PHYSICAL REVIEW A, 66, 032319 共2002兲
Geometric picture of entanglement and Bell inequalities
R. A. Bertlmann, H. Narnhofer, and W. Thirring
Institut für Theoretische Physik, Universität Wien, Boltzmanngasse 5, A-1090 Wien, Austria
共Received 22 November 2001; published 27 September 2002兲
We work in the real Hilbert space Hs of Hermitian Hilbert-Schmidt operators and show that the entanglement witness which shows the maximal violation of a generalized Bell inequality 共GBI兲 is a tangent functional
to the convex set S傺Hs of separable states. This violation equals the Euclidean distance in Hs of the entangled
state to S and thus entanglement, GBI, and tangent functional are only different aspects of the same geometric
picture. This is explicitly illustrated in the example of two spins, where also a comparison with familiar Bell
inequalities is presented.
DOI: 10.1103/PhysRevA.66.032319
PACS number共s兲: 03.67.Hk, 03.65.Ta, 03.65.Ca
I. INTRODUCTION
The importance of entanglement 关1,2兴 of quantum states
became quite evident in the last ten years. It is the basis for
such physics, like quantum cryptography 关3– 6兴 and quantum
teleportation 关7,8兴, and it triggered a new technology: quantum information 关9,10兴. Entangled states lead to a violation
of Bell inequalities 共BI兲 which distinguish quantum mechanics from 共all兲 local realistic theories 关11兴. Much effort has
been made in studying the mathematical structure of entanglement, especially the quantification of entanglement
共see, for instance, Refs. 关12,13兴兲. There exist different kinds
of measures of entanglement indicating somehow the difference between entangled and separable states, which is usually related to the entropy of the states 共see, e.g., Refs. 关14 –
19兴兲. In this paper we define a simple and quite natural
measure for entanglement, a distance of certain vectors in
Hilbert space which has as elements both observables and
states, and we relate it to the maximum violation of a generalized Bell inequality 共GBI兲. We work with a bipartite system in a finite-dimensional Hilbert space but generalizations
are possible.
The Hilbert-Schmidt distance D of a state to the set of
separable states has previously been proposed as a measure
of entanglement 关20,21兴. Our point is that if one admits all of
B(HA 丢 HB ) as entanglement witnesses then the maximal
violation B of the corresponding GBI equals the distance D
numerically. Since D can be written as a minimum and B as
a maximum, upper and lower bounds are readily available. In
fact, in some standard examples one can make them coincide
and thus calculate B⫽D exactly.
Though distinct from the entropic entanglement descriptions, the Hilbert-Schmidt distance D as a quantitative description of entanglement is insofar reasonable; as considered as functional of the state it is convex and invariant
under local unitary transformations. This implies that states
more mixed in the sense of Uhlmann 关22兴 have a lower entanglement. However, D is not monotonic decreasing under
arbitrary completely positive maps in HA or HB but only if
they have norm one. Thus whether they satisfy monotonicity
in ‘‘local operations and classical communication’’ depends
on the exact definition of this term.
We consider a finite-dimensional Hilbert space H⫽CN ,
where observables A are represented by all Hermitian matri1050-2947/2002/66共3兲/032319共9兲/$20.00
ces and states w by density matrices. It is useful to regard
these quantities as elements of a real Hilbert space Hs
2
⫽RN with scalar product
共 w 兩 A 兲 ⫽Tr wA
共1.1兲
and corresponding norm
储 A 储 2 ⫽ 共 Tr A 2 兲 1/2
共1.2兲
共we identify quantities with their representatives in H). Both
density matrices and observables are represented by vectors
in Hs , a density matrix is positive and has trace unity.
Unitary operators U in H induce via UAU * ⫽OA orthogonal operators O in Hs , but the homomorphism U→O
is neither injective nor surjective.
II. SPIN EXAMPLES
Let us begin with two examples which will be of our
interest.
Example I: one spin. Generally an observable can be written as
ជ
A⫽ ␣ 1⫹aជ • ␴
with
␣ 苸R, aជ 苸R3 .
共2.1兲
The operator A is a density matrix iff ␣ ⫽1/2 and 储 aជ 储
⭐1/2, it gives a pure state iff 储 aជ 储 ⫽1/2 or A 2 ⫽A. If the state
is
w⫽
1
ជ • ␴ជ 兲
共 1⫹w
2
共2.2兲
the expectation value of A is
ជ.
共 w 兩 A 兲 ⫽ ␣ ⫹aជ •w
共2.3兲
For us the important structural element is a tensor product
H⫽HA 丢 HB which defines the set S of separable 共classically
correlated兲 states ␳ Ai , ␳ Bj ,
66 032319-1
再
S⫽ ␳ ⫽
c i j ␳ Ai 丢 ␳ Bj
兺
i, j
冏
0⭐c i j ⭐1,
c i j ⫽1
兺
i, j
冎
.
共2.4兲
©2002 The American Physical Society
PHYSICAL REVIEW A 66, 032319 共2002兲
R. A. BERTLMANN, H. NARNHOFER, AND W. THIRRING
ជ A and ␴ជ B , ‘‘Alice and Bob.’’ An
Example II: two spins ␴
observable A can be represented by
A⫽ ␣ 1⫹a i ␴ Ai 丢 1B ⫹b i 1A 丢 ␴ Bi ⫹c i j ␴ Ai 丢 ␴ Bj ,
1
储 A 储 22 ⫽ ␣ 2 ⫹
4
c 2i j .
