±’·±— ’·2 /2
d2 ·e· e χN (·)
W (±) =
π2
1 —
’±— ) ’· — (β’±) ’·2 /2
d2 βP (β) d2 · e·(β
=2 e e . (5.194)
π
The ·integral is readily done by converting to real variables, and the relation between
the Wigner distribution and the P distribution is
2
d2 βe’2β’± P (β) .
2
W (±) = (5.195)
π
An interesting consequence of this relation is that a classical state automatically
yields a positive Wigner distribution, i.e.
P (±) 0 implies W (±) 0, (5.196)
but the opposite statement is not true:
W (±) 0 does not imply P (±) 0. (5.197)
This is demonstrated by exhibiting a single example”see Exercise 5.7”of a state with
a positive Wigner function that is not classical.
It is natural to wonder why P (±) 0 should be chosen as the de¬nition of a
classical state instead of W (±) 0. The relations (5.196) and (5.197) give one reason,
since they show that P (±) 0 is a stronger condition. A more physical reason is that
counting rates are described by expectation values of normalordered products, rather
than Weyl products. This means that P (±) is more directly related to the relevant
experiments than is W (±).
Multimode phase space—
5.6.4
In Section 5.5 we de¬ned multimode coherent states ± by aκ ± = ±κ ± , where aκ
is the annihilation operator for the mode κ and
± = (±1 , ±2 , . . . , ±κ , . . .) . (5.198)
For states in which only a ¬nite number of modes are occupied, i.e. aκ ± = 0 for
κ > κ , the characteristic functions de¬ned previously have the generalizations
†
’· — a
χW · = Tr ρe··a , (5.199)
† —
χN · = Tr ρe··a e’· a
, (5.200)
½ Coherent states
where · ≡ (·1 , ·2 , . . .), and
· · a† = ·κ a† . (5.201)
κ
κκ
The corresponding distributions are de¬ned by multiple Fourier transforms. For ex
ample the P distribution is
⎡ ¤
2
d ·κ ¦ ’··±— +·— ·±
⎣
P (±) = e χN · , (5.202)
π2
κκ
and the density operator is given by
⎡ ¤
⎣ d2 ±κ ¦ ± P (±) ± .
ρ= (5.203)
κκ
All this is plain sailing as long as κ remains ¬nite, but some care is required to
get the mathematics right when κ ’ ∞. This has been done in the work of Klauder
and Sudarshan (1968), but the κ ’ ∞ limit is not strictly necessary in practice. The
reason is to be found in the alternative characterization of coherent states given by
± ’ w , where
A(+) [v] w = (v, w) w , (5.204)
and the wave packets w, v, etc. are expressed as expansions in the chosen modes,
w (r) = ±κ wκ (r) . (5.205)
κ
The vector ¬elds v and w belong to the classical phase space “em de¬ned in Section
3.5.1, so the expansion coe¬cients ±κ must go to zero as κ ’ ∞. Thus any real ex
perimental situation can be adequately approximated by a ¬nite number of modes.
With this comforting thought in mind, we can express the characteristic and distribu
tion functions as functionals of the wave packets. In this language, the normalordered
characteristic function and the P distribution are respectively given by
χN (v) = Tr ρ exp A(+) [v] exp ’A(’) [v] (5.206)
and
D [v] exp {(w, v) ’ (v, w)} χN (v) .
P (w) = (5.207)
The symbol D [v] stands for a (functional) integral over the in¬nitedimensional
space “em of classical wave packets; but, as we have just remarked, it can always
be approximated by a ¬nitedimensional integral over the collection of modes with
nonnegligible amplitudes.
Gaussian states— ½
Gaussian states—
5.7
In classical statistics, the Gaussian (normal) distribution has the useful property that
the ¬rst two moments determine the values of all other moments (Gardiner, 1985, Sec.
