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1210 Part F Quantum Optics

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Part F 80

1215

Quantum Info

81. Quantum Information

For many years atomic physicists had used

quantum mechanics very successfully to cal-

culate energy levels, cross sections and other

practical quantities, and for the most part, left

the philosophical issues of interpretation to oth-

ers. But after the work of Bell in the 1960’s

showed that the peculiarly nonlocal nature of

quantum correlations could be tested in the

lab, a number of atomic physicists turned to

the experimental study of entanglement and

quantum measurement. A second phase be-

gan at the start of the 1990’s when it was

realized that correlations and quantum su-

perpositions could be exploited in quantum

information processing and secure communi-

cation. This has led to an explosive growth of

the subject over the past 10 years, fuelled by

the long-term prospects of quantum computing

and the nearer goal of quantum cryptography.

We review some of these developments in this

chapter.

81.1 Quantifying Information ......................1216

81.1.1 Separability Criterion .................1216

81.1.2 Entanglement Measures.............1217

81.2 Simple Quantum Protocols....................1218

81.2.1 Quantum Key Distribution .......... 1219

81.2.2 Quantum Teleportation ..............1219

81.2.3 Dense Coding............................1220

81.3 Unitary Transformations.......................1221

81.3.1 Single-Qubit Operations.............1221

81.3.2 Two-Qubit Operations................ 1221

81.3.3 Multi-Qubit Gates and Networks . 1222

81.4 Quantum Algorithms............................ 1222

81.4.1 Deutsch–Jozsa Algorithm ...........1222

81.4.2 Grover’s Search Algorithm .......... 1223

81.5 Error Correction ...................................1223

81.6 The DiVincenzo Checklist ......................1224

81.6.1 Qubit Characterization, Scalability1224

81.6.2 Initialization ............................1224

81.6.3 Long Decoherence Times ............ 1224

81.6.4 Universal Set of Quantum Gates ..1225

81.6.5 Qubit-Speciﬁc Measurement.......1225

81.7 Physical Implementations ....................1225

81.7.1 Linear Optics.............................1225

81.7.2 Trapped Ions ............................1226

81.7.3 Cavity QED ................................1226

81.7.4 Optical Lattices, Mott Insulator....1227

81.8 Outlook .............................................. 1227

References .................................................. 1228

Quantum information theory is regarded as a mainly

mathematics-based subject area which straddles the

ﬁelds of theoretical physics (quantum mechanics and

statistics), mathematics, and theoretical computer sci-

ence. Its success stems from the introduction of novel

methods into both physics and mathematics.

The fundamental quantity and resource in many

applications in quantum information processing is

quantum-mechanical entanglement between spatially

separated subsystems. Entanglement is a purely

quantum-mechanical effect and has led to numerous

speculations about the validity of quantum mechan-

ics itself for its apparent paradoxical implications.

Most, if not all, of these difﬁculties have been re-

solved and can be mostly attributed to the simple

fact that paradoxical behaviour is incompatible with

common sense or everyday experience. This initial up-

setting seems to be common to all revolutionary theories

and has occurred most notably in Einstein’s theory of

relativity [81.1].

These quantum-mechanical correlations have nu-

merous applications in quantum cryptography [or rather

quantum key distribution (QKD)], quantum communi-

cation, dense coding, and act as the main resource in

quantum computing. We will brieﬂy touch upon some

mathematical issues concerning separability, quantiﬁ-

cation of entanglement and channel capacities before

describing how quantum key distribution, teleportation

and dense coding work. After that, a brief discus-

sion of single-qubit and two-qubit quantum gates

Part F 81

1216 Part F Quantum Optics

follows before we describe the simplest quantum al-

gorithms. The issues of error correction and fault

tolerant computation as well as DiVincenzo’s check-

list (which any realization should satisfy) provide the

background for the discussion of some physical imple-

mentations.

We are acutely aware of the fact that we can give

only a brief introduction into what has become a major

ﬁeld of investigation over the last decade. There are al-

ready a number of review articles and textbooks on the

market that cover the vast literature on this emerging

subject. Amongst those are the ﬁrst quantum comput-

ing compendium by Gruska [81.2] and the quantum

information textbooks by Nielsen and Chuang [81.3]

and Stolze and Suter [81.4]. A regularly updated anno-

tated bibliography on this subject, compiled by Cabello,

forms an invaluable resource for those interested in the

subject of this chapter [81.5, 6].

