PHYSICAL REVIEW A 68, 022318 共2003兲
Entanglement, mixedness, and spin-flip symmetry in multiple-qubit systems
Gregg Jaeger,1 Alexander V. Sergienko,1,2 Bahaa E. A. Saleh,1 and Malvin C. Teich1,2
1
Quantum Imaging Laboratory, Department of Electrical and Computer Engineering, Boston University,
Boston, Massachusetts 02215, USA
2
Department of Physics, Boston University, Boston, Massachusetts 02215, USA
共Received 14 May 2003; published 27 August 2003兲
A relationship between a recently introduced multipartite entanglement measure, state mixedness, and spinflip symmetry is established for any finite number of qubits. It is also shown that, for those classes of states
invariant under the spin-flip transformation, there is a complementarity relation between multipartite entanglement and mixedness. A number of example classes of multiple-qubit systems are studied in light of this
relationship.
DOI: 10.1103/PhysRevA.68.022318
PACS number共s兲: 03.67.⫺a, 03.65.Ta, 42.79.Ta
I. INTRODUCTION
State entanglement and mixedness are properties central
to quantum information theory. It is therefore important,
wherever possible, to relate them. The relationship between
these quantities has been previously investigated for the simplest case, that of two-qubits 共see, for example, Refs. 关1– 4兴兲,
but has been much less well studied for multiple-qubit states
because multipartite entanglement measures 关5,6兴 have only
been recently given in explicit form 共see, for example, Refs.
关7–10兴兲. Here, we find a relationship, for any finite number
of qubits, between a multipartite entanglement measure and
state mixedness, through the introduction of a multiple-qubit
measure of symmetry under the spin-flip transformation.
Here, several classes of two, three, and four qubit states are
examined to illustrate this relationship. Since quantum decoherence affects the mixedness and entanglement properties
of multiple-qubit states, our results provide a tool for investigating decoherence phenomena in quantum information
processing applications, which utilize just such states.
The purity P of a general quantum state can be given in
all cases by the trace of the square of the density matrix ␳
and the mixedness M by its complement:
P共 ␳ 兲 ⫽Tr ␳ 2 ,
M 共 ␳ 兲 ⫽1⫺Tr ␳ 2 ⫽1⫺P共 ␳ 兲 .
共1兲
Entanglement can be captured in several ways, which may or
may not be directly related, with varying degrees of coarseness 共see, for example, Ref. 关9兴兲, depending on the complexity of the system described 共see, for example, Refs. 关11–14兴兲.
Here, we will measure entanglement by a recently introduced
measure
of
multipartite
entanglement,
the
SL(2,C) ⫻n -invariant quantity
2
˜ 兲,
⫽Tr共 ␳␳
S (n)
共2兲
where the tilde indicates the spin-flip operation 关7兴. The multiple qubit spin-flip operation is defined as
␳ →˜␳ ⬅ ␴ 2丢 n ␳ * ␴ 2丢 n ,
1050-2947/2003/68共2兲/022318共5兲/$20.00
共3兲
where ␳ * is the complex conjugate of the n-qubit density
matrix ␳ and ␴ 2 is the spin-flipping Pauli matrix 关15兴. Its
connection to the n-tangle, another multipartite entanglement
measure, defined in Ref. 关10兴, will be discussed below. We
now discuss this measure in relation to the well-established,
bipartite measures of entanglement 关1兴.
In the simplest case of pure states of two qubits (A and
B), entanglement has most commonly been described by the
entropy of either of the one-qubit reduced density operators,
which is obtained by tracing out the variables of one or the
other qubits from the total system state described by the
projector P 关 兩 ⌿ AB 典 ]⬅ 兩 ⌿ AB 典具 ⌿ AB 兩 . For mixed two-qubit
states ␳ AB , the entanglement of formation, E f , is given by
the minimum average marginal entropy of all possible decompositions of the state as a mixture of subensembles. Alternatively, one can use a simpler measure of entanglement,
the concurrence C, to describe two-qubit entanglement 关11兴.
