The standard fair sampling assumption is not necessary to test local realism
Almost all Bell-inequality experiments to date have used postselection, and therefore relied on the fair sampling assumption for their interpretation. The standard form of the fair sampling assumption is that the loss is independent of the measurement settings, so the ensemble of detected systems provides a fair statistical sample of the total ensemble. This is often assumed to be needed to interpret Bell inequality experiments as ruling out hidden-variable theories. Here we show that it is not necessary; the loss can depend on measurement settings, provided the detection efficiency factorises as a function of the measurement settings and any hidden variable. This condition implies that Tsirelson’s bound must be satisfied for entangled states. On the other hand, we show that it is possible for Tsirelson’s bound to be violated while the CHSH-Bell inequality still holds for unentangled states, and present an experimentally feasible example.
When quantum mechanics was first developed it was argued that it might only be an approximation of an underlying classical “hidden-variable” theory (1). This was put on a testable basis by the development of Bell inequalities (2); (3); (4), which should be obeyed by any local hidden variable (LHV) theory. Experimental violation (5); (6); (7); (8); (9); (10); (11); (12); (13); (14); (15) of Bell inequalities provides strong evidence against LHV theories, but almost all of these experiments (with the exceptions of those in Refs. (13); (15)) have the loophole that the violation of the Bell inequality could, in principle, be caused by loss (17); (16); (18); (19); (20); (21). The interpretation of these experiments as ruling out LHV theories therefore relies on the assumption that the sampling is “fair” (3); (22). For the interpretation of these experiments it is therefore vital to establish what constitutes fair sampling. Here we put fair sampling on a rigorous basis by determining exactly what forms of loss can lead to violation of Bell inequalities.
The standard form of the fair sampling assumption is that the detection efficiency is independent of measurement settings (3); (22). Here we find that assumption is unnecessary. The efficiency can depend on the measurement settings, provided the efficiency factorises as a function of the measurement settings and any hidden variable. Most experimental tests are of the Clauser, Horne, Shimony and Holt (CHSH) form of the Bell inequality (3), and we therefore concentrate on the CHSH-Bell inequality in this work. Our condition is both necessary and sufficient for the CHSH-Bell inequality to be satisfied for LHV theories. An alternative sufficient condition was previously found by Ref. (23). We also establish the necessary and sufficient condition for the CHSH-Bell inequality to be satisfied for unentangled states, and show that if the sampling is fair it will also prevent violation of Tsirelson’s bound (24) with entangled states.
This result means that, in order to obtain violation of Tsirelson’s bound (24) with entangled states, but not of the CHSH-Bell inequality with unentangled states, it is necessary to examine the specific measurement, and not just the form of the loss. We present a scheme that violates Tsirelson’s bound with entangled states, but not the CHSH-Bell inequality with unentangled states. The CHSH-Bell inequality is still violated with LHV theories, but the scheme can tolerate greater loss than the bound derived in Ref. (18). Other methods of constructing Bell inequalities with greater resistance to loss have been proposed in Refs. (20); (25); (26); (27); (28).
This manuscript is organised as follows. First the fair sampling assumption is explained in more detail in Sec. II. General Bell inequalities are presented in Sec. III, then postselection for local hidden variable theories is analysed in Sec. IV. Postselection for quantum mechanics, including Tsirelson’s bound, is analysed in Sec. V. The use of postselection to enhance violation of Bell inequalities beyond Tsirelson’s bound for entangled states is analysed in Sec. VI. We conclude in Sec. VII.
Ii The fair sampling assumption
The first work to give a form of the fair sampling assumption was that of Clauser, Horne, Shimony and Holt (3). They assumed, in deriving their inequality, that the detection efficiency is independent of the measurement settings. Pearle expressed the assumption alternatively, that “the data recorded [is] representative of the accepted data” (16). That is, that the sample of detected pairs provides a fair statistical sample of all the pairs. These forms of the fair sampling assumption, which superficially appear different, are effectively equivalent.
