Bell scenarios in which nonlocality and entanglement are inversely related
We show that for two-qubit chained Bell inequalities with an arbitrary number of measurement settings, nonlocality and entanglement are not only different properties but are inversely related. Specifically, we analytically prove that in absence of noise, robustness of nonlocality, defined as the maximum fraction of detection events that can be lost such that the remaining ones still do not admit a local model, and concurrence are inversely related for any chained Bell inequality with an arbitrary number of settings. The closer quantum states are to product states, the harder it is to reproduce quantum correlations with local models. We also show that, in presence of noise, nonlocality and entanglement are simultaneously maximized only when the noise level is equal to the maximum level tolerated by the inequality; in any other case, a more nonlocal state is always obtained by reducing the entanglement. In addition, we observed that robustness of nonlocality and concurrence are also inversely related for the Bell scenarios defined by the tight two-qubit three-setting inequality, and the tight two-qutrit inequality .
Nonlocality and entanglement are two core concepts in quantum information. If is the joint probability that Alice obtains and Bob on a system prepared in state , nonlocality is the impossibility of expressing as , where are preestablished classical correlations bell64ph (). Entanglement is the impossibility of expressing a quantum state as a convex combination of separable states. Nonlocality and entanglement are related concepts in the sense that, to have nonlocality, entanglement is needed gisin91pla (). The difference between both concepts has been pointed out before. First, it was noticed that there are entangled states which do not violate specific Bell inequalities wern00pra (). Then, in Ref. acin05prl (), the statistical strength of Bell tests was studied, showing that stronger tests (for a given family of Bell inequalities) require nonmaximally entangled states. Similarly, it was shown in acin02pra () that nonmaximally entangled states allow for larger violations (or equivalently a stronger resistance to noise) of the two-qutrit inequality CGLMS02 (). In junge10ar () it was demonstrated that, for general bipartite Bell inequalities with inputs, outputs, and -dimensional Hilbert spaces, the entropy of entanglement of the state is essentially irrelevant in obtaining large violation. Finally, in vidi10ar (); lian11pra (), it is shown that, for certain inequalities, weakly entangled states outperform maximally entangled ones of arbitrary dimension.
One difficulty in reaching a general conclusion about the relationship between nonlocality and entanglement is that of finding a general scenario where incontrovertible measures of nonlocality and entanglement can be compared. Bipartite scenarios have the advantage that any of the many measures of entanglement assign zero entanglement to product states and maximum entanglement to maximally entangled states woot98prl (); IconcPRA (). Nonlocality is a more delicate issue since different restrictions on the number of measurement settings usually lead to different measures of nonlocality. This suggests that to study such relationship, one needs to consider a general scenario in which each party can perform an arbitrary number of local measurements.
The structure of the paper is the following: In Sec. II we define a measure of nonlocality called robustness of nonlocality that will be used through all the paper. In Sec. III we discuss a general bipartite scenario in which both parties have the same number of settings and prove that, no matter the number of settings, robustness of nonlocality and entanglement are inversely related. We then study how noise affects this conclusion. In Sec. IV we numerically explore the second simplest tight bipartite Bell inequality CG04 (), which has three settings per party, each with two outcomes. In Sec. V, we study a tight two-qutrit Bell inequality CGLMS02 (). In all cases considered we observe the same behavior, namely, that entanglement and robustness of nonlocality are inversely related.
Ii Robustness of nonlocality
For an ensemble of entangled particles in a state and a given Bell inequality, we define the robustness of nonlocality (RN) against loss of local information as the maximum fraction of random particles per observer that can be lost such that the remaining ones can violate the Bell inequality. The robustness of nonlocality is related to the minimum detection efficiency, , required for a loophole-free violation of the Bell inequality pear70prd () as RN.
