A.2 Some operations on polytopes
A.2.3 Another tasks : finding facets lying under an inequality
An inequality satisfied by a polytope is not a facet of the polytope if its dimension is lower thand−1. When this is the case, one can show that the inequality can be expressed as the convex combination of a number of tighter inequalities, the tighter of which are facets of the polytope, sharing an intersection of dimensiond−1 with the polytope. Thus, if a point violates a non-facet inequality, it can only violate some of these tighter ones by a larger amount. Given a non-facet inequality, it can thus be interesting to look for these tighter facets.
A method for finding the facets underlying an inequality given a V-description of the polytope is presented in paper [E]. The idea is that the rank of the set of extremal points saturating the inequality can be augmented by adding more extremal points of the polytope to this set. When the achieved rank is sufficient, and if the hyperplane passing through these points does not cut the polytope into two parts, then it describes a facet of the polytope. By construction, the intersection of this facet with the polytope coincides with the intersection of the original inequality with the polytope since it is generated by the same set of extremal points.
Appendix B
Memoryless attack on the 6-state protocol – proof
Here we provide a proof for the bound (4.1.4) used in the main text.
Proof. Without loss of generality, Eve’s POVM elements can be written as:
Fk =A†kAk =ak11+ (bk−ak)Pk (B.0.1) whereAk =√akUkPk+√
bkUkPk is a Kraus operator associated to the elementFk, ak, bk ≥ 0, Uk is a unitary operator, Pk = 11−Pk and Pk is a one-dimensional projector.
Eve’s information on Alice’s bit. We consider a given run k of the protocol for which Alice and Bob used the same basisb =bk, and denote by A Alice’s bit, B Bob’s bit and E the result of Eve’s POVM measurement. The total information gained by Eve on Alice’s bit after the sifting procedure is given byI(A: (E, b)). Since bis independent of A andE, and since the state produced by Alice ρ=ρ(A, b) is a function ofAandb, we have:
I(A: (E, b)) =H(A) +H(E, b)−H(A, E, b)
=H(A) +H(E) +H(b)−H(A, E, b)
=H(A, b) +H(E)−H(A, E, b) =I((A, b) :E)
=I(ρ:E) = log(6)−I(ρ|E)
(B.0.2)
Where H is Shannon’s entropy, and we assumed that the six states are chosen by Alice with the same probability 16. We thus need to bound the quantity I(ρ|E) which expresses the information that Eve is missing after she learns the result of her measurement, to know which state ρ was prepared by Alice. Using the fact that Prob(E=k) = tr(Fk)/2, this quantity can be expressed as
I(ρ|E) =I(E|ρ) +X
k
log(tr(Fk)) + log(3) (B.0.3) whereI(E|ρ) =−P
a,s,kProb(A=a, b=s, E=k) log(Prob(E=k|A=a, b=s)).
For every stateρthat Alice can produce,ρ= 11−ρcan also be produced by Alice.
An attack described by the POVM elements{Fk} thus provides Eve with the same information as the attack {F˜k} where ˜Fk = Ak11+ (bk −ak)Pk. Indeed, as shown above, this information is a symmetric function ofP(E=k|ρ=ρ(a, b)) and
tr(ρFk) =ak+ (bk−ak)tr(ρPk)
=ak+ (bk−ak)tr(ρF˜k) = tr(ρF˜k). (B.0.4)
Memoryless attack on the 6-state protocol – proof
Moreover, the information that can be extracted from a mixture of measurements M1 applied with probabilityp1andM2 applied with probabilityp2is
I([(p1, M1),(p2, M2)]) =p1I(M1) +p2I(M2), (B.0.5) whereI(M1,2) is the information that can be extracted by using measurement M1,2
only. We can thus write
where the factor 1/(ak +bk) is a normalization coefficient. Thus, the information gained by performing an arbitrary POVM measurement can also be achieved by mix-ing measurement strategies consistmix-ing of only two POVM elements. Let us thus consider a POVM measurement for Eve consisting only of the two elements aFk
k+bk
and aF˜k
k+bk.
