Entanglement without hidden nonlocality

Entanglement without hidden nonlocality Flavien Hirsch Département de Physique Théorique, Université de Genève, 1211 Genève, Switzerland

Marco T´ ulio Quintino

arXiv:1606.02215v1 [quant-ph] 7 Jun 2016

Département de Physique Théorique, Université de Genève, 1211 Genève, Switzerland

Joseph Bowles Département de Physique Théorique, Université de Genève, 1211 Genève, Switzerland

Tam´ as V´ ertesi Institute for Nuclear Research, Hungarian Academy of Sciences, H-4001 Debrecen, P.O. Box 51, Hungary

Nicolas Brunner Département de Physique Théorique, Université de Genève, 1211 Genève, Switzerland Abstract. 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.

1. Introduction Nonlocality is one of the most startling predictions of quantum mechanics. It allows two distant observers to obtain experimental statistics that cannot be described by any classical common cause (given a few reasonable physical assumptions) [1, 2]. Recently confirmed in loophole-free experiments [3, 4, 5] nonlocality has been proven to be useful for many tasks, such as device-independent cryptography [6] and randomness certification [7, 8]. This effect is enabled by the genuinely quantum phenomenon of entanglement. However, it is still unclear which entangled quantum states lead to nonlocality [9]. While for pure states it is known that all entangled states can display nonlocal correlations

Entanglement without hidden nonlocality

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[10, 11], mixed states exhibit a more intricate behaviour, as initially shown by Werner [12]. Namely, there exist entangled mixed states that never lead to nonlocality when submitted to any local measurements, even taking general POVMs into account [13]. This result has recently been shown to hold true in the general multipartite case as well: for any number of parties, there exist genuinely multipartite entangled states which admit a local hidden-variable (LHV) model [14]. However, these results have been derived in the scenario considered initially by Bell, i.e., in each run of the experiment, non-sequential local measurements are performed on a single copy of an entangled state. Going beyond this standard Bell scenario allows one to reveal the nonlocality of some entangled states which admit a LHV model (in the standard Bell scenario). For instance, one could allow for local filtering, i.e. local filters applied by each party before the standard Bell test, hence being a pre-processing of the entangled state. This was first proposed by Popescu [15], who concluded that some entangled Werner states which admit a LHV model (for all projective measurements) display some ‘hidden nonlocality’, that is, violate a Bell inequality after well-chosen local filters. This phenomenon has been shown to exist even for entangled states admitting a LHV model for general measurements (POVMs) [16]. Hence, local filtering allows one to reveal the nonlocality of some entangled states which are always local in the standard Bell scenario. Following Ref. [15], several aspects of local filtering in Bell tests have been discussed. For the two-qubit case, Ref. [17] studied how local filtering can increase entanglement and it was shown that local filtering can ‘activate’ CHSH-violation [18, 19], for which a necessary and sufficient condition was derived [20]. Ref. [21] generalized hidden nonlocality to many copies and showed a strong link to entanglement distillability. Ref. [22] showed that all entangled states display some kind of hidden nonlocality, in the sense that any entangled state can help to activate the CHSH violation of another entangled state. Refs [23, 24] discussed the general scenario of Bell tests with sequential measurements, of which local filtering is a particular case. Finally, local filtering was shown to reveal genuine multipartite nonlocality [14]. Altogether local filtering has been shown to be a powerful way of activating nonlocality from entangled states which admit LHV models. A natural question is therefore whether local filtering can reveal the nonlocality of all entangled states. That is, do all entangled states display hidden nonlocality? Here we answer this question in the negative, by showing that some entangled states cannot exhibit hidden nonlocality, considering the scenario of Popescu. Specifically, we show that some entangled two-qubit Werner states admit a LHV model after local filtering on a single copy of the state. Our model takes into account any local filters, and holds for all local POVMs performed on the state after filtering. Our result can also be interpreted as follows: some entangled two-qubit Werner states cannot violate any Bell inequality, even after arbitrary stochastic local operations and classical communication (SLOCC) performed before the Bell test on a single copy of the state[21]. We conclude with some open questions.


