Drake G.W.F. (editor) Handbook of Atomic, Molecular, and Optical Physics
Подождите немного. Документ загружается.
Quantum Information 81.3 Unitary Transformations 1221
two-qubit state is again in one of the four Bell states
(81.12). Since they are mutually orthogonal to each
other, the receiver can tell them apart by measuring
in the Bell basis. In that way, two classical bits of
information (the information about the single-qubit uni-
taries) can be transmitted using only a single qubit at
a time.
An experiment using entangled photon pairs was
reported in [81.77], which demonstrated dense coding
in practise.
81.3 Unitary Transformations
As in classical information theory, one has to define
a certain set of allowed operations or maps between
states of an information-carrying system. Classical in-
formation processing allows operations such as the
NAND (Not-AND), which is defined by the Boolean
operation
X
1
∧ X
2
∧···∧X
n
on the Boolean variables
X
1
, X
2
,... ,X
n
. This operation is not reversible in the
sense that, given the outcome of the operation, there is no
unique way of determining what the input was. Hence,
such types of classical gates destroy information during
the course of their operation.
Loss of information or irreversibility of an opera-
tion is accompanied by an increase in entropy of the
state that has been operated on. For an initially pure
quantum state having zero entropy, this means mix-
ing the state and destroying its superposition nature and
hence its quantum-mechanical entanglement. Therefore,
the advantages of parallelism inherent in the superpo-
sition is lost. Thus, valid quantum operations in this
sense can only be those that preserve the purity of
states, hence unitary operations and partial projective
measurements. Of course, in order to reset a quan-
tum register, information has to be erased [81.78, 79].
This erasure procedure is described by a completely
positive map (see, e.g. [81.3]). Note that any allowed
quantum operation is completely positive; unitary op-
erations are a special class of these. An example
of how this constrains operations in quantum rather
than classical information theory is the absence of
a NOT operation in the former, as a NOT operation
cannot be described in terms of completely positive
maps [81.80].
81.3.1 Single-Qubit Operations
It is instructive to give an example of how to classify
all possible unitary operations that can act on a single
qubit. A unitary operation acting upon the basis states
{|0, |1} can be represented by a unitary (2 × 2)-matrix,
hence a matrix that represents an element of the unitary
group U(2). This group has four generators, the identity
matrix and the three Pauli matrices. Hence, all unitary
(2 × 2)-matrices are linear combinations of those four
matrices. Given the way they act upon basis states, they
can be written as
ˆ
I =|00|+|11| ,
(81.14)
ˆ
σ
x
=|01|+|10| , (81.15)
ˆ
σ
z
=|00|−|11| , (81.16)
and, by virtue of the commutation rules for U(2)-
generators,
ˆ
σ
y
= i
ˆ
σ
x
ˆ
σ
z
. Sometimes, the short-hand
notation X ≡
ˆ
σ
x
, etc., is used.
A particularly useful single-qubit gate which is not
just one of the Pauli operators is the Hadamard gate H.
In terms of Pauli operators it is defined as H =(X +Z)/
√
2. Its purpose is to transform each basis state into equal
superpositions of basis states, i. e., |0→(|0+|1)/
√
2
and |1→ (|0−|1)/
√
2. The Hadamard gate is used to
initialize an equal superposition of all possible N-qubit
basis states from the state |0
⊗N
. Hence,
N
i=1
H
i
|0
⊗N
=
1
√
N!
k
|x
k
, (81.17)
where the |x
k
are all N! possible words of length N
containing 0’s and 1’s.
81.3.2 Two-Qubit Operations
Similarly to the single-qubit case, one can write down
all possible unitary operations on two qubits by noting
that they constitute a representation of U(4). We will
not give an exhaustive list of all 16 generators of this
group since they can be found in the literature. Instead,
we give examples of particularly useful two-qubit gates.
Trivially, the group U(4) contains an 8-parameter sub-
group U(2)×U(2) which consists of operations such as
X
1
⊗X
2
,etc.
Particular examples of nontrivial two-qubit gates
are the controlled-NOT and the controlled-phase gate,
Part F 81.3
1222 Part F Quantum Optics
Control
Target
00典
01典
11典
10典11典
10典
01典
00典
Fig. 81.4 Symbol and truth table of the controlled-NOT
gate. The target qubit is flipped depending on the state of
the control qubit
defined in terms of Pauli operators as
C
12
=|0
1
0
1
|⊗I
2
+|1
1
1
1
|⊗X
2
, (81.18)
Π
12
=|0
1
0
1
|⊗I
2
+|1
1
1
1
|⊗Z
2
, (81.19)
where in both cases qubit 1 acts as the ‘control’ and
qubit 2 acts as the ‘target’ (Fig. 81.4). The net ef-
fect of the controlled-NOT gate is to interchange the
states |10↔|11, whereas the controlled-phase gate
changes the phase of the basis state |11 by π and
leaves all other states unchanged. The controlled-NOT
gate has an interpretation as a sum gate in that it
performs a mapping |x, y →|x, x ⊕y where the ad-
dition has to be taken modulo 2. Moreover, it acts as
an entangling gate when acting on tensor products of
superpositions.
81.3.3 Multi-Qubit Gates and Networks
To realize a unitary operation on many qubits for
a particular algorithm one would need a network of
single-particle and multi-particle quantum gates. Quan-
tum networks enable a prepared input state to be
transformed by the appropriate unitary operator to a final
state which is then measured. Deutsch’s model of quan-
tum networks enables us to decompose the network into
component gates in diagrammatic form [81.81]. The task
is then to optimize the sequence of gates. One can treat
quantum gates acting on N qubits as being elements of
the group U
N
2
which has N
4
generators. This, how-
ever, is not a particularly transparent or useful way of
looking at these gates. Much more useful, and of im-
mense practical importance, is a result essentially from
linear algebra which states that every N-qubit gate can
be decomposed into a network of single-qubit and two-
qubit operations [81.82]. As a matter of fact, there is an
even deeper result which says that every N-qubit gate
can be generated by a network that consists only of very
few elementary building blocks, the so-called universal
set of quantum gates [81.83–85]. This set contains all
possible single-qubit rotations and one nontrivial two-
qubit gate, such as the above-mentioned controlled-NOT
or controlled-phase gate.