For pure states, this quantity can be written as
* 典 兩 ⫽ 兩 具 ⌿ AB 兩 ⌿̃ AB 典 兩 , 共4兲
C 共 兩 ⌿ AB 典 )⫽ 兩 具 ⌿ AB 兩 ␴ 2 丢 ␴ 2 兩 ⌿ AB
where 兩 ⌿̃ AB 典 is the spin-flipped state vector. It has been
shown that the concurrence of a mixed two-qubit state,
C( ␳ AB ), can be expressed in terms of the minimum average
pure-state concurrence C( 兩 ⌿ AB 典 ), where the minimum is
taken over all possible ensemble decompositions of ␳ AB and
that, in general, C( ␳ )⫽max兵 0,␭ 1 ⫺␭ 2 ⫺␭ 3 ⫺␭ 4 其 , where ␭ i
are the square roots of the eigenvalues of the product matrix
˜ , the ‘‘singular values,’’ all of which are non-negative real
␳␳
quantities 关11兴. It has also been shown that the entanglement
of formation of a mixed state ␳ of two qubits can be expressed in terms of the concurrence as
E f 共 ␳ 兲 ⫽h„C 共 ␳ 兲 …,
共5兲
where h(x)⫽⫺x log2 x⫺(1⫺x) log2 (1⫺x) 共see Ref. 关11兴兲.
Here, we measure multipartite entanglement involving n
2
共see Ref. 关7兴兲, which is invariant under the
qubits by S (n)
group corresponding to stochastic local operations and classical communications 共SLOCC兲 关16兴. For two-qubit pure
states, this measure 共with n⫽2) coincides with the squared
concurrence 共or tangle兲:
68 022318-1
©2003 The American Physical Society
PHYSICAL REVIEW A 68, 022318 共2003兲
JAEGER et al.
2
S (2)
共 P 关 兩 ␺ 典 兴 )⫽ ␶ (2) 共 P 关 兩 ␺ 典 兴 )⫽C 2 共 P 关 兩 ␺ 典 兴 ),
共6兲
where P 关 兩 ␺ 典 ]⬅ 兩 ␺ 典具 ␺ 兩 is the projector corresponding to its
state-vector argument 兩 ␺ 典 .
2
for
In Sec. II, after providing another expression for S (n)
all values of n for both pure and mixed states, we show that
for pure states this length coincides with the multipartite generalization of the tangle defined for pure states of any finite
number of qubits, n, that is, the pure state n-tangle 关10,8兴. We
2
and of the purity P
discuss the geometrical properties of S (n)
that allow us to find the general relationship between multipartite entanglement and mixedness in terms of a spin-flip
symmetry measure. In Secs. III and IV, we examine a range
of classes of two-, three-, and four-qubit states to illustrate
the results of the Sec. II.
II. DEFINITIONS AND THE GENERAL CASE
The multiple-qubit SLOCC invariant given in Eq. 共2兲 is
most naturally defined in terms of n-qubit Stokes parameters
关17,18兴,
S i 1 . . . i n ⫽Tr共 ␳␴ i 1 丢 ••• 丢 ␴ i n 兲 ,
再
⫹
k
k...0
兲2
3
兺 兺
k,l⫽1 i k ,i l ⫽1
共 S 0 . . . ik . . . il . . . 0 兲2
3
⫺•••⫹ 共 ⫺1 兲 n
兺
i 1 , . . . ,i n ⫽0
S i2
1 . . . in
共10兲
D HS共 ␳ ⫺ ␳ ⬘ 兲 ⬅ 冑 12 Tr关共 ␳ ⫺ ␳ ⬘ 兲 2 兴 .
共11兲
In particular, the multipartite entanglement and mixedness
are related by the square of the Hilbert-Schmidt distance between the state ␳ and its corresponding spin-flipped counterpart ˜␳ :
2
˜ 兲兴
D HS
共 ␳ ⫺˜␳ 兲 ⫽ 21 Tr关共 ␳ ⫺˜␳ 兲 2 兴 ⫽ 21 关 Tr ␳ 2 ⫹Tr ˜␳ 2 ⫺2Tr共 ␳␳
˜ 兲 ⫽P共 ␳ 兲 ⫺S 2n 共 ␳ 兲 .
