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Linking a distance measure of entanglement to its convex roof
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New Journal of Physics
The open-access journal for physics
Linking a distance measure of entanglement to its convex roof
Alexander Streltsov, Hermann Kampermann and Dagmar Bruß
Heinrich-Heine-Universität Düsseldorf, Institut für Theoretische Physik III,
D-40225 Düsseldorf, Germany
E-mail: streltsov@thphy.uni-duesseldorf.de,
kampermann@thphy.uni-duesseldorf.de and bruss@thphy.uni-duesseldorf.de
New Journal of Physics 12 (2010) 123004 (19pp)
Received 23 June 2010 Published 6 December 2010 Online at http://www.njp.org/
doi:10.1088/1367-2630/12/12/123004
Abstract. An important problem in quantum information theory is the quantification of entanglement in multipartite mixed quantum states. In this work, a connection between the geometric measure of entanglement and a distance measure of entanglement is established. We present a new expression for the geometric measure of entanglement in terms of the maximal fidelity with a separable state. A direct application of this result provides a closed expression for the Bures measure of entanglementof two qubits. We also prove that the number of elements in an optimal decomposition w.r.t. the geometric measure of entanglement is bounded from above by the Caratheodory bound, and we find necessary conditions for the structure of an optimal decomposition.
New Journal of Physics 12 (2010) 123004 1367-2630/10/123004+19$30.00
© IOP Publishing Ltd and Deutsche Physikalische Gesellschaft
Contents
1. Introduction 2
2. Definitions 3
3. Geometric measure of entanglement for mixed states 5
4. Entanglement measures for two qubits 8
4.1. Bures measure of entanglement....................................................8
4.2. Measures induced by the geometric measure of entanglement....................10
5. Optimal decompositions w.r.t. geometric measure of entanglement and consequences for closest separable states 11
5.1. Equivalence between closest separable states and optimal decompositions ... 12
5.2. Caratheodory bound................................................................13
6. Structure of optimal decomposition w.r.t. geometric measure of entanglement 13
6.1. n-partite states......................................................................14
6.2. Bipartite states ....................................................................15
6.3. Qubit-qudit states..................................................................15
6.4. Nonoptimal stationary decompositions............................................15
6.5. Stationarity on the original subspace..............................................16
7. Concluding remarks 16 Acknowledgments 17 Appendix A. Geometric measure of a convex set 17 References 19
1. Introduction
Entanglement [1] is one of the most fascinating features of quantum mechanics, and allows a new view on information processing. In spite of the central role of entanglement there does not yet exist a complete theory for its quantification. Various entanglement measures have been suggested—for an overview see [2, 3].
A composite pure quantum state ) is called entangled iff it cannot be written as a product state. A composite mixed quantum state p on a Hilbert space H = ®n=1 Hj is called entangled iff it cannot be written in the form [2, 4]
p = E Pi (®n=1 ) i) (1)
with pi > 0, i Pi = 1, and where n ^ 2 and j)) e H j.
The degree of entanglement can be captured in a function E (p) that should fulfil at least the following criteria [2]:
• E (p) ^ 0 and equality holds iff p is separable1,
• E cannot increase under local operations and classical communication (LOCC), i.e. E(A(p)) ^ E(p) for any LOCC map A.
1 Note that the distillable entanglement ED does not satisfy this criterion, i.e. it can be zero on entangled states. However it is also accepted as a measure of entanglement [2].
Figure 1. S denotes the set of separable states within the set of all quantum states Q. The state a is the closest separable state to p, w.r.t. the distance D.
These criteria are satisfied by all measures of entanglement presented in this paper. One possibility to define an entanglement measure for a mixed quantum state p is via its distance to the set of separable states [5]; for an illustration see figure 1. Another possibility to define an entanglement measure for a mixed quantum state p is the convex roof extension, in which the entanglement is quantified by the weighted sum of the entanglement measure of the pure states in a given decomposition of p, minimized over all possible decompositions. There is no a priori reason why these two types of entanglement measures should be related. In this paper, we will establish a link between them, by showing the equality between the convex roof extension of the geometric measure of entanglement for pure states and the corresponding distance measure based on the fidelity with the closest separable state. Using this result, we will also study the properties of the optimal decompositions of the given state p and its closest separable state.