81.4 Quantum Algorithms
The search for algorithms that would run faster on
a quantum computer than on any classical computer
is a formidable task. When we say faster, we actu-
ally mean that the temporal complexity in performing
a given task should be drastically reduced. The hope is
that eventually one will find algorithms that provably run
exponentially faster on a quantum computer compared
to a classical computer.
81.4.1 Deutsch–Jozsa Algorithm
Let us give a particularly instructive example known
as the Deutsch–Jozsa algorithm [81.86]. Let us sup-
pose one is given a string of N bits and a Boolean
N-bit function f(x) such that |x|y →|x|y ⊕ f(x)
for x ∈
0,... ,2
N
−1
.From f(x) is known that it
either returns always 0 or 1 (in which case one calls
it ‘constant’) or returns 0 and 1 with equal probabil-
ity (in which case it is ‘balanced’). The task is to find
out whether f(x) is constant or balanced. Classically,
one needs at least 2
N−1
+1 strings to find the answer.
Quantum-mechanically, one prepares the trial input in
a superposition of all possible computational basis states
using the Hadamard gate from (81.17) and uses one
function evaluation on all basis states simultaneously
(Fig. 81.5). A measurement outcome other than 0 on any
of the N query qubits then tells that the function f(x) is
balanced. If it were constant, the measurement outcome
would be 0 in all query qubits. This quantum paral-
lelism is at the heart of the increase in speed that occurs
in quantum computation. Versions of the Deutsch–Jozsa
0典
N
N
H
N
H
H
1典
xx
yy
U
f
f(x)
Fig. 81.5 Gate network for implementing the Deutsch–
Jozsa algorithm
Part F 81.4
Quantum Information 81.5 Error Correction 1223
algorithm involving a few qubits have been implemented
in nuclear magnetic resonance systems [81.87, 88]as
well as ion traps [81.89].
81.4.2 Grover’s Search Algorithm
In contrast to the preceding example, which always gives
the desired answer after exactly one trial, the quantum
search algorithm by Grover [81.90] uses a procedure
that amplifies the sought after result by a method called
‘inversion about average’. The goal of Grover’s algo-
rithm is to search an unsorted database with 2
N
entries
out of which only one fulfils a given criterion. As in the
Deutsch–Jozsa algorithm described above, the query is
simultaneously run on all 2
N
possible N-qubit basis
states, being prepared in an equal superposition. It is
assumed that the state that satisfies the search criterion
will acquire a phase shift of π. After this step, the in-
version about average is carried out. It is represented by
a diffusion operator
ˆ
D = 2
ˆ
P −
ˆ
I ,
(81.20)
where
ˆ
I is the identity operator and
ˆ
P a projection op-
erator that averages each input vector with respect to its
components. Compared to the previous average value
of probability amplitudes, after each of these steps the
magnitude of the desired state increases by O
2
−N/2
.
This procedure is repeated, and after only O
2
N/2
steps
a projective measurement yields the desired result with
probability of O(1), or more precisely, of more than
a half. This is a quadratic increase in speed compared to
classical search algorithms, which need O
2
N
steps.
We have seen in these two examples that the use of
quantum-mechanical superpositions can lead to a speed
increase compared to the best classical algorithms. The
most prominent example of such increases in speed
is found in Shor’s algorithm for factoring large num-
bers [81.91,92]. Its core element is essentially a quantum
Fourier transform to find the period of a Boolean
string. This algorithm provides an exponential speed in-
crease over any known classical algorithm. It should
be noted, however, that the fastest known classical
algorithm has not yet been proven to be optimal. Im-
plementations in nuclear magnetic resonance systems
with a few qubits have been reported for the quantum
Fourier transform [81.93] as well as Shor’s factoring
algorithm [81.94].
It turns out that there exists a whole class of algorith-
mic problems, the hidden subgroup problems [81.95–
97], whose quantum-mechanical analogues can lead
to exponential increases in speed over their classical
counterparts, a particular example of which is Shor’s
factoring algorithm. Another instance of quantum-
mechanically exponentially faster processes is found in
quantum random walks on hypercubes where hitting
times, i. e., the traversal time of an excitation across the
cube’s diagonal, can be exponentially faster than for
classical random walks [81.98].
81.5 Error Correction
As we have discussed, the essence of quantum infor-
mation processing is the use of quantum superpositions,
interference, and entanglement. But quantum interfer-
ence is fragile. It appears in practice that it is very
difficult to maintain a superposition of states of many
particles in which each particle is physically separated
from all the others. Entanglement turns out to be in-
credibly delicate. The reason for this is that all systems,
quantum or classical, are not isolated; they interact with
everything around them: local fluctuating electromag-
netic fields, the presence of impurity ions, coupling to
unobserved degrees of freedom of the system contain-
ing the qubit, etc. These fluctuations destroy quantum
interference. A simple analogy is the interference of op-
tical waves in Young’s double slit experiment. In that
apparatus waves from two spatially separated portions
of a beam are brought together. If the two parts of the
beam have the same phase, then the fringe pattern re-
mains stable. But if the phase of one part of the beam
is drifting with respect to the other, then the fringe pat-
tern will be washed out. And the more slits there are in
the screen, the lower the visibility for the same amount
of phase randomization per pair of slits. The sensitiv-
ity of an N-qubit register to decoherence is even worse,
as a maximally entangled N-particle state decoheres at
arateN times faster than a single particle [81.99], one
of the reasons why the world around us appears so clas-
sical. A single bit of information lost to an unobserved
degree of freedom will result in the reduction of the
quantum superposition to a mixed state. Yet correcting
errors due to environmental interactions is essential if
a quantum computer is to be constructed: to do ‘fault
tolerant’ computing we need to be able to execute many
gate operations coherently within the decoherence time
if we are to have a chance of building a scalable quantum
register [81.100].