⫽Tr ␳ 2 ⫺Tr共 ␳␳
共12兲
Thus, we have the following relation between the chosen
measure of multipartite state entanglement and the state purity:
2
S 2n 共 ␳ 兲 ⫹D HS
共 ␳ ⫺˜␳ 兲 ⫽P共 ␳ 兲 ,
共13兲
3
兺 i兺⫽1 共 S 0 . . . i
n
3
共see Ref. 关7兴兲.
The multiple-qubit state purity 关Eq. 共10兲兴 and the entanglement 关Eqs. 共8兲 and 共9兲兴 can be related by the 共renormalized兲 Hilbert-Schmidt distance in the space of density
matrices 共arising from the Frobenius norm 关21兴兲:
共7兲
n
1
共 S 0 . . . 0 兲2⫺
2n
k⫽1
1
P共 ␳ 兲 ⫽Tr ␳ ⫽ n
2
2
i 1 , . . . ,i n ⫽0,1,2,3,
where ␴ ␮2 ⫽1, ␮ ⫽0,1,2,3, are the three Pauli matrices together with the identity ␴ 0 ⫽I 2⫻2 , and 21 Tr( ␴ ␮ ␴ ␯ )⫽ ␦ ␮ ␯ .
These directly observable parameters form an n-particle generalized Stokes tensor 兵 S i 1 , . . . ,i n 其 . Under SLOCC transformations 关5兴, density matrices undergo local transformations
described by the group SL(2,C), while the corresponding
Stokes parameters undergo local transformations described
by the isomorphic group O0 (1,3), the proper Lorentz group
关19兴, that leave the Minkowskian length unchanged. The
Minkowskian squared-norm of the Stokes tensor 兵 S i 1 . . . i n 其
provides this invariant length 关7兴 共here renormalized by the
factor 2 ⫺n for convenience兲:
2
⬅
S (n)
subscripts on these matrices either will be suppressed or replaced by more descriptive labels, the number n of qubits
being otherwise specified.
Like this measure of entanglement, the state purity P is
also naturally captured in terms of the generalized Stokes
parameters. The subgroup of deterministic local operations
and classical communications 共LOCC兲 关20兴 on qubits,
namely, the unitary group 关SU共2兲兴 of transformations on density matrices, corresponds to the subgroup of ordinary rotations 关SO共3兲兴 of Stokes parameters that preserve the Euclidean length derived from these parameters. The purity for a
general n-qubit state is the Euclidean length in the space of
multiple-qubit Stokes parameters:
兺
i 1 , . . . ,i n ⫽1
冎
共 S i1 . . . in兲2 .
共8兲
As mentioned above, this quantity can be compactly expressed in terms of density matrices, as
2
⫽Tr共 ␳ 12 . . . n˜␳ 12 . . . n 兲 ,
S (n)
共9兲
where ␳ 12 . . . n is the multiple-qubit density matrix and
˜␳ 1 . . . n ⫽( ␴ 2丢 n ) ␳ 1* . . . n ( ␴ 2丢 n ) is the spin-flipped multiplequbit density matrix. Here, we consider these positive, Hermitian, trace-one matrices as operators acting in the Hilbert
space (C 2 ) 丢 n of n qubits. For the remainder of this paper,
2
where D HS
( ␳ ⫺˜␳ ) can be understood as a measure of distinguishability between the n-qubit state ␳ and the corresponding spin-flipped state ˜␳ 关as defined in Eq. 共3兲兴.
The relation of Eq. 共13兲 can be recast as the following
simple, entirely general, relation between multipartite en2
( ␳ ) and mixedness M ( ␳ )⫽1⫺P( ␳ ):
tanglement S (n)
2
S (n)
共 ␳ 兲 ⫹M 共 ␳ 兲 ⫽I 共 ␳ ,˜␳ 兲 ,
共14兲
2
where we have introduced I( ␳ ,˜␳ )⬅1⫺D HS
( ␳ ⫺˜␳ ), which
measures the indistinguishability of the density matrix ␳
from the corresponding spin-flipped state ˜␳ ; I( ␳ ,˜␳ ) is clearly
also a measure of the spin-flip symmetry of the state.
