Lower bounds on entanglement measures from incomplete information
Abstract
How can we quantify the entanglement in a quantum state, if only the expectation value of a single observable is given? This question is of great interest for the analysis of entanglement in experiments, since in many multiparticle experiments the state is not completely known. We present several results concerning this problem by considering the estimation of entanglement measures via Legendre transforms. First, we present a simple algorithm for the estimation of the concurrence and extensions thereof. Second, we derive an analytical approach to estimate the geometric measure of entanglement, if the diagonal elements of the quantum state in a certain basis are known. Finally, we compare our bounds with exact values and other estimation methods for entanglement measures.
pacs:
03.65.w, 03.65.Ud, 03.67.aI Introduction
Entanglement is a key phenomenon in quantum information science and the quantification of entanglement is one of the major problems in the field. For this quantification many entanglement measures have been proposed plenio (); eof (); wei (); hororeview (); meyerwallach (); vidalwerner (); concurrence (); osterloh1 (); mintertreview (). However, a central problem in most of these proposals is the actual calculation of a given measure: entanglement measures are typically defined via optimization procedures, which may consist of a maximization over certain protocols or the minimization over all decompositions of a state into pure states. For some remarkable cases it happens that such minimizations can be performed analytically wootters1 (); terhalvollbrecht (); rungta (); osterloh2 (), however, in the general case these problems are not solved. Therefore, to take a realistic point of view, one can try to estimate entanglement measures, and many proposals for the estimation of entanglement measures have been presented cab1 (); cab2 (); mintert1 (); vicente1 (); datta ().
In an experimental setting, the situation becomes even more complicated: since quantum state tomography for multiparticle systems requires an exponentially increasing effort, the state is often not completely known. Typically, one measures socalled entanglement witnesses, special observables, for which a negative expectation value signals the presence of entanglement horo (); terhal (); mohamed (); kiesel (); haeffner (); lu (); tothguehne (). In this sense, entanglement witnesses allow to detect entanglement, but the question arises, whether they also allow to quantify entanglement. This question has been addressed from several perspectives horoold (); wolf (); brandao (); plenio2 (); mintert2 () and in Refs. wir (); eisert () a general recipe for this problem was found. There it has been shown how one can derive the optimal lower bound on a generic entanglement measure from the expectation value of a witness or another observable. The estimate uses Legendre transforms to give lower bounds on the convex entanglement measure, and the main task in this scheme is to compute the Legendre transform of a given entanglement measure.
In this paper, we extend this method into several directions. First, we derive a simple algorithm for the calculation of the Legendre transform for the concurrence concurrence () and extensions thereof. Then, we present analytical results for the Legendre transform for certain witnesses for the geometric measure of entanglement wei (). Finally, we discuss examples and compare our results to other methods for entanglement estimation. But before presenting the new results, let us shortly review the method presented in Refs. wir (); eisert ().
Ii The method
Let us consider the following situation: in an experiment, an entanglement witness has been measured and the mean value has been found. The task is now to derive from this single expectation value a quantitative statement about the entanglement present in the quantum state. In our case, we aim at providing a lower bound on the entanglement inherent in the state . That is, we are looking for statements like
(1) 
where denotes an arbitrary convex and continuous entanglement measure. We do not specify it at this point further. Naturally, we aim to derive an optimal bound and an estimate is optimal, if there is a state with and
In order to derive such lower bounds, let us consider the socalled Legendre transform of for the witness , defined via the maximization
(2) 
As this is defined as the maximum over all we have for any fixed that hence
(3) 
which is known as Fenchel’s inequality or Young’s inequality. The point is that the first term on the right hand side are the given measurement data, while the second term can be computed. Therefore, a measurable bound on has been obtained.
In order to improve this bound, note that knowing the data is, of course, equivalent to knowing for any Therefore, we can optimize over all and obtain
(4) 
This is a better bound than Eq. (3) and, as we will see, already the optimal bound in Eq. (1).