Part F 81.5
1224 Part F Quantum Optics
It might appear that the problem of stabilizing a reg-
ister of qubits is hopeless, like trying to balance several
pencils on their tips while on the deck of a ship in
a storm. But, amazingly, quantum mechanics provides
a way to solve this problem, through the use of even
higher levels of entanglement. Shor and coworkers,
and independently Steane showed in the mid-1990’s
that encoding information in entangled sets of qubits
offered the opportunity to execute quantum error correc-
tion [81.101, 102]. That one can do this is a remarkable
consequence of entanglement. In classical information
processing, inevitable environmental noise is dealt with
by error correction. In its simplest form, this involves
repeating the message transmission or calculation until
a majority result is obtained.
But there are more efficient ways, for example, use
of a parity check on a block of bits. It turns out that
a similar notion can be applied to a quantum register.
However, the application is not straight forward be-
cause the contents of a register cannot be measured
without destroying the superposition state encoded in
it. The problem then is to determine what errors might
be present in a quantum register without looking at the
qubits. The elegant solution is to entangle the qubits in
question with those in an ancillary register and meas-
ure the ancillary register. Because the two registers are
correlated, the results of the measurement of the an-
cilla reveal any errors present in the processing register
without destroying any coherent superpositions in the
processing register itself.
The first experimental demonstration of quantum
error correction used NMR techniques [81.103–105],
but given the inherently mixed nature of NMR quan-
tum computing [81.106], this has had limited impact
on quantum information processing (QIP). However,
very recently, Wineland’s group in Boulder has suc-
ceeded in implementing quantum error correction using
laser-cooled trapped ions [81.107].
Another way to prevent the register coherence from
falling apart is to know a little about the sort of noise
that is acting on it. If the noise has some very slow com-
ponents (or those with very long wavelengths), then it
is sometimes possible to find certain combinations of
qubit states for which the noise on one qubit exactly
cancels the noise on another. These qubit states live
in a ‘decoherence-free subspace’ (DFS) [81.108–111].
A computer will then be immune to environmental per-
turbations if all the computational states lie in this DFS.
The connection between DFS and quantum error correc-
tion codes has been shown in [81.112]. Kwiat [81.113],
using photonic qubits, and Wineland’s group [81.114],
using trapped ions, have demonstrated the use of DFS
experimentally.
Although these results are encouraging, we are still
a long way from the figure of merit for gate time to
decoherence time needed for fault tolerance.
81.6 The DiVincenzo Checklist
DiVincenzo gave a list of requirements that a phys-
ical implementation must fulfil in order to qualify as
a sensible candidate for an implementation of quantum
information processing [81.115].
81.6.1 Qubit Characterization, Scalability
Each physical implementation must be tested upon how
qubits should be encoded. For a qubit being essentially
a two-level system, this task is generally not too diffi-
cult. Several candidates, such as electronic or nuclear
spin, photon polarization, choice of path in an inter-
ferometer, degenerate ground states of an atom or ion,
charge or flux states in superconducting quantum inter-
ference devices (SQUIDs) or exciton population, have
all been recently explored. Much more challenging will
be the question as to whether there are fundamental
or technological limitations of having many of those
qubits being operated upon seperately, hence whether
the system can be scaled up to contain potentially many
qubits.
81.6.2 Initialization
Once the qubits have been specified, each quantum infor-
mation processing or quantum computation task needs
to be able to start from a well-defined state. This can be
basically any quantum state of the many-qubit system
as long as it is a product state and can be prepared error-
free. Commonly, this state is then called the ground state
and denoted by |0
⊗N
.
81.6.3 Long Decoherence Times
In order to ensure error-free computation without loss of
purity of quantum superpositions, the decoherence times
that are relevant for the quantum operation should be
much longer than the gate operation time itself. In most
Part F 81.6
Quantum Information 81.7 Physical Implementations 1225
situations, decoherence limits the number of qubits that
can be worked on simultaneously, thus affecting the scal-
ability of the system. Typical examples of decoherence
processes are heating mechanisms in ultracold systems,
such as ion trap or atom chip experiments, spin relax-
ation in NMR-type experiments, or absorption in linear
optical elements.
Generally, decoherence is unavoidable due to the
basic principles upon which quantum information pro-
cessing is supposed to work. Avoiding decoherence
means isolating the system from the outside world, the
environment. But controlling the interaction between
subsystems always has the negative effect of bringing
the system in contact with the environment and there-
fore necessarily introduces decoherence. Once one has
accepted that decoherence is unavoidable, ways have
to be found to guard against it. Several error-correction
schemes have been proposed that can correct for certain
small amounts of decoherence as described in Sect. 81.5.
81.6.4 Universal Set of Quantum Gates
A necessary prerequisite for quantum information pro-
cessing and quantum computing is the ability to generate
a set of quantum gates that can be considered universal.
With such a set it will then be possible to generate all
other quantum gates by concatenating them to form suit-
ably arranged networks. The choice of which set out of
the many possible is taken, depending on the physic-
al implementation itself. Basically, it is determined by
the operations that are intrinsically simple for the given
interaction Hamiltonian. In some applications, such as
the ion trap experiments, the controlled-NOT gate is pre-
ferred as the nontrivial two-qubit gate, whereas in linear
optical networks one rather works with the controlled-
phase gate.
81.6.5 Qubit-Specific Measurement
The last requirement is to be able to read out the
result of the computation. That is, there has to be
a way of providing a selective projective measure-
ment. This proves to be a major challenge in most
proposals for implementing quantum computing. Ex-
amples of the challenges involved are the necessity
to provide photon-number resolving photodetectors,
single-electron charge measurement devices, or single-
spin measurements.
81.7 Physical Implementations
Quantum information theory regards relevant objects
as abstract quantities in a Hilbert space of a certain
dimension. The different strands in its development
can be roughly divided into generalized spin sys-
tems (qubits, qudits, the d-dimensional generalization
of qubits) living in finite-dimensional Hilbert spaces,
and harmonic oscillator systems which naturally live in
infinite-dimensional Hilbert space. The latter are hence
called continuous-variable (cv) systems. Examples for
generalized spin systems are polarization states of a pho-
ton, magnetic sublevels of atomic hyperfine states or,
to a good approximation, electronic levels of atoms
and ions. Harmonic oscillator systems can be found in
atomic populations in optical lattices, electromagnetic-
field modes or indeed any excitation of a bosonic
quantized field.
