One way to study nonlocal correlations is to try to simulate them with a measureable amount of nonlocal resources. This allows one to put an upper bound on the power of these correlations. For instance, it is well known that correlations created upon measurement of a singlet state can be simulated by the use of shared randomness supplemented by 1 bit of communication [55], or by 1 use of a PR box [56]. Thus no correlations found upon measurement of a singlet state can achieve a task that 1 bit of communication, or 1 PR box, cannot.
Here, we consider the simulation of then-partite Greenberger-Horne-Zeilinger (GHZ) state
|GHZi= 1
√2(|00. . .0i+|11. . .1i). (2.6.1) Since this state is genuinely tripartite nonlocal (it can violate the Svetlichny inequality (2.1.1) or its n-partite generalization [46, 47]), it cannot be simulated with just shared randomness and interaction between a subset of the parties. Nevertheless, we consider the task of simulating it with the aid of bipartite resources only.
2.6.1 Nonlocal resources
Let us allow the parties to share nonlocal boxes of the following kinds in addition to pre-established randomness.
PR box. A Popescu-Rohrlich (PR) box [57] is a nonlocal box that admits two bits x, y ∈ {0,1} as inputs and produces locally random bits a, b ∈ {0,1}, which satisfy the binary relation
a+b=xy. (2.6.2)
M box. A Millionaire box [58] is a nonlocal box that admits two continuous inputs x, y∈[0,1[ and produces locally random bitsa, b∈ {0,1}, such that the following relation is satisfied:
a+b= sg(x−y) (2.6.3)
where addition is modulo 2 and the sign function is defined as sg(x) = 0 if x > 0 and sg(x) = 1 if x≤0.
Note that none of these boxes is signaling: the outcomes produced by the boxes are locally random and thus carry no information on the other party’s choice of input.
2.6.2 Simulation
A protocol to simulate measurements on the GHZ state (2.6.1) with nonlocal boxes runs as follows: before letting the parties choose their measurement settings, they are allowed to share any information, plus a number of boxes (as shown in Figure 2.4). The parties can then choose their measurement settings (which we represent by vectors ~a,~b, ~c on the Bloch sphere). They are allowed to locally process this setting together with the pre-established shared randomness and accesses to their boxes. The parties then output the result of this process, which we denote by α,β,γ ∈ {−1,1}.
Using the above boxes, we considered the simulation of the correlations found by measuring the GHZ state in the equatorial plane, i.e. with~a= (cosφa,sinφa,0), etc. In this case the correlations take the form
hαi=hβi=. . .=hαβi=. . .= 0 (2.6.4)
2.6 Simulating projective measurements on the GHZ state
Bob Alice
M box
M box
Charlie
PR box
PR box
Figure 2.4: Setup for the simulation of the tripartite GHZ state in the x-y plane : two Millionaire boxes are shared between Alice and Bob and each of them shares a PR box with Charlie.
Bob Alice e
M box
M box
Dave Charlie
M box
M box
PR box
PR box PR box
PR box
Figure 2.5: Distribution of bipartite no-signalling boxes that allows for the simulation of equatorial von Neumann measurement on the 4-partite GHZ state.
for all marginal correlations, and
hαβ . . .i= cos(φa+φb+. . .) (2.6.5) for the fulln-partite correlation term. Here are the results that we could show (proofs in paper [C]):
Theorem. Equatorial von Neumann measurements on the tripartite GHZ state can be simulated with 2 M boxes and 2 PR boxes distributed as in Figure 2.4.
Theorem. Equatorial von Neumann measurements on the 4-partite GHZ state can be simulated with 4 M boxes and 4 PR boxes distributed as in Figure 2.5.
Moreover, one can show that a PR box can be simulated with one bit of classical communication transmitted from one end of the box to the other one, and an M box with an average of 4 bits. Each of these protocols can thus be translated into communication models with a finite-average communication cost. Namely, and average of 10 bits allow for the simulation of equatorial measurements on the tripartite GHZ state, whereas 20 bits suffice on average for the four-partite case.
2.6.3 Conclusion
We showed that bipartite no-signalling resources are enough to reproduce the nonlocal character of these GHZ correlations, even though these correlations are genuinely mul-tipartite nonlocal. Moreover, we provided models to reproduce these correlations with a finite amount of communication on average. Note that this latter result was recently improved for the tripartite case [59].