Our paper is organized as follows: in section 2, we provide the definitions of the used entanglement measures. In section 3, we derive the main result of this paper, namely the equality between the convex roof extension of the geometric measure of entanglement and the fidelity-based distance measure. In section 4, we study the simplest composite quantum system, namely two qubits, give an analytical expression for the Bures measure of entanglement and consider other measures that are based on the geometric measure of entanglement. In section 5, we characterize the optimal decomposition of p (i.e. the one that reaches the minimum in the convex roof construction) from knowledge of the closest separable state and vice versa. Finally, in section 6, we derive a necessary criterion that the states in an optimal decomposition have to fulfil. We conclude in section 7.
2. Definitions
Two classes of entanglement measures are considered in this paper. The first class consists of measures based on a distance [5, 6],
Ed (p) = inf D (p, a), (2)
where D(p, a) is the 'distance' between p and a and S is the set of separable states. This concept is illustrated in figure 1. Following [2], we do not require a distance to be a metric. In this paper, we will consider for example the Bures measure of entanglement [6]
Eb (p) = min(2 - VF(p, a)), (3)
a e S _ 2
where F(p, a) = (Tr [y7^pa^p]) is Uhlmann's fidelity [7]. A very similar measure is the Groverian measure of entanglement [8, 9], defined as
Ear (p) = min^l - F(p,a). (4)
As it can be expressed as a simple function of EB, we will not consider it explicitly. Another important representative of the first class is the relative entropy of entanglement defined as [6]
Er (p) = min S (p||a), (5)
where S(p ||a) is the relative entropy,
S (p||a) = Tr [p log2 p] - Tr [p log2 a]. (6)
The second class of entanglement measures consists of convex roof measures [10]
E (p) = min ^ pE (|f», (7)
where J]i pi = 1, pi ^ 0, and the minimum is taken over all pure state decompositions of p = i pi |f) (f |. An important example of the second class is the geometric measure of entanglement EG, defined as follows [11]:
Eg (|f)) = 1 - max |(0|f)|2 , (8)
|0 )e S
Eg (p) = min £ pi Eg (|f)), (9)
where the minimum is taken over all pure state decompositions of p. Entanglement measures of this form were considered earlier in [12, 13]. Another important representative of the second class for bipartite states pAB is the entanglement of formation EF, which is for pure states p = |f) (f | defined as the von Neumann entropy of the reduced density matrix,
Ef (|f)) = -Tr [pA log2 pA], (10)
where pA = TrB[|f) (f|]. For mixed states this measure is again defined via the convex roof construction [14]:
Ef (p) = min F piEf (|f)). (11)
{pi,|fi)}
For two-qubit states analytic formulae for EF and EG are known; both are simple functions of the concurrence [11, 15].
Remember that the concurrence for a two-qubit state p is given by [15]
C(p) = max{§1 - §2 - §3 - §4, 0}, (12)
where §i, with i e {1, 2, 3, 4}, are the square roots of the eigenvalues of p • p in decreasing order, and p is defined as p = (a^ ® a^)p*(a^ ® a^).
The entanglement of formation for a two-qubit state p as a function of the concurrence is expressed as [15]
Ef(p) = h(1 + 2/1 - C(p)2), (13)
where h(x) = — x log2x — (1 — x)log2(1 — x) is the Shannon entropy. The geometric measure
of entanglement for a two-qubit state p as a function of the concurrence was shown in [11] to be /
Eg (p) = 2 (1 — \Z 1 — C (p)2). (14)
This formula was already found in [16] in a different context. For bipartite states, it is furthermore known that [6]
Ef (p) ^ Er (p), (15)
where for bipartite pure states the equal sign holds [6].
The geometric measure of entanglement plays an important role in the research on fundamental properties of quantum systems. Recently it has been used to show that most quantum states are too entangled to be used for quantum computation [17]. In [18] the authors have shown how a lower bound on the geometric measure of entanglement can be estimated in experiments. A connection to Bell inequalities for graph states has also been reported [19].
3. Geometric measure of entanglement for mixed states
In this section, we will show the main result of our paper: the geometric measure of entanglement, defined via the convex roof, see equation (9), is equal to a distance-based alternative. We introduce the fidelity of separability
Fs (p) = max F (p, a), (16)
where the maximum is taken over all separable states of the form (1).
Theorem 1. For a multipartite mixed state p on a finite dimensional Hilbert space H = ®nj=i H j the following equality holds:
Fs (p) = max EpiFs )), (17)
{pi,lfi)}
where the maximization is done over all pure state decompositions of p = J]i pi )
Proof. Remember that according to Uhlmann's theorem [20, p 411],
F(p, a) = max )|2 (18)
holds for two arbitrary states p and a, where ) is a purification of p and the maximization is done over all purifications of a, which are denoted by ).
