Three-tangle for mixtures of generalized GHZ and generalized W states

We give a complete solution for the three-tangle of mixed three-qubit states composed of a generalized GHZ state, a|000>+b|111>, and a generalized W state, c|001>+d|010>+f|100>. Using the methods introduced by Lohmayer et al. we provide explicit expressions for the mixed-state three-tangle and the corresponding optimal decompositions for this more general case. Moreover, as a special case we obtain a general solution for a family of states consisting of a generalized GHZ state and an orthogonal product state.


Introduction
The occurrence of entanglement in multipartite systems is one of the most important and distinctive features in quantum theory [1,2]. With the ever-increasing number of applications of entanglement, its quantification has become one of the foremost topics in contemporary quantum information research.
While entanglement of pure and mixed states of two qubits is already well understood [3][4][5][6][7], to date there is no generally accepted theory for classification and quantification of entanglement in multipartite qubit systems. For three-qubit systems, numerous interesting results have been found [8][9][10][11][12][13][14][15][16][17][18][19]. A complete SLOCC characterization of three-qubit entanglement has been achieved only for pure states [9,10]. It leads to a schematic characterization for mixed states [14]. A crucial concept for this is the so-called three-tangle, a polynomial invariant for three-qubit states that quantifies the three-partite entanglement contained in a pure three-qubit state (the three-tangle is equal to the modulus of the hyperdeterminant [20,21]). However, even for the simplest case of rank-2 mixed states, no general expression is known for its three-tangle.
Recently, Lohmayer et al. [18] have provided an analytic quantification of the threetangle for a representative family of rank-2 three-qubit states, namely for mixtures of a symmetric GHZ state and an orthogonal symmetric W state. In this article we show that by applying the methods of [18,22] these results can be extended to rank-2 mixtures of a generalized GHZ state and an orthogonal generalized W state. This article is organized as follows. In Section 2, we introduce some basic terminology and give a precise formulation of the problem whose general solution we outline in Section 3. In Section 4 we discuss special cases of this solution, in particular we find the three-tangle for rank-2 mixtures of generalized GHZ states and certain orthogonal product states.
The three-tangle of a mixed state can be defined as convex-roof extension [25] of the pure state three-tangle, A given decomposition {q k , π k : ρ = k q k π k } with τ 3 (ρ) = k q k τ 3 (π k ) is called optimal. We note that τ 3 (ρ) is a convex function on the convex (and compact) set Ω of density matrices ρ.
In this paper, we determine three-tangle and optimal decompositions for the family of mixed three-qubit states composed of a generalized GHZ state and a generalized W state We note that τ 3 (gW c,d,f ) = 0 and τ gGHZ 3 := τ 3 (gGHZ a,b ) = 4|a 2 b 2 |. For the symmetric GHZ and W state (a = b = 1/ √ 2 and c = d = f = 1/ √ 3) the problem and results of [18] are recovered.

The generic case
In this section it is assumed that none of the coefficients is zero, i.e. a, b, c, d, f = 0. The opposite case corresponds to either a rank-2 mixture of a generalized GHZ and a biseparable state, or to a mixture of a generalized W and a completely factorized state and will be studied in the next section.
In the following, we will apply the methods developed in [18,22]. There it was shown that in order to find the convex roof of an entanglement measure for rank-2 mixed states it is useful to study the pure states that are superpositions of the eigenstates of ρ The three-tangle of these states is The phases of the coefficients in |gGHZ a,b and |gW c,d,f merely produce different offsets for the relative phase ϕ in the expression for the three-tangle, Eq. (8). Therefore it suffices to consider the case where all coefficients are positive real numbers. In the following, it will be beneficial to introduce the definition If we factor out the three-tangle τ gGHZ 3 of the generalized GHZ state, the three-tangle of the superposition (7) can be written as Since τ gGHZ 3 is just a constant factor, the behaviour of this function of p and ϕ is completely determined by the value of the parameter s.
As a first step, we identify the zero-simplex containing all mixed states ρ(p) with τ 3 (ρ(p)) = 0. Its corner states are obtained as the zeros of Eq. (10). One obvious solution is p = 0, which corresponds to a pure generalized W state. Therefore, in the calculation of the other solutions we can assume p > 0 and the zeros are determined by Since p and s are real and positive, this implies ‡ ϕ = n 2π 3 , n ∈ N .
(12) ‡ Note that the 2π/3-periodicity is due to the fact that this relative phase is induced by the local transformation diag{exp(i2π/3), 1} on each qubit.
For p, we then get the solution This means that in addition to the state |gW c,d,f the three-tangle vanishes for |p 0 , n · 2π/3 , n = 0, 1, 2. All mixed states whose density matrices are convex combinations of those four states have zero three-tangle. On the Bloch sphere with gGHZ and gW at its poles, this corresponds to a simplex with those four states at the corners. All ρ(p) with p < p 0 are inside this set, and therefore τ 3 (ρ(p)) = 0 for 0 ≤ p ≤ p 0 . In order to determine the mixed three-tangle of ρ(p) for p > p 0 , we note that for any fixed p, τ 3 (p, ϕ) takes a minimum at ϕ 0 = 0 which due to the symmetry of τ 3 is repeated at ϕ 1 = 2π/3 and ϕ 2 = 4π/3. Consequently, for any value of p the state ρ(p) can be decomposed into the three states |p, ϕ i , i = 0, 1, 2. Therefore the characteristic curve τ 3 (p, 0) is an upper bound to τ 3 (ρ(p)). Moreover is it known to give the correct values for the three-tangle at p = p 0 (at the top face of the zero simplex) and p = 1 (ρ(1) = |gGHZ a,b gGHZ a,b |). However, if there is a range of values where τ 3 (p, 0) is a concave function, there are decompositions for ρ(p) with a lower average threetangle [18]. Therefore it is important to examine where the function τ 3 (p, 0) is concave for p ≥ p 0 .
For ϕ = 0 and p ≥ p 0 , the term inside the absolute value bars in (10) is real and positive, and the characteristic curve τ 3 (p, 0) is equal to Concavity of t(p) is indicated by a negative sign of its second derivative The limit p → 1 (p = 1 − ε) in (15) gives that is, t(p) is concave close to p = 1. On the other hand, for small p That is, close to p = 0 we find that t(p) is convex (note that due to the absolute value, τ 3 (p, 0) is actually concave close to p = 0). Due to continuity, there must be at least one zero of t ′′ (p) in between. Moreover we note that the third derivative is negative for all values of p. Thus t ′′ (p) is strictly monotonous and has precisely one zero, implying that t(p) is convex before and concave after that point. As the mixed state three-tangle is convex, the characteristic curve needs to be convexified where it is concave in the interval [p 0 , 1]. Since the concavity extends up to p = 1, corresponding to the state |gGHZ a,b , that state has to be part of the optimal decomposition [22] in this interval. The symmetry and the results in [18] suggest that a good ansatz for the optimal decomposition is where p 1 is chosen such that the mixed-state three-tangle becomes minimal. The value of α is fixed by p and p 1 : The average three-tangle for this decomposition is (p > p 0 ) This describes a linear interpolation between τ 3 (p 1 , 0) and τ gGHZ 3 . Note that for p < p 1 , (19) ceases to be a valid decomposition because α becomes negative.
To find the minimum in p 1 for given p, we look for the zeros of the derivative ∂τ conv 3 /∂p 1 . The resulting equation has the solution p noabs Note that for s > 2 √ 2 we get p noabs 1 < p 0 . In that case the minimum is reached at the border p 1 = p 0 of the considered interval [p 0 , 1], and therefore Putting it all together, we present the central result of this article where p 0 is given by (13), p 1 by (23), τ 3 (p, 0) by (8) and τ conv 3 (p, p 1 ) by (21). The corresponding optimal decompositions are and π j as defined in (2) The curve (24) is convex, and for all p and ϕ: τ 3 (ρ(p)) ≤ τ 3 (p, ϕ). Therefore it is a lower bound to the three-tangle of ρ(p). On the other hand, for each p we have given an explicit decomposition realizing this lower bound. Thus it represents also an upper bound and hence coincides with the three-tangle of ρ(p).

Special cases
In this section we will discuss various special cases of our general solution (24).
First, we briefly demonstrate that the results for the symmetric GHZ state and the symmetric W state in [18] are reproduced. Indeed, the general behaviour described in Section 3 (that is, analytic properties of the three-tangle, optimal decompositions) matches the one found in [18], so we only have to check the values of p 0 and p 1 . In the symmetric case we have Inserting this in (13) and (23) leads to (29) as found in [18]. Next, we consider the limiting cases where at least one of the coefficients is 0. Those require extra care as the calculations above have been done under the assumption of non-vanishing coefficients. However, since we are dealing with continuous functions, one should expect that the results still apply, although possibly in a degenerate form.
The first case we consider is when the generalized GHZ state degenerates into a pure three-party product state. This corresponds to the limit s → ∞. However note that at the same time τ gGHZ 3 → 0 such that (10) remains regular. This can be seen by looking at the explicit form (8). It is clear that in this case τ 3 (ρ(p)) = 0 for all p.
There are two non-equivalent ways to achieve this. One possibility is b = 0 which reduces the generalized GHZ state to |000 . In this case, the three-tangle (8) vanishes for all superpositions (7), and therefore also all mixed states anywhere inside the Bloch sphere have vanishing three-tangle.
The other way to get s → ∞ is a = 0 where the generalized GHZ state is reduced to |111 . While ρ(p) as a mixture of product and gW state again has no three-tangle, unlike in the case b = 0 the three-tangle does not vanish everywhere on the Bloch sphere. Equation (8) reduces to which is independent of ϕ and concave for all p ∈ [0, 1]. Thus the zero simplex degenerates into a zero axis. As long as cdf > 0, outside of this axis the three-tangle  never vanishes. If both a = 0 and cdf = 0, the three-tangle is zero everywhere inside the Bloch sphere.
The opposite limiting case is s = 0, that is, when at least one of the coefficients c, d, f vanishes. Note that for the three-tangle it does not matter whether only one of them vanishes, resulting in a product of a single qubit state with a generalized Bell state, or two of them, resulting in a product of three single-qubit states: In all cases (10) reduces to which is convex for all p ∈ [0, 1]; indeed, (13) and (23) yield p 0 = 0 and p 1 = 1 at s = 0. Consequently, for all p. Even more, τ 3 (ρ) = τ gGHZ 3 p 2 for any mixed state ρ inside the Bloch sphere with gGHZ a,b | ρ |gGHZ a,b = p. We would like to point out that this result reminds of the situation both for two-qubit superpositions [26] and for two-qubit mixtures of an arbitrary entangled state and an orthogonal product state.

Conclusion
In this paper, we have given explicit expressions for the three-tangle of mixtures ρ(p) according to (4) of arbitrary generalized GHZ and orthogonal generalized W states, including the limiting cases where those states are reduced to product states. We have found that the qualitative pattern described in [18] for mixtures of symmetric GHZ and W states holds also more generally. Up to a certain value p 0 given by (13), the mixed three-tangle vanishes. The optimal decomposition for those states consists of the pure states (7) for which the three-tangle is zero. One is always the generalized W state at the bottom of the Bloch sphere. The other three form an equilateral horizontal triangle at the height of p 0 . Note that those states do not depend on p as long as p ≤ p 0 .
For p > p 0 , there may follow a region up to some value p 1 given by Eq. (23), where the mixed state three-tangle follows the minimal pure state three-tangle (10) with the same value for p (which for positive real coefficients is achieved at ϕ = 0). In this region, the optimal decomposition consists of the three states with this property, which form a horizontal eqilateral triangle with corners on the Bloch sphere and ρ(p) in the center. If s ≥ 2 √ 2, p 1 and p 0 coincide and this region with "leaves" of constant three-tangle in the convex roof (cf. [18], Figure 2) is absent. This can be viewed as contraction of this middle region into one point.
For p > p 1 , the three-tangle grows linearly up to its maximum value at p = 1. The optimal decomposition in this case consists of the three pure superposition states for p = p 1 with minimal three-tangle and the generalized GHZ state. That is, the convex roof in the Bloch sphere is affine for an entire simplex whose corners are given by the four pure states that form the optimal decomposition. Moreover, we have demonstrated how the results of this work connect to the findings for the special case of mixtures of a symmetric GHZ and a symmetric W state [18].
In principle, the scheme of three regions for p values as outlined above holds also in the limiting cases when some of the coefficients in the states vanish, except that in this situation the "outer regions" may shrink away. A common feature of these limits is a ϕ-independent characteristic curve. If the generalized GHZ state degenerates into a product state, τ 3 (ρ(p)) = 0 for all p. On the other hand, for s = 0 (i.e., at least one of the coefficients in the generalized W states vanishes), both "outer" affine regions disappear and the whole range of p is covered by the "middle region" with a strictly convex characteristic curve. This case corresponds to a mixture of a generalized GHZ state and an orthogonal product state and the exact convex roof of the three-tangle is obtained everywhere inside the Bloch sphere.