Both strands have their own virtues and disad-
vantages. Harmonic oscillator systems are naturally
abundant and, in their materialization as photons, rather
easily accessible and manipulable. However, due to their
Hilbert space dimensionality, the nonlinear operations
that are required for quantum gates are generally hard
to achieve. In contrast, spin-like systems (apart from
photon polarization) require more experimental effort in
preparing them but, unlike harmonic oscillator systems,
they can show effective nonlinearities due to the finite
dimensionality of their Hilbert space (the nonlinearities
appear when coupled to another physical system which
can then be traced out).
81.7.1 Linear Optics
The use of photons as carriers of quantum informa-
tion seems to be a straightforward matter for several
reasons. First, they are easy to produce and to manip-
ulate, and second, they both show spin-like behaviour
(polarization) and can be treated as continuous-variable
systems (Gaussian states). There exists, however, yet
another possibility to store and manipulate quantum
information in photons, namely when encoding in-
formation in number states or Fock states. But as
noted before, for photons being bosonic systems, there
are no natural nonlinearities (at least none which is
strong enough) on the level of single or few photons.
The trick here is to use conditional measurements or
measurement-induced nonlinearities. The idea was first
Part F 81.7
1226 Part F Quantum Optics
Ancilla photon
Control qubit
Target qubit
PBS
PBS 45
Fig. 81.6 Schematic setup of an all-optical controlled-NOT
gate. Control and target qubits are encoded in the polariza-
tion of single photons. These are fed into polarizing beam
splitters (PBS), one of which is rotated by 45 degrees
put forward in [81.116, 117] and realized experimen-
tally in [81.118,119], where a polarization-encoding was
used. Figure 81.6 shows the schematic setup of a sim-
plified version of a controlled-NOT gate with one single
ancilla photon (after [81.120]).
Measurement-induced nonlinearities make use of
the fact that unitary transformations in a larger
Hilbert space, e.g., with added auxiliary photon modes
combined with photodetection, can yield effective non-
linearities [81.121]. The drawback is, however, that the
wanted nonlinearity is conditioned on the appearance
of a certain measurement pattern which means that
these schemes work only with a certain probability.
Bounds for certain classes of gates have been reported
in [81.122–125].
The set of quantum gates that can be considered
fundamental differs slightly from most other physical
implementations. Within the qubit encoding in photon-
number states the gates that can actually be implemented
efficiently are those that act within Fock layers (sub-
spaces of fixed total photon number), such as Z,the
controlled-phase gate, or the swap gate. Other gates that
do not fall into this class require excessively more re-
sources unless other types of qubit encodings are used
simultaneously.
Gate operation times can be very fast and are only
limited by the gating times of the photodetectors. How-
ever, a major experimental challenge is mode-matching
in larger networks and interferometric set-ups.
A complementary approach to the gate model is
based on so-called cluster states which were originally
introduced to describe the properties in 3D optical lat-
tices [81.126] (Sect. 81.7.4). In the cluster-state model
the computational process is not described by a succes-
sion of elementary gates that act upon an (in principal
arbitrary) input state, but by a well-defined sequence of
single-qubit measurements performed on a maximally
entangled ‘cluster’ of qubits. It has been realized that
the cluster-state model represents an alternative model
for quantum computing, the so-called ‘one-way quan-
tum computer’ [81.127]. It was later found that there
also exists a linear optical realization of the cluster-based
approach [81.128, 129] which has been experimentally
verified [81.130].
81.7.2 Trapped Ions
So far, the most advanced method in terms of the
number of qubits and the number of gates that have
been generated is by using ultracold ions stored in
linear Paul traps (Chapt. 75) in which radio-frequency
fields are used to generate confining potentials. The
ions are trapped in the radial direction by electric
quadrupole fields and in the axial direction by a static
repulsive Coulomb force [81.131, 132]. The ions are
cooled into their motional ground states by Doppler
cooling [81.133, 134] and further cooled by resolved
sideband cooling [81.135]. The qubits are encoded into
two metastable electronic states. Various groups have
used either transitions to metastable states or Raman
coupling to avoid the decoherence that an upper-state
lifetime would generate. The coupling between qubits is
provided by the common vibrational motion in the Paul
trap [81.136,137]. For reviews of the dynamics of laser-
cooled ions [81.138–141]. To date, a few qubits have
been entangled and coherently manipulated. Simple
quantum algorithms have been demonstrated [81.142]
and teleportation achieved [81.143,144].
81.7.3 Cavity QED
Cavity QED provided the very first examples of atom-
field entanglement. Single atoms interacting with single
cavity-field modes are well described by the Jaynes–
Cummings model [81.145]. In this model, excitation
is transferred periodically between atoms and field
provided the Q-factor of the cavity is high enough
(Chapt. 79). The Rabi flopping can be used to gener-
ate controlled superpositions, and the cavity field used
as a catalyst to entangle atoms [81.146]. Although it is
possible to coherently manipulate single or few qubits
in cavity QED, scaling to large numbers of qubits would
Part F 81.7
Quantum Information 81.8 Outlook 1227
Fig. 81.7 Counter-propagating laser beams induce a peri-
odic spatially varying trapping potential through the AC
Stark shift
seem very difficult. Nevertheless, trapped atoms within
cavity QED environments offer great potential as local
processors linked by quantum communication chan-
nels [81.147]. Progress towards this has been reported
by several groups [81.148–151].
81.7.4 Optical Lattices, Mott Insulator
Another possible way of implementing quantum com-
putation is with cold atom technology. This includes the
application of optical lattices in a sufficiently cold cloud
of atoms showing Bose–Einstein condensation (BEC).