兺i 共 a 2i ⫹b 2i 兲 ⫹ 兺
i, j
共2.5兲
共2.6兲
Note that c i j can be diagonalized by two independent orthogonal transformations on ␴ Ai and ␴ Bj 关23兴. The operator A
is a density matrix if ␣ ⫽1/4 and the operator norm 储 储 ⬁ of
A⫺1/4 is ⭐1/4. Since 储 储 2 ⭓ 储 储 ⬁ this is satisfied if
c 2i j ⭐1/16.
兺i 共 a 2i ⫹b 2i 兲 ⫹ 兺
i, j
共2.7兲
For pure states 储 储 2 ⫽ 储 储 ⬁ and 储 ␳ 储 2 ⫽1 is necessary and sufficient for purity. A pure separable state has the form
␳⫽
1
共 1⫹n i ␴ Ai 丢 1B ⫹m i 1A 丢 ␴ Bi ⫹n i m j ␴ Ai 丢 ␴ Bj 兲 ,
4
共2.9兲
States that are not separable are called entangled w苸S c ,
the complement in the set of states. We introduce as a measure of entanglement D(w) the Hs distance of w to the set S
of separable states,
␳ 苸S
B共 w 兲⫽
共2.8兲
III. GENERALIZED BELL INEQUALITY
D 共 w 兲 ⫽ min 储 ␳ ⫺w 储 2 .
共 ␳兩A 兲⬎共 w兩A 兲
␳ 苸S
A max ⫽
᭙ w.
共3.7兲
␳ 0 ⫺w⫺ 共 ␳ 0 兩 ␳ 0 ⫺w 兲 1
苸At .
储 ␳ 0 ⫺w 储 2
冉 冏
min ␳ ⫺w
we generally have
共3.3兲
␳ ⬘ ⫺w
储 ␳ ⬘ ⫺w 储 2
冊
共3.9兲
共For an illustration, see Fig. 1兲
Remark. The proof of the Theorem does not use the product structure of the Hilbert space H but only the geometric
properties of the Euclidean distance in Hs . It can be illustrated already with one spin where the set of separable states
S is replaced by S z ,
再
冎
共3.10兲
储 w 储 2 ⭐1
共3.11兲
1
S z ⫽ ␳ ⫽ 共 1⫹␭ ␴ z 兲 , 兩 ␭ 兩 ⭐1 ,
2
Thus ᭚ w such that
w苸S c .
共3.8兲
⭐B 共 w 兲 ⭐ 储 ␳ ⬘ ⫺w 储 2
᭙ ␳ ⬘ 苸S.
共3.2兲
Usually the Bell inequality refers to an operator in the tensor
product where by classical arguments only some range of
expectation values can be expected whereas quantum mechanics permits other values. A Bell inequality in a generalized sense is given by an observable A⭓
” 0 for which
for some
共3.6兲
共ii兲 The min of D is attained for some ␳ 0 and the max of
B for
␳ 苸S
᭙ ␳ 苸S.
关 min 共 ␳ 兩 A 兲 ⫺ 共 w 兩 A 兲兴 ,
B 共 w 兲 ⫽D 共 w 兲
⭐Tr共 ␳ 2 ⫹ ␻ 2 兲 ⭐2
共 w 兩 A 兲 ⬍0
max
储 A⫺ ␣ 储 2 ⭐1
共iii兲 For B⫽D the following two-sided variational principle holds:
储 ␳ ⫺w 储 22 ⫽Tr共 ␳ 2 ⫹ ␻ 2 ⫺2 冑␳␻ 冑␳ 兲
共 ␳ 兩 A 兲 ⭓0
共3.5兲
we find the following result.
Theorem.
共i兲 The maximal violation of the GBI is equal to the distance of w to the set S,
共3.1兲
Since
0⭐D 共 w 兲 ⭐ 冑2.
᭙ ␳ 苸S.
Considering now the maximal violation of the GBI,
ជ 2 ⫽1, and gives the expectation value of A,
with nជ 2 ⫽m
ជ •bជ ⫹n i m j c i j .
共 ␳ 兩 A 兲 ⫽ ␣ ⫹nជ •aជ ⫹m
Horne, Shimony, and Holt兲 inequality 关26兴. But the number
of summands is not restricted in AW . The operator A苸At
becomes a tangent functional if in addition ᭚ ␳ 0 苸S such
that ( ␳ 0 兩 A)⫽0. Since S is a convex subset of the state space
such tangential A’s always exist. Even more, the set S is
characterized by the tangent functionals and the ␳ 0 ’s with
( ␳ 0 兩 A)⫽0, for some A苸At , are the boundary ⳵ S of S.
Frequently a bigger set than S is considered as classically
explainable in a local hidden variable theory. Bell inequalities are those which contradict even those sets. To avoid
misunderstandings we call generalized Bell inequalities expectation values which contradict the predictions from S, the
set of separable states.
Thus the GBI 共3.3兲 is violated by an entangled state w,
Eq. 共3.4兲, and we get the following inequality for some A
苸AW :
共3.4兲
and
Such elements A苸AW are called entanglement witnesses
关24,25兴. A product operator can never be 苸AW but already
the sum of two products serves for the CHSH 共Clauser,
032319-2
w⫽
1
ជ • ␴ជ 兲 ,
共 1⫹w
2
GEOMETRIC PICTURE OF ENTANGLEMENT AND BELL . . .
