Francoise J.-P., Naber G.L., Tsun T.S. (editors) Encyclopedia of Mathematical Physics
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W
1, p
0
(); z = 0} provided that f satisfies the com-
patibility condition
hf ;’i¼ g;
@’
@n
for all ’ 2 A
p
0
0
ðÞ
The kernel can be characterized in the following way:
it is reduced to {0} if p = 2orp < N and if not, then
A
p
0
ðÞ¼fCð 1Þ; C 2 Rg if p N 3
where is (unique) solution in W
1,2
0
() \ W
1,p
0
()of
the problem = 0in and = 1on@,and
A
p
0
ðÞ¼fCð u
0
Þ; C 2 Rg if p > N ¼ 2
where u
0
(x) = (2jj)
1
R
log jy xjd
y
and is the
only solution in W
1, 2
0
() \ W
1, p
0
() of the problem
= 0in and = u
0
on .
Remark 24 Similar results exist for the Neumann
problem in an exterior domain (see Amrouche et al.
(1997)). The framework of the spaces W
m, p
(R
N
þ
) for
the Dirichlet problem in R
N
þ
was also considered in
the literature. For a more general theory see Kozlov
and Maz’ya (1999).
Elliptic Systems
The Stokes System
The Stokes problem is a classical example in the
fluid mechanics. This system models the slow
motion with the field of the velocity u and the
pressure , satisfying
ðSÞ
u þr ¼ f in
div u ¼ h in
u ¼ g on ¼ @
where >0 denotes the viscosity, f is an exterior force,
g is the velocity of the fluid on the domain boundary,
and h measures the compressibility of the fluids (if
h = 0, it is an incompressible fluid). The functions h and
g must satisfy the compatibility condition
Z
hðxÞdx ¼
Z
g n d ½39
Theorem 25 Let be a Lipschitz bounded domain
in R
N
, N 2. Let f 2 H
1
()
N
, h 2 L
2
(), and g 2
H
1=2
()
N
satisfy [39]. Then the problem (S) has a
unique solution (u, ) 2 H
1
()
N
L
2
()=R satisfy-
ing the a priori estimate
kuk
H
1
ðÞ
þkk
L
2
ðÞ=R
C
kf k
H
1
ðÞ
þkhk
L
2
ðÞ
þkgk
H
1=2
ðÞ
In order to prove Theorem 25, one can start with a
homogeneous problem. The procedure of finding u is a
simple application of the Lax–Milgram theorem.
Application of de Rham’s theorem gives the pressure .
We introduce the space
V¼fv 2DðÞ
N
; div v ¼ 0g
and define F 2 H
1
()
N
by
hF; vi
H
1
H
1
¼ 0 for all v 2V
Moreover, there exists 2 L
2
(), unique up to an
additive constant, and such that F = r. The
problem (S), which we transform to the homoge-
neous case (h = 0, g = 0), can be formulated on an
abstract level. Let X and M be two real Hilbert
spaces and consider the following variational pro-
blem: Given L 2 X
0
and X 2 M
0
, find (u, ) 2 X M
such that
Aðu; vÞþB½v;¼LðvÞ; v 2 X
B½u; q¼XðqÞ; q 2 M
½40
where the bilinear forms A, B and the linear form L
are defined by
Aðu; vÞ¼
Z
ru rv
B½v; q¼
Z
½qrv
LðvÞ¼
Z
f v
Theorem 26 If the bilinear form A is coerc ive in
the space
V ¼fv 2 X; B½v; q¼0g for all q 2 M
that is, if there exists >0 such that
Aðv; vÞkvk
2
X
; v 2 V
then the problem [40] has a unique solution (u, )
if and only if the bilinear form B satisfies the
‘‘inf–sup’’ condition:
there exists >0 such that
inf
q2M
sup
v2X
Bðv; qÞ
kvk
X
kqk
M
As for the Dirichlet problem, the regularity result
is the following:
Theorem 27 Let be a bounded domain in R
N
, of
the class C
mþ1,1
if m 2 N and C
1,1
if m = 1. Let f 2
W
m,p
()
N
, h 2 W
mþ1,p
() and g 2 W
mþ21=p,p
()
N
satisfy condition [39]. Then the problem (S) has a
unique solution (u, ) 2 W
mþ2,p
()
N
W
mþ1,p
()=R.
226 Elliptic Differential Equations: Linear Theory
Remark 28 It is possible to solve (S) under weaker
assumption, for instance, if f 2 W
1=p
(
0
), h = 0 and
g 2 W
1=p,p
()
N
. We can prove that then (u, ) 2
L
p
()
N
W
1,p
().