In recent years it has been realized that Bose–
Einstein condensates can undergo a phase transition
if loaded into a three-dimensional periodic potential
which, for example, can be realized by standing-wave
optical fields [81.152, 153]. That is, one starts off
with a BEC in its superfluid phase, in which the
relative phases (or rather correlations) between the
atoms are well-defined such that the whole ensemble
of atoms can be described by a single macroscopic
wave function (in first approximation). By loading this
condensate into the optical lattice (Fig. 81.7) the num-
ber of atoms per lattice site is undetermined and can
vary widely. However, when increasing the strength
of the potential by increasing the power of the laser
beams that create the standing-wave potential, even-
tually there will be a phase-transition to a state of
the condensate in which each lattice site is occu-
pied by a fixed and well-defined number of atoms
(ideally we would like to have exactly one atom
per site). In this so-called Mott-insulator phase, the
relative phases (or correlations) between neighboring
lattice sites are undetermined. Experimental evidence
of this phase-transition has been obtained in the beau-
tiful experiments described in [81.154–156]. Although
a Bose–Einstein condensate really exists only in three
dimensions (since only there one finds a phase transition
from a thermal cloud to a condensate), there are anal-
ogous systems, such as the quasi-condensate [81.157]
and the Tonks–Girardeau gas [81.158] (for recent exper-
iments, see [81.159, 160]), in one dimension that have
similar properties.
Such a system is well described by the Bose–
Hubbard Hamiltonian [81.152, 153], in which the
collisional interaction between atoms at the same lattice
site provides the necessary nonlinearity. Atoms trapped
in a one-dimensional optical lattice could serve as an
atomic register that promises well controlled single-
qubit and two-qubit manipulability. A universal set of
quantum gates can be realised by manipulations of the
lattice potential with additional laser fields [81.161]. The
different types of quantum gates could, for example,
be realized if the atoms possess two degenerate ground
states that are used for the qubit encoding (as sketched in
Fig. 81.7). A Raman transition between the ground states
would result in single-qubit operations, whereas con-
trolled collisions between atoms in neighboring lattice
sites would produce two-qubit gates. Recently, three-
qubit gates [81.162] and global adressing of strings of
qubits have been proposed [81.163].
The estimated gate evolution times in the adiabatic
regime are roughly O(100 ms), which is just one to
two orders of magnitude below the trapping lifetime
measured in recent experiments [81.164–166]. The dom-
inant loss effect is thereby a thermally induced spin flip
mechanism that causes the atoms to leave the trapping
region [81.167–169]. The gate evolution times could be
reduced by several orders of magnitude in non-adiabatic
regimes. The price to pay is that the temporal evolu-
tion of the laser pulse envelope has to be controlled
much more precisely. Although quantum information
processing using cold atom technology is still in its in-
fancy, it promises to provide relatively long decoherence
times. Moreover, scalability seems possible as rather
long one-dimensional strings of atoms could be formed.
Experimental evidence for this has of course yet to be
shown.
81.8 Outlook
We have discussed the very basic ideas behind quantum
information and described a few possible applications.
However, the immense wealth of ideas and possible
routes have barely been touched upon. Quantum key
distribution is already at a stage where private compa-
nies are selling component parts to set up commercial
Part F 81.8
1228 Part F Quantum Optics
QKD systems. Only a few qubits have been maxi-
mally entangled and manipulated experimentally, so far.
But a great number of qubits has been partially en-
tangled within optical lattices [81.170]orinatomic
vapours [81.171]. The dynamics of qubit interactions
in such systems is closely related to systems studied in
phase-transition theory, pointing to yet another appli-
cation of the subject. The simulation of many-particle
quantum systems is of course intrinsically difficult and
could well require a quantum computer for its anal-
ysis [81.172–175]. If and when a quantum computer
can be built remains shrouded in mist. However, the
ideas and methods that have already come out of quan-
tum information theory provide useful tools for tackling
other, seemingly unrelated, problems. One direction of
current research regards many-body problems in con-
densed matter systems and quantum field theory (see,
for example, [81.176]).
We have touched upon only a small part of a rapidly
developing subject – one in which quantum effects are
the enablers of new technology [81.177]. We are confi-
dent that much more remains to be discovered.
References
81.1 B. Russell: The ABC of Relativity (Allen Unwin, Lon-
don 1969)
81.2 J. Gruska: Quantum Computing (McGraw-Hill, Lon-
don 1999)
81.3 M. A. Nielsen, I. L. Chuang: Quantum Computation
and Quantum Information (Cambridge Univ. Press,
Cambridge 2000)
81.4 J. Stolze, D. Suter: Quantum Computing: A Short
Course from Theory to Experiment (Wiley-VCH,
Weinheim 2004)
81.5 A. Cabello: quant-ph/0012089
81.6 A. Cabello: URL www.adancabello.com
81.7 A. Einstein, B. Podolsky, N. Rosen: Phys. Rev. 47,
777 (1935)
81.8 J. S. Bell: Physics 1, 195 (1964)
81.9 J. S. Bell: Speakable and unspeakable in quan-
tum mechanics (Cambridge Univ. Press, Cambridge
1987)
81.10 S. J. Freedman, J. F. Clauser: Phys. Rev. Lett. 28,938
(1972)
81.11 A. Aspect, P. Grangier, G. Roger: Phys. Rev. Lett. 47,
460 (1981)
81.12 A. Aspect, J. Dalibard, G. Roger: Phys. Rev. Lett. 49,
1804 (1982)
81.13 J. G. Rarity, P. R. Tapster: Phys. Rev. Lett. 64,2495
(1990)
81.14 W. Tittel, J. Brendel, H. Zbinden, N. Gisin: Phys.
Rev. Lett. 81, 3563 (1998)
81.15 T. M. Cover, J. A. Thomas: Elements of Information
Theory (Wiley, New York 1991)
81.16 W. H. Zurek, W. K. Wootters: Nature (London) 299,
802 (1981)
81.17 M. Hillery: Phys. Rev. A 54, 1844 (1996)
81.18 A. Peres: Phys. Rev. Lett. 77, 1413 (1996)
81.19 M. Horodecki, P. Horodecki, R. Horodecki: Phys.
Lett. A 223, 1 (1996)
81.20 P. Horodecki: Phys. Lett. A 232, 333 (1997)
81.21 M. Horodecki, P. Horodecki, R. Horodecki: Phys.
Rev. Lett. 80, 5239 (1998)
81.22 B. Terhal: Phys. Lett. A 271, 391 (2000)
81.23 M. Lewenstein, B. Kraus, J. I. Cirac, P. Horodecki:
Phys. Rev. A 62, 052310 (2000)
81.24 E. Zeidler: Applied functional analysis: main
principles and their applications, AMS, Vol. 109
(Springer, Berlin, Heidelberg 1995)
81.25 C. C. Gerry, P. L. Knight: Introductory Quantum Op-
tics (Cambridge Univ. Press, Cambridge 2005)