We start the proof with equation (16). In order to find Fs (p), we have to maximize | (^ ) |2 over all purifications ) of all separable states a = J]j qj |0j)(0j |, where all |0j) are separable. The purifications of p and a can in general be written as
0 = ^ Jp'i |#)®|i), (19)
|07) = E Vqj\0 j ) ® u f i j >,
where {p/, is a fixed decomposition of p, (k|Z> = 4/ and U is a unitary on the ancillary Hilbert space spanned by the states {|i>}. To see whether all purifications of a separable state a = J]j qj |0j >(0j | are of the form given by |07>, we start with an arbitrary purification |0/7> = J]k V^k |ak> ® |k>, such that a = rk |ak> (ak| and (k|Z> = Sk,/. Further the following holds Vk |ak> = j ukj Vqj 10i>' with ukj being elements of a unitary matrix [21]. Using the last relation we get |0/7> = j 10i> ® Ii >, with |j7> = ^2kukj |k>. Thus we brought an arbitrary purification of a to the form given by |07>.
In order to find Fs (p) in the above parametrization we have to maximize the overlap 7|07>I2 over all unitaries U, all probability distributions {q«} and all sets of separable states {|0i >}.
We will now show that we can also achieve Fs(p) by maximizing the overlap |0>|2 of the purifications
>=E VP >®|i >,
|0 > = E Vqj\0. i) ®| i >
where now the maximization has to be done over all decompositions {pi, |tyi)} of the given state p, all probability distributions {qi} and all sets of separable states {|0i)}. To see how this works we write the matrix U in its elements, U = J2u Uki |k) (lI, and apply it in the overlap I (tyi|0i) I2, thus noting th/t the action of the unitary is equivalent to a transformation of the set of unnormalized states {y^ty/)} into the new set {^pi|tyi)}. The connection between the two
sets is given by the unitary: |tyi) = ^j Uij^pp'j |tyj), which is a transformation between two
decompositions of the state p, see also [20, p 103f]. The advantage of this parametrization is that now both purifications have the same orthogonal states on the ancillary Hilbert space. We now do the maximization of the overlap
|0 >| =
^ VqVP (ft 10«>
starting with the separable states {|0i)}. The optimal states can be chosen such that all terms (tyi ) are real, positive and equal to VFs dtyi)) = max|^)eS Ktyi |0)|; it is obvious that this choice is optimal. We also used the fact that for pure states |ty) it is enough to maximize over pure separable states: Fs(|ty)) = max|^)eS |(ty |0)|2. To see this, note that F(|ty) (ty |, a) = (ty |a | ty). Suppose now the closest separable state to | ty) is the mixed state a with the separable decomposition a = jqj |0j)(0j |, all |0j) being separable. Without loss of generality let |(ty|01 )| ^ |(ty|0j)| be true for all j. Then the following holds: F (|ty) (ty|, a) = (ty|a|ty) = j qj |(ty |0j)|2 ^ 12j qj |(ty |01 )|2 = |(ty |01 )|2, and thus |01) is a closest separable state to |ty).
The maximization over {10« >} gives us
max |0 >| = ^ Vq« VPt^Fs dft >). {|0j >} «
7 IOP Institute of Physics <J>deutsche physikalische Gesellschaft
Now we do the optimization over qt. Using Lagrange multipliers we obtain
^ = Jff VMffii) , (25)
\/Ek PkF (\fk >)
with the result
max |<^>|2 = pF ». (26)
{q; >} ,
It is easy to understand that this choice of {qi} is optimal when one interprets the right-hand side of equation (24) as a scalar product between a vector with entries (VptVFs (|f 1)), Vp2VFs (|f 2)), ...) and a vector with entries (Vq7, Vq2, ...). The scalar product of two vectors with given length is maximal when they are parallel.
In the last step, we do the maximization over all decompositions {pi, |fi)} of the given state p which leads to the end of the proof, namely
Fs (p) = max |(f |0)|2 = max VptFs (|f)). (27)
{pi ,|fi)}
We can generalize theorem 1 for arbitrary convex sets; the result can be found in appendix A. Using theorem 1 it follows immediately that the geometric measure of entanglement is not only a convex roof measure, but also a distance-based measure of entanglement:
Proposition 1. For a multipartite mixed state p on a finite dimensional Hilbert space H = ®n=1 H j the following equality holds:
Ea (p) = 1 - max F(p, a). (28)
Proposition 1 establishes a connection between EG and distance-based measures such as the Bures measure EB and Groverian measure EGr. All of them are simple functions of each other. In [22] the authors found the following connection between ER and EG for pure states:
Er (|f)) ^ - log2(1 - Eg (|f))). (29)
This inequality can be generalized to mixed states as follows:
Er (p) ^ max{0, - log2(1 - Eg (p)) - S (p)}, (30)
where S(p) = -Tr[p log2 p] is the von Neumann entropy of the state. Inequality (30) is a direct consequence of the following proposition.
