½ Coherent states

where U is a unitary transformation. We must ensure that eqn (5.215) goes over into

eqn (5.214) in the classical limit, and the easiest way to do this is to assume that the

unitary transformation has the same form:

a = U aU † = µa + νa† , (5.216)

where µ and ν are c-numbers. The unitary transformation preserves the commutation

relations, so the c-number coe¬cients µ and ν are constrained by

2 2

|µ| ’ |ν| = 1 . (5.217)

Since the overall phase of a is irrelevant, we can choose µ to be real, and set

µ = cosh r , ν = e2iφ sinh r . (5.218)

The relation between a and a is an example of the Bogoliubov transformation ¬rst

introduced in low temperature physics (Huang, 1963, Sec. 19.4).

The condition that the transformed Gaussian state is thermal-like is

†

ρG = U ρG U † = N e’g0 a a

, (5.219)

where the constant g0 is to be determined. The ansatz (5.212) shows that this is

equivalent to

U GU † = g0 a† a , (5.220)

and taking the commutator of both sides of this equation with a produces

1 1

a, La† a + M a†2 + M — a2 = g0 a . (5.221)

2 2

Evaluating the commutator on the left by means of eqn (5.216) will produce two

terms, one proportional to a† and one proportional to a. No a† -term can be present

if eqn (5.221) is to be satis¬ed; therefore, the coe¬cient of a† must be set to zero. A

little careful algebra shows that the free parameter φ in eqn (5.218) can be chosen to

cancel the phase of M . This is equivalent to assuming that M is real and positive to

begin with, so that φ = 0. With this simpli¬cation, setting the coe¬cient of a† to zero

imposes tanh 2r = ’L/M ,√ and using this relation to evaluate the coe¬cient of the

a-term yields in turn g0 = L2 ’ M 2 .

We will now show that the Gaussian state has the properties claimed for it by

applying the general de¬nition (5.125) to ρG , with the result

†

’· — a

χG (·) = Tr ρG e·a

W

†

’· — a

= Tr U ρG U † U e·a U†

† †

’· — a

= N Tr e’g0 a a e·a . (5.222)

The remaining a-dependence can be eliminated with the aid of the explicit form

(5.216), so that

Gaussian states— ½

† †

’ζ — a

χG (·) = N Tr e’g0 a a eζa , (5.223)

W

where

ζ = ·µ ’ · — ν = · cosh r ’ · — sinh r . (5.224)

The parameter g0 in eqn (5.219) plays the role of ω/kT for the thermal state, so

comparison with eqns (2.175)“(2.177) shows that N = [1 ’ exp (’g0 )]. An application

of eqn (5.148) then yields the Wigner characteristic function

1 2

χG (·) = exp ’ nG + |ζ|

W

2

1

|· cosh r ’ · — sinh r|

2

= exp ’ nG + (5.225)

2

for the Gaussian state, where nG = 1/ (eg0 ’ 1) is the analogue of the thermal average

number of quanta. The Wigner distribution is given by eqn (5.126), which in the

present case becomes

1 1

—

±’·±— 2

exp ’ nG + |ζ|

d2 ·e·

WG (±) = . (5.226)

π2 2

After changing integration variables from · to ζ, this yields

1 1

—

β’ζβ —

|ζ|2 ,

exp ’ nG +

d2 ζeζ

WG (±) = (5.227)

π2 2

where

β = µ± ’ ν±— = cosh r ± ’ sinh r ±— . (5.228)

According to eqn (5.149), this means that

|β|2

1 1

exp ’

WG (±) =

π nG + 1/2 nG + 1/2

|cosh r ± ’ sinh r ±— |

2

1 1

exp ’

= . (5.229)

π nG + 1/2 nG + 1/2

It is encouraging to see that the Wigner distribution for a Gaussian state is itself

Gaussian, but we previously found that positivity for the Wigner distribution does not

guarantee positivity for P (±). In order to satisfy ourselves that P (±) is also Gaussian,

we use the relation (5.182) between the normal-ordered and Wigner characteristic

functions to carry out a rather long evaluation of P (±) which leads to

1 1

PG (±) =

π 2

(nG + 1/2) ’ (nG + 1/2) cosh 2r + 1/4

sinh 2r ±2 + ±—2

2

|±| cosh 2r ’ 1

— exp ’ 2

. (5.230)

nG cosh2 r + (nG + 1) sinh2 r

Thus all Gaussian states are classical, and both the Wigner function WG (±) and the

PG (±)-function are Gaussian functions of ±.

