Book chapter ; Quantum technology, quantum computing, quantum sensing, quantum cryptography

1. 14
14.1 Entangled states 296
14.2 Generation of entangled
photon pairs 298
14.3 Single-photon
interference
experiments 301
14.4 Bell’s theorem 304
14.5 Principles of
teleportation 310
14.6 Experimental
demonstration of
teleportation 313
14.7 Discussion 316
Further reading 317
Exercises 318
Entangled states and
quantum teleportation
The third branch of quantum information processing is quantum telepor-
tation. This is a very new subject, and the aim of researchers working
in the field at present is to achieve proof-of-principle demonstrations at
the few-particle level. As we shall see, teleportation relies heavily on the
properties of entangled states. We therefore begin by describing the con-
cept of entangled photon states and explaining how they are generated
in the laboratory. This will enable us to describe some recent experi-
ments testing fundamental ideas of interference at the single-photon
level. We shall then discuss the Einstein–Podolsky–Rosen (EPR) para-
dox and Bell’s theorem, which will allow us to explain the principles
of teleportation, and describe how they have been demonstrated in the
laboratory. Finally, we shall briefly discuss a few of the wider issues that
arise from the EPR paradox and Bell’s theorem.
14.1 Entangled states
See A. Einstein, B. Podolsky, and N.
Rosen, Phys. Rev. 47, 777 (1935),
and E. Schrödinger, Die Naturwis-
senschaften 23, 807, 823, 844 (1935).
An English translation of the latter
is available in Proc. Am. Philos. Soc.
124, 323 (1980). Bohm’s variant on the
EPR experiment was originally devel-
oped in his book Quantum Theory,
published in 1951 by Prentice-Hall,
New Jersey. Bohm actually proposed
to make spin measurements on pairs
of atoms, but the version we present
here is the optical equivalent involving
polarization measurements on pairs of
photons.
Entanglement is one of the most counter-intuitive aspects of the
quantum world. The concept is linked to two famous papers in the histor-
ical development of quantum theory, and has come to the fore in recent
years with the advent of quantum information science. In 1935 Einstein,
Podolsky and Rosen published the ‘EPR’ paper on the properties of an
entangled two-particle system formed from the decay of a radioactive
source. Soon afterwards, Schrödinger coined the term ‘entanglement’ in
his cat paradox paper that has fuelled the imagination of students and
teachers alike for many years.
Let us first consider the EPR paper. We will present the argument in
the ‘EPRB’ form introduced by David Bohm in 1951. The scheme for
an optical EPRB experiment is shown in Fig. 14.1. A source S emits a
pair of photons arbitrarily labelled 1 and 2, with photon 1 going one
way and photon 2 going another. The polarization of each photon is
measured with a beam-splitter/detector arrangement similar to the one
presented in Fig. 12.2. We designate the polarization states | and |↔
as |1 and |0, respectively, according to the BB84 scheme in the ⊕ basis
given in Table 12.1.
The subtlety in the experiment occurs when we use a source that emits
correlated photon pairs. Correlated photon pairs have the following

2. 14.1 Entangled states 297
Fig. 14.1 Apparatus for an EPRB experiment. The source S emits two correlated
photons arbitrarily labelled 1 and 2 towards polarization detectors involving a polar-
izing beam splitter (PBS) and single-photon detectors D. The detectors are given
a subscript 1 and 2 to identify the photon and the results are designated 0 and 1
according to the scheme presented in Table 12.1 for the ⊕ basis.
properties:
1. The polarization of either photon 1 or photon 2 measured independ-
ently of the other is random.
2. The polarization of the pair of photons is perfectly correlated; that is,
if D1(0) fires, then D2(0) always fires, and if D1(1) fires, then D2(1)
always fires. Alternatively if D1(0) fires, then D2(1) always fires, and
vice versa.
The second property follows from internal conservation laws of the source
that will be discussed in Section 14.2.
Fig. 14.2 Schrödinger’s cat. A live cat
is put into a sealed box containing a
radioactive atom. The radiation emit-
ted by the decay of the atom is detected
by a Geiger counter, which activates
a relay on registering a count. The
relay is connected to a hammer which
smashes a sealed flask of cyanide, and
hence kills the cat.
A multi-particle system is described as being in an entangled state
if its wave function cannot be factorized into a product of the wave func-
tions of the individual particles. The mutual dependence of the results
of the polarization measurements on the correlated photon pair means
that the wave function has to be written in the form:
|Φ±
=
1
√
2
(|01, 02 ± |11, 12) , (14.1)
for the case of perfect positive correlation, and
|Ψ±
=
1
√
2
(|01, 12 ± |11, 02) , (14.2)
for perfect negative correlation, with the subscripts referring to the indi-
vidual photons. The wave functions in eqns 14.1 and 14.2 are thus
examples of entangled states. They are also called Bell states for
reasons that will become clear in Section 14.4.
The entangled form of the wave functions in eqns 14.1 and 14.2 implies
that a measurement of the polarization of one photon determines the
result of a polarization measurement on the other. Thus for the wave
function given in eqn 14.1 we will obtain either the result (0,0) or (1,1),
each with equal probability. Similarly, eqn 14.2 implies results of (0,1)
or (1,0) each with 50% probability. In both cases a measurement on one
photon allows us to predict the result of the measurement on the other
with 100% certainty.
The Schrödinger cat paradox illustrates the concept of entangled
states in a graphic way by considering the state of a live cat put into a
sealed box containing a radioactive atom as shown Fig. 14.2. The box