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2. Preliminaries 2.1. Bell nonlocality and quantum steering Consider two distant observers, say Alice and Bob, sharing an entangled quantum state P ρ. Alice performs one measurement chosen in a set {Ma|x } (Ma|x ≥ 0 and a Ma|x = 1), and Bob performs a measurement chosen in a set {Mb|y } (with similar conditions). Given that they choose the measurements labelled by x and y, respectively, the resulting statistics are given by p(ab|xy) = Tr(Ma|x ⊗ Mb|y ρ).

(1)

The state ρ is said to be local for measurements {Ma|x }, {Mb|y } if the distribution (1) admits a Bell-local decomposition Z p(ab|xy) = π(λ) pA (a|x, λ) pB (b|y, λ) dλ. (2) That is, the quantum statistics can be reproduced using a LHV model consisting of a shared local variable λ, distributed with density π(λ), and local response functions given by distributions pA (a|x, λ) and pB (b|y, λ). If for some sets of measurements {Ma|x } and {Mb|y } a decomposition of the form (2) cannot be found, the distribution p(ab|xy) violates (at least) one Bell inequality [2]. In this case, we conclude that ρ is nonlocal for the measurements {Ma|x } and {Mb|y }. Another concept that will be useful here is that of EPR-steering [25]; see [26] for a review. It is a weaker form of nonlocality which captures the fact that if Alice makes a measurement on her half of the state ρ she remotely steers Bob’s state. This nonlocal effect can be detected in the statistics of the experiment if Bob measures his part of the state as well. Specifically, if Alice and Bob perform measurements {Ma|x } and {Mb|y }, respectively, we say that ρ is ‘unsteerable’ (from Alice to Bob) if Z p(ab|xy) = π(λ) pA (a|x, λ) Tr(Mb|y σλ ) dλ. (3) That is, the quantum statistics can be reproduced by a so-called local hidden state model (LHS), where σλ denotes the local quantum state and pA (a|x, λ) is Alice’s response function. If such a decomposition cannot be found, ρ is said to be ‘steerable’ for the set {Ma|x }; note that one would usually consider here a set of measurements Mb|y that is tomographically complete, and thus focus the analysis on the set of conditional states of Bob’s system σa|x = TrA (Ma|x ⊗ 1 ρ),

(4)

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 {Ma|x } and {Mb|y } we say that ρ is local, i.e. ρ admits a LHV model. Similarly if ρ admits a decomposition (3)


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for all measurements {Ma|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 [12]. This result was later extended to general POVMs [13]. Similarly, certain steerable states were shown to admit a LHV model for projective measurements [25], and the same hold considering general POVMs [27]. LHV and LHS models for different classes of entangled states were also constructed, see e.g. [28, 29, 16, 30, 31, 32, 33, 34]. 2.2. 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 [15], 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 † FA = {FA , F¯A } and FB = {FB , F¯B }, where FA† FA + F¯A F¯A = 1 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 FA and Bob the outcome corresponding to FB .

Figure 1. The hidden nonlocality scenario: Alice and Bob share an entangled state ρ and perform local filters FA and FB , 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.


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The state they hold in that case is given by 0

ρ =

FA ⊗ FB ρ FA† ⊗ FB† tr(FA ⊗ FB ρ FA† ⊗ FB† )

.

(5)

