1. Distance and time dependence of an atomic entangled state

We may write a (spatial) electron wave function of a hydrogen-like molecule as (Griffiths, Reference Griffiths2008):

(2a) $$\Psi = 2^{ - 1/2} [\phi _{\rm A} ({\bf r}_1 )\phi _{\rm B} ({\bf r}_2 ) \pm \phi _{\rm B} ({\bf r}_1 )\phi _{\rm A} ({\bf r}_2 )$$

where ϕ _A(r ₁) is the spatial wave function for the first electron (position vector r ₁) and ϕ _B(r ₂) for the second electron etc. The symmetric (plus sign) form of Eq. (2a) typically represents the lower energy, bound state of an electron, enhancing the electron density in the middle between the two atoms A and B. To include spin we must enforce the requirement that the overall product wave function for space and spin is antisymmetric under particle exchange. This means that Eq. (2a) must be multiplied by the antisymmetric (minus) or symmetric (plus) combination of electron spin wave functions:

(2b) $$2^{ - 1/2} [\alpha (1)\beta (2) \pm \alpha (2)\beta (1)]$$

where α means spin +1/2 and β means spin −1/2. Using perturbation theory, the exchange interaction (‘exchange’ meaning interchange of spatial or spin coordinates of identical particles) in the hydrogen molecule is found to have two possible energies (van Vleck Reference Van Vleck1932):

(3) $$E_ \pm = E_0 + (C \pm J_{ex} )/(1 \pm B^2 )$$

where E ₊ is the spatially symmetric case and E ₋ the spatially antisymmetric case. B is the overlap integral 〈ϕ _A(r ₁)ϕ _B(r ₁)〉, C =  〈ϕ _A(r ₁)² Η^exɸ _B(r ₂)²〉 is the Coulomb integral and J _ex  = 〈ϕ _A(r ₁)ϕ _B(r ₂)Η^exɸ _A(r ₁)ϕ _B(r ₂)〉 the exchange integral, where the latter two depend on the inverse distance between the two nuclei (R _AB ⁻¹).

When spin is included, the total wave function is either the (+)(−) or the (−)(+) combinations of wave functions (2a) and (2b), corresponding to the spin singlet and triplet states of the hydrogen molecule, respectively. The triplet state is triply degenerate in the absence of external fields, with the three spin wave functions being α(1)α(2), β(1)β(2), and 2^−1/2[α(1)β(2) + α(2)β(1)]. The energy separation (Dirac, Reference Dirac1926; White, Reference White2007)

(4) $$E_ + - E_ - = 2(J_{{\rm ex}} - CB^2 )/(1-B^4 ) $$

falls off linearly with J _ex and thus approximately as R _AB ⁻¹ with the proton–proton separation. J _ex in fact decreases even more rapidly as the centers of ϕ _A and ϕ _B move apart and experience less interactive overlap with each other – C and B then also vanish. Note that while this description, which is the (covalent) Valence Bond model, is only approximate at short R _AB, it becomes a more exact description at large inter-nuclear separations, which is the case we are interested in.

When the energy difference between the singlet and the triplet states of the two separate hydrogen radicals goes to zero, any perturbation due to the surrounding universe could lead to a complete contamination of states, and annihilation of any entanglement.

In the complete absence of any other matter or external fields the zero angular momentum of the singlet state 2^−1/2[α(1) β(2) − α(2) β(1)] will be conserved so that each hydrogen radical will keep its respective spin angular momentum, whether α or β, for ever. However, with a QM interpretation, i.e. preserved full wave function, it is first at measurement (collapse of wave function) it is decided which particle will have α and which β. In all practical experiments there will be some surrounding matter (gas, liquid or solid state) that could cause admixture of singlet and triplet states: any external perturbation may thus give rise to a spin change, for example, via some spin-lattice kind of relaxation mechanism (Atkins et al. Reference Atkins, McLauchlan and Percival1973; Ardavan et al. Reference Ardavan, Rival, Morton, Blundell, Tyryshkin, Timco and Winpenny2007). Here temperature, reflecting somehow the system's degree of excitation, and the fact that there is generally a radiation field present associated with a certain temperature, can be an important parameter (Ninham & Daicic, Reference Ninham and Daicic1998; Wennerstrom et al. Reference Wennerstrom, Daicic and Ninham1999) – higher temperature observed to shorten the relaxation time. Typically in a molecular magnet system an electronic T ₁ relaxation time of 10⁶ ns at 2 K may drop to 10² ns at 10 K and so on, relaxation time empirically varying as T ⁻¹ (Bogani & Wernsdorfer, Reference Bogani and Wernsdorfer2008). In the solid state, the characteristic time for establishing thermal equilibrium between the lattice and the spin of ferromagnetic gadolinium was estimated to 0·1 ns at 45 K (Vaterlaus et al. Reference Vaterlaus, Beutler and Mair1991). Experiments with nitrogen-vacancy (N–V) centers in diamond indicate that N–V center ensembles may have spin-coherence times as long as 50 µs at room temperature (Epstein et al. Reference Epstein, Mendoza, Kato and Awschalom2005), but generally in liquids relaxation varies as T ⁻¹, typically from 0·1 ns at −40 °C to 0·01 ns at 50 °C (Zager & Freed, Reference Zager and Freed1982).