3. 298 Entangled states and quantum teleportation
also contains a devious mechanism such that the decay of the atom
triggers a device to smash a sealed flask of poison, thereby killing the
cat. The state of the cat is therefore entangled with the state of the atom.
If we wait for a time such that the probability of the atom decaying is
equal to 50%, then we can write the wave function of the system in the
form:
|Ψ =
1
√
2
(|live, 1 + |dead, 2) , (14.3)
where |1 and |2 represent the state of the undecayed and decayed
atom, respectively. This seems to imply that we have a state inside the
box where the cat is both dead and alive at the same time, in clear
contrast to our common experience. On opening the box, we would, of
course, find the cat dead or alive with probability equal to 50%.
Much to the relief of cat-lovers, there is no need to perform the
Schrödinger cat experiment in the laboratory. Paradoxes of this type
are not found in the macroscopic world, because large systems consist-
ing of many particles lose their quantum coherence through interactions
with the noisy macroscopic environment. (See Section 13.4.) Things are
different, however, at the microscopic level of isolated atoms and pho-
tons in a well-controlled environment. Entangled photon states of the
type required for the EPRB experiment can readily be generated in the
laboratory, and photon Schrödinger cat states have been demonstrated.
Quantum entanglement is not restricted to the case of two-particle
polarization that we have considered here. Two-particle photon states
with time or momentum entanglement can also be generated, and entan-
gled states involving three or more particles have many interesting
properties. However, we shall restrict our attention exclusively to two-
particle polarization states for simplicity’s sake. The reader is referred
to the bibliography for details of other types of entangled states.
14.2 Generation of entangled
photon pairs
Many of the early optical experiments on entangled states employed
atomic cascades in calcium to generate the correlated photon pairs. The
experiment consists of a pair of detectors arranged to collect the pho-
tons emitted in an atomic cascade from the 4p2 1
S0 excited state of
calcium as shown in Fig. 14.3(a). Figure 14.3(b) shows the correspond-
ing level scheme for the transitions involved. The cascade occurs by
allowed transitions at 551.3 and 422.7 nm via the 4p4s 1
P1 intermediate
level. Narrow-band interference filters F1 and F2 in front of the photo-
multiplier tube (PMT) detectors selected these photon wavelengths from
others produced by alternative decay routes. In the initial experiment
by Kocher and Commins in 1967, the calcium atoms were excited to
See C. A. Kocher, and E. G. Commins,
Phys. Rev. Lett. 18, 575 (1967).
the 4p2 1
S0 level by absorption of ultraviolet photons from a hydrogen
arc lamp. Photons at 227.5 nm from the lamp first excited the atoms
from the 4s2 1
S0 ground state to the 3d4p 1
P1 level, and the atoms then