In terms of state transformation the operation which transforms ρ into ρ0 can be seen as a stochastic local operation (SLO). Note that FA and FB are any linear operators acting on HA (respectively HB ) 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 ρ0 , that is they perform a second round of local measurements MA = {Ma|x } and MB = {Mb|y }. Repeating the process many times one can access the statistics p(ab|xy) = Tr(Ma|x ⊗ Mb|y ρ0 ), and check whether it admits a Bell-local decomposition (2). This scenario is illustrated in Figure 1. Can ρ0 be nonlocal although ρ admits a LHV model (in the standard Bell scenario)? Ref [15] showed that indeed this can be the case for certain Werner states admitting LHV models for projective measurements, while Ref. [16] 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 FA and FB such that the resulting state ρ0 is nonlocal? We answer this question in the negative by constructing an explicit counter-example. 3. Main result Consider the two-qubit Werner state: ρW (α) = α φ+ φ+ + (1 − α)1/4 (6) √ where |φ+ i = (|00i + |11i)/ 2 is the maximally entangled two-qubit state and 1/4 is the maximally mixed state. The state ρW (α) is entangled if and only if α > 1/3. While Werner originally constructed a LHS model for α = 1/2 and projective measurements, this was later extended. Indeed, local models were presented, for all projective measurements and α . 0.66 [28], and for POVMs for α ≤ 5/12 [13]. The state is steerable for α > 1/2 [25], and nonlocal for α > 0.7055 [35, 36]. Our main result is that ρW (α) remains local, considering arbitrary POVMs, after any local filtering for α . αc = 0.3656. Hence the entangled state ρW (αc ) displays no hidden nonlocality. This can be formalized with the following theorem: Theorem 1. For α ≤ αc the state 0

ρ =

FA ⊗ FB ρW (α) FA† ⊗ FB† Tr(FA ⊗ FB ρW (α) FA† ⊗ FB† )

(7)


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is local for all POVMs. Here, FA and FB represent any possible local filters; FA , FB : H2 → Hd . 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 ρW (α) defined in (6). Alice applies a local filtering FA = {FA , F¯A }. If ρW (α) passes the filter, Alice and Bob hold the state ρFA given by: ρFA =

FA ⊗ 1 ρW (α) FA† ⊗ 1 Tr(FA ⊗ 1 ρW (α) FA† 1)

(8)

where 1 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 ρ(α, θ) = α |ψθ i hψθ | + (1 − α)ρA ⊗ 1/2

(9)

is unsteerable from Alice to Bob (POVMs), for all θ ∈ [0, π/4]. Here, |ψθ i = cos θ |00i + sin θ |11i is the partially entangled state and ρA = TrB (|ψθ i hψθ |) its partial trace. To prove this claim consider first the unnormalized filtered state ρFA = FA ⊗ 1 ρW (α) FA† ⊗ 1.

(10)

Using the singular value decomposition one can write FA = U DV † , where U, V are unitary matrices and D is diagonal and positive. Note that since FA is a d × 2 matrix, U is d × d, D is d × 2 and V is 2 × 2. We thus have ρFA = U DV † ⊗ 1 ρW (α) V DU † ⊗ 1 = U (DV † ⊗ 1ρW (α)V D ⊗ 1)U † .

(11)

We can then use the fact that if the state ρ is unsteerable so is UA ρUA† (for an arbitrary unitary UA acting on Alice’s subspace) as the statistics coming from a measurement {Ma } is tr(Ma UA ρUA† ) = tr(UA† Ma UA ρ) = tr(Ma0 ρ), where {Ma0 } is another (valid) measurement. We can therefore focus on the unormalized state DV † ⊗ 1 ρW (α) V D ⊗ 1.

(12)

By the same observation as above we can apply any unitary UB on Bob’s side. Choosing UB = V we get D(V † ⊗ V ρW (α) V ⊗ V † )D = D ⊗ 1ρW (α)D ⊗ 1

(13)

using the U ⊗ U † symmetry of ρW (α). Finally, note that the normalization Tr(ρFA ) = Tr(D2 )/2 is independent of α. The one-side filtered Werner state ρFA / Tr(ρFA ) is thus equivalent (up to local unitaries) to ρ(α, θ), as stated above. Note also that for k ∈ N ρ(α, θ + kπ/4) is equivalent to ρ(α, θ), up to local unitaries. Therefore, we can focus only on the interval θ ∈ [0, π/4].