By direct computation one finds that I(Ek|ρ) =1
3(h(c0k) +h(d0k) +h(e0k)) (B.0.7) withc0k = ak+tr(ρa2kP+bk)(bk k−ak), d0k = ak+tr(ρa4kP+bk)(bkk−ak), e0k = ak+tr(ρa6kP+bk)(bk k−ak). Since this function is convex inc0, d0, e0, its minimum lies on the boundary of the admissible region
All in all, this gives the following bound on Eve’s information about Alice’s bit:
I(A: (E, b))≤1−X
Perturbation on Bob’s system.The attack of Eve delivers the stateρ0i=P
kAkρiA†k to Bob instead of the expectedρi. This creates some errors in the outcomes of Bob, which are measured by the QBER:
Q= 1−
Note, that the attack {Ak} has the same effect as {A˜k} for ˜Ak =√akUkPk + To bound the second part of this expression, one can show by direct computation that
Putting the two bounds together. Let us consider equations (B.0.10) and (B.0.16) together. Keeping the sumak+bk constant for allk, we choose two values of k if possible: k1 andk2 such thatk1< k2. Following [85] one can show that increasing k1 in such a way that keeps the bound on the QBER (B.0.16) unchanged, can only decreasek2 and increase Eve’s information as given by (B.0.10). It is thus always better to havek=∀k. In this case both bounds become:
Tightness of the boundTo show that the above bound is tight, consider the attack in which Eve uses the two POVM elementsFk = 1−γ2 11+γ|kihk| fork = 0,1. This Moreover, the QBER induced by this attack isQ= 1−
√1−γ2
3 . Thus Eve can choose an attack that saturates equation (4.1.4): the bound is tight.
Memoryless attack on the 6-state protocol – proof
Papers
List of published papers:
[A] Testing a Bell inequality in multipair scenarios, J.-D. Bancal, C. Branciard, N. Brun-ner, N. Gisin, S. Popescu and C. Simon, Phys. Rev. A78, 062110 (2008).
[B] Quantifying Multipartite Nonlocality, J.-D. Bancal, C. Branciard, N. Gisin and S. Pironio, Phys. Rev. Lett. 103, 090503 (2009).
[C] Simulation of Equatorial von Neumann Measurements on GHZ States Using Non-local Resources, J.-D. Bancal, C. Branciard and N. Gisin, Adv. in Math. Phys., vol.
2010, Article ID 293245 (2010).
[D] Guess Your Neighbor’s Input: A Multipartite Nonlocal Game with No Quantum Ad-vantage, M. L. Almeida, J.-D. Bancal, N. Brunner, A. Ac´ın, N. Gisin and S. Pironio, Phys. Rev. Lett. 104, 230404 (2010).
[E] Looking for symmetric Bell inequalities, J.-D. Bancal, N. Gisin and S. Pironio, J. Phys. A: Math. Theor. 43, 385303 (2010).
[F] Detecting Genuine Multipartite Quantum Nonlocality: A Simple Approach and Gen-eralization to Arbitrary Dimensions, J.-D. Bancal, N. Brunner, N. Gisin and Y.-C. Liang, Phys. Rev. Lett. 106, 020405 (2011).
[G] Extremal correlations of the tripartite no-signaling polytope, S. Pironio, J.-D. Bancal and V. Scarani, J. Phys. A: Math. Theor. 44, 065303 (2011).
[H] Practical private database queries based on a quantum-key-distribution protocol, M. Jakobi, C. Simon, N. Gisin, C. Branciard, J.-D. Bancal, N. Walenta and H. Zbinden, Phys. Rev. A83, 022301 (2011).
[I] Various quantum nonlocality tests with a commercial two-photon entanglement source, E. Pomarico, J.-D. Bancal, B. Sanguinetti, A. Rochdi and N. Gisin, Phys. Rev. A 83, 052104 (2011).
[J] Device-Independent Witnesses of Genuine Multipartite Entanglement, J.-D. Bancal, N. Gisin, Y.-C. Liang and S. Pironio, Phys. Rev. Lett. 106, 250404 (2011).
[K] Loophole-free Bell test with one atom and less than one photon on average, N. San-gouard, J.-D. Bancal, N. Gisin, W. Rosenfeld, P. Sekatski, M. Weber and H. Wein-furter, Phys. Rev. A84, 052122 (2011).
