluis Companys 23, 08010 Barcelona, Spain 4H. H. Wills Physics Laboratory, University of Bristol, Tyndall Avenue, Bristol BS8 1TL, United Kingdom 5Departement´ de Physique Theorique,´ Universite´ de Geneve,` 1211 Geneve,` Switzerland 6Institute for Nuclear Research, Hungarian Academy of Sciences, P.O. Box 51, H-4001 Debrecen, Hungary (Received 29 September 2012; published 8 April 2013) Understanding the relation between the different forms of inseparability in quantum mechanics is a longstanding problem in the foundations of quantum theory and has implications for quantum information processing. Here we make progress in this direction by establishing a direct link between quantum teleportation and Bell nonlocality. In particular, we show that all entangled states which are useful for teleportation are nonlocal resources, i.e., lead to deterministic violation of Bell’s inequality. Our result also extends the phenomenon of superactivation of quantum nonlocality, recently proven by C. Palazuelos [Phys. Rev. Lett. 109, 190401 (2012)], and suggests that the latter might in fact be more general than initially thought.
DOI: 10.1103/PhysRevA.87.042104 PACS number(s): 03.65.Ud
I. INTRODUCTION exists a finite number k such that k copies of ρ, i.e., ρ⊗k, The fact that quantum mechanics is at odds with the deterministically violate a Bell inequality. As a by-product, our results also provide examples of principle of locality, although once viewed as evidence of the superactivation of quantum nonlocality. The fact that, by failure of the model, is now recognized as a fundamental aspect combining several copies of a state that admits a local model, of quantum theory. The notion of inseparability in quantum it becomes possible to violate a Bell inequality is known mechanics appears under several forms. At the algebraic level as superactivation of quantum nonlocality. This represents is the concept of quantum entanglement. Entanglement is an example of the phenomenon of activation in quantum the resource for numerous quantum information protocols, mechanics, through which the judicious combination of several in particular for quantum teleportation [1], which plays a quantum entities becomes more powerful than the sum of the central role in quantum communications and computation. The parts. Celebrated examples of activation were demonstrated in strongest notion of quantum inseparability is Bell nonlocality entanglement theory [9] and quantum channel theory [10]. In [2]. Distant observers sharing an entangled state can, by the case of nonlocality, some forms of activation in specific performing suitably chosen local measurements, generate contexts were demonstrated, for instance, when postselection correlations that cannot be reproduced by any local hidden- is considered [11–14], when several copies of the quantum variable model [3], as witnessed by the violation of a Bell state are distributed in a quantum network [15–18], when inequality. the number of measurements considered are restricted [19], Understanding how these various forms of inseparability when taking the amount of violation of a Bell inequality as relate to each other is a long-standing problem, important from a figure of merit [20], or in the case of general nonsignaling both fundamental and applied points of view. Although entan- probability distributions [21]. However, the first example of glement and nonlocality were first thought to be equivalent, it superactivation of quantum nonlocality in the most natural was shown that there exist mixed entangled states which are scenario consisting of two parties who do not perform any local, that is, admit a local hidden variable model [4], even local preprocessing of their quantum state is due to Palazuelos for the most general type of local measurements [5]. It was [22]. As shown below, our findings also allow us to extend then shown by Popescu [6] that some of these local entangled Palazuelos’s result and demonstrate additional examples of states are nevertheless useful for teleportation, which led him superactivation of quantum nonlocality in the bipartite and to argue that teleportation and Bell nonlocality are inequivalent multipartite scenarios, suggesting that this phenomenon might forms of inseparability. Later, it was shown [7] that all be more general than initially thought. two-qubit states violating the simplest Bell inequality, due to Clauser, Horne, Shimony, and Holt (CHSH), are useful for II. CONNECTING TELEPORTATION AND NONLOCALITY teleportation, reviving the hope for a link between teleportation and nonlocality. In what follows we say that a state ρ is k-copy nonlocal if Our main result is to show that a strong link between ρ⊗k is nonlocal for some k. Given a state ρ, we focus on its teleportation and nonlocality does indeed exist. In particular, entanglement fraction, defined by the maximum overlap of ρ we prove that all entangled states useful for teleportation are with a maximally entangled state. The main result of this paper nonlocal resources, that is, can be used to violate a Bell is the following: Every quantum state acting on Cd ⊗ Cd with inequality deterministically. The key point in our work is entanglement fraction higher than 1/d is k-copy nonlocal. to allow the parties to use several copies of the state. More It turns out that having an entanglement fraction higher precisely, for all states ρ useful for teleportation [8], there than 1/d is a necessary and sufficient condition for a state to
