aa r X i v : . [ qu a n t - ph ] O c t Parametrization of spin-1 classical states
Olivier Giraud , Petr Braun , and Daniel Braun Univ. Paris-Sud, CNRS, LPTMS, UMR 8626, Orsay, F-91405, France Fachbereich Physik, Universität Duisburg–Essen, 47048 Duisburg, Germany Institute of Physics, Saint-Petersburg University, 198504 Saint-Petersburg, Russia Laboratoire de Physique Théorique, Université de Toulouse, CNRS, 31062 Toulouse, France (Dated: September 30, 2011)We give an explicit parametrization of the set of mixed quantum states and of the set of mixedclassical states for a spin–1. Classical states are defined as states with a positive Glauber-SudarshanP-function. They are at the same time the separable symmetric states of two qubits. We explorethe geometry of this set, and show that its boundary consists of a two-parameter family of ellipsoids.The boundary does not contain any facets, but includes straight-lines corresponding to mixtures ofpure classical states.
PACS numbers:
I. INTRODUCTION
The rise of quantum information theory has led to alarge interest in the geometry of specific sets of quantumstates [1]. The most general quantum state of a quantumsystem with d -dimensional Hilbert space H is given by adensity operator ρ that acts on H . The density opera-tor is a Hermitian, semi-definite positive operator withtrace 1. Diagonalization shows immediately that it canalways be written as a convex sum of projectors onto itseigenstates. The set N of all physical density operatorsis therefore the convex hull of projectors onto all purestates in H . Certainly the most popular set of states inquantum information theory is the set of separable states,defined for a physical system that can be partitioned intoat least two subsystems. If ρ can be written as a convexsum of tensor products of projectors onto pure states ofthe subsystems, that state is called “separable”, and “en-tangled” otherwise [2]. Clearly, the set S of all separablestates is a subset of N . Knowing the geometry, and inparticular the surface ∂ S of the set S , is an importantbut difficult problem, as it would allow to determine im-mediately whether a given state is inside or outside S ,or, in other words, whether it is entangled or not. In thecase of two qubits, ∂ S was shown to be smooth in theinterior of N [3]. Furthermore, for a general bipartitestate, it was shown that ∂ S is not a polytope [4] and, us-ing non-linear entanglement witnesses, that it does notcontain any facets [5].Recently we introduced the convex set C ⊆ N of “clas-sical states” of a spin (or angular momentum) with totalangular momentum j [6]. It is defined as the convex hullof projectors onto coherent states of SU (2) , which havethe physical interpretation of having minimal quantumuncertainty of the angular momentum vector, i.e. theyresemble as much as possible a point in classical phasespace. The interest of classical states is that they are de-fined even for a single spin, i.e. when the question of en-tanglement does not even arise. Furthermore, they allowa definition of what a genuinely ”quantum” state mightbe. Indeed, one may define a measure of “quantumness” [7] of a spin state by measuring its distance from C , justas the distance to S provides a measure of entanglement(see [8] for an overview of this type of entanglement mea-sure). If distance from C is measured through the Buresdistance [1, 9], quantumness of symmetric multi-qubitstates becomes essentially equivalent to their geometri-cal entanglement [10]. Also note that the set of classicalstates of a spin–1 is identical to the set of separable sym-metric states (under the exchange of particles) of twoqubits.States of a spin with maximal quantumness with giventotal angular momentum j (i.e. with Hilbert space di-mension j + 1 ), the “Queens of Quantum”, can alwaysbe found among pure states [7]. However, if ∂ C containsfacets, there might exist mixed states with the same max-imal quantumness. Knowing the form of the surface ofthe set of classical states is therefore important. In [7] itwas shown that for spin– states maximal quantumnessis reached only for pure states, but for larger j maximallyquantum states might comprise mixed states. After whatwas said above about how little is known about the sur-face ∂ S of the set of separable states, one might expectthat determining ∂ C is a difficult problem as well. Thisis indeed the case, but nevertheless, here we give a com-plete characterization of ∂ C for the case of a qutrit (i.e. athree state system, corresponding to a pseudo-angularmomentum j = 1 ). We show that in this case C , as S ,is not a polytope either, but rather a continuous familyof ellipsoids. We also show that the surface of C containsfamilies of straight lines. II. SPIN– CASE
Let us first consider the trivial case of a spin– system.In this case it was shown in [6] that the set C of classicalstates coincides with N . Any × density matrix canbe expanded over the basis of Pauli matrices σ a as ρ = 12 + X a u a σ a (1)with the × identity matrix and u a are real numberswith a = x, y, z . The matrix ρ given by (1) is Hermitianand has trace 1, therefore it belongs to N if and only ifit is positive. The characteristic polynomial of ρ can beput under the form det ( x − ρ ) = x − x + 1 − P a u a . (2)Its roots are positive if and only if X a u a ≤ . (3)Equation (3) is thus the necessary and sufficient condi-tion for ρ ∈ N in terms of the coordinates u a whichparametrize ρ . The boundary ∂ N of N correspondsto points where one of the eigenvalues of ρ vanishes.In terms of coordinates u a it is given by the equa-tion P a u a = 1 . The parametrization of ∂ N is theparametrization of a sphere.The results above correspond to the usual picture ofthe Bloch sphere for spin– . The vector u is the Blochvector, and the boundary of (classical) states is theboundary of the sphere, corresponding to rank-one ma-trices, or pure states, with Bloch vector of length 1. Sucha simple picture does not exist for higher spins. Let usnow consider the case of spin– states. III. CLASSICALITY CRITERION FOR SPIN–1STATES
We start with the expansion of a mixed spin–1 stateover the basis formed by the × angular momentum ma-trices, J a , a = x, y, z , together with the ( J a J b + J b J a ) / and the × identity matrix . We define a vector u and a matrix W through coefficients of this expansion,as ρ = 13 + 12 u . J + 12 X a,b = x,y,z (cid:18) W ab − δ ab (cid:19) J a J b + J b J a . (4)The coefficients u and W are related with ρ through u a = tr ( ρJ a ) , W ab = tr ρ ( J a J b + J b J a ) − δ ab . (5)Note that u is real and that W is a real symmetric matrix,with tr W = 1 .The expression (4) ensures that ρ is Hermitian with tr ρ = 1 . Thus the set N of density matrices is the set ofmatrices of the form (4) with ρ ≥ . According to [6], ρ is a density matrix associated with a classical state if andonly if the real symmetric × matrix Z with matrixelements Z ab = W ab − u a u b (6)is non-negative, thus the set C of classical density matri-ces is the set of matrices of the form (4) with Z ≥ . IV. SET N OF DENSITY MATRICES
The class of density matrices N comprises Hermitiannon-negative matrices with trace 1. Its parametrizationis important in many applications and can be achievedin several ways. One of these is based on the represen-tation ρ = U diag[ λ ..λ j +1 ] U − , λ i ≥ , P λ i = 1 ,where U runs over a subset of the unitary group cho-sen such that each ρ is obtained once and only once. Aparametrization for the case j = 1 using Gell-Mann ma-trices is considered in [11]; see also [12] for the closelyrelated problem of × coherence matrices of nonparax-ial light. Another method uses the factorization ρ = V V † where V is upper triangular [13]. Here we shall give analternative representation based on the formula (4).Parameters u and W have the nice feature, similarto the Bloch picture in the two-dimensional case, thatunder rotation of the coordinate system with an orthog-onal rotation matrix O , ρ is transformed into a matrixwith parameters O u and OW O T , i.e. u and W trans-form with the same rotation O . Thus it will be conve-nient to express them in a basis where W is diagonal, W = diag[ µ x , µ y , µ z ] ; we shall write the result as ρ = ρ ′ + 12 u . J ,ρ ′ = 12 (cid:0) µ x J x + µ y J y + µ z J z (cid:1) . (7)Considering that tr W = 1 and that, in a state ρ withangular momentum 1, we have ≤ tr ρJ a ≤ , ≤| tr ρ J | ≤ we obtain the necessary conditions on theparameters in (7), X a = x,y,z µ a = 1 , − ≤ µ a ≤ , a = x, y, z, (8) u x + u u + u z ≤ . (9)For the “truncated” matrix ρ ′ = ρ | u =0 conditions (8)are also sufficient to guarantee that ρ ′ ∈ N . Indeed,direct calculation shows that the eigenvalues of ρ ′ are λ ′ a = 1 − µ a ≥ , a = x, y, z, (10)while the corresponding eigenvectors | v a i are eigenvectorsof J a with eigenvalue zero. Since h v a | J | v a i = , we