Download Explicit product ensembles for separable quantum states

Explicit product ensembles
for separable quantum states
(a,b)
arXiv:quant-ph/9904109v2 19 May 1999
Rüdiger Schack∗
(a)
and Carlton M. Caves†
(a)
Center for Advanced Studies, Department of Physics and Astronomy,
University of New Mexico, Albuquerque, NM 87131–1156, USA
(b)
Department of Mathematics, Royal Holloway,
University of London, Egham, Surrey TW20 0EX, UK
February 1, 2008
Abstract
We present a general method for constructing pure-product-state representations
for density operators of N quantum bits. If such a representation has nonnegative expansion coefficients, it provides an explicit separable ensemble for the density
operator. We derive the condition for separability of a mixture of the GreenbergerHorne-Zeilinger state with the maximally mixed state.
1
Introduction
The state of a quantum system composed of N subsystems is separable if it can be written as a classical mixture or classical ensemble of tensor product states. Although it is
straightforward to decide whether a pure state is separable, the same question is in general
unsolved for mixed states. A simple separability criterion is known only for two subsystems,
and even then only if one of the subsystems is a two-level system (qubit) and the other one
is at most a three-level system (qutrit) [1, 2]
Recently it has been shown that there exists a neighborhood of the maximally mixed
state in which all states are separable [3]. For N qubits the size of this neighborhood
decreases exponentially with the number of qubits [4]. These results are interesting, among
other reasons, because NMR quantum computers [5, 6, 7] operate with states near the
maximally mixed state.
In this paper we develop a general method for constructing pure-product-state representations for N-qubit density operators; when the expansion coefficients in such a representation are nonnegative, it provides an explicit product ensemble for the density operator,
∗
†
E-mail: [email protected]
E-mail: [email protected]
1
and the state is separable. In Sec. 2.1, we introduce a superoperator formalism that allows us to treat different discrete and continuous representations on the same footing. In
Sec. 2.2, we apply the formalism to a discrete overcomplete operator basis, leading to a
simple rederivation of the separability bound found in [4]. Section 2.3 introduces a “canonical” continuous representation for an arbitrary density operator, and Sec. 2.4 develops a
method for deriving discrete representations from the continuous representation. In Sec. 3,
we show that the canonical continuous representation is the unique representation containing only scalar and vector spherical-harmonic components; all other representations can
be obtained by adding arbitrary higher-order spherical harmonics. Section 3.2 applies the
canonical continuous representation to mixtures of the N-qubit “cat state” with the maximally mixed state. In Sec. 4, we derive the conditions for separability of a mixture of the
Greenberger-Horne-Zeilinger state [8] with the maximally mixed state. Section 5 concludes
with a brief discussion.
2
2.1
Product-state representations and separable ensembles
Superoperator formalism
The set of linear operators acting on a D-dimensional Hilbert space H is a D 2 -dimensional
complex vector space L(H). Let us introduce operator “kets” |A) = A and “bras” (A| = A† ,
distinguished from vector kets and bras by the use of round brackets [9]. Then the natural
inner product on L(H), the trace-norm inner product, can be written as (A|B) = tr(A† B).
The notation S = |A)(B| defines a superoperator S acting like
S|X) = |A)(B|X) = tr(B † X)A .
(1)
Now let the set {|Nj )} constitute a (complete or overcomplete) operator basis; i.e., let
the operator kets |Nj ) span the vector space L(H). It follows that the superoperator G
defined by
X
|Nj )(Nj |
(2)
G≡
j
is invertible. The operators
Qj ≡ G −1 |Nj )
(3)
form a dual basis, which gives rise to the following resolutions of the superoperator identity:
1=
X
j
|Qj )(Nj | =
X
j
|Nj )(Qj | .
(4)
An arbitrary operator A can be expanded as
A=
X
j
|Nj )(Qj |A) =
X
j
|Qj )(Nj |A) =
(5)
Qj tr(Nj† A) .
(6)
j
and
A=
Nj tr(Q†j A)
X
X
j
These expansions are unique if and only if the operators Nj are linearly independent.
2
2.2
A discrete representation for the 6 cardinal directions
The Hermitian operators Pjµ , j = 1, 2, 3, µ = 0, 1, defined by
Pjµ ≡
1
1 + (−1)µ σj ,
2
(7)
are the pure-state projectors corresponding to the 6 cardinal directions on the Bloch sphere.
