Werner States 843

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Werner States

Antonio Acı́n

In our macroscopic world, correlations are established by means of a set of clas-sical instructions, that could be agreed in advance or come from a source. Usingthese pre-established instructions, distant parties that are unable to communicatecan behave in a correlated manner. Assume for instance a scenario where two dis-tant parties are asked different questions from a set of m possible questions withn possible answers. We denote by x and y the question asked to Alice and Bob,while a and b label their responses. The correlations between the parties will bedescribed by a joint probability distribution p (a, b|x, y). If the parties received inadvance correlated instructions, denoted by λ, but are not able to communicate, theprobability distributions can generically be written as

pc(a, b|x, y) =∑λ

p (λ)p (a|x, λ)q(b|y, λ). (1)

In what follows, correlations of this form are called local, since they can be repro-duced by means of a (local) model that uses only classical correlations, given by λ,and local responses, namely p (a|x, λ) and q(b|y, λ).

Are these correlations modified if the parties share a quantum state of two par-ticles, ρAB , instead of classical instructions? Here, after receiving the question, theparties apply a local measurement, which depends on the question, on each particleand decide the response depending on the obtained result. Any probability distribu-tion that can be obtained in this way can be written, using the standard � Born rulefor probabilities, as

pq(a, b|x, y) = Tr(ρABMxa ⊗M

y

b ), (2)

where Mxa and My

b are the operators describing the measurements by Alice and Bob.Interestingly, not all the probability distributions having this quantum origin can bewritten as (1), which means that � correlations in quantum mechanics are morepowerful than their classical counterparts.

All this discussion is nothing but a reformulation of the well-known fact thatquantum states violate Bell’s inequalities [1]. Indeed, beyond their clear fundamen-tal importance, Bell’s inequalities can also be understood as constraints satisfiedby all probability distributions achievable by means of shared classical correlations(1). � Bell’s Theorem, then, represents a seminal result for the understanding ofquantum mechanics, but also shows that quantum states can be used to establishcorrelations between distant parties that are not achievable by classical means. Aquantum state is said to display non-local correlations when it leads to the violationof a Bell’s inequality.

844 Werner States

A natural question then emerges: Do all quantum states contain non-localcorrelations? It is relatively easy to see that (i) all entangled pure states (� states,pure and mixed) that are not of product form, |ψ〉 �= |α〉 |β〉 violate a Bell’s inequal-ity [2], while (ii) measurements on separable states, i.e. states that can be writtenas a mixture of product states ρAB = ∑i pi |αi〉 |βi〉 〈αi | 〈βi |, always allow a localdescription. Remarkably, there exist entangled mixed states, i.e. states that are notseparable, whose measurement correlations can also be described by a local model.Thus, these states, despite being entangled, do not violate any Bell’s inequality. Thefirst examples of such states were derived in 1989 by Werner [3]. These states arenow known as Werner states and play a fundamental role in foundations of quantummechanics and quantum information theory.

Werner states, ρw, are those states belonging to a composite space Cd ⊗ Cd thatremain unchanged when the two parties apply the same unitary operation, (U ⊗U)ρw(U ⊗U)† = ρw. For the sake of simplicity, we restrict here the considerationsto the simplest case of two-dimensional systems, d = 2. In this case, Werner statesare given by the mixture of a singlet state,

∣∣ψ−⟩ = (|01〉−|10〉)/√2, and completelydepolarized noise,

ρw = p∣∣ψ−⟩ ⟨ψ−∣∣+ (1− p)

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4. (3)

Werner proved that these states are entangled whenever p > 1/3. If Alice andBob perform local � spin measurements on directions n̂A and n̂B, the obtainedcorrelations read

p (a, b|n̂A, n̂B) = 1− p × ab× n̂A · n̂B

4. (4)

Here, n̂A and n̂B represent the labels for the local measurements by Alice and Bob,while the measurement outcomes are a, b = +1,−1. The goal is to be able toreproduce this probability distribution by means of classical correlations. Wernerbuilt a local model achieving this. It works as follows: the classical correlationsare given by normalized real vectors, n̂λ ∈ R3. Alice’s response is governed bythe overlap between the received vector and the vector defining her measurement,pw(+1|n̂A, λ) = (1 + n̂A · n̂λ)/2, as in the quantum case. Bob’s response is equalto +1 if n̂A · n̂λ < 0, otherwise is −1. Putting all these things together, one can seethat the obtained correlations are the same as in the quantum case (4) with p = 1/2.Therefore, Werner states with 1/3 < p � 1/2 have a local description despite beingentangled.

It is clear that Werner’s result represents a seminal and surprising achievement:the fact that a state is entangled is not sufficient to display non-local correlations.Since Werner’s original derivation, a few results have been able to generalize hisfindings to other situations. Among them, there is the extension of Werner’s modelto completely general measurements [4] or to tripartite states [5]. At this point, it isworth mentioning that even if the correlations between measurement outcomes on aquantum state admit a local description, this state may have some hidden forms of� non-locality: for instance, it may display non-local correlations after sequences

Which-Way or Welcher-Weg-Experiments 845

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of local measurements [6,7] or be useful when performing quantum teleporation [8]� quantum communication. To conclude, the relation between � entanglement andnon-locality is fascinating and full of open questions!

Literature

1. J. S. Bell: On the Einstein Podolsky Rosen paradox. Physics 1, 195 (1964)2. N. Gisin: Bell’s inequality holds for all non-product states. Phys. Lett. A 154, 201 (1991)3. R. F. Werner: Quantum states with Einstein–Podolsky–Rosen correlations admitting a hidden-

variable model. Phys. Rev. A 40, 4277 (1989)4. J. Barrett: Nonsequential positive-operator-valued measurements on entangled mixed states do

not always violate a Bell inequality. Phys. Rev. A 65, 042302 (2002)5. G. Toth, A. Acı́n: Genuine tripartite entangled states with a local hidden-variable model. Phys.

Rev. A 74, 030306 (2006)6. S. Popescu: Bell’s inequalities and density matrices: revealing “hidden” nonlocality. Phys. Rev.

Lett. 74, 2619 (1995)7. N. Gisin: Hidden quantum nonlocality revealed by local Filters. Phys. Lett. A 210, 151 (1996)8. S. Popescu: Bells inequalities versus teleportation: What is nonlocality? Phys. Rev. Lett. 72, 797

(1994)

Which-Way or Welcher-Weg-Experiments

Paul Busch and Gregg Jaeger

The issue of the � wave-particle duality of light and matter is commonly illus-trated by the � double-slit experiment, in which a quantum object of relativelywell defined momentum (such as a photon, electron, neutron, atom, or molecule)is sent through a diaphragm containing two slits, after which it is detected at a cap-ture screen. It is found that an interference pattern characteristic of wave behaviouremerges as a large number of similarly prepared quantum objects is detected on thescreen. This is taken as evidence that it is impossible to ascertain through whichslit an individual quantum object has passed; if that were known in every individ-ual case and if the quantum objects behaved as free classical particles otherwise, aninterference pattern would not arise.

The notion that a description of atomic objects in terms of definite classical par-ticle trajectories is not in general admissible is prominent in Werner Heisenberg’sseminal paper [1] of 1927 on the � Heisenberg uncertainty principle; there he notes:“I believe that one can fruitfully formulate the origin of the classical ‘orbit’ in thisway: the ‘orbit’ comes into being only when we observe it.” In the same year, inhis famous Como lecture, Niels Bohr introduced the � complementarity princi-ple, which entails that definite particle trajectories cannot be defined or observed