Theorem 6.3 A pure state is separable if and only if the quantum ¬‚uctuations of all

observables A and B are uncorrelated.

See Exercise 6.2 for a suggested proof. Combining this result with the fact that entan-

gled states are not separable leads easily to the following theorem.

Theorem 6.4 A pure state |Ψ is entangled if and only if there is at least one pair of

observables A and B with correlated quantum ¬‚uctuations.

Thus the observation of correlations between measured values of A and B is experi-

mental evidence that the pure state |Ψ is entangled.

B Mixed states

Since the density operator ρ is simply a convenient description of a probability distri-

bution Pe over an ensemble, {|Ψe }, of normalized pure states, the analysis of entan-

glement for mixed states is based on the previous discussion of entanglement for pure

states.

¾¼ Entangled states

From this point of view, it is natural to de¬ne a separable mixed state by an

ensemble of separable pure states, i.e. |Ψe = |ζe |‘e for all e. The density operator

for a separable mixed state is consequently given by a convex linear combination,

Pe |ζe |‘e ζe | ‘e | ,

ρ= (6.45)

A BA B

e

of density operators for separable pure states. By writing this in the equivalent form

Pe (|ζe ζe |) — (|‘e ‘e |) ,

ρ= (6.46)

AA BB

e

we ¬nd that the reduced density operators are

Pe |ζe ζe |

ρA = TrB (ρ) = (6.47)

e

and

Pe |‘e ‘e | .

ρB = TrA (ρ) = (6.48)

e

In the special case that both sets of vectors are orthonormal, i.e.

ζe |ζf = ‘e |‘f = δef , (6.49)

the reduced density operators have the same spectra, so that”just as in the discussion

following Theorem 6.2”the two subsystems have the same purity and von Neumann

entropy. In the general case that one or both sets of vectors are not orthonormal, the

statistical properties can be quite di¬erent. An entangled mixed state is one that

is not separable, i.e. the ensemble contains at least one entangled pure state. De¬ning

useful measures of the degree of entanglement of a mixed state is a di¬cult problem

which is the subject of current research.

The clear experimental tests for separability and entanglement of pure states, pre-

sented in Theorems 6.3 and 6.4, are not available for mixed states. To see this, we

begin by writing out the correlation function and the averages of the observables A

and B as

C (A, B) = δA δB = Tr ρδA δB

Pe Ψe |δA δB| Ψe ,

= (6.50)

e

and

A= Pe Ψe |A| Ψe , B= Pe Ψe |B| Ψe . (6.51)

e e

We will separate the quantum ¬‚uctuations in each pure state from the ¬‚uctuations

associated with the classical probability distribution, Pe , over the ensemble of pure

states, by expressing the ¬‚uctuation operator δA as

δA = A ’ A = A ’ Ψe |A| Ψe + Ψe |A| Ψe ’ A . (6.52)

¾¼

Entanglement for identical particles

The operator

δe A = A ’ Ψe |A| Ψe (6.53)

represents the quantum ¬‚uctuations of A around the average de¬ned by |Ψe , and the

c-number

δ A e = Ψe |A| Ψe ’ A (6.54)

describes the classical ¬‚uctuations of the individual quantum averages Ψe |A| Ψe

around the ensemble average A . Using eqns (6.52)“(6.54), together with the analo-

gous de¬nitions for B, in eqn (6.50) leads to

C (A, B) = Cqu (A, B) + Ccl (A, B) , (6.55)

where

Cqu (A, B) = Pe Ψe |δe A δe B| Ψe (6.56)

e

represents the quantum part and

Ccl (A, B) = Pe δ A e δ B (6.57)

e

e

represents the classical part.

For a separable mixed state, the quantum correlation functions for each pure state

vanish, so that

C (A, B) = Ccl (A, B) = Pe δ A e δ B e . (6.58)

e

Thus the observables A and B are correlated in the mixed state, despite the fact that

they are uncorrelated for each of the separable pure states. An explicit example of this

peculiar situation is presented in Exercise 6.4. As a consequence of this fact, observing

correlations between two observables cannot be taken as evidence of entanglement for

a mixed state.