They form an overcomplete basis in the space of operators acting on a qubit. The superoperator
!
X
X
1
|σj )(σj |
(8)
3|1)(1| +
|Pjµ)(Pjµ | =
G=
2
j
j,µ
√
√
has normalized eigenoperators |1)/ 2 and |σj )/ 2, with eigenvalues 3 and 1, respectively,
from which it follows that
G
−1
!
X
1 1
|σj )(σj | .
|1)(1| +
=
2 3
j
(9)
The last equation allows us to find the dual-basis operators,
Qjµ = G −1 |Pjµ) =
1
1 + 3(−1)µ σj .
6
(10)
~ · ~σ ) for one qubit can be represented in the form
Any density operator ρ = 21 (1 + S
ρ=
X
j,µ
|Pjµ )(Qjµ |ρ) =
1 X
Pjµ 1 + 3(−1)µ Sj .
6 j,µ
(11)
For an N-qubit density operator, the analogous representation is
ρ=
X
j1 ,µ1 ,...,jN ,µN
w can (j1 , µ1, . . . , jN , µN )Pj1 µ1 ⊗ · · · ⊗ PjN µN ,
(12)
where
w can (j1 , µ1 , . . . , jN , µN ) ≡ tr(ρ Qj1 µ1 ⊗ · · · ⊗ QjN µN )
1
µN
µ1
.
= N tr ρ 1 + 3(−1) σj1 ⊗ · · · ⊗ 1 + 3(−1) σjN
6
(13)
This representation exists for arbitrary density operators, entangled or separable. If the
coefficients w can (j1 , µ1 , . . . , jN , µN ) are nonnegative, the density operator ρ is separable,
and the representation (12) provides an explicit product-state ensemble for the density
operator. The expansions (11) and (12) are not unique because the 6 operators Pjµ are
not linearly independent. Throughout the paper we use the superscript “can,” standing
for canonical, to denote the natural, but generally not unique expansion coefficients that
come from using the procedure of Sec. 2.1.
3
For any N-qubit density operator ρ, the coefficients (13) obey the bound
w
can
(j1 , µ1 , . . . , jN , µN ) ≥
smallest eigenvalue of
Qj1 µ1 ⊗ · · · ⊗ QjN µN
=−
22N −1
.
6N
(14)
This follows from the fact that Qjµ has eigenvalues 2/3 and −1/3. The most negative
eigenvalue of the product operator Qj1 µ1 ⊗ · · · ⊗ QjN µN is therefore (−1/3)(2/3)N −1 =
−22N −1 /6N , from which Eq. (14) follows.
Consider now an N-qubit density operator that is a mixture of the maximally mixed
density operator, 1/2N , and an arbitrary density operator ρ1 :
ρǫ =
1−ǫ
1 + ǫρ1 .
2N
(15)
Any density operator can be written in this form [4]. The coefficients (13) for ρǫ are given
by
1−ǫ
+ ǫw1can (j1 , µ1, . . . , jN , µN )
N
6
1 − ǫ(1 + 22N −1 )
≥
,
6N
wǫcan (j1 , µ1 , . . . , jN , µN ) =
which is nonnegative for
ǫ≤
1
.
1 + 22N −1
(16)
(17)
This bound, derived in [4] using a different route, implies that, for ǫ ≤ 1/(1 + 22N −1 ), all
density operators of the form (15) are separable. It sets a lower bound on the size of the
separable neighborhood surrounding the maximally mixed state.
2.3
Canonical continuous representation
The set of all pure-state projectors on the Bloch sphere,
1
P~n ≡ |~nih~n| = (1 + ~σ · ~n) ,
2
(18)
forms an overcomplete basis. The corresponding superoperator
G=
Z
1X
|σj )(σj |
dΩ |P~n )(P~n | = π |1)(1| +
3 j
!
(19)
is proportional
to
√
√ the superoperator of Eq. (8). The normalized eigenoperators of G are
|1)/ 2 and |σj )/ 2, with eigenvalues 2π and 2π/3, respectively. Thus G −1 can be expressed
as


X
1
|1)(1| + 3
G −1 =
|σj )(σj | .