For pure states, M ( ␳ )⫽M ( 兩 ␺ 典具 ␺ 兩 )⫽0, and we have
2
S (n)
␺ 典具 ˜␺ 兩 兲 .
共 兩 ␺ 典具 ␺ 兩 兲 ⫽I 共 兩 ␺ 典具 ␺ 兩 , 兩 ˜
共15兲
2
( ␳ ) is also seen to
In this case, the Minkowskian length S (n)
be equal to the n-tangle ␶ (n) , as mentioned above:
022318-2
PHYSICAL REVIEW A 68, 022318 共2003兲
ENTANGLEMENT, MIXEDNESS, AND SPIN-FLIP . . .
where w苸 关 0,1兴 , and mixtures of three Bell states,
2
S (n)
␺ 典具 ˜␺ 兩 兲兴 ⫽ 具 ␺ 兩 共 兩 ˜␺ 典具 ˜␺ 兩 兲 兩 ␺ 典
共 兩 ␺ 典具 ␺ 兩 兲 ⫽Tr关共 兩 ␺ 典具 ␺ 兩 兲共 兩 ˜
⫽ 兩 具 ␺ 兩 ˜␺ 典 兩 2 ⫽ ␶ (n) ,
␳ 3 ⫽w 1 P 关 兩 ⌽ ⫹ 典 ]⫹w 2 P 关 兩 ⌽ ⫺ 典 ]⫹w 3 P 关 兩 ⌿ ⫹ 典 ],
共16兲
共21兲
where 兩 ˜␺ 典 ⬅ ␴ 2丢 n 兩 ␺ * 典 and ␶ (n) ⫽ 兩 具 ␺ 兩 ˜␺ 典 兩 2 is the pure state
n-tangle. Thus, we also see that for pure states
where w i 苸 关 0,1兴 , and w 1 ⫹w 2 ⫹w 3 ⫽1, were both considered in Ref. 关4兴. Since both cases fall within the same larger
class, the Bell-decomposable states
␶ (n) ⫽I 共 ␳ ,˜␳ 兲 ,
␳ BD ⫽w 1 P 关 兩 ⌽ ⫹ 典 ]⫹w 2 P 关 兩 ⌽ ⫺ 典 ]⫹w 3 P 关 兩 ⌿ ⫹ 典 ]
共17兲
that is, the n-tangle coincides with the degree of spin-flip
symmetry.
From Eq. 共14兲, one obtains an exact complementarity relation for those classes of n-qubit states, whether pure or
mixed, for which ␳ ⫽˜␳ 关and thus I( ␳ ,˜␳ )⫽1]:
2
S (n)
共 ␳ 兲 ⫹M 共 ␳ 兲 ⫽1,
共18兲
since then D HS( ␳ ⫺˜␳ )⫽0. For the special case of pure states
M ( ␳ )⫽0; this result can be understood as expressing the
fact, familiar from the set of Bell states, that pure states of
more than one qubit invariant under n-qubit spin flipping
have full n-qubit entanglement. The Bell state 兩 ⌿ ⫹ 典 is the
most obvious example from this class. By contrast, for those
states ␳ that are fully distinguishable from their spin-flipped
counterparts, ˜␳ , i.e., for which I( ␳ ,˜␳ )⫽0, one finds instead
2
S (n)
共 ␳ 兲 ⫹M 共 ␳ 兲 ⫽0,
III. TWO-QUBIT SYSTEMS
Relations between entanglement and mixedness have previously been found for limited classes of two-qubit states
using the tangle ␶ (2) as an entanglement measure 关2– 4兴. In
particular, for two important classes of states, the Werner
states and the maximally entangled mixed states, it was
found analytically that as mixedness increases, entanglement
decreases 关2兴. By exploring more of the Hilbert space of
two-qubit systems by numerical methods, it was also found
that a range of other states exceed the ratio of entanglement
to mixedness present in the class of Werner states including,
in particular, the maximally entangled mixed states. We now
use our results above to provide further insight into the relationship between entanglement and mixedness in two-qubit
systems, before going on to examine larger multiqubit systems.
