Loopholes in Bell inequality tests of local realism

The EPR setting is as follows: consider a (small) physical system on which we intend to measure position Q or momentum P. The physical measurement devices associated with these measurements are mutually exclusive, and furthermore, the quantum-mechanical description for this physical system tells us that the measurements Q and P do not commute. The standard way to interpret this is that the system does not possess the properties of position or momentum, only probabilities are possible to obtain from quantum mechanics.

EPR now propose using a combined system of two subsystems of the above type, in a combined state so that measurement of the position difference gives ${{Q}_{1}}-{{Q}_{2}}={{q}_{0}}$ (constant) and measurement of the momentum sum gives ${{P}_{1}}+{{P}_{\,2}}=0$. These two combinations are jointly measurable, even though the individual positions and momenta are not, which implies that it is possible to produce a joint state with these properties. Letting the two subsystems separate, usually very far, EPR consider individual measurement of position or momentum. The system is such that the position difference and momentum sum is preserved under the separation process, which means that if the position of one subsystem has been measured, the position of the remote subsystem can be predicted. Therefore, EPR argue, the position of the remote subsystem must exist as a property of that subsystem. EPR write:

If, without in any way disturbing a system, we can predict with certainty (i.e., with a probability equal to unity) the value of a physical quantity, then there exists an element of physical reality corresponding to this physical quantity.

Likewise, if the momentum of one subsystem has been measured, the momentum of the remote subsystem can be predicted. In this case, the momentum of the remote subsystem must exist as a property of that subsystem. EPR continue to argue that both position and momentum must simultaneously exist as properties of the remote subsystem, otherwise

... the reality of P and Q [will] depend upon the process of measurement carried out on the first system, which does not disturb the second [remote] system in any way. No reasonable definition of reality could be expected to permit this.

Bohrʼs response (Bohr 1935) is that

... we are not dealing with an incomplete description characterized by the arbitrary picking out of different elements of physical reality at the cost of sacrifying other such elements, but with a rational discrimination between essentially different experimental arrangements and procedures which are suited either for an unambiguous use of the idea of space location, or for a legitimate application of the conservation theorem of momentum.

Bohr was considered to have won the debate: the consensus at the time was that there would seem to be no reality to describe. It was not until much later that Bell realized that Bohrʼs argument was incomplete. The exact system that EPR use for their argument does admit a more complete description (Bell 1986) but perhaps, as Bell writes, it was implicitly anticipated already in 1935 that there exist systems that do not. In any case, the argument needs more support.

The next step forward is the setup proposed in Bohm (1951), see figure 1, that uses a system combined of two spin-1/2 subsystems. The systems are produced in a total spin 0 state, so that the spins ${{A}_{a}}+{{B}_{b}}=0$ when measured along equal directions a = b. The subsystems are allowed to separate and a spin measurement is made on one of the subsystems, and the choice of measurement directions is a continuous choice instead of the dichotomic choice in the original EPR setup. Also in this system, the remote outcome at a given measurement setting can be predicted with certainty based on the local outcome at that setting. And because the reality of the spin measurement outcome in the remote system cannot depend on the local choice, the spin along any direction is an EPR element of reality. Note that also here, a more complete model exists for the system than quantum mechanics, when only requiring prediction with certainty for equal settings at the two sites. But this complete model does not give the full quantum predictions, a fact Bell made use of.

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A much stronger and more stringent argument for the non-existence of a more complete theory than quantum mechanics was provided by J S Bell in 1964. He took EPRʼs premise as a starting point and from the assumptions derived a statistical inequality that is violated by the quantum-mechanical predictions. This inequality is derived for EPR–Bohm setup, and must be fulfilled by any mathematical model that is 'local realist'. Realism is here motivated by the spin component along any axis being an EPR element of reality, and locality is motivated by the finite speed of light, or more specifically, because local measurement is made 'without in any way disturbing' the remote system.

In what follows we will use notation from probability theory. Realist models (or hidden variable models) are probabilistic models that use three building blocks. The first building block is a sample space $\Lambda$, which is the set of possible hidden variable values. The second is a family of event subsets of $\Lambda$, e.g., the sets of hidden variable values where specific measurement outcomes occur. The third and final building block is that these event sets must be measurable using a probability measure P, so that each event has a well-defined probability. If the sample (hidden variable) $\lambda$ is reasonably well-behaved, its distribution $\rho$ can be constructed from P. The outcomes of experiment are now described by random variables (e.g., $A(\lambda )$) that are maps from the sample to the possible outcomes of the experiment, and the expected (average) outcome of an experiment can be calculated as

$$\begin{eqnarray}&&E(A)={{\int }_{\Lambda }}A(\lambda ){\rm{d}} P(\lambda )={{\int }_{\Lambda }}A(\lambda )\rho (\lambda ){\rm{d}} \lambda ,\end{eqnarray} \tag{ 1 }$$

the latter if $\rho$ can be constructed. In this notation, the original Bell inequality can be written as follows.

Theorem 1. The following four prerequisites are assumed to hold except at a set of zero probability:

• (i)  
  Realism. Measurement outcomes can be described by two families of random variables (A for site 1 with local setting a, B for site 2 with local setting b):

  $$\begin{eqnarray}&&A(a,b,\lambda )\ {\rm and}\ B(a,b,\lambda ).\end{eqnarray} \tag{ 2 }$$

  The dependence on the hidden variable $\lambda$ is usually suppressed in the notation.
• (ii)  
  Locality. Measurement outcomes are independent of the remote setting:

  $$\begin{eqnarray}\begin{array}{rcl} A(a,\lambda ) & \mathop{=}\limits^{{\rm def}} & A(a,{{b}_{1}},\lambda )=A(a,{{b}_{2}},\lambda ) \\ B(b,\lambda ) & \mathop{=}\limits^{{\rm def}} & B({{a}_{1}},b,\lambda )=B({{a}_{2}},b,\lambda ). \\ \end{array}\end{eqnarray} \tag{ 3 }$$

  For brevity denote ${{A}_{i}}(\lambda )=A({{a}_{i}},\lambda )$ and ${{B}_{j}}(\lambda )=B({{b}_{j}},\lambda )$.
• (iii)  
  Outcome restriction. Measurement outcomes are $\pm 1$:

  $$\begin{eqnarray}&&|A(a,\lambda )|=|B(b,\lambda )|=1.\end{eqnarray} \tag{ 4 }$$
• (iv)  
  Complete anticorrelation. At equal settings, measurement outcomes are opposite:

  $$\begin{eqnarray}&&{\rm If}\;a=b\ {\rm then}\ A(a,\lambda )=-B(b,\lambda ).\end{eqnarray} \tag{ 5 }$$

Then, if ${{a}_{1}}={{b}_{1}}$,

$$\begin{eqnarray}&&\left| E\left( {{A}_{2}}{{B}_{1}} \right)-E\left( {{A}_{2}}{{B}_{2}} \right) \right|\leqslant 1+E\left( {{A}_{1}}{{B}_{2}} \right).\end{eqnarray} \tag{ 6 }$$

Proof. Since ${{a}_{1}}={{b}_{1}}$ we have ${{B}_{1}}{{A}_{1}}=-1$ (with probability 1), and

$$\begin{eqnarray}\begin{array}{rcl} |E\left( {{A}_{2}}{{B}_{1}} \right)-E\left( {{A}_{2}}{{B}_{2}} \right)| & = & \left| E\left( {{A}_{2}}{{B}_{1}}+{{A}_{2}}{{B}_{1}}{{A}_{1}}{{B}_{2}} \right) \right| \\ {} & \leqslant & E\left( \left| {{A}_{2}}{{B}_{1}}\left( 1+{{A}_{1}}{{B}_{2}} \right) \right| \right)=1+E\left( {{A}_{1}}{{B}_{2}} \right). \\ \end{array}\end{eqnarray} \tag{ 7 }$$

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Bell used deterministic realism in the sense of EPR elements of reality: perfect predictability because of the complete anticorrelation. However, it is never possible to predict with certainty in an experiment. There will always be a probability strictly less than unity that the remote site has the predicted value, because of experimental imperfections. This will be our first example of a loophole: the loophole of complete anticorrelation. The proof in inequality (7) relies on perfect anticorrelation, so assumption (iv) must be true, otherwise the equality does not hold. It is true that the assumption is explicit and therefore not 'an unexpected circumstance,' but one could argue that it is unexpected that a slightly lowered anticorrelation would make the proof invalid. This was realized by Clauser, Horne, Shimony, and Holt (CHSH 1969) who 'present a generalization of Bellʼs theorem which applies to realizable experiments': the CHSH inequality. The removal of the anticorrelation assumption was done in the CHSH paper while the weaker outcome restriction used below first appeared in Bell (1971).

Theorem 2. The prerequisites (i) and (ii) of theorem 1 and the following third prerequisite are assumed to hold except at a set of zero probability.

• (iii)  
  Outcome restriction. Measurement outcomes are bounded in absolute value by 1:

  $$\begin{eqnarray}&&|A(a,\lambda )|\leqslant 1\ {\rm and}\ |B(b,\lambda )|\leqslant 1.\end{eqnarray} \tag{ 8 }$$

Then,

$$\begin{eqnarray}&&\left| E\left( {{A}_{1}}{{B}_{1}} \right)+E\left( {{A}_{1}}{{B}_{2}} \right) \right|+\left| E\left( {{A}_{2}}{{B}_{1}} \right)-E\left( {{A}_{2}}{{B}_{2}} \right) \right|\leqslant 2.\end{eqnarray} \tag{ 9 }$$

Proof. 

$$\begin{eqnarray}&&\begin{array}{l} \left| E\left( {{A}_{1}}{{B}_{1}} \right)-E\left( {{A}_{1}}{{B}_{2}} \right) \right|=\left| E\left( {{A}_{1}}{{B}_{1}}\pm {{A}_{1}}{{B}_{1}}{{A}_{2}}{{B}_{2}}-({{A}_{1}}{{B}_{2}}\pm {{A}_{1}}{{B}_{1}}{{A}_{2}}{{B}_{2}}) \right) \right| \\ \quad \leqslant E\left( \left| {{A}_{1}}{{B}_{1}}\left( 1\pm {{A}_{2}}{{B}_{2}} \right) \right|+\left| {{A}_{1}}{{B}_{2}}\left( 1\pm {{A}_{2}}{{B}_{1}} \right) \right| \right)\leqslant 2\pm \left( E\left( {{A}_{2}}{{B}_{2}} \right)+E\left( {{A}_{2}}{{B}_{1}} \right) \right). \\ \end{array}\end{eqnarray} \tag{ 10 }$$

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It was also noted in Bell (1971) that if ${{A}_{1}}{{B}_{1}}=-1$ holds, then the inequality of Theorem 1 is a simple corollary of Theorem 2. If instead $E({{A}_{1}}{{B}_{1}})$ is larger than $-1$, Theorem 2 still holds. Therefore, there are no severe effects of the loophole. Taking the loophole into account changes the inequality, but the change is not large, and there will still be a violation. Somewhat surprisingly, the CHSH inequality is actually better than the original Bell inequality, giving a higher violation for different settings a_i and b_j. We will adapt this inequality for other loopholes below but will not obtain higher violation in any other case. However, it is almost always possible to derive a useful adapted inequality, as we shall see.