The Linearized Elasticity
The equations governing the displacement u =
(u
1
, u
2
, u
3
) of a three-dimensional structure
subjected to an external force field f are written as
( is a bounded open subset of R
3
and =@)
u ð þ Þrðr uÞ¼f in
u ¼ 0on
0
X
3
j¼1
ij
ðuÞ
j
¼ g
i
on
1
¼
0
where >0 and >0 are two material character-
istic constants, called the Lame´ coefficients, and
(v = (v
1
, v
2
, v
3
))
ij
ðvÞ¼
ji
ðvÞ
¼
ij
X
3
k¼1
"
kk
ðvÞþ2"
ij
ðvÞ
with "
ij
ðvÞ¼"
ji
ðvÞ¼
1
2
ð@
j
v
i
þ @
i
v
j
Þ
½41
where
ij
denotes the Kronecker symbol, that is,
ij
= 1, for i = j and
ij
= 0, for i 6¼ j. These equations
describe the equilibrium of an elastic homogeneous
isotropic body that cannot move along
0
;along
1
,
surface forces of density g = (g
1
, g
2
, g
3
) are given. The
case
1
= ; physically corresponds to clamped struc-
tures. The matrix with entries "
ij
(u)isthelinearized
strain tensor while
ij
(u) represents the linearized
stress tensor; the relationship [41] between these
tensors is known as Hooke’s law. We refer for
example to Ciarlet and Lions (1991) and Nec
ˇ
as and
Hlava´c
ˇ
ek (1981) (and references therein) for most of
the results stated in this paragraph. The variational
formulation of this problem is
to find u 2 V such that
Aðu; vÞ¼LðvÞ for all v 2 V
½42
where the bilinear form A and the linear form L are
given by
Aðu; vÞ¼
Z
½ðr uÞðr vÞ
þ 2
X
3
i;j¼1
"
ij
ðuÞ"
ij
ðvÞðxÞdx; ½43a
LðvÞ¼
Z
f ðxÞvðxÞdx þ
Z
1
gðÞvðÞd ½43b
The functional space V is defined as
V ¼fv ¼ðv
1
; v
2
; v
3
Þ2½H
1
ðÞ
3
;
0
v
i
¼ 0on
0
; 1 i 3g
To prove the ellipticity of A, one needs the following
Korn inequality: There exists a positive constant C()
such that, for all v = (v
1
, v
2
, v
3
) 2 [H
1
()]
3
,wehave
kvk
1;
CðÞ
X
3
i;j¼1
k"
ij
ðvÞk
2
L
2
ðÞ
þ
X
3
i¼1
kv
i
k
2
L
2
ðÞ
"#
1=2
½44
The following result holds true:
Theorem 29 Let be a bounded open in R
3
with a
Lipschitz boundary, and let
0
be a measurable
subset of , whose measure (with respect to the
surface measure d(x)) is positive. Then the mapping
v 7!
X
3
i;j¼1
k"
ij
ðvÞk
2
L
2
ðÞ
"#
1=2
is a norm on V, equivalent to the usual norm k.k
1,
.
As a consequence, we get:
Theorem 30 Under the above assumptions, there
exists a unique u 2 V solving the variational
problem [42]–[43]. This solution is also the unique
one which minimizes the energy functional
EðvÞ¼
1
2
Z
ðr vÞ
2
þ 2
X
3
i;j¼1
"
ij
ðvÞ
2
ðxÞdx
Z
f ðxÞvðxÞdx þ
Z
1
gðÞvðÞd
over the space V.
Acknowledgmnt
M Krbec and S
ˇ
Nec
ˇ
asova´ were supported by the
Institutional Research Plan No. AV0Z10190503
and by the Grant Agency of the Academy of
Sciences No. IAA10019505.
See also: Evolution Equations: Linear and Nonlinear;
-Convergence and Homogenization; Image Processing:
Mathematics; Inequalities in Sobolev Spaces; Partial
Differential Equations: Some Examples; Schro
¨
dinger
Operators; Separation of Variables for Differential
Equations; Viscous Incompressible Fluids: Mathematical
Theory.
Further Reading
Adams RA and Fournier JJF (2003) Sobolev Spaces, 2nd edn.
Amsterdam: Academic Press.
Agmon S (1965) Lectures on Elliptic Boundary Value Problems.
Princeton: D. van Nostrand.
Elliptic Differential Equations: Linear Theory 227
Amrouche C, Girault V, and Giroire J (1997) Dirichlet and
Neumann exterior problems for the n-dimensional Laplace
operator. An approach in weighted Sobolev spaces. Journal de
Mathe´matiques Pures et Applique´es IX. Se´rie 76: 55–81.
Ciarlet PG and Lions JL (1991) Handbook of Numerical Analysis,
Vol. 2, Finite Element Methods. Amsterdam: North-Holland.
Dautray R and Lions JL (1988) Analyse mathe´matique et calcul
nume´rique pour les sciences et les techniques. Paris: Masson.
Friedman A (1969) Partial Differential Equations. New York:
Holt, Rinehart and Winston.
Gilbarg D and Trudinger N (1977) Elliptic Partial Differential
Equations of Second Order. Berlin: Springer.
Grisvard P (1980) Boundary Value Problems in Non-Smooth
Domains, Lecture Notes 19. University of Maryland, Depart-
ment of Mathematics.
Ho¨ rmander L (1964) Linear Partial Differential Operators.
Berlin: Springer.
Kozlov V and Maz’ya V (1999) Differential Equations with
Operators Coefficients with Applications to Boundary Value
Problems for Partial Differential Equations. Berlin: Springer.
Ladyzhenskaya O and Uraltseva N (1968) Linear and Quasilinear
Elliptic Equations. New York: Academic Press.
Lions J-L and Magenes E (1968) Proble`mes aux limites non
homoge`nes et applications, vol. 1. Paris: Dunod.
Nec
ˇ
as J (1967) Les me´thodes directes en e´quations elliptiques.
Prague: Academia.
Nec
ˇ
as J and Hlava´c
ˇ
ek I (1981) Mathematical Theory of Elastic and
Elastoplastic Bodies: An Introduction. Amsterdam: Elsevier.
Renardy M and Rogers RC (1992) An Introduction to Partial
Differential Equations. New York: Springer.
Stein EM (1970) Singular Integrals and Differentiability Proper-
ties of Functions. Princeton, NJ: Princeton University Press.
Triebel H (1978) Interpolation Theory, Function Spaces, Differ-
ential Operators, VEB Deutsch. Verl. Wissenschaften, Berlin.
Sec. rev. enl. ed. (1995): Barth, Leipzig.
Triebel H (2001) The Structure of Functions. Basel: Birkha¨user.
