In a recent paper, Rungta et. al. [Phys. Rev. A, 64, 042315, 2001] introduced
a measure of mixed-state entanglement called the I-concurrence for arbitrary
pairs of qudits. We find an exact formula for an entanglement measure closely
related to the I-concurrence, the I-tangle, for all mixed states of two qudits
having no more than two nonzero eigenvalues. We use this formula to provide a
tight upper bound for the entanglement of formation for rank-2 mixed states of
a qubit and a qudit.Comment: 5 pages, uses amsthm and mathrsf

G. Vidal

W. K. Wootters

English

arXiv

We provide an easily computable formula for a bipartite mixed-state entanglement measure. Our formula can be applied to readily calculate the entanglement for any rank-2 mixed state of a bipartite system. We use this formula to provide a tight upper bound for the entanglement of formation for rank-2 states of a qubit and a qudit. We also outline situations where our formula could be applied to study the entanglement properties of complex quantum systems

Osborne, T. M.

University of Queensland eSpace

Entanglement measure for rank-2 mixed states
Tobias J. Osborne*
Centre for Quantum Computer Technology and Department of Physics, The University of Queensland 4072,
Brisbane, Queensland, Australia
Received 7 May 2002; published 12 August 2005
We provide an easily computable formula for a bipartite mixed-state entanglement measure. Our formula can
be applied to readily calculate the entanglement for any rank-2 mixed state of a bipartite system. We use this
formula to provide a tight upper bound for the entanglement of formation for rank-2 states of a qubit and a
qudit. We also outline situations where our formula could be applied to study the entanglement properties of
complex quantum systems.
DOI: 10.1103/PhysRevA.72.022309 PACS numbers: 03.67.Mn, 03.65.Ud
Quantum entanglement is now commonly believed to be a
type of physical resource whose manipulation is critical for
the success of a majority of quantum-information processing
tasks. There is also some evidence that the theory of en-
tanglement may provide additional insights into the physics
of complex quantum systems. For this reason, the formula-
tion of a good way to measure entanglement has become a
guiding problem in quantum-information science.
For bipartite quantum systems, it is relatively straightfor-
ward to propose good mixed-state entanglement measures.
However, the evaluation of such measures typically involves
difficult minimizations over high-dimensional spaces. As a
result, the development of an easily computable formula for
a good entanglement measure has become an immediate pri-
ority.
In this paper we provide an easily computable formula for
a good bipartite mixed-state entanglement measure, the I
tangle proposed by Rungta et al. 1. In particular, we de-
velop a simple procedure to calculate the mixed-state en-
tanglement for general rank-2 mixed states of an arbitrary
bipartite quantum system.
The structure of this paper is as follows. We begin by
reviewing two entanglement measures, the concurrence and
the tangle, for a pair of qubits. The main result of this paper,
a formula for the I tangle for rank-2 mixed states, is then
established. We also prove a corollary of the main result, an
upper bound for the entanglement of formation of a rank-2
mixed state of a qubit and a qudit. We conclude by outlining
situations where our formula may be applied to study the
entanglement for complex quantum systems.
Before we discuss the I tangle, we introduce the concur-
rence, a mixed-state entanglement measure for states of a
pair of qubits AB 2–4. The definition of the concurrence
makes use of a specific transformation on density operators,
the spin-flip operation, which is defined as follows. Consider
an arbitrary mixed state  of AB. We define the spin-flip of 
to be
˜ tr†I  I − A
†
 I − I  B
† + †, 1
where A=trB and B=trA denote the reduced density
operators for subsystems A and B, respectively. We have
included the trace and Hermitian adjoint terms so that the
spin-flip operation is defined for arbitrary operators acting
on AB. The formula for the spin flip is applicable to arbi-
trary bipartite systems, in which case it is called the univer-
sal state inverter 1.
The spin-flip operation Eq. 1 on a pair of qubits is an
example of an antilinear operation. To be more precise, con-
sider a pure state = . The spin-flip operation, when
applied to this state, is equivalent to the expression ˜
= ˜ ˜ , where
˜  = y  y*, 2
and where y is expressed in the computational basis as
i
0
0
−i, and the complex conjugation is taken in the computa-
tional basis. The operation in Eq. 2 is clearly an antilinear
operator 5. The definition of the spin flip as an antilinear
operator extends, via linearity, to all mixed states, ˜= ,
where we have added the arrows above the antilinear opera-
tor  representing the spin flip to indicate the direction in
which it acts. It is worth noting that the description of the
spin flip Eq. 1 in terms of an antilinear operator  is spe-
cific to two qubits.
For pure states = , the concurrence C of  is de-
fined to be C= 	 ˜ 	=
˜. When the state  of the
two qubits is mixed, the concurrence C is defined to be a
minimum over all pure-state decompositions pi , i of :
C = min
pi,i

