Quiltlet – Mathematical Connections


Mathematical Connections

Quantum key distribution scheme BB84

The small quantum cryptography quiltlet, attached to the large Cryptoquilt, illustrates the quantum key distribution scheme BB84, which was the first quantum cryptography protocol. The Brassard-Bennett 84 protocol was first published in a 1984 conference; here is a copy of that first paper. 

In this scheme, two parties have to communicate over a distance, using a channel where signals may be intercepted by an eavesdropping third party, to agree on a secret binary key which they plan to use for further encrypted communication, beyond the protocol. 

It is customary in cryptography to call the two parties Alice and Bob (rather than just A and B).

In the protocol depicted on the quiltlet, Alice sends linearly polarized photons to Bob, of which Bob then measures the polarization. Alice has several sources for her polarized photons: in the “straight” system S, the polarization is either Vertical or Horizontal, and in the “diagonal” system D the polarization vector is diagonal, pointing up to either the Right or the Left. (This will be illustrated below.)

To produce these photons, Alice directs light from her firefly sources to traverse two calcite crystals. 

About the calcite crystals

Calcite is a naturally occurring mineral; its crystals have striking birefringent properties that have been used for optics and navigation for centuries. 

A perfect calcite crystal is transparent and has faces that are rhombus- or parallellogram-shaped; in particular, the angles at its vertices are a bit off from the straight angles at the vertices of a cube. 

Source : Wikipedia. Work own by Jan Pavelka 

Calcite crystals with normals

When a beam of unpolarized light falls on the face of one of these crystals, perpendicular to the face, then part of the beam will traverse the crystal, and exit the opposite face in a perfect straight extension of the incident beam, as is usual in refraction experiments. But another beam is also formed at the incidence point, *at an angle* with the incident beam — this is *not* usual in a refraction set-up when the incident beam is perpendicular to the surface, and this second beam is appropriately called the extraordinary refracted beam. On exiting the crystal on the other side, this beam will bend again, and it will then continue in a path parallel to the other exiting beam. Inside the crystal, and also after they have exited, both beams are linearly polarized. The polarizations of the two beams are perpendicular to each other, with the polarization on the extraordinary beam lying in the plane spanned by the two beams, and the polarization for the ordinary beam perpendicular to this plane. 

Alice standard set-up

In the quiltlet illustration of the BB84 quantum key distribution protocol, Alice uses two calcite crystals, and four fireflies. Before going into the communication protocol itself, we describe the geometric setup.

The fireflies each provide lightbeams that we assume to be monochromatic, with identical frequencies for all four. 

(The whole protocol uses light of only that one wavelength.) 

The first two fireflies are positioned and their emitted lightbeams are windowed so that only a narrow beam from each can hit the first facet of the first crystal, at a straight angle. Each of these two possible incident beams produces an ordinary beam, proceeding through and out of the crystal without deviating, and an extraordinary beam that goes off at an angle from the incidence point. 

The light sources are positioned so that the extraordinary beam from one source exits the crystal at exactly the same spot as the ordinary beam from the other; the other beam from each pair is disregarded (e.g. absorbed by a well-placed patch on the crystal at their exit spots).

The incident beams are parallel to each other, causing the paths of the two exiting beams to coincide after they exit the crystal. Because of the set-up, the beams pertaining to this first crystal all lie in one plane; depending on whether the light exiting from the first crystal came from the ordinary beam of one of the fireflies or the extraordinary beam of the other, it will be polarized perpendicular to that plane, or parallel to it; let's call these directions, respectively, V and H. Let's call the set-up with this first crystal the 'Standard' side. 

Alice 3D set-up

The set-up for the second crystal is similar: two firefly sources, incident beams perpendicular to the first facet of the crystal, so that the extraordinary beam from one source will concide with the ordinary beam from the other, once they exit the crystal. The two crystals are put in a very precise geometric relationship to each other: on the one hand, they are arranged so that their exiting lightbeams will intersect, making a right angle with each other at the intersection point. On the other hand, if one views the lightbeam exiting from the second crystal as a rotation axis, then the set-up with the second crystal is rotated in such a way around this axis that the two linear polarization arrows that arrive on its lightbeam make angles of 45 degrees with the polarization arrows of the lightbeam exiting from the first crystal. We call this side of the setup the 'Diagnonal' part, and label the corresponding polarization direction LD and RD (for pointing up along the Left Diagonal or the Right Diagonal, with respect to the plane of the lightbeams exiting from the two crystals).

At the intersection point of the two lightbeam paths coming from the two crystals, Alice places a half-silvered mirror perpendicular to the plane of the two lightbeams, and bisecting their angle. Light from either of the beams arriving at the mirror has a 50% chance to pass uninterrupted or to be reflected. If the light is reflected, then the position of the mirror guarantees that its new path direction will coincide with the uninterrupted path from the other beam.