(20)
4π
j
4
The dual-basis operators are then given by
1
(1 + 3~σ · ~n) .
4π
Q~n = G −1 |P~n ) =
(21)
~ · ~σ ) for one qubit can be represented in the form
Any density operator ρ = 21 (1 + S
ρ=
Z
1 Z
~ · ~n) .
dΩ P~n (1 + 3S
dΩ |P~n )(Q~n |ρ) =
4π
(22)
For N qubits, we define the pure-product-state projector
P (ñ) ≡ P~n1 ⊗ · · · ⊗ P~nN =
1
(1 + ~σ · ~n1 ) ⊗ · · · ⊗ (1 + ~σ · ~nN ) ,
2N
(23)
where ñ stands for the collection of unit vectors ~n1 , . . . , ~nN . Any N-qubit density operator
can be expanded as
ρ=
Z
dΩñ w can (ñ)P (ñ) ≡
Z
dΩn1 · · · dΩnN w can (ñ)P (ñ) ,
where
w can (ñ) ≡ tr ρQ(ñ)
is the canonical expansion function, with
Q(ñ) ≡ Q~n1 ⊗ · · · ⊗ Q~nN =
(24)
(25)
1
(1 + 3~σ · ~n1 ) ⊗ · · · ⊗ (1 + 3~σ · ~nN ) .
(4π)N
(26)
If the coefficients w can (ñ) are nonnegative, the density operator ρ is separable, and the
representation (24) provides an explicit product-state ensemble for the density operator.
Using the same argument as in the preceding subsection, one can show that for any
density operator ρ [4],
w
can
(ñ) ≥
smallest eigenvalue
of Q(ñ)
=−
22N −1
.
(4π)N
(27)
Applying this result to density operators of the form (15), one arrives again at the lower
bound (17) on the size of the separable neighborhood of the maximally mixed density
operator [4].
2.4
Other discrete representations
The expansion coefficients (13) for the discrete representation of Sec. 2.2 can be obtained
by evaluating the expansion coefficients (25) for the canonical continuous representation
along the cardinal directions for each qubit and then renormalizing the resulting function.
This is no accident. It is easy to characterize a class of discrete representations whose
expansion coefficients are similarly related to the continuous representation.
5
Consider a discrete set of projection operators,
1
P~nα = |~nα ih~nα | = (1 + ~σ · ~nα ) ,
2
α = 1, . . . , K.
(28)
The corresponding superoperator
G =
K
X
α=1
|P~nα )(P~nα |


X
X
X
1
=
~nα · |~σ)(1| + |1)(~σ| · ~nα +
|σj )(σk | (nα )j (nα )k 
K|1)(1| +
4
α
α
j,k
(29)
generates dual-basis operators and expansion coefficients proportional to those for the continuous representation if and only if G is proportional to the superoperator (19) for the
continuous representation, i.e., if and only if
0 =
X
~nα ,
(30)
α
1 X
1
δjk =
(nα )j (nα )k .
3
K α
(31)
The trace of Eq. (31) is satisfied by any collection of unit vectors.
When these conditions are satisfied, the superoperator (29) simplifies to
!
1X
K
|1)(1| +
|σj )(σj | ,
G=
4
3 j
(32)
!
(33)
with an inverse
G
−1
X
1
=
|1)(1| + 3
|σj )(σj |
K
j
that generates dual-basis operators
Q~nα = G −1 |P~nα ) =
1
(1 + 3~σ · ~nα ) .
K
(34)
~ · ~σ ) for one qubit can be represented in the form
Any density operator ρ = 21 (1 + S
ρ=
X
α
|Pα )(Qα |ρ) =
1 X
~ · ~nα ) .
Pα (1 + 3S
K α
(35)
The conditions (30) and (31) are satisfied by unit vectors that point to the vertices of
any regular polyhedron inscribed within the Bloch sphere—i.e., a tetrahedron, octahedron,
cube, icosahedron, or dodecahedron. The 4 projectors for a tetrahedron are linearly independent, making the corresponding tetrahedral representation (35) unique. An octahedron
gives rise to the 6-cardinal-direction representation of Sec. 2.2. The regular polyhedra by no
means exhaust the possibilities for representations of this sort. One can use simultaneously
the vertices from any number of polyhedra. Moreover, one can produce an appropriate set
6
of unit vectors by the following procedure: starting with any set of unit vectors in the first
octant, reflect the set successively through the three Cartesian planes to produce a set of
vectors that occupies all eight octants.