Note that we are not, strictly speaking, using EPR elements of reality anymore, but the weaker notion of stochastic realism (Clauser and Shimony 1978) that can be formulated as follows:

If, without in any way disturbing a system, we can predict with high probability (i.e., with a probability close to unity) the value of a physical quantity, then there exists an element of physical reality corresponding to this physical quantity.

Here, 'close to unity' is more a matter of taste; the most commonly used quantum system has prediction with probability $1/\sqrt{2}$, see below. The question of deterministic versus stochastic realism is important and subtle, but we will here use notation and theorems for deterministic realism without loss of generality: outcome random variables $A(\lambda ,a)$ whose value are determined directly from the hidden variable $\lambda$ rather than conditional outcome probabilities $P(A|\lambda ,a)$ that may be less than one so that the outcomes are not completely determined by $\lambda$. The latter stochastic model can be described by a probabilistic model that adds more hidden variables, that must be local, which means that the obtained inequalities do not change. Another way to say this is that the assumption of determinism in the theorems is not a loophole, since stochastic realist models are equivalent to mixtures of deterministic ones (Fine 1982).

The quantum-mechanical predictions violates the CHSH inequality, and the largest violation occurs for a total spin zero state, that gives the correlation

$$\begin{eqnarray}&&\left\langle {{A}_{a}}{{B}_{b}} \right\rangle =-{\rm cos} ({{\phi }_{ab}})\end{eqnarray} \tag{ 11 }$$

with ϕ_ab being the angle between the two directions a and b. Choosing the four directions $\pi /4$ apart in a plane in the order b₂, a₁, b₁, a₂, one obtains

$$\begin{eqnarray}&&\left| \left\langle {{A}_{1}}{{B}_{1}} \right\rangle +\left\langle {{A}_{1}}{{B}_{2}} \right\rangle \right|+\left| \left\langle {{A}_{2}}{{B}_{1}} \right\rangle -\left\langle {{A}_{2}}{{B}_{2}} \right\rangle \right|=2\sqrt{2},\end{eqnarray} \tag{ 12 }$$

and this violates inequality (10). Therefore, the quantum-mechanical predictions for such a state cannot be obtained from a local realist model.

The predictions of quantum mechanics are rotationally symmetric, and it has been common to simply assume rotational invariance of the experimental setup. For example, Clauser and Shimony (1978) discuss this, saying that

no harm is done in assuming [symmetry relations], since they are susceptible to experimental verification.

They proceed to simplify inequality (9) into $\left| 3E(\phi )-E(3\phi ) \right|\leqslant 2$, which is simpler to maximize and to measure since only one setting is used at the first site (and both at the second). While it is true that almost 'no harm is done' when using this assumption, it is important that the experimental verification really is performed. Otherwise, the assumption can fail and give a loophole in the experiment. In a sense, failure of the assumption is a systematic error, and the size of this error needs to be controlled. Even a good (but non-ideal) experiment will have small deviations, and the size of these deviations must be taken into account when calculating the size of the violation. Checking the size of the errors will need to be done by measuring at the settings that were eliminated by the assumption, so it seems better to avoid symmetry assumptions that can be avoided, even if they are relatively harmless.

Another simplification that can be made is to measure the outcomes from the two subsystems jointly: directly measure the product of ${{A}_{a}}{{B}_{b}}$ rather than the individual A_a and B_b. But this enables a loophole of 'individual existence': the individual values must be assumed to exist since they are not realized in experiment. It is no longer possible to use the EPR argument for their existence, nor make predictions to the remote site, since one does not have the local value to make predictions from. It is not even possible to use quantum mechanics to argue for the individual existence since the individual measurements are not performed, and 'unperformed experiments have no results' in quantum mechanics (Peres 1978). A restricted version of this loophole is present in Rowe et al (2001) who measure brightness of two ions in an ion trap. They perform a CHSH experiment, but use joint output values that are three-valued: no bright ions, one bright ion, or two bright ions. Assuming that the individual outcomes are realized, these correspond to $(+1,+1)$, $\{(-1,+1)$ or $(+1,-1)\}$, and $(-1,-1)$. The obtained correlations do violate the CHSH inequality, but with the loophole of individual existence (Danforth 2014). In foundational experiments, this loophole can and should be avoided by measuring the individual outcomes, whose (local) realism the experiment attempts to disprove. (The experiment by Rowe et al is important from the point of view of other loopholes, and we will return to it in section 3.)

Finally, accidental detections at the measurement sites are sometimes handled as a systematic deviation from the ideal quantum-mechanical predictions. Photonic experiments in particular are prone to this, e.g., because of stray light that enters the detectors, or so-called 'dark counts'—false detection events that are due to thermal noise within the detector. These accidentals could now appear to be valid measurement results, that due to their random nature will increase the noise in the experiment. An experimenter could be tempted to remove this extra noise, by 'subtracting the accidentals'. It is simple to estimate the size of the extra noise, and therefore also easy to remove it. But one should be aware that this opens up a loophole, that the accidentals and the higher correlation that remains after subtracting the accidentals both together are given by a local realist model. The loophole can be avoided by making sure that such accidentals occur rarely enough to give a violation in the raw data, including accidentals. An early example of such an experiment is Freedman and Clauser (1972), who report a violation after subtracting accidentals, but in a footnote add

In fact, our conclusions are not changed if accidentals are neglected entirely; the signal-to-accidental ratio with polarizer removed is about 40 to 1 for the data presented.

This eliminates the loophole; subtraction of accidentals should be avoided.

We now turn to random variations rather than systematic errors. Because of random variations in experimental data, such data should not be confused with the quantum-mechanical predictions: experimental data can violate the bound even under local realism. This is perhaps unexpected, but completely natural because of the randomness. For example, independent fair coin tosses for the outcomes gives

$$\begin{eqnarray}&&P\left( {{A}_{i}}{{B}_{j}}=\pm 1 \right)=\frac{1}{2},\end{eqnarray} \tag{ 13 }$$

and there certainly exists a local realist model that gives $E({{A}_{i}}{{B}_{j}})=0$, so the predictions do not give a violation. Nonetheless, the probability is non-zero for an apparent violation in individual outputs: even with independent fair coins, the prediction for four independent experiments (superindex 1–4) at the four setting combinations is

$$\begin{eqnarray}&&P\left( A_{1}^{(1)}B_{1}^{(1)}+A_{1}^{(2)}B_{2}^{(2)}+A_{2}^{(3)}B_{1}^{(3)}-A_{2}^{(4)}B_{2}^{(4)}=4 \right)=\frac{1}{16}.\end{eqnarray} \tag{ 14 }$$

Individual experimental outcomes are simply not bounded by the inequality. This is natural because of the random nature of the experiment, but could be thought of as an unexpected circumstance, a loophole. The problem here is the finite¹ sample size used in experiment, above a single sample for each setting combination. A common name for the problem is 'finite statistics', meaning that the sample mean can and will deviate from the mean of the distribution.

This loophole has also been observed under a different guise, because of the following observation: The different terms in

$$\begin{eqnarray}&&A_{1}^{(1)}B_{1}^{(1)}+A_{1}^{(2)}B_{2}^{(2)}+A_{2}^{(3)}B_{1}^{(3)}-A_{2}^{(4)}B_{2}^{(4)}\end{eqnarray} \tag{ 15 }$$

use different values of the hidden variable ${{\lambda }^{(i)}}$, while the CHSH proof uses the same $\lambda$ for all four measurement setting combinations. Therefore the CHSH inequality does not apply to that sum of measurement values—which is completely true, and indeed the reason for the non-zero probability in equation (14). It is in some sense the reason for the loophole of finite statistics. However, although the CHSH inequality does not apply to the above combination of individual samples, it does apply to the distribution mean. And the sample mean will tend to the distribution mean as the sample size grows; the adherence to the CHSH inequality will become better and better. To determine to what degree the inequality applies for a given sample size, we need to use methods from statistics.

Because of the above, obtaining a high value of the CHSH right-hand side from experiment is not enough. Even a seemingly high violation could be too low if the sample size is small. This unexpected circumstance could be thought of as a loophole: the loophole of low violation. We must determine how strong the evidence is against local realism, given our finite-sized sample of experimental data. The CHSH inequality applies to the correlations, but from experiments we can only estimate the correlations. The quality of the estimate is simplest to judge if we can assume that the outcomes $A_{i}^{(n)}$ and $B_{j}^{(n)}$ are independent identically distributed (IID; this assumption will give another loophole which we will return to in 2.2). Sometimes this is called 'Poisson statistics' although that term is more suitable when looking at a continuously pumped experiment, where events happen at random times at a constant rate. Assuming IID successive outcomes, the standard unbiased point estimate of the correlation $E({{A}_{i}}{{B}_{j}})$ using data from N_ij experimental runs is the sample mean

$$\begin{eqnarray}&&{{E}^{*}}\left( {{A}_{i}}{{B}_{j}} \right)=\frac{1}{{{N}_{ij}}}\mathop{\sum }\limits_{n=1}^{{{N}_{ij}}}A_{i}^{(n)}B_{j}^{(n)}.\end{eqnarray} \tag{ 16 }$$

Perhaps it is appropriate to point out that this is a random variable, since it is a sum of random variables. The estimate can now be used on each of the four terms in the left-hand side of the CHSH inequality to give

$$\begin{eqnarray}&&{{\beta }^{*}}={{E}^{*}}\left( {{A}_{1}}{{B}_{1}} \right)+{{E}^{*}}\left( {{A}_{1}}{{B}_{2}} \right)+{{E}^{*}}\left( {{A}_{2}}{{B}_{1}} \right)-{{E}^{*}}\left( {{A}_{2}}{{B}_{2}} \right).\end{eqnarray} \tag{ 17 }$$

Note that experiments at different setting combinations use different samples (superindices), as in equation (14). The above is now an unbiased point estimate of the distribution mean

$$\begin{eqnarray}&&\beta =E\left( {{A}_{1}}{{B}_{1}} \right)+E\left( {{A}_{1}}{{B}_{2}} \right)+E\left( {{A}_{2}}{{B}_{1}} \right)-E\left( {{A}_{2}}{{B}_{2}} \right),\end{eqnarray} \tag{ 18 }$$

which is unknown but less than 2 under local realism.