For our later discussion, it is important to note that this estimation has a clear geometrical meaning [see Fig. (1)]. As is convex, the minimal compatible with denoted by is convex, too remark1 (). Let us consider a generic affine lower bound on i.e., We have and in order to make the bound for a fixed as good as possible, we have to choose as small as possible. This leads to which is exactly the optimization in Eq. (2).
For a fixed slope we obtain by this method an affine bound (characterized by ) which is already optimal for a certain value , as it touches in this point. For any other mean value it delivers a valid, but not necessarily optimal bound. To obtain the optimal bound for any given we have to vary the slope This corresponds to the optimization in Eq. (4). Since is convex, we obtain for each the tight linear bound, showing that this optimization procedure gives indeed the best possible bounds in Eq. (1).
Three remarks are in order at this point. First, it was not needed that is an entanglement witness, we may consider an arbitrary observable instead. Second, we can also consider a set of observables at the same time. We just have to introduce a vector and replace by then all formulas remain valid. Finally, also for a nonconvex the method delivers valid bounds, however, then it is not guaranteed that the bounds are the optimal ones.
In any case, the method relies vitally on the ability to compute the Legendre transform in Eq. (2). The difficulty of this task clearly depends on the witness and on the measure chosen. At first sight, the task may seem hopeless, as the calculation of for mixed states is for many measures already impossible.
For instance, a large class of entanglement measures is defined via the convex roof construction. For that, one first defines the measure for pure states, and then defines for mixed states
(5) 
where the infimum is taken over all possible decompositions of i.e., over all and with Clearly, this infimum is very difficult to compute.
However, for the computation of the Legendre transform, this is not relevant: as one can easily prove (see Ref. wir () for details), for convex roof measures the maximization has to run only over pure states
(6) 
which simplifies the calculation significantly. In fact, this shows that for convex roof measures, which are by construction rather difficult to compute, the Legendre transform is rather simple to compute.
In Ref. wir () we have considered the entanglement of formation and the geometric measure of entanglement as entanglement measures, which are both convex roof measures. We have provided simple algorithms for the calculation of the Legendre transform, and for special witnesses, we calculated the Legendre transform of the geometric measure also analytically. In this paper, we will consider the concurrence and extensions thereof, and we will also present new analytical results for the geometric measure.
Finally, it should be noted that the optimization in Eq. (4) can be completely skipped. Any delivers already a valid bound. However, the optimal can easily be found numerically.
Iii The concurrence
As a first entanglement measure, let us discuss the concurrence. This quantity is defined for pure states as concurrence ()
(7) 
where the reduced state of for Alice. For mixed states this definition is extended via the convex roof construction in Eq. (5). For the special case of two qubits, the concurrence is a monotonous function of the entanglement of formation, moreover, the minimization in the convex roof construction can be explicitly performed wootters1 (). In the general case, it is not so directly connected to the entanglement of formation. The concurrence is an entanglement monotone vidal (), but it does not fulfill all desired axioms for an entanglement measure, e.g. it is not additive.
Here, we want to calculate the Legendre transform of for a generic witness Similar as it has been done in Ref. wir () for the entanglement of formation, we will construct an iterative algorithm for the optimization, which converges to the maximum as a fixed point.
First, as the concurrence is defined via the convex roof, it suffices in Eq. (2) to optimize over pure states only. Then, using the fact that for
(8) 
we can rewrite the Legendre transform as
(9) 
The idea is to write this maximization as an iteration that optimizes and in turn. For a fixed we perform the maximization over analytically and similarly we can find the optimal for a fixed Concerning the first step, note that for a fixed the optimal is simply given by
(10) 
Concerning the second step, if is fixed, we have essentially to solve an optimization problem like
(11) 
where is proportional to the original witness
Here, the second term is nothing but the qentropy, investigated in detail by Havrda, Charvat, Daróczy and Tsallis havrda (); daroczy (); wehrl (); tsallis (),
(12) 
for the case Therefore, we can try to write as an infimum via the Gibbs principle, similarly as it has been done for the von Neumann entropy in Ref. wir ().