For more than one qubit, the possibilities for discrete representations that follow from
the continuous representation are even more diverse. The simplest possibility is to use the
same set of pure-state projectors for each qubit, but this is not necessary. One can use a
different set of projectors for each qubit, or more generally, one can use a “tree structure,”
in which there is a different set of projectors for the nth qubit for each projector for
the first n − 1 qubits. There are possibilities more general even than those with a tree
structure, analogous to the “domino states” introduced for two qutrits in [10]. We do not
dwell here on the myriad of discrete representations of this sort, merely noting that since
the expansion coefficients in such representations follow from the expansion coefficients
for the continuous representation, all such representations lead to the same general lower
bound (17) on the size of the separable neighborhood surrounding the maximally mixed
state. Both the continuous and these discrete representations can, however, give better
bounds on the separability of particular states of the form (15), a point to which we return
in Sec. 3.2.
3
Spherical-harmonic expansions
In this section we explore the relationship between product-state representations and spherical-harmonic expansions.
3.1
Spherical-harmonic expansions and the Pauli representation
We begin by noting that if ~n is a unit vector, then
~σ · ~n = σ3 cos θ + σ1 sin θ cos φ + σ2 sin θ sin φ
s
4π 1
1
∗
∗
∗
∗
∗
σ3 Y10 + σ1 √ (−Y1+1 + Y1−1 ) − iσ2 √ (Y1+1 + Y1−1 )
3
2
2
0 0∗
+1 +1 ∗
−1 −1 ∗
= σ1 Y1 + σ1 Y1 + σ1 Y1 ,
=
where
s
s
(37)
2π
(σ1 ± iσ2 ) .
3
(38)
3
3
1
cos θ , Y1±1 = ∓
sin θ e±iφ
Y00 = √ , Y10 =
4π
8π
4π
are spherical harmonics and, in the last line, we define
σ00 ≡
√
4π 1 , σ10 ≡
s
4π
σ3 ,
3
s
σ1±1 ≡ ∓
(36)
We can then write any pure-state projector on the Bloch sphere in the form
l
1 X
1 X
1
P~n = |~nih~n| = (1 + ~σ · ~n) =
σ m Y m∗ .
2
2 l=0 m=−l l l
7
(39)
Any N-qubit density operator can be written in the form
Z
ρ=
dΩñ w(ñ)P (ñ) ,
(40)
where the expansion coefficients w(ñ) ≡ w(~n1 , . . . , ~nN ) are not necessarily nonnegative [the
expansion coefficients w(ñ) need not be those of the canonical representation (25)]. A
density operator ρ is separable if and only if it can be written in the form (40) for some
nonnegative probability density w(ñ). Now expand the product projector (23) in terms of
spherical harmonics,
P (ñ) =
1
2N
X
{lj ≤1,mj }
where
X
{lj ≤1,mj }
N ∗ m1
Yl1m1 ∗ · · · Ylm
σl1 ⊗ · · · ⊗ σlmNN ,
N
1
X
≡
l1
X
l1 =0 m1 =−l1
···
1
X
lN
X
(41)
(42)
lN =0 mN =−lN
denotes a sum that contains only scalar and vector terms (l = 0, 1), and also expand the
expansion coefficients,
X
1 ...mN
w(ñ) =
alm1 ...l
Yl1m1 · · · YlNmN ,
(43)
N
{lj ,mj }
where
X
{lj ,mj }
≡
∞
X
l1
X
l1 =0 m1 =−l1
···
∞
X
lN
X
(44)
lN =0 mN =−lN
denotes an unrestricted sum over spherical harmonics. Inserting Eqs. (41) and (43) into
Eq. (40) and using the orthonormality of the spherical harmonics yields the (unique) Paulioperator representation of ρ:
ρ=
1
2N
X
{lj ≤1,mj }
1 ...mN m1
alm1 ...l
σl1 ⊗ · · · ⊗ σlmNN .
