Entanglement without hidden nonlocality

We consider Bell tests in which the distant observers can perform local filtering before testing a Bell inequality. Notably, in this setup, certain entangled states admitting a local hidden variable model in the standard Bell scenario can nevertheless violate a Bell inequality after filtering, displaying so-called hidden nonlocality. Here we ask whether all entangled states can violate a Bell inequality after well-chosen local filtering. We answer this question in the negative by showing that there exist entangled states without hidden nonlocality. Specifically, we prove that some two-qubit Werner states still admit a local hidden variable model after any possible local filtering on a single copy of the state.

That is, the quantum statistics can be reproduced by a so-called local hidden state model (LHS), where s l denotes the local quantum state and ( | ) l p a x, A is Alice's response function. If such a decomposition cannot be found, ρ is said to be 'steerable' for the set { } M ; a x note that one would usually consider here a set of measurements M b y that is tomographically complete, and thus focus the analysis on the set of conditional states of Bob's system a x a x A referred to as an assemblage. Note that any LHS model is also a LHV model, although the converse does not hold.
If a state ρ admits a decomposition (2) for all measurements { } M a x and { } M b y we say that ρ is local, i.e. ρ admits a LHV model. Similarly if ρ admits a decomposition (3) for all measurements { } M a x we say that ρ is unsteerable. With these definitions we have that entanglement, steering and nonlocality are strictly different concepts, even taking all possible POVMs into account. More precisely, one can show that there exist entangled states which are unsteerable (hence local) states, as well as entangled local states which are steerable. Indeed, Werner showed that some entangled quantum states admit a LHS model (3) for all projective measurements [13]. This result was later extended to general POVMs [14]. Similarly, certain steerable states were shown to admit a LHV model for projective measurements [28], and the same hold considering general POVMs [30]. LHV and LHS models for different classes of entangled states were also constructed, see e.g. [19,[31][32][33][34][35][36][37].

Hidden nonlocality
One could conclude from the above that local entangled states are somehow classical, as they can always be replaced by classical variables λ (with no noticeable difference in any Bell experiment). Nevertheless the nonlocality of some local entangled quantum states can in fact be revealed in more complex ways than the traditional Bell scenario. As first imagined by Popescu [18], one could submit a quantum state to a sequence of measurements. Indeed in quantum mechanics a measurement generally changes the state, leading to different statistics when a second round of measurements is applied. Note that a measurement does not necessarily break the entanglement of the quantum state. On the contrary, for a given measurement outcome, entanglement can be increased.
The simplest way to implement this idea is that of local filtering: Alice and Bob first perform local filters on their shared state ρ, given by a set of Kraus operators and similarly for Bob. Alice and Bob keep the post-filter state only when ρ 'passes' the filter, meaning Alice obtained the outcome corresponding to F A and Bob the outcome corresponding to F B . The state they hold in that case is given by In terms of state transformation the operation which transforms ρ into r¢ can be seen as a stochastic local operation. Note that F A and F B are any linear operators acting on  A (respectively  B ) and can in particular increase or decrease the dimension of the Hilbert space of the quantum state. Next Alice and Bob can perform a standard Bell test on r¢, that is they perform a second round of local measurements . Repeating the process many times one can access the statistics , and check whether it admits a Bell-local decomposition (2). This scenario is illustrated in figure 1.
Can r¢ be nonlocal although ρ admits a LHV model (in the standard Bell scenario)? Reference [18] showed that indeed this can be the case for certain Werner states admitting LHV models for projective measurements, while [19] extended this result by considering a state ρ admitting a LHV model for general POVMs. Hence there are entangled states that cannot lead to nonlocality in the standard Bell-scenario (even taking general POVMs into account), but nevertheless violate a Bell inequality after local filtering.
These examples open the question of whether all entangled states can lead to hidden nonlocality. That is, for any entangled state ρ, can we find local filters  A and  B such that the resulting state r¢ is nonlocal? We answer this question in the negative by constructing an explicit counter-example.