The idea behind this measure of nonlocality is simple: A violation of a Bell inequality with perfect detection efficiency implies that no local model can reproduce the observed joint probabilities. If the minimum detection efficiency is , this means that no local model exists, even if one locally rejects a fraction RN of the events. Therefore, the larger RN, the harder it is to reproduce the observed results with local models. Therefore, RN may be taken as a measure of nonlocality. As a measure of entanglement we will use the concurrence woot98prl (); IconcPRA ().
Any bipartite Bell inequality involving and dichotomic () observables and on Alice’s and Bob’s sides, respectively, can be written in the following form:
where is the expectation value of in the state and
In the previous expression, are the joint probabilities of detecting the eigenvectors and of the observables and . If the observables have and outcomes, any Bell inequality can be expressed in a similar way by using only the first and outcomes.
Let us now evaluate the effect of detection inefficiency. For inequalities involving only outcomes such as (2), it is customary to assume that no-detection events do not contribute to the inequality (they can be seen as detection on the “” outcome). However, in order to compute the robustness of non-locality, it is necessary to optimize over all possible strategies for the no-detection events; for instance, whenever Alice does not get a detection, she can choose to always output for observables and output for all other observables brun08pla (). It is worth noting that, instead of grouping inconclusive events with one of the outcomes, different strategies can be used. For instance, a further outcome, corresponding to nondetections, can be added to the observables, mass02pra (), or one can also choose to treat nondetections as simply “undefined” lars98pra (). However, these strategies will require a modification of the Bell inequality. In the present paper we will study the robustness of nonlocality by assigning one of the observable outcomes to inconclusive events.
Each strategy giving a definite output to each observable is completely equivalent to relabeling the inputs or outputs of a Bell inequality and using the “” outcome in the case of no-detection for any observable. To give an example, the inequality (2) with Alice giving output only for observable in the case of no detection is equivalent to replacing and and using the outcome in the case of a no-detection event for any observable.
It can also be noted that from the experimental viewpoint, assigning outcomes for non-detection gives a simple way to handle these events. This is because with this assignment, no-detection events do not contribute to the inequality, so that there is no need to distinguish whether there was a pair produced but no detection, or if there was no pair produced. Distinguishing these are sometimes nontrivial, for example in a continuously pumped experiment, but this is not needed with the suggested assignment.
Thus, since any no-detection strategy is equivalent to rewrite the Bell inequality, the robustness of nonlocality RN can be evaluated by optimizing over all possible ways of rewriting the inequality and using the outcome in the case of non detection (in the case of observable with outcomes, the last outcome is typically used in the case of non-detection). In order to violate a Bell inequality written as (2), in the case of detection efficiencies and , the following relation must hold:
Eberhard first showed that states with lower entanglement allow a violation of the Clauser-Horne-Simony-Holt (CHSH) inequality clau69prl () with lower required detection efficiency eber93pra () with respect to maximally entangled states. Low entangled states tolerate smaller efficiencies when one of the two particles is always detected cabe07prl (); brun07prl (). The same occurs in the -site Clauser-Horne inequality lars01pra (). In vert10prl (), it was noticed that nonmaximally entangled states of two qudits can lower the required detection efficiency with respect to maximally entangled states. Recently, it was shown that states with low entanglement can be also useful for EPR-steering with low detection inefficiencies wise07prl (); vall13pra ().
In the following sections we will demonstrate that states with low entanglement can tolerate lower detection inefficiency for the violation of different Bell inequalities. In particular, we will show that the robustness of non-locality RN and the entanglement are inversely correlated for the studied inequalities.
Iii Robustness of nonlocality vs concurrence for chained Bell inequalities
Pearle pear70prd () and Braunstein and Caves (BC) brau89klu (); brau90ap () introduced a generalization of the CHSH clau69prl () and Clauser-Horne (CH) clau74prd () Bell inequalities, known as chained Bell inequalities, in which Alice and Bob choose among settings. Chained Bell inequalities have some interesting applications: The case fixes a loophole that occurs in some experiments based on the CHSH inequality aerts99prl (). Besides, it reduces the number of trials needed to rule out local hidden variable theories peres00fp (), and improves the security of some quantum key distribution protocols barr05prl (). In the case in which tends to infinity, the inequality allows one to discard nonlocal hidden variable theories with a nonzero local fraction barr06prl (). Chained Bell inequalities have been experimentally tested using pairs of photons, with vall11pra (), bosc97prl (), and barb05pla (). It was recently shown than they can be used for randomness expansion dhar13pra ().