PHYSICAL REVIEW A 66, 032319 共2002兲
FIG. 1. Illustration of Theorem 共3.7兲. The maximal violation of
GBI B(w), Eq. 共3.6兲, which is equal to the Hs distance D(w), Eq.
共3.1兲, of an entangled state w to the set S of separable states, is
shown together with the tangent plane defined by A max 共3.8兲.
FIG. 2. For illustration we have drawn the vectors used in the
Proof of Theorem 共3.7兲.
Here the observable
is considered as the analog of an entangled state, if w x or
w y ⫽0.
The observables A with 储 A 储 2 ⫽1 are of the form
A⫽
␣ 1⫹aជ • ␴ជ
and
冑2 共 ␣ 2 ⫹a 2 兲 1/2
a⫽ 储 aជ 储 , aជ 苸R3 .
共3.12兲
For the Hs distance D, our measure of entanglement, we
calculate
1
ជ • ␴ជ 储 22
min储 ␳ ⫺w 储 22 ⫽min 储 ␭ ␴ z ⫺w
4
␳
␭
A max ⫽⫺
M i j⫽
1
⫽ 共 w 2x ⫹w 2y 兲
2
attained for ␭⫽w z , so that we have
1
冑2
共 w 2x ⫹w 2y 兲 1/2.
共3.13兲
Otherwise, we find for the maximal violation of the GBI,
冉
1
ជ • ␴ជ
B 共 w 兲 ⫽max min ␭ ␴ z ⫺w
2
ជ
␭
a, ␣
⫽max
aជ , ␣
⫽
1
冑2
冏
␣ 1⫹aជ • ␴ជ
冑2 共 ␣
2
⫹a 兲
2 1/2
冊
共 w 2x ⫹w 2y 兲 1/2
共3.15兲
冉 冊
0
1
1
0
.
Proof of the Theorem: Eq. (3.7). D(w)⫽min␳苸S储␳⫺w储2 is
attained for some ␳ 0 since 储 储 2 is continuous and S is compact. Now take for A⫺ ␣ ⫽( ␳ 0 ⫺w)/ 储 ␳ 0 ⫺w 储 2 in the definition of B and use the orthogonal decomposition with respect
to this unit vector, Hs 苹 v ⫽ v 储 ⫹ v⬜ , ( v⬜ 兩 ␳ 0 ⫺w)⫽0. Therefore we can apply simple Euclidean geometry and decompose the vector ␳ ⫺w in the above sense.
We also remember that ␳ 0 ⫺w is the normal to the tangent
plane to S, which means
储共 ␳ ⫺w 兲 储 储 2 ⭓ 储共 ␳ 0 ⫺w 兲 储 储 2 ⫽ 储 ␳ 0 ⫺w 储 2
ជ •aជ
⫺1 兩 a z 兩 ⫹w
冑2 共 ␣ 2 ⫹a 2 兲 1/2
共 w 2x ⫹w 2y 兲 1/2.
冑2
is the tangent functional ᭙ ␳ 苸S z , ⳵ S z ⫽S z .
Note that for the maximal violation of the GBI 共3.14兲 the
min␳苸S is attained for 21 (1⫺ ␴ z ) if a z ⬎0 and not for 21 (1
⫹w z ␴ z ) as in case of the distance 共3.13兲. It means that for D
the min␳ is not necessarily attained for a pure state but for B
it is since it is effectively a max. Thus the equality B⫽D,
Theorem part 共3.7兲, is not so trivial since the extrema may be
attained at disjointed sets. Then min max may be bigger than
max min as can be seen already in mini and max j for the
matrix
1
⫽min 关共 ␭⫺w z 兲 2 ⫹w 2x ⫹w 2y 兴
2
␭
D共 w 兲⫽
w x ␴ x ⫹w y ␴ y
共3.14兲
since S is convex, see Fig. 2. We can prove this in the following way. The tangent A max divides the state space into
Hw ⫽ 兵 ␳ : 储 ( ␳ ⫺w) 储 储 2 ⬍ 储 ( ␳ 0 ⫺w) 储 储 2 其 , which contains w, and
H wc , the complement to Hw . If Hw were to contain ␳ 苸S
then because of the convexity of S it would contain all ␳ ␭
⫽(1⫺␭) ␳ 0 ⫹␭ ␳ , ␭苸 关 0,1兴 . Since ␳ ␭ would have an angle
of less than 90° with ␳ 0 ⫺w there would be a ␳ ␭ inside the
032319-3
PHYSICAL REVIEW A 66, 032319 共2002兲
R. A. BERTLMANN, H. NARNHOFER, AND W. THIRRING
ball 储 ( ␳ ⫺w) 储 2 ⬍ 储 ( ␳ 0 ⫺w) 储 2 ⫽D(w) and ␳ 0 would not be
the point of S of minimal distance to w. Therefore S傺H wc
and 储 ( ␳ ⫺w) 储 储 2 ⭓ 储 ( ␳ 0 ⫺w) 储 2 ᭙ ␳ 苸S.
