7.1. Entanglement
Entanglement is a quantum property that may be easily defined within the HS formalism, but the definition does not provide any intuitive picture. It appears in systems with several degrees of freedom when the total state vector of the system cannot be written as a product of vectors associated with one degree of freedom each. In formal terms, a typical entangled state is the following
where 1 and 2 correspond to two different degrees of freedom, usually belonging to different subsystems that may be placed far from each other, and
are complex numbers. The essential condition is that the state Equation (
54) cannot be written as a single product, that is the sum cannot be reduced to just one term via a change of basis in the Hilbert space. Entanglement appears as a specifically quantum form of correlation, which is claimed to be dramatically different from the correlations that appear in all other branches of science, including classical physics.
The relevance of entanglement was stressed in 1935 by Schrödinger [
29] 1935, who wrote that it is not one but
the characteristic trait of quantum mechanics. He also pointed out the difficulty to understand entanglement with his celebrated example of the cat suspended between life and death. Indeed, if one assumes that quantum mechanics is complete, i.e., that a state-vector like Equation (
54) represents a pure state, then a realistic interpretation is impossible because we are confronted with consequences in sharp contradiction with both the intuition and a well established pardigm, namely that complete information about the whole requires complete information about every part. In fact, we are compelled to believe that a state-vector like Equation (
54) represents complete information about the state of the system, but incomplete information about every one of the subsystems. Indeed, according to quantum theory, the state of the first subsystem should be obtained by taking the partial trace with respect to the second subsystem, leading to the following density matrix (assuming all state-vectors normalized)
The density matrix represents a mixed state, where the information is incomplete, that is we only know the probabilities, for the first subsystem to be in the different states .
An important result is that entanglement is a necessary condition for the violation of Bell inequalities, as discussed in the following [
30].
7.2. Bell Inequalities
It is common wisdom that any correlation between two events, say
A and
B, is either a causal connection or it derives from a common cause. There is causal connection if
A is the cause of
B or
B the cause of
A, and a common cause means that there is another event
C that causes both
A and
B. In formal terms, we may write either
or
for causal connection,
and
for common cause. In 1965, John Bell allegedly proved that the said common wisdom is not true according to quantum mechanics. In fact, he derived inequalities [
31] that he claimed to be necessary conditions for the existence of a common cause, and pointed out possible experiments where the inequalities would be violated.
Typical experimental tests of the Bell inequalities consist of preparing a system that produces pairs of signals, one of them going to an observer Alice and the other one to observer Bob. Alice may measure a dichotomic property
on her signal with the possible results
and in another run of the experiment she may measure
also with the possible results
Similarly, Bob may measure either
or
with the possible results
Alice and Bob may perform coincidence measurements of
and
After many runs of the experiment with identical preparations of the system, Alice may obtain from the frequencies the single probability,
, that the result in her measurement is 1, and similarly Bob may get the probability
. They may also obtain the probability
that both results are 1 in a coincidence measurement. Then, the following Bell inequality [
32]
should hold true.
The relevant fact is that quantum mechanics predicts violations of Bell inequalities in some cases. The contradiction has been named “Bells theorem”:
quantum mechanics is not compatible with local realism. Local realism is the assertion that all correlations in nature are either causal connections or derive from a common cause. The word “local” is introduced because a direct communication between Alice and Bob could produce results violating Equation (
56) which would invalidate the test. The possible communication is named local if any possible information travels with velocity not greater than the speed of light, whence locality should be better named “relativistic causality”. As a consequence, the crucial experiments must be performed so that coincidence measurements by Alice and Bob take place both within a time window
smaller than the distance between their measuring devices divided by the velocity of light, that is with spacelike separation in the sense of relativity theory.
Many experiments have been performed in the last 50 years in order to test Bell inequalities, with results that generally agree with quantum predictions, but there are loopholes for the proof that local realism is refuted. In particular, in most of the performed experiments, the spacelike separation is not guaranteed. The reader should consult the vast literature on the subject. See e.g., [
30,
33].