N
(45)
1 ...mN
We see that ρ depends on and determines only the scalar and vector parts of alm1 ...l
;
N
the higher-order-spherical-harmonic (HOSH) content of w(ñ) corresponds to the freedom
in constructing pure-product-state representations of ρ. It is straightforward to generate
all pure-product-state representations: given the Pauli-operator representation (45) of ρ,
1 ...mN
any w(ñ) is given by Eq. (43), where the coefficients alm1 ...l
are arbitrary except for
N
l1 , . . . , lN ≤ 1.
The canonical continuous representation w can (ñ) of Eq. (25) is the unique pure-productstate representation given by the l = 0 and l = 1 components. We can see this directly in
the following way. Write the Pauli representation of ρ in the form
ρ=
1
2N
X
α1 ,...,αN
cα1 ...αN σα1 ⊗ · · · ⊗ σαN ,
(46)
where
cα1 ...αN = tr(ρσα1 ⊗ · · · ⊗ σαN ) = hσα1 ⊗ · · · ⊗ σαN i .
8
(47)
Here σ0 = 1, and the sums over Greek indices run from 0 to 3. Now we have that
E
1 D
(1
+
3~
σ
·
~
n
)
⊗
·
·
·
⊗
(1
+
3~
σ
·
~
n
)
1
N
(4π)N
E
3 N X D
(n1 )α1 σα1 ⊗ · · · ⊗ (nN )αN σαN
=
4π
α1 ,...,αN
w can (ñ) =
=
3
4π
N
X
α1 ,...,αN
(48)
cα1 ...αN (n1 )α1 · · · (nN )αN ,
where (nj )αj ≡ 1/3 if αj = 0 and (nj )αj is a Cartesian component of ~nj if αj = 1, 2, or
3. Thus any pure-product-state representation of a density operator ρ can be obtained
by adding HOSH to the canonical continuous representation w can (ñ). The discrete representations discussed in Sec. 2.4 are examples—though by no means the only examples—of
representations with HOSH content.
The continuous canonical representation can be generated directly from the Pauli representation of ρ by making the substitutions
2N ρ −→ w can (ñ) ,
1
3
, σj −→
nj .
4π
4π
These rules are equivalent to σlm −→ Ylm and to σα −→ (3/4π)nα.
1 −→
(49)
(50)
The ǫ-cat state
3.2
As an example, consider the following N-qubit generalization of the Werner [11] state,
ρǫ =
where
1−ǫ
1 + ǫ|ψcat ihψcat | ,
2N
1 |ψcat i ≡ √ |0 . . . 0i + |1 . . . 1i
2
(51)
(52)
is the N-qubit “cat state.” We call the mixed state (51) the ǫ-cat state.
Now let |0iand |1i be the eigenstates of σ3 , with |0i at the top of the Bloch sphere and |1i
at the bottom, so that σ1 = |0ih1| + |1ih0|, σ2 = −i|0ih1| + i|1ih0|, and σ3 = |0ih0| − |1ih1|.
It follows that the Pauli representation of the ǫ-cat state is
ρǫ
1
= N (1 − ǫ)1
2
ǫ
ǫ
(53)
+ (1 + σ3 ) ⊗ · · · ⊗ (1 + σ3 ) + (1 − σ3 ) ⊗ · · · ⊗ (1 − σ3 )
2
2
ǫ
ǫ
+ (σ1 + iσ2 ) ⊗ · · · ⊗ (σ1 + iσ2 ) + (σ1 − iσ2 ) ⊗ · · · ⊗ (σ1 − iσ2 ) .
2
2
9
Applying the rules (49) and (50) yields the continuous canonical representation of ρǫ :
wǫcan (ñ)
1
=
(1 − ǫ)
(4π)N
ǫ
ǫ
+ (1 + 3 cos θ1 ) · · · (1 + 3 cos θN ) + (1 − 3 cos θ1 ) · · · (1 − 3 cos θN )
2
2 N
+ ǫ 3 sin θ1 · · · sin θN cos(φ1 + . . . + φN ) .
(54)
The minimum value of wǫcan (ñ) occurs either (i) when one qubit is evaluated at the
north (south) pole, and the other N − 1 qubits are evaluated at the south (north) pole, or
(ii) when all the qubits are evaluated on the equator with cos(φ1 + . . . + φN ) = −1. For
N = 2 and N = 4, both the poles and the equator give the same minimum value, which
shows that the ǫ-cat state is separable for ǫ ≤ 1/9 for N = 2 and for ǫ ≤ 1/81 for N = 4.