i
pi	i˜ i	 . 3
It is convenient to introduce another entanglement measure
closely related to the concurrence, the tangle  6, which is
also defined as a minimization over pure-state decomposi-
tions:
 = min
pi,i

i
pi	i˜ i	2. 4
The squared concurrence satisfies the inequality C2,
which follows from the convexity of 	i ˜ i	2=C2i. It
turns out that the reverse inequality also holds, so that the
tangle is equal to the square of the concurrence, 
=C2 7. The reverse inequality may be established by*Electronic address: tjo@maths.uq.edu.au
PHYSICAL REVIEW A 72, 022309 2005
1050-2947/2005/722/0223094/$23.00 ©2005 The American Physical Society022309-1
noting that there exists a decomposition pi , i achieving
the minimum in Eq. 3 which has the property that Ci
=C j 3. The inequality follows from substituting this
decomposition into the expressions for  and C2. A simple
formula for the concurrence of two qubits is known 3,
C = max0,1 − 2 − 3 − 4 , 5
where the i are the square roots of the singular values, in
decreasing order, of the matrix ˜.
We now focus our attention on the general case of two
d-dimensional quantum systems or qudits. For a pair of qu-
dits AB we use a variant of the I concurrence of Rungta et al.
1 to measure the entanglement for mixed states of A and B.
The I concurrence is defined via Eq. 1 1,
C = min
pi,i

i
pi
i˜ii , 6
where i= ii. The entanglement measure we use is a
generalization of the tangle, the I tangle, defined by
 = min
pi,i

i
pii˜ii . 7
The I concurrence and the I tangle are good mixed-state
entanglement measures because they satisfy the standard
properties usually regarded as essential for a good entangle-
ment measure see, for example, 8,9. The inequality C2
 may be established, by convexity, as for two qubits. Be-
cause the equal-entanglement decomposition only exists for
pairs of qubits, the I tangle is not, in general, equal to the
square of the I concurrence. Based on the results of this
paper, and the calculations of the I concurrence for isotropic
states 10, we feel that the I tangle, as defined by a minimi-
zation, is the proper generalization of the tangle Eq. 4.
The universal state inverter Eq. 1 may be expressed in
terms of another formula which will be most useful in the
following. Before we write down this formula, however, we
need to introduce some definitions. Let iA and jB denote
the computational basis states for subsystems A and B, with
dimensions dA and dB, respectively. For an arbitrary pair
iA , iA, jB , jB of the computational basis states of
AB we set up the projectors PAii= iAi+ iAi, PBj j
= jBj+ jBj, and Q= PAii PBj j, where 
= i , i , j , j. Consider the object =QQ. The operator
 is a positive operator supported on a 2	2 subspace of the
Hilbert space of AB spanned by ij , ij , ij , ij. In this
way we can think of  as a subnormalized state of two
qubits. The two-qubit spin flip, when applied to , gives
˜ = 