In the last decades, most tests of the inequalities have used entangled photon pairs produced via spontaneous parametric down conversion (SPDC). In
Section 7.6, I shall analyze a representative test similar to those providing for the first time the loophole-free violation of a Bell inequality [
34,
35]. The empirical violation is interpreted as a refutation of local realism, but I will show that the commented experiments may be interpreted as locally realistic even if a Bell inequality is violated. That is, I will prove that the Bell inequalities are not necessary conditions for local realism, contrary to common wisdom.
7.3. Spontaneous Parametric Down Conversion (SPDC)
SPDC has been the main source of entangled photon pairs from about 1980. In the following, I will study, within the quantum Hilbert space formalism (HS), the SPDC process and a simple experiment involving entangled photon pairs. I shall work in the Heisenberg picture where the obvervables evolve, see Equation (
58) below, but the state vector is fixed, in our case the vacuum state
In
Section 7.4, I shall pass to the WW formalism, which suggests an interpretation of SPDC experiments in terms of random variables and stochastic processes without any reference to photons.
SPDC is produced when a pumping laser impinges a crystal possessing nonlinear electric susceptibility. Radiation with several colors may be observed going out from the opposite side of the crystal. By means of appropriate apertures, two beams of the radiation may be selected, which in quantum language consist of a set of entangled photon pairs, one photon of every pair in each beam.
The HS theory of the process is as follows, with the simplification of taking only two radiation modes into account, having amplitudes
. Avoiding a detailed study of the physics inside the crystal, that may be seen elsewhere [
36,
37], we might describe the phenomenon with a model interaction Hamiltonian [
38], that is
when the laser is treated as a classically prescribed, undepleted and spatially uniform field of frequency
The interaction of the pumping laser with the incoming vacuum mode,
within the crystal produces a new field with amplitude
named “signal”. If the beams have been adequately chosen, that signal travels superposed to the vacuum field
after exiting the crystal. Similarly, the vacuum field
produces a field
named “idler”, that travels superposed to the vacuum field
.
As a result, the radiation fields at the crystal exit may be represented by
where the wavevectors
and
form a finite angle amongst them.The parameter
D is proportional to the interaction coefficient
A Equation (
57) and it depends also on the crystal size. In practice, it fulfils
The following equality holds for the frequencies of the selected beams
which is usually interpreted assuming that the signal and idler photons, with energies
and
were the result of the division of one laser photon with energy
. That is Equation (
59) is viewed as “energy conservation” in the splitting of laser photons. However, I interpret it as a condition of frequency matching, induced by the nonlinear susceptibility, with no reference to photons.
In the following, I will ignore the spacetime dependence, whence Equation (
58) will be written
These equations are the formal representation of entangled photon pairs in the Heisenberg picture of the HS formalism, and they show a strong correlation between the fields and .
As a simple application, I shall derive the quantum prediction for an experiment that consists of measuring the single and coincidence detection rates when the beams with fields Equation (
60) arrive at Alice and Bob detectors, respectively. It is convenient to get the quantum prediction in terms of the probability of detection,
P, in a given time window. Thus, if we divide the unit of time in a number
n of windows, the detection rate would be
. In the quantum HS formalism Alice, single detection probability is given by the following vacuum expectation (to order
where only one out of four terms contributes, but I have written two of them explicitly for clarity, and similar for Bob. It is easy to prove that the spacetime factors, explicit in Equation (
58), cancel.
The quantum prediction for the coincidence detection probability is
In our case, taking into account that
and
commute, both terms are equal and we have
The quantum predictions Equations (
61) and (
62) show that the correlation is the maximum possible, that is the coincidence detection rate equals the single rate of either Alice or Bob. In contrast, if there was no correlation we should have
. In any case, single and detection probabilities obviously must fulfil
and
In actual experiments, the predictions for real detectors should take into account the detection efficiency. If it is
equal for both detectors the prediction would be
confirmed in actual experiments.
Entanglement of the form Equation (
54) may be exhibited if we pass to the Schrödinger picture, where the evolution goes in the state. The appropriate representation of the joint quantum state of the radiation at Alice and Bob detectors is
which may be interpreted saying that the state of the radiation is entangled and consists of two terms Alice and Bob having one photon each in the second term and none of them having photons in the first term. I stress that in the HS of quantum theory, Equation (
63) represents a pure state, not a statistical mixture. It cannot be interpreted as a probability
of having two photons and a probability
of no photons. If
and
are the photon number (operator) observables for Alice and Bob in a given time window, the detection single probability will be
and a similar for
. From the two terms of
Equation (
63) we get four terms for the expectation but three of them do not contribute. The coincidence probability also consists of four terms, but only one contributes, namely
In summary, Equation (
63) exhibits entanglement between the vacuum and the two- photon state, as has been pointed out [
39].