Using above arguments we obtain
冉 冏 冊
冉 冏 冊
冉 冏 冊
B 共 w 兲 ⭓min ␳ ⫺w
␳
␳
⭓ ␳ 0 ⫺w
共i兲 B(w) and D(w) are continuous,
兩 共 w⫹ ␦ ⫺ ␳ 兩 A 兲 ⫺ 共 w⫺ ␳ 兩 A 兲 兩 ⭐␧
᭙ 储 ␦ 储 2 ⭐␧, 储 A 储 2 ⭐1
⇒ 兩 共 B or D 兲共 w⫹ ␦ 兲 ⫺ 共 B or D 兲共 w 兲 兩 ⭐␧
␳ 0 ⫺w
储 ␳ 0 ⫺w 储 2
⭓min 共 ␳ ⫺w 兲 储
Proofs of the B, D properties
᭙ 储 ␦ 储 2 ⭐␧.
共ii兲 B(w) is convex,
␳ 0 ⫺w
储 ␳ 0 ⫺w 储 2
B
冉兺 冊
i
␭ i w i ⫽max
␳ 0 ⫺w
储 ␳ 0 ⫺w 储 2
⫽ 储 ␳ 0 ⫺w 储 2 ⫽D 共 w 兲 .
However, D and B can be written as min␳maxA and
maxAmin␳ of ( ␳ ⫺w 兩 A) and generally we have minmax
⭓max min. So a priori we know D(w)⭓B(w) and we conclude D(w)⫽B(w).
IV. PROPERTIES OF THE GENERALIZED
BELL INEQUALITY
Now we discuss the properties of D(w), Eq. 共3.1兲, the Hs
distance of w to the set S of separable states, which is equal
to B(w), Eq. 共3.6兲, the maximal violation of the GBI.
Properties of D„w…
D(w) has following properties.
共i兲 D(w) is convex.
共ii兲 D(w) is continuous.
共iii兲 D(w)⫽D(U A 丢 U B w U A* 丢 U B* ) ᭙ unitary operators
U A,B .
共iv兲D(w) is monotonic decreasing under mixing enhancing maps; see, e.g., Ref. 关27兴.
A
共 ␳ 兩 A 兲 ⫺ 共 w i 兩 A 兲兴
兺i ␭ i 关 min
␳ 苸S
⭐
兺i ␭ i 兵 max
A
⫽
兺i ␭ i B 共 w i 兲 .
关 min共 ␳ 兩 A 兲 ⫺ 共 w i 兩 A 兲兴 其
␳ 苸S
共iii兲 D(w)⫽D(U A 丢 U B w U A* 丢 U B* ) follows from the invariance of S under U A 丢 U B .
共iv兲 The monotonic decrease under mixing enhancing
maps is a consequence of points 共i兲 and 共iii兲.
The ‘‘most’’ separable state w tr⫽1/dim H is a convex
combination of most entangled states. From the properties of
D(w) we get the following artistic impression. In the state
space there is a plane around w tr with D(w)⫽0. From it
emerge valleys with D(w)⫽0 to the pure separable states on
the boundary. In their neighborhood are entangled states,
thus D slopes up in such a way that the regions D⭐c, with
0⭐c⭐D max , are convex. On the boundary of the state space
also sit the states with D⫽D max , forming a rim. Since U A
丢 U B act continuously in a neighborhood of maximally entangled states there are others with D⫽D max but also some
with D⬍D max which one gets by mixing in a little bit with
the separable states.
This somewhat poetic description is mathematically
supplemented by considering S as a subset of the state space
S艛S c 傺Hs , so the boundary ⳵ S are those elements of S
where in each neighborhood there are entangled states.
V. GEOMETRY OF SEPARABLE STATES
What is the geometric structure of the set S of separable
states? Let us investigate its properties.
Properties of S
Corresponding remarks
共i兲 It means that by mixing the entanglement decreases
and the maximally entangled states must be pure. This is to
be expected since the tracial state w tr⫽1/(dim HA dim HB ) is
separable ⇔D(w)⫽0. Furthermore, the set 兵 w 兩 D(w)⬍c 其
is convex.
共ii兲 It tells us that the neighborhood of an entangled state
is also entangled, provided that it is small enough. Actually a
neighborhood of the tracial state is also separable.
共iii兲 The state space decomposes into equivalence classes
of states with the same entanglement. All pure separable
states are in the same equivalence class.
共iv兲 Mixing enhancing maps are essentially a combination
of unitary transformations and convex combinations.
共i兲 The dimensions of both S and S c are N 2 ⫺1.
共ii兲 Pure separable states belong to the boundary ⳵ S and
convex combinations of two of them are still on ⳵ S.
n
␮ i ␳ i is on ⳵ S then there is a
共iii兲 If a mixture ␳ ⫽ 兺 i⫽1
face, i.e.,
n
¯␳ ⫽
兺
i⫽1
n
¯ i␳ i苸 ⳵ S
␮
¯ i ⭓0,
᭙ ␮
兺 ␮¯ i ⫽1.
i⫽1
共iv兲 If HA ⫽HB (⫽C冑N ) then ⳵ S contains at least N dimensional faces.
共v兲 S is invariant under T A 丢 1B , with T A any positive map
B(HA )→B(HA ).
共vi兲 If A⭓
” 0 but (T A 丢 1B )A⭓0 then A苸AW and if
᭚ ␳ 0 苸S such that ( ␳ 0 兩 A)⫽0 then A苸At .