Main result
Consider the two-qubit Werner state: 2 is the maximally entangled two-qubit state and  4 is the maximally mixed state. The state ( ) r a W is entangled if and only if a > 1 3. While Werner originally constructed a LHS model for a = 1 2 and projective measurements, this was later extended. Indeed, local models were presented, for all projective measurements and  a 0.66 [31], and for POVMs for  a 5 12 [14]. The state is steerable for a > 1 2 [28], and nonlocal for a > 0.7055 [38,39].
Our main result is that ( ) r a W remains local, considering arbitrary POVMs, after any local filtering for  a a = 0.3656 c . Hence the entangled state ( ) r a W c displays no hidden nonlocality. This can be formalized with the following theorem: Figure 1. The hidden nonlocality scenario: Alice and Bob share an entangled state ρ and perform local filters F A and F B , respectively. When the filtering is successful, they perform a standard Bell test. If the resulting statistics ( | ) p ab xy violate a Bell inequality, the state ρ displays hidden nonlocality. Here we ask if this effect is possible for all entangled states, and show that this is not the case.
is local for all POVMs. Here, F A and F B represent any possible local filters; Proof. We will proceed in two steps. First we characterize the filtered state when only Alice applies a local filter.
Then we show that this state remains local over all operations applied locally by Bob.
Consider again the Werner state ( ) r a W defined in (6). Alice applies a local filtering passes the filter, Alice and Bob hold the state r F A given by: where  is the identity operator in Bob's Hilbert space. One can show that this state is unsteerable from Alice to Bob (for all POVMs) if and only if the state cos 00 sin 11 is the partially entangled state and its partial trace. To prove this claim consider first the unnormalized filtered state We can therefore focus on the unormalized state By the same observation as above we can apply any unitary U B on Bob's side. Choosing , , as stated above. Note also that for , , up to local unitaries. Therefore, we can focus only on the interval After Alice has applied her filter, we are left with the state (9), on which Bob will now apply his filter F B . A possible approach to deal with Bob's filter is to use the concept of steering, introduced above. Indeed if a state ρ is unsteerable from Alice to Bob, then the state remains unsteerable (hence local) after any local operation on Bob's side. For a proof see [30], lemma 2. In our case, this implies that if ( ) r a q , is unsteerable (from Alice to Bob), then the state p Î 0, 4 and for all POVMs. Note that if we restricted ourselves to projective measurements on Alice's side we could use the family of LHS models presented in [36], but the requirement of general POVMs forces us to find another approach. In particular the restriction of projective measurements would prevent Alice's filter from increasing the dimension of the Hilbert space, and is thus not general enough.
In order to construct a LHS model for states of the form ( ) r a q , we use several methods, in particular the algorithmic method presented in [40,41]. In principle this method allows us to find a LHS model for any given unsteerable state. However, here we need to prove that the entire class of states Here θ is in the neighbourhood of p 4. We decompose the target state ( ) r a q , as a mixture of states admitting a LHS model for POVMs. Specifically, we search for which values of θ and α, we can find a convex combination of the form: W admits a LHS model for POVMs [14,30]. Here σ is an unspecified two-qubit state, and as long as σ admits a LHS model, this implies that ( ) r a q , is unsteerable, as one can write it as a probabilistic mixture of two unsteerable states. A simple solution is to demand that , we obtain a diagonal matrix σ (for all α and θ). To verify that σ represents a valid state, we only need to ensure that its eigenvalues are positive. One can check that this is the case when Now we focus on the interval I 2 . In this regime we essentially use the technique presented in [40]. More precisely we choose finitely many values q Î I k 2 ( = k n 1 .. ). For each of them, a slightly improved version of Protocol 1 of [40] allows us to find a value a k such that ( ) r a q , k k admits a LHS model for POVMs (more details in appendix A). The obtained values of a k and q k are shown on the blue dashed curve of figure 2. In order to extend the result to the continuous interval I 2 , we use the following lemma, which is proven in appendix B: , is unsteerable from Alice to Bob (for POVMs) then the state ( ) , is also unsteerable from Alice to Bob (for POVMs) as long as , which is close to a k if q k and q + k 1 are close. Therefore, we have to choose q + k 1 sufficiently close to q k , such that the value of α does not drop below a c .
We are thus left with the interval I 1 , where θ is in the neighbourhood of 0. We cannot use the same method as in I 2 as whatever smallest { } q q = min s k k we choose, lemma 1 only allows us to say something about some Parameter region for which we could prove that the states ( ) r a q , of equation (9)   q q s , leaving the interval [ ] q 0, s unsolved. Note also that by setting q = 0 s we get a separable state and consequently mixing it with another separable state cannot give rise to an entangled one (note that setting q = 0 in lemma 1, one obtains a¢ = 0).
However, in this region, an explicit LHS model for projective measurements is known [36]. The model holds for ( ) r a q , as long as To take POVMs into account we use a method developed in [19], Protocol 2. Starting from an entangled state ρ admitting a local model for projective measurements, we can construct another entangled state r¢ admitting a local model for POVMs. Note that while the method was originally developed for LHV, we use it here for LHS models (i.e. only on Alice's side). Specifically, we now apply this method to the state ( ) r a q , where condition (19) is fulfilled, thus ensuring that the state admits a LHS model for projective measurements. We obtain the class of states . This state is therefore unsteerable from Alice to Bob, for all POVMs. The last step consists in showing that ( ) r a q , , where a a > c can be written as a convex combination of r q and a separable state, for all q Î I 1 . This proof is given in appendix C for a = 0.4.
Finally, we summarize these results in figure 2.