For N = 3 and N = 5, the equator gives the minimum value, which shows that the ǫ-cat
state is separable for ǫ ≤ 1/27 for N = 3 [4] and for ǫ ≤ 1/243 for N = 5. For N ≥ 6, the
poles give the minimum value, which shows that ρǫ is separable for
ǫ≤
1±
2N
1
,
+ 22N −2
(55)
the upper (lower) sign applying when N is even (odd). Notice that except for N = 2, all
these bounds are better than the general-purpose bound of Eq. (17).
These bounds on the separability of the ǫ-cat state can be improved by using tetrahedral
ensembles of the sort discussed in Sec. 2.4. The tetrahedral ensembles are chosen to have a
tree structure such that the vertices avoid the places where wǫcan (ñ) has its smallest values.
We do not discuss these tetrahedral bounds here, preferring instead to consider in detail
the separability of the ǫ-GHZ state.
4
The ǫ-GHZ state
In this section we consider the 3-qubit version of the ǫ-cat state introduced in Sec. 3.2.
In the case of 3 qubits, the cat state of Eq. (52) is called the Greenberger-Horne-Zeilinger
(GHZ) state [8],
1 |ψGHZ i = √ |000i + |111i ;
(56)
2
we call the mixture ρǫ of Eq. (51) the ǫ-GHZ state. We show in this section that the ǫ-GHZ
state is separable if and only if ǫ ≤ 1/5. The class of ǫ-GHZ states has been considered
previously in [12].
4.1
Separability of the Werner state
Before proceeding to the proof of the separability of the ǫ-GHZ state, we review a similar,
but simpler proof for the Werner state [11], the 2-qubit version of the ǫ-cat state:
ρW =
ǫ
1−ǫ
1 ⊗ 1 + |00i + |11i h00| + h11|
4
2
10
=
1
1 ⊗ 1 + ǫ (σ3 ⊗ σ3 + σ1 ⊗ σ1 − σ2 ⊗ σ2 ) .
4
(57)
The Werner state is separable if and only if ǫ ≤ 1/3 [13, 14], a result we now show in the
following way.
A density operator ρ for 2 qubits, A and B, is separable if and only if it can be written
as
Z
ρ = dΩ~nA dΩ~nB w(~nA , ~nB )P~nA ⊗ P~nB ,
(58)
with w(~nA , ~nB ) being a nonnegative probability distribution. If a state is separable, the
coefficients (47) in the Pauli representation (46) of ρ become classical expectations over the
distribution w(~nA , ~nB ), expectations that we denote by E[·]:
cαβ = tr(ρ σα ⊗ σβ ) =
Z
dΩ~nA dΩ~nB (nA )α (nB )β w(~nA , ~nB ) ≡ E[(nA )α (nB )β ] .
(59)
For the Werner state, the coefficients cαβ can be read off the Pauli representation in
Eq. (57):
E[(nA )j ] = E[(nB )j ] = 0 ,
E[(nA )j (nB )k ] = 0 , for j 6= k,
E[(nA )1 (nB )1 ] = −E[(nA )2 (nB )2 ] = E[(nA )3 (nB )3 ] = ǫ .
(60)
(61)
(62)
If w(~nA , ~nB ) is a probability distribution, we have that
1
ǫ = E[(nA )j (nB )j ] ≤ E[(nA )2j + (nB )2j ] .
2
(63)
Adding over the three values of j gives
1
3ǫ ≤ E[(nA )21 + (nA )22 + (nA )23 + (nB )21 + (nB )22 + (nB )23 ] = 1 ,
2
(64)
which shows that if ǫ > 1/3, the Werner state is nonseparable.