6.5 Entanglement for identical particles

6.5.1 Systems of identical particles

In this section, we will be concerned with particles having nonzero rest mass”e.g. elec-

trons, ions, atoms, etc.”described by nonrelativistic quantum mechanics. In quantum

theory, particles”as well as more complex systems”are said to be indistinguish-

able or identical if all of their intrinsic properties, e.g. mass, charge, spin, etc., are

the same. In classical mechanics, this situation poses no special di¬culties, since each

particle™s unique trajectory provides an identifying label, e.g. the position and mo-

mentum of the particle at some chosen time. In quantum mechanics, the uncertainty

principle removes this possibility, and indistinguishability of particles has radically

new consequences.2

2A more complete discussion of identical particles can be found in any of the excellent texts on

quantum mechanics that are currently available, for example Cohen-Tannoudji et al. (1977b, Chap.

XIV) or Bransden and Joachain (1989, Chap. 10).

¾¼ Entangled states

For identical particles, we will replace the previous labeling A and B by 1, 2, . . . , N ,

for the general case of N identical particles. Since the particles are indistinguishable,

the labels have no physical signi¬cance; they are merely a bookkeeping device. An

N -particle state |Ψ can be represented by a wave function

Ψ (1, 2, . . . , N ) = 1, 2, . . . , N |Ψ , (6.59)

where the arguments 1, 2, . . . , N stand for a full set of coordinates for each particle.

For example, 1 = (r1 , s1 ), where r1 and s1 are respectively eigenvalues of r1 and s1z .

The permutations on the labels form the symmetric group SN (Hamermesh,

1962, Chap. 7), with group multiplication de¬ned by successive application of permu-

tations. An element P in SN is de¬ned by its action: 1 ’ P (1) , 2 ’ P (2) , . . . , N ’

P (N ). Each permutation P is represented by an operator ZP de¬ned by

1, 2, . . . , N |ZP | Ψ = P (1) , P (2) , . . . , P (N ) |Ψ , (6.60)

or in the more familiar wave function representation,

ZP Ψ (1, 2, . . . , N ) = Ψ (P (1) , P (2) , . . . , P (N )) . (6.61)

It is easy to show that ZP is both unitary and hermitian. A transposition is a permu-

tation that interchanges two labels and leaves the rest alone, e.g. P (1) = 2, P (2) = 1,

and P (j) = j for all other values of j. Every permutation P can be expressed as a prod-

uct of transpositions, and P is said to be even or odd if the number of transpositions

is respectively even or odd. These de¬nitions are equally applicable to distinguishable

and indistinguishable particles.

One consequence of particle identity is that operators that act on only one of the

particles, such as A and B in Theorems 6.3 and 6.4, are physically meaningless. All

physically admissible observables must be unchanged by any permutation of the labels

for the particles, i.e. the operator F representing a physically admissible observable

must satisfy

†

(ZP ) F ZP = F . (6.62)

Suppose, for example, that A is an operator acting in the Hilbert space H(1) of one-

particle states; then for N particles the physically meaningful one-particle operator

is

A = A (1) + A (2) + · · · + A (N ) , (6.63)

where A (j) acts on the coordinates of the particle with the label j.

The restrictions imposed on admissible state vectors by particle identity are a

bit more subtle. For systems of identical particles, indistinguishability means that a

physical state is unchanged by any permutation of the labels assigned to the particles.

For a pure state, this implies that the state vector can at most change by a phase

factor under permutation of the labels:

ZP |Ψ = eiξP |Ψ . (6.64)

¾¼

Entanglement for identical particles

By using the special properties of permutations, one can show that the only possibilities

P P

are eiξP = 1 or eiξP = (’1) , where (’1) = +1 (’1) for even (odd) permutations.3

In other words, admissible state vectors must be either completely symmetric or com-

pletely antisymmetric under permutation of the particle labels. These two alternatives

respectively de¬ne orthogonal subspaces (HC )sym and (HC )asym of the N -fold tensor

product space HC = H(1) (1)— · · ·—H(1) (N ). It is an empirical fact that all elementary

particles belong to one of two classes: the fermions, described by the antisymmetric

states in (HC )asym ; and the bosons, described by the symmetric states in (HC )sym .