Weinberger HF (1965) A First Course in Partial Differential
Equations. New York: Wiley.
Ziemer WP (1989) Weakly Differentiable Functions. New York:
Springer.
Entanglement
R F Werner, Technische Universita
¨
t Braunschweig,
Braunschweig, Germany
ª 2006 Elsevier Ltd. All rights reserved.
Introduction
Entanglement is a type of correlation between
subsystems, which cannot be explained by the action
of a classical random generator. It is a key notion
of quantum information theory and corresponds
closely to the possibility of channels which transmit
quantum information, and cannot be simulated by
classical channels. In this article, we consider the
development of the concept, and its qualitative
aspects. The quantitative aspects are treated in a
separate article (see Entanglement Measures).
Historical Development
The first realization that quantum mechanics comes
with new, and perhaps rather strange, correlations
came in the famous 1935 paper by Einstein, Podolsky,
and Rosen (EPR) (Einstein et al. 1935), in which they
set up a paradox showing that the statistics of certain
quantum states could not be realized by assigning
wave functions to subsystems. It was in response to
this paper that Schro¨ dinger (1935), in the same year,
coined the term ‘‘entanglement,’’ as well as its German
equivalent ‘‘Verschra¨nkung.’’ The subject lay dormant
for a long time, since Bohr, in his reply, completely
ignored the entanglement theme, and there was a
widespread reluctance in the physics community to
consider problems of interpretation. The leaf turned
slowly with Bohm’s reduced model of the EPR
paradox using spins rather than continuous variables,
and decisively with Bell’s 1964 strengthening of the
paradox (Bell 1964). He showed that not only wave
functions assigned to individual systems failed to
describe the correlations predicted by quantum
mechanics, but any set of classical parameters assigned
to the subsystems. This eliminated all reference to a
possibly dubious quantum ontology and all reference
to the quantum formalism from the argument. Bell
derived a set of inequalities from the assumption that
each subsystem could be described in terms of classical
variables, and that these (possibly hidden) variables
would not be changed by the mere choice of a
measurement for the distant correlated system. The
only relation to quantum mechanics was then the
simple quantum calculation showing, in certain situa-
tions, such as the state described by EPR, quantum
mechanics predicted a violation of Bell’s inequalities.
This immediately suggested an experiment, and
although it was difficult at first to find an efficient
source of suitably quantum-correlated pairs of parti-
cles, the experiments that have been made since
then have supported the quantum-mechanical result
beyond reasonable doubt. This came too late for
Einstein, whose research program in quantum
mechanics had been precisely to build a ‘‘local
hidden-variable theory’’ of the type seen in contra-
diction with Bell’s inequality. But at least the EPR
paper had finally received the response it deserved.
In Schro¨ dinger’s work, entanglement was a purely
qualitative term for the strange way the subsystems
228 Entanglement
seemed to be intertwined as soon as one insisted on
discussing their individual properties. After Bell’s
work, the favored mathematical definition of entan-
glement would probably have been the existence of
measurements on the subsystems, such that Bell’s
inequality (or some generalization derived on the same
assumptions) is violated. However, around 1983
another notion of (the lack of) entanglement was
independently proposed by Primas (1983) and Werner
(1983). According to this definition, a quantum state
is called unentangled if it can be written as
¼
X
p
1
2
½1
where the
i
are arbitrary states of the subsystems
(i = 1, 2), which depend on a ‘‘hidden variable’’ ,
drawn by a classical random generator with prob-
abilities p
. Such states are now called separable,
which is a bit awkward, since the notion is typically
applied to systems which are widely separated.
However, the term is so firmly established that it is
hopeless to try to improve on it.
In any case, it was shown by Werner (1989) that
there are nonseparable states, which nevertheless
satisfy Bell’s inequalities and all its generalizations.
The next step was the observation by Popescu
(1994) that entanglement could be distilled: this is
a process by which some number of moderately
entangled pair states is converted to a smaller
number of highly entangled states, using only local
quantum operations, and classical communication
between the parties. For some time it seemed that
this might close the gap, that is, that the failure of
separability might be equivalent to ‘‘distillability’’
(i.e., the existence of a distillation procedure produ-
cing arbitrarily highly entangled states from many
copies of the given one). However, this turned out to
be false, as shown by the Horodecki family in 1998
(Horodecki et al. 1998), by explicitly exhibiting
bound entangled, that is, nonseparable, but also not
distillable states. In 2003 Oppenheim and the
Horodeckis introduced a further distinction, namely
whether it is possible to extract a secret key from
copies of a given quantum state by local quantum
operations and public classical communication
(Horodecki et al. 2005). This task had hitherto
been viewed as an application of entanglement
distillation, but it turned out that secret key can be
distilled from some bound entangled (but never from
separable) states.
For the entanglement theory of multipartite states,
that is, states on systems composed of three or more
parts, between which no quantum interaction takes
place, one key observation is that new entanglement
properties must be expected with any increase of the
number of parties. As shown by Bennett et al.
(1999), there are states of three parties which cannot
be written in the three-party analog of [1], but are
nevertheless separable for all three splits of the
system into one vs. two subsystems.
The crucial advance of entanglement theory,
however, lies not so much in the distinctions
outlined above, but in the quantitative turn of the
theory. With the discovery of the teleportation and
dense coding processes (Bennett and Wiesner 1992,
Bennett et al. 1993), entanglement changed its role
from a property of counterintuitive contortedness to
a resource, which is used up in teleportation and
similar processes. Distillation is then seen as a
method to upgrade a given source to a new source
of highly entangled states suitable for this purpose,
and it is not just the possibility of doing this, but the
rate of this conversion, which becomes the focus of
the investigation. All the tasks in which entangle-
ment appears suggest quantitative measures of
entanglement. In addition, there are many entangle-
ment measures, which appear natural from a
mathematical point of view, or are introduced
simply because they can be estimated relatively
easily and in turn give bounds on other entangle-
ment measures of interest. The current situation is
that there is no shortage of entanglement measures
in the literature, but it is not yet clear which ones
will be of interest in the long run. Some of these
measures are described in Entanglement Measures.