81.26 L. M. Duan, G. Giedke, J. I. Cirac, P. Zoller: Phys.
Rev. Lett. 84, 2722 (2000)
81.27 R. Simon: ibid. 84, 2726 (2000)
81.28 H. Araki, E. Lieb: Commun. Math. Phys. 18,160
(1970)
81.29 E. Schmidt: Math. Annalen 63,433(1906)
81.30 A. Ekert, P. L. Knight: Am. J. Phys. 63, 415 (1995)
81.31 K. W. Chan, C. K. Law, J. H. Eberly: Phys. Rev. A 68,
022110 (2003)
81.32 V. Vedral, M. B. Plenio: Phys. Rev. A 57, 1619 (1998)
81.33 M. B. Plenio, V. Vedral: Contemp. Phys. 39,431
(1998)
81.34 M. Horodecki, P. Horodecki, R. Horodecki: Phys.
Rev. Lett. 84, 2014 (2000)
81.35 C. H. Bennett, D. P. DiVincenzo, J. A. Smolin,
W. K. Wootters: Phys. Rev. A 54, 3824 (1996)
81.36 S. Hill, W. K. Wootters: Phys. Rev. Lett. 78,5022
(1997)
81.37 W. K. Wootters: Phys. Rev. Lett. 80, 2245 (1998)
81.38 M. M. Wolf, G. Giedke, O. Krüger, R. F. Werner,
J. I. Cirac: Phys. Rev. A 69, 052320 (2004)
81.39 C. H. Bennett, H. Bernstein, S. Popescu, B. Schu-
macher: Phys. Rev. A 53, 2046 (1996)
81.40 G. Vidal, R. F. Werner: Phys. Rev. A 65, 032314
(2002)
81.41 J. Eisert, M. B. Plenio: Int. J. Quant. Inf. 1,479
(2003)
81.42 N. Linden, S. Popescu: Fortschr. Phys. 46, 567 (1998)
81.43 P. van Loock, S. L. Braunstein: Phys. Rev. Lett. 84,
3482 (2000)
81.44 S. Wu, Y. Zhang: Phys. Rev. A 63, 012308 (2001)
81.45 A. S. Holevo: Proceedings of the Second Japan–
USSR Symposium on Probability Theory (Springer,
Berlin, Heidelberg 1973) pp. 104–119
81.46 A. S. Holevo: IEEE Trans. Inf. Theory 44, 269 (1998)
81.47 B. Schumacher, M. D. Westmoreland: Phys. Rev. A
56, 131 (1997)
Part F 81
Quantum Information References 1229
81.48 C. H. Bennett, G. Brassard: Proceedings of IEEE
Int. Conf. on Computers Systems, Signal Processing
(IEEE, Bangalore, India 1984) pp. 175–179
81.49 A. K. Ekert: Phys. Rev. Lett. 67, 661 (1991)
81.50 C. H. Bennett, F. Bessette, G. Brassard, L. Salvail,
J. Smolin: J. Crypt. 5, 3 (1992)
81.51 P. D. Townsend, J. G. Rarity, P. R. Tapster: Electron.
Lett. 29, 1291 (1993)
81.52 A. Muller, J. Breguet, N. Gisin: Europhys. Lett. 23,
383 (1993)
81.53 W. Tittel, J. Brendel, H. Zbinden, N. Gisin: Phys.
Rev. Lett. 84, 4737 (2000)
81.54 N. Lütkenhaus: Phys. Rev. A 54, 97 (1996)
81.55 N. Lütkenhaus: Phys. Rev. A 59, 3301 (1999)
81.56 N. Lütkenhaus: Phys. Rev. A 61, 052304 (2000)
81.57 N. Lütkenhaus: Photonics News Optics 3, 24 (2004)
81.58 M. Bourennane, F. Gibson, A. Karlsson, A. Hening,
P. Jonsson, T. Tsegaye, D. Ljunggren, E. Sundberg:
Opt. Express 4, 383 (1999)
81.59 H. Kosaka, A. Tomita, Y. Nambu, T. Kimura,
K. Nakamura: Electr. Lett. 39, 1199 (2003)
81.60 C. Gobby, Z. L. Yuan, A. J. Shields: Appl. Phys. Lett.
84, 3762 (2004)
81.61 N. Gisin, G. Ribordy, W. Tittel, H. Zbinden: Rev.
Mod. Phys. 74, 145 (2002)
81.62 C. H. Bennett, G. Brassard, C. Crepeau, R. Josza,
A. Peres, W. K. Wootters: Phys. Rev. Lett. 70,1895
(1993)
81.63 L. Vaidman: Phys. Rev. A 49, 1473 (1994)
81.64 S. L. Braunstein, A. Mann: Phys. Rev. A 51,R1727
(1995)
81.65 S. L. Braunstein, A. Mann: Phys. Rev. A 53,630
(1996)
81.66 D. Bouwmeester, J. W. Pan, K. Mattle, M. Eibl,
H. Weinfurter, A. Zeilinger: Nature (London) 390,
575 (1997)
81.67 D. Boschi, S. Branca, F. DeMartini, L. Hardy,
S. Popescu: Phys. Rev. Lett. 80, 1121 (1998)
81.68 D. Bouwmeester, K. Mattle, J. W. Pan, H. Wein-
furter, A. Zeilinger, M. Zukowski: Appl. Phys. B 67,
749 (1998)
81.69 S. L. Braunstein, H. J. Kimble: Phys. Rev. Lett. 80,
869 (1998)
81.70 A. Furusawa, J. L. Sorensen, S. L. Braunstein,
C. A. Fuchs, H. J. Kimble, E. S. Polzik: Science 282,
706 (1998)