As a consequence of the antisymmetry of the state vectors, two fermions cannot oc-

cupy the same single-particle state; however the symmetry of bosonic states allows

any number of bosons to occupy a single-particle state. For large numbers of parti-

cles, these features lead to strikingly di¬erent statistical properties for fermions and

bosons; the two kinds of particles are said to satisfy Bose“Einstein or Fermi sta-

tistics. This fact has many profound physical consequences, ranging from the Pauli

exclusion principle to Bose“Einstein condensation.

In the following discussions, we will often be concerned with the special case of two

identical particles. In this situation, a basis for the tensor product space H(1) — H(1)

is provided by the family of product vectors {|χmn = |φm 1 |φn 2 }, where {|φn } is a

basis for the single-particle space H(1) . A general state |Ψ in H(1) — H(1) can then be

expressed as

|Ψ = Ψmn |χmn , (6.65)

m n

where

Ψmn = χmn |Ψ . (6.66)

The symmetric (bosonic) and antisymmetric (fermionic) subspaces are respectively

characterized by the conditions

Ψmn = Ψnm (6.67)

and

Ψmn = ’Ψnm . (6.68)

6.5.2 E¬ective distinguishability

There must be situations in which the indistinguishability of particles makes no di¬er-

ence. If this were not the case, explanations of electron scattering on the Earth would

have to take into account the presence of electrons on the Moon. This would create

rather serious problems for experimentalists and theorists alike. The key to avoiding

this nightmare is the simple observation that experimental devices have a de¬nite

position in space and occupy a ¬nite volume. As a concrete example, consider a mea-

suring apparatus that occupies a volume V centered on the point R. Another fact

of life is that plane waves are an idealization. Physically meaningful wave functions

are always normalizable; consequently, they are localized in some region of space. In

many cases, the wave function falls o¬ exponentially, e.g. like exp (’ |r ’ r0 | /Λ), or

3 Thisis generally true when the particle position space is three dimensional. For systems restricted

to two dimensions, continuous values of ξP are possible. This leads to the notion of anyons, see for

example Leinaas and Myrheim (1977).

¾¼ Entangled states

2

exp ’ |r ’ r0 | /Λ2 , where r0 is the center of the localization region. In either case,

we will say that the wave function is exponentially small when |r ’ r0 | Λ. With this

preparation, we will say that an operator F ”acting on single-particle wave functions

in H(1) ”is a local observable in the region V if F ·s (r) is exponentially small in V

whenever the wave function ·s (r) is itself exponentially small in V .

Let us now consider two indistinguishable particles occupying the states |φ and

|· , where |φ is localized in the volume V and |· is localized in some distant region”

possibly the Moon or just the laboratory next door”so that ·s (r) = rs |· is expo-

nentially small in V . The state vector for the two bosons or fermions has the form

1

|Ψ = √ {|φ |· ± |· |φ 2 } , (6.69)

1 2 1

2

and a one-particle observable is represented by an operator F = F (1) + F (2). Let Z12

be the transposition operator, then Z12 |Ψ = ± |Ψ and Z12 F (2) Z12 = F (1). With

these facts in hand it is easy to see that

Ψ |F | Ψ = 2 Ψ |F (1)| Ψ

= φ |F | φ + · |F | · ± φ |F | · · |φ ± · |F | φ φ |· . (6.70)

The ¬nal two terms in the last equation are negligible because of the small overlap be-

tween the one-particle states, but the term · |F | · is not small unless the operator F

represents a local observable for V . When this is the case, the two-particle expectation

value,

Ψ |F | Ψ = φ |F | φ , (6.71)

is exactly what one would obtain by assuming that the two particles are distinguish-

able, and that a measurement is made on the one in V .