032319-4
PHYSICAL REVIEW A 66, 032319 共2002兲
GEOMETRIC PICTURE OF ENTANGLEMENT AND BELL . . .
Corresponding remarks
共i兲 It means that both S and S c are everywhere thick and
do not have pieces of lower dimensions.
共ii兲 Clearly the convex combination of two pure states lies
共for N⬎2) on the boundary of the state space since in each
neighborhood there are not positive functionals. Here we
have the stronger statement that in each neighborhood there
are entangled states.
共iii兲 If ⳵ S has an n dimensional flat part this means that
mixtures of n pure states are on ⳵ S. Point 共iii兲 affirms the
converse in the sense that in the decomposition the ␳ i ’s span
a face.
共iv兲 It says that n⫽N actually occurs.
共v兲 Strangely, the tensor product of two positive maps is
not necessarily positive, but applied to separable states it is.
continuous path from the entangled w̄ to the separable w tr
formed from states with corresponding invertible density matrices. When this path passes the boundary ⳵ S then according
to property 共iii兲 we obtain a separable state embedded in a
N-dimensional face of ⳵ S.
共v兲 Follows from the results in Ref. 关24兴.
共vi兲 Follows from 共v兲 and the definitions of AW and At .
VI. GEOMETRY OF ENTANGLED AND SEPARABLE
STATES OF SPIN SYSTEMS
We focus again on the two spin example and calculate the
entanglement of the following quantum states.
Example: Alice and Bob, the ‘‘Werner states.’’ Let us consider Werner states 关30兴 which can be parametrized by
Proofs of the properties of S
共i兲 S has the full dimension of N since a neighborhood of
the tracial state w tr⫽1/N is separable and as a convex set it
has the same dimension everywhere. The complement S c ,
the set of entangled states, has the full dimension since D is
continuous and if D(w)⬎0 it is so for a neighborhood of w.
共ii兲 ␳ is pure and separable, i.e., ␳ 苸 ⳵ S. If
w ␣⫽
再
1 共 1A ⫺ ␴ Ax 兲 丢 共 1B ⫹ ␴ Bx 兲 共 1A ⫹ ␴ Ax 兲 丢 共 1B ⫺ ␴ Bx 兲
⫹
2
2
2
⫽
⫹ 共 1⫺␭ 兲 兩 ␾ 2 丢 ␺ 2 ⫹␧ ⬘ ␾ 1 丢 ␺ 1 典具 ␾ 2 丢 ␺ 2 ⫹␧ ⬘ ␾ 1 丢 ␺ 1 兩 .
⇒
冉兺 冏 冊
⇒ 共 ␳ i 兩 A 兲 ⫽0 ᭙ i
兺
1⫺ ␴ Ax 丢 ␴ Bx
4
␳ 0⫽
For ␧, ␧ ⬘ →0 it comes arbitrarily close to ␳ ␭ but in the twodimensional Hilbert subspace spanned by ␾ i 丢 ␺ i (i⫽1,2) the
only separable pure states are of the form ␳ 1,2 . Thus a state
that is not a linear combination of ␳ 1 and ␳ 2 needs for its
decomposition into pure states at least one pure entangled
state, and is therefore entangled itself. Therefore we have an
entangled state arbitrarily close to ␳ ␭ ⇒ ␳ ␭ 苸 ⳵ S 共compare
with Refs. 关28,29兴兲.
共iii兲 For a tangent functional A at ␳ ⫽ 兺 ␮ i ␳ i , ␳ i 苸S, we
have
¯ i ␳ i A ⫽0 ⇒
␮
冎
and then with x→y, x→z, finally
␭ 兩 ␾ 1 丢 ␺ 1 ⫹␧ ␾ 2 丢 ␺ 2 典具 ␾ 1 丢 ␺ 1 ⫹␧ ␾ 2 丢 ␺ 2 兩
兺 ␮ i共 ␳ i兩 A 兲
共6.1兲
and they are possible density matrices for ⫺1/3⭐ ␣ ⭐1 since
␴ជ A 丢 ␴ជ B has the eigenvalues ⫺3,1,1,1. To calculate the entanglement we first mix product states to get
␳ ⫽ 兩 ␾ 丢 ␺ 典具 ␾ 丢 ␺ 兩
共pure and separable兲 then
兩 ␾ 丢 ␺ ⫹␧ ␾ ⬘ 丢 ␺ ⬘ 典具 ␾ 丢 ␺ ⫹␧ ␾ ⬘ 丢 ␺ ⬘ 兩
comes for ␧→0 arbitrarily close and is ᭙ ␧ pure and not a
product state, it is entangled, i.e., ␳ 苸 ⳵ S. ␳ i is pure and
separable, i.e., ␳ ␭ ⫽␭ ␳ 1 ⫹(1⫺␭) ␳ 2 苸 ⳵ S. Let us take ␳ i
⫽ 兩 ␾ i 丢 ␺ i 典具 ␾ i 丢 ␺ i 兩 and consider
0⫽ 共 ␳ 兩 A 兲 ⫽
ជ A 丢 ␴ជ B
1⫺ ␣ ␴
,
4
冊
共6.2兲
This seems a good ␳ 0 for w ␣ if 1/3⬍ ␣ ⭐1; and we use it for
␳ ⬘ in the Theorem part 共iii兲, Eq. 共3.9兲. With ␳ 0 ⫺w ␣ ⫽ 41 ( ␣
ជ A 丢 ␴ជ B and 储 ␴ជ A 丢 ␴ជ B 储 2 ⫽2 冑3 we get
⫺1/3) ␴
D共 w␣兲⭐
冑3
2
共 ␣ ⫺1/3兲 .