Another problem is that it is difficult to imagine how the dissociation energy, hugely exceeding the spin flip energy, may be delivered gently without affecting the spin. A vanishing entanglement may not exclude the possibility of some residual ‘magnetic memory’ of the original spin remaining, for example, if implanted into some solid state material, and potentially possible to pick up by sensitive instrumentation. It would, in principle, be anti-correlated, between the two hydrogen fragments for each split event. However, only in a passive, exclusive way, will there be any residual memory and, furthermore, spin-ups and spin-downs will statistically be at random for an ensemble of split ‘unsorted’ hydrogen atoms. The ensemble picture contrasts the putative dynamic correlation of an entangled pair of spin particles, which was the origin of Einstein's concern. Entanglement with 30 s lifetime of trapped ions has been demonstrated by Blatt & Wineland (Reference Blatt and Wineland2008).

Wennerström and Westlund have considered the problem of randomizing spin during passage of a Stern-Gerlach magnet (Wennerström & Westlund, Reference Wennerström and Westlund2013). They draw attention to an issue with Bell's theorem, the derivation of which was based on the assumption of only two possible spin values after exiting the analyzer. Their arguments for a wide spin distribution are supported by considering the T ₂ spin relaxations due to spin angular momentum transfer between particles and magnet, results that are found to be in agreement with the blurred pattern in original Stern-Gerlach data.

Some ground breaking discoveries on atoms and ions should be noted. Wineland and his group (Rowe et al. Reference Rowe, Kielpinski, Meyer, Sackett, Itano, Monroe and Wineland2001; Turchette et al. Reference Turchette, Wood, King, Myatt, Leibfried, Itano, Monroe and Wineland1998) demonstrated violation of Bell's inequality with material particles, using ⁹Be⁺ ions prepared in Bell-like singlet and triplet states, driven by stimulated Raman processes from pairs of laser beams in one of two geometries, an important parallel to the photon experiments that we shall discuss later. In a topical review Leggett (Reference Leggett2002) discusses limits of QM and specifically over what length scales QM has been tested – he points into some directions where QM might potentially break down, one being very small length scales (very high energies) and another, which could be also chemically and biophysically interesting, being short times and space-like separation in the sense of special relativity.

2. How Einstein (possibly) was thinking

The problem of entanglement is closely connected with the experiment. The entangled state has no meaning until somebody tries to probe it – then it is like opening a box to find out what spin one of the hydrogen radicals has, and then, puff!, the wave function collapses into one of the eigenstates. With this view, it is the act of making a measurement that forces the system to react irrespective of how far away the two parts are from each other. This is what Einstein saw as a spooky action at a distance! So where is the problem?

Already during the 5th Solvay Conference in 1927, Einstein expresses a view that clearly lies behind the 1935 EPR paper (Jammer, Reference Jammer1974). He considers two viewpoints of behavior for a particle passing a slit, and then hitting a screen:

• Viewpoint 1: ‘the de Broglie-Schrödinger waves do not represent one individual particle but rather an ensemble of particles distributed in space’…‘Thus |Ψ(r)|² expresses the probability density that there exists at r some particle of the ensemble.’

• Viewpoint 2 ‘QM is considered as a complete theory: each particle moving towards the screen is described as a wave packet which, after diffraction, arrives at a certain point on the screen where |Ψ(r)|² expresses the probability that one and the same particle shows its presence at r. As long as no localization has been effected, the particle must be considered as potentially present over the whole area of the screen; however, as soon as it is localized, a peculiar action-at-a-distance must be assumed to take place, which prevents the continuously distributed wave in space from producing an effect at two places on the screen.’ He also said: ‘the interpretation 2 of |Ψ(r)|², I think, contradicts the postulate of relativity.’