2.8.1). For a Gaussian distribution over N real variables”with the averages of single
variables arranged to vanish”all odd moments vanish and the even moments satisfy
(2q)!
x1 · · · x2q = xk xl · · · xm xn ]sym ,
[ xi xj (5.208)
q!2q
where i, j, k, l, m, n range over 1, . . . , 2q and the subscript sym indicates the average
over all ways of partitioning the variables into pairs. Two fourthorder examples are
4! 1
x1 · · · x4 = { x1 x2 x3 x4 + x1 x3 x2 x4 + x1 x4 x2 x3 }
2!22 3
= x1 x2 x3 x4 + x1 x3 x2 x4 + x1 x4 x2 x3 (5.209)
and
x4 = 3 x2 x2 . (5.210)
1 1 1
This classical property is shared by the coherent states, as can be seen from the
general identity
m
± a†m an ± = ±—m ±n = ± a† ±
n
( ± a ± ) . (5.211)
A natural generalization of the classical notion of a Gaussian distribution is to de¬ne
Gaussian states (Gardiner, 1991, Sec. 4.4.5) as those that are described by density
operators of the form
ρG = N exp ’G a, a† , (5.212)
where
1 1
G a, a† = La† a + M a†2 + M — a2 , (5.213)
2 2
L and M are free parameters, and the constant N is ¬xed by the normalization
condition Tr ρ = 1.
For the special value M = 0, the Gaussian state ρG has the form of a thermal state,
and we already know (see eqn (5.148)) how to calculate the Wigner characteristic
function for this case. We would therefore like to transform the general Gaussian state
into this form. If the operators a and a† were replaced by complex variables ± and ±— ,
this would be easy. The cnumber quadratic form G (±, ±— ) can always be expressed
as a sum of squares by a linear transformation to new variables
± = µ± + ν±— ,
(5.214)
±— = µ— ±— + ν — ± .
What is needed now is the quantum analogue of this transformation, i.e. the new and
old operators are related by
a = U aU † , (5.215)
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 cnumbers. The unitary transformation preserves the commutation
relations, so the cnumber 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 thermallike 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
aterm 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 adependence 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 —
ζ2 ,
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
β2
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 normalordered 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 coordinatespace 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 operatorvalued 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 Numberphase 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 cnumber
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 antinormallyordered characteristic function—
5.8
The argument in Section 5.6.1B 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 antinormallyordered 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) ζ2 ,
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
allimportant 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 onedimensional
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 twoparticle 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 twoparticle 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 actionatadistance”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 momentumspace
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 twoparticle system. Furthermore, any interaction
½
Extensions of the notion of entanglement
between the particles will typically cause the wave function for a twoparticle 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 fourdimensional 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 twoparticle 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 twoparticle 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 twoparticle 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
Schmidt decomposition. In the following sections, we will give a de¬nition of entan
glement for the general case of two distinguishable quantum objects, and then extend
this de¬nition to indistinguishable particles and to the electromagnetic ¬eld.
6.3.1 Tensor product spaces
In Section 4.2.1, the Hilbert space HQED for quantum electrodynamics was constructed
as the tensor product of the Hilbert space Hchg for the atoms and the Fock space HF for
the ¬eld. This construction only depends on the Born interpretation and the superposi
tion principle; consequently, it works equally well for any pair of distinguishable phys
ical systems A and B described by Hilbert spaces HA and HB . Let {φ± } and {·β }
be basis sets for HA and HB respectively, then for any pair of vectors (ψ A , ‘ B ) the
product vector Λ = ψ A ‘ B is de¬ned by the probability amplitudes
φ± , ·β Λ = φ± ψ ·β ‘ . (6.7)
Since {φ± } and {·β } are complete orthonormal sets of vectors in their respective
spaces, the inner product between two such vectors is consistently de¬ned by
Λ1 Λ2 = Λ1 φ± , ·β φ± , ·β Λ2
±β
ψ1 φ± ‘1 ·β φ± ψ2 ·β ‘2
=
±β
= ψ1 ψ2 ‘1 ‘2 , (6.8)
where the inner products ψ1 ψ2 and ‘1 ‘2 refer respectively to HA and HB . The
linear combination of two product vectors is de¬ned by componentwise addition, i.e.
the ket
¦ = c1 Λ1 + c2 Λ2 (6.9)
is de¬ned by the probability amplitudes
φ± , ·β ¦ = c1 φ± , ·β Λ1 + c2 φ± , ·β Λ2
= c1 φ± ψ1 ·β ‘1 + c2 φ± ψ2 ·β ‘2 . (6.10)
The tensor product space HC = HA — HB is the family of all linear combinations of
product kets. The family of product kets,
{χ±β = φ± , ·β = φ± A ·β B } , (6.11)
forms a complete orthonormal set with respect to the inner product (6.8), i.e.