[L] A framework for the study of symmetric full-correlation Bell-like inequalities, J.-D. Bancal, C. Branciard, N. Brunner, N. Gisin and Y.-C. Liang, J. Phys. A: Math.
Theor45, 125301 (2012).
Papers
Preprints:
[M] Quantum nonlocality based on finite-speed causal influences leads to superluminal signaling, J.-D. Bancal, S. Pironio, A. Ac´ın, Y.-C. Liang, V. Scarani and N. Gisin, arXiv:1110.3795
[N] The definition of multipartite nonlocality, J. Barrett, S. Pironio, J.-D. Bancal and N. Gisin, arXiv:1112.2626
[O] Imperfect measurements settings: implications on quantum state tomography and entanglement witnesses, D. Rosset, R. Ferretti-Sch¨obitz, J.-D. Bancal, N. Gisin and Y.-C. Liang, arXiv:1203.0911
[P] Useful multipartite correlations from useless reduced states, L. E. W¨urflinger, J.-D. Bancal, T. V´ertesi, A. Ac´ın and N. Gisin, arXiv:1203.4968
Paper A
Testing a Bell inequality in multipair scenarios
J.-D. Bancal, C. Branciard, N. Brunner, N. Gisin, S. Popescu and C. Simon
Physical Review A78, 062110 (2008)
Testing a Bell inequality in multipair scenarios
Jean-Daniel Bancal,1Cyril Branciard,1Nicolas Brunner,1Nicolas Gisin,1Sandu Popescu,2,3and Christoph Simon1
1Group of Applied Physics, University of Geneva, 20 rue de l’Ecole-de-Médecine, CH-1211 Geneva 4, Switzerland
2H.H. Wills Physics Laboratory, University of Bristol, Tyndall Avenue, Bristol BS8 1TL, United Kingdom
3Hewlett-Packard Laboratories, Stoke Gifford, Bristol BS12 6QZ, United Kingdom 共Received 8 October 2008; published 15 December 2008兲
To date, most efforts to demonstrate quantum nonlocality have concentrated on systems of two共or very few兲 particles. It is, however, difficult in many experiments to address individual particles, making it hard to highlight the presence of nonlocality. We show how a natural setup with no access to individual particles allows one to violate the Clauser-Horne-Shimony-Holt inequality with many pairs, including in our analysis effects of noise and losses. We discuss the case of distinguishable and indistinguishable particles. Finally, a comparison of these two situations provides insight into the complex relation between entanglement and nonlocality.
DOI:10.1103/PhysRevA.78.062110 PACS number共s兲: 03.65.Ud
I. INTRODUCTION
Entanglement is the resource that allows one to establish quantum nonlocal correlations 关1兴. These correlations have been the center of a wide interest, because of their fascinat-ing nature and of their impressive power for processfascinat-ing in-formation. Experimentally, quantum nonlocality has been demonstrated in so-called Bell experiments, which have to date all confirmed the quantum predictions关2兴.
Most theoretical works on Bell experiments and Bell in-equalities have focused on the case where the source emits a single entangled pair of particles at a time. Indeed, this is the simplest situation to study. From the experimental point of view, most experiments have been designed in order to match this theoretical model. For example, in photonic ex-periments, the source, usually based on parametric down-conversion 共PDC兲, is set in the weak regime; i.e., when the source emits something, it is most likely a single pair of entangled photons.
However, there are experimental situations, such as in many-body systems, where producing single entangled pairs is rather difficult. For instance, in Ref. 关3兴 many entangled pairs共⯝104兲of ultracold atoms have been created, but can-not be addressed individually. So, while entanglement has definitely been created in this system, one still lacks an effi-cient method for demonstrating its quantum nonlocality through the violation of some Bell inequality. The goal of the present paper is to discuss techniques for testing Bell in-equalities in such multipair scenarios, where the particles on Alice’s and Bob’s side cannot be individually addressed, and must therefore be measured globally 共see Fig.1兲. What we mean here by global measurements is that each particle is submitted to the same measurement. Note that the case of more general measurements共collective measurements on all particles兲has been considered in Ref.关4兴.