Proposition 2. For two arbitrary quantum states p and a holds
S (p||a) > Tr [p log2 p] - log2 F(p, a). (31)
Proof. With p = Y]ipi |fi)(fi | we will estimate -Tr[p log2 a] from below:
-Tr [p log2 a] = - ^ pi(fi | log2 a |f) (32)
> - V pi log2 (fi|a|fi). (33)
Here we used concavity of the log function
log2 (fi|a|fi) ^ (fi| log2 a|fi). Using concavity again we obtain J]ipi log2 (f i |a |f i) ^ log2 J]ipi (f i |a |f i) and thus
-Tr [p log2 a] ^ - log^ pi (fi |a |fi)
= - log2 Tr [pa ]. The fidelity can be bounded from below as follows:
F (p,a) = Tr
X2 = Tr [Vpa Vp] = Tr [pa ],
where ki are the eigenvalues of the positive operator ^ Vpa Vp.
(38) □
Inequality (30) becomes trivial for states with high entropy. As a nontrivial example we consider the two-qubit state
p = p|f )(f | + (1 - p)|01)(01|, (39)
with |f) = +fa |01) + V1 - a 110). This state was called the generalized Vedral-Plenio state in [23], where the authors showed that the closest separable state a w.r.t. the relative entropy of entanglement is given by
a = (1 - p + pa) |01><01| + p (1 - a) |10><10|.
In figures 2 and 3, we show the plot of EF (dotted curve), ER (solid curve) and E = max{0, - log2 (1 - Eg (p)) - S (p)} (dashed curve) as a function of a for p = H0 and p = 10, respectively. It can be seen that E drops quickly with increasing entropy of the state and thus is nontrivial only for states close to pure states with high entanglement.
In [24, 25], the authors gave lower bounds for the relative entropy of entanglement in terms of the von Neumann entropies of the reduced states, which provide better lower bounds for Er than (30). Thus, the inequality (30) should be seen as a connection between the two entanglement measures ER and EG, and not as an improved lower bound for ER.
4. Entanglement measures for two qubits
4.1. Bures measure of entanglement
We can use proposition 1 to evaluate entanglement measures for two qubit states. From [11, 16] we know the geometric measure for two-qubit states as a function of the concurrence, see equation (14). Using this together with equation (28), we find the fidelity of separability as a function of the concurrence:
Fs (p)=max f (p, a) = 2 +v 1 -c (p)2j. (41)
Now we are able to give an expression for the Bures measure of entanglement for two-qubit states, remember its definition in equation (3).
Figure 2. Entanglement of formation EF (dotted curve), relative entropy of entanglement ER (solid curve) and E = max{0, — log2(1 — EG (p)) — S (p)} (dashed curve) of the state p = p ) (^ | + (1 — p) |01) <011 with > = |01) + V1 — a j 10) for p = 100 as a function of a.
Figure 3. Entanglement of formation EF (dotted curve), relative entropy of entanglement ER (solid curve) and E = max{0, — log2 (1 — EG (p)) — S (p)} (dashed curve) of the state p = p j^> (^ j + (1 — p) |01) (01 j with j^> = |01) + V1 — a j 10) for p = 10 as a function of a.
Proposition 3. For any two-qubit state p the Bures measure of entanglement is given by
Eb (p) = 2 - 2i
1 + yi - c (p)2
Note that for a maximally entangled state, EG = 1 and EB = 2 — V2. In order to compare these measures we renormalize them such that each of them becomes equal to 1 for maximally entangled states. We show the result in figure 4. There we also plot the Groverian measure of entanglement, see equation (4).
Figure 4. Plot of the geometric measure of entanglement EG, Bures measure of entanglement EB and Groverian measure of entanglement EGr as a function of the concurrence C for two qubit states. All measures were renormalised such that they reach 1 for maximally entangled states.
4.2. Measures induced by the geometric measure of entanglement
We consider now any generalized measure of entanglement for two-qubit states p which can be written as a function of the geometric measure of entanglement:
Proposition 4. Let f (x) be any convex function that is non-negative for x ^ 0 and obeys f (0) = 0. Then for two qubits Ef (p) = f (EG (p)) is equal to its convex roof, that is,
where the minimization is done over all pure state decompositions of p.