4. 14.2 Generation of entangled photon pairs 299
Fig. 14.3 Correlated photon pair generation by atomic cascade in calcium. (a) Experimental arrangement employing two linear
polarizers (P) and photomultiplier tube (PMT) detectors. (b) Atomic level scheme. The narrow-band interference filters F1
and F2 used in the experiment were chosen to select the photons at 551.3 and 422.7 nm, respectively. (After C. A. Kocher and
E. G. Commins, Phys. Rev. Lett. 18, 575 (1967).)
dropped to the desired 4p2 1
S0 level by spontaneous decay. In the subse-
quent experiments by Aspect et al. described in Section 14.4.3, the atoms
were excited directly to the 4p2 1
S0 level by two-photon absorption of
photons at 406 and 581 nm from separate laser beams.
The initial and final states for the cascade are both J = 0 states
with no net angular momentum. This demands that the photon pairs
emitted in the cascade carry no net angular momentum. In addition, the
rotational invariance of J = 0 states, and the fact that the initial and
final levels are both of the same even parity, requires that the photon
pairs have the polarization correlation properties required for the EPRB
experiments. This correlation was confirmed by placing linear polarizers
in front of both detectors and checking for coincidences. The experiments
clearly demonstrated that the coincidences only occur when the axes of
the polarizers are aligned parallel to each other, indicating that Bell
states of the type given in eqn 14.1 are being produced.
Fig. 14.4 Schematic representation of
a down-conversion process within a
nonlinear crystal. A single photon of
angular frequency ω0 simultaneously
generates a pair of signal and idler pho-
tons of angular frequencies ω1 and ω2
subject to the phase-matching condi-
tions set out in eqns 14.4 and 14.5.
In the 1980s and 1990s new sources of correlated photon pairs with
higher flux rates were developed by techniques of nonlinear optics. (See
Section 2.4.) The correlated photon pairs were generated by the down-
conversion process in which a single photon from a pump laser at
angular frequency ω0 is converted into a pair of signal and idler photons
at angular frequencies ω1 and ω2, as shown in Fig. 14.4. Conservation
of energy and momentum, respectively, require that:
ω0 = ω1 + ω2, (14.4)
and
k0 = k1 + k2, (14.5)
where ki is the wave vector of the photon in the crystal. The second
of these conditions is equivalent to requiring that the nonlinear waves
and the fundamental beam all remain in phase throughout the nonlinear
medium. For this reason, the circumstances in which eqns 14.4 and 14.5
are satisfied simultaneously are called phase-matching conditions. The
down-conversion process is called degenerate when ω1 = ω2 = ω0/2, and
non-degenerate otherwise.
At first sight, it might seem that there would be many combinations of
frequencies and wave vectors that can be phase-matched. However, this is

5. 300 Entangled states and quantum teleportation
not the case because of the dispersion in the nonlinear crystal. (See Exer-
cise 14.4.) Dispersion is a general property of all optical materials and
refers to the variation of the refractive index with frequency. This means
that the refractive indices at the three different frequencies are in general
different, making it impossible under normal circumstances to satisfy
the phase-matching conditions. Fortunately, the nonlinear crystals are
also birefringent, which means that the refractive index depends on the
direction of the polarization of the light with respect to the crystal axes.
This allows us to balance birefringence against dispersion, and achieve
two different types of phase matching. In type-I phase matching the
polarizations of the down-converted photons are parallel to each other
and orthogonal to the pump photon, while in type-II phase matching
the down-converted photons have orthogonal polarizations.
Figure 14.5 illustrates the generation of entangled photon pairs by
degenerate down-conversion with type-II phase matching. The principle
of the technique in shown in Fig. 14.5(a). Ultraviolet photons from a
pump laser are focussed into a β-barium borate (BBO) crystal and are
down-converted to two red photons at half the frequency. The phase-
matching requirements determine that the down-converted photons
emerge in cones of opposite polarization, leading to a double ring pat-
tern with two intersection points, as shown in Fig. 14.5(b). Equation 14.5
demands that if we find a vertically polarized photon at one of the inter-
section points, then the photon at the other intersection point must be
horizontally polarized, and vice versa. However, the photon at each inter-
section point might have originated from either of the two oppositely
polarized rings and can therefore be horizontal or vertical with equal
probability. The arrangement therefore produces states of the type:
|Ψ =
1
√
2
| ↔1, 2 + eiφ
| 1, ↔2
, (14.6)
where φ is an optical phase that can be altered with compensator plates.
By setting φ equal to 0 to π we can then produce either of the Bell
Fig. 14.5 Generation of polarization entangled photon pairs by degenerate down-
conversion with type-II phase matching. (a) Experimental arrangement employing
an ultraviolet pump laser and a BBO crystal. The phase-matching conditions require
that the beams emerge in cones of opposite polarization. (b) Degenerate type II
down-conversion as seen through a narrow band filter. The two entangled photons cor-
respond to the intersection points of the rings. (After P. G. Kwiat, et al., Phys. Rev.
Lett. 75, 4337 (1995), c
American Physical Society, reproduced with permission.)