共6.3兲
The observable which according to Eq. 共3.8兲 violates the
ជ A 丢 ␴ជ B /2冑3. In fact,
GBI 共3.5兲 maximally is A⫽⫺ ␴
冉 冏
w␣
¯ i ␳ i 苸 ⳵ S.
␮
共iv兲 For a given tangent functional A t ⫽A 1 ⫺A 2 , A i ⭓0,
储 A 2 储 2 ⫽1 there exists an entangled state w with (w 兩 A t )⫽
⫺(w 兩 A 2 )⭐⫺1⫹␧. The homotopic state w̄⫽(1⫺␧/2)w
⫹␧/2w tr is also entangled since D(w̄) is continuous, and the
corresponding density matrix is invertible and needs N components to be decomposed into pure states. There exists a
冉
1
1
ជ 丢 ␴ជ B 苸S.
1⫺ ␴
4
3 A
⫺
␴ជ A 丢 ␴ជ B
2 冑3
冊
⫽␣
冑3
2
共6.4兲
ជ A 丢 ␴ជ B )⫽nជ •m
ជ . Since
and a pure product ␳ gives ( ␳ 兩 ␴
ជ
ជ
兩 n •m 兩 ⭐1 and this cannot be increased by mixing, we have
proved B(w ␣ )⭓( 冑3/2)( ␣ ⫺1/3). But D and B can be written as min␳maxA and maxAmin␳ of ( ␳ ⫺w 兩 A) and generally
min max⭓max min so a priori we know D(w)⭓B(w).
Therefore the above inequalities imply
032319-5
PHYSICAL REVIEW A 66, 032319 共2002兲
R. A. BERTLMANN, H. NARNHOFER, AND W. THIRRING
D 共 w ␣ 兲 ⫽B 共 w ␣ 兲 ⫽
冑3
2
␳ z⫽
共 ␣ ⫺1/3兲 ᭙ 1/3⭐ ␣ ⭐1. 共6.5兲
Furthermore, the minimizing ␳ 0 is given by Eq. 共6.2兲 and the
ជ A 丢 ␴ជ B /2冑3. Considering the
maximizing observable is ⫺ ␴
state with ␣ ⫽1 we finally get
ជ A 丢 ␴ជ B 兲 ⭐1
共 ␳ 兩⫺␴
1
共 1⫹ ␴ Az 丢 1B ⫹1A 丢 ␴ Bz ⫹ ␴ Az 丢 ␴ Bz 兲
4
we easily see within 4⫻4 matrices that the operators
␳ z⫽
᭙ ␳ 苸S
and
1
4
and
ជ A 丢 ␴ជ B 兲 ⫽3,
共 w ␣ ⫽1 兩 ⫺ ␴
共6.6兲
1
ជ A 丢 ␴ជ B 兲 .
共 1⫹ ␴
4
共6.7兲
It is not positive but applying the transposition operator T,
defined by T( ␴ i ) kl ⫽( ␴ i ) lk , on Bob it turns into a positive
operator
共 1A 丢 T B 兲 A t ⫽
1
共 1⫹ ␴ Ax 丢 ␴ Bx ⫺ ␴ Ay 丢 ␴ By ⫹ ␴ Az 丢 ␴ Bz 兲 ,
4
共6.8兲
A t⫽
1
A t⫽
4
冉 冊
0
0
0
0
0
2
0
0
2
0
0
0
0
0
2
1
共 1A 丢 T B 兲 A t ⫽
4
冉 冊
2
0
0
2
0
0
0
0
0
0
0
0
2
0
0
2
共6.9兲
.
0
0
0
0
0
0
0
0
0
0
0
0
0
1
a ⫹b
2
0
2
a
0
共 1A 丢 T B 兲 A t ⫽
1
a 2 ⫹b 2
2
0
0
ab
0
0
b
2
0
冉 冊
0
0
0
0
0
0
2
ab
0
0
ab
b
2
0
0
0
0
0
a
共6.12兲
⬎ 0
0
共6.13兲
satisfy the requirement of a tangent functional. For the state
␳ x 关let z→x in Eq. 共6.11兲兴, however, we have ( ␳ x 兩 A t )⫽1.
Remark. At this stage we would like to compare our approach to generalized Bell inequalities with the more familiar type of inequalities 共compare also with Refs. 关25,31兴兲.
Usually the BI is given by an operator in the tensor product,
where by classical arguments only some range of expectation
values can be expected, whereas the quantum case permits
another range. In our case, classically we would expect
because the expectation value of the individual spin is maximally 1 and the largest 共smallest兲 value should be obtained
when they are parallel 共antiparallel兲. This range of expectation values can exactly be achieved by all separable states
␳ 苸S, whereas we can find an entangled quantum state, the
spin singlet state w ␣ ⫽1 共6.1兲, which gives
ជ A 丢 ␴ជ B 兲 ⫽⫺2
共 w ␣ ⫽1 兩 1⫹ ␴
or
ជ A 丢 ␴ជ B 兲 兩 ⫽3.