In what follows, we only consider the reflected part of the Standard lightbeam and the uninterrupted part of the Diagonal lightbeam -- absorbing patches can eliminate the other parts of the lightbeams. 

Bob 3D set-up

On Bob's side, the set-up mirrors that of Alice. The light beam carrying Alice's information is directed to a half-silvered mirror that sends photons on either an uninterrupted path or one that is reflected by 90 degrees, each photon retains its polarization in this process (it is reflected as well, for reflected photons -- use the toy car analogy again to see this). 

The two resulting beam paths lead to calcite crystals in either the Standard or the Diagonal set-up. If a photon arriving at one of Bob's calcite crystals has a polarization that "fits" its set-up (i.e. if it is a H or V-polarized photon arriving at the S crystal, or if it is a LD or RD-polarized photon at the D crystal), then it is automatically sent on according to its polarization, e.g. in the S case, on an uninterrupted straight line if it is V-polarized, or along the refracted extraordinary path if it is H-polarized. If an arriving photon's polarization is not matched with the crystal to which it is directed, then it has, for each of the two available polarization directions, a 50% chance of its own polarization to be reoriented in that available direction, and then to proceed along the corresponding path. 

Full set-up

Once the photon exits the crystal at which it arrived, on Bob's side, it is detected (with some non-zero probability) by the frog at the end of its path; this frog is able to perceive single photons and trained to react by jumping up (because, in training, it usually gets a fly when it detects a photon). 

The full set-up is what is illustrated in the quiltlet.  

Please don't go out to catch poor unsuspecting fireflies or amphibians to replicate this at home. It would be hard to set up the firefly side so that Alice really has sources that let only one photon imping on a crystal at a time. 

Ingrid Daubechies

Single-photon detecting frogs also haven't been found in nature yet. It has been demonstrated that frog retinas, at cold temperatures (alluded to by the ice cubes in the frogs' basins on the quiltlet) can indeed perceive the arrival of a single photon and induce a neuronal firing in about 30% of the cases. 

But this is a far cry from having a stable of trained single-photon detecting frogs to provide to Bob ...

Quantum protocol explanation

How then does all this work? How can Alice and Bob use this set-up to agree on a secret binary key sequence, and how can it be secure against interception? 

Alice will send her photons one by one (and not in beams); she sends those at a sequence of times that have been pre-arranged with Bob (and which Eve may well know). Whether or not she sends them from the S or the D side is picked completely randomly, with equal probability for both sides, for each individual photon.

For each side she picks equally randomly, each time, one of the two available polarizations (V or H for the S side, LD or RD for the D side); Bob and she have agreed to give the label 1 to a V photon on the S side or to a LD photon on the D side, and the label 0 to H photons on side S, and RD photon on side D.  

Since there are several stages where the photon may, with some non-zero probablity, continue in a different direction than on the desired path, and also a non-zero probability that even a photon that arrives at Bob's end in one of the four possible channels is not detected by the frog sitting there,  Bob needs to let Alice know for which of those prearranged times one of his frogs jumped up. He also lets her know (again without shielding it from Eve's possible eavesdropping), for each of these times, whether the reacting frog was on the S or D side. Alice responds, still openly, by telling Bob for which of those time slots she had picked the same side (S or D) as that where Bob's frogs detected it.

If Eve didn't intervene (see below for when she does), then for those particular photons where Alice and Bob have identical sides, the polarization that Alice picked (H or V for the Standard crystal, LD or RD for the Diagonal crystal) will have survived the whole trajectory unscathed and Bob will know the exact polarization choice that Alice made by checking his records of which frog jumped for each instance. By translating the polarizations into binary symbols, using the agreed-upon convention, Bob and Alice have then at their disposal identical *secret* binary sequences that they can start using for communication that requires a secret binary key. 

But what if Eve tried, over the open bit of the channel, to detect the polarizations of the photons coming through? Since she doesn't know whether they are S or D photons at that point, any measurement she makes will, on average, lead to a rearranging of the photon from S to D, or vice versa, in about 50% of all the cases. The photon of which she has measured the polarization will thus be still "correct" (i.e. have a polarization identical to what Alice sent) in only 50% of the cases. In the 50% of the cases where her interference has changed the polarization, the further processing by Bob's half-silvered mirror may nevertheless send it, half the time, to the crystal corresponding to Alice's original choice (S or D); this results in it still being a candidate for a "matched" photon once Alice and Bob have compared the S/D status of Bob's detected photons with Alice's records. Its polarization, however, has a 50% chance of being in either of the two possible choices available, regardless of the original polarization with which Alice sent it on its way.

In order to detect whether some interference is happening, Alice and Bob can split the secret bit sequence on which they have "agreed" into two parts -- one part to preserve as a secret key, and one that they sacrifice to check the security of their quantum key sharing channel. They communicate the bits in this last part to each other; if these supposedly identical bit sequences differ from each other, then Eve must have interfered, and their attempt at obtaining a common secure secret key has failed. But if they are truly identical, then they have succeeded!

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