For the sample to be completely fair, the probability of sampling a pair (i.e. the efficiency) needs to be independent of the pair. For this to be the case, the efficiency needs to be independent of any quantity that varies between the individual pairs. The only quantities that can vary between the individual pairs are the internal state (the quantum state or any hypothetical hidden variable) and the measurement setting that is used in measuring the pair. Therefore the requirement that the sample is fair is exactly equivalent to requiring the efficiency to be independent of the internal state and measurement setting.
This requirement is slightly stronger than the requirement given by Ref. (3), because it requires that the efficiency is also independent of the state of the pair. However, it is easily seen that the efficiency need not be independent of the state. This is because, if the efficiency is dependent on the state, but independent of the measurement setting, then the postselection simply changes the probabilities for the internal state, yielding a different postselected state. That is, the sampling may not be fair, but the loss is simply yielding a postselected state on which the sampling is fair.
This is taken advantage of in Procrustean entanglement concentration (29), which has been demonstrated to enhance the violation of Bell inequalitites (30). Procrustean entanglement concentration gives loss that depends on the state, but because this loss is independent of the measurement settings of the Bell measurement, it does not invalidate the Bell inequalities for unentangled states. Although the sampling is not entirely fair for the initial state, it is fair for the state produced by the Procrustean entanglement concentration.
It is important to note that the condition that the efficiency is independent of the measurement setting means that it must be completely independent of the measurement setting. That is, it can not be a function of the measurement setting, so it is independent of the measurement setting for any internal state. On the other hand, if the efficiency is independent of the measurement setting for just one state, then it is easy to provide examples where the Bell inequality is violated with unentangled states or hidden variables. For a simple example, see Appendix A.
Iii General Bell inequalities
In a general Bell inequality experiment with multiple parties, these parties each share one component of a state , and each performs one of a number of different measurements. The measurement settings for party are denoted , and the measurement results are denoted . We denote the vectors of measurement settings and measurement results and . One obtains a set of measurement probabilities , and can define a Bell quantity as a linear combination of these probabilities. A Bell inequality is then an upper bound on the value of this quantity for LHV theories. With loss, we denote the probability of a successful measurement for settings by . The postselected probabilities are then given by
We do not consider complete loss, which would make the postselected probabilities undefined.
In the specific case of CHSH-Bell inequalities, there are two parties, Alice and Bob, and each performs one of two dichotomic measurements. Now using the notation and for the measurement settings for Alice and Bob, respectively, we have , , and . The CHSH-Bell quantity without postselection is defined by
The postselected form of the CHSH-Bell quantity is
Iv Local hidden-variable theories
First we prove that Bell inequalities must be satisfied for LHV theories provided the detection efficiency factorises as a function of the measurement setting and a function of the hidden variable. For a stochastic local hidden-variable theory, probabilities are given as
where is the hidden variable. The state simply controls the probability distribution for . With loss, the probabilities can sum to less than 1. Summing over the measurement results gives the efficiencies as
where is the single-party efficiency for party with measurement setting and hidden variable .
We also use the notation for the efficiency due to measurement setting for party , and for the efficiency due to the hidden variable for party . With this notation defined, we can now state our condition rigorously.
When the efficiency for each party factorises as
the set of postselected probabilities that can be obtained is identical to that which can be obtained without postselection (provided no efficiency is zero).
This result means that this form of loss does not change the type of probability distributions that are possible with LHV models, and in particular any Bell inequality must still be satisfied with postselection.
The proof follows by showing that there exists a measurement scheme without postselection that yields the same probabilities as the for the postselected scheme. The postselected probabilities may alternatively be given by
This means that, by using a different LHV model with different probabilities, we may obtain exactly the same probability distribution for the measurement results as for the postselected case. ∎
This result shows that our condition (7) is sufficient for Bell inequalities to hold with postselection. We also have a necessity proof in the case of CHSH-Bell inequalities. Before showing this result, we first show the general form of the CHSH-Bell inequality with postelection.