We still have the task of designing a product ensemble that gives the Werner state for
ǫ ≤ 1/3. Defining three density operators
1
(P~e ⊗ P~e3 + P−~e3 ⊗ P−~e3 ) =
2 3
1
≡
(P~e ⊗ P~e1 + P−~e1 ⊗ P−~e1 ) =
2 1
1
(P~e ⊗ P−~e2 + P−~e2 ⊗ P~e2 ) =
≡
2 2
ρ1 ≡
ρ2
ρ3
1
(1 ⊗ 1 + σ3 ⊗ σ3 ) ,
4
1
(1 ⊗ 1 + σ1 ⊗ σ1 ) ,
4
1
(1 ⊗ 1 − σ2 ⊗ σ2 ) ,
4
(65)
(66)
(67)
we see that each takes care of one of the correlations in Eq. (62). An equal mixture gives
the ǫ = 1/3 Werner state:
1
1
1
1 ⊗ 1 + (σ3 ⊗ σ3 + σ1 ⊗ σ1 − σ2 ⊗ σ2 )
(ρ1 + ρ2 + ρ3 ) =
3
4
3
.
(68)
Smaller values of ǫ can be handled by adding on a product ensemble for the maximally
mixed state.
11
4.2
Separability of the ǫ-GHZ state
Consider now the ǫ-GHZ state
1−ǫ
ǫ
1 ⊗ 1 ⊗ 1 + |000i + |111i h000| + h111|
8
2
1
1 ⊗ 1 ⊗ 1 + ǫ 1 ⊗ σ3 ⊗ σ3 + σ3 ⊗ 1 ⊗ σ3 + σ3 ⊗ σ3 ⊗ 1
=
8
ρǫGHZ =
(69)
+ σ1 ⊗ σ1 ⊗ σ1 − σ1 ⊗ σ2 ⊗ σ2 − σ2 ⊗ σ1 ⊗ σ2 − σ2 ⊗ σ2 ⊗ σ1
.
A density operator ρ for 3 qubits, A, B, and C, is separable if and only if it can be
written as
Z
ρ = dΩ~nA dΩ~nB dΩ~nC w(~nA , ~nB , ~nC )P~nA ⊗ P~nB ⊗ P~nC ,
(70)
with w(~nA , ~nB , ~nC ) being a nonnegative probability distribution. If a state is separable,
the coefficients (47) in the Pauli representation (46) of ρ become classical expectations over
the distribution w(~nA , ~nB , ~nC ):
cαβγ = tr(ρ σα ⊗ σβ ⊗ σγ ) = E[(nA )α (nB )β (nC )γ ] .
(71)
For the ǫ-GHZ state, the coefficients cαβγ can be read off the Pauli representation in
Eq. (69):
E[(nA )j ] = E[(nB )j ] = E[(nC )j ] = 0 ,
E[(nA )j (nB )k ] = E[(nA )j (nC )k ] = E[(nB )j (nC )k ] = ǫ δj3 δk3 ,
E[(nA )j (nB )k (nC )l ] = ǫ (δj1δk1 δl1 − δj1 δk2 δl2 − δj2 δk1 δl2 − δj2 δk2 δl1 ) .
(72)
(73)
(74)
Now define a “vector”
~ ≡ (nB )1 (nC )1 − (nB )2 (nC )2 ~e1 − (nB )1 (nC )2 + (nB )2 (nC )1 ~e2 + (nB )3~e3 ,
N
whose magnitude satisfies
~ ·N
~ =
N
=
(nB )1 (nC )1 − (nB )2 (nC )2
(nB )21 + (nB )22
≤ ~nB · ~nB = 1 .
2
+ (nB )1 (nC )2 + (nB )2 (nC )1
(nC )21 + (nC )22 + (nB )23
2
(75)
+ (nB )23
(76)
If w(~nA , ~nB , ~nC ) is a probability distribution, we have that
h
i
i
h
h
i
~ ≤ E ~nA · N
~ ≤ 1 E ~nA · ~nA + N
~ ·N
~ ≤1,
5ǫ = E ~nA · N
2
which shows that if ǫ > 1/5, the ǫ-GHZ state is nonseparable.