Basically one should distinguish two cases: independent pairs and indistinguishable pairs. In the first case, the pairs are created independently, but cannot be addressed individu-ally; therefore, they must be measured globally共on both Al-ice’s and Bob’s sides兲. During this global measurement, the classical information about the pairing is lost: there is no
way to tell which particle is entangled with which. The cor-responding loss of entanglement has been derived in Ref.关5兴. In the second case, the pairs are indistinguishable; so in some sense the information about the pairing is here lost in a co-herent way.
Reidet al.关6兴have considered the case of indistinguish-able pairs共with global measurement兲in optics. More specifi-cally, these authors, extending on a previous work of Drum-mond 关7兴, showed how Bell inequalities can be tested 共and violated兲when many pairs are created via PDC. In this case the pairs are indistinguishable because of the process of stimulated emission. In Ref.关8兴, Joneset al.have considered a related scenario; there, entangled pairs are delivered via an inept delivery service, but at the end only a single pair is measured. Also considering multiparticle entanglement in such scenario is an interesting problem: see, for example, Refs.关9,10兴.
In this paper we will study the violation of Bell inequali-ties in a general multipair scenario. We start by treating the case of independent pairs 共Sec. II兲. We argue that the resis-tance to noise is here the relevant measure of nonlocality, evaluating it. The consequences of particle losses are also investigated. Next, we move to the case of indistinguishable pairs共Sec. III兲after a brief review of the results of Ref.关6兴, we present an analysis of the influence of noise and losses in this case. In Sec. IV, we compare the entanglement and
non-FIG. 1. Setup: a source produces M independent pairs 共or equivalentlyMindependent sources each produce a pair兲, the pair-ing between Alice’s and Bob’s particles is lost durpair-ing their trans-mission, and each party measures all their incoming particles in the same basis. The total numbern+共−兲of particles detected in the⫹共⫺兲 outcome is tallied on both sides.
PHYSICAL REVIEW A78, 062110共2008兲
locality in both cases. This leads us to a surprising result:
while the state of indistinguishable pairs contains more en-tanglement than the state of independent pairs共after the clas-sical mixing兲, the latest appears to be more nonlocal. In other words, the incoherent loss of information provides more non-locality, but less entanglement, than the coherent loss of in-formation共indistinguishable pairs兲. This provides a novel ex-ample共here in the case of multipairs兲of the complex relation between entanglement and nonlocality. Finally we provide some experimental perspectives共Sec. V兲and conclusions.
II. INDEPENDENT PAIRS
We consider a source emittingMentangled pairs, each of them being in the same entangled two-qubit state. Thus the global state is
M=丢M=丢丢 ¯ 丢
Mtimes
.
共1兲 Each pair being independent, Alice and Bob receive M uncorrelated particles. Since Alice and Bob are unable to address single particles in their ensemble, they perform a global measurement on theirMparticles; i.e., allM particles are measured in the same basis共we shall consider here only von Neumann measurements兲 or, equivalently, in the same direction on the Bloch sphere. After the measurement appa-ratus, two detectors count the number of particles,n+andn−, in each output mode 共see Fig. 1兲. If the detectors are per-fectly efficient 共= 1兲, one hasM=n++n−.
A. Testing the CH inequality
Our goal is to test a Bell inequality. Here we shall focus on the simplest Bell inequality, the Clauser-Horne-Shimony-Holt 共CHSH兲inequality关11兴, which involves two inputs on Alice and Bob’s sides, A1, A2 and B1,B2, and two outputs re-spectively. Recall that under the hypothesis of no-signaling both CH and CHSH inequalities are equivalent关13兴. Now, in order to test inequality 共2兲, Alice and Bob must transform their data, basically n+ and n−, into a binary result “⫹” or or any intermediate possibility—for instance, 2/3 or 3/4 ma-jority. For each voting method and given M corresponds a
thresholdN=M/2,2M/3, . . . ,M such that the outcome is + iffn+艌N.
At this point the two relevant questions are the following:
first, is it possible to violate the CH inequality with any of these voting procedures? Second, if yes, which strategy yields the largest violation? To address these questions one must compute the joint and marginal probabilities entering the CH inequality for each procedure.