81.1 Quantifying Information

As already noted, entanglement comes about if

a quantum-mechanical system can be divided into

several parts. As an example, consider a two-photon

emission process from a spin-zero particle by which

two photons escape in opposite directions. Given that

the photons are spin-one particles, their spin projections

onto some axis must be mutually opposite. As there is

no prior information about the actual orientation of the

spin, the part of the photon wave function associated

with the spin degree is therefore

|ψ=

√

(

| ↑↓−| ↓↑

)

(81.1)

The striking feature of this type of quantum state is

that it describes correlations of two spatially separated

particles. If the polarization state of one photon is meas-

ured, the state of the other particle, which can be far

away, is then instantly predetermined. These (nonlo-

cal) correlations that exist between the particles are

of purely quantum origin and are called entanglement.

Note, however, that no information can be transmitted

faster than the speed of light with this type of set up

because the (classical) information concerning the meas-

urement result on one particle needs to be transmitted

via a necessarily causal classical channel.

The issue of nonlocality has been seen as a vital part

in understanding the foundations of quantum mechan-

ics itself (Chapt. 80). In 1935, Einstein, Podolsky,and

Rosen argued on the basis of entangled states that quan-

tum mechanics is incomplete [81.7]. They were most

concerned about the existence of elements of reality

within strongly correlated quantum systems and initiated

the debate on quantum nonlocality. The non-existence

of so-called local hidden variable theories for the de-

scription of states like (81.1) was ﬁnally demonstrated

by Bell [81.8, 9]. He showed that maximally entangled

states violate certain inequalities (now called Bell’s in-

equalities) which local hidden variable models would

have obeyed. Later experiments showed the correctness

of Bell’s demonstration [81.10–14].

In classical information theory, the unit of informa-

tion is called a bit, which can be deﬁned as the amount of

information contained in a yes–no question. As a mat-

ter of fact, ‘bit’ is the abbreviation for ‘binary digit’ and

refers to Boolean algebra in which the allowed states of

a system are the logical 0 and the logical 1. Therefore, by

abuse of language, one bit (as a unit) is the information

carried by one bit (as a binary digit) [81.15].

In quantum mechanics, however, due to its inherent

linearity, two ‘quantum bits’ (qubits for short) can be

in superpositions of the logical states |01 and |10,or

| ↑↓ and | ↓↑, as in the example above. This typical

example of an entangled state shows that quantifying

the amount of information contained in a quantum state

is different from what is known in classical information

theory because of the superposition property. The very

same linearity prohibits us from copying an arbitrary

quantum state. This effect is known as the no-cloning

theorem [81.16]. However, universal copying machines

can be constructed within the constraints of quantum

mechanics [81.17].

81.1.1 Separability Criterion

From the above it is clear that entangled states play

a major role in deﬁning the differences between clas-

sical and quantum information. Let us begin by asking

under which circumstances a particular given quantum

state is entangled or not. For this, we need to give a cri-

terion which allows one to decide this crucial question.

Consider a bipartite quantum state, i. e., a state which is

decomposed into two distinct, albeit possibly correlated,

subsystems A and B. Note that these subsystems them-

selves might consist of ensembles of particles, in which

Part F 81.1

Quantum Information 81.1 Quantifying Information 1217

case we are looking at a bipartite cut through the whole

system. Then we say that a bipartite state is not entan-

gled and hence separable if its density operator can be

written as a convex combination of tensor product states,

viz.

 =





⊗





= 1 . (81.2)

The range of summation in (81.2) is limited by a theo-

rem due to Caratheodory which states that every point

in a convex set can be reached by suitable convex

combinations of its extreme points. All states that can-

not be written in the form of (81.2) are said to be

entangled. Note that the set of separable states form

a convex subset of the convex set of all possible

states.

We will now give a simple criterion which de-

cides whether a given state is actually separable or

not. For this, one notes that by transposing the part

of the density operator associated with the subsystem

B, an operation which is called partial transposition,

the resulting operator will not necessarily stay pos-

itive. However, if the density operator is separable,

then its partial transpose is again a positive operator,

and hence is a valid density operator. This condi-

tion of a state possessing a positive partial transpose

is a necessary separability criterion [81.18]butsuf-

ﬁcient only in the case of density matrices having

Hilbert space dimensions 2 × 2 or 2 × 3 [81.19]. In

higher-dimensional Hilbert spaces there exist states with

positive partial transposes (PPT) which are neverthe-

less inseparable. This phenomenon is called bound

entanglement [81.20, 21].