Mixtures of two Bell states,
共20兲
共22兲
where w i 苸 关 0,1兴 and w 1 ⫹w 2 ⫹w 3 ⫹w 4 ⫽1, we consider
them via this general case. For these states, the general relation, Eq. 共14兲, tells us that, within this class, as the entanglement increases, the mixedness must decrease, as expressed
by Eq. 共18兲, since all of these states are symmetric under the
2
( ␳ BD ).
spin-flip operation: ␳ ⫽˜␳ . That is, M ( ␳ BD )⫽1⫺S (n)
These states can be studied in particular cases by performing
Bell-state analysis, say on qubit pairs within pure states of
three particles, and can characterize, for example, the ensemble of states transmitted during a session of quantum
communication using quantum dense coding. The Werner
states 关22兴, such as
␳ Werner⫽w P 关 兩 ⌽ ⫹ 典 ]⫹
共19兲
implying that both the entanglement and the mixedness are
2
⫽0 and M ( ␳ )⫽0, since both quantities are nonzero, S (n)
negative and sum to zero. An arbitrary n-fold tensor product
of states 兩 0 典 and 兩 1 典 is an n-qubit example from this class. A
third noteworthy case is that when the mixedness and indistinguishability are nonzero and equal: M ( ␳ )⫽I( ␳ ,˜␳ ). In that
2
is obviously zero. The fully
case the entanglement S (n)
mixed n-qubit state 共described by identity matrix normalized
to trace unity兲 is an example from this last class.
␳ 2 ⫽w P 关 兩 ⌽ ⫹ 典 ]⫹ 共 1⫺w 兲 P 关 兩 ⌽ ⫺ 典 ],
⫹w 4 P 关 兩 ⌿ ⫺ 典 ],
1⫺w
I 丢I ,
4 2 2
共23兲
which describe weighted mixtures of Bell states and fully
mixed states, where w苸 关 0,1兴 , also have the spin-flip symmetry property, that is, I( ␳ Werner ,˜␳ Werner)⫽1. Because of
their symmetry, all the above examples obey the exact
complementarity relation, Eq. 共18兲: entanglement and mixedness are seen to be strictly complementary. However, not all
two-qubit states possess full symmetry under spin flips.
Therefore, only the full three-way relation 关Eq. 共14兲兴 will
hold in general.
Let us now proceed further by considering two example
classes where the state is not invariant under the spin-flip
operation, one class of pure states and one class of mixed
states. For the two-qubit generalized Schrödinger cat states
defined by
兩 ⌽ 共 ␣ 兲 典 AB ⫽ ␣ 兩 00典 ⫹ 冑1⫺ ␣ 2 兩 11典 ,
共24兲
where ␣ 苸 关 0,1兴 , considered in Ref. 关4兴, clearly ˜␳ ⫽ ␳ in general. For them, one has for the state distinguishability
D HS( ␳ ⫺˜␳ )⫽ 兩 2 ␣ 2 ⫺1 兩 , and for the indistinguishability,
2
⫽4 ␣ 2 (1⫺ ␣ 2 ). Since these states are pure,
I( ␳ ,˜␳ )⫽1⫺D HS
the mixedness M ⫽0 and the relation given by Eq. 共14兲 共with
n⫽2) reduces to an expression for state entanglement 关Eq.
共15兲兴, that is,
2
S (2)
共 ␳ 兲 ⫽ ␶ (2) ⫽4 ␣ 2 共 1⫺ ␣ 2 兲 ,
共25兲
in accordance with Eq. 共17兲. This shows the entanglement 共in
this case coinciding with the tangle ␶ (2) ) to be parametrized
by ␣ 2 , reaching its maximum at the maximally entangled
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PHYSICAL REVIEW A 68, 022318 共2003兲
JAEGER et al.
pure state of this class, 兩 ⌿ ⫹ 典 , as it should. We will see below
that similar expressions obtain for larger integral-n
Schrödinger cat states.