The current state of entanglement theory is marked
firstly by some long-standing open problems in the
basic bipartite theory on the one hand (additivity of
the entanglement of formation, the existence of NPT
bound entangled states, and more recently the
existence of entangled states with vanishing key
rate). Secondly, there is significant effort to try to
compute some of the entanglement measures, at least
for simple subclasses of states. This is so difficult,
because many definitions involve an optimization
over operations on an asymptotically large system.
Thirdly, there is a new trend in multipartite entangle-
ment theory, namely looking specifically at entangle-
ment in lattice structures such as spin systems of
harmonic-oscillator lattices. Here one can expect very
fruitful interaction with the statistical mechanics and
solid-state physics in the near future.
Qualitative Entanglement Theory
Setup
Throughout this section, we will consider density
operators on a Hilbert space split in some fixed way
into a tensor product of a Hilbert space H
A
for
Entanglement 229
Alice’s system and a Hilbert space H
B
for Bob’s
system, that is, H= H
A
H
B
. For simplicity, we will
mostly consider finite-dimensional spaces, and if a
dimension parameter d < 1 appears, it is under-
stood that d = dim H
A
= dim H
B
.ByB(H) we will
denote the set of bounded operators on a Hilbert
space, and by B
(H) the set of trace-class operators.
We distinguish these even in the finite-dimensional
case, because of their different norms. By S we will
denote the state space of the combined system, that
is, the set of positive elements of B
(H) with trace 1.
For such a density operator =
AB
we denote by
A
and
B
the restrictions to the subsystems, defined
by the partial trace over the other system, or by
tr(
A
F) = tr(
AB
(F 1)). We denote by the opera-
tion of matrix transposition, and by id the
partial transposition, applied only to the second
tensor factor. Since transposition is not completely
positive (see Channels in Quantum Information
Theory) partial transposition may take positive
operators to non-positive operators. The relative
entropy (see Entropy and Quantitative Transvers-
ality) of two density operators , will be used with
the convention S(jj) = tr ( log log ).
Witnesses and the Criterion of Positivity
of Partial Transpose
A state is called separable iff it is of the form [1],
and entangled otherwise. The set of separable states
C is a convex subset of the set S of all states. Its
extreme points are obvious from the representation
[1], namely the pure product states = j
A
B
ih
A
B
j. Since C , like S , is a convex set in
(d
4
1) dimensions, Caratheodory’s theorem asserts
that the sum can be taken to be a decomposition
into d
4
such terms. For a given , deciding whether
it is separable or entangled, hence, involves a
nonlinear search problem in roughly 4d
5
real
parameters, namely the vector components of the
A
,
B
appearing in the sum.
Dually, the convex set C can be described by a set
of linear inequalities. Here is a simple way of
generating such inequalities: let T : B
(H
B
) !B(H
A
)
be a positive linear map, that is, a map taking
positive matrices to positive matrices. Then for
A
,
B
0 the expression tr(
A
T(
B
)) is positive. It
is also bilinear, so we can find a Hermitian operator
T
\
2B(H
A
H
B
) such that
trð
A
Tð
B
ÞÞ ¼ trðð
A
B
ÞT
\
Þ
Since the left-hand side is positive, we see by taking
convex combinations that tr(T
\
) 0 for all separ-
able states . Hence, if we find a state with a negative
expectation of T
\
, we can be sure it is entangled.
Therefore, such operators T
\
are called entanglement
witnesses. This is often a useful criterion, especially
when one has some additional information about the
state, allowing for an intelligent choice of witness. It
is known from the theory of ordered vector spaces
and their tensor products that the set of witnesses
constructed above is complete. Hence, in principle,
checking all such witnesses provides a necessary and
sufficient criterion for entanglement. However, in
practice this remains a difficult task, because the
extreme points of the set of positive maps are only
known for some low dimensions.
By restricting T to completely positive maps, we
get a useful necessary criterion. It can be seen that it
is equivalent to
ðid ÞðÞ0
that is, to the positivity of the partial transpose
(PPT). States with this property are called ‘‘PPT
states’’ in current jargon.
Pure States, Purification
For pure states, that is, for the extreme points of S ,
separability is trivial to decide: since for pure states
the sum [1] can only be a single term, a pure state is
separable iff it factorizes.
A useful observation is that, for pure states
= jihj, all information about entanglement is
contained in the spectrum of the reduced states.
Consider a vector 2H
A
H
B
of the form
¼
X
ffiffiffiffiffi
r
p
A
B
½2
where
A
2H
A
and
B
2H
B
are orthonormal
systems, r
> 0, and
P
r
= 1. Then it is easy to
check that
A
=
P
r
j
A
ih
A
j is the spectral resolu-
tion of the restriction. Conversely, by diagonalizing
the restriction of a general unit vector , we find a
biorthogonal decomposition of the from [2], also
known as the Schmidt decomposition. The Schmidt
spectrum {r
1
, ..., r
d
} hence classifies vectors up to
local basis changes in H
A
and H
B
.
Since any
A
can appear in this construction, we
see that any mixed state can be considered as the
restriction of a pure state, which is essentially
unique, namely up to the choice of basis in the
purifying system B, and up to perhaps adding or
deleting some irrelevant dimensions in H
B
. The
resulting vector is known as the purification of
A
.