The lesson to be drawn from this calculation is that the indistinguishability of two

particles can be ignored if the relevant single-particle states are e¬ectively nonover-

lapping and only local observables are measured. This does not mean that an electron

on the Earth and one on the Moon are in any way di¬erent. What we have shown is

that the large separation involved makes the indistinguishability of the two electrons

irrelevant”for all practical purposes”when analyzing local experiments conducted on

the Earth. On the other hand, the measurement of a local observable will be sensitive

to the indistinguishability of the particles if the one-particle states have a signi¬cant

overlap. Consider the situation in which the distant particle is bound to a potential

well centered at r0 . Bodily moving the potential well so that the original condition

|r0 ’ RA | Λ is replaced by |r0 ’ RA | Λ restores the e¬ects of indistinguishability.

6.5.3 De¬nition of entanglement

For identical particles, there are no physically meaningful operators that can single out

one particle from the rest; consequently, there is no way to separate a system of two

identical particles into distinct subsystems. How then are we to extend the de¬nitions

of separability and entanglement given in Section 6.4.1 to systems of identical particles?

Since de¬nitions cannot be right or wrong”only more or less useful”it should not be

too surprising to learn that this question has been answered in at least two di¬erent

¾¼

Entanglement for identical particles

ways. In the following paragraphs, we will give a traditional answer and compare

it to another de¬nition that is preferred by those working in the ¬eld of quantum

information processing.

For single-particle states |ζ 1 and |· 2 , of distinguishable particles 1 and 2, the

de¬nition (6.40) tell us that the product vector

|Ψ = |ζ |· (6.72)

1 2

is separable, but if the particles are identical bosons then |Ψ must be replaced by the

symmetrized expression

|Ψ = C {|ζ |· + |· |ζ 2 } , (6.73)

1 2 1

where C is a normalization constant. Unless |· = |ζ , this has the form of an en-

tangled state for distinguishable particles. The traditional approach is to impose the

symmetry requirement on the de¬nition of separability used for distinguishable parti-

cles; therefore, a state |Ψ of two identical bosons is said to be separable if it can be

expressed in the form

|Ψ = |ζ 1 |ζ 2 . (6.74)

In other words, both bosons must occupy the same single-particle state.

It is often useful to employ the de¬nition (6.66) of the expansion coe¬cients Ψmn

to rewrite the de¬nition of separability as

Ψmn = Zm Zn , (6.75)

where

Zn = φn |ζ . (6.76)

Thus separability for bosons is the same as the factorization condition (6.75) for the

expansion coe¬cients. From the original form (6.74) it is clear that eqn (6.75) must

hold for all choices of the single-particle basis vectors |φn .

Entangled states are de¬ned as those that are not separable, e.g. the state |Ψ

in eqn (6.73). This seems harmless enough for bosons, but it has a surprising result

for fermions. In this case eqn (6.72) must be replaced by

|Ψ = C {|ζ |· ’ |· |ζ 2 } , (6.77)

1 2 1

and setting |· = |ζ gives |Ψ = 0, which is simply an expression of the Pauli exclusion

principle. Consequently, extending the distinguishable-particle de¬nition of entangle-

ment to fermions leads to the conclusion that every two-fermion state is entangled.

An alternative transition from distinguishable to indistinguishable particles is based

on the observation that the symmetrized states

|Ψ = C {|ζ |· ± |· |ζ 2 } (6.78)

1 2 1

for identical particles seem to be the natural analogues of product vectors for distin-

guishable particles. From this point of view, states that have the minimal form (6.78)

imposed by Bose or Fermi symmetry should not be called entangled (Eckert et al.,

¾½¼ Entangled states

2002). For those working in the ¬eld of quantum information processing, this view is

strongly supported by the fact that states of the form (6.78) do not provide a useful

resource, e.g. for quantum computing. This argument is, however, open to the objec-

tion that utility”like beauty”is in the eye of the beholder. We will illustrate this

point by way of an example.