1050-2947/2013/87(4)/042104(4)042104-1 ©2013 American Physical Society CAVALCANTI, ACIN,´ BRUNNER, AND VERTESI´ PHYSICAL REVIEW A 87, 042104 (2013) provide a quantum advantage for teleportation [8]. Thus, our ⊗k ⊗k k for which (ρiso ) > 1, which implies that ρiso is nonlocal. result implies that every state that is useful for teleportation is Since F = 1/d is the separability bound for the isotropic state, a nonlocal resource. we conclude point (i), i.e., that every entangled isotropic state The proof of our result proceeds as follows: (i) we first is k-copy nonlocal. show that every entangled isotropic state is k-copy nonlocal We now proceed to show our main result. In order to see and then (ii) show that point (i) implies our main result. this, first notice that any quantum state ρ0 can be depolarized In order to prove point (i) we essentially follow and into an isotropic state by the application of randomly chosen slightly improve the proof by Palazuelos for the existence unitaries as of superactivation of nonlocality [22]. This was based on the = ⊗ ∗ ⊗ ∗ −1 Khot-Visnoi Bell inequality [23,24], ρiso(F0) pi (Ui Ui )ρ0(Ui Ui ) (8) i N n while keeping the entanglement fraction F0, i.e., F0 = βKV = cx,y,a,bP (ab|xy) κloc, (1) max |ρ |, where the maximum is taken over all d × d x,y=1 a,b=1 0 maximally entangled states [26,27]. Since the initial state where κloc is the local bound and cx,y,a,b are positive coeffi- has an entanglement fraction higher than 1/d, the resulting cients. The following upper bound is known for κloc: isotropic state is entangled and, as we have just shown, k-copy nonlocal. That is, the state κloc C/n, (2) = n ρ (F )⊗k = p ···p U ρ⊗k U −1 , (9) where N 2 /n is the number of measurements per party, iso 0 i1 ik i1,...,ik 0 i1,...,ik n is the number of measurement outcomes [25], and C is i1,...,ik a universal constant. Moreover, it is known that there exist where U = (U ⊗···⊗U ) ⊗ (U ∗ ⊗···⊗U ∗ ) , i1...ik i1 ik Alice i1 ik Bob local measurements on a maximally entangled state of local violates the Khot-Visnoi Bell inequality. This means that dimension n which produce a probability distribution such that ⊗k there is a Bell operator B such that tr[B ρiso(F0) ] >L(L 2 βKV C /(ln n) , (3) is the local bound of the corresponding Bell inequality). But then there is (at least) one term in the sum of (9) such that where C is another universal constant. Since both the local tr(BU ρ⊗kU −1 ) >L. Since the local unitaries can be i1,...,ik 0 i1,...,ik bound and quantum violation depend on the dimension, it is absorbed into the choice of local measurements, it follows that useful to define the nonlocality fraction, given by ⊗k the original state ρ0 is also k-copy nonlocal. This ends the = βKV /κloc. (4) proof of our main result. So we have a nonlocal probability distribution if >1. In what follows we sometimes denote by (ρ) the nonlocal fraction III. SUPERACTIVATION OF NONLOCALITY of the probability distribution obtained by local measurements In Ref. [22], Palazuelos studied the nonlocal properties applied to state ρ. of many copies of the isotropic state (5). It is known that Consider now the isotropic states, given by there exist ranges of the entanglement fraction F , given by 1 −| | F Floc, such that this state is local; i.e., the probability ρ (F ) = F | |+(1 − F ) d d , (5) iso d d d2 − 1 distributions of the results of local measurements applied to (5) = | | can be explained by a (classical) local model [28]. However, where F d ρiso(d) d is the entanglement fraction [26] Palazuelos showed that sufficiently many copies of a local of the state ρiso(F ). ⊗k isotropic state, i.e., ρiso(FL) , can violate a Bell inequality We can write k copies of the isotropic state as if d 8[22]. This shows that quantum nonlocality can be −| | ⊗k ⊗k k k (1 d d ) superactivated. ρ = F | k k |+···+(1 − F ) , iso d d (d2 − 1)k Point (i) extends the result by Palazuelos in two senses. (6) First, it shows superactivation of nonlocality for the whole range of the parameter F for which the isotropic state is where we have used the fact that the tensor product of entangled and has a local model (see Fig. 1). Second, it is k maximally entangled states of local dimension d is a valid for any dimension, while the minimum local dimension maximally entangled state of local dimension dk. One can used in the proof of Ref. [22]isd = 8. use bounds (1) and (3) to show that the nonlocal fraction of Furthermore our main result allows us to construct addi- ⊗ ρ k satisfies tional examples of superactivation of nonlocality. For instance, iso ⊗k k consider mixtures of an arbitrary pure state and the maximally ρ = F (| k k |) +··· iso d d mixed noise, (1 −| |)⊗k + (1 − F )k d d 1 2 − k σ = p|ψψ|+(1 − p) . (10) (d 1) d2 k | | F ( dk dk ) An arbitrary bipartite pure state |ψ reads, in its Schmidt C dk decomposition, |ψ= d λ |ii. Its entanglement fraction F k . (7) i=1 i C (k ln d)2 can be written as d 2 Thus, in the case F>1/d the right-hand side of (7) increases = λi F =| |ψ|2 = i 1 . (11) with k. This implies that there will be a given number of copies ψ d d
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Useful for teleportation direct and general link between teleportation and nonlocality, two central forms of inseparability in quantum mechanics Separable k-copy nonlocal previously thought to be unrelated [6]. Our results also suggest Super-activation that the superactivation of quantum nonlocality might be of nonlocality Local Bell violation more general than expected, as any entangled state useful for F teleportation that has a local model provides an instance of F F 1 Fs L NL this phenomenon. In fact, we use this connection to provide explicit examples of superactivation. FIG. 1. Nonlocal properties of the isotropic state (5). This state Our paper raises several questions. First, it would be is separable for F Fs , has a local model for F FL [28], and interesting to see if the converse of our main result holds, is known to violate a Bell inequality for FL
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