The extreme cases of [2] are pure product states
on the one hand, and vectors, for which
A
= 1=d is
the totally chaotic state. These are known as
maximally entangled and embody, in the most
extreme way, the observation that in quantum
230 Entanglement
mechanics, as opposed to classical probability, the
restriction of a pure state may be mixed.
Let us fix a maximally entangled vector , and
the matching Schmidt bases, so that
¼
1
ffiffiffi
d
p
X
k
jkki½3
where we have used the simplified ket notation, in
which only the basis label is written. Then, an
arbitrary vector can be written as =(X 1)
=(1 X
T
), where X
T
denotes the matrix trans-
pose of X.Clearly,thisvectorisagainmaximally
entangled iff X is unitary. Hence, the set of maximally
entangled vectors is a single orbit under unilateral
unitary transformations, and we even have the choice
to which side we apply the unitaries.
Teleportation
Suppose we have an orthonormal basis of maximally
entangled vectors
2H
A
H
B
. By the remarks
above, this is equivalent to choosing unitaries
U
, = 1, ..., d
2
such that
= (U
1) , and
tr(U
U
) = d
. For example, a finite Weyl system
constitutes such a system of unitaries, which shows
that we can find realizations in any dimension d.
Suppose that Alice and Bob each own part of a
system prepared in the state then they can
transmit perfectly the state of a d-dimensional
system, using only classical communication. Classi-
cal communication by itself would never suffice to
transmit quantum information, and the entangled
resource by itself does not allow the transmission
of any signal. But the combination of these resources
does the trick: Alice measures the observable
associated with the basis
on the combined system
formed by the unknown input and her part of the
entangled pair. The result is then transmitted to
Bob, who performs a U
-rotation on his part of the
entangled pair, producing the output state of the
teleportation. One can show by direct calculation
that this is exactly equal to the input state.
Note that the resource is destroyed in this
process, so that for every transmission we need a
fresh entangled pair. Less than maximally entangled
states instead of lead to less-than-perfect transmis-
sion, which can be extended to quantitative relations
between entanglement and channel capacity.
Special Systems
Qubits
For qubit pairs, there is a special basis of maximally
entangled vectors, which has some amazing
properties. It consists of the vectors
0
=, and
k
= i(
k
1), where
k
, k = 1, 2, 3, denotes the
Pauli matrices. Then a vector is maximally entangled
iff its components are real in this basis, up to a
common phase. A unitary matrix of determinant 1
factorizes into U
1
U
2
iff its matrix elements are
real, up to a common phase.
For qubit pairs, and also for dimensions 2 3, the
partial transposition criterion for entanglement is
necessary and sufficient, as shown by Woronowicz
and the Horodecki family.
Orthogonally Invariant States
Astate on C
d
C
d
is called orthogonally invariant
if, for any orthogonal matrix U (with respect to some
fixed product basis) [, U U] = 0. This leaves a
three-dimensional space of operators, spanned by the
identity, the permutation F =
P
i, j
jijihjij, and its
partial transpose
b
F =
P
i, j
jiiihjjj,whichisd times the
projection onto the maximally entangled vector .
Figure 1 shows the plane of Hermitian operators
with the described symmetry and tr = 1. Convenient
coordinates are tr F and tr
b
F. Note that these are
defined for any density operator, and are also invariant
under the ‘‘twirl’’ operation 7!
R
dU(U U)(U
U)
, using the Haar measure dU, which projects onto
the orthogonally invariant states. Hence, the diagram
provides a section as well as a projection of the state
space. The intersection of the positive operators with
those having positive partial transpose is the set of PPT
states, which in this case coincides with the separable
states. The thin lines correspond to states of higher
symmetry, namely on the one hand the ‘‘isotropic
states’’ commuting with U
U,with
U the complex
conjugate of U, and the ‘‘Werner states’’ commuting
with all unitaries U U . Their intersection point is the
normalized trace.
1
1
tr(ρF )
ˆ
tr(ρF
)
Figure 1 The plane of orthogonally invariant unit trace
Hermitian operators of a 3 3-system. The upright triangle
gives the positive operators, and the dashed one those with
positive partial transpose. The shaded area gives the PPT
states.
Entanglement 231
Gaussians
In general, the entanglement in systems with infinite-
dimensional Hilbert spaces is more difficult to
analyze. However, if the system is characterized by
variables satisfying canonical commutation rela-
tions, like positions and momenta, or the compo-
nents of the free quantum electromagnetic field,
there is a special class of states, which is again
characterized by low-dimensional matrices. This
allows the discussion of entanglement questions, in
a way largely parallel to the finite-dimensional
theory.
Let R
1
, ..., R
2f
denote the canonical operators,
where f is the number of degrees of freedom. The
commutation relations can be summarized as
i[R
, R
] =
1, where is the symplectic matrix.
Operators R
have a common set of analytic
vectors, and generate the unitary Weyl operators
W(a) = exp (ia
R
), which describe the phase space
displacements. Gaussian states are those making
a 7!tr W() a Gaussian function or, equivalently,
those with Gaussian Wigner function. Up to a gloal
displacement, they are completely characterized by
the covariance matrix
¼ tr ðR
R
þ R
R
Þ½4
The only constraint for a real symmetric matrix to
be a covariance matrix of a quantum state is that
þ i is a positive semidefinite matrix, which is a
version of the uncertainty relations.
Now for entanglement theory, we take some of
the degrees of freedom as Alice’s and some as Bob’s.
Separability can be characterized in terms of ,
namely by the condition that
0
, where
0
is the
covariance matrix of a Gaussian product state.
Similarly, partial transposition can be implemented
as an operation on covariance matrices, which
allows a simple verification of the PPT condition.