Set of all

density

matrices

Separable

states

␳

(

)

Fig. 81.1 Convex set of bipartite density matrices; the inner

convex set represents the separable states. The witness op-

erator

W forms a hyperplane that separates



from the

set of separable states

Because of the convexity of the set of separable

states, one can construct an operator (a hyperplane)

that separates an entangled state from the disentangled

states,





< 0 ⇔



inseparable ,





≥ 0 ⇔



separable . (81.3)

Such an operator is called an entanglement wit-

ness [81.22, 23], and its existence is ensured by

a consequence of the Hahn–Banach theorem [81.24].

A similar separability criterion can be found for

a particularly interesting class of quantum states

in inﬁnite-dimensional Hilbert spaces, the Gaussian

states. Gaussian states are most frequently encoun-

tered in quantum optics. They comprise all coherent,

squeezed and thermal states, and combinations of them.

Although being inﬁnite-dimensional, these states per-

mit a complete description in terms of their ﬁrst

and second moments. The characteristic function of

a single-mode Gaussian state with λ

= (x, p) is given

by [81.25]

χ(λ) = exp



λ −

V λ



, (81.4)

where m is a vector containing the ﬁrst moments

and V is the covariance matrix containing the

second moments. A necessary and sufﬁcient crite-

rion for separability of a bipartite Gaussian state

is that the partially transposed covariance matrix

still possesses positive symplectic eigenvalues [81.26,

27].

81.1.2 Entanglement Measures

Once one has checked for inseparability, the obvious

question to ask concerns the amount of entanglement,

hence the amount of nonclassical correlations in the

given state. For bipartite pure states the answer is unique

and given by the von Neumann entropy of one subsys-

tem, viz.,

E(|ψ

) = S

(



) = S

(



), (81.5)

with S

(



) =−tr



where



A(B)

= tr

B(A)

|ψ

ψ

The second equality in (81.5) follows from the left-hand

side of the Araki–Lieb inequality [81.28]

−S

|≤S

≤ S

(81.6)

Part F 81.1

1218 Part F Quantum Optics

when noting that the entropy of a pure state vanishes. Ob-

viously, since S

= 0, no information can be extracted

from the total state, all information is contained in the

correlations between the subsystems A and B which are

revealed by performing a measurement on one of the

subsystems.

For bipartite quantum systems prepared in mixed

states the answer is not so obvious. However, some in-

sight can already be gained by looking at the Schmidt

decomposition of the state (which, for bipartite states,

always exists) [81.29, 30], in particular, the number

of elements in the decomposition, named the Schmidt

number [81.31].

For a more precise deﬁnition of mixed bipartite

entanglement, something more is needed. Recall that

the set of separable density matrices forms a con-

vex subset of all feasible density matrices. It therefore

makes sense to look for a distance-type measure be-

tween the given state and the convex hull of product

states. Note that the possibility of deﬁning such a meas-

ure is provided by the convexity of the separable

states and a consequence of the Hahn–Banach theo-

rem [81.24]. Generally, agreement has been reached on

what properties any feasible entanglement measure must

fulﬁl [81.32–34]. Let E(



) be a real-valued functional

over the tensor-product Hilbert space of bipartite dens-

ity matrices. If in addition E(



) has the following

properties:

1. E(



) = 0 for all separable states;

2. E(



) is invariant under local unitary trans-

formations, viz., E



⊗





†

⊗

†





);

3. E(



) is non-increasing under general local op-

erations assisted by classical communication, viz.,





⊗



i†



≤ E(



);

4. E(



) reduces to the reduced von Neumann en-

tropy for pure states,

then E(



) is called an entanglement measure.

Important examples of widely used entanglement

measures are the entanglement of formation [81.35]

(

) = min





|ψ

ψ



E(|ψ

), (81.7)

and the relative entropy of entanglement [81.32, 33]

(



) = min

σ=



⊗





(ln



−ln

σ)



(81.8)

In general, both of these measures are hard to eval-

uate. Analytical formulas are known only in special

cases. For qubits, the entanglement of formation is also

a monotonic function of the concurrence [81.36,37]. The

deﬁnition of the entanglement of formation, (81.7), can

also be extended to cover Gaussian states [81.38].

The number of singlets, i. e., states of the form (81.1),

that can be distilled from an ensemble of non-maximally

entangled states is called the entanglement of distilla-

tion [81.39]. The entanglement of formation and the

entanglement of distillation differ by the amount of

bound entanglement (Sect. 81.1.1).