 = 
y
 yQQ*y  y , 8
where =Q is the antilinear operator representing the
spin-flip operation on the 2	2 subspace, and y is naturally
defined on the two-dimensional subspaces of A and B, re-
spectively. Using these definitions we can write an alterna-
tive formula for the universal state inverter,
˜ = 

, 9
where the sum over  runs over all of the dAdA−1 /
2dBdB−1 /2 possible choices of pairs of computational
basis states. The reader may verify that Eq. 9 follows from
the expression of the universal state inverter as a tensor prod-
uct of two superoperators of the form P T. See 1 for fur-
ther details.
It is convenient, at this point, to introduce two quantities
that will simplify the statement of our main result. Let  be a
density operator for a pair of qudits having no more than two
nonzero eigenvalues. We may write  in terms of its eigen-
vectors,
 = pv1v1 + 1 − pv2v2 . 10
Using these eigenvectors we construct the tensor
Tijkl = tr
ij
˜kl , 11
where 
ij = viv j. We also construct the real symmetric 3
	3 matrix Mij whose independent entries are given by
M11 =
1
4
T1221 +
1
2
T1122 +
1
4
T2112,
M12 =
i
4
T1221 −
i
4
T2112,
M13 =
1
4
T1121 −
1
4
T2122 +
1
4
T1112 −
1
4
T1222,
M22 = −
1
4
T1221 +
1
2
T1122 −
1
4
T2112,
M23 =
i
4
T1121 −
i
4
T1112 +
i
4
T2122 −
i
4
T1222,
M33 =
1
4
T1111 −
1
2
T1122 +
1
4
T2222. 12
The entries of M will be shown to be real in the following.
We now have all the necessary ingredients required for
the statement of our main result.
Theorem 1. Let  be any density operator for a pair AB of
qudits, of dimensions dA and dB, respectively, having no
more than two nonzero eigenvalues. The I tangle  between
A and B is given by the expression
 = tr˜ + 2min1 − tr2 , 13
where min is the smallest eigenvalue of the matrix M de-
fined by Eq. 12.
It is worth noting that the formula Eq. 13 for the I tangle
is easy to compute for all rank-2 mixed states of a pair of
qudits.
Proof. The method we use to prove this theorem is similar
to that employed by Hill and Wootters 2.
Consider an arbitrary pure state  which can be written
as a linear combination of the two eigenvectors of , 
TOBIAS J. OSBORNE PHYSICAL REVIEW A 72, 022309 2005
022309-2
=c1v1+c2v2. The I tangle of  is given by the expression
 = ˜ = 

 , 14
where = , and we have used Eq. 9 to rewrite the
spin flip in terms of the antilinear operators . Each of the
terms in the sum over  may be written as a trace,
 = 

tr** , 15
where ij =cicj
* is the density matrix of  expressed in the
v1 , v2 basis, and ij

= viv j.
The function on the right-hand side RHS of Eq. 15 can
be extended via linearity to a function f of all 2	2 density
matrices  expressed in terms of the v1 , v2 basis, i.e.,
f=tr**. The function f has the property that it
is equal to the I tangle for all pure states , f=.
Any 2	2 density operator  may be expressed in terms
of the Pauli matrices via the operator expansion, =1/2I
+r ·, where ri=tri. Substituting this expansion into
the expression for f gives the quadratic form
f = 1
4
tr + 
j
rjLj + 
j,k
rjrkMjk, 16
where =*,
Lj = tr j , 17
and
Mjk = 