It turns out that as long as one partner has only a
single degree of freedom, the PPT condition is
necessary and sufficient for separability, but this
fails for larger systems.
The pure Gaussian states allow a normal form
with respect to local symplectic transformations
analogous to the Schmidt decomposition. For the
minimal case of one degree of freedom on either
side, one obtains a one-parameter family of
‘‘two mode squeezed states.’’ Its limit for infinite
squeezing parameter is the state used by EPR
(Einstein et al. 1935), which, however, makes
rigorous mathematical sense only as a singular
state, that is, a linear functional on B(H), which
can no longer be represented as the trace with a
density operator.
Multipartite Stars
A key feature of entanglement in a multipartite
system is usually referred to as ‘‘monogamy’’: when
Alice shares a highly entangled state with Bob, her
system cannot also be highly entangled with Bill.
More formally, suppose that a multipartite state for
systems A, B
1
, ...,B
n
is given, such that the restric-
tion to each pair AB
k
is the same bipartite state .
Then as n becomes larger, the existence of such a
star-shaped extension constrains to become less
and less entangled. In fact, as n !1, this condition
is equivalent to the separability of .
Open Problems
Recall from the introduction the following chain of
inclusions:
separable states states with vanishing key rate
PPT state
undistillable states
all states
The second and fourth inclusions are strict, but for
the first and third one might have equality, for all
we know. Especially for the third inclusion, this is a
long-standing problem.
Finally, we would like to point out that qualita-
tive and conceptual aspects of entanglement are
surveyed by Bub (2001), Popescu and Rohrlich
(1998), and Horodecki et al. (2001). For quantita-
tive aspects see Entanglement Measures.
See also: Capacities Enhanced by Entanglement;
Capacity for Quantum Information; Channels in Quantum
Information Theory; Entanglement Measures; Entropy
and Quantitative Transversality.
Further Reading
Bell JS (1964) Physics 1: 195.
Bennett CH and Wiesner S (1992) Communication via one- and
two-particle operators on Einstein–Podolsky–Rosen states.
Physical Review Letters 69: 28812884.
Bennett CH, Brassard G, Cre´peau C et al. (1993) Teleporting an
unknown quantum state via dual classical and Einstein–Podolsky–
Rosen channels. Physical Review Letters 70: 1895–1899.
Bennett CH, et al. (1999) Unextendible product bases and bound
entanglement. Physical Review Letters 82: 5385.
Bub J (2001) Quantum Entanglement and Information, The
Stanford Encyclopedia of Philosophy, URL = http://plato.stan-
ford.edu/entries/qt-entangle/.
Einstein A, Podolsky B, and Rosen N (1935) Can quantum-
mechanical description of physical reality be considered
complete? Physical Review 47: 777–780.
232 Entanglement
Horodecki K, Horodecki M, Horodecki P, and Oppenheim J
(2005) Secure key from bound entanglement. Physical Review
Letters 94: 160502.
Horodecki M, Horodecki P, and Horodecki R (1998) Mixed-state
entanglement and distillation: is there a ‘‘bound’’ entangle-
ment in nature? Physical Review Letters 80: 5239–5242.
Horodecki M, Horodecki P, and Horodecki R (2001) Mixed-state
entanglement and quantum communication. In: Alber G et al.
(eds.) Quantum Information, Springer Tracts in Modern
Physics, vol. 173.
Popescu S (1994) Teleportation versus Bell’s inequalities. What is
nonlocality? Physical Review Letters, 72: 797.
Popescu S and Rohrlich D (1998) The joy of entanglement. In:
Lo H-K, Popescu S, and Spiller T (eds.) Introduction to
Quantum Computation and Information.Singapore:World
Scientific.
Primas H (1983) Verschra¨nkte Systeme und Quantenmechanik. In:
Kanitscheider B (ed.) Moderne Naturphilosophie.Wu¨rzburg:
Ko¨nigshausenþNeumann.
Schro¨ dinger E (1935) Die gegenwa¨rtige Situation in der Quan-
tenmechanik. Naturwiss 23: 807–812, 823–828, 844–849.
See the server for open problems at URL = http://www.imaph.
tu-bs.de/qi/problems.
Werner R (1983) Bell’s inequalities and the reduction of statistical
theories. In: Balzer W, Pearce DA, and Schmidt H-J (eds.)
Reduction in Science, vol. 175, Synthese Library, (Reidel 1984).
Werner RF (1989) Quantum states with Einstein–Rosen–Podolsky
correlations admitting a hidden-variable model. Physical
Review A 40: 4277–4281.
Entanglement Measures
R F Werner, Institut fu
¨
r Mathematische Physik,
Braunschweig, Germany
ª 2006 Elsevier Ltd. All rights reserved.
Introduction
Entanglement, or quantum correlation, is one of
the central concepts in quantum information
theory. Its theory can be roughly separated into
three parts. The first is qualitative ,thatis,it
addresses the question ‘‘Is this state entangled or
not?’’ The second, comparative part asks ‘‘Is this
state more entangled than that state?,’’ and finally
the quantitative theory asks ‘‘How entangled is this
state?,’’ and gives its answers in the form of
entanglement measures assigning a number to
every state. Quantitative questions come up natu-
rally whenever entanglement is used as a resource
for tasks of quantum information processing. For
example, entangled states are in a way the fuel for
the processes of teleportation and dense coding: in
each transmission step a maximally entangled pair
system is required, and cannot be used for a further
transmission. The process also works with less than
maximally entangled states, but then it also
becomeslessefficient.Since entangled states
created in the laboratory typically have imperfec-
tions, it becomes important to understand the rates
at which imperfectly entangled states may be
distilled to maximally entangled ones, and this
rate is a direct measure of the usefulness of the
given state for many purposes. The quantitative,
task related turn is a new development in the study
of the foundations of quantum mechanics. It has
been imported from classical information theory,
where this way of thinking has been standard for a
long time. The combination makes the particular
flavor of quantum information theory.