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After Alice has applied her filter, we are left with the state (9), on which Bob will now apply his filter FB . 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 [27], Lemma 2. In our case, this implies that if ρ(α, θ) is unsteerable (from Alice to Bob), then the state ρFB (α, θ) =

1 ⊗ FB ρ(α, θ) 1 ⊗ FB† Tr(1 ⊗ FB ρ(α, θ) 1 ⊗ FB† )

(14)

is unsteerable, hence local. Thus we can prove the theorem by showing that the state ρ(αc , θ) is unsteerable (from Alice to Bob) for all θ ∈ [0, π/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 Ref. [33], 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 ρ(α, θ) we use several methods, in particular the algorithmic method presented in [37, 38]. 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 ρ(αc , θ) admits a LHS model, for a fixed value αc and the whole interval θ ∈ [0, π/4]. To do so we first choose finitely many angles θk ∈ [0, π/4] in order to get pairs (αk , θk ) such that ρ(αk , θk ) admits a LHS model. Then we consider convex combinations of these states with separable states in order to extend the model to the whole interval. To finish the proof, we must treat the interval boundaries, i.e. the two limits θ → 0 and θ → π/4, for which we use different techniques. We consequently break the interval [0, π/4] in three sub-intervals: I1 = [0, θs ], I2 = [θs , θl ] and I3 = [θl , π/4], where θs = 0.1 and θl = 0.7354. Let us start with I3 . Here θ is in the neighbourhood of π/4. We decompose the target state ρ(α, θ) 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: ρ(α, θ) = q ρW (5/12) + (1 − q)σ

(15)

with 0 ≤ q ≤ 1. Recall that the Werner state ρW (5/12) admits a LHS model for POVMs [13, 27]. Here σ is an unspecified two-qubit state, and as long as σ admits a LHS model, this implies that ρ(α, θ) is unsteerable, as one can write it as a probabilistic mixture of two unsteerable states. A simple solution is to demand that σ=

ρ(α, θ) − qρW (5/12) 1−q

(16)

be separable. By setting q = 12/5α sin(2θ), 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


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are positive. One can check that this is the case when α≤

1 . 17/5 cot θ − 1

(17)

Now we focus on the interval I2 . In this regime we essentially use the technique presented in [37]. More precisely we choose finitely many values θk ∈ I2 (k = 1..n). For each of them, a slightly improved version of Protocol 1 of Ref. [37] allows us to find a value αk such that ρ(αk , θk ) admits a LHS model for POVMs (more details in Appedix A). The obtained values of αk and θk are shown on Fig. 2. In order to extend the result to the continuous interval I2 , we use the following lemma, which is proven in Appendix B: Lemma 1. If the state ρ(α, θ) is unsteerable from Alice to Bob (for POVMs) then the state ρ(α0 , θ0 ), with θ0 ≥ θ, is also unsteerable from Alice to Bob (for POVMs) as long as α0 α tan(θ ) ≤ tan(θ) . (1 + α0 ) (1 + α) 0

(18)

0.43 0.42 0.41 0.4

α

0.39 0.38 0.37 0.36 0.35 0.34 0.33 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

θ Figure 2. Parameter region for which we could prove that the states ρ(α, θ) of Eq. (9) admit a LHS model for all POVMs. As explained in the main text, we separate the interval θ ∈ [0, π/4] in three regions. In interval I1 = [0, 0.1], the existence of a LHS model is guaranteed below the black dashed horizontal line. In interval I2 = [0.1, 0.7354], a LHS model is demonstrated for all points below the blue dashed curve. In interval I3 = [0.7354, π/4], a LHS model exists below the green dash-dotted curve. Overall, this guarantees that the state ρ(α, θ) admits a LHS model for α . 0.3656 and for all values of θ ∈ [0, π/4], i.e. all points below the red solid horizontal line.


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Therefore, for any pair of points (θk , αk ) and (θk+1 , αk+1 ), the above lemma allows us to ensure that the state ρ(αc , θ) is unsteerable for θ ∈ [θk , θk+1 ], as long as θk and θk+1 are sufficiently close to each other. We are thus left with the interval I1 , where θ is in the neighbourhood of 0. We cannot use the same method as in I2 as whatever smallest θs = mink {θk } we choose, Lemma 1 only allows us to say something about some θ ≥ θs , leaving the interval [0, θs ] unsolved. Note also that by setting θs = 0 we get a separable state and consequently mixing it with another separable state cannot give rise to an entangled one (note that setting θ = 0 in Lemma 1, one obtains α0 = 0). However, in this region, an explicit LHS model for projective measurements is known [33]. The model holds for ρ(α, θ) as long as cos(2θ)2 =