Any local hidden variable theory must satisfy
For given there exist probabilities that saturate this inequality.
Here we have used the same notation as for general Bell inequalities, except we have used and for Alice and Bob’s measurement settings, respectively. We use the superscript on and to indicate that these are chosen as a function of the hidden variable .
For a LHV theory, the postselected CHSH-Bell quantity can be rewritten as
Because the probabilities are non-negative, we have . Considering arbitrary measurements for a given form of loss, we can choose the such that takes any value in the range .
Because is linear in each of the , it is maximised (or minimised) by taking extreme values where . By appropriately choosing the signs, it is possible to make any one of the terms in the curly brackets in Eq. (IV) negative, and the rest positive. Changing the sign preserves the parity of the number of negative terms, so it is not possible to make all terms positive.
In particular, taking makes the last term negative. Then changing to makes only the third term negative. Alternatively, changing to makes only the second term negative, or changing to and to makes only the first term negative.
To maximise , we take three terms positive and one negative for each value of . The value of will be maximised with the smallest term taken to be negative. That is,
Summing the first four terms in Eq. (IV) gives equal to
To change the overall sign, so one term is positive and the remaining are negative, we can change the sign of both and . Using this, we can ensure that the largest three terms in Eq. (IV) positive, and the smallest is positive. Doing this for each value of , we simply obtain the negative of what was obtained before, and so obtain . We therefore find that the generalisation of the CHSH-Bell inequality for the case of postselection is , as given in Eq. (10). As the above argument is constructive, it shows how to choose probabilities in order to saturate this inequality. ∎
Using this result, we can prove the necessary and sufficient condition for the CHSH-Bell inequality to hold for LHV theories with postselection. We first prove the necessary and sufficient condition in an alternative form.
The condition that
is independent of and is necessary and sufficient for to be satisfied for all probabilities .
Thus the condition is sufficient.
To prove necessity, we show that if the condition in Lemma 2 is not satisfied, then may be violated. If the condition is not satisfied for , then select and such that
We than have
The condition that the efficiency for each party factorises as
is necessary and sufficient for to be satisfied for all probabilities (provided no efficiency is zero).
which is independent of and .
It remains to show that these definitions satisfy (18). We find that
where in the second line we use the condition of Lemma 2, and may be arbitrary. Hence we find that is independent of , and we therefore have
Rearranging Eq. (25) to isolate gives
Taking the maximum over then yields
where we have used . Hence
Using this expression in Eq. (25) gives
V Quantum mechanics
Next we consider the restriction on the loss for quantum mechanics, rather than LHV theories. It might be thought that the case of quantum mechanics is equivalent, because a LHV theory can be thought of as an unentangled state, with the labeling orthogonal basis states. However, the case of quantum mechanics is slightly different, because we also need to consider all linear combinations of orthogonal basis states.
For quantum theory the probabilities of local measurement results are obtained via a positive operator-valued measure with elements , which corresponds to successful measurement result for measurement setting for party . We also use the notation
for the operator corresponding to a successful measurement. Here the sum is over all successful measurement results ; the measurement operator for failure is .
The restriction on the loss in terms of hidden variables (7) implies that
This expression may be taken to be the definition of the restriction on the loss for quantum mechanics. This restriction may alternatively be expressed as
Using a similar method as for hidden variables, we can show that postselection with this form of loss cannot change the form of probability distributions obtained, either for the case of entangled or unentangled states. In particular, the result is as in the following theorem.
For a Bell experiment on a quantum mechanical system, provided the loss is restricted by
the set of postselected probabilities it is possible to obtain without entanglement is identical to that which can be obtained without postselection or entanglement, and the set of postselected probabilities it is possible to obtain with entanglement is identical to that which can be obtained without postelection but with entanglement (provided no efficiency is zero).
We define new measurement operators as
These measurement operators now give unit efficiency (so there is no postselection). We also define
where is defined as in Eq. (33). Because we restrict to the case of incomplete loss, the measurement operators are positive and can be inverted. The new measurement operators and states give exactly the same probabilities without postselection as the original measurement operators and states did with postselection.