So we make the ansatz
(13)  
(14) 
For the case of the von Neumann entropy these formulas just express the Gibbs variational principle, where is the free energy, and the inverse temperature was set to
We are mainly interested in the first minimization and have to compute and the where the minimum is attained. The point is that the second minimization has been solved already by Guerberoff and Raggio guerberoff () and the unique thermal state minimizing Eq. (14) for an arbitrary Hamiltonian and consequently is known. For the first minimization, it remains to find the Hamiltonian for which the given state is the thermal state.
In order to do this in practice, let us first recall the results of Ref. guerberoff (). For the case and the results of this reference state the following:
Let be a Hamiltonian with ground state energy , ground state degeneracy and let the energy of the first excited state be Define
(15) 
and the monotonically increasing function as
(16) 
where denotes the positive part of the operator Let be the inverse function to
Then, for the unique thermal state is given by
(17) 
where denotes the normalization, and for the thermal state is just the normalized projector onto the eigenspace corresponding to the lowest energy From this, it is clear that and are diagonal in the same basis.
Coming back to the minimizations in Eqs. (13, 14) let us assume that a density matrix with decreasing eigenvalues () is given, and the task is to compute the corresponding Hamiltonian with increasing eigenvalues Without loosing generality, we can choose
Let us first consider the case that is non degenerate and has full rank. Then, Eq. (17) implies that the eigenvalues of have to fulfill for all Since we are considering the case we have for due to Eq. (16) that From Eq. (17) is follows first that , and then hence
(18) 
For this Hamiltonian, we have also
(19) 
As a remark, first note that this solution fulfills as requested at the beginning. Second, it delivers justifying the ansatz in Eq. (17). Finally, it is easy to see that if is not of full rank or degenerated, the recipe for above works also and delivers a correct solution.
In summary, we can write the problem of the computation of the Legendre transform for the concurrence as:
(20) 
If and are fixed, we can compute the optimal as shown in the beginning. If and are fixed, we can determine the optimal as above. Finally, if and are fixed we choose as the eigenvector corresponding to the maximal eigenvalue of Therefore, we have an iterative optimization, which delivers an monotonously increasing sequence of values for the Legendre transform, with the actual value as a fixed point.
This algorithm can be implemented with few lines of code, examples will be discussed in Section V. It should be noted that in principle we cannot prove that the algorithm converges always to the global optimum, as the final fixed point may depend on the starting values of , and leading to an overestimation of the entanglement. In practice, however, the convergence behavior is quite good natured and the algorithm delivers a well suited tool for obtaining sharp bounds on the concurrence.
Finally, let us add that with the same method also extensions of the concurrence may be treated. For instance, one may consider quantities for pure states as With the convex roof extension, these are also entanglement monotones vidal (), and using the results of Ref. guerberoff (), the Legendre transform can be calculated in a similar manner as above.
In addition, there are several multipartite entanglement monotones, which can be seen as multipartite extensions of the concurrence meyerwallach (); brennen (); mintertreview (). For instance, for qubit states one can define the Meyer Wallach measure
(21) 
where the are all reduced onequbit states. Clearly, due to the similar structure as in Eq. (7) the methods from above can also be used to compute the Legendre transformation for these measures.
Iv The geometric measure
As the second entanglement measure, let us consider the geometric measure of entanglement wei (). This is an entanglement monotone multipartite systems, quantifying the distance to the separable states. The geometric measure is defined for pure states as
(22) 
i.e., as one minus the maximal overlap with pure fully separable states, and for mixed states via the convex roof construction.
The geometric measure is a multipartite entanglement measure, as it is not only a summation over bipartite entanglement properties. Despite of its abstract definition, it has turned our that can be used to quantify the distinguishability of multipartite states by local means hayashi (). As the geometric measure is one of the few measures for multipartite systems, which have a reasonable operational meaning and are at the same time proved to fulfill all of the conditions for entanglement monotones, it has been investigated from several perspectives, for instance it has been used to study multipartite entanglement in condensed matter systems geospin ().
In Ref. wir () the problem of calculating the Legendre transform for the geometric measure was already considered and the following results were obtained:
First, an iterative algorithm (in the same spirit as the algorithm for the concurrence in the previous section) has been derived for calculating the Legendre transform of arbitrary witnesses. Second, for the important case that the witness is of the form (or, equivalently ) an analytic formula of the Legendre transform has been derived, reading,
(23) 
Here, we want to generalize this result by determining analytically for the case that is diagonal in some special basis, e.g. the GHZstate basis. We first consider the special case of the GHZ state basis, then the formula for the general case can directly be written down.
So let us assume that
(24) 
is the witness for an qubit system, where with and is the GHZstate basis. Without loosing generality we assume that the are decreasingly ordered, i.e., but not necessarily positive.
The Legendre transform is given by
(25)  
where denotes the maximal eigenvalue of the operator In order to compute this maximal eigenvalue, we write the operator in the GHZ basis. is diagonal there and since the maximal overlap between the fully separable state and any of the GHZ states is (i.e., the geometric measure for GHZ states is wei ()), the matrix representation of has matrix elements with absolute values not larger than
Our claim is now that the optimal choice of is to take as a matrix with all entries and acting on the twodimensional space corresponding to the largest eigenvalues and To prove that this choice is really optimal, we show that the above mentioned choice is also optimal if we consider the more general class of all which have an overlap smaller or equal with all the GHZ states, but which are not necessarily product states.
We prove it by contradiction. Let us assume a different optimal solution and a corresponding eigenvector to the maximal eigenvalue The vectors can be written as and We assume without loosing generality that and The function to maximize is given by
(26) 
Since we are interested in the maximum, we can, without restricting generality assume that and are real and positive. The interesting terms for the discussion are
(27) 
is a scalar product, which is maximal if the vectors and are parallel. For given values of this may be prohibited by the constraint Then, however, it is clearly optimal to take an at the border of the domain, which has for one leading to a contradiction to the assumption on the form of Otherwise, we can choose the two vectors parallel, and is not at the border. Then, however, we can enlarge by increasing in (and, simultaneously in in order to keep the vectors parallel). This leads to another contradiction concerning the optimality of
Note that the class of the considered is strictly larger than the class of product vectors, since not for any pair of GHZ states and we can find a product vector such that has an overlap of with both of the However, we obtain an upper bound on the Legendre transform from this ansatz, which can be used for a valid lower bound on the entanglement measure. Further, one can check whether this bound is tight by direct inspection of and afterwards.
Having shown that the simplest choice of is optimal, the calculation of the Legendre transform reduces to a calculation of eigenvalues of a matrix, and we have:
(28)  
The generalization to other states besides GHZ states is straightforward: if the overlap is bounded by some other number (e.g. 1/4 for fourqubit cluster states), we only have to calculate the eigenvalues of some larger matrix (e.g. a matrix for fourqubit cluster states), in order to derive an analytical upper bound on We can summarize:
Observation. Let be an operator, where for all eigenvectors the overlap with fully separable states is bounded by Then the Legendre transform is bounded by
(29) 
where is a matrix with the entries on the diagonal, and offdiagonal entries The question whether this bound is the exact value, can be decided by direct inspection of the
This Observation allows for a simple calculation of a lower bound on if the fidelities of the basis states are known. First, the estimation is much simpler and faster compared with the iteration algorithm for arbitrary witnesses, since the optimization runs only over the largest eigenvalues of the possible witnesses . For the iteration algorithm, it would be necessary to consider all witnesses , which amounts to a variation over parameters . Second, the bounds on may become significantly better, compared with the estimation from a single fidelity, according to Eq. (23).
For example, the may be a graph state basis, where the fidelities have been determined from the expectation values of the stabilizer operators kiesel (). In Section V we will discuss an example for fourqubit states.
V Examples
In this section, we present several examples for the presented method and compare it with other estimation methods as well as with exact values of the entanglement measures. A Mathematica file with the used algorithms for the calculation of the Legendre transforms is available from the authors.
v.1 Concurrence for isotropic states
As a first example, let us consider isotropic states in a system, defined by
(30) 
which are a convex combination of a maximally entangled state and the totally mixed state. The parameter encodes the fidelity of i.e., For these states, the concurrence is known to be rungta ()
(31) 
In order to test our methods, we consider the standard witness for states of the for states of the from in Eq. (30), namely
(32) 
and estimate from its expectation value the concurrence, using our algorithm. The results for the case are shown in Fig. (2). It turns out that for this case, our lower bounds are sharp and reproduce the exact value of the concurrence.
v.2 Comparison with other estimation methods
Let us compare the presented estimation method with other methods of estimating entanglement measures. For this aim, we consider twoqubit states of the form
(33) 
where is a pure entangled state and is unknown and random separable noise. As entanglement measure we consider the entanglement of formation. This is, for the case of two qubits, equivalent to the concurrence, since it is a monotonous function of it. We consider six different methods for estimating the entanglement of formation:

For the twoqubit case we can exactly calculate the entanglement of formation due to the formula of Wootters wootters1 (). Clearly, since this requires complete knowledge of the density matrix, for an experimental implementation state tomography is needed.

In Refs. cab1 (); cab2 () a method to estimate the entanglement of formation or the concurrence from the separability criterion of the positivity of the partial transpose (PPT) or the computable cross norm or realignment criterion (CCNR) was presented. Experimentally, this approach requires again state tomography.

We can also take the witness which is proper for the states of the form This witness might not be the optimal one for the state under investigation, since the noise is not known and in general white. However, we can use the Legendre transform with the algorithm of Ref. wir () to estimate the entanglement of formation from it. Equivalently, we can use the algorithm of Section III to estimate the concurrence. Experimentally, this method does not require state tomography, only three local measurements are needed for the measurement of the witness jmo ().

In Ref. mintert1 () lower bounds on the concurrence from measurements on two copies of the state have been derived. For the case of two qubits we can use them to bound also the entanglement of formation. A measurement of a single observable on two copies can always be expressed as a function of mean values of local observables on a single copy. In this case, however, for such an implementation effectively state tomography is needed.

In Ref. mintert2 () a method for estimating the concurrence from special entanglement witnesses has been derived. Namely, it was shown that
(35) where is a witness belonging to the reduction criterion. We use this method with the witness for the state With this choice, the method automatically reproduces the exact value for the case and it can be seen that Eq. (35) is nothing but the Legendre transform for a special choice of the slope Experimentally, this method would also require three local measurements mintert2 (); jmo ().
As an example, we considered in Eq. (33) the pure state and the state was randomly (in the HilbertSchmidt measure) chosen from the set of separable states zs (). For a fixed we calculated all the above mentioned values depending on the noise level and finally averaged over hundred realizations of The results are shown in Fig. (3) and Table 1.
Method  CAF1  WIT  CAF2  2COP  RWIT 

# of measurements  9  3  3  9  3 
for  96.0 %  94.8 %  94.8 %  70.1 %  63.0 % 
for  86.0 %  72.0 %  72.0 %  5.9 %  7.0 % 
One can clearly see that the methods two, three and four result in bounds on the entanglement of formation, which are very close to the exact result. The second method is the best bound, however, it requires complete knowledge of the state. The third method is by construction better than the fourth method (as the Legendre transform delivers by construction the best possible bounds from a given witness). In this example, however, they are practically equivalent. The fifth and sixth method deliver good results if is close to a pure state.
v.3 Geometric measure for fourqubit states
As a third example we discuss the geometric measure of entanglement. In order to demonstrate the method in Section IV, we consider an experimental situation similar to the one in Ref. kiesel (). In this experiment, a fourphoton cluster state
(36) 
has been prepared using parametric down conversion and a controlled phase gate. The fidelity of the target state was then determined by the measurement of all stabilizer operators. These operators are given by the local observables , , , and , and products of these observables. The cluster state is an eigenstate (with eigenvalue ) of all these observables, and from their expectation values the fidelity can be determined kiesel (); tothguehne ().