2α − 1 . (2 − α)α3

(19)

To take POVMs into account we use a method developed in Ref. [16], Protocol 2. Starting from an entangled state ρ admitting a local model for projective measurements, we can construct another entangled state ρ0 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 ρ(α, θ) where condition (19) is fulfilled, thus ensuring that the state admits a LHS model for projective measurements. We obtain the class of states 1 ρθ = (ρ(α, θ) + |0i h0| ⊗ ρB ) 2

(20)

where ρB = TrA (ρ(α, θ)). This state is therefore unsteerable from Alice to Bob, for all POVMs. The last step consists in showing that ρ(α, θ), where α > αc can be written as a convex combination of ρθ and a separable state, for all θ ∈ I1 . This proof is given in Appendix C for α = 0.4. Finally, we summarize these results in Fig. 2. This implies that the state ρ(α, θ) is unsteerable for α ≤ αc = 0.3656, and for all θ ∈ [0, π/4]. Therefore the Werner state ρW (αc ) displays no hidden nonlocality. 4. 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 two-qubit 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 [24], beyond local filters, and consider the entire statistics of the measurements. Here, one chooses between several possible measurements


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(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 [39]. 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 1/d where d is the local Hilbert space dimension) can be super activated [40]. In fact, it turns out that the class of states we considered, i.e. two-qubit Werner states, are always useful for teleportation [41] 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 [42, 43] where stronger notions of nonlocality could be considered [44, 45, 46]. Acknowledgements We acknowledge financial support from the Swiss National Science Foundation (grant PP00P2 138917, Starting grant DIAQ) and from the Hungarian National Research Fund OTKA (K111734). [1] J. S. Bell, “On the Einstein-Poldolsky-Rosen paradox,” Physics 1, 195–200 (1964). [2] N. Brunner, D. Cavalcanti, S. Pironio, V. Scarani, and S. Wehner, “Bell nonlocality,” Reviews of Modern Physics 86, 419–478 (2014), arXiv:1303.2849 [quant-ph]. [3] B. Hensen, et al,“Experimental loophole-free violation of a Bell inequality using entangled electron spins separated by 1.3 km,” Nature 526, 682–6868 (2015), arXiv:1508.05949 [quant-ph]. [4] M. Giustina, et al, “Significant-Loophole-Free Test of Bell’s Theorem with Entangled Photons,” Physical Review Letters 115, 250401 (2015), arXiv:1511.03190 [quant-ph]. [5] L. K. Shalm,et al, “Strong Loophole-Free Test of Local Realism∗ ,” Physical Review Letters 115, 250402 (2015), arXiv:1511.03189 [quant-ph]. [6] A. Ac´ın, N. Brunner, N. Gisin, S. Massar, S. Pironio, and V. Scarani, “Device-Independent


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Appendix A. Details about the algorithmic construction of LHS models As stated in the main text we used the algorithmic construction of [37] to find LHS models for states ρ(α, θ). More precisely we note that for a fixed θ = θf the state is


13

linear with respect to α and we can thus use Protocol 1 of [37] to find an α = αf such that ρ(αf , θf ) admits a LHS model. We have run a slightly improved version of Protocol 1, which requires to choose a finite set of measurements {Ma|x }, a quantum state ξA , and to run the following SDP: Protocol 1. (Improved version) find q ∗ = max q s.t. T rA (Ma|x ⊗ I χ) =

(A.1) X

σλ Dλ (a|x) ∀a, x, σλ ≥ 0 ∀λ

λ

ρ(q, θf ) − ηχ + (1 − η)ξA ⊗ χB ≥ 0 (ρ(q, θf ) − ηχ + (1 − η)ξA ⊗ χB )TB ≥ 0 T r(χ) ≥ 0 where TB stands for the partial transposition on Bob’s side and the optimization variable are (i) the positive matrices σλ and (ii) a d × d hermitian matrix χ . This SDP must be performed considering all possible deterministic strategies for Alice Dλ (a|x), of which there are N = (kA )mA (where mA denotes the number of measurements of Alice and kA the number of outcomes); hence λ = 1, ..., N . For the answer to hold (i.e. ensuring that ρ(q ∗ , θf ) admits a LHS model) the parameter η must be smaller or equal to the ‘shrinking factor’ of the set of all POVMs with respect to the finite set {Ma|x } (and given state ξA ), that is, the largest η = η ∗ such that any shrunk POVM {Maη } with elements defined by Maη = ηMa + (1 − η) Tr[ξA Ma ]1