Proof. From [11] we know that the geometric measure of entanglement is a convex non-negative function of the concurrence, see also (14) and figure 4. As shown in [11], from convexity follows that Eg and EF have identical optimal decompositions, and every state in this optimal decomposition has the same concurrence. This observation led directly to expression (14) for Eg of two qubit states.
As f is convex, Ef also is a convex function of the concurrence. To see this we note that
Ef (p) = f (Eg(p)).
Ef (p) = min Ç piEf » = f (i(l 1 - C (p)2^
convexity of EG implies
where we defined EG (C) = 1 (1 — V1 — C2). As f (x) is convex, non-negative and f (0) = 0, it also must be monotonically increasing for x ^ 0. Thus we have
f E pC,^ < f pEg(C^.
Now we can use convexity of f to obtain
f (eg^E PiC^ ^ E Ptf (Eg (Ci )).
Defining Ef (C) = f (EG (C)) = f ( 1 (1 - V1 - C2)) the inequality above becomes
Ef (EPC^j < EP-Ef (Ci).
This proves that Ef (C) is a convex function of the concurrence. Using the same argumentation as was used in [11] to prove expression (14) we see that (44) must hold. □
As an example consider the Bures measure of entanglement, which can be written as EB (p) = Ef (p) with the convex function f = 2 — 2^1 — EG (p). Using proposition 4, we see that for two qubits the Bures measure of entanglement is equal to its convex roof.
However, this might not be the case for a general higher-dimensional state p. To see this assume that EB(p) is equal to min i piEB (|tyi)). This means that VFs (p) is equal to max i p^Fs (|tyi)). On the other hand, from theorem 1 we know that
Fs (p) = max^piFs (ty)), (49)
and using monotonicity and concavity of the square root, we find
VF (p) = max^^ pF (|tyi)) ^ max ^ p^Fs (ty)). (50)
The Bures measure of entanglement is equal to its convex roof if and only if the inequality (50) becomes an equality for all states p.
Finally we note that any entanglement measure Eh defined as Eh(p) = minaGSh(F(p, a)) with a monotonically decreasing non-negative function h, h (1) = 0, becomes Eh (p) = h(Fs (p)), and can be evaluated exactly for two qubits using proposition 1. An example of such a measure is the Bures measure of entanglement.
5. Optimal decompositions w.r.t. geometric measure of entanglement and consequences for closest separable states
Let p be an n-partite quantum state acting on a finite-dimensional Hilbert space H = ®ni=iHi of dimension d. A decomposition of a mixed state p is a set {pi, )} with pi > 0, iPi = 1 and p = J]ipi |. Throughout this paper, we will call a decomposition optimal if it minimizes
the geometric measure of entanglement, i.e. if EG(p) = J]i piEG(|^i)). A separable state a is a closest separable state to p if E G (p) = 1 - F (p, a). In the following, we will show how to find an optimal decomposition of p, given a closest separable state.
5.1. Equivalence between closest separable states and optimal decompositions
In the maximization of F(p, a), we can restrict ourselves to separable states a acting on the same Hilbert space H. To see this, note that this is obviously true for pure states, as we can always find a pure separable state |0) e H such that |0)|2 is maximal. (Extra dimensions cannot increase the overlap with the original state.) Let now a = J]i qt |0i) | be the closest separable state with purification ) such that Fs(p) = )|2, where ) is a purification of p. We can again write the purifications as
i^)=ifi
^ ) = E /qj ^ j )i j)
with separable pure states |0j) such that VFs (|^i)) = (^ |0i). As the states |0j) are elements of H, the reduced state a = Tra[|0) (01] is a bounded operator acting on the same Hilbert space H, Tra denotes partial trace over the ancillary Hilbert space spanned by the orthonormal basis {|i)}.
Now we are in a position to prove the following result:
Proposition 5. Let p be an n-partite quantum state acting on H = ®ni=1Hi. The separable state a = sj=1 qj |0j)(0j | with s ^ d separable pure states |0j) and Y]=1 qj = 1 qi ^ 0, is the closest separable state if and only if there exists an optimal decomposition {pi, )}si=1 with s ^ d elements such that the following holds: J Fs (|^i)) = (^ |0i) and qi = ^iFs^ .