In this article we consider the comparative and
quantitative aspects of entanglement. The historical
aspects and qualitative theory are treated in a separate
article (see Entanglement), to which we refer for basic
notions and notations. The example of teleportation
suggests close links between quantitative entanglement
theory and the theory of capacity Bennett et al. (1996),
which is the transfer rate of quantum information
through a given channel. These connections are
described in Quantum Channels: Classical Capacity.
We follow the notations of the basic article on
entanglement (see Entanglement). In particular,
denotes the transpose operation, and (id ) the
partial transpose. A state is called ‘‘PPT’’ if its
partial transpose is positive. The two physicists
operating the laboratories in which the two parts
of a bipartite system are kept are called Alice and
Bob, as usual. The restriction of a state to Alice’s
subsystem is denoted by
A
.
Comparative Entanglement
and Protocols
Protocols
In this section we introduce relations of the kind ‘‘state
1
is more entangled than
2
.’’ We take this to mean
that
2
can be obtained by applying to
1
some
operations which ‘‘cannot create entanglement.’’ The
definition of a class of operations of which this can be
claimed then defines the comparison. It turns out that
there are different choices for the class of such
operations, depending on the resources available for
the transformation steps. The class of operations is
usually referred to as a protocol.
Certainly local operations performed separately
by Alice and Bob cannot increase entanglement.
Alice and Bob might have to make some choices,
and even if they make these according to a
Entanglement Measures 233
prearranged scheme, by using a shared table of
random numbers, entanglement will not be gener-
ated. In this restrictive protocol, which we abbre-
viate by LO, for ‘‘local operations,’’ no
communication is allowed. It is clear that by just
discarding the initial state, and prepari ng a new one,
based on the random inst ruction allows Alice and
Bob to make any separable state, so these states
come out as the ‘‘least entangled’’ ones for this and
any richer protocol.
Next we might allow classical communication
from Alice to Bob. That is, Bob’s decision to
perform some operation in his laboratory is allowed
to depend on measuring results obtained by Alice in
an earlier stage. Of course, Alice is not allowed to
send quantum systems, since in this case she might
just send a particle entangled to one of her own, and
any state could be generated. This protocol is
referred to as ‘‘local operations and one-way
classical communication’’ (LOWC). Obviously, we
might also allow Bob to talk back, arriving at ‘‘local
operations and classical communication’’ (LOCC).
This is the protocol underlying most of the work in
entanglement theory.
The drawback of the LOCC protocol is that its
operations are extremely difficult to characterize: an
LOCC operation can take many rounds, and there is
no way to simplify a genera l operation to some kind
of standar d form. This is the main reason why other
protocols have been considered. For example, it is
obvious that an LOCC operation can be written as a
sum of tensor products of local operations, in a form
reminiscent of the definition of separability. How-
ever, such ‘‘separable superoperators’’ may fall
outside LOCC. Another property easily checked for
all LOCC operations is that PPT states go into PPT
states. The protocol ‘‘PPT-preserving operations’’
(PPTP) can also be characterized as the set of
channels T for which (id )T(id ) is positive
(although not necessarily completely positive). This
condition is relatively easy to handle mathemati-
cally, so that the best way to show that some
1
cannot be converted to
2
by LOCC is often to show
that this transition is impossible under PPTP. The
drawback of the PPTP protocol is that it may create
some entanglement after all, namely arbitrary PPT
states. So it properly belongs to a modi fied
entanglement theory in which separability is
replaced by the PPT condition.
Converting Pure States and Majorization
The entanglement ordering is exactly known for
pure states due to a famous theorem by Nielsen
(1999): a pure state
1
is more entangled than a pure
state
2
under the LOCC protocol iff the restriction
A
1
is more mixed than the restriction
A
2
in the sense
of majorization of spectra (i.e., for every k the sum
of the k largest eigenvalues of
A
1
is less than the
corresponding sum for
A
2
). Equivalently, there is a
doubly stochastic channel (completely positive linear
map preserving both the identity and the trace
functional) taking
A
2
to
A
1
.
An interesting aspect of this theory is the
phenomenon of catalysis: It may happen that
although
1
cannot be converted by LOCC to
2
,
1
can be converted to
2
. The ‘‘catalyst’’
is a resource borrowed at the beginning of the
transformation, and is returned unchanged after-
wards. The order relation allowing such catalysts is
yet to be fully characterized.
Asymptotic Conversion
In many applications we are not interested in exact
conversion of one state to another, but are quite
satisfied if the transformation can be done with a
small controlled error. In particular, when we ask
for the achievable conversion rate between many
copies of the states involved, we allow small errors,
but require the errors to go to zero. Given any
protocol, and states
1
,
2
, we say that
1
can be
converted to
2
with rate r if, for all sufficiently
large n, there is a channel of the protocol, which
takes n copies of
1
, that is, the state
n
1
,toa
state
0
which approximates roughly m rn copies
of
2
, in the sense that m rn, and the trace norm
k
0
m
2
k goes to zero.
Of course, one is usually interested in the
supremum of the achievable conversion rates, which
we call simply the maximal conversion rate. In
particular, when
2
is the maximally entangled pure
state of a qubit pair (usually called the ‘‘singlet’’), the
maximal rate is called the distillable entanglement
E
D
(
1
). In the other direction, when
1
is the singlet,
we call the inverse of the maximal conversion rate the
entanglement cost E
C
(
2
). These are two of the key
entanglement measures to be discussed below.