The minimum detection efficiency required for a loophole-free violation of chained Bell inequalities for any using maximally entangled states has been obtained in cabe09pra (). The fact that the maximum quantum violation of chained Bell inequalities is always achieved with maximally entangled states wehn06pra () might suggest that the minimum detection efficiency occurs for maximally entangled states, but no proof exists of whether the detection efficiency for the chained Bell inequalities can indeed be reduced when one considers more general classes of entangled states. Indeed, for case , corresponding to the CH inequality (that is equivalent the CHSH), the minimum detection efficiency occurs for almost product states eber93pra (); lars01pra ().
In the following we will show that, in absence of noise (e.g., considering pure states), the states with higher robustness of nonlocality (or the minimum detection efficiency) for any chained Bell inequality written in the form of (5) are almost product states for which the robustness of nonlocality tends to
The important point here is that this value is larger than the maximum value of RN for maximally entangled states cabe09pra (), namely,
Moreover, for Bell inequalities of the form (4) with fixed , we will show that the value in (6) is the maximum achievable robustness of nonlocality for any quantum state. This shows that, for all chained Bell inequalities, entanglement and nonlocality of pure states are inversely related.
Theorem: The maximum of the robustness of nonlocality of inequality (4) is and can be obtained by almost product state.
Proof: Assuming the same detection efficiency for every party and setting, i.e., , the value of becomes
where is the expectation value of in the state . Therefore, inequality (4) is violated when , with
Since , it is easy to show that . Then,
and the lowest possible bound of the right-hand side of (10) is obtained when for and not both equal to 1. We obtain
which cannot be achieved exactly, but arbitrarily close by the following procedure: Any generic two-qubit pure states , can be written (in a suitable basis) as
with . Let us consider the following eigenstates:
and choose such that . Then, and the critical efficiency becomes
which, when tends to zero (i.e., when the state tends to a product state), tends to
concluding our proof.
We have numerically obtained, by using the method of conjugate gradient, RN as a function of the pure state used to violate the inequality and compared it with the corresponding maximal achievable violation of the Bell inequality . Moreover, through exhaustive numerical searches, we have obtained that the form (5) gives the maximum RN for any given state (in the specific case of a maximally entangled state this is analytically demonstrated in the Appendix section). Note that, for nonmaximally entangled states such as (13), the concurrence is given by . The results for are shown in Fig. 1. We observe that larger violations of correspond to lower values of RN. From Fig. 1 one can clearly see see that nonlocality (measured by RN) and entanglement (measured by ) are inversely related: Larger concurrence, allowing larger violation of the inequality, implies lower RN.
iii.1 Adding noise
How does noise affect this conclusion? In the presence of white noise, the state becomes and the threshold detection for the chained Bell inequalities efficiency is changed to
In Fig. 2 we show, for three different values of noise (, , and ), the dependence of RN and RN and the maximum values of and with the degree of entanglement of the initial pure state. We observe that, when the noise is different from , the best quantum state giving the lowest threshold is not an almost separable state, but a nonmaximally entangled state depending on . However, the lower the noise , the smaller the entanglement required to obtain the optimal threshold.
Furthermore, in Fig. 2(b) we observe that, the lower is, the more resistant to noise is the violation of the Bell inequality. In fact, it is possible to calculate the maximum tolerated noise to violate the chained Bell inequalities. Given and the maximal violation of defined as , the maximum tolerated noise is .