12
(77)
To finish the proof, we need to construct a product ensemble for the ǫ-GHZ state when
ǫ ≤ 1/5. We begin by defining five density operators
ρ1 ≡
=
ρ2 ≡
=
ρ3 ≡
=
ρ4 ≡
=
ρ5 ≡
=
1
P~e3 ⊗ P~e3 ⊗ P~e3 + P−~e3 ⊗ P−~e3 ⊗ P−~e3
2
1
1 ⊗ 1 ⊗ 1 + 1 ⊗ σ3 ⊗ σ3 + σ3 ⊗ 1 ⊗ σ3 + σ3 ⊗ σ3 ⊗ 1 ,
(78)
8
1
P~e1 ⊗ P~e1 ⊗ P~e1 + P~e1 ⊗ P−~e1 ⊗ P−~e1 + P−~e1 ⊗ P~e1 ⊗ P−~e1 + P−~e1 ⊗ P−~e1 ⊗ P~e1
4
1
(1 ⊗ 1 ⊗ 1 + σ1 ⊗ σ1 ⊗ σ1 ,
(79)
8
1
P~e1 ⊗ P~e2 ⊗ P−~e2 + P~e1 ⊗ P−~e2 ⊗ P~e2 + P−~e1 ⊗ P~e2 ⊗ P~e2 + P−~e1 ⊗ P−~e2 ⊗ P−~e2
4
1
(1 ⊗ 1 ⊗ 1 − σ1 ⊗ σ2 ⊗ σ2 ,
(80)
8
1
P~e2 ⊗ P~e1 ⊗ P−~e2 + P−~e2 ⊗ P~e1 ⊗ P~e2 + P~e2 ⊗ P−~e1 ⊗ P~e2 + P−~e2 ⊗ P−~e1 ⊗ P−~e2
4
1
1 ⊗ 1 ⊗ 1 − σ2 ⊗ σ1 ⊗ σ2 ,
(81)
8
1
P~e2 ⊗ P−~e2 ⊗ P~e1 + P−~e2 ⊗ P~e2 ⊗ P~e1 + P~e2 ⊗ P~e2 ⊗ P−~e1 + P−~e2 ⊗ P−~e2 ⊗ P−~e1
4
1
1 ⊗ 1 ⊗ 1 − σ2 ⊗ σ2 ⊗ σ1 ,
(82)
8
each of which takes care of one of the nonzero correlations in Eqs. (73) and (74). An equal
mixture gives the (ǫ = 1/5)-GHZ state:
1
(ρ1 + ρ2 + ρ3 + ρ4 + ρ5 )
5
1
1
1 ⊗ 1 ⊗ 1 + 1 ⊗ σ3 ⊗ σ3 + σ3 ⊗ 1 ⊗ σ3 + σ3 ⊗ σ3 ⊗ 1
=
8
5
+ σ1 ⊗ σ1 ⊗ σ1 − σ1 ⊗ σ2 ⊗ σ2 − σ2 ⊗ σ1 ⊗ σ2 − σ2 ⊗ σ2 ⊗ σ1
(83)
,
thus completing the proof.
For both the Werner state and the ǫ-GHZ states, we are able to construct an optimal ensemble based on the 6 cardinal directions, but notice that this ensemble is not the
canonical representation of Sec. 2.2.
5
Conclusion
In this paper we introduce a canonical procedure for constructing representations of density operators in terms of complete or overcomplete sets of pure product projectors. The
procedure is useful for investigating the separability of density operators: if the expansion
coefficients in such a representation are nonnegative, they can be interpreted as a probability distribution, with the result that the representation then provides an explicit product
13
ensemble for the density operator. Generally, however, the canonical representations have
negative values even when the density operator is separable.
The canonical continuous representation is the unique representation that contains only
scalar and vector spherical-harmonic terms, and we outline how to obtain all possible pureproduct-state representations from the canonical continuous representation by adding arbitrary higher-order-spherical-harmonic content. Searching the space of spherical-harmonic
expansions for nonnegative representations would be quite difficult, but we provide an optimal ensemble in the particular case of the ǫ-GHZ state, thereby identifying the separability
boundary for this class of states.
Acknowledgments
Thanks to G. J. Milburn and R. Pranaw for discussions of the method used to prove the
separability of the ǫ-GHZ state. This work was supported in part by the US Office of Naval
Research (Grant No. N00014-93-0-0116) and by the UK Engineering and Physical Sciences
Research Council.
Note added
The separability bound for the ǫ-GHZ state derived in Sec. 4 has also been found by Dür,
Cirac, and Tarrach [15] as part of a more general result. They prove that the N-qubit ǫ-cat
state (51) is separable if and only if ǫ < 1/(1 + 2N −1 ), which supersedes our Eq. (55).
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