7.4. Stochastic Interpretation of the Correlation Experiment
The quantum–mechanical prediction for the experiment commented in the previous section may be easily worked in the WW formalism. The Weyl transform of the field operators Equation (
60) are
Vacuum expectation in HS corresponds in WW to an average weighting by the vacuum probability distribution Equation (
20). However, the detection probabilities in WW cannot be obtained just taking averages of Equation (
64), but should be obtained from the Weyl transform of the HS vacuum expectations. For Alice single detection probability, the Weyl transform of Equation (
61) is
where Equation (
53) has been taken into account. I ignore two terms that do not contribute and are not relevant for the interpretation, similarly for Bob detection probability
. The result agrees with the prediction using HS, as it should because the WW formalism is an equivalent form of quantum theory for the radiation field.
The different signs in front of 1/2 in the two terms of Equation (
65) may seem strange. Of course, they appear in the Weyl transform of Equation (
61) because the former comes from the vacuum expectation of
which is zero but the latter from the vacuum expectation of
which is unity. However, in the WW formalism we are working with commuting amplitudes and the different ordering should not make any difference. We may understand intuitively the reason for the signs taking into account that the second term of Equation (
65) corresponds to the signal (it contains
) but the first term corresponds to vacuum modes that should not contribute to the detection and therefore should be removed. The addition of 1/2 in the signal term effectively multiplies the detection probability by 2. This is more difficult to understand intuitively and I will not comment further.
In order to derive the coincidence detection probability,
, we might proceed translating to the WW formalism the calculation made in
Section 7.3 using the HS formalism, which led to Equation (
62) (see [
28]). However, I will not do that but make a direct stochastic derivation of single and coincidence probabilities, which may allow greater understanding of the physics of the experiment. I will start from the fields Equation (
64) and proceed using classical laws and plausible assumptions for the correlations.
I shall start by proposing a model of detection. According to our assumptions, any photodetector in free space is immersed in an extremely strong stochastic radiation, infinite if no cut-off existed, see Equation (
20). Thus, how might we explain that detectors are not activated by the vacuum radiation? Firstly, the strong vacuum field is effectively reduced to a weaker level if
we assume that only radiation within some (small) frequency interval is able to activate a photodetector, that is the interval of sensitivity
. Actually, the frequency selection is quite common in radiation detection, for instance, when tuning radio or TV. The theoretical explanation of this fact is easy, that is detection takes place via resonance with some oscillator having the same characteristic frequency than the radiation to be detected. For instance, an appropriate electric circuit in case of radiowaves or a molecular resonator for visible light (e.g., molecules with a appropriate frequency of excitation inside the elements of color vision in our retina).
However, the problem is not yet solved because the signals involved in experiments may have intensities of the order of vacuum radiation in the said frequency interval, whence the detector would be unable to distinguish a signal from ZPF noise. Our assumption is that a detector may be activated only when the Poynting vector (i.e., the directional energy flux) of the incoming radiation is different from zero, including both signal and vacuum fields. To make a trivial comparison, we live immersed in air but its pressure is almost unnoticed except when there is strong wind producing an unbalanced force that pushes us towards a given direction.
Thus, a plausible hypothesis is that light detectors possess an active area, the probability of a photocount depending on the integrated energy flux crossing that area during some activation time, T. The assumption allows understanding why the signals, but not the vacuum fields, activate detectors. Indeed, the ZPF arriving at any point (in particular the detector) would be isotropic on average, hence the associated energy flux integrated over a large enough time would be very small because fluctuations are averaged out. Therefore, only the signal, which is directional, would produce a large integrated energy flux during the activation time, thus giving rise to photocounts. A problem remains because the integrated flux would not be strictly zero. Indeed, the integrated flux during a time integral T, divided by T would go to zero when T . Hence, we may predict the existence of some dark rate induced by vacuum fluctuations even at zero Kelvin. In summary, we are assuming that photocounts are not produced by an instantaneous interaction of the radiation field with the detector, but the activation requires some time interval, a fact well known by experimentalists but sometimes ignored by theoretitians.