If we apply Einstein's Viewpoint 1 to the dissociating hydrogen molecules, the wave function does not refer to any specific pair of hydrogen atoms but to an ensemble of identically prepared pairs. For instance, if a beam of hydrogen atoms is passed through a Stern-Gerlach apparatus, one of the two emerging beams will be deflected up and contain only atoms with S _z  = (+1/2)h/2π while the other beam will only contain atoms with S _z  = (−1/2)h/2π. This experiment, which may better be called ‘state preparation’, as suggested by Bransden & Joachain (Reference Bransden and Joachain2000), can be immediately repeated by letting, e.g. the +1/2 beam pass a second Stern-Gerlach magnet oriented in the same way as the first. Then all +1/2 particles will pass, but none of those corresponding to −1/2. What is essential to note here, and which applies to how the Bell inequality is interpreted (see below), is that in our experiment we have prepared the beams of atoms in a reproducible way, and the measurement can be repeated again and again. However, by way of contrast, if we were to perform a true measurement on an unprepared object, to assess its energy, position, momentum etc., one has a different situation: about the former (prepared) system we may say something about its future, while for the latter (unknown) system we might only tell something about its immediate past.

Einstein's probabilistic view of reality will require more parameters to be considered in a statistical ensemble system (such as, e.g. temperature). ‘Ensemble’ is here considered a series of repeated preparations of close to identical individual quantum systems measured sequentially one by one. Assume, for example, that we consider Eq. (1) as a linear superposition of the two wave functions Ψ₊(1) Ψ₋(2) and Ψ₋(1) Ψ₊(2). We may continue by including additional pairs of (non-interacting) particles and in this way get some wave function for the whole system

$$ \Psi ({\rm total}) = N^{ - 1/2} \Sigma [\Psi _ + (1)\Psi _ - (2) - \Psi _ - (1)\Psi _ + (2)]$$

whose statistical meaning is represented by the square of its modulus:with δ representing some phase shift reflecting e.g. how the different entanglements were created, their coherence etc. The interference term can obviously have very drastic effects on the total and may represent the statistical distribution of further (hidden?) parameters of the experiment.

Therefore, let us elaborate on the experiment a little further along the lines of Bransden & Joachain (Reference Bransden and Joachain2000). Let us say we want to measure the value of an observable A of our system. The result of a single measurement must thus be one of the eigenvalues a _n of A corresponding to some eigenvector |ϕ _n >. However, the system cannot be viewed in isolation. The analyzer (Stern-Gerlach apparatus or light polarizer, see below) should be combined with the system we wish to analyze:

$$\left\{ {{\rm system} + {\rm analyzer}} \right\}$$

Suppose the initial wave function for the ensemble of states of the two-particle system is ∑b _n |ϕ _n > with the expansion running over all pure states of a complete set of states, |ϕ _n >. The state function for the combined, system plus analyzer, may then be written as the direct product:

(5a) $$ \vert \Psi ({\rm total},\,t)\!\! \gt = \Sigma _n \,b_n \vert \phi _n \!\!\gt \!\otimes \,\vert \Phi_0 \!\!\gt $$

with |Φ₀> being the initial eigenvector of the analyzer. The total state function, we presume, will satisfy some version of the time-dependent Schrödinger equation and evolve so that, after the measurement at time t′, it will become |Ψ(total, t′)>:

(5b) $$ \vert \Psi ({\rm total}, t^{\prime}) \gt = \Sigma _m \, b_m \vert \phi _m \gt \otimes \vert \Phi_m \gt. $$

Now Einstein's (putative) interpretation of |Ψ(total, t′)> could be that within a large ensemble of identically prepared states, one should look for the fraction in which the value a _n is found, which will be given by |b _m |², all state vectors assumed normalized to unity. This view is self-consistent and any attempt to interpret Eq. (5b) in terms of a single-particle system would seem inconsistent and unphysical. The consistency arises from taking the ensemble view. A statistical interpretation of QM by Ballentine (Reference Ballentine1970), using a minimum of assumptions, arrives at similar recommendations. Below we shall use a function, Malus’ law, to introduce a probabilistic view of a many-particle ensemble to entangled photons.

An example of the dissipative evolution of a prepared set of states is the study of femtosecond dynamics of dissociation of sodium-iodide molecules by Zewail and coworkers (Engel et al. Reference Engel, Metiu, Almeida, Marcus and Zewail1988; Rose et al. Reference Rose, Rosker and Zewail1988; Zewail, Reference Zewail1988, Reference Zewail1990): during the cycle of a vibration of the molecule after being excited to its first excited state by a femtosecond pulse, the passage of the Landau-Zener crossing point (at inter-nuclear separation 6·93 Å) could occur either when the covalent quantum wave packet is on its way out (then leading to dissociation into Na and I atoms) or the ionic wave packet on its way in (leading to trapping of NaI in the bound ground state). Whereas, a single-particle experiment would encounter a problem to resolve the wave packet motion because of the uncertainty principle, the quantum statistics of the large number of molecules coherently excited by a femtosecond pulse, to vibrate all of them in phase, will over-rule the uncertainty and provide macroscopic features to the oscillations, possible to experimentally follow as either activated complex or free atoms, appearing as ringing (resonance) particle emission intensities with coinciding temporal peak positions. An analogous classical case is the ringing that DNA molecules display during migration through an electrophoresis gel, if ‘excited’ in phase by the sudden application of an electric field (Nordén et al. Reference Nordén, Elvingson, Jonsson and Åkerman1991). In both cases the ringing is sharp in the beginning but eventually blurred out as different molecules (NaI and DNA, respectively) encounter various perturbations and get out of phase.

3. Bell's theorem and photon entanglement

The EPR paradox has been elaborated on in Gedankenexperiments by Bohm (Reference Bohm1952) and Bell (Reference Bell1966) to show that no hidden-variable theory may reproduce all predictions from QM. Note again, that Einstein believed the predictions of QM to be correct, but only as consequences of appropriate statistical averaging over distributions of (possibly hidden) further variables.

Bell's argument proceeds as follows. In the EPR-Bohm Gedankenexperiment a source emits pairs of spin ½ particles from singlet state systems. Alternatively, pairs of photons are produced in an analogous, non-factorizing state. After the particles have separated, one performs correlated measurements of their spins (or polarizations for photons). The detection devices – hereafter polarizers – can have arbitrary settings a and b, being vectors in the planes, on the respective sides of the source, perpendicular to the photon propagation direction, with any direction in the plane allowed. Let us denote the observed results as ‘+’ if the polarization is found parallel with a, and ‘−’ if the polarization is found perpendicular, etc. Contributing to the total probability P _{± ±} (a, b), we have the following four possibilities:

(6) $$P_{ + +} ({\bf a},{\bf b}),P_{ + -} ({\bf a},{\bf b}),P_{ - +} ({\bf a},{\bf b}),P_{ - -} ({\bf a},{\bf b})$$

The function

(7) $$E({\bf a},{\bf b}) = P_{ + +} ({\bf a},{\bf b}) + P_{ - -} ({\bf a},{\bf b}) - P_{ + -} ({\bf a},{\bf b}) - P_{ - +} ({\bf a},{\bf b})$$

representing the correlation of the particles’ polarizations, was studied by Bell. Considering the particles in each pair and adding locality assumptions he arrived at certain inequalities that could not always be fulfilled by the predictions of QM.

Defining the correlation function

(8) $$S = \left \vert {E({\bf a},{\bf b}) - E({\bf a},{\bf b^{\prime}})} \right \vert + \left \vert {E({\bf a^{\prime}},{\bf b}) + E({\bf a^{\prime}},{\bf b^{\prime}})} \right \vert $$

his inequality could be written (Carg & Mermin, Reference Carg and Mermin1987; Clauser et al. Reference Clauser, Horne, Shimony and Holt1969; Larsson, Reference Larsson1998):

(9) $$S \les (4/\eta ) - 2$$

with η the detection efficiency and where S thus involves four series of repeated measurements involving the polarizers in four different combinations of orientations: (a, b), (a, b′), (a′, b) and (a′, b′) (Clauser et al. Reference Clauser, Horne, Shimony and Holt1969).

Two critical questions emerge from this analysis:

1. (1) whether true correlation may be experimentally achieved between the two emitted photons (i.e. no time lag between their emissions), and

2. (2) whether their polarizations may be considered known as a result of the design of the source.

In the experiments by Freedman & Clauser (Reference Freedman and Clauser1972) and later Aspect et al. (Reference Aspect, Dalibard and Roger1982a, Reference Aspect, Grangier and Roger b ), a Ca⁴⁰ cascade is exploited to generate pairs of photons of different wavelength, assumed to be entangled. This was an EPR type experiment first made by Kocher & Commins (Reference Kocher and Commins1967), which unfortunately only set the polarizers at 0° and 90° relative to each other – a new experiment and inequality was proposed by Clauser et al. in 1969. A problem, however, was the finite lifetime of the intermediate ¹P₁ state (approximately 5 ns) making the second photon lag behind the first one, possibly also depolarized and not entangled. Optical Parametric Amplifiers (OPA) are used today to generate coherent states of photons as developed by Mandel and followers (Friberg et al. Reference Friberg, Hong and Mandel1985; Hong et al. Reference Hong, Ou and Mandel1987). The temporal and spatial resolution may still be a problem if the two photons are not created at the same time and spot in the crystal. Potentially the high photon flux, required to get high electric fields in the non-linear crystal, could present another problem: which photon forms an entangled pair with which? Two entangled photons should be typically created at times differing less than, say, 10⁻¹⁶ s, corresponding to a flight time of 10% the ‘size’ of a 300 nm photon (my tentative suggestion). This aspect has as far as I know not been addressed, but it is possible that the small probability (say, 1 in roughly 10¹² pump photons) for this to occur may be compensated for by the extreme intensities of signal and idler photons in the diffraction-limited volume of the crystal. An attractive aspect of the parametric down conversion experiment is that it emits trains of highly correlated photons in one process – photons that may be measured later far apart.

As for polarization conditions these may vary, but generally the photons created in the non-linear crystal by high-intensity electric laser fields will have a well-defined polarization. We will in our own Gedankenexperiment below assume that the two photons have the same linear polarization parallel with some laboratory reference axis of the photon-generating crystal. If we were to challenge Bell's theorem it is enough to find a set of (possible) photon parameters that will violate the statement that ’classical’ (see Section 7) and quantum results are different at certain angles of polarizer settings.

4. Classical versus QM correlations

Following the analysis of Aspect et al. (Reference Aspect, Dalibard and Roger1982a, Reference Aspect, Grangier and Roger b ), who essentially repeated the experiments and analysis by Freeman & Clauser, we may now calculate the expected classical correlation function assuming that all photons, whether quantum entangled or not, all have initially parallel (say vertical) polarization. If we are going to question the validity of the Bell inequality, we can disregard the issue of whether we measure on correlated photons. First let us consider the probabilities P _ij (a, b) for the case of the outcoming photon being vertically polarized using classical optical theory.

According to Malus’ empirical, but also theoretically justified law of polarization (discovered by Étienne-Louis Malus in 1808) the intensity of a beam of linearly polarized light is attenuated by an inserted polarizer by the factor cos² ϕ so that the light intensity before (S ₀) and after (S) the polarizer are related as:

(10) $$S/S_0 = {\rm cos}^2 \phi $$

with ϕ the angle between the plane of linear polarization and the polarizer axis.

Malus’ law immediately tells us the (classical) probability for transmission of a photon through a polarizer. At 0° and 180° there is maximum probability (one) for transmission, whereas at 90° and 270° the probability that a photon should come through is identically zero. At all angles the probability is given by Eq. (10).

If the polarizer settings (vectors a and b) are set at angles α _a and α _b , respectively, we can now calculate the classical correlation function:

(11) $$E({\bf a},{\bf b}) = P_{ + +} ({\bf a},{\bf b}) + P_{ - -} ({\bf a},{\bf b}) - P_{ + -} ({\bf a},{\bf b}) - P_{ - +} ({\bf a},{\bf b})$$

(12) $$\eqalign{E({\bf a},{\bf b}) &= [\langle {\rm cos}^2 \alpha _a {\rm cos}^2 \alpha _b \rangle + \langle {\rm sin}^2 \alpha _a {\rm sin}^2 \alpha _b \rangle \cr & \quad- \langle {\rm cos}^2 \alpha _a {\rm sin}^2 \alpha _b \rangle - \langle {\rm sin}^2 \alpha _a {\rm cos}^2 \alpha _b \rangle ] \cr & \quad/[\langle {\rm cos}^2 \alpha _a {\rm cos}^2 \alpha _b \rangle + \langle {\rm sin}^2 \alpha _a {\rm sin}^2 \alpha _b \rangle + \langle {\rm cos}^2 \alpha _a {\rm sin}^2 \alpha _b \rangle \cr & \quad+ \langle {\rm sin}^2 \alpha _a {\rm cos}^2 \alpha _b \rangle ]\;} $$

If we assume that each and every count is averaged for each E(a, b) at a time, we have (the normalizing denominator obviously becomes equal to unity):

(13) $$\langle E({\bf a},{\bf b})\rangle = \langle {\rm cos}^2 \alpha _a ({\rm cos}^2 \alpha _b - {\rm sin}^2 \alpha _b ) - {\rm sin}^2 \alpha _a ({\rm cos}^2 \alpha _b - {\rm sin}^2 \alpha _b )\rangle \; $$

(14) $$\langle E({\bf a},{\bf b})\rangle = \langle {\rm cos}2\alpha _a {\rm cos}2\alpha _b \rangle $$

The biggest difference between the classical and the quantum predictions according to Bell's theorem is noted to be at α = 22·5° and α  = 67·5° (Clauser et al. Reference Clauser, Horne, Shimony and Holt1969). This is so because Bell assumes a linear dependence of the correlation on the angle – see Fig. 1. Setting the a polarizer to 0° and the b one to 22·5°, calculating: E(a, b) = 2^−1/2 and getting S = E(a, b) – E(a, b′) + E(a′, b) + E(a′, b′) = 1·414 if a′ is set at 45° and b′ at 67·5° as originally suggested, a result that is not in conflict with Bell's inequality. Higher S values can be obtained ‘artificially’ if other settings of a and b are allowed, surmounting to S = 2·828 for (a, b) = (a′, b) = (a′, b′) = (0, 22·5°) in obvious conflict with the Bell inequality, Eq. (9), where S is predicted to be <+2 for the classical case. One could question the interpretation of some of these experiments as the delay of the second photon may obstruct entanglement. However, in an experiment by Aspect et al. (Reference Aspect, Dalibard and Roger1982a) a light switch with 10 ns period seems to have taken care of the latter problem, at least in principle. The corresponding case of orbital angular momentum (circularly polarized photons) may be treated similarly – yielding similar result – using the classical light-polarization vector of Stokes (Reference Stokes1852) together with a (4 × 4) Mueller matrix representing the analyzers (see e.g. Nordén, Reference Nordén1978).

Fig. 1. Photon correlation functions as function of the angle θ between the polarizer settings, shown together with Bell's inequality (diagonal line). (a) Correlation function E(a, b) according to the classical and quantum mechanical Eqs. (15) and (16). (b) Correlation function R(θ)/R ₀ in inequality (17) for classical and quantum mechanical Eqs. (18) and (19).

If one assumes that the two photons, or light flashes, are uncorrelated but have the same polarization, which is not unreasonable if they are generated in the same photoselection process by an orthogonal exciting laser beam, Eq. (14) becomes simply:

(15) $$\langle E({\bf a},{\bf b})\rangle = (1/2)\ {\rm cos}2\theta \qquad\qquad({\rm classical})$$

with θ = (α _a  – α _b ) the angle between polarizers a and b. This result is obtained from Eq. (14) by integrating 〈cos 2α cos 2(α + θ)〉 over all angles α. This classical correlation has the same functional dependence on the angle θ between the polarizers as the QM correlation:

(16) $$\langle E({\bf a},{\bf b})\rangle = {\rm cos}2\theta \qquad\qquad ({\rm QM}) $$

but obviously does not violate the Bell inequality (Fig. 1 a). Note, however, that η the detection efficiency in Eq. (9) introduces an uncertainty and for less than η = 0·83 (83% detection efficiency) S values from Eq. (16) do not any longer violate the Bell inequality.

Historically, one of the first Bell tests, using single channel polarizers (Freedman & Clauser, Reference Freedman and Clauser1972), was applying the inequality:

(17) $$\Delta = \vert R(22.5^{\circ})/R_0 - R(67.5^{\circ} )/R_0 \vert \les \displaystyle1 / 4 $$

with R(θ) the coincidence rate with angle θ between the polarizers and R₀ the rate with polarizers removed. Δ = 0·35 was obtained experimentally (Freedman & Clauser, Reference Freedman and Clauser1972; Shih & Alley, Reference Shih and Alley1988), thus clearly violating the inequality. Integrating 〈cos² α cos²(α + θ)〉 over all angles α gives us the corresponding classically expected correlation function:

(18) $$R(\theta )/R_0 = (1/8) \; ({\rm cos}2\theta + 2)\qquad\qquad ({\rm classical}) $$

to be compared with the QM and experimentally observed one (Fig. 1 b):

(19) $$R(\theta )/R_0 = (1/4) \; ({\rm cos}\,2\theta + 1) \qquad\qquad ({\rm QM}) $$

Equation (19) reproduces the experimental result Δ = 0·35 obtained by Freedman & Clauser (Reference Freedman and Clauser1972) of Eq. (17). The difference between Eqs. (18) and (19) is (1/8) cos 2θ, the Bell limit given by R(θ) = ½ – θ/π. The similarities between Eqs. (15) and (16) and their relation to the Bell line in Fig. 1 a, as well as the similarity between Eqs. (18) and (19) are remarkable algebraic features whose interpretation in the respective experimental contexts seems to have been rarely addressed. Ou & Mandel (Reference Ou and Mandel1988a) depicted classical and quantum correlation curves like those in Fig. 1 b.

5. The ‘Interaction Hypothesis’

The EPR paradox inspired two different strands of experiments, neither of which appears to have given yet an unambiguous answer to the question whether two quantum-entangled particles (or photons) may remain in contact at very large distance from each other so that a collapse of entanglement, induced by interaction with one of them, is immediately propagated to the other one. The first group of experiments were those just mentioned based on the work of Bell (Reference Bell1966); Bohm (Reference Bohm1952), Freedman & Clauser (Reference Freedman and Clauser1972) and others (Aspect et al. Reference Aspect, Dalibard and Roger1982a, Reference Aspect, Grangier and Rogerb; Weihs et al. Reference Weihs, Jennewein, Simon, Weinfurter and Zeilinger1998), focusing on photon statistics and Bell inequality. Fundamental experiments proving the non-locality of our description of the physical world were due to Wu & Shaknov (Reference Wu and Shaknov1950) and Wineland (Turchette et al. Reference Turchette, Wood, King, Myatt, Leibfried, Itano, Monroe and Wineland1998).

A second line of important experiments were conducted in the interference regime where Mandel and coworkers were pioneers both in method development and to observe nonlocal effects (Gosh & Mandel, Reference Gosh and Mandel1987; Hong et al. Reference Hong, Ou and Mandel1987; Ou & Mandel, Reference Ou and Mandel1988a, Reference Ou and Mandelb). The discovery by Kimble et al. (Reference Kimble, Dagenais and Mandel1977) of photon anti-bunching in atomic fluorescence was the first example of a new type of phenomena that could not be described without a complete quantum approach. It gave rise to new technologies such as squeezed and entangled light. The work culminated with the first observation of the vacuum Rabi splitting of an atom in the optical domain – basis for a number of applications of quantum logic of single atoms or photons. Here also the demonstrations by Moerner and Orrit of absorption and fluorescence of single molecules trapped in a solid matrix should be mentioned as they led to the also important discovery of antibunching of emitted photons (Basché et al. Reference Basché, Moerner, Orrit and Talon1992).

Franson (Reference Franson1989) and Kwiat et al. (Reference Kwiat, Steinberg and Chiao1993) have in two-photon interference experiments shown sinusoidal fringes with 80% visibility, such as predicted by QM, violating Bell inequality by 16 standard deviations. More recently Salart et al. (Reference Salart, Baas, Branciard, Gisin and Zbinden2008) and Yin et al. (Reference Yin, Cao, Yong, Ren, Linag, Liao, Zhou, Liu, Wu, Pan, Zhang, Peng and Pan2013), following a proposal by Eberhart (Reference Eberhart1989) for a 12-h continuous space-like Bell inequality exploiting the Earth's self-rotation, discuss the possibility of superluminal interference at ca 20 km distance. Already Grangier et al. (Reference Grangier, Roger and Aspect1986), using a Mach-Zehnder interferometer and the atom cascade source of Clauser, showed photon anticorrelation effects at a beamsplitter in agreement with QM-described single-photon states. Also using spontaneous parametric downconversion at two wavelengths, a ‘heralded’ single photon has been used to clock the event when the other was a photon thrown at a beamsplitter after which it is detected in either of two arms of the optics: it is shown that the probability of coincidence in the two detectors A and B, P _AB is much less than the probabilities of detection on each side P _A × P_B, as would be expected in the case of uncorrelated events (Guerreiro et al. Reference Guerreiro, Sanguinetti, Zbinden, Gisin and Suarez2012). This space-like antibunching experiment beautifully corresponds to Albert Einstein's Gedankenexperiment of the 5th Solvay Conference in 1927 and provides strong evidence for the non-locality of a single photon.

For atoms and molecules, quantum entanglement may be viewed as basis of both chemical bonding as well as excitonic stationary quantum states (Kasha Reference Kasha1963; Bayer et al. Reference Bayer, Hawrylak, Hinzer, Fafard, Korkusinsky, Wasilewki, Stern and Forchel2000; Nordén et al. Reference Nordén, Rodger and Dafforn2010; Halpin et al. Reference Halpin, Johnson, Tempelaar, Murphy, Knoester, Jansen and Miller2014). The latter are important in many photophysical and biophysical contexts and can be regarded the result of coupling between molecular subsystems in which virtual excitations via attractive or repulsive interactions lead to eigenstates of the total Hamiltonian, without eigenstates for the separate subsystems. Typically, an excitonic state arising from interactions between degenerate excited states in a dimeric molecular structure may be described by the wave function

(20) $$\Psi = 2^{ - 1/2} [\phi _{\rm A} ^{\circ} \phi _{\rm B} ^{\prime} \pm \phi _{\rm B} ^{\circ} \phi _{\rm A} ^{\prime} ]\; $$

where |ϕ _A°> and |ϕ _A′> are, respectively, ground and excited states in fragment A, etc., and eigenstates to the Hamiltonian H _A of that separate fragment. The total excitonic wave function (20) corresponds to entanglement between fragments A and B and is an eigenstate of a total Hamiltonian:

(21) $$H = H_{\rm A} + H_{\rm B} + V_{{\rm AB}} $$

with V _AB the interaction energy between A and B. Without entanglement, V _AB = 0, the exciton Ψ collapses into the product states of the free subsystems, |ϕ _A°>|ϕ _B’> etc. The impact of the intermolecular interaction V _AB is to ensure indistinguishability of A and B as to what molecule actually carries the exciton. In the absence of V _AB, symmetry breaking analogous to Jahn-Teller distortion commonly takes place, manifestly localizing the exciton to one of the two molecules. Upon increasing the intermolecular coupling strength though, the relaxation is eventually avoided and the entangled state becomes preferred. If we adopt the same formalism for the entanglement of two photons A and B, the question arises what significance has the interaction operator V _AB? By analogy with the entangled atom pair, we end up with the following options: either applying the conservation of momentum law for the separate particles or applying the wave function (20) and assigning the interaction V _AB to the entanglement. While the former case is just the rather uninteresting memory effect that we have already discussed for split hydrogen, the latter corresponds to the entanglement that we want to address. For the entanglement between a pair of photons V _AB will be probably the field interaction operator described by quantum electrodynamics and Maxwell's equations. It should satisfactorily explain any experiment performed in laser cavities or other confined optical systems where the photons are ‘in sight’ of each other, so we predict, when V _AB = 0 (or in level with thermal noise), there will be no entanglement. In one stroke this Interaction Hypothesis will dissolve the ‘spooky-action-at-a-distance’ problem as it clearly defines when entanglement can occur but also how fast it may collapse – a change of the fields that connect the photons cannot generally propagate faster than speed of light.Footnote ²

6. Future experiments

The author suggests an experiment to decide whether or not photons may be entangled when no longer in field contact with each other (see Fig. 2). Consider a pair of (presumed entangled) photons, one propagating along the positive (right) x-axis and one along the negative (left) x-axis. They can be produced at a beam-splitter from photon beams of an OPA crystal (Friberg et al. Reference Friberg, Hong and Mandel1985; Kim, Reference Kim2003; ‘heralding’ photons may be used to keep track of the photon pair). Introduce then a (black absorbing) beamstop at x ₀, just after that the right-hand photon has passed, and another beamstop at −2x ₀ just before the left-hand photon is to pass. Will we be able to tell a difference, by probing the right-hand photon, when no longer able of communicating with its left partner, whether the latter is still entangled or has been absorbed? Based on absence of interaction due to beamstop x ₀, one is inclined to say: no!

Fig. 2. Proposed experiment to determine whether two photons can be entangled if not ‘in sight’ of each other (see text).

As far as I know no experiment has yet been performed along this line on single photons, but those experimentalists who pioneered the field of quantum optics could have made it in passing. Kimble has been prolific in squeezed and other quantum states in cavity using the technique of parametric down-conversion and the demonstration by his group (Ou et al. Reference Ou, Pereira, Kimble and Peng1992a, Reference Ou, Pereira and Kimbleb) realizing the EPR paradox for continuous variables using these techniques inspired and paved way for the later establishment of a theory for model-independent continuous-variable entanglements (Duan et al. Reference Duan, Giedke, Cirac and Zoller2000; Simon, Reference Simon2000). The group has demonstrated how entangled photons may be created and are not separable. Experiments demonstrating that entangled photon beams may be ‘exchanged’ into atomic quantum memory, and then mapped back to photonic state again (Choi et al. Reference Choi, Deng, Laurat and Kimble2008), will put some special demands on the interaction field hypothesis to be able to adopt to new ‘currency’ whether photons or atoms. Besides asking whether our ‘interaction postulate’ holds, we may also ask: is ‘instantaneous’ collapse of entanglement in conflict with special relativity? While many theorists have dealt with the tension between QM and relativity, only few experiments have been done. Zbinden et al. (Reference Zbinden, Brendel, Gisin and Tittel2001) have studied two-photon interference fringes with detectors at 10 km separation and conclude that introduction of ‘passive’ detectors (absorbers) does not affect the QM-expected result. Salart et al. (Reference Salart, Baas, Branciard, Gisin and Zbinden2008) and Yin et al. (Reference Yin, Cao, Yong, Ren, Linag, Liao, Zhou, Liu, Wu, Pan, Zhang, Peng and Pan2013) as mentioned suggest potential superluminal interference over ca 20 km distance. Gosh & Mandel (Reference Gosh and Mandel1987) and Ou & Mandel (Reference Ou and Mandel1988a, Reference Ou and Mandelb) early demonstrated the superiority of interference-beating as a technique for joint two-photon detection. An experiment, representing entanglement of orbital momentum states, should have instead of linearly polarized photons a pair of circularly polarized photons of opposite circularity (Leggett, Reference Leggett2003; Mair et al. Reference Mair, Vaziri, Weihs and Zeilinger2001). In order to get deeper understanding of the nature of entanglement, systematic studies of dependence on distance and surrounding media are needed – it is also important to quantify the degree of entanglement as has been critically prescribed by Kimble (van Enk et al. Reference Van Enk, Luthenhaus and Kimble2007).