 χ±β = φ±  φ± ·β ·β = δ±± δββ ,
χ± (6.12)
β
½
Extensions of the notion of entanglement
and a general vector ¦ in HC can be expressed as
¦ = ¦±β χ±β = ¦±β φ± ·β . (6.13)
A B
± ±
β β
The inner product between any two vectors is
Ψ— ¦±β .
Ψ ¦ = (6.14)
±β
± β
One can show that choosing new basis sets in HA and HB produces an equivalent
basis set for HC . This notion can be extended to composite systems composed of N
distinguishable subsystems described by Hilbert spaces H1 , . . . , HN . The composite
system is described by the N fold tensor product space
HC = H1 — · · · — HN , (6.15)
which is de¬ned by repeated use of the twospace de¬nition given above.
It is useful to extend the tensor product construction for vectors to a similar one
for operators. Let A and B be operators acting on HA and HB respectively, then the
operator tensor product, A — B, is the operator acting on HC de¬ned by
(A — B) ¦ = ¦±β A φ± A B ·β B . (6.16)
± β
This de¬nition immediately yields the rule
(A1 — B1 ) (A2 — B2 ) = (A1 A2 ) — (B1 B2 ) (6.17)
for the product of two such operators. Since the notion of the outer or tensor product
of matrices and operators is less familiar than the idea of product wave functions, we
sometimes use the explicit — notation for operator tensor products when it is needed
for clarity. The de¬nition (6.16) also allows us to treat A and B as operators acting
on the product space HC by means of the identi¬cations
A ” A — IB ,
(6.18)
B ” IA — B ,
where IA and IB are respectively the identity operators for HA and HB . These relations
lead to the rule
AB ” A — B , (6.19)
so we can use either notation as dictated by convenience.
As explained in Section 2.3.2, a mixed state of the composite system is described
by a density operator
Pe Ψe Ψe  ,
ρ= (6.20)
e
where Pe is a probability distribution on the ensemble {Ψe } of pure states. The
expectation values of observables for the subsystem A are determined by the reduced
density operator
½ Entangled states
ρA = TrB (ρ) , (6.21)
where the partial trace over HB of a general operator X acting on HC is the operator
on HA with matrix elements
φ± TrB (X) φ± = χ± β X χ±β . (6.22)
β
This can be expressed more explicitly by using the fact that every operator on HC can
be decomposed into a sum of operator tensor products, i.e.
An — Bn .
X= (6.23)
n
Substituting this into the de¬nition (6.22) de¬nes the operator
An TrB (Bn )
TrB (X) = (6.24)
n
acting on HA , where the cnumber
·β Bn  ·β
TrB (Bn ) = (6.25)
β
is the trace over HB . The average of an observable A for the subsystem A is thus given
by
Tr (ρA) = TrA (ρA A) . (6.26)
In the same way the average of an observable B for the subsystem B is
Tr (ρB) = TrB (ρB B) , (6.27)
where
ρB = TrA (ρ) . (6.28)
6.3.2 The Schmidt decomposition
For ¬nitedimensional spaces, the general expansion (6.13) becomes
dA dB
Ψ = Ψ±β χ±β , (6.29)
±=1 β=1
where Ψ±β = χ±β Ψ . In the study of entanglement, it is useful to have an alternative
representation that is speci¬cally tailored to a particular state vector Ψ .
For our immediate purposes it is su¬cient to explain the geometrical concepts
leading to this special expansion; the technical details of the proof are given in Section
6.3.3. The basic idea is illustrated in Fig. 6.1, which shows the original vector, Ψ ,
and the normalized product vector, ζ1 A ‘1 B , that has the largest projection Y1
onto Ψ .
½
Extensions of the notion of entanglement
Ψ>

Fig. 6.1 A qualitative sketch of the procedure
for deriving the Schmidt decomposition, given
by eqn (6.30). The heavy arrow represents the
original vector Ψ and the plane represents
ζ ‘ the set of all product vectors ζ ‘ . The light
 
> >
1 1
arrow denotes the projection of Ψ onto the
plane.
