You're basically asking the same question that puzzles many people in quantum mechanics. The fact that the state is (apart from normalization) |HV> + |VH> is not the same as saying that first photon is either in |H> or |V> and second photon is either in |V> or |H>, respectively (this latter case corresponds to the mixture you wrote). Entanglement is a global property of the system, and the whole state is pure, i.e. if you measure many copies of it in the Bell basis, you always get the same result with probability one. Things are completely different in the mixed case, where, no matter which basis you use, when measuring it the result is uncertain, i.e. cannot get the same result with probability one, unless the state is pure.

In the case of type-2 parametric down-conversion, the photons are entangled, so their global state is well defined and known (just measure multiple copies of it in Bell basis and you'll always get the same result), however one cannot talk about definite values for the polarization of the individual photons. This latter fact is not viewed as an additional lack of knowledge as is the case for mixed states. It is just saying that for an entangled state it makes no sense to assign definite values to its individual parts, however the overall state is well defined and of maximal knowledge one can possibly have about the system.

Hi Vlad, thanks for helping me out with the issue but it is still not clear to me.

Is the fact that generated photon pair is entangled only an experimental fact? Why is it that the photons are entangled at the intersection points. I don't understand how the wavefunction is defined for the generated photon pair. Please give a brief explanation.

They phenomenon of entanglement arises because any linear superposition of pure states is allowed in quantum mechanics, so coherent superpositions of |00> and |11> of the form (|00>+|11>)/sqrt(2) (Bell states) are perfectly valid. Your 2-photon state happens to be entangled because of the interaction between the incident photon and the non-linear crystal (I don't know more experimental details). So the fact that the pair is entangled is both an experimental fact (testable in the lab), and also a theoretical one (the state is described by the linear superposition of |HV> and |VH>). 

The question of "why" is a bit more complicated, but think about this way: you can always start with a product state, say |HH>, then apply a unitary evolution (i.e., there exist a physical Hamiltonian which satisfies Schrodinger's equation) on it which will transform it to (|HV>+|VH>)/sqrt(2). The interaction between the crystal and the incident pair is described by this kind of Hamiltonian, so at the end of the day you end up with an entangled pair. 

A mixed state (|HV>, whereas if the result is tails, prepare it in |VH>. The measurement statistics on this latter state are completely different from the measuring statistics obtained by measuring an entangled state, and in the lab it was indeed observed that the pair of photons generated via parametric down-conversion obey the statistics of an entangled pair and not of a mixture.

Hi Sumanth Kumar,

Just to add some more information regarding experimentally generating polarization entangled photon states using non-collinear type-II parametric down-conversion (PDC) in a nonlinear birefringent crystal: photons originating from PDC process can be put in an entangled state and are *not* entangled upon creation. You can understand that by remembering that ordinary and extraordinary waves traversing a birefringent media propagate at different group velocities due to the difference in their refractive indices. This fact results in timing information that prevents orthogonally-polarized photons from the two spatial modes (defined by the cones' intersection points) to be entangled upon emerging the crystal. This is because in each of the two spatial modes, one of the two polarization modes will always arrive first to the detector (depending on the properties of the birefringent crystal).  This phenomenon is known as temporal walk-off (or longitudinal walk-off), and its maximal value (for photons created at the beginning of the generating crystal) is deltaT=L*(1/ve-1/vo), where L is the crystal thickness and ve (vo) are the group velocities of the extraordinary (ordinary) polarization modes, respectively. If the value of the time delay deltaT is larger than the coherence length of the pump pulse (which can be around 150 fs for a standard femtosecond pulsed Ti:sapphire laser), then the two terms |H1V2> and |V1H2> are distinguishable and therefore do not interfere, and no entanglement is formed (here 1 and 2 represent the two spatial modes defined by the cones' intersection points).