Conclusion
We proved that there exist entangled quantum states which do not display hidden nonlocality, i.e. which remain local after any local filtering on a single copy of the state. Specifically we showed this to be the case for some twoqubit Werner states. This consequently proves that local filters (or equivalently SLOCC procedures before the Bell test) are not a universal way to reveal nonlocality from entanglement.
The natural question now is wether the use of even more general measurement strategies could help to reveal the nonlocality of the states we consider. In particular, one could look at sequential measurements [27], beyond local filters, and consider the entire statistics of the measurements. Here, one chooses between several possible measurements (or filters) at each round. In order to show that a quantum state is local, one should now construct a LHV model that is genuinely sequential. That is, the distribution of the local variable should be fully independent of the choice of the sequence of measurements. Indeed, this is not the case in our model, as the distribution of the local variable depends on Alice's filter. Nevertheless, as our model is of the LHS form, the distribution of local states does not depend on the choice of local filter for Bob, and more generally covers any possible sequence of measurements on Bob's side. It would therefore be interesting to see if one can find a model that holds for arbitrary sequences of measurements on Alice's side as well. If this is not the case, then a sequence of measurements should be considered strictly more powerful than local filtering in Bell tests.
There exists however another possible extension of the Bell scenario, where Alice and Bob share many copies of the state. Here, in each run of the experiment, the two observers can now perform joint measurements on the many copies they hold. It has been shown that some local entangled states (in the standard Bell scenario) produce nonlocal statistics in this setup [42]. This phenomenon is known as 'super-activation' of quantum nonlocality. While it is not known whether super activation is possible for all entangled states, it was nevertheless shown that any entangled state useful for teleportation (or equivalently, with entanglement fraction greater than d 1 where d is the local Hilbert space dimension) can be super activated [43]. In fact, it turns out that the class of states we considered, i.e. two-qubit Werner states, are always useful for teleportation [44] and can thus be super activated. Our result thus demonstrates that quantum nonlocality via local filtering or many-copy Bell tests are inequivalent.
Finally, the main open question is still whether there exists an entangled state that would display no form of nonlocality, considering arbitrary sequential measurements on many copies, or in quantum networks [45,46] where stronger notions of nonlocality could be considered [47][48][49]. x . The exact value * h is in general hard to evaluate, but [19] gives a general procedure to obtain arbitrary good lower bounds on * h , which is therefore enough for us to make sure that *  h h .
The method requires the choice of a finite set of measurements { } M a x which should 'approximate well' the set of all POVMs. We considered a set of projective measurements, the directions of which were given by the vertices of the icosahedron, that is, 12 Bloch vectors forming an icosahedron. More precisely we consider all relabellings of { } + -P P , , 0, 0 for + P being a projector onto a vertex of the icosahedron and -P onto the opposite direction. In addition we consider the four relabellings of the trivial measurement { }  , 0, 0, 0 2 , which comes for free as it cannot help to violate any steering or Bell inequalities and consequently does not even need to be inputed in Protocol 1. The set thus have 76 elements, but we need to take into account only six of them when running the Protocol, corresponding to the vertices in the upper half sphere of the icosahedron.
There To prove this lemma we show that ( ) r a q ¢ ¢ , can be written as a convex combination of ( ) r a q , and a separable state, as long as condition (B.1) holds. We want: where S is a separable state. Inverting this relation we get: That is: Under this condition we are then left to show that S is a valid state, i.e. a semi-definite positive trace-one matrix.
First one can check that its trace is always equal to 1, second its eigenvalues are just given by the diagonal elements, this gives us the four following conditions for positivity: and again using that ℓ ℓ  1 2 we see that that the inequality (B.5) is more constaining than the inequality (B.8). We are thus left with the two inequalities (B.5) and (B.6), but one can show that the inequality (B.5) is more constraining. One has  where S is a separable state. Inverting this relation we get: where the non-zero elements of S are given by: