Evaluating convex roof entanglement measures

We show a powerful method to compute entanglement measures based on convex roof constructions. In particular, our method is applicable to measures that, for pure states, can be written as low order polynomials of operator expectation values. We show how to compute the linear entropy of entanglement, the linear entanglement of assistance, and a bound on the dimension of the entanglement for bipartite systems. We discuss how to obtain the convex roof of the three-tangle for three-qubit states. We also show how to calculate the linear entropy of entanglement and the quantum Fisher information based on partial information or device independent information. We demonstrate the usefulness of our method by concrete examples

Quantum entanglement plays a central role in quantum information science and quantum optics [1].There are now efficient methods to detect entanglement, that have even been used in many experiments [2].These mostly answer the yes or no question: "Is the quantum state entangled?"or "Is the quantum state genuine multipartite entangled?"After verifying the presence of entanglement, the next step is quantifying it.Calculating measures is becoming increasingly important in experiments in quantum information science [3][4][5] and it also plays a crucial role in investigations in quantum statistical physics, e.g., in studying phase transitions [6].
Most entanglement measures are based on the convex roof of a quantity on pure states such as the entropy of the reduced state [7][8][9].Measures of this type can also be used to classify states according to their membership in some convex sets, for example, based on their Schmidt rank [10,11].They play a central role in quantum information theory, however, in most of the cases they are not computable as there are no efficient ways to calculate convex roofs.Most importantly, the simplest multipartite entanglement measure, the three-tangle for three-qubits, cannot be computed for a general state.Thus, for obtaining entanglement measures in theory and experiments, it would be crucial to find methods to calculate convex roof constructions efficiently, at least for not too large systems.This seems to be a very difficult task since straightforward numerical search means an optimization over an infinite number of convex decompositions of the density matrix.Such an approach will lead to an upper bound on the measure, since a multivariable numerical optimization is not guaranteed to find the global optimum [12].Upper bounds, however, are often not very useful as the amount of entanglement can be much lower or even zero even if the procedure signals considerable entanglement.
In this paper, we present a method that produces a series of very good lower bounds on important entanglement measures.Our method has the following characteristics: (i) It is based on semidefinite programming.The series of bounds obtained converge in a controllable way to the true value.Even the first lower bound in the series is non-trivial.(ii) We have a clear physical picture for what states our method yields a nonzero value for the measures.(iii) The set of separable states is used in the optimization procedure.This way we connect calculating convex roofs to the separability problem, which might help to find applications of the separability problem in other areas of physics.We will demonstrate the use of our method with the example of computing bipartite entanglement measures for bound entangled states, computing the convex roof of the tangle for various three-qubit states, and even quantities outside of quantum information science.Our method can also be used to compute a lower bound from incomplete data of the quantum state or in device independent scenarios [13][14][15][16].
Convex roof of linear entropy.For pure states, the linear entropy of entanglement is given as where we used the definition of the linear entropy S lin (̺) = 1 − Tr(̺ 2 ).Hence, the linear entropy of entanglement for pure states equals also C 2 /2, where C is the concurrence [7], and it is also equal to the I-tangle [17].The definition (1) can be extended to mixed states by a convex roof construction as where {p k , |Ψ k } is a decompositon to pure states It can be shown that E lin (̺) does not increase under local operations and classical communication (LOCC) on average, hence it is an entanglement monotone [18].Consequently, E lin (̺) has also been used to characterize entanglement even in the multipartite setting [19].
Next, we will show a method to compute Eq. (2).For this aim, first we write the liner entropy of entanglement as an expectation value of an operator acting on two copies of a bipartite pure state |Ψ as [20] (4) Here, A and B denote the parties of the first copy while A ′ and B ′ denote the parties of the second copy.Moreover, the projector to the antisymmetric space is defined as A AA ′ := (1 1 − F ) AA ′ , F is the flip operator, and we explicitly wrote out 1 1 BB ′ for clarity [21].
Next, we will consider mixed states.Let us assume that {p k , | Ψk } is the decomposition attaining the convex roof.Then, for a state with such a decomposition we obtain where the state on the two-copy space is defined as The density matrix ω 12 has three important properties.
It is a mixture of product states, i.e., a separable state [22].Moreover, all the pure product components are symmetric.Thus, ω 12 is supported on the symmetric subspace.In fact, any symmetric separable states can be written in the form (6) [23].Finally, Tr 2 (ω 12 ) = ̺.Hence, we arrive at our first main result.Observation 1.-The convex roof of the linear entropy can be written as s.t.ω 12 symmetric, separable, where ω 1 ≡ Tr 2 (ω 12 ).Observation 1 connects the separability problem of symmetric bipartite states, i.e., answering the question "Is the state entangled?"mentioned in the introduction, to entanglement quantification.In principle, to obtain a lower bound on E lin (̺), any necessary condition for separability could be used.We will consider the method based on the positivity of partial transpose (PPT) [24] 0 0.2 0.4 0.6 0. and obtain a lower bound as s.t.ω 12 symmetric, PPT, Next, we will demonstrate that our method can be used to quantify the entanglement of states not detected by the PPT condition, called bound entangled states [25][26][27].
Horodecki state.-Wetest our method to calculate entanglement measures for the one-parameter family of the 3 × 3 bound entangled state ̺ PH a introduced by P. Horodecki [26].We mix the state with white noise according to ̺ a (p) = p̺ PH a + (1 − p)1 1/9 and calculate the entanglement as a function of a and p.The results can be seen in Fig. 1.The critical noise for which E (ppt) lin (̺) = 0 agrees with the calculations of Ref. [21] and Ref. [28].We note that we made the computer program calculating E (ppt) lin (̺), with all other programs used for this publication, publicly available [29].Other methods for calculating entanglement measures are in Refs.[30,31].
It is a surprise that, while the bound relies on the PPT criterion, the method is still able to detect PPT entangled states.In order to obtain more information on what kind of states are detected, we need to know the separability criterion based on symmetric extensions [32].A given bipartite state ̺ AB is said to have a n : m symmetric extension if it can be written as the reduced state of a multipartite state ̺ A1..AnB1..Bm , which is symmetric under A k ↔ A l and B k ↔ B l for all k = l.If we also require that the state is PPT for all bipartitions, then it is a PPT symmetric extension.Separable states have such extensions for arbitrarily large n and m, while the lack of such an extension signals the presence of entanglement.Observation 2.-For all non-PPT states and for all states that do not have a 2 : 2 symmetric extension we have E (ppt) lin (̺) > 0.Moreover, for all states having a 2 : 2 PPT symmetric extension E (ppt) lin (̺) = 0 holds.The proof can be found in the Supplement [33].
Before we continue let us point out that we can also obtain a lower bound on E lin (̺) if we choose any other entanglement condition, such as the method based on local uncertainty relations [46], the covariance matrix criterion [47], or the computable cross norm or realignment criterion (CCNR) [48].However, for symmetric states these are all equivalent to the PPT condition [49].
Therefore, to strengthen the bound a stronger criterion must be employed.Here again the method of PPT symmetric extensions can be used [32].Rather than approximating ω 12 by PPT states, we demand that ω 12 has an n : 1 PPT symmetric extension [50].In this way we obtain a sequence of lower bounds E (n) lin with increasing accuracies.The corresponding optimization can similarly be carried out by semidefinite programming.Note that the PPT symmetric extensions converge to the set of separable states in a controlled way [51].Finally, note also that semidefinite programs not only detect entanglement, but through solving the dual problem, it is possible to find entanglement witnesses [32].In our case, these witnesses can even bound entanglement measures, as explained in the Supplement [33].
Generalization and further examples.-Theprevious ideas can straightforwardly be generalized to compute the convex roof of any quantity that can be written as a polynomial of expectation values for pure states as where A m are operators and c mn are constants (see e.g., [52,53]).It is possible to define an operator , whose expectation value on several copies reproduces Eq. ( 9).Then, the convex roof of Eq. ( 9) can be obtained as an optimization over Ncopy symmetric fully separable states [23] E(̺) = min ω12..N Tr(Lω 12..N ), (10) s.t.ω 12..N symmetric, fully separable, Three-tangle.-Our next example is the calculation of the three-tangle, a three-qubit entanglement monotone [54].For pure states, it has been defined by Coffmann, Kundu and Wootters [9].Remarkably, it can be written as a fourth-order polynomial in expectation values [52].Hence, for mixed states, the tangle can be defined through a convex roof extension, which we can now map to the optimization problem s.t.ω 1234 symmetric, fully separable, where T is an operator acting on four copies of the threequbit state [55].Note that if we know τ (̺), we can decide whether a three-qubit fully entangled state is in the W or in the GHZ class [10].
The optimization can be carried out for symmetric multiqubit states that are PPT with respect to all bipartions rather than symmetric separable states, leading to the lower bound τ (ppt) .The results are shown in Fig. 2 for states of the form where Note that a lower bound for the convex roof of the tangle for general states, which is exact for states with certain symmetries, has been developed [56].
As a practical comment, we add that the numerical computation is challenging, but τ (ppt) can be computed on a standard laptop with standard free packages for semidefinite programming [57], if the state has some symmetries, or has a rank up to six.Calculations for general three-qubit states of rank eight are realistic with computer clusters and professional packages.
Schmidt rank.-Letus consider the quantities R r that are nonzero for states with a Schmidt rank larger than , where λ k are the eigenvalues of the reduced state.The R k quantities are proven to be entanglement monotones [58].We can calculate the convex roof of R r with our method.Convex roofs for such quantities allow us to bound the dimensionality of the entanglement from below.A powerful bound can be obtained by carrying out the optimisation for rqudit symmetric states that are PPT with respect to all bipartitions.An alternative is computing the negativity [60,61].In particular, N (̺) − 1/2 > 0 signals that the Schmidt number is larger than 2. We show that our method outperforms the negativity as a dimension witness in Fig. 3(a) for the family of states with |Ψ S = (|00 + |11 + |22 )/ √ 3 and colored noise We add that we checked several random 3 × 3 edge states to test the conjecture of Sanpera, Bruß and Lewenstein claiming that all bound entangled states in such systems have a Schmidt rank 2, and did not find a counter example [11].
Evaluation of entanglement measures based on incomplete information.-Experimentallyit is very important that entanglement measures can be evaluated based on incomplete knowledge on the quantum state.There are efficient methods to bound entanglement measures based on an operator expectation value from below [3][4][5].The current method can be adapted straightforwardly to the partial information case by replacing the condition ω 1 = ̺ with the set of linear constraints Tr(ω where O i are the measured observables and v i are the corresponding expectation values.As an example, see Fig. 3(b), where the entanglement is bounded from below based on complete information and based on σ x ⊗ σ x and σ z ⊗ σ z measurements for the state where |Φ + 3×3 is a two-qubit Bell state (|00 + |11 )/ √ 2 embedded in the 3 × 3 system and σ l ⊗ σ l acts on this two-qubit system.
Device independent scenario.-Theamount of entan-glement can be bounded exclusively from the observed data but independent of the quantum description.Depending whether only one or both sides are untrusted one distinguishes between a steering-type or a Bell-type scenario.The necessary steps to lift the method using only partial information to such device independent scenarios employs the translation idea highlighted in Ref. [15] and is explained in more detail in the supplement [33].As an example, in Fig. 3(c) we plot a lower bound on the linear entropy of entanglement given as a function of the violation of the CHSH Bell inequality [1].
Concave roof.-Besidesconvex roofs, concave roofs can also be computed.For example, if in Eq. ( 2) a concave roof is used instead of a convex roof, then we compute the linear entanglement of assistance [62], which is the maximal entanglement available if the mixed state is given as a purification to us, and a third party which holds the ancilla needed for the purifucation is assisting us.In this case, in our method minimum must be replaced by maximum.In this way, we obtain a converging series of upper bounds on the entanglement of assistance.The results are shown in Fig. 3(d) for the family of 3 × 3 states where and ǫ = 0.3.As a reference, the linear entropy of entanglement is also shown for the same state.
Conclusions.-We have shown a general framework for calculating convex roof-based entanglement measures.We demonstrated its use in calculating the entanglement for bipartite systems, as well as, the three-tangle for three-qubits.We discussed several other quantities for which it can be applied.In the future, we would like to explore further possibilities of using our algorithm to compute convex roofs, in calculating the linear Holevo capacity [63,64], the quantum Fisher information based on incomplete information [65], or the convex or concave roofs of sums of several variances, as outlined in the Supplement [33].

Supplemental Material
In this supplemental material, we give some further details of our derivations.
To prove the second part, note that based on the discussion above ω 12 is a 2:2 symmetric extension of ̺.It is not necessarily a PPT symmetric extension since for the A : BA ′ B ′ partition it can also be non-PPT.
Note that Theorem 2 can be generalized to states that have E (n) lin (̺) > 0, involving PPT symmetric extensions and symmetric extensions to several parties.

Quantitative entanglement witnesses
In this section, we describe how our method can be used to construct quantitative entanglement witnesses.
As an example, we present a condition for entanglement witnesses, such that the expectation value of all witnesses satisfying the condition gives a lower bound on E (ppt) lin defined in Eq. (8).We also prove that for every state ̺ AB there is a witness of this type that gives not only a lower bound, but gives the value of E (ppt) lin exactly.For the linear entropy of entanglement we needed to minimize the expectation value of the operator M = A AA ′ ⊗½ BB ′ over all symmetric separable states ω 12 with a fixed reduced marginal Tr 2 (ω 12 ) = ρ AB .Consider now an operator W = W AB that acts on the original bipartite Hilbert space.We require that is satisfies where P, Q ≥ 0.Here P is an operator acting only on the symmetric subspace of the two copies Sym(H ⊗2 AB ), while Q acts on the full tensor product H ⊗2 AB but we only used the projected symmetric part of the partial transpose.For such a decomposition, it can be shown that its expectation value for ω 12 is The projectors onto the symmetric subspace Π sym can be dropped in the third line since ω 12 is supported only on it.In the last line we used Tr(XY T1 ) = Tr(X T1 Y ), while nonnegativity holds because all occuring operators are positive semidefinite.Hence, Eq. (S4) can be rewritten as where we have further simplified the right-hand side using that ω 12 has a fixed reduced density matrix.Since Eq. (S5) holds for any valid state ω 12 , it holds in particular for the one yielding the linear entropy of entanglement, thus we arrive at Hence the expectation value of our witness provides a lower bound on the linear entropy of entanglement.
Next, we will show that for a given quantum state ρ AB , if we optimize over all such witness operators, it is always possible to find one that saturates the inequality (S6).
Observation 3.-For the linear entropy of entanglement we obtain with W being the set of all operators W of Eq. (S3).
Proof.The proof is given by applying the dual form of a semidefinite program [57], which has been employed in a variety of different quantum information problems.In particular we refer to Ref. [32] which explains such a procedure very nicely for the separability criterion based on symmetric extensions.We have structured the proof in two parts: In the first part, we show an equivalent formulation on the two-copy level.Afterwards we further simplify this dual problem to interpret it as an operator acting on a single density operator using techniques that were introduced in Ref. [32].
In the first part, we parse the original problem as given in Observation 1 into the form of a semidefinite program and invoke its dual, which provides the same solution.In order to achieve this one should note that the two i w i v i , where v i are the corresponding expectation values O i ρ .
Finally, if one also wants quantitative entanglement witness for the other tasks one can proceed similarly.For instance, if one likes to bound the tangle one demands that T − Π sym (W ⊗ ½ ⊗3 )Π sym is a non-negative on all fully separable states, thus it is an entanglement witness to test against full separability.
Other quantities that can be calculated by our approach Convex roof of the Meyer-Wallach measure.-TheMeyer-Wallach measure is an entanglement measure for pure states defined as [34] where ̺ n is the reduced state of the n th qubits.This measure can be generalized to include the reduced states of multi-qubit groups [35].Our method can calculate the convex roof of the measure (S15) and the generalized measures as well.
Holevo capacity.-The linear Holevo χ capacity is defined as [63,64] (S16) It is a capacity measure for a channel Λ.For qubit channels, explicit formula is given in Ref. [64].
Convex and concave roofs in entanglement conditions with the quantum Fisher information and the variance.-First,let us see simple entanglement conditions with the quantum Fisher information and the variance.We start from the fact that for pure N -qubit states holds.Next, we need the fundamental properties of the quantum Fisher information F Q [̺, A] in our criteria [65]: (i) For pure states F Q [̺, A] equals four times the variance (∆A) 2  ̺ .(ii) For mixed states, it is a convex function of the state.Hence, for separable states follows [36] Due to the concavity of the variance, we can obtain a similar entanglement condition with variances as [37] (∆J Any state that violates Eq. (S18) or Eq.(S19) is entangled.
The conditions (S18) and (S19) can be improved if we take the concave and convex roofs, respectively, of the left-hand sides of Eq. (S17).Hence, alternative separability conditions arise min Any state that violates these is entangled.Numerical evidence shows that Eq. (S20) is stronger than Eq.(S18).Moreover, numerical evidence shows also that Eq. (S21) is stronger than Eq.(S19).These ideas can be extended to improve other entanglement conditions based on variances [38].We note that Ref. [39] shows that 2 × 2 covariance matrices C ̺ (A, B) can always be decomposed as the where ̺ has the decomposition as in Eq. ( 3).Hence, we know that the bound on the sum of two variances cannot be improved this way.However, Ref. [40] demonstrates that such a decomposition is not always possible for 3 × 3 covariance matrices.This is connected to the fact that the bound for separable states for the sum of three variances can be improved.Quantum Fisher information based on incomplete data.-Thequantum Fisher information can be bounded from below from partially known data.That is, we know the expectation value of some operators, and want to find a lower bound for the quantum Fisher information.The problem can be mapped to a semidefinite optimization in the two-copy space.A very good lower bound can be obtained if we optimize over PPT states.
For that we can use that the quantum Fisher information is, apart from a constant factor, the convex roof of the variance [41] F The variance of a pure state |Ψ can be expressed on two copies as Hence, a lower bound on the quantum Fisher information can be obtained as Quantum Fisher information from collective measurements where the constraints are given with the expectation values v i = O i ̺ .The optimization (S25) can straightforwardly be carried out with semidefinite programming.In Fig. S1(a), we present a simple example where a lower bound on the quantum Fisher information F Q [̺, J z ] is shown based on measurements of the fidelity with respect to the GHZ state.Below a fidelity of 1/2, the bound for F Q [̺, J z ] is zero.This is due to the fact that the product state |11..111 reaches this fidelity value, while F Q [̺, J z ] is zero for this state.If the fidelity is 1, we obtain F Q [̺, J z ] = N 2 and F Q [̺, J x ] = N, as expected [36].
In Fig. S1(b), we present a bound on the quantum Fisher information based on collective measurements, relevant to spin squeezing.Note that for well polarized ensembles, increasing J 2 x leads to decreasing F Q [̺, J y ].On the other hand, for small J z , increasing J 2 x leads to increasing F Q [̺, J y ].Some of the curves have points only in certain ranges of J 2 x , as there are no physical states corresponding to measurement results outside of these ranges, assuming a given value for J z and J x = 0.
Similar methods can be used for bounding the variance of an observable from above based on the expectation value of other observables.We can use that the variance is the concave roof of itself [41] (∆A) 2 = max The difference between the two cases is that for the quantum Fisher information we have to look for the minimum, while for the variance we have to look for the maximum.Genuine multipartite entanglement.-It is possible to define quantities that detect true multipartite entanglement and can be evaluated with our method.Let us define where S linn (|Ψ ) is the linear entropy for the n th bipartition of the qudits.To be more precise, S linn (|Ψ ) is the linear entropy of the reduced state of the qudits in one of the two partitions for the n th bipartion.If G = 0 then the state is biseparable, otherwise it is genuine multipartite entangled.
Similar idea can work such that only a sum of entropies must be computed by defining given in Eq. ( 8) then Eq. (S28) can be obtained via a semidefinite program.The advantage of Eq. (S28) is that only two copies of the original state are needed to calculate the value with our approach, while for the formula (S27) we need much more copies.The formalism of Eq. (S28) is in the spirit of the PPT mixer detecting genuine multipartite entanglement [42].
Note that a three-qubit state mixed from states that are PPT with respect to some partitions have been found that is genuine multipartite entangled [43].Thus, detecting genuine multipartite entanglement is a non-trivial task.

FIG. 2 :
FIG.2:(Color online) Three-tangle of a family of states(12) as a function of the parameters x and y.Light color indicates the region where the tangle is zero, darker color indicates a nonzero value.

3 NegFIG. 3 :
FIG.3:(Color online) (a) Schmidt-number witness vs. negativity for a state of the type (13) as a function of pcn.As the inset shows, even when we consider negativity of the normal form, obtained through stochastic local operations and classical communications (SLOCC) such that all local matrices are fully mixed[59], our numerical method is superior.(b) E (ppt) lin (̺), given in Eq. (8), based on partial information for the state(14).(c) Estimation of E (ppt) lin (̺) as a function of the violation of the CHSH inequality.(d) E (ppt) lin(̺) and the corresponding bound for the entanglement of assistance (defined with the linear entropy) for a state of the type (15) as a function of p.
Minimal quantum Fisher informationFidelity with respect to the GHZ state Quantum Fisher information from a fidelity measurement FIG. S1: (a) Lower bound on FQ[̺, Jz] based on the fidelity with respect to the three-qubit GHZ state.(b) Lower bound on FQ[̺, Jy] based on J 2x for various values of Jz and for Jx = 0, for N = 3 qubits.
where E linn is linear entropy for the n th bipartition.If H = 0 then the state is biseparable, otherwise it is genuine multipartite entangled.If, instead of E lin (̺), we calculate E