B. Pure states
Let us first consider the pure entangled states
兩典= cos兩00典+ sin兩11典, 共4兲 so that in Eq. 共1兲, =兩典具兩. We will also write 兩⌿M典
=兩典丢M. For detectors with a perfect efficiency = 1, all M particles are detected on both Alice’s and Bob’s sides. The marginal and joint probabilities entering the CH inequality for a vote with given thresholdNare
P+A共A兲=n
兺
joint and marginal probabilities for a single pair. Alice and Bob’s outputs are denoted␣,苸兵+ , −其,n␣is the number of pairs which gave detections ␣ and , and ⌶=兵n␣苸N+兩兺n␣=M,n+A=n+++n+−艌N,n+B=n+++n−+艌N其 is the set of all events yielding the result “⫹⫹” after voting.
Next, one can choose the state兩典and the measured set-tings. For the maximally entangled state of two qubits 共
=/4兲 one may choose the standard optimal 共for the case M= 1兲 settings for the CH inequality—i.e., A0=z, A1=x, B0=x冑+2z, and B1=−冑x+2z. Doing so with majority voting 共N=M/2兲, the CH inequality can be violated for any value ofM; the maximal amount of violation is numerically found to decrease with the number of emitted pairs asM−1.
Remarkably, a higher violation is found for different mea-surement settings, given by optimal. In this case the decrease ofICH is onlyM−1/2 关see Fig.2共a兲兴. The state leading to the largest violation is always the maximally entangled one共=/4兲for majority voting.
The one-parameter planar settings共7兲were already used by several authors 关6,7兴; for example, in Bell experiments
BANCALet al. PHYSICAL REVIEW A78, 062110共2008兲
using a state we shall look at共10兲later with the unanimous vote for any M.
Performing numerical optimizations, we also found that a violation can be obtained for any voting strategy with any number of emitted pairsM关see Fig.2共a兲兴. We optimized the state共兲and the four measurement settings, each time find-ing optimal settfind-ings of the form共7兲. For the unanimous vote, for instance, the optimal state is less and less entangled as the number of emitted pairs,M, increases, as described in关14兴, and the violation decreases exponentially with M. Thus for pure entangled two-qubit states, the largest amount of viola-tion is obtained with majority voting.
C. Resistance to noise
We now compute the resistance to noise that these viola-tions could bear, which is the relevant measure of nonlocality considering experimental perspectives—the amount of viola-tion being basically just a number, without much significance in the present case as we shall see.
In a practical Einstein-Podolsky-Rosen共EPR兲experiment, imperfect detectors, noisy sources, or disturbing channels in-troduce noise in the measurement results. To first order, this
noise can be modeled at the level of the source, supposing that the produced pairs are not in the pure state兩典具兩, but instead in a Werner state of the form
=w兩典具兩+共1 −w兲1
4. 共8兲
The resistance to noise of a given violation is then defined by the maximal amount ⑀= 1 −w of white noise that can be added to the pure state兩典具兩such that the resulting state still violates a Bell inequality共CH here兲.
Considering now that the sources of Fig.1 produce the state共8兲, we look for the largest value of⑀which still gives a positive value ofICH, using settings of the form 共7兲 and optimizing on the state共兲. For all voting strategies we find a resistance to noise decreasing like ⑀⬃M−1, the majority voting being still the best choice关see Fig.2共b兲兴. Unlike when maximizing ICH in the absence of noise, here the optimal state is always the maximally entangled one共=/4兲, even for intermediary voting strategies, for which the ICH viola-tion with this state decreases exponentially with M. This shows that appropriate figures of merit need to be used when examining practical situations.
These results are encouraging, but just as detectors might not be perfect, maybe the source cannot guarantee an exact number of pairs,M, as needed here. To show that these vio-lations are relatively robust towards this issue, we now look at the case of sources producing a number of entangled pairs which follow a Poissonian distribution.
D. Poisson sources
A Poissonian source produces a stateMofM pairs with a Poissonian probability
p共M兲=e−M
M!, 共9兲
where is the mean number of photon pairs. With such a source, a different number of pairs is created every time. So
where is the mean number of photon pairs. With such a source, a different number of pairs is created every time. So