Spectral decomposition based separability
criteria: a numerical survey
J. Batle1, M. Casas1, A. R. Plastino1,2 and A. Plastino1,3
1 - Departament de Física, UIB, Spain E-mail: [email protected]
2 - Faculty of Astronomy and Geophysics, UNLP, and CONICET,
La Plata, Argentina.
3 - Department of Physics, UNLP, La Plata, Argentina
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 r 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 qentropic, based upon a spectral decomposition and quantify the relations that link
them by means of a Monte Carlo exploration in volving the (NM2-1)-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 rAB, or some
operator that includes the reduced matrices rA=TrB[rAB] and rB=TrA[rAB] 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.
The criteria
i) PPT
rm ,n  m rˆ n
r separable
r PT   n rˆ m  0
m ,n
This is a spectral criterion indeed (the information about non-separability is encoded
in the negative eigenvalue of rPT). Simple to operate, remains still the strongest
necessary condition for separability. It is also sufficient for 2x2 and 2x3 systems.
ii) Reduction
oˆ1 ( r AB )  r A  I B  r AB  0 & oˆ2 ( r AB )  I A  rB  r AB  0
A bit more complicated spectral criterion [4] (matrices obtained from operators O1
and O2 contain the reductions rA=TrB[rAB] and rB=TrA[rAB]). It is implied by PPT,
that is, if r has PPT then
must comply with reduction. The properties related to
distillability will be discussed later on.

iii) Majorization: if rAB in N=N1xN2 is separable or classically correlated, then
 ( r AB )   ( r A ) and  ( r AB )   ( rB )  vector eigenvalues in
decreasing order
k1

j 1
k1
( j)
AB
   , for all k1  N1  1 &
j 1
( j)
A
k2

j 1
( j)
AB
k2
  (Bj ) , for all k 2  N 2  1
j 1
Majorization is closely related to reduction.
iv) q-entropies: if rAB is separable, the classical entropic inequalities hold [5]
Rq ( A | B)  Rq ( r AB )  Rq ( r B ), Rq ( B | A)  Rq ( r AB )  Rq ( r A )
0  ) B | A ( qR 


0  )A | B ( qR 

Rq is the Rényi entropy, with wq=Tr[r q], q real, which reads as Rq = ln (wq)/(1 - q).
Conversely, one also can use the Tsallis entropy Sq=(1-wq) /(q - 1). We focus the very
interesting case of q  
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 (NM2-1)
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 r following a recent work by
Zyczkowski et al. [6,7]. A general state is expressed as r = 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 RN-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 r 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.
3xN2 majorization and q=infinity nearly coincide
2xN2 q=infinity
2xN2 majorization
2xN2 PPT
and reduction
Notice the exponential decay
of PPT 2xN2 and 3xN2
3xN2 reduction
All criterions except PPT
decay in a linear fashion
3xN2 PPT
Probability that a state r complies with one of the several criterions considered
2xN2
PPT-majorization
PPT-q=infinity
3xN2
PPT-reduction
The recovering means that
as we increase N, all
criterions will lead to the
same conclusion about r
3xN2
Probability that a state r complies or violates PPT and some other critrerion
3xN2
2xN2
3xN2
2xN2
Majorization-q= 
reduc-major
reduc-q=infinity
In 3xN2 , the recovering
may occur for high N
Probability of coincidence for reduction and majorization, and reduction and entropic
q=infinity (a). This last one is also considered together with majorization (b).
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 r 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]).

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 2xN2 reduction case, to be compared with the linear behavior for majorization.
The 3xN2 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 2xN2 states are distillable, whereas
for 3xN2 states more than 50% can be distilled.
2xN2
violation of reduction
violation of majorization
3xN2
2xN2
3xN2
Probability of finding a state that by violationg reduction or majorization is distillable
Conclusions
By performing a Monte Carlo calculation, we compute the a priori probability that
a mixed state rAB of a bipartite system in any dimension N=N1xN2 complies with a
given criterion.
All criteria rely in one way or another in the spectra of some operator involving the
state rAB and/or its reductions rA=TrB[rAB] and rB=TrA[rAB]. 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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