Using the fact that the maximal overlap of the cluster state with fully separable states equals (i.e., the geometric measure is markham1 ()) Eq. (23) can be used to bound from this fidelity. There are, however, also other common eigenstates of the with different eigenvalues. These states are orthogonal to the cluster state and form the socalled cluster state basis. All states in this basis share the same entanglement properties and their fidelities can also be determined from the mean values of the In fact, the knowledge of all is equivalent to the knowledge of all fidelities.
In order to investigate how the information about the fidelities of all states in the cluster state basis can be used for the estimation of entanglement, we consider the simple case that only three fidelities are larger than zero, , and For a given triple of fidelities we may first consider the maximal fidelity and then use Eq. (23) to obtain a lower bound on Alternatively, we can use the methods of Section IV and consider all three fidelities at the same time. In practice, this gives a lower bound on by the optimization problem
(37) 
where is defined as in Eq. (29) for Any set of delivers already a valid lower bound, and the optimum over all is easily found.
The results are plotted in Fig. (4). One can clearly see, that taking all fidelities into account, improves the lower bounds significantly.
Vi Conclusion
In conclusion, we have investigated how entanglement measures can be estimated from incomplete experimental data. We have shown that the method of Legendre transforms can successfully be applied to the concurrence and extensions thereof. Furthermore, we have presented an analytical way to estimate the geometric measure if the fidelities of certain basis states are known. Extending the presented methods to other entanglement measures is an interesting task for further study.
We thank J. Eisert, N. Kiesel, A. Osterloh, J. Siewert and W. Wieczorek for discussions. This work has been supported by the FWF, the DFG, and the EU (OLAQUI, PROSECCO, QUPRODIS, QICS, SCALA).
References
 (1) M.B. Plenio and S. Virmani, Quant. Inf. Comp. 7, 1 (2007).
 (2) R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, quantph/0702225.
 (3) C.H. Bennett, D.P. DiVincenzo, J.A. Smolin, and W.K. Wootters, Phys. Rev. A 54, 3824 (1996).
 (4) P. Rungta, V. Buzek, C. M. Caves, M. Hillery, and G. J. Milburn, Phys. Rev. A 64, 042315 (2001).
 (5) D.A. Meyer and N.R. Wallach, J. of Math. Phys. 43, 4273 (2002).
 (6) G. Vidal and R.F. Werner, Phys. Rev. A 65, 032314 (2002).
 (7) T.C. Wei and P.M. Goldbart, Phys. Rev. A 68, 042307 (2003).
 (8) A. Osterloh and J. Siewert, Phys. Rev. A 72, 012337 (2005).
 (9) F. Mintert, A.R.R. Carvalho, M. Kus, and A. Buchleitner, Phys. Rep. 415, 207 (2005).
 (10) W.K. Wootters, Phys. Rev. Lett. 80, 2245 (1998).
 (11) B.M. Terhal and K.G.H. Vollbrecht, Phys. Rev. Lett. 85, 2625 (2000).
 (12) P. Rungta and C.M. Caves, Phys. Rev. A 67, 012307 (2003).
 (13) R. Lohmayer, A. Osterloh, J. Siewert, and A. Uhlmann, Phys. Rev. Lett. 97, 260502 (2006).
 (14) K. Chen, S. Albeverio and S.M. Fei, Phys. Rev. Lett. 95, 040504 (2005).
 (15) K. Chen, S. Albeverio and S.M. Fei, Phys. Rev. Lett. 95, 210501 (2005).
 (16) F. Mintert and A. Buchleitner, Phys. Rev. Lett. 98, 140505 (2007).
 (17) J.I. de Vicente, Phys. Rev. A 75, 052320 (2007).