Furthermore, because is a local operator, it can not produce entangled states from unentangled states. Therefore the postselected Bell experiment with an unentangled state gives postselected probabilities identical to the non-postselected probabilities for a different Bell experiment with different measurement operators and a different unentangled state . ∎
This result means that, for an unentangled state, all Bell inequalities must still be satisfied. This is exactly as we expect, because unentangled states may be regarded as equivalent to hidden variables. Similarly, inequalities that hold for entangled states (such as Tsirelson’s bound) will be unaffected by the postselection.
For CHSH-Bell inequalities, it would be expected that Eq. (32) is also a necessary condition due to the result for hidden-variable theories. However, it does not directly follow, and necessity needs to be proven separately. This is because one cannot arbitrarily choose the probabilities for given efficiencies. Therefore it is not necessarily possible to achieve the maximum value of Eq. (10), as in the case of a general hidden-variable theory. Nevertheless, it is possible to show the result in the following theorem.
For a quantum mechanical system, the condition that the efficiency for each party factorises as
is necessary and sufficient for to be satisfied for all probabilities (provided no efficiency is zero).
The sufficiency follows immediately from Theorem 3, so it only remains to show necessity. The condition in the theorem is equivalent to Eq. (32), so we show necessity for Eq. (32). First consider the case where Eq. (32) is violated. In that case, either is dependent on , or is dependent on (or both). We omit the subscripts and in the notation and for simplicity. The party is simply indicated by the symbol used for the measurement setting ( for party 1 and for party 2).
If (for example) is not independent of , then there exist orthogonal states and such that
This result may be proven in the following way. Let us assume that all orthogonal states give equality in (38). Then, for any orthogonal basis , we have
for some . Therefore . Now we can take to be the basis which diagonalises . Then we have
so . Because is nonzero (these are positive operators), we have . Therefore the diagonal elements of must be zero in the basis . As this is the basis which diagonalises , we must have . Hence we find that equality in (38) implies , so if is not proportional to , then there must exist orthogonal states and such that (38) is satisfied.
We obtain the exact equivalent result if is not independent of . Therefore, if Eq. (32) is violated, we can select and such that
with strict inequality in at least one of these cases. We therefore have
Now consider the density operator
For this density operator
where , and indicates the expectation value using the state (for Alice) or (for Bob). Given the , consider the measurement operators for the individual results given by
Using these measurement operators gives
This gives the postselected Bell quantity as
Equation (V) may be rearranged to give
We therefore obtain .
Thus we find that, if Eq. (31) is not satisfied, for given there exists a separable state and a set of measurement operators such that . Eq. (31) is therefore a necessary condition for the CHSH-Bell inequality to be satisfied for separable states. As Eq. (31) is equivalent to the condition in the theorem, we have proven both necessity and sufficiency as required. ∎
For entangled states, is limited by Tsirelson’s bound of (24). As explained above, the restriction given by Eq. (31) is sufficient for Tsirelson’s bound to be satisfied due to the result in Theorem 3. However, it turns out that it is not necessary. In particular, we find that Tsirelson’s bound is not violated if there is no loss on one side, and moderate loss on the other side. Let , and let
We have performed numerical maximisations over the measurements and states for a range of values of , and the results are shown in Fig. 1. For small values of , below about , there is violation of Tsirelson’s bound, but for larger values no violation of Tsirelson’s bound is achieved. This indicates that there is not a simple necessary and sufficient condition in the case of Tsirelson’s bound. Whether Tsirelson’s bound can be violated depends on the particular value of the loss.
Vi Postselection that violates Tsirelson’s bound
A central motivation for this work was to determine forms of loss that can violate Tsirelson’s bound while ensuring that the CHSH-Bell inequality is still valid, in order to generalise Ref. (31). The result of the previous section is that the form of postselection that is necessary for the CHSH-Bell inequality to hold for unentangled states (as shown in Theorem 4) also implies that Tsirelson’s bound holds for entangled states (as follows from Theorem 3).