After that, I will obtain the detection probabilities as averages of intensities derived from the fields Equation (
64). I will assume that the detection probability is proportional to the mean intensity arriving at the detector, taking the proportionality coefficient as a unit for simplicity. For the single detection by Alice, we get the detection probability as the average of the intensity arriving at her detector, that is
According our previous analysis, we should use time averages but we may assume that they are equal to ensemble averages, a kind of ergodic property. Equation (
66) has two intensity contributions, the former
coming from the signal and the latter
from the ZPF. On the other hand, if the laser pumping on the crystal was switched off, then the total intensity arriving at the Alice detector should be zero on the average, that is
where
is the intensity arriving at the detector from the source in place of the signal when there is no pumping. The intensity
comes from the vacuum fields and it may be derived from Equation (
64) putting
. From Equation (
67) the probability Equation (
66) becomes
where the second term correspond to the ZPF subtraction. This means that we should not expect any detection if there is no signal, a quite plausible result.
The radiation intensities may be obtained form the fields taking Equation (
64) into account, as follows
Equations (
66) to (
68) lead to
where we take into account that
and we have calculated the expectation of
taking the vacuum probability distribution Equation (
20) into account. The same is obtained for Bob single detection.
The result Equation (
69) agrees with both the HS and WW results, Equations (
61) and (
65), except for a factor 1/2. It is caused by our choice, unity, for the proportionality constant between field intensity and detection probability made in Equation (
66).
The coincidence detection probability for a given time window will be the average of the product of the field intensities whence the detection probability per time window is obtained as follows
As in the derivation of Equation (
69), the detection probability
should be zero when the pumping is off, whence we get
From Equations (
67), (
70) and (
71) and the plausible assumption that
and
are uncorrelated with the signals we obtain
The average of single intensities in Equation (
68) may be easily obtained taking the vacuum distribution Equation (
20) into account. We get
For the average of products of intensities we have
The terms with an odd number of amplitudes do not contribute (see Equation (
53)) whence we get the following sum of averages to order
We will show that the last term does not contribute whence, collecting all terms, Equation (
72) becomes
The reason why the last term of Equation (
76) does not contribute is that we cannot ignore the spacetime phase factors in this case, see Equation (
58). In fact,
comes from the intensity arriving at Alice, but
from the Bob intensity. In the former, we should include a phase
and in the latter
, these phases being uncorrelated. Therefore, in the average of the last term of Equation (
76) the phases give a nil contribution. In contrast, all other terms contain absolute values whence the phases disappear.
The results derived from a Equations (
69) and (
77), that have been obtained from the fields via our stochastic approach, reproduce the relevant result of the experiment, namely that there is a maximum positive correlation shown by the equality
which is also predicted by the HS results Equations (
61) and (
62) (with a factor 1/2 with respect to the latter as explained above).
The picture of the experiment in our approach is quite different from the picture in terms of photons suggested by the HS formalism. In HS, a few photons in the (usually pulsed) laser beam are assumed to split by the interaction with the nonlinear crystal, giving two photons each. The probability of producing an entangled photon pair by the splitting within a detection time is assumed to be of order , whence the simultaneous arrival of entangled photons at Alice and Bob happens for a small fraction of laser pulses. However, the detection of the photons conditional to the photon production, is assumed to occur with probability of order unity (say . The probability is named detection efficiency.
In our approach, the probability of photocounts by Alice or Bob does not factorize that way. Furthermore, the concept of photon does not appear at all, but there are continuous fluctuating fields including a real ZPF arriving at the detectors, which are activated when the radiation intensity is big enough.
7.5. Understanding Entanglement
The strong correlation exhibited by the comparison of Equations (
69) and (
77) is a consequence of the phenomenon of entanglement and it is labeled strange from a classical point of view. In our stochastic interpretation, it is due to the fact that the signal field
produced in the crystal is correlated with the ZPF field
that had entered the crystal, see Equation (
64); similarly for the correlation between the signal
and the ZPF field
. That is, the strong correlation appears because the same normal modes of the radiation appear in both fields,
and
, that go to Alice and Bob, respectively.
