1. J. Batle 1, M. Casas 1, A. R. Plastino 1,2 and A. Plastino 1,3 1 - Departament de Física, UIB, Spain E-mail: vdfsjbv4@uib.es 2 - Faculty of Astronomy and Geophysics, UNLP, and CONICET, La Plata, Argentina. 3 - Department of Physics, UNLP, La Plata, Argentina Spectral decomposition based separability criteria: a numerical survey

2. Introduction A complete characterisation of quantum entanglement [1] has not yet been obtained, the associated separability problem being a difficult one indeed. The goal is to be in a position to assert whether a given state  describing a quantum system is useful or not for quantum information processing purposes. For bipartite Hilbert spaces of low dimensionality (d=NxM=2x2 and 2x3) the Positive Partial Transpose (PPT) criterion [2,3] is the strongest one, providing a necessary and sufficent condition for quantum separability and being only necessary for N>6. In the present endeavour we revisit the application of different separability criteria, such as the reduction, majorization and q- entropic, based upon a spectral decomposition and quantify the relations that link them by means of a Monte Carlo exploration in volving the (NM 2 -1)-dimensional space of pure and mixed states. All these criterions rely, in one way or another, on the spectra of matrices that involve either the partial transposition of the general density operator  AB, or some operator that includes the reduced matrices  A =Tr B [  AB ] and  B =Tr A [  AB ] of both subsystems, e.g the reduction criterion. Particular instances are i) the q-entropic criterion and ii) the majorization criterion. As we will see, the latter constitutes a lower bound to the former in the volume set of all mixed states. Although the separability issue is common to all these criterions, distillability is very important and so is reflected in the cases of reduction and majorization.

4. The criteria i) PPT This is a spectral criterion indeed (the information about non-separability is encoded in the negative eigenvalue of  PT ). Simple to operate, remains still the strongest necessary condition for separability. It is also sufficient for 2x2 and 2x3 systems. ii) Reduction A bit more complicated spectral criterion [4] (matrices obtained from operators O1 and O2 contain the reductions  A =Tr B [  AB ] and  B =Tr A [  AB ]). It is implied by PPT, that is, if  has PPT then must comply with reduction. The properties related to distillability will be discussed later on.  separable

5. iii) Majorization: if  AB in N=N 1 xN 2 is separable or classically correlated, then Majorization is closely related to reduction. vector eigenvalues in decreasing order iv) q-entropies: if  AB is separable, the classical entropic inequalities hold [5] R q is the Rényi entropy, with  q =Tr[  q ], q real, which reads as R q = ln (  q )/(1 - q). Conversely, one also can use the Tsallis entropy S q =(1-  q ) /(q - 1). We focus the very interesting case of

6. The numerical method A better understanding of these criteria can be quantified by computing in the space of the volume set of mixed states in arbitrary dimensions the number of states that comply with a given criterion. This is a very complex space with (NM 2 -1) dimensions and presents very interesting features. In other words, we find the a priori probability that a state complies with a given criterion. Even in the case of two-qubits this is clearly a non-trivial task. To such an end we make a survey on the space S of all states  following a recent work by Zyczkowski et al. [6,7]. A general state is expressed as  = U D[{ i }] U +, where the group of unitary matrices U(N) is endowed with a unique, uniform Haar [6-8] measure and the diagonal N-simplex D[{ i }] is naturally given by the standard Leguesbe measure on R N-1. Also, a systematic comparison between all criterions in any dimension allow us to describe and quantify the way they are related one to the other. By generating using a Monte Carlo method the states  according to this measure, we numerically compute those probabilities. The implication chain discussed previously translates into probabilities being smaller if a given criterion is weaker. Because all criterions are necessary for separability, the lesser the probability (or volume occupied), the stronger the criterion is. This is clearly seen in the next picture. There we plot the probabilities of a state complying with PPT, reduction, majorization or q-entropic.

7. Notice the exponential decay of PPT 2xN 2 and 3xN 2 Probability that a state  complies with one of the several criterions considered 2xN 2 PPT and reduction 3xN 2 PPT 3xN 2 reduction 2xN 2 majorization 2xN 2 q=infinity 3xN 2 majorization and q=infinity nearly coincide All criterions except PPT decay in a linear fashion

8. 2xN 2 3xN 2 PPT-reduction PPT-majorization PPT-q=infinity Probability that a state  complies or violates PPT and some other critrerion The recovering means that as we increase N, all criterions will lead to the same conclusion about 

9. 2xN 2 3xN 2 reduc-major reduc-q=infinity 2xN 2 3xN 2 Majorization-q= Probability of coincidence for reduction and majorization, and reduction and entropic q=infinity (a). This last one is also considered together with majorization (b). In 3xN 2, the recovering may occur for high N

10. Distillability Distillation is the process by which one concentrates the entanglement contained in an inseparable mixed state and converts it to active singlets by means of LOCC. Not all inseparable states can be distilled (bound entanglement). In 2x2 and 2x3 systems, all states  can be distilled. The problem comes in for higher dimensions. There is no general criterion to discriminate whose states can be distilled. However, two criterions are sufficient: reduction [4] and majorization. Their violation is sufficient for distillability. The implication reduction majorization with regard to separability also holds for distillability (recently shown in [9]). lower bound The computation using the present numerical procedure of the proportion of states that violate reduction and majorization can help us quantify giving a lower bound to the volume set occupied by those states that can be distilled, and thus useful for quantum information processing purposes. From figure one notices the steep growth in the 2xN 2 reduction case, to be compared with the linear behavior for majorization. The 3xN 2 reduction case is not that spectacular, and goes almost lineal with N. The implication reduction majorization is patent in the probability plot: the volume occupied by states violating reduction is greater than the one by majorization. It is worth stressing that around N=20 nearly all 2xN 2 states are distillable, whereas for 3xN 2 states more than 50% can be distilled.

11. 2xN 2 3xN 2 violation of reduction violation of majorization Probability of finding a state that by violationg reduction or majorization is distillable

12. Conclusions By performing a Monte Carlo calculation, we compute the a priori probability that a mixed state  AB of a bipartite system in any dimension N=N 1 xN 2 complies with a given criterion. All criteria rely in one way or another in the spectra of some operator involving the state  AB and/or its reductions  A =Tr B [  AB ] and  B =Tr A [  AB ]. Very special cases are the majorization criterion and the q= -entropic criterion. Computing explicitely the volumes occupied by states according to a given criterion, and bearing in mind that the PPT is only a necessary separability criterion for N>6, one notices that PPT is the strongest criterion, implying all the others. The relations between several criterions are quantified by computing the ratios of coincidence in the volume set of states. We have numerically verified the assertion made in [10] that majorization is not implied by the relative entropic criteria. In general, majorization probabilities constitute lower bounds for relative q-entropic positivity. Finally, the issue of distillability is considered in the cases of reduction and majorization criterions, providing explicit lower bounds for any dimension to the volume of states that can be distilled.

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