/ ,k pkFs (|^k))
Proof. In the following, {|i)} denotes a basis on the ancillary Hilbert space Ha. The closest separable state a = ] = qj |0j )(0j | can be purified by
) = E V7 |0j )| j). (53)
We write a purification of the state p as
)= E^A |Ai) U \i), (54)
where are the eigenvalues and ) are the corresponding eigenstates of p, with = 0 for i ^ d, and U is a unitary acting on the ancillary Hilbert space Ha. According to Uhlmann's theorem [7, 20] it holds
|0)|2 ^ F (p, a) = Fs (p). (55)
In the following, let U be a unitary such that equality is achieved in (55); its existence is assured by Uhlmann's theorem. Writing U = J2k1=1 ukl |k)(l | in (54), we obtain
) = E |Al )|k ) = E vpk |fk )|k) (56)
k ,l=1 k=1
with vpk |^k) = YH= 1 uk/VA ). Note that {pk, |^k)}k=1 is a decomposition of p.
We will now show that {pk )}sk=1 is an optimal decomposition by showing that |0)|2 = ^2iPiFs)). As we chose the purifications such that |0)|2 = Fs(p), this will complete the proof. Computing the overlap | (^) |2 using (53) and (56) we obtain
E VPîq
As in the proof of theorem 1, maximality of (57) implies that |0i)| = VFTGW)) and qi = ^FkFtlfk)). Then we immediately see that {pk, )}sk=1 is optimal, because Fs (p) =
)|2 = 1 piFs (№)), which is exactly the optimality condition.
So far, we proved the existence of an optimal decomposition {pi, )} with the property VFs (|^i)) = (fa |0i) starting from the existence of the closest separable state a = Y^j = qj )(0j |. Now we will prove the inverse direction. Given an optimal decomposition {pi, )}i=1, we will find the closest separable state. We again define the purifications of p and a as
)= E Vp № i)®|i), (58)
^ ) = E vqyl^j} ®| j), (59)
where we define the states |0j) to be separable and to have maximal overlap with ), i.e. (^j|0j) = yFs(|^j)). The real numbers qj are defined as follows: qj = j))). Now we
note that |0)|2 = Fs(p) because the decomposition {pi, )} was defined to be optimal. Thus, we see that there exists no purification |07) such that |07)| > |0)|. Together with Uhlmann's theorem this implies that F(p, a) = Fs(p). □
5.2. Caratheodory bound
Now we are in a position to show that the number of elements in an optimal decomposition (w.r.t. the geometric measure of entanglement) is bounded from above by the Caratheodory bound.
Corollary 1. For any state p acting on a Hilbert space of dimension d there always exists an optimal (w.r.t. the geometric measure of entanglement) decomposition {pi, )}s=1 such that s < d2.
Proof. Let a be the closest separable state. From Caratheodory's theorem [6, 26] follows that a can be written as a convex combination of s ^ d2 pure separable states. According to proposition 5 the state a can be used to find an optimal decomposition with s elements. □
6. Structure of optimal decomposition w.r.t. geometric measure of entanglement
In this section, we will show that the optimal decomposition of p w.r.t. the geometric measure of entanglement has a certain symmetric structure.
6.1. n-partite states
First, we derive the structure of an optimal decomposition {pi, |f )} for a general n-partite state. Proposition 6. Every optimal decomposition {pi, f )}S= 1 must have the following structure,
VF (|fk )) <f i |0* ) = 7 F (|f )) <0i |fk ) (60)
for all 1 ^ i, k ^ s. Here the states |0i) are separable and have the property <0i |f) =
VF (|fi)).
Equation (60) represents a nonlinear system of equations. Finding all solutions of it is equivalent to computing the optimal decomposition of p. For pure states our result reduces to the nonlinear eigenproblem given in equations (5a) and (5b) in [11].
Proof. Let the states |i) denote an orthonormal basis on the ancillary Hilbert space Ha. Let |f ) = Yi VP f ) |i ) and ) = Y j Vqjj ) | j ) be purifications of p and a, respectively, such that {pi, |f)} is an optimal decomposition of p, <f) = VFs (|f)) and qt = ))'
This implies that
Fs (p) = |<f|0)|2 = E |<№| (№)®|i))|2 •
Optimality implies that |<>|2 is stationary under unitaries acting on the ancillary Hilbert space Ha (for stationarity under unitaries acting on the original space see subsection 6.5), that is,
ddt |eitHa |< >|2=o = 0 (62)
for any Hermitian Ha = Hj acting on Ha and the derivative is taken at t = 0. Using (61) we can write
|<f |eitHa)|2 = £|<f| (|&) eitHa |k))|2.