81.71 W. P. Bowen, N. Treps, B. C. Buchler, R. Schnabel,
T. C. Ralph, H. A. Bachor, T. Symul, P. K. Lam: Phys.
Rev. A 67, 032302 (2003)
81.72 I. Marcikic, H. deRiedmatten, W. Tittel, H. Zbinden,
N. Gisin: Nature (London) 421, 509 (2003)
81.73 M. Zukowski, A. Zeilinger, M. A. Horne, A. K. Ekert:
Phys. Rev. Lett. 71, 4287 (1993)
81.74 J.-W. Pan, D. Bouwmeester, H. Weinfurter,
A. Zeilinger: Phys. Rev. Lett. 80, 3891 (1998)
81.75 C. H. Bennett, S. J. Wiesner: Phys. Rev. Lett. 69, 2881
(1992)
81.76 R. F. Werner: J. Phys. A 34, 7081 (2001)
81.77 K. Mattle, H. Weinfurter, P. G. Kwiat, A. Zeilinger:
Phys. Rev. Lett. 76, 4656 (1996)
81.78 R. Landauer: IBM J. Res. Develop. 5,183(1961)
81.79 M. B. Plenio, V. Vitelli: Contemp. Phys. 42, 25 (2001)
81.80 M. Hillery, R. F. Werner: Phys. Rev. A 60, R2626
(1999)
81.81 D. Deutsch: Proc. R. Soc. London A 425, 73 (1989)
81.82 M. Reck, A. Zeilinger, H. J. Bernstein, P. Bertani:
Phys. Rev. Lett. 73, 58 (1994)
81.83 A. Barenco: Proc. R. Soc. London A 449, 679 (1995)
81.84 T. Sleator, H. Weinfurter: Phys. Rev. Lett. 74,4087
(1995)
81.85 D. P. DiVincenzo: Phys. Rev. A 51, 1015 (1995)
81.86 D. Deutsch, R. Jozsa: Proc. R. Soc. London A 439,
553 (1992)
81.87 J. A. Jones, M. Mosca: J. Chem. Phys. 109,1648
(1998)
81.88 I. L. Chuang, L. M. K. Vandersypen, X. L. Zhou,
D. W. Leung, S. Lloyd: Nature (London) 393,143
(1998)
81.89 S. Gulde, M. Riebe, G. P. T. Lancaster, C. Becher,
J. Eschner, H. Häffner, F. Schmidt-Kaler, I. L.
Chuang, R. Blatt: Nature (London) 421, 48 (2003)
81.90 L. Grover: Phys. Rev. Lett. 79, 325 (1997)
81.91 P. W. Shor: Proceedings, 35th Annual Symposium
on Foundations of Computer Science, 1994, Los
Alamitos (IEEE Computer Society Press, New York
1994) pp. 124–134
81.92 P. W. Shor: SIAM J. Comput. 26, 1484 (1997)
81.93 Y. S. Weinstein, M. A. Pravia, E. M. Fortunato,
S. Lloyd, D. G. Cory: Phys. Rev. Lett. 86, 1889
(2001)
81.94 L. M. K. Vandersypen, M. Steffen, G. Breyta,
C. S. Yannoni, M. H. Sherwood, I. L. Chuang: Nature
(London) 414, 883 (2001)
81.95 D. R. Simon: Proceedings, 35th Annual Symposium
on Foundations of Computer Science, 1994, Los
Alamitos (IEEE Computer Society Press, New York
1994) pp. 116–123
81.96 D. R. Simon: SIAM J. Comput. 26, 1474 (1997)
81.97 R. Jozsa: quant-ph/9707033
81.98 J. Kempe: Proc. 7th Intern. Workshop Random-
ization, Approximation Techniques Comput. Sci.
(RANDOM’03) (Springer, Berlin, Heidelberg 2003)
pp. 354–369
81.99 W. H. Zurek: Rev. Mod. Phys. 75, 715 (2003)
81.100 J. Preskill: Proc. R. Soc. London A 454, 385 (1998)
81.101 A. R. Calderbank, P. W. Shor: Phys. Rev. A 54,1098
(1996)
81.102 A. Steane: Proc. R. Soc. London A 452, 2551 (1996)
81.103 D. G. Cory, M. D. Price, W. Maas, E. Knill,
R. Laflamme, W. H. Zurek, T. F. Havel, S. S. Somaroo:
Phys. Rev. Lett. 81, 2152 (1998)
81.104 D. Leung, L. Vandersypen, X. Zhou, M. Sherwood,
C. Yannoni, M. Kubinec, I. Chuang: Phys. Rev. A 60,
1924 (1999)
81.105 E. Knill, R. Laflamme, R. Martinez, C. Negrevergne:
Phys. Rev. Lett. 86, 5811 (2001)
Part F 81
1230 Part F Quantum Optics
81.106 S. L. Braunstein, C. M. Caves, R. Jozsa, N. Linden,
S. Popescu, R. Schack: Phys. Rev. Lett. 83,1054
(1999)
81.107 J. Chiaverini, D. Leibfried, T. Schaetz, M. D. Barrett,
R. B. Blakestad, J. Britton, W. M. Itano, J. D. Jost,
E. Knill, C. Langer, R. Ozeri, D. J. Wineland: Nature
(London) 432, 602 (2004)
81.108 G. A. Palma, K.-A. Suominen, A. K. Ekert: Proc. R.
Soc. London A 452, 567 (1996)
81.109 L.-M. Duan, G.-C. Guo: Phys. Rev. Lett. 79,1953
(1997)
81.110 P. Zanardi, M. Rasetti: Phys. Rev. Lett. 79, 3306
(1997)
81.111 D. A. Lidar, I. L. Chuang, K. B. Whaley: Phys. Rev.
Lett. 81, 2594 (1998)
81.112 D. A. Lidar, D. Bacon, K. B. Whaley: Phys. Rev. Lett.
82, 4556 (1999)
81.113 P. G. Kwiat, A. J. Berglund, J. B. Altepeter,
A. G. White: Science 290, 498 (2000)
81.114 D. Kielpinski, C. Monroe, D. J. Wineland: Nature
(London) 417, 7097 (2002)
81.115 D. P. DiVincenzo: Fortschr. Phys. 48, 7718 (2000)
81.116 E. Knill, R. Laflamme, G. J. Milburn: Nature (Lon-
don) 409, 46 (2001)
81.117 M. Koashi, T. Yamamoto, N. Imoto: Phys. Rev. A 63,
030301 (2001)