Now consider the class of two-qubit ‘‘maximally entangled mixed states,’’ as defined in Ref. 关3兴 as the states
possessing the greatest possible amount of entanglement
among those states having a given degree of purity. These
can be written as
␳ mems共 ␥ 兲 ⫽ 21 关 2g 共 ␥ 兲 ⫹ ␥ 兴 P 关 兩 ⌽ ⫹ 典 ]⫹ 21 关 2g 共 ␥ 兲 ⫺ ␥ 兴 P 关 兩 ⌽ ⫺ 典 ]
兩 W g 典 is manifestly distinguishable from 兩 W̃ g 典 , so the indistinguishability I( ␳ Wg ,˜␳ Wg ) is clearly zero as well, in accordance with Eq. 共14兲 with M ( ␳ Wg )⫽0. Since this three-qubit
state is pure, this is an instance of Eq. 共19兲, the uniquely
three-qubit entanglement ␶ (3) is seen to be zero. Furthermore, for the two-qubit subsystems of this three-qubit system, which are in mixed states, the entanglement measures
2
, calculated using their two-qubit reduced
for n⫽2, S (2)
states, ␳ (AB) , ␳ (BC) , ␳ (AC) , are
共26兲
2
S (AB)
共 ␳ (AB) 兲 ⫽4 兩 ␣ 兩 2 兩 ␤ 兩 2 ⫽C 2 共 ␳ AB 兲 ,
where g( ␥ )⫽ 31 for 32 ⬎ ␥ ⬎0 and g( ␥ )⫽ ␥ /2 for ␥ ⭓ 32 关3兴.
2
For these states, we find that I( ␳ ,˜␳ )⫽1⫺D HS
( ␳ ⫺˜␳ )
⫽4g( ␥ ) 关 1⫺g( ␥ ) 兴 , so that
2
S (BC)
共 ␳ (BC) 兲 ⫽4 兩 ␤ 兩 2 兩 ␥ 兩 2 ⫽C 2 共 ␳ BC 兲 ,
⫹ 关 1⫺2g 共 ␥ 兲兴 P 关 兩 01典 ],
2
S (2)
共 ␳ mems兲 ⫹M 共 ␳ mems兲 ⫽4g 共 ␥ 兲关 1⫺g 共 ␥ 兲兴 .
共27兲
The formal similarity of the expressions for degree of symmetry I( ␳ ,˜␳ ) in the above two cases, the generalized Schrödinger cat and maximally entangled mixed states, is noteworthy, with the degree of spin-flip symmetry of the state being
of the same form but with different arguments, ␣ 2 and g( ␥ ),
respectively. This suggests that the maximally entangled
mixed states of Eq. 共26兲 could substitute for Schrödinger cat
states in situations requiring such entanglement and symmetry, but where decoherence is unavoidable.
The above example classes have previously been used to
explore the relationship between entanglement and mixedness of states for a given ability to violate the Bell inequality
关3,4兴. The behavior of these properties for Belldecomposable states was previously shown 关4兴 to differ in
that context from that of the Werner and maximally entanglement mixed states 关3兴. Therefore, no general conclusion
could be drawn about the relationship of entanglement and
mixedness for the general two-qubit state for a given amount
of Bell inequality violation. That is presumably because Bell
inequality violation indicates a deviation from classical behavior 共see, for example, Ref. 关23兴兲, rather than an inherent
property of the state. The results here show instead that the
degree of state symmetry governs the relationship of entanglement and mixedness.
2
S (AC)
共 ␳ (AC) 兲 ⫽4 兩 ␣ 兩 2 兩 ␥ 兩 2 ⫽C 2 共 ␳ AB 兲 ,
each taking the value 49 for the traditional W state 关that is, the
state of Eq. 共28兲 with ␣ ⫽ ␤ ⫽ ␥ ⫽ 冑 13 ].
On the other hand, for generalized GHZ 共or three-cat兲
states, defined by
兩 GHZg 典 ⫽ ␣ 兩 000典 ⫹ 冑1⫺ ␣ 2 兩 111典 ,
兩 W g 典 ⫽ ␣ 兩 100典 ⫹ ␤ 兩 010典 ⫹ ␥ 兩 001典 ,
2
⫽Tr共 ␳ GHZg˜␳ GHZg 兲 ⫽4 兩 ␣ 兩 2 共 1⫺ 兩 ␣ 兩 2 兲 ⫽ ␶ (3)
S (3)
共32兲
关from Eq. 共16兲, and analogously for two-cat states兴, while
2
2
2
2
⫽C (BC)
⫽C (AC)
⫽0 for all values of ␣ and S (3)
⫽ ␶ (3)
C (AB)
can reach the maximum value, 1 关for the three-cat state with
␣ ⫽ 冑 12 , see Eq. 共25兲兴.