We now need to quantify how close the estimate is to the true mean. Experimental papers often measure the strength of the test as the 'number of standard deviations' of violation. One should be aware that the distribution standard deviation is not known, but the sample standard deviation is used instead, as the square root of the unbiased point estimate of $V({{A}_{i}}{{B}_{j}})$,

$$\begin{eqnarray}&&s_{ij}^{2}=\frac{1}{{{N}_{ij}}-1}\mathop{\sum }\limits_{n=1}^{{{N}_{ij}}}{{\left( A_{i}^{(n)}B_{j}^{(n)}-{{E}^{*}}\left( {{A}_{i}}{{B}_{j}} \right) \right)}^{2}}=\frac{{{N}_{ij}}}{{{N}_{ij}}-1}\left( 1-{{\left( {{E}^{*}}\left( {{A}_{i}}{{B}_{j}} \right) \right)}^{2}} \right).\end{eqnarray} \tag{ 19 }$$

The last equality holds here because of the $\pm 1$ outcomes of our experiment. If the independence assumption holds, the variance of the sum in equation (15) can be estimated by

$$\begin{eqnarray}&&{{s}^{2}}=s_{11}^{2}+s_{12}^{2}+s_{21}^{2}+s_{22}^{2}.\end{eqnarray} \tag{ 20 }$$

An unbiased point estimate for $V({{E}^{*}}({{A}_{i}}{{B}_{j}}))$ (the variance of the sample mean) would now be $s_{ij}^{2}/{{N}_{ij}}$, and consequently an estimate for the variance for ${{\beta }^{*}}$ is

$$\begin{eqnarray}&&s_{{{\beta }^{*}}}^{2}=\frac{s_{11}^{2}}{{{N}_{11}}}+\frac{s_{12}^{2}}{{{N}_{12}}}+\frac{s_{21}^{2}}{{{N}_{21}}}+\frac{s_{22}^{2}}{{{N}_{22}}}.\end{eqnarray} \tag{ 21 }$$

If N_ij are close to N, then $s_{{{\beta }^{*}}}^{2}\approx {{s}^{2}}/N$ with equality if ${{N}_{ij}}=N$. The sample standard deviation for the CHSH estimate ${{\beta }^{*}}$ is then $s/\sqrt{N}$, which is the usual behaviour of the estimated standard deviation of the mean as the sample size grows. Some papers use another approach, and coarse-grain their data into several decent-sized subsamples, calculate a subsample mean for each, and use those to calculate ${{\beta }^{*}}$ and ${{s}_{{{\beta }^{*}}}}$. Either method can be used to obtain the number of standard deviations of violation

$$\begin{eqnarray}&&k=\frac{{{\beta }^{*}}-2}{{{s}_{{{\beta }^{*}}}}}.\end{eqnarray} \tag{ 22 }$$

While this is the statistic presented in many papers, a statistician would prefer to do a hypothesis test. The natural null hypothesis is local realism, and a statistician would decide on the level of significance of the test before doing the experiment. The level of significance is then compared to the probability of obtaining the observed violation or higher, under local realism. The experiment is a Binomial trial, but for simplicity here we assume that our sample is large enough to use the central limit theorem, which allows us to approximate the distribution of ${{\beta }^{*}}$ with the normal distribution. An often quoted number for 'large enough' in a Binomial trial is $Np\gt 5$ where p is the smallest probability of one outcome. The quantum-mechanical smallest predicted probability of ${{A}_{i}}{{B}_{j}}=\pm 1$ is $p=(2-\sqrt{2})/4$, so that $N\gtrsim 35$ would be large enough. Since we do not know p, this only gives a rough idea on when Normal approximation could be expected to work, the curious should look into the standard literature to learn more. However, modern experiments are typically run with N into the thousands. Also, the variance of ${{\beta }^{*}}$ is not known, so we should use $({{\beta }^{*}}-\beta )/{{s}_{{{\beta }^{*}}}}$ which has the Student t-distribution with $N-1$ degrees of freedom, but again if N is large we can approximate with the Normal distribution. Note that if the data is coarse-grained as mentioned above, the number of degrees of freedom is the number of subsamples minus one, so that the Student t-distribution may need to be used. If Normal approximation can be used and local realism holds so that $\beta \leqslant 2$, then

$$\begin{eqnarray}&&P\left( {{\beta }^{*}}\geqslant 2+k{{s}_{{{\beta }^{*}}}} \right)\leqslant P\left( \frac{{{\beta }^{*}}-\beta }{{{s}_{{{\beta }^{*}}}}}\geqslant k \right)=1-\Phi \left( k \right)\lt \frac{1}{k\sqrt{2\pi }}{{e}^{-\frac{{{k}^{2}}}{2}}},\end{eqnarray} \tag{ 23 }$$

(where $\Phi$ is the Normal cumulative distribution function) and if ${{N}_{ij}}=N$ this reads

$$\begin{eqnarray}&&P\left( {{\beta }^{*}}\geqslant 2+k\frac{s}{\sqrt{N}} \right)\lt \frac{1}{k\sqrt{2\pi }}{{e}^{-\frac{{{k}^{2}}}{2}}}.\end{eqnarray} \tag{ 24 }$$

The number k can be chosen arbitrarily, and can be used to translate the number of standard deviations of violation into a probability of getting the observed violation under local realism, essentially behaving as exp$(-{{k}^{2}}/2)$. This enables a proper hypothesis test, closing the loophole of finite statistics.

Different Bell tests have different statistical properties. There is now, some fifty years after Bellʼs paper, a long list of Bell inequalities with different properties. Examples include higher dimension, more sites, and different distribution of entanglement between the sites. The above analysis can be further refined to compare the statistical strength of different Bell inequality tests (van Dam et al 2005), but as the authors of that paper point out, statistical strength is not all, one has to look at how difficult the experiments are to implement well so that the statistical strength is useful, and also the rate at which the samples are obtained so that a large sample can be generated. Currently, it seems that low dimension and few sites is the most popular alternative.

It is certainly true that if communication between the sites is possible (so that locality is not enforced), then local hidden variable models are possible. If measurement settings are chosen and set long before an experimental run, there is nothing that prevents a signal from traveling from one site to another, a signal that can carry influence from the remote setting to the local outcome. The importance of changing the measurement settings quickly was first stressed in Bohm and Aharonov (1957), in the context of EPR–Bohm experiments, noting that

... particle B (which does not interact with A or with the measuring apparatus) realizes its potentiality for a definite spin in precisely the same direction as that of A. ... Such an interaction ... would have to be instantaneous, because the orientation of the measuring apparatus could very quickly be changed, and the spin of B would have to respond immediately to the change.

Bell (1964), considers this in conjunction with his inequality, but not as a loophole in the proof of the inequality. He is well aware that if locality is not enforced in an experimental test, then violation does not say anything about properties of the model. Instead, he wonders if there will be a violation at all if locality is enforced; perhaps nature will simply deviate from the quantum-mechanical predictions, allowing an underlying local realist theory. After all, at the time the quantum-mechanical predictions were not known to hold if the settings are chosen and set so that locality is enforced. While this was certainly expected, already Furry (1936) had theorized that the quantum wave function would separate into factors at large distances, which would mean there would be no violation. An experimental test was still missing, except for using static settings at parallel and perpendicular directions (Wu and Shaknov 1950; see also Kocher and Commins 1967). And, as Bell (1964) writes:

In that connection, experiments of the type proposed by Bohm and Aharonov [1957], in which the settings are changed during the flight of the particles, are crucial.

In this sense, the loophole is not so much the possibility of a hidden subluminal communication from one site to the other in slow-changing measurement setups, but rather that the quantum violation might disappear in fast-changing setups. This would constitute the unexpected circumstance, that could cause failure of the conclusion that local realist theories cannot give the same correlations as found in nature. The correlation could be low enough to allow local realism in a fast-changing setup. Then, closing the loophole is not the actual process of enforcing fast changes of measurement settings, but instead ensuring that the violation remains while doing so.

It is possible to view the loophole differently: that the possibility of slower-than-light communication invalidates the inequality. The unexpected circumstance would in that case be the unknown mechanism for communication. An influence is in principle allowed by special relativity, but it does not correspond to any known force or influence in any established physical theory. In this form of the loophole, prevention of communication is the focus, but of course one should make sure that the violation remains. Since prevention of communication is important here, one can ask what correlations a bounded but non-zero amount of communication would allow, and some early papers on this are Brassard et al (1999), Steiner (2000), Toner and Bacon (2003).

The first experiment to close the locality loophole was Aspect et al (1982), which really focuses on the loophole in the first sense above. This groundbreaking experiment is the first to show that even with fast-changing settings, the quantum-mechanical predictions remain, and therefore also the violation of the Bell inequality remains. We will continue here to look at some weaknesses of this experiment, but bear in mind that it is the first experiment that tests quantum mechanics in a new regime, that rules out suggestions of the type that Furry (1936) made, and answers Bellʼs (1964) worries that the quantum-mechanical predictions might not hold when the measurement settings change rapidly. Bell was acutely aware that such an experiment was needed and urged Aspect and his co-workers to finish the experiment (see e.g., Bell 1981). The experiment by Aspect, Dalibard, and Roger has had an enormous impact on the field, and rightly so.

In Aspect et al (1982), the distance from each measurement site to the source is 6 m, or 20 ns. The settings are switched every 10 ns, and the coincidence window used is 18 ns, closing the locality loophole as already stated. It is still vulnerable to the efficiency loophole (section 3), but another weakness of the setup remains: the settings are switched periodically using an ultrasonic standing wave. Quantum mechanics does not care how the settings are chosen, but a local realist model might. Two problems arise from this, the first slightly less important problem (Zeilinger 1986) has to do with an unfortunate property of the geometry of the physical setup used: twice the period of the switching coincides with the light-distance between the measurement sites. This means that a light-speed signal carrying the remote setting from the previous period would arrive when the local measurement is performed. Avoiding this problem is simply to vary the period of the switching, or use properly random measurement settings, see below. The second problem is that the setting sequence is predictable, and is more serious and enables a loophole in its own right, the memory loophole.

It was already noted in the proposal of the experiment (Aspect 1976) that a periodic or predictable sequence of settings is problematic in itself. He notes that if predictable settings are used,

a supplementary assumption should be exhibited: the polarizers have no 'memory', i.e., they can be influenced by signals received at a certain time from the [remote site] (with a certain delay) but they cannot store all this information for a long time and extrapolate in the future even if there is some regularity in the [settings].

In short: a local realist model can give a violation if it can remember earlier settings and from that predict what the current remote setting is. The assumption of no memory is needed in the Aspect experiment, while presence of memory would be an unexpected circumstance, a loophole.

The assumption would not be needed if properly random or unpredictable measurement settings are used, and this is done in the experiment by Weihs et al (1998). There, the measurement setting was applied through fast electro-optic modulators, so that the setting can be chosen independently for each experimental run (how to count these is discussed in section 3). It now remains to choose the settings randomly, and this is done via a physical random number generator built from a light-emitting diode, a beam-splitter, and two photomultipliers. Signals from the two photomultipliers are then used to select the local setting, in fast electronics to ensure locality of the setting choice. In this experiment, the distance between the measurement sites is 400 m or 1.3 μs, and the total delay from the random number generation to the measurement has finished is less than 100 ns. The coincidence time window is 6 ns. It is now reasonable to think that the setting choice is no longer predictable, and then the assumption of no memory is no longer needed. This setup is often regarded as having conclusively closed the locality loophole.

We shall return to the question of randomness shortly, but there remains something to be said about memory. Accepting that the settings are unpredictable, the Weihs et al experiment closes the above memory loophole, because predicting the current remote setting using the previous sequence of settings is no longer possible. However, there is another issue here. In the standard statistical analysis of subsection 1.5, one assumes that the experimental runs are independent. If this is not the case, then one experiment in principle has access to what settings were used and what the results were in previously performed experiments, or even the local hidden variable values. These values might enter into the process of determining the outcome of the present experiment. Then, the standard statistical analysis cannot be used, resulting in another memory loophole (Gill et al 2003a, Gill 2003b, Barrett et al 2002). This particular memory loophole is not about prediction of the measurement settings, but rather about dependence of outcomes within a sequence of experiments.