Nonlocality with three and more parties
Chapter 3
Device-independent entanglement detection
Entanglement is one of the most intriguing feature of quantum physics. It allows several particles to be in a state which cannot be understood as a concatenation of the sate of each particle. Experimental demonstration of entanglement is generally performed with one of the two following techniques: tomography of the full quantum state, or evaluation of an entanglement witness.
In the first case, the state ρ of the system is characterized by performing a number of complementary measurements on it [64]. For instance, on two qubits, measurement of the product of all Pauli operatorsσi⊗σj, with j = 0,1,2,3, and σ0 = 11 allows one in principle to deduceρ by solving the set of linear equations
tr(ρ σi⊗σj) =fij (3.0.1)
where fij is the observed frequency for the corresponding measurements. In practice however, experimental imperfections typically lead to a solution for the former set of equations which is unphysical so that more complicated techniques are generally used instead of the linear inversion, like maximum likelihood estimation [65]. Still, once the reconstructed state is found, theoretical analyses can be performed on it to check directly whether the quantum state is entangled or not.
In contrast, an entanglement witness is an observableW such that tr(ρW)≥0 when-everρis separable [35]. Any decomposition ofW in terms of local observables allows one to evaluate it by performing local measurements on the state under consideration. If a value tr(ρW)<0 is found, the measured state is then said to be entangled.
3.1 Imperfect measurements
Any experimental manipulation is affected by imperfections, be it only the finiteness of the number of times measurements are repeated in order to accumulate sufficient statistics.
Interestingly, the effect that statistical uncertainties on the frequencies fij can have on tomographically reconstructed states was analysed rigorously only very recently [66, 67].
Still, even in absence of statistical uncertainties, which can in principle be avoided by performing enough measurements in a random order, systematic errors in the measure-ment process can possibly affect the conclusion of a test. While this problem is known, it is seldom discussed in the literature. Let us show what kind of effects these errors can have in the detection of entanglement.
Device-independent entanglement detection
ε ε
ε
~n3 m~3
~ m2
~n2
~n1
~ m1
Figure 3.1: Intended and actual measurement directions for tomography on a qubit. The actual measurement directions~n are distant from the desired onesm~ by an angle smaller thanε.
3.1.1 Effects of systematic errors on tomography
In order to evaluate the effect of systematic errors on the process of tomography recon-struction, we consider the situation in which each measurement can be slightly misaligned.
Namely, if m~ ·~σ denotes the desired measurement on a qubit, the actual measurement performed can be written as~n·~σwith m~ ·~n≥cos(ε) (c.f. Figure 3.1). However, since~n is unknown, results measured along ~n are interpreted during the reconstruction process as coming from measurements alongm.~
Considering qubit states|ψi, we looked for the maximum effect that these errors could have on the reconstructed stateρ by performing the following optimization :
|ψi,nmini
hψ|ρ|ψi
subject to m~i·~ni ≥cos(ε) ∀measurement i
(3.1.1) The results of this numerical optimization are shown in Figure 3.2a. For small errors ε in the definition of the measurement bases, the uncertainty of the reconstructedn-qubit state increases at least as √n
2ε (c.f. paper [O] for more details). Thus, if measurements are done with linear polarizers having a precision of 1o in real space, for instance, the precision of the reconstructed state can decrease by 2.5% per qubit in the system.
Interestingly, we found that entangled states are usually more robust to systematic errors than the worst bound shown in Figure 3.2a (see paper [O]). Nevertheless, imper-fect measurements on separable states can sometimes lead to an entangled reconstructed state [68]. Entanglement can thus be wrongly witnessed through tomography because of systematic errors.
3.1.2 Effects of systematic errors on entanglement witnesses In a similar fashion, we analysed the witness
W = 1
211− |GHZihGHZ| (3.1.2)
which detects genuine multipartite entanglement [35]. For this we used the decomposition ofW in terms of local operators given in [69]. Allowing again all measurement operators to differ from the prescribed ones by at mostε, we looked for the smallest value tr(ρW) that could be achieved by measuring a biseparable stateρ.