½¼ Coherent states

5.8 Exercises

Are there eigenvalues and eigenstates of a† ?

5.1

The equation

a† |φβ = β |φβ ,

where β is a complex number, is apparently analogous to the eigenvalue problem

a |± = ± |± de¬ning coherent states.

(1) Show that the coordinate-space representation of this equation is

1 d

√ ωQ ’ φβ (Q) = βφβ (Q) .

dQ

2ω

(2) Find the explicit solution and explain why it does not represent an eigenvector.

Hint: The solution violates a fundamental principle of quantum mechanics.

Expectation value of functions of N

5.2

Consider the operator-valued function f (N ), where N = a† a and f (s) is a real func-

tion of the dimensionless, real argument s.

(1) Show that f (N ) is represented by

∞

dθ

f (θ) eiθN ,

f (N ) =

2π

’∞

where f (θ) is the Fourier transform of f (s).

(2) For any coherent state |± , show that

± eiθN ± = exp |±|2 eiθ ’ 1 ,

and use this to get a representation of ± |f (N )| ± .

5.3 Approach to orthogonality

By analogy with ordinary vectors, de¬ne the angle ˜±β between the two coherent

states by cos (˜±β ) = | ± |β |. From a plot of ˜±β versus |± ’ β| determine the value

at which approximate orthogonality sets in. What is the physical signi¬cance of this

value?

5.4 Number-phase uncertainty principle

Assume that the quantum fuzzball in Fig. 5.1 is a circle of unit diameter.

(1) What is the physical meaning of this assumption?

(2) De¬ne the phase uncertainty, ∆φ, as the angle subtended by the quantum fuzzball

at the origin. In the semiclassical limit |±0 | 1, show that ∆φ∆n ∼ 1, where ∆n

is the rms deviation of the photon number in the state |±0 .

½½

Exercises

5.5 Arecchi™s experiment

What is the relation of the fourth and second moments of a Poisson distribution?

Check this relation for the data given in Fig. 5.4.

5.6 The displacement operator

(1) Show that eqn (5.47) follows from eqn (5.46).

(2) Derive eqn (5.56) and explain why ¦ (±, β) has to be real.

(3) Show that exp [’i„ K (±)], with K (±) = i±a† ’ i±— a, satis¬es

‚

exp [’i„ K (±)] = ±a† ’ ±— a exp [’i„ K (±)] ,

‚„

and that exp [’i„ K (±)] = D („ ±).

(4) Let ± ’ „ ± and β ’ „ β in eqn (5.56) and then di¬erentiate both sides with

respect to „ . Show that the resulting operator equation reduces to the c-number

equation

‚¦ („ ±, „ β)

= 2„ Im (±β — ) ,

‚„

and then conclude that ¦ (±, β) = Im (±β — ).

5.7 Wigner distribution

(1) Show that the Wigner distribution W (±) for the density operator

ρ = γ |1 1| + (1 ’ γ) |0 0| ,

with 0 < γ < 1, is everywhere positive.

(2) Determine if the state described by ρ is classical.

The antinormally-ordered characteristic function—

5.8

The argument in Section 5.6.1-B begins by replacing the exponential in the classical

† —

de¬nition (5.114) by e·a ’· a , but one could just as well start with the classically

— †

equivalent form e’· a e·a , which is antinormally ordered. This leads to the de¬nition

— †

χA (·) = Tr ρe’· a e·a

of the antinormally-ordered characteristic function.

(1) Use eqn (5.72) to show that

—

’· — ±)

d2 ±e(·±

χA (·) = Q (±) .

(2) Invert this Fourier integral, e.g. by using eqn (5.127), to ¬nd

1 —

’· — ±)

d2 ·e’(·±

Q (±) = χA (·) .

π2

½¾ Coherent states

5.9 Classical states

(1) For classical states, with density operators ρ1 and ρ2 , show that the convex com-

bination ρx = xρ1 + (1 ’ x) ρ2 with 0 < x < 1 is also a classical state.