Removing the loophole requires a statistical analysis that does not use the independence assumption. If local realism holds, ${{\beta }^{*}}-2$ is a supermartingale (Gill 2003b), and then a bound can be obtained for the probability of a large violation of the CHSH inequality, by using the Hoeffding (1963) inequality. In the simplified situation of having equal number of samples in each term of ${{\beta }^{*}}$ the bound reads

$$\begin{eqnarray}&&P\left( {{\beta }^{*}}\geqslant 2+k\frac{4}{\sqrt{N}} \right)\leqslant {{e}^{-\frac{{{k}^{2}}}{2}}}.\end{eqnarray} \tag{ 25 }$$

Again, the constant k can be chosen arbitrarily, just as in the bound (23). There is a small correction when the number of samples differ for different terms in ${{\beta }^{*}}$ (Barrett et al 2002). The difference between the bounds (23) and (25) is not so large, the estimated standard deviation s changes to the number 4 (half the range of expression (15); the quantum prediction is IID with $s\approx 2$), and the factor $1/(k\sqrt{2\pi })$ disappears on the right-hand side. The same bound applies simultaneously to all earlier ${{\beta }^{*}}$, with less than N samples in each term (see Gill 2003b). This enables a hypothesis test that closes the memory loophole in a good experiment, meaning a sizable violation and a large sample.

So far we have talked about locality of the random variables, that these are independent of the remote setting. The local result is given by the hidden variable and the local setting only. However, to enable the inequality, it is not enough to have the random variables independent of the remote setting. The probability measure P (of the hidden variable $\lambda$) also cannot depend on either local setting a and b. In a sense, this is more related to our reality assumption than to the locality assumption. The reality assumption (theorem 1 : i) is intended to capture that the model describes an underlying reality independent of measurement. In the theorem, the local hidden variable $\lambda$ is a mathematical representation of the underlying reality, while the settings a and b are free parameters that are under control of the experimenter. Formally, the value of $\lambda$ should be independent of the external parameters a and b.

This was discussed in an exchange between Bell (1976), Shimony et al (1976) and Bell (1977) (reprinted in Bell, Clauser, Horne and Shimony 1985), where the consensus is that this independence is important. The one difference seems to be stressing that the hidden variable does not depend on the settings (Shimony, Horne, and Clauser) or stressing that the settings does not depend on the hidden variable (Bell). From a theoretical point of view, the difference is not so important, because what really matters is that the hidden variable and the local parameters do not share a common cause.

In one particular type of experiments, the second kind of independence is important and actually an essential assumption: so-called passive choice experiments. In this kind of experiment, one uses the randomness inherent in the system itself to choose the setting. One example is, for a photonic experiment, to set up separate detection stations for each setting at each local site, two at each site for CHSH, and using a beamsplitter to decide which of the two detection stations (measurement settings) will be used. Now, the hidden variable itself might influence the path through the beam-splitter, the setting choice, so it is difficult to argue for independence. It would be possible to outright assume independence in a passive choice experiment, but this would constitute a loophole in the experiment.

It is argued in Gisin and Zbinden (1999) that when the switches are lossy, passive switching is almost on equal basis with active switching. The corresponding assumption for lossy active switching is the fair sampling assumption (see section 3 below); the assumption that losses in the switch do not depend on the hidden variable. But this argument is weak. An active switch has an external input for an external parameter, the measurement setting choice, which is an essential part of the Bell Theorem. A passive switch does not have an input, so the theorem itself does not apply. It is an explicit assumption that there is a random choice that does not depend on the hidden variable. There is, on the face of it, a big difference between using the fair sampling assumption (see below) and assuming that a passive switch is independent of the hidden variable. Both are assumptions and lead to loopholes, but they are conceptually different. The only way to avoid the assumption of passive switch hidden-variable independence is to use active choice of the measurement setting.

Now, even in an actively switched experiment one needs to make sure that measurement settings are chosen independently of the hidden variables generated at the source. And using locality to argue for such independence is possible only if the setting choice is made at space-like separation from the emission event. It is not enough to have the setting choice inside the future light cone of the emission event; this only prohibits influence from the setting choice to the hidden variable, but does not prohibit the reverse influence. Such an influence would prevent free choice of the measurement settings, and constitute a 'freedom-of-choice' loophole, so named in Scheidl et al (2010) who report an experiment where the setting choice is properly space-like separated from the emission event, enabling a locality-based argument for independence (a three-particle variation is presented in Erven et al 2014). However, the possibility of a common cause remains, see below.

The question of settings as free parameters is deep; it could be discussed if it is even in principle possible to have free parameters. The concept is linked to the concept of free will in the mentioned exchange between Bell; Shimony, Horne, and Clauser, and as Bell (1977) put it,

Here I would entertain the hypothesis that the experimenters have free will.

For this reason, it is sometimes suggested that an experiment with an actual human that chooses the experiment settings would be desirable. But one should be aware of the limitations of such a trial. The sample rate would need to be low, because a human needs time to produce the parameter values. Similarly, the distance between source and measurement site needs to be large, at least on the order of light-seconds (say, the distance to the moon) for the events to be space-like separated. Finally, a human subject is typically not a good source of randomness (see e.g., Wagenaar 1972), despite having a free will, or so we believe. While such an experiment would certainly be interesting, these considerations cast doubt on the added strength that it would give to the argument against local realism.

Ensuring independence between the source and the settings is a grand task. One could in principle believe that events that are close in space depend on each other, because their recent past light-cones intersect, and that large separation would decrease the dependence. Already at the conference 'Thinkshops on Physics: Experimental Quantum Mechanics' in Erice, Italy, 1976, it was suggested that one could generate random settings from astronomical objects outside each others' light cones, e.g., quasars on opposite sides of the universe (Zeilinger 2014). The benefit of such an experiment would be to enlarge the separation of the setting choice. However, it is impossible for us to know if the backward light-cones from the two distant emission events overlap, and if that overlap contains a common cause for the emissions. The most popular model of the origins of our universe is as a very small space-time region, so there may still be a common cause. Even remote quasar emissions must be assumed not to have a common cause. It is possible that all events in the universe share common causes, a philosophical view called superdeterminism (see e.g., Shimony et al 1976, Bell 1977, Brans 1988). This constitutes a loophole, but if superdeterminism holds, there is no point in discussing what mathematical models could be used to model nature, be it local realist or quantum or any other model. We can never rule out this possibility using scientific methods, because (Shimony et al 1976):

In any scientific experiment in which two or more variables are supposed to be randomly selected, one can always conjecture that some factor in the overlap of the backward light cones has controlled the presumably random choices. But, we maintain, skepticism of this sort will essentially dismiss all results of scientific experimentation. Unless we proceed under the assumption that [superdeterminism does not hold], we have abandoned in advance the whole enterprise of discovering the laws of nature by experimentation.

The loophole of superdeterminism cannot be closed by scientific methods; the assumption that the world is not superdeterministic is needed to do science in the first place.

The simplest situation to analyze is when local events are missing from the data record, as already mentioned. The problem here is that not all values of the hidden variable gives a detection, and that the obtained correlation is really the conditional correlation

$$\begin{eqnarray}&&E\left( {{A}_{i}}{{B}_{j}}|{{A}_{i}}\ {\rm det} .{\rm and}\ {{B}_{j}}\ {\rm det} . \right).\end{eqnarray} \tag{ 26 }$$

Conditional correlations do not add if the conditioning is over different subsets, so it is not immediately possible to write

$$\begin{eqnarray}&&\begin{array}{l} \qquad E\left( {{A}_{1}}{{B}_{1}}|{{A}_{1}}\;{\rm{det}} {\rm{.}} \;{\rm{and}} \;{{B}_{1}}{\rm det} . \right)+E\left( {{A}_{1}}{{B}_{2}}|{{A}_{1}}\;{\rm{det}} {\rm{.}} \;{\rm{and}} \;{{B}_{2}}{\rm det} . \right) \\ \qquad \quad ={\mkern 1mu} E\left( {{A}_{1}}{{B}_{1}}+{{A}_{1}}{{B}_{2}}|\ldots \right). \\ \end{array}\end{eqnarray} \tag{ 27 }$$

This is only visible when explicitly writing out the conditioning on detection, which is rarely done in practice—resulting in an unexpected circumstance, a loophole. The proof of the Bell inequality will not go through as is, since it needs addition of the correlations, but there are two ways of re-establishing (modified) inequalities. One is to add auxiliary assumptions, which will give back the original inequality, and the other is to take the missing events into account in the analysis, which will give an inequality with a modified bound.

The first paper that discusses this in earnest is Pearle (1970), where the observation is made that 'data rejection' enables a deterministic local hidden-variable model to give the same predictions as quantum mechanics for the coincident events. This is substantiated by an explicit model that does just that, so that the conditional correlations (seemingly) violate the Bell inequality. However, the model has a coincidence probability that varies with the angular difference of the two remote settings, which makes it less appealing because the quantum-mechanical description predicts constant detection probability. Pearle derives a bound for the conditional detection efficiency for models of the type he uses, and the lowest bound he finds for his varying efficiency is $2\sqrt{2}-2\approx 82.84$%. This number equals the generic bound for the CHSH inequality, which later has been derived under increasingly general assumptions, see below.

The CHSH inequality from theorem 2 is slightly earlier (1969), and is intended for an experimental setup that uses filters, followed by detectors that assigns $+1$ to detection and $-1$ to non-detection. The authors note that photonic experiments at that time with polarization filters would not violate the CHSH inequality 'because available photoelectric efficiencies are rather small'. This is not so much noticing a loophole but simply that there is no violation since the overwhelming amount of $-1$ outcomes would wash the violation out completely. Low photoelectric efficiencies is a problem but the preceding polarization filters are better, so they consider applying the inequality to emergence (here labeled $+1$) and non-emergence ($-1$) of the photon from the filter instead.

It now remains to connect emergence to detection, and CHSH (1969) do this by assuming that detected photons are a fair sample of the emerging photons,

that if a pair of photons emerges from [the polarization filters] the probability of their joint detection is independent of [the settings].

In the notation used here, the assumption is that there is a local realist model that describes emergence and non-emergence by ${{A}_{i}}=\pm 1$ and ${{B}_{j}}=\pm 1$, and that

$$\begin{eqnarray}&&P\left( {{A}_{i}}={{B}_{j}}=+1\cap {{A}_{i}}\ {\rm det} .{\rm and}\ {{B}_{j}}\ {\rm det} . \right)=cP\left( {{A}_{i}}={{B}_{j}}=+1 \right),\end{eqnarray} \tag{ 28 }$$

where c does not depend on i or j. Since the emergence correlations on the right-hand side must obey the CHSH inequality, a similar inequality can now be derived for the detection rates R at the $+1$ detectors,

$$\begin{eqnarray}&&R\left( {{A}_{1}}{{B}_{1}} \right)+R\left( {{A}_{1}}{{B}_{2}} \right)+\left| R\left( {{A}_{2}}{{B}_{1}} \right)-R\left( {{A}_{2}}{{B}_{2}} \right) \right|\leqslant \mathop{{\rm max} }\limits_{i,j}\left( R\left( {{A}_{i}} \right)+R\left( {{B}_{j}} \right) \right).\end{eqnarray} \tag{ 29 }$$

The problem is now reduced to finding sufficiently efficient polarizers, which is simpler; the paper mentions polarizing efficiencies of 0.92–0.95, which would give a violation.