After determining this ¬rst product vector, we de¬ne a new vector, Ψ1 = Ψ ’
Y1 ζ1 A ‘1 B , that is orthogonal to ζ1 A ‘1 B . The same game can be played with
Ψ1 ; that is, we ¬nd the normalized product vector ζ2 A ‘2 B that has the maximum
projection Y2 onto Ψ1 and is orthogonal to ζ1 A ‘1 B . Since the spaces HA and HB
are ¬nite dimensional, this process must terminate after a ¬nite number r of steps, i.e.
when Yr+1 = 0. The orthogonality of the successive product vectors implies that they
are linearly independent; therefore, the largest possible number of steps is the smaller
of the two dimensions, min (dA , dB ). The ¬nal result is the Schmidt decomposition
r
Ψ = Yn ζn ‘n , (6.30)
A B
n=1
where the Schmidt rank r min (dA , dB ). The density operator for this pure state
is therefore
r r
—
Ym Yn ζm ‘m ζn  ‘n 
ρ= A BA B
m=1 n=1
r r
—
ζn ) — (‘m ‘n ) .
= Ym Yn (ζm (6.31)
AA BB
m=1 n=1
The minimum value (r = 1) of the Schmidt rank occurs when Ψ is a product vector.
The product vectors ζn A ‘n B are orthonormal by construction, i.e. ζn ζm =
‘n ‘m = δnm , and the coe¬cients Yn satisfy the normalization condition
r
2
Yn  = 1 . (6.32)
n=1
In applications of the Schmidt decomposition (6.30), it is important to keep in mind
that the basis vectors ζn A ‘n B themselves”and not just the coe¬cients Yn ”are
uniquely associated with the vector Ψ . The Schmidt decomposition for a new vector
¦ would require a new set of basis vectors.
Proof of the Schmidt decomposition—
6.3.3
We o¬er here a proof”modeled on one of the arguments given by Peres (1995, Sec.
53)”that the expansion (6.30) exists. For normalized vectors ζ1 A and ‘1 B : set
¾¼¼ Entangled states
ζ1 , ‘1 = ζ1 A ‘1 B , and consider the projection operator P1 = ζ1 , ‘1 ζ1 , ‘1 . The
identity Ψ = P1 Ψ + (1 ’ P1 ) Ψ can then be written as Ψ = Y1 ζ1 , ‘1 + Ψ1 ,
where Y1 = ζ1 , ‘1 Ψ and the vector Ψ1 = (1 ’ P1 ) Ψ is orthogonal to ζ1 , ‘1 . By
applying the general expansion (6.29) to the vectors Ψ and ζ1 , ‘1 , one can express
Y1 2 as
2
dA dB
Ψ— x± yβ
Y1 2 = 1, (6.33)
±β
±=1 β=1
where x± = φ± ζ1 , yβ = ·β ‘1 , and the upper bound follows from the normaliza
tion of the vectors de¬ning Y1 .
From a geometrical point of view, Y1  is the magnitude of the projection of ζ1 , ‘1
2
onto Ψ . In quantum terms, Y1  is the probability that a measurement of P1 will
result in the eigenvalue unity and will leave the system in the state ζ1 , ‘1 . The
next step is to choose the product vector ζ1 , ‘1 ”i.e. to ¬nd values of x± and yβ ”
that maximizes Y1 2 . This is always possible, since Y1 2 is a bounded, continuous
function of the ¬nite set of complex variables (x1 , . . . , xdA , y1 , . . . , ydB ). The solution
is not unique, since the overall phase of ζ1 , ‘1 is not determined by the maximization
procedure. This is not a real di¬culty; the undetermined phases can be chosen so that
Y1 is real. In general, there may be several linearly independent solutions for ζ1 , ‘1 ,
but this is also not a serious di¬culty. By forming appropriate linear combinations of
the degenerate solutions it is always possible to make them mutually orthogonal. We
will therefore simplify the discussion by assuming that the maximum is always unique.
Note that the maximum value of Y1 2 can only be unity if the original vector is itself
a product vector.