81.118 T. B. Pittman, B. C. Jacobs, J. D. Franson: Phys. Rev.
A 64, 062311 (2001)
81.119 T. B. Pittman, B. C. Jacobs, J. D. Franson: Phys. Rev.
Lett. 88, 257902 (2002)
81.120 T. B. Pittman, M. J. Fitch, B. C. Jacobs, J. D. Franson:
Phys. Rev. A 68, 032316 (2003)
81.121 S. Scheel, K. Nemoto, W. J. Munro, P. L. Knight:
Phys. Rev. A 68, 032310 (2003)
81.122 E. Knill: Phys. Rev. A 68, 064303 (2003)
81.123 S. Scheel, N. Lütkenhaus: New J. Phys. 6,51
(2004)
81.124 J. Eisert: quant-ph/0409156
81.125 S. Scheel, K. M. R. Audenaert: New J. Phys. 7,149
(2005)
81.126 H. J. Briegel, R. Raussendorf: Phys. Rev. Lett. 86,
910 (2001)
81.127 R. Raussendorf, H. J. Briegel: Phys. Rev. Lett. 86,
5188 (2001)
81.128 M. A. Nielsen: Phys. Rev. Lett. 93, 040503 (2004)
81.129 D. E. Browne and T. Rudolph: quant-ph/0405157
81.130 A. N. Zhang, C. Y. Lu, X. Q. Zhou, Y. A. Chen, Z. Zhao,
T. Yang, J. W. Pan: quant-ph/0501036
81.131 W. Paul, O. Osberghaus, E. Fischer: Forschungs-
berichte des Wirtschafts- und Verkehrsministeri-
ums Nordrhein-Westfalen Nr. 415 (Westdeutscher
Verlag, Köln 1958)
81.132 W. Neuhauser, M. Hohenstatt, P. E. Toschek,
H. G. Dehmelt: Phys. Rev. A 22, 1137 (1980)
81.133 D. J. Wineland, R. Drullinger, D. F. Walls: Phys. Rev.
Lett. 40, 1639 (1978)
81.134 W. Neuhauser, M. Hohenstatt, P. E. Toschek,
H. G. Dehmelt: Phys. Rev. Lett. 41,233(1978)
81.135 D. J. Wineland, H. G. Dehmelt: Bull. Am. Phys. Soc.
20,637(1975)
81.136 J. I. Cirac, P. Zoller: Phys. Rev. Lett. 74, 4094 (1995)
81.137 C. Monroe, D. M. Meekhof, B. E. King, W. M. Itano,
D. J. Wineland: Phys. Rev. Lett. 75, 4714 (1995)
81.138 J. F. Poyatos, J. I. Cirac, P. Zoller: Fortschr. Phys. 48,
785 (2000)
81.139 J. F. Poyatos, J. I. Cirac, P. Zoller: J. Mod. Opt. 49,
1593 (2002)
81.140 D. Leibfried, R. Blatt, C. Monroe, D. J. Wineland:
Rev. Mod. Phys. 75, 281 (2003)
81.141 D. J. Wineland, M. Barrett, J. Britton, J. Chiaverini,
B. DeMarco, W. M. Itano, C. Langer, D. Leibfried,
V. Meyer, T. Rosenband, T. Schätz: Philos. Trans. R.
Soc. London A 361, 1349 (2003)
81.142 S. Gulde, M. Riebe, G. P. T. Lancaster, C. Becher,
J. Eschner, H. Häffner, F. Schmidt-Kaler, I. L. Chuang,
R. Blatt: Nature (London) 421, 48 (2003)
81.143 M. Riebe, H. Häffner, C. F. Roos, W. Hänsel, J. Ben-
helm, G. P. T. Lancaster, T. W. Körber, C. Becher,
F. Schmidt-Kaler, D. F. V. James, R. Blatt: Nature
(London) 429, 734 (2004)
81.144 M. Barrett, J. Chiaverini, T. Schaetz, J. Brit-
ton, W. M. Itano, J. D. Jost, E. Knill, C. Langer,
D. Leibfried, R. Ozeri, D. J. Wineland: Nature (Lon-
don) 429, 737 (2004)
81.145 E. T. Jaynes, F. W. Cummings: Proc. IEEE 51, 89 (1963)
81.146 S. Haroche: Philos. Trans. R. Soc. London A 361, 1339
(2003) and references therein
81.147 C. Monroe: Nature (London) 416, 238 (2002)
81.148 T. Pellizzari, S. A. Gardiner, J. I. Cirac, P. Zoller: Phys.
Rev. Lett. 75, 3788 (1995)
81.149 J. I. Cirac, P. Zoller, H. J. Kimble, H. Mabuchi: Phys.
Rev. Lett. 78, 3221 (1997)
81.150 A. Kuhn, M. Hennrich, G. Rempe: Phys. Rev. Lett.
89, 067901 (2002)
81.151 J. McKeever, A. Boca, A. D. Boozer, R. Miller,
J. R. Buck, A. Kuzmich, H. J. Kimble: Science 303,
1992 (2004)
81.152 D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner,
P. Zoller: Phys. Rev. Lett. 81, 3108 (1998)
81.153 S. R. Clark, D. Jaksch: Phys. Rev. A 70,043612
(2004)
81.154 M. Greiner, I. Bloch, O. Mandel, T. W. Hänsch,
T. Esslinger: Phys. Rev. Lett. 87, 160405 (2001)
81.155 M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch,
I. Bloch: Nature (London) 415, 39 (2002)
81.156 C. Orzel, A. K. Tuchman, M. L. Fenselau, M. Yasuda,
M. A. Kasevich: Science 291, 2386 (2001)
81.157 D. S. Petrov, G. V. Shlyapnikov, J. T. M. Walraven:
Phys. Rev. Lett. 85, 3745 (2000)
81.158 V. Dunjko, V. Lorent, M. Olshanii: Phys. Rev. Lett.
86, 5413 (2001)
81.159 B. Paredes, A. Widera, V. Murg, O. Mandel,
S. Folling, J. I. Cirac, G. V. Shlyapnikov, T. W. Hän-
sch, I. Bloch: Nature (London) 429, 277 (2004)
81.160 T. Kinoshita, T. Wenger, D. S. Weiss: Science 305,
1125 (2004)
Part F 81