6. 14.3 Single-photon interference experiments 301
states given by eqn 14.2. Down-conversion sources of this type have now
generally supplanted atomic cascade sources for practically all of the
experiments that require polarization-entangled photon states.
Example 14.1 A correlated pair of photons is generated by non-
degenerate parametric down conversion using a laser at 502 nm. Given
that the wavelength of one of the photons is 820 nm, calculate the
wavelength of the other.
Solution
We use eqn 14.4 with ω = 2πc/λ, which implies:
2πc
λ0
=
2πc
λ1
+
2πc
λ2
.
Hence:
1
λ2
=
1
λ0
−
1
λ1
=
1
502
−
1
820
nm−1
,
giving λ2 = 1294 nm.
14.3 Single-photon interference
experiments
The first measurement of a single-
photon interference pattern was made
as early as 1909, when a Young’s slit
experiment was performed with only
one quantum of energy within the
apparatus at a given instant. See G. I.
Taylor, Proc. Camb. Phil. Soc. 15, 114
(1909).
The main reason for introducing correlated photons pairs in this chapter
is to explain how they can be used to test Bell’s theorem and to imple-
ment quantum teleportation. However, the use of correlated photon pair
sources has also enabled the testing of several fundamental ideas about
the nature of photon interference, and it is worthwhile to consider some
of these briefly here.
Consider first the experimental arrangement shown in Fig. 14.6. Signal
and idler beams of the same polarization and frequency are generated by
type-I degenerate down conversion in a nonlinear crystal and are made to
Fig. 14.6 Experimental arrangement for demonstrating single-photon interference
effects using correlated photon pairs. M1 and M2 are mirrors, D1 and D2 are single-
photon counting detectors, and BS is a 50 : 50 beam splitter. The path difference
between the signal and idler beams can be adjusted by translating BS up and down.
(Adapted from C. K Hong, Z. Y. Ou, and L. Mandel, Phys. Rev. Lett. 59, 2044
(1987).)

7. 302 Entangled states and quantum teleportation
interfere at a beam splitter BS. The path difference between the beams
can be adjusted by translating the beam splitter up and down. With
high-intensity classical beams, we would expect to see bright and dark
fringes appearing at the output ports as the beam splitter is translated.
The total signal on detectors D1 and D2 would be constant, but the
magnitude of the signal on the individual detectors would oscillate in
anti-phase as BS is translated.
The experiment becomes more interesting when we operate at the
single-photon level. The signal and idler beams now contain correlated
photon pairs. When the path lengths of the beams are identical, the
two photons arrive at the beams splitter at the same time and inter-
fere. When single photons interfere at a 50 : 50 beam splitter, destructive
interference prevents the possibility that the two photons go to different
output ports, and both photons therefore emerge at the same output.
(See Exercise 8.11.) Hence the only possible results are that both pho-
tons go to D1 or both go to D2, leading to no coincidence events on the
detectors.
The absence of coincidences when the path lengths are equal was ver-
ified experimentally in 1987 by Hong, Ou, and Mandel. For this reason,
See C. K. Hong, Z. Y. Ou, and
L. Mandel, Phys. Rev. Lett. 59, 2044
(1987).
the arrangement shown in Fig. 14.6 is sometimes called a Hong–Ou–
Mandel interferometer. An argon ion laser operating at 351.1 nm was
used as the pump laser and potassium dihydrogen phosphate (KDP) as
the nonlinear crystal. When the difference in the path lengths of the sig-
nal and idler beams was larger than the coherence length, no interference
occurred. In this situation, each photon randomly exits at either output
port, producing coincidences on D1 and D2 for 50% of the events. How-
ever, when the path difference was smaller than the coherence length,
no coincidences were recorded, confirming the single-photon interference
effect.
Consider now the interference experiment shown in Fig. 14.7. The
interferometer incorporates two down-converting nonlinear crystals NL1
and NL2, both driven by photons derived from a single pump laser. The
overall down-conversion efficiency is rather small, so that it is extremely
unlikely that correlated photon pair generation occurs simultaneously
in the two nonlinear crystals. The crystals are arranged so that the
paths of the two idler beams i1 and i2 are coincident. (This is possible
because the nonlinear crystal NL2 is transparent at the idler frequency.)
A detector Di registers the combined signal of these two idler beams.
At the same time, the signal beams s1 and s2 are combined at a 50 : 50
beam splitter BS2 and the signal at one of the output ports is registered
by the detector Ds.
From a classical perspective, the two signal beams s1 and s2 should
interfere at BS2, and we would therefore expect to observe interference
fringes on Ds as the path difference is varied by translating BS2. At the
single-photon level, the photon emerges at either of the output ports of
BS2 with a probability determined by the classical interference pattern
of s1 and s2. However, when working with single photons, it is natural to