In general, E
D
() < E
C
(), so the asymptotic
conversion between different states is usually not
reversible. However, this is the case for pure states,
and one finds
E
D
ðÞ¼E
C
ðÞ¼Sð
A
Þ½1
where S () = tr log
2
() denotes the v on Neu-
mann entropy (see Entropy and Quantitative Trans-
versality) based on the binary logarit hm.
Since one can do the conversion between different
pure states via singlets, it is clear that the maximal
conversion rate from a pure state
1
to a pure state
2
234 Entanglement Measures
equals S(
A
1
)=S(
A
2
). Hence, in contrast to the ordering
given by Nielsen’s theorem, all pure states are
interconvertible, and the ordering is described by a
single number. For this simplification, the allowance
of small errors is crucial. Without asymptotically
small but nonzero errors, it would also be impossible
to obtain singlets from any generic mixed state.
Entanglement Measures
Properties of Interest
We now consider more systematically functions
E : S ! R defined on the states spaces of arbitrary
bipartite quantum systems. When can we regard this
as a measure of entanglement? The minimal require-
ments are that E() 0 for all ,andE() = 0for
separable states. Since the choice of local bases
should be irrelevant, we will require E((U
A
U
B
)(U
A
U
B
)
) = E() for unitaries U
A
, U
B
.We
also normalize all entanglement measures so that
E() = 1, when is the maximally entangled state of
a pair of qubits. Beyond that, consider the following:
1. V (Convexity E(
P
p
)
P
p
E(
)) Start-
ing from any E, possibly defined only on a subset
containing the pure stat es, we can enforce this
property by taking the convex hull (or ‘‘roof’’)
coE, defin ed as the largest convex function,
which is E wherever it is defined.
2. M (Monotonicity) Suppose that some LOCC
protocol applied to returns some classical
parameter with probability p
, and in that
case a bipartite state
.Then
P
p
E(
) E().
3. A
(Subadditivity E(
1
2
) E(
1
) þ E(
2
))
In this and the following, the tensor products of
bipartite states are to be reordered from
A
1
B
1
A
2
B
2
to (A
1
A
2
)(B
1
B
2
), so the separation
into Alice’s and Bob’s subsystems is respected.
4. A
þ
(Superadditivity E(
1
2
) E(
1
) þ E(
2
))
5. A
þþ
(Strong sup eradditivity E(
12
) E(
1
)þ
E(
2
)) Here
i
denotes the restriction of a general
state
12
to the ith subsystem.
6. A
1
(Weak additivity E(
n
) = nE()) This can
be enforced by regularization, going from E to
E
1
ðÞ¼lim
n
1
n
Eð
n
Þ
Note that this is implied by additivity, which is
the conjunction of A
þ
and A
.
7. C (Continuity) Here it is crucial to postulate the
right kind of dimensional dependence. A good
choice is to demand that jE(
1
) E(
2
)j
log df (k
1
2
k), where f is some funct ion with
lim
t !0
f (t) = 0.
8. L (Lockability) A property related to, but not
equal to, discontinuity: a measure is called
lockable, if the loss (i.e., the tracing out) of a
single qubit by Alice or Bob can make E() drop
by an arbitrarily large amount.
The Collection of Entanglement Measures
The following are the main entanglement measures
discussed in the liter ature. Note that all measures
defined by conversion rates in principle depend on
the protocol used. Unless otherwise stated, we will
only consider LOCC. For every function we list in
brackets the properties which are known.
1. E
F
(Entanglement of formation [V, M, A
, C, L])
This is defined as the convex hull of the
entanglement of pure states given by eqn [1].
For qubit pairs, there is a closed formula due
to Wootters (1998), orthogonally invariant states
(Vollbrecht and Werner 2001) (see 00510), and
permutation symmetric 2-mode Gaussians. One
of the big open questions is whether E
F
is
additive. This is equivalent to E
F
satisfying A
þþ
,
and also to the additivity of Holevo’s -capacity
of quantum channels (see Quantum Channels:
Classical Capacity).
2. E
C
(Entanglement cost [V, M, A
, A
1
, C, L]) This
was already defined in the section ‘ ‘Asymptotic
conversion.’ ’ It has been shown to be equal to the
regularization of E
F
,thatis,E
C
= E
1
F
.IfE
F
would
turn out to be additive, we would thus have E
C
= E
F
.
3. E
D
(Distillable entanglement [M, A
þþ
, A
1
])
Again, see the section ‘‘Asymptotic convers ion.’’
This is one of the important measures from the
practical point of view, but notoriously difficult
to compute explicitly. Convexity of E
D
is an open
problem related to the existence of bound
entangled, but not PPT states.
4. E
!
(One-way distillable entanglement [M, A
þþ
,
A
1
]) Same as E
D
, but restricting to the LOWC
protocol. Obviously, E
!
() E
D
(). There are
examples of proper inequality ‘‘<’’ (Bennett et al.
1996). E
!
is more directly linked to quantum
capacity than E
D
, which in turn corresponds to the
quantum capacity, allowing classical backwards
communication as a resource.
5. E
N
(Logarithmic negativity [M, A
, A
þ
, L]) This
is a quantitative companion of the PPT criterion: one
sets E
N
() = log
2
k(id )()k, where the norm is
the trace norm. For PPT states, this is equal to the
trace, and E
N
() = 0. If the partial transpose has
negative eigenvalues, the sum of their absolute
values is >1, and E
N
() > 0. E
N
is an easily
computed upper bound to E
D
, but gives the wrong
value for nonmaximally entangled pure states.
Entanglement Measures 235