The derivative at t 0 becomes
dt l<f |eltHa № )|2=o = Tra
Ha Tra
E( a+A )
with Ak = i(|<k>(<k| ® |k>(k|) >(^| and Tra means partial trace over all parts except for the ancillary space Ha. Using ((<k|(k|)\^> = (|^k>), we can write Ak as
Ak = VPkFs dfk>) |(k>|k>(^|. (65)
Expression (64) has to be zero for all Hermitians Ha, which can only be true if Tra k (Ak + Aj)] = 0, which is equivalent to
ETra [VPkFs (|fk)) |0k) |k) <f |] = ETra [^PkF (|fk)) |f ) <k| <^k|] .
With |f ) = VPi |fi) |i) we obtain
Ev7 Pk P' F s (|fk )) <f |0k ) |k ) <11 = E^ Pi PkFs (|fk )) |f ) |i ) <k |. (67)
i,k i,k Using orthogonality of {|i)} completes the proof.
6.2. Bipartite states
Let us illustrate the structure of an optimal decomposition with the example of bipartite states. We consider expression (60) for a bipartite mixed state p with optimal decomposition {pi, )}. In this case it is possible to write the Schmidt decomposition of the pure states ) as follows:
) = £ a, } | j(1) )| j
with£ j k2j = 1, and the Schmidt coefficients are in decreasing order, i.e. Ai1 ^ A,-.2 ^ • • • > 0. The separable states ) that have the highest overlap with ) are given by
|0i ) = |1(1) )|1(2)) ,
and VFs (|^i)) = ki, 1. With this in mind, expression (60) reduces to
A , 1 (ft |l[1} )|lf )= A , 1 (1(1) |(1(2) №) (69)
for all i , k.
6.3. Qubit-qudit states
Let now the first system be a qubit, that is, dl = 2. In this case, we can set kk,i = cos ak and kk,2 = sinak, with cosak ^ sinak. With )= cosak 111 ) + sinak |22), we get from equation (69)
cos ak sin ai ((2^ |l[1} )(2(2) |lf )) = cos ai sin a* «l(1} |2[1} )(l(2) |2f )). (70)
Noting that |(2(1) | l£1})| = |(lP |2*1})| it follows that
tan ai tan ak
It is interesting to mention that in the case d2 = 2, we can simplify (7l) to tan ai = tan ak. This means that in the optimal decomposition {pi, )} of a two-qubit state all states ) have the same Schmidt coefficients, a result already known from [l5].
(1(2) |2f:
(2(2) |1f>:
6.4. Nonoptimal stationary decompositions
Note that expression (60) is necessary, but not sufficient for a decomposition to be optimal. To prove this we will give two nonoptimal decompositions that satisfy (60).
6.4.1. Bell diagonal states. Consider the state
p = 1 №+) W +1 + 2 №+)(0 +1 , (72)
with +) = V (|01) + 110)) and |0+) = V (|00) + 111)). It is well known that the state (72) is separable, and thus the decomposition into Bell states cannot be optimal. On the other hand, it is easy to see that this decomposition satisfies (60).
6.4.2. Separable states. Now we will give a more complicated example. We call a decomposition {pi, )}s=1 s-optimal if for a given number of terms s there is no decomposition
{q, \0i)K=i such that J2¿=1 qt^G 00)) < s=1Pi^G \№). It is known [2] that there exist separable states p of dimension d with the property that any d-optimal decomposition is not separable and thus not optimal. Let {pi, \№ )}d= 1 be a d-optimal decomposition of such a state p.
We write a purification of p as \№) = Ya=1 Vp )\i). Further, we define separable states \0i) such that (№) = VF (\№)), qt = E)) and \0) = j V57 ) \ j) Then it holds
K^|0>|2 = E >)2 .
From d-optimality of \(№\0)\2 it follows that for all Hermitian matrices acting on a d-dimensional Hilbert space Ha
d IW |elH |0 >12=0 = 0
holds. We will now show that ^ K^ |e^a>|2=0 = 0 also holds for dim(Ha) ^ d. This means that adding more dimensions to the ancillary Hilbert space will not help. Performing the same calculation as in the proof of proposition 6 we obtain
dd IW ieliHa № >l2=o = Tr
Ha -Ttâ
'd (ha )
with Ak = i^/pkFs(\№)) \0k)\k)(№\. Note that Ak is non-zero only for k ^ d, because pk = 0 otherwise. Thus, we can restrict ourselves to k ^ d in the calculation, which is equivalent to setting dim(Ha) = d. Then (74) implies Tra[Ed=Ha)(Ak + A)] = 0 and it follows that (74) holds for arbitrary d (Ha) ^ d.