Using the method of conjugate gradient to minimize Eq. (17), it is also possible to obtain the threshold and the required entanglement for any value of the noise . The results are shown in Fig. 3. We observe that, for chained Bell inequalities, nonlocality and entanglement are simultaneously maximized only in the case of extreme noise, namely the maximum noise level tolerated by the inequality. A better threshold detection efficiency is obtained by lowering the noise and suitably decreasing the entanglement. From this we conclude that nonlocality and entanglement are synonymous only for extremely noisy scenarios.
Iv Robustness of nonlocality vs concurrence for Bell inequality
After the results presented in the previous section, a natural question is whether or not the same behavior occurs for other bipartite Bell inequalities. In this section we present the results for the second simplest tight bipartite Bell inequality, namely, CG04 (); Froissart81 (); Sliwa03 (), involving three dichotomic measurements on both and sides (the simplest tight bipartite Bell inequality is the CHSH inequality or , studied in the previous section).
The inequality may be written as
where was defined in CG04 () as . However, this form will not lead to the best RN. We have numerically checked that the forms giving the best RN are the following:
that can be obtained from by replacing and
obtained from by and .
Fig. 4 shows the RN and the violation of the inequality as a function of the degree of entanglement measured by . We observe that, in the absence of noise, almost product states are again those that require lower detection efficiencies. For the optimal RN is obtained with , while for the optimal RN is obtained with . The optimality of can be analytically shown for maximally entangled states (see the Appendix).
The minimum required efficiency with maximally entangled states is as reported in Ref. brun08pla (). Indeed, this value can be obtained analytically. For two-qubit systems, the maximum violation of is 1/4, and can be achieved with a maximally entangled state, as was previously shown in CG04 (). Given a maximally entangled state that maximally violates and symmetric efficiencies , to violate the inequality it is necessary that
Remembering that for a maximally entangled state (MES) , we obtain
The maximal robustness of non-locality RN can be achieved for almost product states. If we consider the form of the inequality, the critical efficiency can be written as . Let us choose , , , . By using such that we obtain
which, when tends to zero tends to
We observe that the maximum RN for the is greater than the one for the inequality, which has the same number of local settings.
V Robustness of nonlocality vs concurrence for the two-qutrit inequality
For the two-qubit Bell inequalities discussed above we have observed that nonlocality and entanglement are inversely related. Here we show that this is also true for other bipartite scenarios. For this purpose we repeat our analysis but now for a tight bipartite inequality maximally violated by two-qutrit states, the inequality CGLMS02 ().
The inequality is given by where . Here, and () denote the settings that the parties may choose for the local measurements, and the index denotes each measurement outcome (). The inequality can be rewritten in the form of (2) as , with
In the previous expression and denote the 1-eigenstates of the observables and , while and denote the 2-eigenstates. Note that, no probability containing the 3-eigenstate is present. Moreover, .
In Fig. 5 we show the maximal achievable violation of in function of the concurrence while considering the initial two-qutrit state given by , where . The maximally entangled state is obtained when and . The concurrence for the two-qutrit state described above is given by and ranges from 0 to IconcPRA (). As was first observed in acin02pra (), the maximal violation of is obtained with partially entangled states. The maximal value achievable with MES is and one can clearly see that it does not correspond to the maximum of .
We then numerically optimized the robustness of nonlocality RN in a function of , by obtaining the results shown in Fig. 6: the form of given in (25) is the optimal form for maximizing the RN parameter. The optimality can be analytically shown for maximally entangled states (see the Appendix). Also in this case RN and entanglement are negatively correlated.
The maximal RN is and can be obtained for almost product states, as in the previous inequalities. Let us consider the following entangled state:
and the following measurements , , , , . In the form (25) the threshold efficiency becomes (non-detection events correspond to eigenvalues and thus does not contribute to the inequality):
With the above measurement the critical efficiency becomes
If we choose and let go to zero we get
For maximally entangled states, for which , the RN is given by
We would argue that robustness of nonlocality RN is a good measure of nonlocality, since it marks the border where local hidden variable descriptions become possible: The larger robustness of nonlocality is, the harder it is to express the joint probabilities with local models.