The clever solution that Paul Kwiat et al. found for this problem in their famous work (https://www.researchgate.net/publication/13230395_New_high-intensity_source_of_polarization-entangled_photon_pairs) was adding to each of the two spatial modes a similarly-cut birefringent crystal with half of the thickness. After rotating the polarization of both down-converted photons by 90 deg., pairs of polarization entangled photons emerge from the system, where entanglement is formed between two orthogonally-polarized photons generated at distances of +-x from the center of the generating crystal. Using this technique the timing information is completely erased (deltaT=0) and entanglement is created.

I hope it helps, and I strongly suggest reading Kwiat's original paper to better understand this experiment.

Assaf

Article New High-Intensity Source of Polarization-Entangled Photon Pairs

Hi Asaaf, 

Your explanation is very informative but I have a few questions. To generate entangled photons, I think we need to compensate only the transverse walk-off as this will help us to get the entangled pair at the same time. Is there any need to cancel the longitudinal walk-off? And I understand that we have to choose the length of the crystal in such a way that the time delay is less than the coherence time of the laser. Does the efficiency depend upon the type of input state we are feeding to the crystal, like the coherent state, single photon, etc.? Which one is typically used to produced entangled pairs?

Sumanth

Hi again,

First of all, it is essential to compensate for both transverse and longitudinal walk-offs, since they both label the down-converted photons (spatially and temporally, respectively) and prevent the wave functions to interfere on the detector. This wave function interference generates entanglement between the photons. Compensation of only the transverse walk-off, as you suggested, will leave the photons with different arrival times to the detectors and prevent entanglement generation.

Regarding the other questions you've asked, I'm not sure I understood you, but I'll try anyway: the length of the crystal is one of the parameters that determine the spectral bandwidth of the down-converted photons: the intensity of second-harmonic generation scales as L^2*sinc(deltaK*L/2), where deltaK is the allowed phase mismatch of the pump beam and L is the crystal thickness (see Boyd's Nonlinear Optics for more details). Since parametric down-conversion is basically the reverse process, the same logic can be used  to describe it. So generally using a shorter crystal allows a bigger phase mismatch, but  since the process efficiency scales as L^2, it is desirable to use the longest crystal possible.

Keep in mind that both CW and pulsed lasers are used as the pump beam in these experiments. Usually CW pump is used when the desired state contains only two photons, whereas femtosecond pulsed sources are used for multiphoton entanglement generation (i.e., when more than one photon undergo parametric down-conversion at the same time, thus creating an entangled state containing more than two photons).

Generally, the down-converted spectrum is much broader than the pump spectral bandwidth, since the phase-matching conditions (energy and momentum conservation) are very tolerant with respect to the wavelength in nonlinear birefringent crystals. This applies both to CW pump and also for ultrashort pump pulses. Moreover, using a pulsed pump results in a decrease of the entanglement visibility due to spectral distinguishability, arising from the availability of wavelengths to be down-converted in the pump beam.

Because of their large bandwidths, the coherence time of the down-converted photons is very short (and may even be shorter than the pump pulse duration). Therefore, it is common to incorporate bandpass filters that lengthen the coherence time and increase the entanglement visibility. By inserting narrow filters we effectively increase the transmitted down-converted photons' coherence lengths by narrowing their frequency spectrum (the coherence time is inversely proportional to the bandwidth of the filter). So usually the maximal longitudinal walk-off value I've discussed in my first answer is compared to the coherence time of the bandpass filters that are being used in the experiment (see discussion regarding coherence time in Kwiat's 1995 article I've mentioned in my first answer).

I hope it helps,

Assaf

The conventional story has its problems:

1. An entangled state has never been observed!  Rather just the "collapsed" daughter. Or. so it is said.

2. The space of entanglement for all polarization phenomena ("qubit space") is described by the group SU(2), which homeomorphic to S(3) [3-D rotations], introduced by Stockes ca. 1855, i.e., long before QM.  The noncommutivity here has nothing to do with QM, but geometry.  In other words, the whole story is non-quantum from the start! 

It would seem that any answer to the original question must pass these waypoints.