6.5. Stationarity on the original subspace
In proposition 6, we used the argument that in the optimal case \(№\0)\2 has to be stationary under unitaries acting on the ancillary Hilbert space Ha. In (61), we could rewrite this expression as
Fs (p) = \(№\0 )\2 = E\(№\0i )\i )\2,
where all \0i) are separable. We can also demand i \(№\0i) \i)\2 to be stationary under (separable) unitaries acting on the original Hilbert space of the states \0i). From this procedure we will gain stationary equations describing the states \0i). However, we already know that in the optimal case we can choose \0i) to be the closest separable state to №), that is, (№ \0i) = VFs (\№)), such that this method does not give new results.
7. Concluding remarks
We have shown in this paper that the geometric measure of entanglement belongs to two classes of entanglement measures. Namely it is a convex roof measure and also a distance measure of entanglement. As an application we gave a closed formula for the Bures measure of
entanglement for two qubits. We also note that the revised geometric measure of entanglement defined in [27] is equal to the original geometric measure of entanglement.
We furthermore proved that the problems of finding a closest separable state and finding an optimal decomposition are equivalent. We used this insight to bound the number of elements in an optimal decomposition (w.r.t. the geometric measure of entanglement). It turns out that the bound is exactly given by the Caratheodory bound.
Finally, we obtained stationary equations that ensure optimality of a decomposition. For the case of two qubits these equations lead to the known fact that each constituting state of an optimal decomposition has equal concurrence. Our equations hold for any dimension. However, they are only necessary, not sufficient for a decomposition to be optimal. Given an arbitrary decomposition, they provide a simple test whether the decomposition may be optimal.
Acknowledgments
We acknowledge discussions with M Plenio. AS thanks C Gogolin, H Hinrichsen and P Janotta. This work was partially supported by Deutsche Forschungsgemeinschaft.
Appendix A. Geometric measure of a convex set
In theorem 1 we stated that if S is the set of separable states it holds
Fs (p) = max^ pF )), (A.1)
where Fs is the maximal fidelity between p and the set of separable states: Fs (p) = maxaeSF(p, a) and the maximization is done over all pure state decompositions of p. In the following, we will generalize this result to arbitrary convex sets.
Let X be a set of states {ak} and C be a set containing all convex combinations of the elements of X, these are states a such that it holds
a = E qk ak (A.2)
with qk ^ 0, k qk = 1. We define the quantities FX (p) and FC (p) to be the maximal fidelity between p and an element of X and C, respectively,
FX(p) = max F (p, a), (A.3)
Fc(p) = max F (p,a). (A.4)
Theorem 2. For an arbitrary quantum state p and a convex set of states C it holds
Fc (p) = max Y] pi Fx (pi), (A.5)
p=E k pk pk~^
where the maximization is done over all decompositions of p = J]i pipi, pi ^ 0.
Proof. The proof is a modification of the proof of theorem l. According to Uhlmann's theorem [20, p 4ll] it holds
F (p,a) = max |0)|2 , (A.6)
where > is a purification of p and the maximization is done over all purifications of a denoted by |0>.
In order to find FC(p) we have to maximize |0>|2 over purifications |0> of all states of the form a = J]k qkak, ak g X. Using similar arguments as in the proof of the theorem 1, we see that the purifications can always be written as
i^>=E vp (E VPÛi^,j> ® i-, j> I,
i0 >=e v^^ ^ e V^ i^,i > ® i*,1 > j,
with (i, j ^, l> = 5ik5j/. In the maximization of |0>|2 we are free to choose the states |0k,/> under the restriction that Yi VÔkj |0k,l> ® |k, l> purifies ak g X, the probabilities qk > 0 are restricted only by Ykqk = 1. We are also free to choose j>}, {pi} and {pitj} under the restriction p = i jPi Pij j >(^i, j |. With this in mind we obtain
K^ i0 >i =
with ai;k being the product of the purifications of pi and ak:
aik = ^E Vp^(№i,j \®(i, j \j ^E Vqk!7\0k,i)®\k, I^ .
Now we optimize over {qksl, |0k;l)} with the result
ai,k = VFx (pi)Sk
and thus
max \(№)\ ^JWiy/Fx (pi).
Now we do the optimization over qi. Using Lagrange multipliers we obtain
VPiVFx (Pi ) \/Ek PkFx (Pk) '
with the result
max K^i0>i2 = EP-Fx (Pi)•
(A.10) (A.11)
(A.12) (A.13)
In the last step we do the maximization over all decompositions {pi, pi} of the given state p, which leads to the final result
Fc (p) = max K^ >i2 = max E Pi Fx (Pi).
(A.14) □
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