tr j*k* . 18
Each of the terms in the sum over  in Eq. 18 is a real
symmetric matrix, so that M is a real symmetric matrix. It
may be straightforwardly verified that the entries of M are
given by Eq. 12.
For the rank-2 density operator , the state space of the
system AB can be considered to be the space of all convex
combinations of superpositions of v1 and v2. If a particu-
lar state  of AB is pure, its corresponding 2	2 density
operator in the v1 , v2 basis, =1/2I+r ·, satisfies the
condition r2=1. In this way, we can think of the entire state
space as the Bloch sphere where the poles are the eigenvec-
tors v1 and v2. A particular decomposition of  may be
viewed as the weighted sum of points on the surface of the
Bloch sphere, where  lies at the center of mass of the
weighted sum. The function f is defined on the entire state
space r21.
When the bipartite system AB is a pair of qubits, there is
only one term in the sum Eq. 9, and f reduces to the qua-
dratic form that Hill and Wootters 2 study. In this case, the
eigenvalues of M are given by ±1/2det  and 1/4tr*,
which means that f is convex along two directions and con-
cave along a third. In general, the matrix M will have three
positive eigenvalues, so that f is typically convex.
For the purposes of this proof it is essential that a qua-
dratic form g be constructed which agrees with f on pure
states  which has the additional property that it is convex
along two directions and linear along a third. A function g
which has these properties may be constructed from f as
follows:
g  f − minr2 − 1 , 19
where min is the smallest eigenvalue of the matrix M. This
function is a quadratic form,
g = K + 
j
rjLj + 
j,k
rjrkNjk, 20
where N=M −minI, and K=1/4tr+min. The matrix N
that defines the quadratic form g has two positive eigenval-
ues and one zero eigenvalue so that g is convex along two
directions and linear along the third. The quadratic form g
has the additional property that it is equal to f for pure states
, r2=1.
At this point we recall a theorem due to Uhlmann 11,12,
which concerns functions of density matrices expressed as
minimisations over all pure-state decompositions.
Theorem 2. Let G be a positive continuous real-valued
function defined on pure states. The function G, defined for
all mixed states , given by
G = min
pi,i