This raises the question of how the postselection in Ref. (31) differs from the postselection used here. The difference is that the postselection relies on the experimenter knowing what the state is. This method for constructing Bell inequalities is criticised in Ref. (32), because it introduces an additional assumption beyond locality and realism. If an entangled state is incorrectly assumed, but the actual state is unentangled, then the method of Ref. (31) still yields a violation of the Bell inequality (See Appendix C). We therefore avoid the approach of assuming a state here, and simply allow postselection that may depend on the state and measurement setting.
A crucial subtlety in our results is that the condition (31) is necessary for the CHSH-Bell inequality to be satisfied for unentangled states, provided we consider arbitrary measurement schemes for the given form of loss. This does not eliminate the possibility that there are particular measurement schemes with postselection such that the Bell inequality is satisfied for unentangled states but Tsirelson’s bound may be violated for entangled states. We consider such a scheme in this section.
We emphasise that this scheme does not violate the Bell inequality with unentangled states provided the measurements are acting as expected. But, if there is an underlying hidden variable theory, or equivalently if the measurements are not acting as expected on the underlying quantum state, then the CHSH-Bell inequality can be violated without entanglement.
The advantage of this postselection is that it also increases the Bell quantity that may be obtained with entanglement. The standard result that the efficiency must be at least with maximally entangled states (18) is based upon the assumption that Tsirelson’s bound still holds for entangled states. If the Bell quantity is also enhanced for entangled states, then the efficiency may be lower before the value that is possible with a LHV theory reaches that possible with entangled states.
The example we consider is where Alice and Bob share the two-qubit entangled state
where is the normalisation factor. The states and are assumed to be non-orthogonal, with the real inner product . We define a local transformation which acts as and .
The transformation is non-unitary and cannot be realised deterministically with a nonzero . Alice performs the local operation , and Bob applies . The different measurement settings for Alice and Bob are achieved by using different rotation angles and . This non-deterministically transforms the state to
Alice and Bob then perform orthogonal measurements on their respective qubits using the basis , and there are four possible cases of the combined measurement results. The probabilities for these four cases are
The Bell function is then obtained as
The maximum is obtained for
for some parameter . The Bell function then depends only on :
There does not appear to be an analytic solution for , but a good approximation is given by .
To determine the loss that is required for the operation on and , consider the singular value decomposition (33) of the matrix representing , i.e., (omitting a global phase) where
with , , , and . This therefore gives the minimum detector efficiency as
The maximum value of possible with LHV theories is then (18).
The values of the quantity for the initial state , as well as the maximum values for separable states, and the maximum values for LHV theories, are plotted in Fig. 2. The maximum value for separable states was obtained by numerical optimisation, and the maximum value for LHV theories was obtained using the formula . For , the value for entangled states varies from Tsirelson’s bound for to a maximum of 4 in the limit . On the other hand, initially has a maximum of for separable states. It increases with , but still does not exceed 2 for . For this value of , is almost 3 (2.966) for . On the other hand, the maximum value of for LHV theories increases from 2 for (no loss), and crosses the line for entangled states at . At this value of , , slightly below the limit of derived in Ref. (18).
We have also considered more general measurements that do not have the simple interpretation given above. These measurements are numerically optimised to maximise the value of for entangled states with a given level of loss. The results for this scheme are shown in Fig. 3. For this numerically optimised scheme the value of can be greater than 3 (3.0046) for entangled states before it reaches 2 for separable states. The value of when the value of for LHV theories reaches that for entangled states is slightly lower than before, at .
Methods for general non-unitary transformations such as have been presented in Ref. (34), and methods for performing single-qubit non-unitary transformations in optical systems in Ref. (35). An experiment for our proposal can be performed using current technology with the photon-polarisation qubit basis, namely, and where is the horizontal-polarisation state of a photon and vertical. The two non-orthogonal states and can be defined as and , where is assumed to be a real value larger than . In this case, the orthogonal basis states, and