Now that we have made our choice of ζ1 , ‘1 , we pick a new product vector
ζ2 , ‘2 ”with projection operator P2 = ζ2 , ‘2 ζ2 , ‘2 ”and write the identity Ψ1 =
P2 Ψ1 + (1 ’ P2 ) Ψ1 as
Ψ1 = Y2 ζ2 , ‘2 + Ψ2 , (6.34)
where Y2 = ζ2 , ‘2 Ψ1 and Ψ2 = (1 ’ P2 ) Ψ1 . Since Ψ1 is orthogonal to ζ1 , ‘1 ,
we can assume that ζ2 , ‘2 is also orthogonal to ζ1 , ‘1 . Now we proceed, as in the
2
¬rst step, by choosing ζ2 , ‘2 to maximize Y2  . At this point, we have
Ψ = Y1 ζ1 , ‘1 + Y2 ζ2 , ‘2 + Ψ2 , (6.35)
and this procedure can be repeated until the next projection vanishes. The last re
mark implies that the number of terms is limited by the minimum dimensionality,
min (dA , dB ); therefore, we arrive at eqn (6.30).
6.4 Entanglement for distinguishable particles
In Section 6.3.1 we saw that the Hilbert space for a composite system formed from any
two distinguishable subsystems A and B (which can be atoms, molecules, quantum
dots, etc.) is the tensor product HC = HA — HB . The current intense interest in
quantum information processing has led to the widespread use of the terms parties
¾¼½
Entanglement for distinguishable particles
for A and B, and bipartite system, for what has traditionally been called a two
particle system. Since our interests in this book are not limited to quantum information
processing, we will adhere to the traditional terminology in which the distinguishable
objects A and B are called particles and the composite system is called a twoparticle
or twopart system.
In order to simplify the discussion, we will assume that the two Hilbert spaces have
¬nite dimensions, dA , dB < ∞. A composite system composed of two distinguishable,
spin1/2 particles”for example, impurity atoms bound to adjacent sites in a crys
tal lattice”provides a simple example that ¬ts within this framework. In this case,
HA = HB = C2 , and all observables can be written as linear combinations of the spin
operators, e.g.
OA = C0 I A + C1 n · SA , (6.36)
where C0 and C1 are constants, I A is the identity operator, n is a unit vector, SA =
σ A /2, and σ = (σx , σy , σz ) is the vector of Pauli matrices. A discrete analogue of the
EPR wave function is given by the singlet state
1
= √ {‘
S = 0 “ ’ “ ‘ B} , (6.37)
AB A B A
2
where the spinup and spindown states are de¬ned by
1 1
n · SA ‘ = + ‘ n · SA “ = ’ “
A, , etc. (6.38)
A A A
2 2
The singlet state has total spin angular momentum zero, so one can show”as in
Exercise 6.3”that it has the same expression for every choice of n. If several spin
projections are under consideration, the notation ‘n A and “n A can be used to
distinguish them.
The most important feature of entanglement for pure states is that the result of
one measurement yields information about the probability distribution of a second,
independent measurement. For the twospin system, a measurement of n · SA with the
result ±1/2 guarantees that a subsequent measurement of n · SB will yield the result
“1/2. A discrete version of the unentangled (separable) state (6.4) is
φ = {c‘ ‘ + c“ “ A } {b‘ ‘ + b“ “ B} . (6.39)
A B
In this case, measuring n · SA provides no information at all on the distribution of
values for n · SB .
6.4.1 De¬nition of entanglement
We will approach the general idea of entanglement indirectly by ¬rst de¬ning separable
(unentangled) pure and mixed states, and then de¬ning entangled states as those that
are not separable. Since entangled states are the focus of this chapter, this negative
procedure may seem a little strange. The explanation is that separable states are simple
and entangled states are complicated. We will de¬ne separability and entanglement
in terms of properties of the state vector or density operator. This is the traditional
approach, and it provides a quick entry into the applications of these notions.