In some instances, when it is not necessary to com-

ply with all of the above properties of entanglement

measures, other quantities can be used to assess the

entanglement content of a bipartite state. Particularly

useful is the logarithmic negativity [81.40, 41]

(



) = log





P. T.



, (81.9)

where ·

denotes the trace norm and



P. T.

the par-

tial transpose of



. This measure is often used in

connection with Gaussian states.

In close analogy to classical information theory, the

amount of nonclassical correlations is measured in ebits

when one computes entropies with the dual logarithm

(log

). For example, a pure state with state vector

|ψ=

√

(

|01+|10

)

(81.10)

in an abstract two-particle Hilbert space spanned by

the basis states {|00, |01, |10, |11} contains 1 ebit of

entanglement. It is also a maximally entangled state as-

sociated with this Hilbert space since the von Neumann

entropy of any state in a Hilbert space of dimension N

is bounded from above by log

dim N.

We have concentrated here on bipartite entangle-

ment. The extension to multipartite systems is by no

means trivial and much remains to be done on this

subject [81.42–44].

81.2 Simple Quantum Protocols

In this section we describe the historically ﬁrst and

simplest quantum protocols – quantum key distribu-

tion, quantum teleportation, and super-dense coding –

that make use of inherently ‘quantum’ properties of

quantum-mechanical systems. These are either entangle-

ment or, in the case of the simplest version of quantum

Part F 81.2

Quantum Information 81.2 Simple Quantum Protocols 1219

key distribution, properties of the quantum-mechanical

measurement process. We should mention here the pion-

eering work of Holevo [81.45] who showed that there

are fundamental limits on the amount of information that

can be extracted by measurements. The application of his

ideas to channel capacity and communication [81.46,47]

are well described in [81.3] and space limitations prevent

us from elaborating on it in this chapter.

81.2.1 Quantum Key Distribution

Historically, the earliest protocol that used quantum-

mechanical features in order to realize some speciﬁc task

that could not have been performed classically was a pro-

tocol for secure distribution of a key in cryptography,

known as the BB84-protocol after its inventors Bennett

and Brassard and the year of its invention [81.48]. Al-

though it is commonly referred to as the ﬁrst example of

quantum information processing, it does not make use

of entanglement which was only done some years later,

by Ekert [81.49].

The BB84 protocol works in the following way. The

sender A prepares a random sequence (or string) of sin-

gle photons in a polarization state which is chosen out

of a set of four basis states, horizontally and vertically

(H and V ) polarized, and 45

◦

and 135

◦

(L and R) po-

larized. In each of the two basis sets {H, V }, {L, R}

one of the states is used to encode the logical value 0

(say in H and L) and the other states encode the lo-

gical value 1 (V and R). The random sequence is sent

to the receiver B who performs measurements on the

sequence of signals by randomly choosing analogous

basis states. The result will be another string of 0’s

and 1’s that generically does not coincide completely

with the original string. To rectify this problem, sender

and receiver communicate over a classical public chan-

nel where the sender announces the sequence of basis

sets in which the photon states were prepared. The re-

ceiver compares its sequence of randomly chosen basis

states with the announced string and keeps all measure-

Table 81.1 BB84 protocol for secret key distribution. The

sender A sends information encoded in either of two basis

sets. The receiver B randomly chooses a measurement basis

which is publicly communicated. For those cases when

sender and receiver chose the same basis, the receiver’s

measurement yields a secure bit

Sender A  ↑  →  ↑ →

Receiver B →↑   →↑   →↑

Key 1 1 0 1

ment results for which the choice of basis had been the

same. In that way a common secret key is established

(Table 81.1).

The security against eavesdropping of this simple

protocol comes from the fact that even by knowing the

measurement basis (say {H, V }) no information has been

revealed about the choice of the actual bit value (H or

V ). Hence, it is the quantum-mechanical measurement

process itself that provides security of the protocol. The

ﬁrst quantum key distribution experiments were reported

in [81.50–52]. However, imperfections in the generation

and detection of photons, transmission losses and polar-

ization drift causes an actual experimental realization

to be far from ideal. In practise, encodings other than

polarization may be used (for example a time-binned in-

terferometric basis [81.53]). Despite these error sources,

unconditionally secure quantum key distribution can

be [81.54–57] and has been achieved [81.58]. Some

ﬁber-based systems have reached distances of more than

100 km [81.59, 60], but discussions of their security

continue. For a review of theoretical and experimental

aspects of quantum cryptography, see [81.61].