The restricted fair sampling assumption used by CHSH applies only to the detected ${{A}_{i}}={{B}_{j}}=+1$ events: that these constitute a fair sample of all the ${{A}_{i}}={{B}_{j}}=+1$ events. The fair-sampling assumption as used today is slightly extended, and applies to all combinations of $\pm 1$ outcomes. In short, one uses the same assumption of existence of a local realist model that describes the ideal outcomes, and in addition that the probability of joint detection is independent of the settings, so that

$$\begin{eqnarray}&&P\left( {{A}_{i}}=\pm 1\cap {{B}_{j}}=\pm 1\cap {{A}_{i}}\ {\rm det} .{\rm and}\ {{B}_{j}}{\rm det} . \right)=cP\left( {{A}_{i}}=\pm 1\cap {{B}_{j}}=\pm 1 \right).\end{eqnarray} \tag{ 30 }$$

One could think of this as a constant correction to the probability of emergence from an ideal polarizing beamsplitter, but usually one does not explicitly refer to the internal workings of the measurement devices. With this modern variant of the fair-sampling assumption it is easy to see that

$$\begin{eqnarray}&&P\left( {{A}_{i}}\ {\rm det} .{\rm and}\ {{B}_{j}}\ {\rm det} . \right)=c,\end{eqnarray} \tag{ 31 }$$

so that

$$\begin{eqnarray}&&P\left( {{A}_{i}}=\pm 1\cap {{B}_{j}}=\pm 1|{{A}_{i}}\ {\rm det} .{\rm and}\ {{B}_{j}}{\rm det} . \right)=P\left( {{A}_{i}}=\pm 1\cap {{B}_{j}}=\pm 1 \right).\end{eqnarray} \tag{ 32 }$$

Therefore, since the CHSH inequality applies to the underlying local realist model for the unobserved ideal outcomes, we also have a CHSH inequality for the detected events that reads

$$\begin{eqnarray}&&\begin{array}{l} |E\left( {{A}_{1}}{{B}_{1}}|{{A}_{1}}\ {\rm det} .{\rm and}\ {{B}_{1}}\ {\rm det} . \right)+E\left( {{A}_{1}}{{B}_{2}}|{{A}_{1}}\ {\rm det} .{\rm and}\ {{B}_{2}}\ {\rm det} . \right)| \\ \quad +\left| E\left( {{A}_{2}}{{B}_{1}}|{{A}_{2}}\ {\rm det} .{\rm and}\ {{B}_{1}}\ {\rm det} . \right)-E\left( {{A}_{2}}{{B}_{2}}|{{A}_{2}}\ {\rm det} .{\rm and}\ {{B}_{2}}\ {\rm det} . \right) \right|\leqslant 2. \\ \end{array}\end{eqnarray} \tag{ 33 }$$

This is violated by the quantum-mechanical predictions as soon as the efficiency is non-zero.

One should perhaps point out that fair sampling is not equivalent to constant detector efficiency. There exist relatively simple local realist models with constant lowered efficiency that reproduce the quantum-mechanical predictions (Santos 1996, Larsson 1999, Gisin and Gisin 1999; an earlier example with small deviations is Marshall et al 1983). Of course, these models typically do not contain predictions on the level of emergence from the polarizing filters (or beam-splitters). The few that do contain such predictions do not violate the CHSH inequality at that level, while the conditional correlations after detection do give a violation.

The efficiency problem was again addressed in Bell (1971), where he provided a new proof of the CHSH inequality that does not require the outcomes to be only $\pm 1$, but allows them to take values between $+1$ and $-1$. He uses this to address inefficiency by assigning the value 0 to the missing outcomes (Bell credits J A Crawford for the suggestion). Note that the value $-1$ as used in Clauser et al (1969) is equally suitable, and will give the same correlations and bounds when the local outcomes are equally probable.

Assignment of values to missing outcomes needs a locality assumption: that detection is decided locally, and only depends on the hidden variable and the local setting. This is the natural local hidden-variable description of the detection process; formally, it is the assumption that the events (subsets of $\Lambda$) of detection

$$\begin{eqnarray}&&\left\{ \lambda :A({{a}_{i}},{{b}_{j}}\lambda )\ {\rm det} . \right\}\ {\rm and}\ \left\{ \lambda :B({{a}_{i}},{{b}_{j}}\lambda )\ {\rm det} . \right\}\end{eqnarray} \tag{ 34 }$$

are independent of the remote setting, so that it is meaningful to write

$$\begin{eqnarray}&&\left\{ \lambda :{{A}_{i}}(\lambda )\ {\rm det} . \right\}\ {\rm and}\ \left\{ \lambda :{{B}_{j}}(\lambda )\ {\rm det} . \right\}.\end{eqnarray} \tag{ 35 }$$

If one, like Bell or CHSH, includes the non-detections as events, there is no conditioning and the CHSH inequality is recovered. However, lowered efficiency will mean higher probability of the 0 (or $-1$) 'outcome' which in turn will mean lower violation, and as CHSH already had noted, the available equipment at the time would not violate the inequality.

Value assignment to no-detection events will require the experimenter to recognize no-detection events, so that the value used can be assigned. One way to do this is to change the experimental setup into a so-called 'event-ready' (or 'heralded') setup, in which there is a clear signal when a pair has been emitted at the source. Then, value assignment is possible even for non-detection, and direct calculation of the detection probability can be performed. The need for event-ready setups is not specifically stated in Bell (1971) 'but is clear from the context' (Clauser and Shimony 1978); assigning 0 to a non-detection requires knowledge that a detection should have occurred but did not. This point was later stressed in, for example, Bell (1981). An alternative to a heralded source is heralded detection: to locally certify that the photon is present at the measurement site before actually inserting the measurement setting into the measurement device (Ralph and Lund 2009); such an experiment has yet to be done. Heralded sources, on the other hand, have been built, one example is the already mentioned atomic source of Hofmann et al (2012). Here, the signal-of-emission needs to be generated very carefully, so that the violation remains—in quantum-mechanical language, the entanglement should not be broken.

Since event-ready experiments are demanding to set up, one alternative is to use simplifying assumptions on the distribution of non-detections. Garg and Mermin (1987) assume that non-detections occur at a constant probability

$$\begin{eqnarray}&&P({{A}_{i}}\ {\rm det} .)=P({{B}_{j}}\ {\rm det} .)=\eta ,\end{eqnarray} \tag{ 36 }$$

and that the errors are statistically independent,

$$\begin{eqnarray}&&P({{A}_{i}}\ {\rm det} .{\rm and}\ {{B}_{j}}\ {\rm det} .)=P({{A}_{i}}\ {\rm det} .)P({{B}_{j}}\ {\rm det} .)={{\eta }^{2}}.\end{eqnarray} \tag{ 37 }$$

With these assumptions, the CHSH inequality changes into

$$\begin{eqnarray}&&\begin{array}{l} |E\left( {{A}_{1}}{{B}_{2}}|{{A}_{1}}\ {\rm and}\ {{B}_{2}}\ {\rm det} . \right)+E\left( {{A}_{2}}{{B}_{2}}|{{A}_{1}}\ {\rm and}\ {{B}_{1}}\ {\rm det} . \right)| \\ \quad +\left| E\left( {{A}_{2}}{{B}_{1}}|{{A}_{2}}\ {\rm and}\ {{B}_{1}}\ {\rm det} . \right)-E\left( {{A}_{2}}{{B}_{2}}|{{A}_{2}}\ {\rm and}\ {{B}_{2}}\ {\rm det} . \right) \right|\leqslant \frac{4}{\eta }-2. \\ \end{array}\end{eqnarray} \tag{ 38 }$$

This gives a the generic bound at the same level as the lowest bound of Pearle (1970), and gives a violation from the quantum-mechanical predictions as soon as $\eta \geqslant 2\sqrt{2}-2\approx 82.84\%$. As a comparison, the approach of Bell (1971) with the same assumptions gives a bound of ${{2}^{-1/4}}\approx 84.09\%$ (however, this number is not given in Bellʼs paper).

Lo and Shimony (1981) recognize the need of avoiding extra assumptions and derive a new inequality that explicitly includes lowered efficiency without the symmetry assumptions (36) and (37). Their inequality is violated by the quantum-mechanical predictions as soon as the efficiency exceeds 85.97%, which at the time was the best-known bound. But since they avoid the mentioned assumptions, a different estimate of the efficiency is needed. Lo and Shimony measure the output rate of the source separately, and compare it with the rate of detection in the full experiment. However, this leaves room for a local realist model to behave differently in the source-rate measurement than it does in the violation measurement; this is a loophole. For the proposal of Lo and Shimony, since the used system is built up from material particles (quite heavy Na atoms), it could be argued that there is no room for this kind of changing behavior. But it is an extra restriction on the local realist model, motivated by other physical properties of the system than we normally use in Bell inequality testing. And the experiment seems very difficult to perform. It would arguably be better to have a generic method that can avoid the symmetry assumptions, for all kinds of systems without this type of extra argument.

Fortunately, the symmetry assumptions (36) and (37) can be avoided; they are needed because it is difficult to connect the probability $P({{A}_{i}}$ det. and B_j det.) to the experimentally available conditional detection probabilities $P({{A}_{i}}$ det. $|{{B}_{j}}$ det.) (see e.g., Burnham and Weinberg 1970). In Larsson (1998a) it is shown that the conditional probability can be used directly, by avoiding value assignment to non-detections altogether; the output of the measurement device will remain undefined. In this case, the random variables A_i and B_j are only defined on subsets of the hidden-variable space, on the subsets where detections occur. Then, the original CHSH inequality only applies for the joint subset where all detections would have occurred, where all A_i and B_j have well-defined values.

In this case, the efficiency $\eta$ must now be estimated differently, and the appropriate definition is to use the minimum of the conditional detection probabilities

$$\begin{eqnarray}&&\eta =\mathop{{\rm min} }\limits_{i,j}\left\{ P\left( {{A}_{i}}\ {\rm det} .|{{B}_{j}}\ {\rm det} . \right),\ P\left( {{B}_{j}}\ {\rm det} .|{{A}_{i}}\ {\rm det} . \right) \right\}.\end{eqnarray} \tag{ 39 }$$

These conditional probabilities are well-defined and easy to estimate in experiment, and relieve an experimenter from the difficult task of estimating $P({{A}_{i}}$ det.) and $P({{A}_{i}}$ det. and B_j det.). It suffices to estimate the conditional detection probabilities directly from experimental data and use the lowest value. Remarkably, this gives the same inequality as in (38), and the bound $\eta \geqslant 2\sqrt{2}-2\approx 82.84\%$ is valid for the $\eta$ from equation (39) without the auxiliary assumptions (36) and (37).

In all three approaches: event-ready, with, or without symmetry assumptions, it is the efficiency of whole experiment that is the important number—the loophole is often called 'the detector-efficiency loophole,' but it is important that the number in question concerns the whole experiment. For comparison, the fair sampling assumption that was described earlier essentially allows an experimenter to ignore the detector inefficiency, but still needs the polarizer/beamsplitter efficiency to be included. The one exception to this is if event-ready analyzer stations are used (Ralph and Lund 2009) ensuring that presence of a photon is signaled before the setting has been input into the station. Then, losses in transmission before the event-ready indicator can be ignored, while losses in the analyzer station still needs to be incorporated in the analysis.