 (18) A. Datta, S.T. Flammia, A. Shaji and C.M. Caves, Phys. Rev. A 75, 062117 (2007).
 (19) M. Horodecki, P. Horodecki, and R. Horodecki, Phys. Lett. A 223, 1 (1996).
 (20) B. Terhal, Phys. Lett. A 271, 319 (2000).
 (21) G. Tóth and O. Gühne, Phys. Rev. Lett. 94, 060501 (2005).
 (22) M. Bourennane, M. Eibl, C. Kurtsiefer, S. Gaertner, H. Weinfurter, O. Gühne, P. Hyllus, D. Bruß, M. Lewenstein, and A. Sanpera, Phys. Rev. Lett. 92, 087902 (2004).
 (23) N. Kiesel, C. Schmid, U. Weber, G. Tóth, O. Gühne, R. Ursin and H. Weinfurter, Phys. Rev. Lett. 95, 210502 (2005).
 (24) H. Häffner, W. Hänsel, C.F. Roos, J. Benhelm, D. Chekalkar, M. Chwalla, T. Körber, U.D. Rapol, M. Riebe, P. O. Schmidt, C. Becher, O. Gühne, W. Dür, and R. Blatt, Nature 438, 643 (2005).
 (25) C.Y. Lu, X.Q. Zhou, O. Gühne, W.B. Gao, J. Zhang, Z.S. Yuan, A. Goebel, T. Yang, J.W. Pan, Nature Phys. 3, 91 (2007).
 (26) M. Horodecki, P. Horodecki, and R. Horodecki, Phys. Rev. A 59, 1799 (1999).
 (27) F. Verstraete and M.M. Wolf, Phys. Rev. Lett. 89, 170401 (2002).
 (28) F. Brandão, Phys. Rev. A 72, 022310 (2005).
 (29) K.M.R. Audenaert and M.B. Plenio, New J. Phys. 8, 266 (2006).
 (30) F. Mintert, Phys. Rev. A 75, 052302 (2007).
 (31) O. Gühne, M. Reimpell, and R.F. Werner, Phys. Rev. Lett. 98, 110502 (2007).
 (32) J. Eisert, F. Brandão, and K. Audenaert, New J. Phys. 9, 46 (2007).
 (33) Here, we also use the conditions that is continuous, and that the state space is compact. These conditions may be relaxed by requiring only that the sublevel sets are compact. See Ref. reimpell () for a more detailed discussion.
 (34) M. Reimpell, Quantum Information and Convex Optimization, (PhDthesis, University of Braunschweig, 2007).
 (35) G. Vidal, J. Mod. Optics 47, 355 (2000).
 (36) J. Havrda and F. Charvat, Kybernetika 3, 30 (1967).
 (37) Z. Daróczy, Inform. Control 16, 36 (1970).
 (38) A. Wehrl, Rev. Mod. Phys. 50, 221 (1978).
 (39) C. Tsallis, J. Stat. Phys. 52, 479 (1988).
 (40) G.R. Guerberoff and G.A. Raggio, J. Math. Phys. 37, 1776 (1996).
 (41) G.K. Brennen, Quant. Inf. Comput. 3, 619 (2003).
 (42) D. Markham, A. Miyake, and S. Virmani, New J. Phys. 9, 194 (2007).
 (43) M. Hayashi, D. Markham, M. Murao, M. Owari, and S. Virmani, Phys. Rev. Lett. 96, 040501 (2006).
 (44) T.C. Wei, D. Das, S. Mukhopadyay, S. Vishveshwara and P.M. Goldbart, Phys. Rev. A 71, 060305(R) (2005).
 (45) O. Gühne, P. Hyllus, D. Bruß, A. Ekert, M. Lewenstein, C. Macchiavello and A. Sanpera, J. Mod. Optics 50, 1079 (2003).
 (46) K. Życzkowski and H.J. Sommers, J. Phys. A: Math. Gen. 34, 7111 (2001).