¾¼¾ Entangled states
A Pure states
The de¬nitions we give here are simply generalizations of the examples presented in
Sections 6.1 and 6.2, or rather the ¬nitedimensional analogues given by eqns (6.37)
and (6.39). Thus we say that a pure state Ψ of the twoparticle system described by
the Hilbert space HC = HA — HB is separable if it can be expressed as
Ψ = ¦ Ξ , (6.40)
A B
which is the general version of eqn (6.39), and entangled if it is not separable. This
awkward negative de¬nition of entanglement as the absence of separability can be
avoided by using the Schmidt decomposition (6.30). A little thought shows that the
states that cannot be written in the form (6.40) are just the states with r > 1. With
this in mind, we could de¬ne entanglement positively by saying that Ψ is entangled
if it has Schmidt rank r > 1. The discrete analogue (6.37) of the continuous EPR wave
function is an example of an entangled state.
The de¬nitions given above imply several properties of the state vector which,
conversely, imply the original de¬nitions. Thus the new properties can be used as
equivalent de¬nitions of separability and entanglement for pure states. For ease of
reference, we present these results as theorems.
Theorem 6.1 A pure state is separable if and only if the reduced density operators
represent pure states, i.e. separable states satisfy the classical separability principle.
There are two assertions to be proved.
(a) The reduced density operators for a separable pure state Ψ represent pure states
of A and B.
(b) If the reduced density operators for a pure state Ψ describe pure states of A and
B, then Ψ is separable.
Suggestions for these arguments are given in Exercise 6.1.
Since entanglement is the absence of separability, this result can also be stated as
follows.
Theorem 6.2 A pure state is entangled if and only if the reduced density operators
for the subsystems describe mixed states.
Mixed states are, by de¬nition, not states of maximum information, so this result
explicitly demonstrates that possession of maximum information for the total system
does not yield maximum information for the constituent parts. However, the statistical
properties of the mixed states for the subsystems are closely related. This can be seen
by using the Schmidt decomposition (6.31) to evaluate the reduced density operators:
r
2
Ym  (ζm ζm )
ρA = TrB (ρ) = (6.41)
m=1
and
¾¼¿
Entanglement for distinguishable particles
r
2
Ym  (‘m ‘m ) .
ρB = TrA (ρ) = (6.42)
m=1
Comparing eqns (6.41) and (6.42) shows that the two reduced density operators”
although they act in di¬erent Hilbert spaces”have the same set of nonzero eigenvalues
2 2
Y1  , . . . , Yr  . This implies that the purities of the two reduced states agree,
r
4
P (ρA ) = Ym  = P (ρB ) < 1 ,
TrA ρ2 = (6.43)
A
m=1
and that the subsystems have identical von Neumann entropies,
r
2 2
S (ρA ) = ’ TrA [ρA ln ρA ] = ’ Ym  ln Ym  = S (ρB ) . (6.44)
m=1
An entangled pure state is said to be maximally entangled if the reduced density
operators are maximally mixed according to eqn (2.141), where the number of degen
erate nonzero eigenvalues is given by M = r. The corresponding values of the purity
and von Neumann entropy are respectively P (ρ) = 1/r and S (ρ) = ln r.
We next turn to results that are more directly related to experiment. For observ
ables A and B acting on HA and HB respectively and any state Ψ in HC = HA — HB ,
we de¬ne the averages A = Ψ A—IB  Ψ and B = Ψ IA —B Ψ and the ¬‚uctu
ation operators δA = A ’ A and δB = B ’ B . The quantum ¬‚uctuations are
said to be uncorrelated if Ψ δA δB Ψ = 0. With this preparation we can state the
following.
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
cnumber
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 CohenTannoudji 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 oneparticle 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 singleparticle state; however the symmetry of bosonic states allows
any number of bosons to occupy a singleparticle 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 singleparticle 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 singleparticle 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 oneparticle 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 oneparticle states, but the term · F  · is not small unless the operator F
represents a local observable for V . When this is the case, the twoparticle 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 singleparticle 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 oneparticle 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 singleparticle 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 singleparticle 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 singleparticle 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 distinguishableparticle de¬nition of entangle
ment to fermions leads to the conclusion that every twofermion 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 spinsinglet 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 zcomponent 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 twoparticle state that is entangled in the minimal form (6.78),
required by indistinguishability, is kinematically entangled, and that an entangled
twoparticle 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 positionspace 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 manyparticle system.