We have shown that, for the two-party -setting chained Bell scenario (for any finite), for a tight two-qubit Bell inequality and a tight two-qutrit Bell inequality , robustness of nonlocality and concurrence, are in the absence of noise, inversely related.
The main result of this paper is the observation that, for many distinct types of Bell scenarios, larger nonlocality requires smaller entanglement; in the absence of noise, almost product states are the most nonlocal ones. We analytically showed that the maximal RN can be achieved with almost product state. The maximal values of RN (related to the minimum required detection efficiency as RN) are given by RN, RN and RN for the chained Bell inequality, the inequality and the inequality respectively.
When noise is present, the most nonlocal states acquire some amount of entanglement; however, the smaller the noise is, the lower their entanglement becomes.
Some questions naturally arise: are the nonlocality and entanglement inversely related in any Bell inequality involving , observables with and outcomes? If yes, is there some physical mechanism for such counterintuitive behavior? These questions require further research.
Acknowledgements.GL, ESG and GC were supported by the CONICYT, AGCI, FONDECYT 1120067, MilenioP10-030-F and PIA-CONICYT PFB0824. GV was supported by the Strategic-Research-Project QUINTET of the Department of Information Engineering, University of Padova and the Strategic-Research-Project QUANTUMFUTURE of the University of Padova. PM acknowledge the Chistera EU project QUASAR. AC was supported by Project No. FIS2011-29400 (MINECO, Spain).
Appendix A Optimality of detection strategy for maximally entangled states
In this section we will demonstrate which is the optimal way of rewriting the Bell inequalities analyzed in the main text in case of maximally entangled states. We start by giving the general framework to solve the optimization.
Let us consider a general bipartite Bell inequality involving and observables and on the Alice and Bob side. The observables have and outcomes respectively, and . Any Bell inequality can be written as (2):
In the previous expression are the joint probabilities of detecting the and eigenvectors and of the observables and . Note that only the first and outcomes are involved in the inequality.
When inefficiencies are present it is necessary to give a strategy for the non-detection events. Let us suppose that the strategy on Alice’s side is the following. If Alice is measuring the observable and the particle is not detected, she assigned, with probability , the outcome . Clearly, . The same happens at Bob’s side, with probabilities . If we consider Alice and Bob inefficiencies as and , the Bell inequality is violated if
The sum is taken over and while and : also in the previous expression the outcomes and of each observable are not present.
We start with the chained Bell inequalities, and then analyze the and inequalities.
a.1 Chained Bell inequalities
For the chained Bell inequalities of section III, we have dichotomic observables. Then, in the case of non detection on the observable , Alice chooses to output the outcome with probability and the outcome with probability . The same happens to Bob. Remembering that, for MES, we have:
Since corresponds to the chained Bell inequality applied to the classical probabilities and , we have . In order to maximize the Bell parameter it is necessary to choose the ’s and ’s that maximize . The trivial choice , satisfies this requirement. It is worth noticing that the choice , , corresponds precisely to consider all non-detections as outputs for the inequality written as (4).
Let us consider the inequality written in its original form . In the case of inefficiencies with non maximally entangled states we have
Since the maximal value of with maximally entangled state is 1/4, the Bell parameter in the case of detection inefficiencies becomes
The choice that minimizes the critical efficiency is given by and , giving , and
The choice of the ’s and ’s corresponds to choosing for the non-detection events the outcome for the inequality written as .
a.3 Two-qutrit inequality
For this two-qutrit inequality Alice has three outcomes for each observable . In the case of non-detection she assigns with probability the outcome 1, with probability the outcome 2, and with probability the outcome 3. The same applies to Bob. For maximally entangled states , and we have
The optimal choice of ’s and ’s is the one that maximizes . This term is clearly upper bounded by 0 (it corresponds to the Bell inequality). Then the choice saturates the bound. This choice corresponds to choosing for the non-detection events the outcome 2 for the inequality written as (25).
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