So far we could only demonstrate (`, h) measurement-dependent nonlocality for ` >
0. In the (2,2,2) scenario, we showed that ` = 0 allows to reproduce all quantum and nonsignaling correlations and inequality (3.8) in the (N,2,2) scenario can also not be violated for this case. This is however not a fundamental limit; it is possible to violate (`, h)-MDL inequalities even for ` = 0. To demonstrate this, we work in the (2, n,2) scenario. We can then construct the following inequality, which holds for all (0, h)-MDL correlationsPABXY:
1−(n2−n+ 1)h
P(0000)−h
n−1
X
i=1
P(010i) +P(10i0) +P(00ii)
≤0. (3.10) The inequality is trivially satisfied if h≥ n2−1n+1 since then (1−(n2−n+ 1)h≤0.
We prove that the inequality also holds forh < n2 1
−n+1. In this case, at mostn−2 inputpairs can be excluded, i.e. occur with probability 0 due to the normalization of probabilities. Lett be the number of zero probability entries and set the remaining n2−t probabilities to their maximum of h. Then we have that
(n2−t)h+t·0 = 1
⇔t =n2− 1 h
< n2−(n2−n+ 1)
=n−1
and thus t ≤ n−2. We are therefore guaranteed that ∀v ∈ VMDL(2,n,2)(0, h) with h < n2−1n+1 ∃j ∈ {1. . . n} such that vXY(xy) ≥ 1−(n2 −n+ 1)h > 0 for xy = (0j),(j0),(jj). Recall that all vertices ofMDL(2,n,2)(`, h) are of the formv(abxy) = v(xy)δa,¯axδb,¯by for ¯ax ∈ {0,1}and ¯by ∈ {0,1}. Evaluating the left hand side of (3.10) we thus have
1−(n2−n+ 1)h
v(0000)−h
n−1
X
i=1
v(10i0) +v(010i) +v(00ii)
≤ 1−(n2−n+ 1)h
v(0000)−h v(10j0) +v(010j) +v(00jj)
≤h 1−(n2−n+ 1)h
δ0,¯a0δ0,¯b0 −δ1,¯ajδ0,¯b0 −δ0,¯a0δ1,¯bj −δ0,¯ajδ0,¯bj
≤0
To get from the first to the second line, we simply use that v(abxy)≥ 0 ∀a, b, x, y.
The third line follows fromv(abxy) = v(xy)δa,¯axδb,¯by together with 1−(n2−n+1)h≤ v(xy) for (xy) = (0j),(j0),(jj) and vXY(00)≤h.
This generalized inequality can be violated by quantum mechanics using again the state |Aui defined in (3.4) by performing the measurements|0i+|1i
√2 ,|0i−|√ 1i
2 for
input 0 and
|0i,|1i for all other inputs. The resulting correlations fulfill P(0000) = PXY(00)
12 , P(010i) = P(10i0) =P(00ii) = 0 ∀i
Of course one may argue that if the same measurement is performed then it should not be considered to be different inputs. Nonetheless this example shows that
` >0 is not a necessary condition for quantum mechanics to be (`, h) measurement-dependent nonlocal. Unfortunately, this does not imply that a violation can be found for any (`, h) with [`, h]([0,1] since we only proved measurement dependent nonlocality when either ` > 0 or h < n2 1
−n+1 ≤ 138. The case of ` = 0 and h ≥ 13 remains open. Due to the fact that this was not a full analysis, we do not conjecture that (0, h) measurement dependent local distributions withh ≥ 13 can reproduce all quantum correlations. Further investigations are required.
3.4 Dealing with memory and finite statistics: the non-i.i.d. case
A general problem when implementing experiments for nonlocality is that it is im-possible to determine the exact probability distribution PA, ~~X. Instead the experi-ment consists of a finite number of trials for which a new state is prepared and new measurement inputs are chosen each time. The work of Gill [30] and in this case more specifically Bierhorst [14] allows us to deal with this situation. In conversation with Peter Bierhorst we established that the methods presented in his work on finite statistics and memory effects in nonlocality can be applied to (`, h) measurement-dependent nonlocality as well.
8Note that n ≥ 2. If n = 1 then there is no choice of inputs and nonlocality cannot be demonstrated.
Consider again the physicist and his friend. The friend is testing some devices that the physicist prepared to see if they are (`, h) measurement-dependent nonlocal.
They need to take into account that they can only test the devices finitely many times and that the devices could have an internal memory, allowing them to adapt their potentially local strategy depending on the previous runs. The friend is considering some previously chosen (`, h)-MDL inequality given by
X
~a,~x
β~a,~xP(~a, ~x)≤0 (3.11) where the normalization constraint was used to set the bound to 0. For each run of the test, the friend notes a score of β~a,~x if the inputs and outputs for that run were
~x and ~a respectively. For the i-th run, we denote by A~i the output and by X~i the input random variables for that run. For the score, we define the random variable Si given by
Si =β~a,~x if A~i =~a, ~Xi =~x.
We abuse notation slightly here, technically Si is a function from the outcome set Ωi for that run to the set {β~a,~x}~a,~x such that Si(ωi) = β~a,~x when A~i(ωi) = ~a and X~i(ωi) =~x. We denote by the random variableTi the transcript up to thei-th run, i.e.
Ti = (A~1. . . ~Ai, ~X1. . . ~Xi, S1. . . Si,Λ1. . .Λi),
where the local hidden random variables Λ1. . .Λi that the physicist could have used are included as well.
In each run, the friend assumes that the (`, h) measurement-dependence assump-tion holds, even when past runs are considered, i.e.
` ≤P(~xi|λi, ti−1)≤h. (3.12) This means that the physicist does not gain better control over the random number generators as the test goes on, but remains limited by (`, h)9. Inequality (3.11) then holds for (`, h)−MDL distributions even when conditioning on the transcript,
X
~a,~x
β~a,~xPA~
i, ~Xi|Ti−1(~a, ~x|t)≤0.
For the score variable, this implies E(Si|Ti−1) = max
t
X
~a,~x
β~a,~xPA~i, ~Xi|Ti−1(~a, ~x|t)
≤0,
where E denotes the expectation value. In other words, for local strategies us-ing the transcripts, the total average score Pi
j=1Sj does not increase. A sequence S1, . . . Si. . .fulfilling this property is called a supermartingale10.
9(3.12) is very similar to the Santha-Vazirani assumption (3.2)
10Note thatS1. . . Si is part ofTi
The interesting value in such a setup that the friend wishes to know is the probability that devices that are (`, h)-MDL could achieve the total score that they get at the end of their tests. This is called the p-value. If the p-value is low, then even though the physicist could have adapted their strategy in each run depending on what happened so far and taking into account potential statistical fluctuations, it is unlikely that the observed total score would have occurred. This would constitute good evidence that the physicist indeed has constructed nonlocal devices.
The works of Gill, Bierhorst and others can be used to put bounds on the p-value if the score variable is a supermartingale, which we proved to be the case here. In an actual (`, h)-MDL experiment or potential future (`, h)-MDL applications, their work can thus be directly applied.