For photons, the de¬nitions of Fock space in Sections 2.1.2C 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 twoparticle system, the Fock space approach replaces explicitly
symmetrized vectors like
φm 1 φn 2 + φn 1 φm 2 (6.81)
by Fockspace 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 boxquantized creation operators to express a general twophoton state
as
1
Cks,k s a† a† s 0 ,
Ψ = √ (6.83)
ks k
2 ks,k s
where the normalization condition Ψ Ψ = 1 is
2
Cks,k s  = 1 , (6.84)
ks,k s
and the expansion (6.83) can be inverted to give
1ks , 1k s Ψ
√
Cks,k s = . (6.85)
2
By comparing eqns (6.83) and (6.75), we can see that a twophoton state is sepa
rable if the coe¬cients in eqn (6.83) factorize:
Cks,k s = γks γk s , (6.86)
where the γks s are cnumber coe¬cients. In this case, Ψ can be expressed as
¾½¾ Entangled states
1 2
Ψ = √ “† 0 , (6.87)
2
where
γks a† ,
“† = (6.88)
ks
ks
and the normalization condition (6.84) becomes
γks 2 = 1 . (6.89)
ks
The normalization of the γks s in turn implies “, “† = 1; therefore, “† can be inter
preted as a creation operator for a photon in the classical wave packet:
E (r) = γks Fk eks eik·r , (6.90)
ks
where
ωk
Fk = i . (6.91)
2 0V
Thus the bosonic character of photons implies that a separable state necessarily con
tains two photons in the same classical wave packet, in agreement with the de¬nition
(6.74) for massive bosons.
A twophoton state that is not separable is said to be entangled. This leads in
particular to the useful rule
1ks , 1k s is entangled if ks = k s . (6.92)
The factorization condition (6.86) provides a de¬nition of separable states and entan
gled states that works in the absence of positionspace wave functions for photons, but
the physical meaning of entanglement is not as intuitively clear as it is in ordinary
quantum mechanics. The best remedy is to ¬nd a substitute for the missing wave
function.
6.6.2 The detection amplitude
Let us pretend, for the moment, that the operator Es (r) = e— · E(’) (r) creates a
(’)
s
photon, with polarization es , at the point r. If this were true, then the state vector
(’)
r, s = Es (r) 0 would describe a situation in which one photon is located at r with
polarization es . For a onephoton state Ψ , this suggests de¬ning a singlephoton ˜wave
function™ by
Ψ (r, s) = r, s Ψ
(+)
= 0 Es (r) Ψ
= e— 0 Ej
(+)
(r) Ψ . (6.93)
sj
Now that our attention has been directed to the appropriate quantity, we can discard
this very dubious plausibility argument, and directly investigate the physical signi¬
cance of Ψ (r, s). One way to do this is to use eqn (4.74) to evaluate the ¬rstorder
¾½¿
Entanglement for photons
¬eld correlation function for the onephoton state Ψ . For equal time arguments, the
result is
(1) (’) (+)
Gij (r ; r) = Ψ Ei (r ) Ej (r) Ψ
(’) (+)
Ψ Ei (r ) n n Ej (r) Ψ
=
n
(’) (+)
= Ψ Ei (r ) 0 0 Ej (r) Ψ , (6.94)
where the last line follows from the observation that the vacuum state alone can
contribute to the sum over the number states n . By combining these two equations,
one ¬nds that
G(1) (r s ; rs) = es i e— Gij (r ; r)
(1)
sj
= Ψ (r, s) Ψ— (r , s ) . (6.95)
This result for G(1) (rs; r s ) is quite suggestive, since it has the form of the density
matrix for a pure state with wave function Ψ (r, s). On the other hand, the usual Born
interpretation does not apply to Ψ (r, s), since there is no photon position operator. An
important clue pointing to the correct physical interpretation of Ψ (r, s) is provided by
the theory of photon detection. In Section 9.1.2A it is shown that the counting rate for
a photon detector”located at r and equipped with a ¬lter transmitting polarization
2
es ”is proportional to G(1) (rs; rs). According to eqn (6.95), this means that Ψ (r, s)
is the probability that a photon is detected at r, the position of the detector. In view