(2) Consider the superposition |ψ = C |± + C |’± of two coherent states, where C

and ± are both real.

(a) Derive the relation between C and ± imposed by the normalization condition

ψ |ψ = 1.

(b) For the state ρ = |ψ ψ| calculate the probability for observing n photons,

and decide whether the state is classical.

Gaussian states—

5.10

Apply the general relation (5.182) to the expression (5.225) for the Wigner character-

istic function of a Gaussian state to show that

1

d2 ζ exp [ζ — β ’ ζβ — ] exp |cosh r ζ + sinh r ζ — | /2

2

PG (±) =

π2

— exp ’ (nG + 1/2) |ζ|2 ,

where β is given by eqn (5.228). Evaluate the integral to get eqn (5.230).

6

Entangled states

The importance of the quantum phenomenon known as entanglement ¬rst became

clear in the context of the famous paper by Einstein, Podolsky, and Rosen (EPR)

(Einstein et al., 1935), which presented an apparent paradox lying at the foundations

of quantum theory. The EPR paradox has been the subject of continuous discussion

ever since. In the same year as the EPR paper, Schr¨dinger responded with several

o

1

publications (Schr¨dinger, 1935a, 1935b ) in which he pointed out that the essential

o

feature required for the appearance of the EPR paradox is the application of the

all-important superposition principle to the wave functions describing two or more

particles that had previously interacted. In these papers Schr¨dinger coined the name

o

˜entangled states™ for the physical situations described by this class of wave functions.

In recent times it has become clear that the importance of this phenomenon ex-

tends well beyond esoteric questions about the meaning of quantum theory; indeed,

entanglement plays a central role in the modern approach to quantum information

processing. The argument for the EPR paradox”which will be presented in Chapter

19”is based on the properties of the EPR states discussed in the following section.

After this, we will outline Schr¨dinger™s concept of entanglement, and then continue

o

with a more detailed treatment of the technical issues required for later applications.

6.1 Einstein“Podolsky“Rosen states

As part of an argument intended to show that quantum theory cannot be a com-

plete description of physical reality, Einstein, Podolsky, and Rosen considered two

distinguishable spinless particles A and B”constrained to move in a one-dimensional

position space”that are initially separated by a distance L and then ¬‚y apart like

the decay products of a radioactive nucleus. The particular initial state they used is

a member of the general family of EPR states described by the two-particle wave

functions ∞

dk

F (k) eik(xA ’xB ) .

ψ (xA , xB ) = (6.1)

’∞ 2π

Every function of this form is an eigenstate of the total momentum operator with

eigenvalue zero, i.e.

(pA + pB ) ψ (xA , xB ) = 0 . (6.2)

Peculiar phenomena associated with this state appear when we consider a measure-

ment of one of the momenta, say pA . If the result is k0 , then von Neumann™s projection

1 An English translation of this paper is given in Trimmer (1980).

½ Entangled states

postulate states that the wave function after the measurement is the projection of the

initial wave function onto the eigenstate of pA associated with the eigenvalue k0 .

Combining this rule with eqn (6.1) shows that the two-particle wave function after the

measurement is reduced to

ψred (xA , xB ) ∝ F (k0 ) eik0 (xA ’xB ) . (6.3)

The reduced state is an eigenstate of pB with eigenvalue ’ k0 . Since pA and pB are

constants of the motion for free particles, a measurement of pB at a later time will

always yield the value ’ k0 . Thus the particular value found in the measurement of

pA uniquely determines the value that would be found in any subsequent measurement

of pB .

The true strangeness of this situation appears when we consider the timing of the

measurements. Suppose that the ¬rst measurement occurs at tA and the second at tB >

tA . It is remarkable that the prediction of the value ’ k0 for the second measurement

holds even if (tB ’ tA ) < L/c. In other words, the result of the measurement of pB

appears to be determined by the measurement of pA even though the news of the

¬rst measurement result could not have reached the position of particle B at the time

of the second measurement. This spooky action-at-a-distance”which we will study in

Chapter 19”was part of the basis for Einstein™s conclusion that quantum mechanics

is an incomplete theory.