In Clauser and Horne (1974) another approach is presented, to use probabilities instead of correlations. In the notation we have adapted here, the Clauser–Horne (CH) inequality reads

$$\begin{eqnarray}\begin{array}{rcl} -1 & \leqslant & P({{A}_{1}}={{B}_{1}}=1)+P({{A}_{1}}={{B}_{2}}=1)+P({{A}_{2}}={{B}_{1}}=1)-P({{A}_{2}}={{B}_{2}}=1) \\ {} & {} & -P({{A}_{1}}=1)-P({{B}_{1}}=1)\leqslant 0. \\ \end{array}\end{eqnarray} \tag{ 40 }$$

If detection counts are inserted instead of probabilities, this is sometimes referred to as the Eberhard inequality, having been rediscovered and found to be very important and useful in Eberhard (1993), more on this below. The CH inequality appears to be different from the CHSH inequality, but the two are equivalent. Nonetheless, the specific form of the CH inequality makes it useful for handling the efficiency loophole, because both single and coincidence probabilities enter directly. Then, it is clear that conditional probabilities should not be used. And there is no need to modify the bound for lowered efficiency; instead, a decreased efficiency will immediately lower the probabilities so that the violation is reduced. The coincidence terms will decrease faster than the single-outcome terms, so there will a be violation only for high efficiency.

Clauser and Horne (1974) realized that because of this, the inequality would not be useful for the available experiments at the time, so they added a 'no enhancement' assumption to recover a violation. They assume that, for a given value of the hidden-variable $\lambda$ in a stochastic local realist model,

the probability of a count with a polarizer in place is less than or equal to the probability with the polarizer removed.

Denoting a count at site A with the polarizer removed by ${{A}_{\infty }}=1$, this is the pointwise assumption (in $\lambda$, allowing briefly for stochastic hidden variables)

$$\begin{eqnarray}&&\begin{array}{l} P\left( {{A}_{i}}(\lambda )=1|\lambda \right)\leqslant P\left( {{A}_{\infty }}(\lambda )=1|\lambda \right), \\ P\left( {{B}_{j}}(\lambda )=1|\lambda \right)\leqslant P\left( {{B}_{\infty }}(\lambda )=1|\lambda \right). \\ \end{array}\end{eqnarray} \tag{ 41 }$$

Under this assumption the inequality changes so that it contains only coincidence probabilities, some with the polarizers removed:

$$\begin{eqnarray}\begin{array}{rcl} -P({{A}_{\infty }}={{B}_{\infty }}=1) & \leqslant & P({{A}_{1}}={{B}_{1}}=1)+P({{A}_{1}}={{B}_{2}}=1)+P({{A}_{2}}={{B}_{1}}=1) \\ {} & {} & -P({{A}_{2}}={{B}_{2}}=1)-P({{A}_{1}}={{B}_{\infty }}=1)-P({{A}_{\infty }}={{B}_{1}}=1)\leqslant 0. \\ \end{array}\end{eqnarray} \tag{ 42 }$$

This is more restrictive on both sides. The improvement is large when $P({{A}_{\infty }}={{B}_{\infty }}=1)$ is close to $P({{A}_{i}}$ and B_j det.), i.e., when the polarizers have low loss. Similarly to CHSH with non-ideal polarizing filters and fair sampling, the remaining problem is polarizer efficiency. If the polarizer loss is zero (ideal polarizers), there is a violation as soon as the detection efficiency is greater than zero, under the no-enhancement assumption.

To illustrate this, a CHSH-like inequality can be derived from inequality (42). If the non-detection events are assigned the value 0 and we assume no-enhancement, the inequality reads

$$\begin{eqnarray}&&\left| E\left( {{A}_{1}}{{B}_{1}} \right)+E\left( {{A}_{1}}{{B}_{2}} \right) \right|+\left| E\left( {{A}_{2}}{{B}_{1}} \right)-E\left( {{A}_{2}}{{B}_{2}} \right) \right|\leqslant 2P({{A}_{\infty }}={{B}_{\infty }}=1).\end{eqnarray} \tag{ 43 }$$

The right-hand side is lower than in the usual CHSH inequality, but so are the correlations to the left because of the lowered efficiency. Ideal polarizers correspond to $P({{A}_{\infty }}={{B}_{\infty }}=1)=P({{A}_{i}}$ and B_j det.), and under no-enhancement we arrive at

$$\begin{eqnarray}&&\begin{array}{l} |E\left( {{A}_{1}}{{B}_{1}}|{{A}_{1}}\ {\rm and}\ {{B}_{1}}\ {\rm det} . \right)+E\left( {{A}_{1}}{{B}_{2}}|{{A}_{1}}\ {\rm and}\ {{B}_{2}}\ {\rm det} . \right)| \\ \quad +\left| E\left( {{A}_{2}}{{B}_{1}}|{{A}_{2}}\ {\rm and}\ {{B}_{1}}\ {\rm det} . \right)-E\left( {{A}_{2}}{{B}_{2}}|{{A}_{2}}\ {\rm and}\ {{B}_{2}}\ {\rm det} . \right) \right|\leqslant 2, \\ \end{array}\end{eqnarray} \tag{ 44 }$$

which is violated by the quantum-mechanical predictions as soon as the efficiency is non-zero.

There is a third assumption that can be made: that the subset of observed events does not vary with the settings (Larsson 1998a, Berry et al 2010). If this subset remains constant, the addition in equation (27) can be performed, and the inequality (44) again holds. These three different assumptions are distinct and none of the three implies the others. Comparing all three, the no-enhancement assumption applies pointwise in the sample space, the assumption of constant set of detection applies on the level of events (subsets in sample space), and the fair sampling assumption applies on the output statistics. Pointwise or event-wise assumptions in the sample space may be thought to be more restrictive than the latter statistical assumption, but the difference between these three is small. In recent experiments, the fair sampling assumption seems to be the most popular.

Since 82.83% efficiency is very difficult to achieve in photonic experiments, improvements are needed. One important improvement was found in Eberhard (1993) who rederived the CH inequality (40) for detection counts, now sometimes called the Eberhard inequality. His approach was to assume that the detection efficiency is known (constant, and independent detections), and from that optimize the quantum state to maximize the violation. It turns out that the maximal violation at non-ideal detection probability occurs for a non-maximally entangled state. Such a violation can be obtained as soon as the efficiency is above $2/3\approx 66.67\%$ (Eberhard 1993, Larsson and Semitecolos 2001). One example of a state that violates the CH inequality at efficiency lower than 82.83% is the one used in the 'Hardy paradox' (Hardy 1993). As the efficiency tends to 2/3, the optimal state tends to the product state $\mid {{A}_{1}}={{B}_{1}}=-1\rangle$. The probabilities in the CH inequality concern $+1$ outcomes, so approaching that product state means that all the probabilities will approach zero. This will lower the noise tolerance in the test, so that an experimenter aiming to use a non-maximally entangled state needs to choose it carefully to balance efficiency demands against noise tolerance. Finding the optimal state is a simple matter of numerical optimization, as in Eberhard (1993), or solving a fourth-degree polynomial equation (Lima et al 2012).

Two recent photonic experiments (Giustina et al 2013, Christensen et al 2013) report violation of the CH inequality, closing the efficiency loophole. Both estimate their efficiencies to close to 75%. Still, both are vulnerable to the locality loophole, but this makes photonic systems the only system where both the efficiency and the locality loophole has been closed, in separate experiments (given Weihs et al 1998). It should be added that the experiment in Giustina et al (2013) is also vulnerable to the memory (setting prediction) loophole of subsection 2.2, because the settings are not chosen randomly, nor switched rapidly. There also remains to discuss another loophole, see subsection 3.5 below.

The bound can be improved even further by going to, for example, more sites. A first example of this is the GHZ paradox (Greenberger et al 1989) for which the efficiency bound is 75% (Larsson 1998b). The CH inequality can be generalized to many sites and the corresponding bound tends to 50% in the infinite limit (Larsson and Semitecolos 2001). Using a different family of inequalities, the bound approaches zero for infinitely many sites (Buhrman et al 2003). Another alternative is to go to higher dimension, where the bound also can be made to go to zero (Massar 2002). Unfortunately, these extreme bounds are impractical to use, e.g., the approach of Massar needs 1600-dimensional states to improve over the CHSH inequality, and is quite sensitive to noise at that point. A more practical example is Vértesi et al (2010) where a bound of 61.8% is reached using four-dimensional systems.

One possibility is to use a setup where the system at one site is different from that at the second site. An example would be to use an ion in an ion trap at the first site, giving high detection efficiency ${{\eta }_{A}}$ there, and a photon at the second site, making it possible to separate the two sites well but at the cost of low efficiency ${{\eta }_{B}}$. In this case, the high ${{\eta }_{A}}$ makes the bound lower for ${{\eta }_{B}}$. If ${{\eta }_{A}}=100\%$, the CH inequality can be violated as soon as ${{\eta }_{B}}\gt 50\%$, a significant improvement (Cabello and Larsson 2007). This can be improved further by using more settings, e.g., for the I₃₃₂₂ inequality (Collins and Gisin 2004) a violation is obtained as soon as ${{\eta }_{B}}\gt 43\%$ (Brunner et al 2007). A final interesting variation is to allow different efficiency between the $\pm 1$ channels at each local site (Garbarino 2010). This analysis makes it possible to use more exotic systems like strangeness measurements on kaons, which may enable high efficiency setups.

There are many alternatives to improve the bounds by choosing system properly, but it seems that currently low-dimensional systems with few sites are the most popular, both symmetric and asymmetric with non-maximal entanglement. This is possibly because entanglement generation can be performed efficiently at high rate, giving a high enough violation in these systems.

We now turn to another reason that events are missing from the data used to calculate correlations: determining which local events that form pairs. The problem of pair identification is especially pronounced in continuously pumped photonic experiments, but is in principle present in all experiments that have rapid repetition in the same physical system. Relative timing is often used to identify pairs: if two detections are close in time they are 'coincident,' otherwise they are not. The obtained correlations are again conditional correlations, here conditioned on coincidence,

$$\begin{eqnarray}&&E\left( {{A}_{i}}{{B}_{j}}|{\rm coinc}{\rm .\ for\ }{{A}_{i}}\ {\rm and}\ {{B}_{j}} \right).\end{eqnarray} \tag{ 45 }$$

And again, conditional correlations do not add if conditioning over different subsets. This results in the 'coincidence' loophole first discussed in Larsson and Gill (2004). In that paper a simple local realist model with varying delays is constructed, that has single-particle efficiency of 100% and a conditional coincidence probability of 87.87%, but still gives the quantum value $2\sqrt{2}$ in the CHSH inequality, when conditioning on coincidence. This is quite remarkable, given the general validity of the 82.84% bound of inequality (38).

