2.3 Quantum Key Distribution in Detail
2.3.3 The E91 Protocol
The E91 protocol is yet another implementation of quantum key distribution; it involves the use of entangled pairs of particles. It is so named after Artur Ekert, who published it22 in 1991. The basic setup is the following: users A and B are physically separated and can communicate directly only via classical means. Both A and B independently are linked to a source of entangled spin–12 particles, so that the first particle of each pair is sent to A and the second particle to B. As we have explained at length previously, entangled particles do not have a known individual state; it is only when they are measured that each particle is assigned an individual state. Now, when A performs a measurement on her particle, this will affect B’s particle so that the outcome of B’s measurement is known.
Example 2.11 The source generates an entangled pair of particles in the joint state
jψsi[12]= p1
2(j"#i j#"i) = 1
p
2 j"i[1]j#i[2] j"i[2]j#i[1]
Note that the possible values of spin, when a spin–12 particle is measured, are +}2 (called “spin–up,” corresponding to statej"i) and }2 (called “spin–down,” corresponding to statej#i). If A is given particle 1, and B is given particle 2, then:
2.3. Quantum Key Distribution in Detail ❦ 31 Figure 2.3The B92 Protocol.
1. Transmissions over a Quantum Channel
(a) A generates a private, random binary sequencefdigof lengthm.
(b) A prepares a sequence jΨ(di)iof mpolarised photons, each representing a bit from
fdig, withjΨ(0)i=j0iandjΨ(1)i=j i, so thathΨ(0)jΨ(1)i 6=0.
(c) B receives the photons and chooses an operator, P0or P1 at random, with which to
measure each one. The operators are defined in Equations (2.23) and (2.24). The choices of operators for alliform a sequence
fop(i)jop(i) =P0orop(i) =P1for allig
while the results of B’s measurements form a sequence of bitsfd0ig.
2. Transmissions over a Classical, Possibly Public Channel
(a) B reports to A, for eachi, the choice ofop(i). A then tells him whetherop(i)iscom- patible withjΨ(di)i; if so, A and B know with certainty that, for that particular value ofi,d0i = di. All values ofdi andd0i for which B has used a compatible measurement operator are kept and form thetentative key;other values are discarded.
(b) As for BB84: the tentative key is processed further, in order to correct errors and to eliminate any information which may have “leaked” about the tentative key. This re- sults in afinal key.
The meaning of “compatible” is defined as follows: ifdi =0, thenop(i) =P0is a compatible measure-
ment operator; conversely, ifdi=1, thenop(i) =P1is compatible.
❧
when A measures her particle and obtains “spin–up” (statej"i), then subsequent measurement by B will produce “spin–down” (statej#i);❧
when B measures her particle and obtains “spin–down,” then subsequent measurement by B will produce “spin–up.”This assumes that A and B’s measurements are compatible, as we shall explain shortly.
As the example demonstrates, although the first measurement always produces a random result, the second is completely deterministic. We will take up this issue again in Chapter 3, where we will see how this behaviour can be modelled using a Markov chain.
A certain complication arises with regard to the nature of the second measurement. The sec- ond measurement is completely deterministiconlyif it iscompatiblewith the first. Physicists pre- fer the term “totally anticorrelated” to characterise compatible measurements in this context. But what exactly do these terms mean?
In protocol E91 for key distribution, the two users make measurements of particle spin using analysers whose principal axis is in one of three orientations; for user A, these are at anglesφA1 =
0 ,φ2A=45 andφ3A=90 with respect to the vertical. For user B’s analyser, possible orientations
are at anglesφB1 = 45 ,φ2B = 90 , andφB3 = 180 . When A measures particle 1 by setting her
analyser’s axis at a given angleφ,so must B,when measuring particle 2, in order that they obtain
32 ❦ Chapter 2. A Survey of Quantum Protocols and Security Criteria Sothe axes of the two analysers have to be parallelin order to obtain totally anticorrelated results. There are only two cases in which A and B’s results are guaranteed to be totally anticorrelated: when both analyser orientations are set at 45 or 90 .
Notation 2.3 To quantify the possible measurement results for A and B, physicists use the so–called correlation coefficient E(~αi,~βj); this varies as a function of~αi, the orientation chosen by A to make the first measurement (~αi has an angleφAi with respect to the vertical), and of~βj, the vector chosen by B to make
the second measurement (~βjhas an angleφBj with respect to the vertical).
The value ofE(~αi,~βj)depends on the probabilitiesP (~αi,~βj), which correspond to the out-
comes of measurements with~αiand~βj. By way of example, the symbol
P+ (~αi,~βj)
denotes the probability of the following two events occurring together:
1. The first measurement, along direction~αi, produces “spin–up,” and
2. The second measurement along direction~βj, produces “spin–down.”
Notation 2.4 The subscripts+and refer to “spin–up” (cf. value+}
2), and “spin–down” (cf. value
}
2), respectively.
The definition of the correlation coefficient is thus:
E(~αi,~βj) =P++(~αi,~βj) +P (~αi,~βj) P +(~αi,~βj) P+ (~αi,~βj) (2.25)
Quantum theory requires thatE(~αi,~βj) = ~αi,~βj =~αi ~βj. Importantly for our purposes, the
total anticorrelation of A and B’s results, which occurs when their analysers’ axes are parallel, is expressed as follows:
E(~α2,~β1) =E(~α3,~β2) = 1 (2.26)
But how does all this relate to key distribution? We have just seen that, if two users are sup- plied with a set of spin–entangled particles then, by measuring them, they may obtain totally an- ticorrelated results. Without exchanging any information directly, they can thus obtain sequences of bits which are the inverse of each other. Assuming all measurements are compatible, and no disturbance is produced prior to measurement, it is thus possible to establish a common sequence of bits. Two problems remain:
1. Not all measurements are compatible. The compatible choices of analyser orientations (~α2,~β1
and~α3,~β2) are only two possibilities out of nine in total. It is therefore highly likely that the
second measurement will not be totally anticorrelated with the first.
2.3. Quantum Key Distribution in Detail ❦ 33 The first problem is solved by performingerror correction,as is done in BB84 and B92; non– compatible measurements generate errors with nonzero probability. Ekert23describes this step as
follows:
“ After the transmission has taken place, A and B can announce in public the orientations of the analysers they have chosen for each particular measurement and divide the measurements into two separate groups: a first group for which they used different orientation of the analysers, and a second group for which they used the same orientation of the analysers. They discard all measurements in which either or both of them failed to register a particle at all. Subsequently, A and B can reveal publicly the results they obtained but within the first group of measurements only. ”
The measurements in the “second group” are known to be totally anticorrelated after the above steps, and can be converted to a secret key.
Now, if no disturbance has occurred due to eavesdropping, the quantity
S=E(~α1,~β3) +E(~α1,~β2) +E(~α2,~β3) E(~α2,~β2) (2.27)
will be equal to 2p2, as required by the rules of quantum theory. When eavesdropping does occur, however, the value ofSchanges so that
p
26S6p2 (2.28)
and this allows A and B to detect the eavesdropper. Evaluating this inequality is, effectively, a test for eavesdropping.
The full E91 protocol is listed in two parts, in Figures 2.4 and 2.5.