(A.2)

P can be written as a convex combination of the elements of {Ma|x }, i.e. Maη = x px Ma|x P (∀a) with px = 1 and px ≥ 0. The exact value η ∗ is in general hard to evaluate, but Ref. [16] gives a general procedure to obtain arbitrary good lower bounds on η ∗ , which is therefore enough for us to make sure that η ≤ η ∗ . The method requires the choice of a finite set of measurements {Ma|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 {12 , 0, 0, 0}, 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 6 of them when running the Protocol, corresponding to the vertices in the upper half sphere of the icosahedron. There is two degrees of freedom left: the quantum state ξA and the global orientation of the measurements {Ma|x }. Note indeed that a global rotation of the icosahedron can not change its shrinking factor η ∗ , but can nevertheless improve the value q ∗ given by Protocol 1. We thus computed lower bounds on the shrinking factors for different ξA of

Entanglement without hidden nonlocality p η

14

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.67 0.67 0.66 0.66 0.66 0.66 0.62 0.56 0.47 0.32 Table A1. Lower bounds on shrinking factors of the set of qubit POVMs with respect to the icosahedron and with ξA = p |0i h0| + (1 − p)1/2

the form ξA = p |0i h0| + (1 − p)1/2, and for each of them we then tried to optimize the obtained q ∗ over the different orientations of the icosahedron. The different estimates for η ∗ in function of p are given in Table 1, and the best points obtained are shown on Fig. 2. Appendix B. Proof of Lemma 1 If the state ρ(α, θ) is unsteerable from Alice to Bob (for POVMs) then the state ρ(α0 , θ0 ), with θ0 ≥ θ, is also unsteerable from Alice to Bob (for POVMs) as long as tan(θ0 )

α α0 ≤ tan(θ) . 0 (1 + α ) (1 + α)

(B.1)

To prove this lemma we show that ρ(α0 , θ0 ) can be written as a convex combination of ρ(α, θ) and a separable state, as long as condition (18) holds. We want: ρ(α0 , θ0 ) = qρ(α, θ) + (1 − q)S

(B.2)

where S is a separable state. Inverting this relation we get: (1 − q)S = (ρ(α0 , θ0 ) − qρ(α, θ)).

(B.3)

That is:  1   (1 − q)S = 2 

cos2 θ 0 (1 + α0 ) − qcos2 θ(1 + α) 0 0 α0 cosθ 0 sinθ 0 − qαcosθsinθ

Setting q =

α0 cosθ0 sinθ0 αcosθsinθ

0 cos2 θ 0 (1 − α0 ) − qcos2 θ(1 − α) 0 0

0 0 sin2 θ 0 (1 − α0 ) − qsin2 θ(1 − α) 0

 α0 cosθ 0 sinθ 0 − qαcosθsinθ  0   0 sin2 θ 0 (1 + α0 ) − qsin2 θ(1 + α)

makes S diagonal, thus separable. We need 0 ≤ q ≤ 1, that is:

α0 ≤

αcosθsinθ . cosθ0 sinθ0

(B.4)

Under this condition we are then left to show that S is a valid state, i.e. a semi-definite positive trace-one matrix. This gives us the four following inequalities: α0 (cos2 θ0 − `cos2 θ(1 + α)) + cos2 θ0 ≥ 0

(B.5)

α0 (−cos2 θ0 − `cos2 θ(1 − α)) + cos2 θ0 ≥ 0

(B.6)

α0 (−sin2 θ0 − `sin2 θ(1 − α)) + sin2 θ0 ≥ 0

(B.7)

0

2 0

2

2 0

α (sin θ − `sin θ(1 + α)) + sin θ ≥ 0

(B.8)