A state |Ψ of two electrons is described by a wave function Ψ (r1 , s1 ; r2 , s2 ) which

is antisymmetric with respect to the transposition (r1 , s1 ) ” (r2 , s2 ). For this example,

it is convenient to use the wave function representation for the spatial coordinates and

to retain the Dirac ket representation for the spins. With this notation, we consider

the spin-singlet state

|Ψ (r1 , r2 ) = ψ (r1 ) ψ (r2 ) {|‘ 1 |“ ’ |“ |‘ 2 } , (6.79)

2 1

which is symmetric in the spatial coordinates and antisymmetric in the spins. If Alice

detects a single electron and measures the z-component of its spin to be sz = +1/2,

then an electron detected by Bob is guaranteed to have the value sz = ’1/2. Thus the

state de¬ned in eqn (6.79) displays the most basic feature of entanglement; namely,

that the result of one measurement gives information about the possible results of

measurements that could be made on another part of the system. This establishes the

fundamental utility of the state in eqn (6.79), despite the fact that it does not provide

a resource for quantum information processing. A similar example can be constructed

for bosons, so we will retain the traditional de¬nition of entanglement for identical

particles.

Our preference for extending the traditional de¬nition of entanglement to indistin-

guishable particles, as opposed to the more restrictive version presented above, does

not mean that the latter is not important. On the contrary, the stronger interpreta-

tion of entanglement captures an essential physical feature that plays a central role in

many applications. In order to distinguish between the two notions of entanglement,

we will say that a two-particle state that is entangled in the minimal form (6.78),

required by indistinguishability, is kinematically entangled, and that an entangled

two-particle state is dynamically entangled if it cannot be expressed in the form

(6.78). The use of the term ˜dynamical™ is justi¬ed by the observation that dynamically

entangled states can only be produced by interaction between the indistinguishable

particles. For photons, this distinction enters in a natural way in the analysis of the

Hong“Ou“Mandel e¬ect in Section 10.2.1. For distinguishable particles, there is no

symmetry condition for multiparticle states; consequently, the notion of kinematical

entanglement cannot arise and all entangled states are dynamically entangled.

6.6 Entanglement for photons

Since photons are bosons, it seems reasonable to expect that the de¬nition of entangle-

ment introduced in Section 6.5.3 can be applied directly to photons. We will see that

this expectation is almost completely satis¬ed, except for an important reservation

arising from the absence of a photon position operator.

The most intuitively satisfactory way to understand entanglement for bosons is in

terms of an explicit wave function like

¾½½

Entanglement for photons

1

ψs1 s2 (r1 , r2 ) = √ [ζs1 (r1 ) ·s2 (r2 ) + ·s1 (r1 ) ζs2 (r2 )] , (6.80)

2

where the subscripts describe internal degrees of freedom such as spin. If we recall that

ζs1 (r1 ) = r1 , s1 |ζ , where |r1 , s1 is an eigenstate of the position operator r for the

particle, then it is clear that the existence of a wave function depends on the existence

of a position operator r. For applications to photons, this brings us face to face with the

well known absence”discussed in Section 3.6.1”of any acceptable position operator

for the photon. In Section 6.6.1 we will show that the absence of position-space wave

functions for photons is not a serious obstacle to de¬ning entanglement, and in Section

6.6.2 we will ¬nd that the intuitive bene¬ts of the absent wave function can be largely

recovered by considering a simple model of photon detection.

6.6.1 De¬nition of entanglement for photons

In Section 6.5.1 we observed that states of massive bosons belong to the symmetrical

subspace (HC )sym of the tensor product space HC describing a many-particle system.

For photons, the de¬nitions of Fock space in Sections 2.1.2-C or 3.1.4 can be un-

derstood as a direct construction of (HC )sym that works for any number of photons.

In the example of a two-particle system, the Fock space approach replaces explicitly

symmetrized vectors like

|φm 1 |φn 2 + |φn 1 |φm 2 (6.81)

by Fock-space vectors,

a† a† s |0 , (6.82)

ks k

generated by applying creation operators to the vacuum. Despite their di¬erent ap-

pearance, the physical content of the two methods is the same.

We will use box-quantized creation operators to express a general two-photon state

as

1