81.2.2 Quantum Teleportation

An important utilization of entanglement as a necessary

resource can be found in what is commonly known as

quantum teleportation. The task of teleportation is to

transmit the complete information of an arbitrary un-

known quantum state to a spatially different location

with the aim of re-creating it. The simplest and obvious

way to perform this task would be to take the quantum

object which is prepared in the original state and physi-

cally transport it to a different location. But sometimes

this is not possible because for example an ion needs

to be stored in a trap and cannot be moved. The next

obvious thing to do would be to measure the quantum

state and to re-create it at a different position using the

classical information obtained during the measurement.

However, single measurements on a quantum system

yield only partial information and multiple measure-

ments on many identically prepared copies would have

to be performed.

The protocol, which was originally proposed

in [81.62] for qubits and later generalized to states in

inﬁnite-dimensional Hilbert spaces in [81.63], makes

use of the existence of maximally entangled states. Let

the unknown quantum state which is to be teleported be

a qubit superposition state of the form

|ψ=α|0+β|1 , |α|

+|β|

= 1 . (81.11)

Part F 81.2

1220 Part F Quantum Optics

2 bit classical information

Output state

Pauli

rotation

Bell

measurement

Entangled

resource

Input state

Fig. 81.2 Schematic outline of an ideal teleportation

protocol

Then one prepares a maximally entangled state of the

form (81.10) which is one of the four so-called Bell

states deﬁned by

|Ψ

=

√

(

|01±|10

)

|Φ

=

√

(

|00±|11

)

(81.12)

We then form the tensor product state |ψ|Ψ

 as

|ψ

|Ψ

=

√

(|0

+|0

)

√

(|1

+|1

)



(α|0

+β|1

)





+(α|0

−β|1

)



−



+(α|1

+β|0

)





+(α|1

−β|1

)



−





(81.13)

where we have explicitly indexed the relevant sub-

systems. After performing a joint measurement on

subsystems A and B in the Bell basis (this is called

a Bell-state measurement [81.64, 65]) one obtains one

of four possible results. If the measurement result was

|Ψ

, then the subsystem C is indeed prepared in the

original unknown quantum state |ψ, hence the state

has been ‘teleported’ from subsystem A to C.Forall

other measurement results the outcome is not exactly

the same quantum state as intended, but the difference

is just a unitary transformation which is uniquely de-

termined by the outcome of the Bell measurement. For

example, measuring |Ψ

−

 means one has to perform a

-operation that ﬂips the sign of the state |1, whereas

on obtaining |Φ

 or |Φ

−

 the operations to be applied

have to be

, respectively.

Note that this quantum teleportation protocol works

with perfect ﬁdelity only if a maximally entangled state

has been used, i. e., a state containing 1 ebit of quan-

tum information. In the course of the Bell measurement,

the quantum information is used up, and two classi-

cal bits of information (the measurement result) have

to be communicated to C in order to restore the orig-

inal quantum state. In this sense, entanglement can

be regarded as a resource or ‘fuel’ for certain tasks

in quantum information processing. The ﬁrst experi-

mental demonstrations of teleportation of qubits were

performed in [81.66–68] and of continuous variables

in [81.69–71]. Recently, a teleportation experiment over

2 km standard telecommunication ﬁbre has been re-

ported [81.72]. A generalization of teleportation is

entanglement swapping, in which EPR correlations are

established between previously uncorrelated particles by

Bell-state measurements [81.73, 74].

81.2.3 Dense Coding

The complementary protocol to teleportation is charac-

terized by the name of (super) dense coding [81.75].

The idea here is to transmit two classical bits of

information at the expense of consuming 1 ebit of

quantum information. The similarity to teleportation

is best seen by noting that if the experimental appa-

ratus of sender and receiver are interchanged and the

protocol reversed (Fig. 81.3), then one reduces to the

other. The mathematical equivalence of the teleporta-

tion and dense coding schemes has been beautifully

shown in [81.76]. As in teleportation, sender and re-

ceiver initially share a two-particle maximally entangled

state, i. e., one of the Bell states deﬁned in (81.12).

By acting with one of the four operations

on the qubit on the sender’s side, the total

2 bit classical output

1 qubit transmission

Pauli

rotation

Bell

measurement

Entangled

resource

2 bit classical input

Fig. 81.3 Schematic outline of an ideal superdense coding

protocol

Part F 81.2