Now consider the following class of mixed three-qubit
states:
␳ m3 ⫽w P 关 兩 000典 ]⫹ 共 1⫺w 兲 P 关 兩 111典 ],
共33兲
with w苸 关 0,1兴 . Being mixed, these states obey the general
relation, Eq. 共14兲, rather than its special case, Eq. 共15兲. For
these states the mixedness is
2
M 共 ␳ m3 兲 ⫽1⫺Tr共 ␳ m3
兲 ⫽2w 共 1⫺w 兲 ,
共34兲
and the multipartite entanglement measure takes the value
2
S (3)
共 ␳ m3 兲 ⫽Tr共 ␳ m3˜␳ m3 兲 ⫽2w 共 1⫺w 兲 ,
共35兲
as well. In accordance with Eq. 共14兲, we also have
I 共 ␳ m3 ,˜␳ m3 兲 ⫽4w 共 1⫺w 兲 .
共36兲
共28兲
2
In this case, we see that both M and S (3)
vary proportionally
to the degree of spin-flip symmetry.
Note that the states ␳ m3 can be viewed as three-qubit reduced states arrived at by partial tracing out one of the qubits
from the class of generalized four-qubit Schrödinger cat pure
states 共taking ␣ 2 ⬅w),
共29兲
兩 4catg 典 ⫽ ␣ 兩 0000典 ⫹ 冑1⫺ ␣ 2 兩 1111典 ,
with 兩 ␣ 兩 2 ⫹ 兩 ␤ 兩 2 ⫹ 兩 ␥ 兩 2 ⫽1, using Eq. 共15兲, we have
2
⫽Tr共 ␳ Wg˜␳ Wg 兲 ⫽0,
S (3)
共31兲
we find the entanglement behaves oppositely, that is
IV. THREE-QUBIT AND FOUR-QUBIT SYSTEMS
Let us first use the relation of Eq. 共14兲 with n⫽3 to study
three-qubit states. For pure states, this relation takes the special form given in Eq. 共15兲 and provides the entanglement
2
measure S (3)
directly in terms of the spin-flip symmetry measure I( ␳ ,˜␳ ). For the generalized W state with complex amplitudes,
共30兲
022318-4
共37兲
PHYSICAL REVIEW A 68, 022318 共2003兲
ENTANGLEMENT, MIXEDNESS, AND SPIN-FLIP . . .
2
with ␣ 苸 关 0,1兴 , for which the entanglement measure S (4)
takes the values
2
S (4)
⫽Tr共 ␳ 4catg˜␳ 4catg 兲 ⫽4 ␣ 2 共 1⫺ ␣ 2 兲 .
共38兲
Because these states are pure, the mixedness M ( ␳ ) is zero;
we have that the four-qubit entanglement is the degree of
2
⫽ ␶ (4) ( ␳ 4catg )⫽I( ␳ 4catg ,˜␳ 4catg ), and
spin-flip symmetry S (4)
2
2
that S (3) ( ␳ m3 )⫽S (4) ( ␳ 4catg )/2⫽ 21 ␶ 4 is half that value, as is
the three-qubit mixedness.
tite entanglement, as described by a recently introduced measure, and state mixedness were seen to be complementary
within classes of states possessing the same degree of spinflip symmetry. For pure states, the value of this multipartite
entanglement measure, the degree of spin-flip symmetry, and
the n-tangle were seen to coincide. These results can be expected to be particularly useful for the study of quantum
states used for quantum information processing in the presence of decoherence.
V. CONCLUSIONS
ACKNOWLEDGMENTS
By considering multipartite entanglement and mixedness
together with the degree of symmetry of quantum states under the n-qubit spin-flip transformation, a general relation
was found between these fundamental properties. Multipar-
We are grateful to the National Science Foundation
共NSF兲; the Center for Subsurface Imaging Systems 共CenSSIS兲, an NSF Engineering Research Center; and the Defense Advanced Research Projects Agency 共DARPA兲.
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