The reason for the high coincidence probability is that the coincidence loophole concerns a joint property: if two given events form a pair or not. For comparison, the efficiency loophole concerns a local property: if an event occurs or not. The two belong to the same class of loopholes, because the root of the problem is experimental runs that do not count in the calculated correlations, but the coincidence loophole is more difficult to handle because it concerns a joint property of the two events. This can be made explicit as follows. Under local realism, detection times are random variables $T_{i}^{A}(\lambda )$ and $T_{j}^{B}(\lambda )$ that are real-valued, and only depend on $\lambda$ and the local setting. A coincidence occurs if these are close enough, say at a maximum distance $\tau /2$ from each other (making 'the coincidence window width' equal $\tau$). In terms of $\lambda$, this corresponds to the subensemble

$$\begin{eqnarray}&&\left\{ \lambda :{\rm coinc}{\rm .\ for\ }{{A}_{i}}(\lambda )\ {\rm and}\ {{B}_{j}}(\lambda ) \right\}=\left\{ \lambda :\left| T_{i}^{A}(\lambda )-T_{j}^{B}(\lambda ) \right|\leqslant \frac{\tau }{2} \right\}.\end{eqnarray} \tag{ 46 }$$

The subensemble in the efficiency loophole where both detections occur has the simpler structure

$$\begin{eqnarray}&&\left\{ \lambda :{{A}_{i}}(\lambda )\ {\rm det} .{\rm and}\ {{B}_{j}}(\lambda )\ {\rm det} . \right\}=\left\{ \lambda :{{A}_{i}}(\lambda )\ {\rm det} . \right\}\cap \left\{ \lambda :{{B}_{j}}(\lambda )\ {\rm det} . \right\}.\end{eqnarray} \tag{ 47 }$$

The latter set is an intersection of two local parts, while the former cannot be separated into local parts. Also for the coincidence case, the underlying model is local, but the process of identifying coincidences makes the subensemble (46) depend on both settings i and j in such a way that it cannot be split into local parts.

In this situation, one could make a 'fair coincidence' assumption that would remove the loophole: (paraphrasing the CHSH criterion) that if a pair of photons emerges from the beam-splitters, the probability of them registering as coincident is independent of the settings, or

$$\begin{eqnarray}&&P\left( {{A}_{i}}=\pm 1\cap {{B}_{j}}=\pm 1\cap {\rm coinc}{\rm .\ for\ }{{A}_{i}}\ {\rm and}\ {{B}_{j}} \right)=cP\left( {{A}_{i}}=\pm 1\cap {{B}_{j}}=\pm 1 \right).\end{eqnarray} \tag{ 48 }$$

Again, a possible mental image might be $({{A}_{i}},{{B}_{j}})=(\pm 1,\pm 1)$ denoting coincident emergence from an ideal polarizing beamsplitter, and subsequent delays in the detectors. Using the fair-coincidence assumption, and the same reasoning as above, the CHSH inequality applies to the underlying local realist model for the unobserved ideal outcomes, so that a CHSH inequality applies for the coincidences,

$$\begin{eqnarray}&&\begin{array}{l} |E\left( {{A}_{1}}{{B}_{1}}|{\rm coinc}{\rm .\ for\ }{{A}_{1}}\ {\rm and}\ {{B}_{1}} \right)+E\left( {{A}_{1}}{{B}_{2}}|{\rm coinc}{\rm .\ for\ }{{A}_{1}}\ {\rm and}\ {{B}_{2}} \right)| \\ \quad +\left| E\left( {{A}_{2}}{{B}_{1}}|{\rm coinc}{\rm .\ for\ }{{A}_{2}}\ {\rm and}\ {{B}_{1}} \right)-E\left( {{A}_{2}}{{B}_{2}}|{\rm coinc}{\rm .\ for\ }{{A}_{2}}\ {\rm and}\ {{B}_{2}} \right) \right|\leqslant 2. \\ \end{array}\end{eqnarray} \tag{ 49 }$$

Also this is violated by the quantum-mechanical predictions as soon as the efficiency is non-zero.

Deriving an inequality that is valid without the fair-coincidence assumption is more complicated than for the efficiency loophole. Here, it is not possible to assign an outcome for the missing data, because this will give a non-local assignment. The assigned outcome will depend on the remote setting, because of the reference to the joint property (46) of the two events. Therefore, another technique needs to be employed that avoids value assignment to the outcomes. This is done in Larsson and Gill (2004) and the method is similar to that used in Larsson (1998a) that avoids using assigned values for non-detections. The result is the generally valid CHSH-like inequality

$$\begin{eqnarray}&&\begin{array}{l} |E\left( {{A}_{1}}{{B}_{1}}|{\rm coinc}{\rm .\ for\ }{{A}_{1}}\ {\rm and}\ {{B}_{1}} \right)+E\left( {{A}_{1}}{{B}_{2}}|{\rm coinc}{\rm .\ for\ }{{A}_{1}}\ {\rm and}\ {{B}_{2}} \right)| \\ \quad +\left| E\left( {{A}_{2}}{{B}_{1}}|{\rm coinc}{\rm .\ for\ }{{A}_{2}}\ {\rm and}\ {{B}_{1}} \right)-E\left( {{A}_{2}}{{B}_{2}}|{\rm coinc}{\rm .\ for\ }{{A}_{2}}\ {\rm and}\ {{B}_{2}} \right) \right|\leqslant \frac{6}{\gamma }-4, \\ \end{array}\end{eqnarray} \tag{ 50 }$$

where

$$\begin{eqnarray}&&\gamma =\mathop{{\rm min} }\limits_{i,j}P\left( {\rm coinc}.{\rm for\ }\ {{A}_{i}}\ {\rm and}\ {{B}_{j}} \right).\end{eqnarray} \tag{ 51 }$$

This probability is not so easy to estimate from experimental data, but it has been conjectured (Larsson and Gill 2004) that the same bound holds for the relevant conditional coincidence efficiency

$$\begin{eqnarray}&&\eta =\mathop{{\rm min} }\limits_{i,j}\left\{ P\left( {\rm coinc}.{\rm for\ }\ {{A}_{i}}\ {\rm and}\ {{B}_{j}}|{{B}_{j}}\ {\rm det} . \right),\ P\left( {\rm coinc}.{\rm for\ }\ {{A}_{i}}\ {\rm and}\ {{B}_{j}}|{{A}_{i}}\ {\rm det} . \right) \right\}.\end{eqnarray} \tag{ 52 }$$

This is a measure of the 'apparent efficiency', because it can be less than 1 even if the detectors have ideal efficiency. Again, it is important that the apparent efficiency of the entire experiment is taken into account.

Also here, an event-ready setup can be used to remove the difficulty of estimating $\gamma$. In such a setup, coincidence is determined by closeness to the event-ready indication time, not by small time difference to the remote detection. The result is well-defined experimental runs that are independent of the measurement settings, similar to the detector-efficiency case. Then, event-ready signals from the source that are not coincident with detections at the measurement sites can be interpreted as missing detections, locally assigned some value (say 0), and included in a detector-efficiency-like analysis. In this situation, we recover

$$\begin{eqnarray}&&\begin{array}{l} |E\left( {{A}_{1}}{{B}_{1}}|{\rm coinc}{\rm .\ for\ }{{A}_{1}}\ {\rm and}\ {{B}_{1}} \right)+E\left( {{A}_{1}}{{B}_{2}}|{\rm coinc}{\rm .\ for\ }{{A}_{1}}\ {\rm and}\ {{B}_{2}} \right)| \\ \quad +\left| E\left( {{A}_{2}}{{B}_{1}}|{\rm coinc}{\rm .\ for\ }{{A}_{2}}\ {\rm and}\ {{B}_{1}} \right)-E\left( {{A}_{2}}{{B}_{2}}|{\rm coinc}{\rm .\ for\ }{{A}_{2}}\ {\rm and}\ {{B}_{2}} \right) \right|\leqslant \frac{4}{\eta }-2. \\ \end{array}\end{eqnarray} \tag{ 53 }$$

Another simpler option is to use fixed time slots. This is natural in a pulsed-pump experiment, but also possible in a continuously pumped experiment. The important property is well-defined experimental runs that are independent of the measurement settings. Then, coincidence is determined by presence in the same time-slot, not by small time difference to the remote detection. Slots that are empty can locally be assigned some value (say 0), and the system can be analyzed in a detector-efficiency-like manner (Larsson et al 2014). Note that there actually is something to prove here, to re-establish (53): that delays on both sides do not influence the bound negatively, raising it. A CH-like inequality that includes restriction to coincidences can also be derived by this method, and it reads

$$\begin{eqnarray}&&\begin{array}{l} P({{A}_{1}}={{B}_{1}}=1\cap {\rm coinc}.{\rm for}\ {{A}_{1}}\ {\rm and}\ {{B}_{1}}) \\ \quad +P({{A}_{1}}={{B}_{2}}=1\cap {\rm coinc}.{\rm for}\ {{A}_{1}}\ {\rm and}\ {{B}_{2}}) \\ \quad +P({{A}_{2}}={{B}_{1}}=1\cap {\rm coinc}.{\rm for}\ {{A}_{2}}\ {\rm and}\ {{B}_{1}}) \\ \quad -P({{A}_{2}}={{B}_{2}}=1\cap {\rm coinc}.{\rm for}\ {{A}_{2}}\ {\rm and}\ {{B}_{2}}) \\ \quad -P({{A}_{1}}=1)-P({{B}_{1}}=1)\leqslant 0. \\ \end{array}\end{eqnarray} \tag{ 54 }$$

The third and final method to avoid using the fair-coincidence assumption is to use different-sized time windows for the different coincidence probabilities. This method is most easily applied on inequalities that contain probabilities directly, since it is quite complicated to handle this in correlation-based inequalities. For example, in the CH inequality the time windows can be arranged as follows:

$$\begin{eqnarray}&&\begin{array}{l} {\rm coinc}.{\rm for}\ {{A}_{1}}\ {\rm and}\ {{B}_{1}}:\ \left| T_{1}^{A}-T_{1}^{B} \right|\leqslant \frac{\tau }{2}, \\ {\rm coinc}.{\rm for}\ {{A}_{1}}\ {\rm and}\ {{B}_{2}}:\ \left| T_{1}^{A}-T_{2}^{B} \right|\leqslant \frac{\tau }{2}, \\ {\rm coinc}.{\rm for}\ {{A}_{2}}\ {\rm and}\ {{B}_{1}}:\ \left| T_{2}^{A}-T_{1}^{B} \right|\leqslant \frac{\tau }{2}, \\ {\rm coinc}.{\rm for}\ {{A}_{2}}\ {\rm and}\ {{B}_{2}}:\ \left| T_{2}^{A}-T_{2}^{B} \right|\leqslant \frac{3\tau }{2}. \\ \end{array}\end{eqnarray} \tag{ 55 }$$

Then, the subensemble that gives coincidences for all three of ${{A}_{1}}{{B}_{1}}$, ${{A}_{1}}{{B}_{2}}$, and ${{A}_{2}}{{B}_{1}}$ will also give a coincidence for ${{A}_{2}}{{B}_{2}}$. This will also enable inequality (54).

A few recent experiment address the coincidence loophole explicitly, but because of inequality (53), all performed pulsed-pump optical experiments do this implicitly: they are not vulnerable to the coincidence loophole, but only the efficiency loophole. In Agüero et al (2012) a pulsed-pump optical system is used to further probe four different coincidence properties of local realist models for their system, but they cannot address the coincidence loophole in Bell tests as such because of the pulsed pump: their setup is simply not vulnerable to the loophole. A pulsed-pump setup is used in Christensen et al (2013), so that it is free of the coincidence loophole, in addition to the efficiency loophole. Finally, the continuously pumped optical experiment by Giustina et al (2013) is free of the efficiency loophole, and also of the coincidence loophole as shown in Larsson et al (2014).

Both the efficiency loophole and the coincidence loophole can be viewed as caused by deficiencies in the experimental equipment. Low overall experimental efficiency or coincidence probability can in principle be counteracted by improving the experimental setup. We will now turn to a different setup, for which the problem is built in at a deeper level and cannot so easily be removed.