6.2 Schr¨dinger™s concept of entangled states

o

In order to understand Schr¨dinger™s argument, we ¬rst observe that a product wave

o

function,

φ (xA , xB ) = · (xA ) ξ (xB ) , (6.4)

does not have the peculiar properties of the EPR wave function ψ (xA , xB ). The joint

probability that the position of A is within dxA of xA0 and that the position of B is

within dxB of xB0 is the product

2 2

dp (xA0 , xB0 ) = |· (xA0 )| dxA |ξ (xB0 )| dxB (6.5)

of the individual probabilities, so the positions can be regarded as stochastically inde-

pendent random variables. The same argument can be applied to the momentum-space

wave functions. The joint probability that measurements of pA / and pB / yield values

in the neighborhood dkA of kA0 and dkB of kB0 is the product

dp (kA0 , kB0 ) = |· (kA0 )|2 dkA |ξ (kB0 )|2 dkB (6.6)

of independent probabilities, analogous to independent coin tosses. Thus a measure-

ment of xA tells us nothing about the values that may be found in a measurement of

xB , and the same holds true for the momentum operators pA and pB .

One possible response to the conceptual di¬culties presented by the EPR states

would be to declare them unphysical, but this tactic would violate the superposi-

tion principle: every linear combination of product wave functions also describes a

physically possible situation for the two-particle system. Furthermore, any interaction

½

Extensions of the notion of entanglement

between the particles will typically cause the wave function for a two-particle system”

even if it is initially described by a product function like φ (xA , xB )”to evolve into

a superposition of product wave functions that is nonfactorizable. Schr¨dinger called

o

these superpositions entangled states. An example is given by the EPR wave function

ψ (xA , xB ) which is a linear combination of products of plane waves for the two par-

ticles. The choice of the name ˜entangled™ for these states is related to the classical

principle of separability:

Complete knowledge of the state of a compound system yields complete knowledge

of the individual states of the parts.

This general principle does not require that the constituent parts be spatially sep-

arated; however, experimental situations in which there is spatial separation between

the parts provide the most striking examples of the failure of classical separability. A

classical version of the EPR thought experiment provides a simple demonstration of

this principle. We now suppose that the two particles are described by the classical

coordinates and momenta (qA , pA ) and (qB , pB ), so that the composite system is rep-

resented by the four-dimensional phase space (qA , pA , qB , pB ). In classical physics the

coordinates and momenta have de¬nite numerical values, so a state of maximum pos-

sible information for the two-particle system is a point (qA0 , pA0 , qB0 , pB0 ) in the two-

particle phase space. This automatically provides the points (qA0 , pA0 ) and (qB0 , pB0 )

in the individual phase spaces; therefore, the maximum information state for the com-

posite system determines maximum information states for the individual parts. The

same argument evidently works for systems with any ¬nite number of degrees of free-

dom.

In quantum theory, the uncertainty principle implies that the maximum possi-

ble information for a physical system is given by a single wave function, rather than

a point in phase space. This does not mean, however, that classical separability is

necessarily violated. The product function φ (xA , xB ) is an example of a maximal in-

formation state of the two-particle system, for which the individual wave functions in

the product are also maximal information states for the parts. Thus the product func-

tion satis¬es the classical notion of separability. By contrast, the EPR wave function

ψ (xA , xB ) is another maximal information state, but the individual particles are not

described by unique wave functions. Consequently, for an entangled two-particle state

we do not possess the maximum possible information for the individual particles; or

in Schr¨dinger™s words (Schr¨dinger, 1935b):

o o

Maximal knowledge of a total system does not necessarily include total knowledge

of all its parts, not even when these are fully separated from each other and at the

moment are not in¬‚uencing each other at all.

6.3 Extensions of the notion of entanglement

The EPR states describe two distinguishable particles, e.g. an electron and a proton

from an ionized hydrogen atom. Most of the work in the ¬eld of quantum information

processing has also concentrated on the case of distinguishable particles. We will see

later on that particles that are indistinguishable, e.g. two electrons, can be e¬ectively

distinguishable under the right conditions; however, it is not always useful”or even

½ Entangled states

possible”to restrict attention to these special circumstances. This has led to a con-

siderable amount of recent work on the meaning of entanglement for indistinguishable

particles.

In the present section, we will develop two pieces of theoretical machinery that are

needed for the subsequent discussion: the concept of tensor product spaces and the