兩 共 w ␣ ⫽1 兩 ␴
共6.15兲
ជ A 丢 ␴ជ B
This demonstrates that the tensor product operator ␴
cannot be written as a CHSH operator, where the ratio is
limited by 冑2. If we perturb a pure separable state like
1
Tr关共 1⫹n i ␴ Ai 丢 1B ⫹m i 1A 丢 ␴ Bi
16
1
ជ 兲 ⫽0
共 1⫹nជ •m
4
0
ជ A 丢 ␴ជ B 兲 ⭐2 or 兩 共 ␳ class 兩 ␴ជ A 丢 ␴ជ B 兲 兩 ⭐1
0⭐ 共 ␳ class 兩 1⫹ ␴
共6.14兲
ជ A 丢 ␴ជ B 兲兴
⫹n i m j ␴ Ai 丢 ␴ Bj 兲共 1⫹ ␴
⫽
0
0
Operator A t is not only a tangent functional for the mixed
separable state ␳ 0 共6.2兲 but with
共 ␳兩At兲⫽
0
ab
which can be nicely written as 4⫻4 matrices,
2
1
0
and the GBI is violated by a factor 3. But this ratio is not
significant since by A→A⫹c1 it can be given any value.
B(w) is meaningful since it is not affected by this change.
For the parameter values ⫺1/3⭐ ␣ ⭐1/3 the states w ␣
共6.1兲 are separable, for 1/3⬍ ␣ ⬍1 they are mixed entangled,
and the limit ␣ ⫽1 represents the spin singlet state which is
pure and maximally entangled.
Let us consider next the tangent functionals. From expression 共3.8兲 we get the flip operator 关30兴
A t⫽
冉 冊
冉 冊
共6.11兲
共6.10兲
1
␳ ␧ ⫽ 关 1⫹n i ␴ Ai 丢 1B ⫺n i 1A 丢 ␴ Bi ⫺ 共 n i n j ⫹␧ i j 兲 ␴ Ai 丢 ␴ Bj 兴
4
共6.16兲
then the expectation value
it is a tangent functional for all pure separable states with
ជ ⫽⫺nជ , which is especially the case for those states used for
m
␳ 0 共6.2兲. This illustrates point 共iii兲 of the properties of S.
However, for the pure separable states in this face we can
find other tangent functionals. For example, for the state
ជ A 丢 ␴ជ B 兲 ⫽O 共 ␧ 兲
共 ␳ ␧ 兩 1⫹ ␴
共6.17兲
is of order O(␧), as the operator constructed in Ref. 关32兴,
which shows the sensitivity of At 共6.13兲 as entanglement
witness.
032319-6
PHYSICAL REVIEW A 66, 032319 共2002兲
GEOMETRIC PICTURE OF ENTANGLEMENT AND BELL . . .
In the familiar Bell inequality derived by CHSH 关26兴
共 ␳ 兩 A CHSH 兲 ⭐2,
共6.18兲
with ␳ 苸S 共actually CHSH consider classical states ␳ class , a
generalization of separable states, in their work 关26兴兲, a
rather general observable 共a four parameter family of observables兲
ជ A 丢 共 bជ ⫺bជ ⬘ 兲 • ␴ជ B ⫹aជ ⬘ • ␴ជ A 丢 共 bជ ⫹bជ ⬘ 兲 • ␴ជ B
A CHSH ⫽aជ • ␴
共6.19兲
is used, where aជ ,aជ ⬘ ,bជ ,bជ ⬘ are any unit vectors in R3 .
However, the spin singlet state w ␣ ⫽1 共6.1兲 gives
共 w ␣ ⫽1 兩 A CHSH 兲 ⫽⫺aជ • 共 bជ ⫺bជ ⬘ 兲 ⫺aជ ⬘ • 共 bជ ⫹bជ ⬘ 兲 ,
共6.20兲
which violates the CHSH inequality 共6.18兲 maximally
共 w ␣ ⫽1 兩 A CHSH 兲 ⫽2 冑2,
共6.21兲
for appropriate angles: (aជ ,bជ )⫽(aជ ⬘ ,bជ )⫽(aជ ⬘ , bជ ⬘ )⫽135°,
(aជ ,bជ ⬘ )⫽45°, whereas in this case we find 共for all separable
states ␳ 苸S)
max 共 ␳ 兩 A CHSH 兲 ⫽ 冑2.
␳ 苸S
共6.22兲
Bell in his original work 关33兴 considers only three different directions in space 关which corresponds to the specific
case aជ ⬘ ⫽⫺bជ ⬘ in CHSH 共6.19兲兴 and assumes a strict anticorrelation
ជ A 丢 aជ ⬘ • ␴ជ B 兲 ⫽⫺1.
共 ␳ 兩 aជ ⬘ • ␴
3
max 共 ␳ 兩 A Bell 兲 ⫽ .
4
␳ 苸S
Note that generally ᭙ ␳ 苸S the maximum 共6.28兲 is larger,
namely, 冑3/2 instead of 3/4.
We observe that the maximal violation of the GBI, Eq.
ជ A 丢 ␴ជ B , where the dif共3.6兲, is largest for our observable ⫺ ␴
ference between singlet state and separable state is 2 关recall
Eq. 共6.6兲兴, whereas in case of CHSH it is 冑2 and in Bell’s
original case it is 3/4.