The experiment in question is the Franson interferometer (1989), see figure 2. The setup uses a source that emits time-correlated photons at unknown moments in time, and two unbalanced Mach–Zehnder interferometers. The interferometers should have a path difference that is large enough to prohibit first-order interference, which means that the difference must be larger than the coherence time of the photons emitted from the source. When this is the case, the probability is equal for a photon to emerge from each port of the final beamsplitter.

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Now, the interferometer path-differences should be as equal as possible; the path-difference difference (repetition intended) should be small enough to make photon pairs where both photons 'take the long path' and both photons 'take the short path' indistinguishable. Quotation marks are used here to remind the reader that photons are not particles, but quantum objects and as such, do not take a specific path. Then, there are two possibilities for the photons to reach the detectors, and these interfere at the second beamsplitter.

If the path-difference difference is smaller than the coherence length of the pump photons, the phase difference between the two wave-functions that meet on the second beam splitter is stable. And if the total phase delay $a+b=0$, then a photon emerging in the $+1$ channel on the left is always accompanied by a photon emerging in the $+1$ channel on the right, and the same for the $-1$ channels. As a function of the total phase delay, we have

$$\begin{eqnarray}&&\left\langle {{A}_{a}}{{B}_{b}}|{\rm coinc}{\rm .\ for\ }{{A}_{a}}\;{\rm{and}} \;{{B}_{b}} \right\rangle ={\rm cos} (a+b).\end{eqnarray} \tag{ 56 }$$

Thus, the interference is not visible as a change in output intensity, as in first-order interference, but instead as a sinusoidal correlation of the outputs. And a sinusoidal correlation immediately points to violation of the CHSH inequality.

If the photons are not coincident, the delays are different. Then one photon took the long path and the other took the short, and the emission time can be calculated easily as the early detection time minus the short path length divided by the speed of light. In this case, there is only one possible way that the photons could have reached the detectors, and there will be no interference. Therefore, these events are not useful in our setting and are discarded.

And this is precisely the problem: we postselect events based on coincidence, as in the coincidence loophole above. But here, the selection is built into the setup, rather than a property of the equipment. It is there even with ideal equipment. And indeed, even though the correlation is sinusoidal and seems to violate the CHSH inequality, it does not. Even with a pulsed source at the appropriate rate, the relevant inequality with conditioning on coincidences is (53) with $\eta =50\%$ which means that it reads

$$\begin{eqnarray}&&\begin{array}{l} |E\left( {{A}_{1}}{{B}_{1}}|{\rm coinc}{\rm .\ for\ }{{A}_{1}}\ {\rm and}\ {{B}_{1}} \right)+E\left( {{A}_{1}}{{B}_{2}}|{\rm coinc}{\rm .\ for\ }{{A}_{1}}\ {\rm and}\ {{B}_{2}} \right)| \\ \quad +\left| E\left( {{A}_{2}}{{B}_{1}}|{\rm coinc}{\rm .\ for\ }{{A}_{2}}\ {\rm and}\ {{B}_{1}} \right)-E\left( {{A}_{2}}{{B}_{2}}|{\rm coinc}{\rm .\ for\ }{{A}_{2}}\ {\rm and}\ {{B}_{2}} \right) \right|\leqslant 6, \\ \end{array}\end{eqnarray} \tag{ 57 }$$

which is not a useful bound. Indeed, there exists a local realist model that gives the same outputs as quantum mechanics predicts (Aerts et al 1999), for ideal experimental equipment, at all values of a and b. The root of the problem is the built-in postselection; this is called the 'postselection' loophole, or sometimes the geometric loophole.

As with the detection and coincidence loopholes this can be avoided in a number of ways. The first is to assume that the photon really does take one path through the interferometer (Franson 1999, 2009). This assumption of 'path realism' implies that the photon will travel either through the long path, or through the short path, as decided at the first beamsplitter. The decision needs to be independent of the phase-delay setting to ensure that the subensembles are the same, so that the conditional correlations can be added directly. This independence could be part of the assumption, but it is better to arrange the experiment so that independence is ensured by locality. Then, the choice of path and the choice of phase-delay setting must be space-like separated, and to enforce this the experimenter needs a large distance between the beamsplitter and the phase-delay internally in each interferometer. In short, he or she needs to enforce locality inside the interferometer, instead of between measurement sites and the source, as in the standard Bell setup. If this is done and path realism is assumed, one can re-establish inequality (49) with the bound 2, and a quantum violation.

A second alternative is to use the elements of reality that are at hand. These are: the measurement outcomes $\pm 1$ and the time of emission. The latter is an EPR element of reality because the entire interferometer can be removed at one site without affecting the other site, and the emission time can be calculated from the detection time there. This means that the emission time of the other photon of the pair can be predicted without disturbing it. Furthermore, the delay must be specified by the local realist model (Aerts et al 1999), and there are two equally large subensembles inside the coincidence subensemble: early–early events, and late–late events. The bound for the early–early events cannot be lowered, but the late–late bound for the CHSH inequality is two if the system is set up with some care. In short, the phase-delay needs to be as close to the detectors as possible; for details see Aerts et al (1999), Jogenfors and Larsson (2014). This needs no auxiliary assumptions and no space-like separation of beam-splitter and phase delay in the same interferometer. Since the early–early and late–late events occur with the same probability, we arrive at

$$\begin{eqnarray}&&\begin{array}{l} |E\left( {{A}_{1}}{{B}_{1}}|{\rm coinc}{\rm .\ for\ }{{A}_{1}}\ {\rm and}\ {{B}_{1}} \right)+E\left( {{A}_{1}}{{B}_{2}}|{\rm coinc}{\rm .\ for\ }{{A}_{1}}\ {\rm and}\ {{B}_{2}} \right)| \\ \quad +\left| E\left( {{A}_{2}}{{B}_{1}}|{\rm coinc}{\rm .\ for\ }{{A}_{2}}\ {\rm and}\ {{B}_{1}} \right)-E\left( {{A}_{2}}{{B}_{2}}|{\rm coinc}{\rm .\ for\ }{{A}_{2}}\ {\rm and}\ {{B}_{2}} \right) \right|\leqslant 3, \\ \end{array}\end{eqnarray} \tag{ 58 }$$

which is better than inequality (57), but not good enough. Fortunately, Pearle (1970) and Braunstein and Caves (1990) come to our rescue. These papers contain a class of generalized Bell inequalities known as 'chained' Bell inequalities, with more terms, e.g., the six-term inequality

$$\begin{eqnarray}&&\begin{array}{l} |E\left( {{A}_{1}}{{B}_{1}} \right)+E\left( {{A}_{1}}{{B}_{2}} \right)|+\left| E\left( {{A}_{2}}{{B}_{2}} \right)+E\left( {{A}_{2}}{{B}_{3}} \right) \right| \\ \qquad +\left| E\left( {{A}_{3}}{{B}_{3}} \right)-E\left( {{A}_{3}}{{B}_{1}} \right) \right|\leqslant 4. \\ \end{array}\end{eqnarray} \tag{ 59 }$$

Using this for the late–late subensemble and the trivial bound 6 for the early–early, we obtain

$$\begin{eqnarray}&&\begin{array}{l} \left| E\left( {{A}_{1}}{{B}_{1}}|{\rm coinc}{\rm .\ for\ }{{A}_{1}}\ {\rm and}\ {{B}_{1}} \right)+E\left( {{A}_{1}}{{B}_{2}}|{\rm coinc}{\rm .\ for\ }{{A}_{1}}\ {\rm and}\ {{B}_{2}} \right) \right| \\ \qquad +\left| E\left( {{A}_{2}}{{B}_{2}}|{\rm coinc}{\rm .\ for\ }{{A}_{2}}\ {\rm and}\ {{B}_{2}} \right)+E\left( {{A}_{2}}{{B}_{3}}|{\rm coinc}{\rm .\ for\ }{{A}_{2}}\ {\rm and}\ {{B}_{3}} \right) \right| \\ \qquad +\left| E\left( {{A}_{3}}{{B}_{3}}|{\rm coinc}{\rm .\ for\ }{{A}_{3}}\ {\rm and}\ {{B}_{3}} \right)-E\left( {{A}_{3}}{{B}_{1}}|{\rm coinc}{\rm .\ for\ }{{A}_{3}}\ {\rm and}\ {{B}_{1}} \right) \right|\leqslant 5. \\ \end{array}\end{eqnarray} \tag{ 60 }$$

The quantum-mechanical prediction is $6{\rm cos} \pi /6=5.196$, so this gives a violation. The above inequality is more sensitive to noise than the usual Bell inequality; this is the price one has to pay for avoiding the assumption of path realism. Adding more terms in the chain will decrease the noise sensitivity but the minimum is reached already at ten terms (Jogenfors and Larsson 2014).

A third alternative method to remove the loophole is to change the experiment setup so that path really is an EPR element of reality. There have been three proposals of how to do this. The first by Strekalov et al (1996) uses polarization-entangled photons and a polarizing beamsplitter to force the two paths to be the same: the only alternatives are long–long and short–short pairs, which means all pairs will be coincident. There is no postselection anymore, so of course, there is no postselection loophole (but other loopholes will still apply).

A second proposal by Brendel et al (1999) is to use a pulsed source with a similar unbalanced Mach–Zehnder interferometer preceding the pump, and moving mirrors in place of the first beamsplitters at the analyzer stations. This ensures that photon pairs created by the non-delayed pump pulse take the long path at the analyzers, and photon pairs created by the delayed pump pulse take the short path at the analyzers. The effect is the same as for the polarization scheme above, that all pairs will be coincident. There is no postselection anymore, so consequently, there is no postselection loophole (but other loopholes will still apply).

This is known as time-bin entanglement, but there is one caveat here. Standard time-bin entanglement experiments do not use active mirrors, but instead ordinary beamsplitters, so that the analyzer stations are actually exactly the same as in Fransonʼs scheme. This is suggested already in Brendel et al (1999) which only mentions a 50% loss as detrimental effect, which is true if the loss occurs at the source; then there is no adverse effects besides the loss. However, at the analyzer stations, the effect is random delays in the detections of the photons similar to the original Franson proposal, meaning that the 50% loss is due to postselection and this re-enables the postselection loophole. Experiments that want to be loophole-free must use active moving mirrors, and would constitute 'genuine' time-bin entanglement.

The third and final method to ensure that path is an EPR element of reality was proposed in Cabello et al (2009), and the suggestion is to interchange the long paths of the analyzer stations. Then, the long path from the first beam-splitter is directed towards the opposite station. These first beamsplitters should now be considered part of the source and located at the source. With this configuration, path is an EPR element of reality because it can be remotely predicted from the result of a local measurement. If only one photon emerges from the paths, the other photon of the pair can be predicted to emerge from the corresponding path at the remote station, giving a coincidence and coincident events will behave as in the Franson setup, giving interference. A pair for which there is no coincidence will behave differently: both photons of the pair will appear at one of the analyzer stations, and can be discarded by an entirely local process. There is postselection, but no loophole; this setup gives 'genuine' energy-time entanglement (but other loopholes will still apply). An experimental violation of the CHSH inequality in this system was reported in Lima et al (2010) at a distance of 1 m, while non-maximal entanglement was created in Vallone et al (2011). Finally, in Cuevas et al (2013) one of the interferometers had 1 km long arms, and was still stabilized well enough to give a violation of the CHSH inequality.