Entanglement without hidden nonlocality 0

15

0

cosθ sinθ where ` = αcosθsinθ . Each inequality is of the form Aα0 + C ≥ 0 with C always positive. if A ≥ 0 the inequality is satisfied (since α0 ≥ 0) so the non-trivial case corresponds to A ≤ 0 leading to the solution α0 ≤ − CA . Let us focus on inequalities (B.6) and (B.7). first. We have:

(B.6) =⇒ α0 ≤ 0

1 1 + `1 (1 − α)

(B.7) =⇒ α0 ≤

1 1 + `2 (1 − α)

(B.9)

0

cosθsinθ cosθ sinθ where `1 = αcosθ 0 sinθ , `2 = αcosθsinθ 0 . Now we use the fact that θ0 ≥ θ to get that sinθ0 ≥ sinθ and cosθ ≥ cosθ0 , in the range we are interested in: θ ∈ [0, π/4]. This implies `1 ≥ 1/α ≥ 1 while `2 ≤ 1/α and `1 ≥ `2 meaning that the inequality (B.6) is always stronger than the inequality (B.7) in the range of interest. We can similarly merge the inequalities (B.7) and (B.8) We have

(B.5) =⇒ α0 ≤

1 `1 (1 + α) − 1

(B.8) =⇒ α0 ≤

1 `2 (1 + α) − 1

(B.10)

and again using that `1 ≥ `2 we see that that the inequality (B.5) is stronger than the inequality (B.8). We are thus left with two the inequalities (B.5) and (B.6), but one can show that the inequality (B.5) is always stronger. One has 1 1 ≤ `1 (1 + α) − 1 1 + `1 (1 − α)

(B.11)

`1 (1 + α) − 1 ≥ 1 + `1 (1 − α) 2α`1 ≥ 2 which is true since `1 ≥ 1/α. Finally we can show that the inequality (B.5) is stronger than condition (B.4): 1 1 ≤ `1 (1 + α) − 1 `

(B.12)

`1 (1 + α) ≥ 1 + ` 1 + α ≥ `/`1 + 1/`1 . To see that the last inequality is true one can compare terms by terms: we have that 2 θ0 `/`1 = cos ≤ 1 and 1/`1 ≤ α, once again using θ0 ≥ θ and θ ∈ [0, π/4]. cos2 θ Appendix C. POVM model for small θ Here we give the proof that for θ ∈ [0, 0.1] the state ρ(0.4, θ) can be written as a convex combination of a separable state and ρθ = 12 (ρ(β, θ) + |0i h0| ⊗ ρB ), where 2β−1 ρB = TrA (ρ(β, θ)) and β and θ are linked by cos(2θ)2 ≥ (2−β)β 3. We want: ρ(α, θ) = qρθ + (1 − q)S

(C.1)


16

where S is a separable state. Inverting this relation we get: (1 − q)S = (ρ(α, θ) − qρθ )

(C.2)

where the non-zero elements of S are given by: S(1, 1) = 2cos2 θ(1 + α) − q(2cos2 θ(1 + β) + sin2 θ(1 − β))

(C.3)

S(2, 2) = 2cos2 θ(1 − α) − q(2cos2 θ(1 − β) + sin2 θ(1 + β))

(C.4)

S(3, 3) = 2sin2 θ(1 − α) − qsin2 θ(1 − β)

(C.5)

2

2

S(4, 4) = 2sin θ(1 + α) − qsin θ(1 + β)

(C.6)

S(1, 4) = S(4, 1) = 4αcosθsinθ − q2βcosθsinθ.

(C.7)

To prove that S can be made separable we set α = 0.4, q = 1/2 and β = 3/4, which leads to θ . 0.11, for which value S can be proved positive and separable (via PPT criterion [47]). To extend it one just has to note that the positivity and the PPT constraints of S are of the form sin2 θ(Acos2 θ − Bsin2 θ) ≥ 0, where A, B ≥ 0. This implies that if S is positive and separable for θ, so is it for θ0 ≤ θ, for any θ ∈ [0, 0.1].