i
piGi , 21
where the minimization runs over all pure-state decomposi-
tions of , pi , , is the largest convex function which
agrees with G on pure states =.
The I tangle is expressed as a minimization over all pure-
state decompositions of a density operator, so if we could
find the largest convex function that agrees with f = on pure
states  it is guaranteed to be equal to the I tangle for all
mixed states. We claim that g is precisely this function.
Assume that there is a convex function g which agrees
with f on pure states but which is larger than g for some
density operator . Consider the line running through  along
which g grows linearly. The direction along which g grows
linearly is given by the eigenvector of N with eigenvalue 0.
Let the points on the surface of the Bloch sphere at either end
of this line be the pure states  and , respectively, so that
=q+ 1−q for some q, 0q1. Convexity of g im-
plies that
g qg + 1 − qg = qf + 1 − qf = g ,
22
which is a contradiction. This implies that g is the largest
convex function that takes the values  on the set of all
pure states. Therefore, Theorem 2 shows that g is equal to
the I tangle.
The expression for the I tangle  may be simplified
by noting that, for rank 2 , 1− r2=21−tr2. Note,
also, that f=tr˜. Hence we can write =tr˜
+2min1−tr2. 
The decomposition that achieves the minimum for the I
tangle Eq. 13 consists of two terms. In contradistinction to
the case of two qubits, the minimizing decomposition will, in
general, consist of terms with differing values of . This is
because the surfaces of constant g will typically be curved,
ENTANGLEMENT MEASURE FOR RANK-2 MIXED STATES PHYSICAL REVIEW A 72, 022309 2005
022309-3
so that the trick of Hill and Wootters cannot be applied see
2 for the construction of the minimizing decomposition
when the surfaces of constant g are elliptic cylinders. The
construction of the minimizing decomposition follows from
observing that the function g has the property that it grows
linearly in one direction. Consider the line parallel to the
eigenvector of N, whose associated eigenvalue is zero, which
passes through the density operator . The density operator 
may be written as a convex sum of the two pure states 1
and 2 which lie at either end of the line, =q111
+q222. Because the I tangle is convex, we obtain the
inequality
 q11 + q22 . 23
However, =g varies linearly in this direction, so that the
inequality in Eq. 23 is actually an equality.
When one of the subsystems of the bipartite system AB is
a qubit it is possible to obtain a relation between the I tangle
 and the entanglement of formation F. For pure states  of
AB the entanglement of formation is given in terms of the I
tangle via
F = E„… , 24
where Ex=H1/2+1/2
1−x, and H is the binary entropy
function Hx=−x log x− 1−xlog1−x, where the loga-
rithm is taken to base 2. The function E is concave and
monotone increasing. If we consider the minimizing decom-
position qi , i we constructed in the previous paragraph,
we obtain the chain of inequalities
F q1E„1… + q2E„2… E„… , 25
where the first inequality follows from the definition of the
entanglement of formation, and the second from the fact Eg
is concave along the line passing through the pure states i.
This statement is the content of the following corollary.
Corollary 1. For rank-2 mixed states  of a qubit A and a
qudit B, the entanglement of formation F of  satisfies the
inequality
F H12 + 12
1 −  . 26
Numerical experiments indicate that the expressions on
the LHS and RHS of Eq. 26 usually differ only by about
10−4, so that the inequality is typically very close to an equal-
ity; it is not, however, an equality.
Our formula for the I tangle may be immediately applied
to study the entanglement for a wide class of complex quan-
tum systems. For example, consider a pure state  of an
n+1-partite quantum system AB1B2¯Bn, where A is a qu-
bit and Bj are arbitrary quantum systems. Let  be the state
found by tracing out the qubit A. The I-tangle formula Eq.
13 may be used to study the mixed-state entanglement be-
tween any bipartition of the n parties B1B2¯Bn. This type of
configuration can arise in many situations such as a qubit
interacting with n modes of an electromagnetic field, and
most lattice models in condensed matter physics. In particu-
lar, it may be possible to use Eq. 13 to provide insight into
the scaling of entanglement at a quantum phase transition,
along the lines of 13,14.
I would especially like to thank Carl Caves, Michael
Nielsen, and Bill Wootters for their help and for inspirational
discussions which led to this work. I would also like to thank
Carl Caves for pointing out the substantially simpler proof of
Eq. 9. Thanks also to Michael Bremner, Chris Dawson,
Jennifer Dodd, Alexei Gilchrist, Duncan Mortimer, and
Armin Uhlmann for helpful and stimulating discussions. This
work has been funded, in part, by an Australian Postgraduate
Award.
1 P. Rungta, V. Bužek, C. M. Caves, M. Hillery, and G. J. Mil-
burn, Phys. Rev. A 64, 042315 2001.
2 S. Hill and W. K. Wootters, Phys. Rev. Lett. 78, 5022 1997.
3 W. K. Wootters, Phys. Rev. Lett. 80, 2245 1998.
4 W. K. Wootters, Quantum Inf. Comput. 1, 27 2001.
5 An antilinear operator  is an operator which satisfies
c11+c22=c1
*1+c2
*2.
6 It should be noted that our definition of the tangle differs from
the definition used in the literature, where the tangle is simply
defined to be equal to the squared concurrence, =C2. As we
show, our definition is equivalent to the standard one for two
qubits. We use this alternative definition because it naturally
generalizes to a quantity that is different from the squared
concurrence for pairs of qudits.
7 The equality of the tangle and the squared concurrence was
pointed out by Michael Nielsen private concurrence.
8 V. Vedral and M. B. Plenio, Phys. Rev. A 57, 1619
1998.
9 G. Vidal, J. Mod. Opt. 47, 355 2000.
10 P. Rungta and C. M. Caves private communication.
11 A. Uhlmann, Phys. Rev. A 62, 032307 2000.
12 A. Uhlmann, Open Syst. Inf. Dyn. 5, 209 1998.
13 A. Osterloh, L. Amico, G. Falci, and R. Fazio, Nature
London 416, 608 2002.
14 C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, and W. K.
Wootters, Phys. Rev. A 54, 3824 1996.
TOBIAS J. OSBORNE PHYSICAL REVIEW A 72, 022309 2005
022309-4


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