Now I shall stress the relevance of the vacuum fluctuations in order to understand the difference between the “classical correlation” and “entanglement”. In the evaluation of the averages in Equation (
76), we have taken the distribution of field amplitudes Equation (
20) into account giving
relations typical of a Gaussian distribution of the amplitudes. Now let us assume that we had used, instead of Equation (
20) a sure (i.e., not fluctuating) distribution, e.g.,
being Dirac delta. In this case, we had obtained
In this case, the result for the coincidence probability had been
(or
if we worked to order
Equation (
81) would mean that there was no correlation between Alice and Bob detections. In contrast, a strong positive correlation is obtained if we take into account the fluctuations. This happens when the field is assumed Gaussian, which leads to a stronger correlation, as may be realized comparing Equation (
78) with (
80), the former leading to Equation (
77) and the latter to (
81).
We conclude that the strong positive correlation associated with entanglement requires that
the fluctuations are correlated. That is, the high probability of coincidence detection requires a strong positive correlation between fluctuations of the fields arriving at Alice and Bob, respectively. This leads to a physical (realistic) interpretation as follows:
entanglement is a correlation between fluctuations of fields in distant places. In our example, the correlation of fluctuations involves the vacuum fields and might be labeled entanglement between a signal and the vacuum [
39], see Equation (
63).
7.6. The Violation of Bell Inequalities
The interpretation of SPDC experiments in terms of stochastic processes, including the vacuum fields, allows local models violating a Bell inequality. This contradicts the wide consensus that Bells is the unique local realistic formalism appropriate for experiments measuring correlations between distant parties. Our proof consists of exhibiting a local model for an experiment leading to predictions that violate a Bell inequality. In the construction of the model, we are free to fix the fields produced in the source, but then we should obtain the predictions using classical laws and determining the correlations as in Equations (
66), (
67), (
70) and (
71). I propose a model where the fields produced in the source correspond to vectors as follows
where
and
are unit vectors in two perpendicular directions, say horizontal and vertical.
Now I assume that
goes to Alice, who possesses a polarization analyzer at an angle
with the horizontal in front of her detector. Then the field arriving at the detector will have a component in the direction
given by
plus some amount of ZPF. I also assume that Bob has a polarization analyzer at an angle
with the horizontal in front of his detector. Hence the field arriving at his detector will be
plus some amount of ZPF. We also need the signal fields that would arrive at Alice and Bob detectors if the pumping laser was off, which may be easily derived from Equations (
83) and (
84) putting
. That is
We shall also define the fields carrying the signals by the difference, that is
The intensities will be the squared moduli of the corresponding fields Equations (
83) to (
87). Hence it is easy to get the single detection probability by Alice, that is
and a similar result for Bob, i.e.,
The calculation of the coincidence probability is more involved, although still straightforward. We shall use Equation (
72) that we rewrite for convenience
where from Equation (
85) it is trivial to obtain
In order to obtain the expectation
it is convenient to define partial intensities as follows
and similar for
. Then the desired expectation is
consisting of nine terms. One of them will be cancelled with
Equation (
88), the terms
and
will not contribute taking Equation (
53) into account, and
is of order
therefore negligible. Thus, to order
we have
In this form, it is easy to select the expectations
whence Equation (
92) gives
The term
leads to the same result, and the term
does not contribute by the same reason as the last term of Equation (
76), as commented in the paragraph after Equation (
77). From Equations (
88)–(
93) we finally obtain
The predictions of our model may violate the Bell inequality Equation (
56). In fact, we may consider an experiment where Alice measures with her detector when the polarizer is put at angles
or
and, similarly, Bob at angles
or
. The predicted probability of a single count by either is
The coincidence probability is given by Equation (
94) and the violation of the Bell inequality is produced if the following angles are chosen in the experiment
If this is inserted in Equation (
56) we get on the left side
and on the right side
that violates the Bell inequality.
That is, our model agrees with both the standard quantum predictions and a local realistic view of nature. We conclude that Bell inequalities are not necessary conditions for local realism, contrary to the current wisdom.