Although the violation of BI’s is a manifestation of entanglement, as a criterion for separability it is rather poor.
There exists a class of entangled states which satisfy the
considered BI’s, CHSH 共6.18兲, Bell 共6.24兲, but not our GBI
共3.3兲 or 共6.6兲. For a given entangled state there exists always
some operator 共entanglement witness兲 so that it satisfies the
GBI for separable states but not for this entangled state. The
class of these operators can be obtained by the positivity
condition of Ref. 关24兴. However, as a criterion for nonlocality the violation of the familiar BI’s is of great importance.
Let us finally return again to the geometry of the quantum
states 共see also Ref. 关34兴兲. For two spins there is a one parameter family of equivalence classes of pure states, interpolating between the separable one and the one containing
w ␣ ⫽1 . The latter is quite big and contains four orthogonal
projections, the ‘‘Bell states.’’ They are obtained by rotating
␴ជ A by 180° around each of the axis,
1
w ␣ ⫽1 ⫽ 共 1⫺ ␴ Ax 丢 ␴ Bx ⫺ ␴ Ay 丢 ␴ By ⫺ ␴ Az 丢 ␴ Bz 兲 ⫽: P 0
4
1
→ 共 1⫺ ␴ Ax 丢 ␴ Bx ⫹ ␴ Ay 丢 ␴ By ⫹ ␴ Az 丢 ␴ Bz 兲 ⫽: P 1
4
共6.23兲
Then he derives the inequality
共 ␳ 兩 A Bell 兲 ⭐1,
1
→ 共 1⫹ ␴ Ax 丢 ␴ Bx ⫺ ␴ Ay 丢 ␴ By ⫹ ␴ Az 丢 ␴ Bz 兲 ⫽: P 2
4
共6.24兲
1
→ 共 1⫹ ␴ Ax 丢 ␴ Bx ⫹ ␴ Ay 丢 ␴ By ⫺ ␴ Az 丢 ␴ Bz 兲 ⫽: P 3 .
4
关which clearly follows from Eq. 共6.18兲 under the mentioned
conditions兴, where now the observable is
ជ A 丢 共 bជ ⫺bជ ⬘ 兲 • ␴ជ B ⫺bជ ⬘ • ␴ជ A 丢 bជ • ␴ជ B .
A Bell ⫽aជ • ␴
共6.25兲
The expectation value of Bell’s observable in the spin singlet
state
共 w ␣ ⫽1 兩 A Bell 兲 ⫽⫺aជ • 共 bជ ⫺bជ ⬘ 兲 ⫹bជ ⬘ •bជ
共6.26兲
However there are far more since ␴ A and ␴ B can be rotated
independently.
The matrix c i j in Eq. 共2.5兲 will in general not be diagonalizible but by two independent orthogonal transformations
on both spins it can be diagonalized. Thus the correlation
part of a density matrix w c contains three parameters c i ,
冉
lies 共maximally兲 outside the range of BI 共6.24兲,
3
共 w ␣ ⫽1 兩 A Bell 兲 ⫽ ,
2
3
冊
1
c i ␴ Ai 丢 ␴ Bi .
w c ⫽ 1⫹
4
i⫽1
共6.27兲
for the angles (aជ ,bជ ⬘ )⫽(bជ ⬘ ,bជ )⫽60°, (aជ ,bជ )⫽120°, whereas
now we have for all anticorrelated separable states ␳ a ⫽ 兵 ␳
ជ ⫽⫺1 其
苸S 兩 with nជ •m
共6.28兲
a
兺
共6.29兲
Density matrix w c can be expressed as convex combination
of the projectors onto the four Bell states. Positivity requires
that the c i are contained in the convex region spanned by the
four points (⫺1,⫺1,⫺1), (⫺1,1,1), (1,⫺1,1), (1,1,⫺1).
032319-7
PHYSICAL REVIEW A 66, 032319 共2002兲
R. A. BERTLMANN, H. NARNHOFER, AND W. THIRRING
FIG. 3. In the left figure above
we have plotted the tetrahedron of
states described by the density
matrix w c 共6.29兲 in the cជ space
and to the right the reflected set of
states (1A 丢 T B )w c is shown. In
the left figure below we have plotted the intersection of the two sets
(w c and its mirror image兲 and, finally, to the right the double pyramid of separable states S艚 兵 w c 其 .
This region is screwed and the intersection with its mirror
image—compare with point 共vi兲 of the properties of
3
兩 c i 兩 ⭐1. ReflecS—characterizes the separable states 兺 i⫽1
tion in c space is effected by time reversal on one spin and
not on the other 共‘‘partial transposition’’兲 and the classically
correlated states form the set invariant under this transformation. These properties are illustrated in Fig. 3 共see also Refs.
关35,36兴兲.
Finally, we would like to mention that the quantum states
which are used in the model for decoherence of entangled
systems in particle physics 关37,38兴 also lie in the regions of
the plotted separable and entangled states.
VII. SUMMARY AND CONCLUSION
In this paper we have used tangent functionals on the set
of separable states as entanglement witnesses defining a generalized Bell inequality. The operators are vectors in the Hil-
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ACKNOWLEDGMENTS
We are thankful to Katharina Durstberger for her drawings and to Fabio Benatti, C̆aslav Brukner, Franz Embacher,
Walter Grimus, Beatrix Hiesmayr, and Anton Zeilinger for
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