have h v a | ρ | v a i = λ ′ a . These averages give an upper bound tothe smallest eigenvalue of ρ . It immediately follows thatif ρ belongs to N then so does ρ ′ but not vice versa.In fact, a stronger statement can be made. Let ρ κ = ρ ′ + κ u . J be a density matrix differing from ρ by a pos-itive factor κ in the part linear in J . Then the lowesteigenvalue of ρ κ is a monotonically decreasing functionof κ . Consequently if ρ κ with some κ = κ belongs to N then so do all matrices with ≤ κ < κ . These assertionsfollow from the following theorem of perturbation theory(for a proof, see the Appendix): Let H = H + κV, κ ≥ , be a Hermitian matrix whose spectrum is bounded frombelow, and E ( κ ) , ψ ( κ ) be its lowest eigenvalue andeigenstate. Suppose that E (0) is non-degenerate and h ψ (0) | V | ψ (0) i = 0 . Then E ( κ ) is a monotonically de-creasing function. Setting H → ρ ′ , V → (1 / u . J wecome to the statement above.Let us find the constraints sufficient and necessary toguarantee non-negativity of ρ . The characteristic poly-nomial of ρ written in the form (4) can be presented as det ( x − ρ ) = x − x + ax − det ρ (11)with a = 14 (cid:18) −| u | + 1 − tr W − (cid:19) . (12)Since ρ is Hermitian the three roots of the polynomial arereal. According to Descartes’ rule of signs, a polynomialof the form x − x + ax − b with three real roots has allits roots positive if and only if a and b are positive. Thus ρ ∈ N iff det ρ ≥ and − | u | + 1 − tr W ≥ . (13)The latter condition defines a sphere in the u -space.Since it does not depend on the basis in which u and W is expressed, we can write Eq. (13) in the basis where W is diagonal; in that basis it becomes | u | ≤ µ x µ y + µ x µ z + µ y µ z . (14)One can check that the condition det ρ ≥ can be rewrit-ten h u | W | u i − | u | + 1 − tr W − det W ≥ , (15)which in the basis where W is diagonal becomes X a u a (1 − µ a ) ≤ (1 − µ x )(1 − µ y )(1 − µ z ) . (16)When all µ a differ from 1, it defines an ellipsoid in the u − space lying inside both spheres (9) and (14). Indeed,the squared radius of the ellipsoid along the x -axis forinstance is given by r x = (1 − µ y )(1 − µ z ) = µ x + µ y µ z (17)and using the fact that − µ x ≥ we get r x ≤ − µ x + r x = 1 + µ x µ y + µ x µ z + µ y µ z . (18)On the other hand, the inequality µ y ≥ − µ z (comingfrom µ x ≤ ) yields r x = (1 − µ y )(1 − µ z ) ≤ − µ z ≤ , (19)thus the ellipsoid also lies within the sphere of radius 1.However, when one or two µ a are equal to 1, then Eqs. (9)and (14) have to be taken into account. We finally obtainthat ρ ∈ N if and only if the ( u a , µ a ) verify one of thefollowing conditions: 1. All µ a are such that − ≤ µ a < , and X a u a µ a + µ b µ c ≤ , (20)with P a µ a = 1 and b, c are the two indices whichdiffer from a (this automatically implies (9),(14));2. Exactly one µ a is equal to 1, say µ z = 1 . Then µ y = − µ x with − < µ x < , and u x = u y = 0 .Equation (14) yields u z ≤ − µ x and is obviouslymore restrictive than (9);3. Two of the µ a are equal to 1, say µ y = µ z = 1 ,then µ x = − , u x = u y = u z = 0 . This corre-sponds, up to rotation, to the state | , ih , | (in | j, m i notation).The boundary ∂ N of N corresponds to points whereone of the inequalities (13) or (15) becomes an equality.For µ a = 1 (case 1 above), Eq. (15), equivalent to theequation of the ellipsoid (16), is more restrictive than(13) so that ∂ N coincides with the surface of the ellip-soid (20). The cases when one or two µ a are equal to 1correspond to cases 2 and 3, where equality is reached in(16). Therefore, the surface ∂ N is the union of pointscorresponding to case 1 when (20) is an equality, and ofpoints corresponding to cases 2 and 3.Points of ∂ N corresponding to case 1, with equality in(20), belong to a two-parameter set of ellipsoids that canbe parametrized by µ a ∈ [ − , and u = √ µ x + µ y µ z cos θ cos ϕ √ µ y + µ x µ z cos θ sin ϕ √ µ z + µ x µ y sin θ , θ ∈ [0 , π ] , ϕ ∈ [0 , π [ . (21)Any density matrix ρ ∈ N for spin 1 is parametrizedby 8 real numbers. Thus ∂ N should be parametrized by 7numbers. For points corresponding to case 1, besides µ , µ , θ and ϕ , the three remaining parameters correspondto the three angles that parametrize the orthogonal ma-trix required to diagonalize W . These orthogonal trans-formations also include transpositions of axes x, y, z ; toget each matrix once and only once we must introducerestrictions on µ a , say, µ z ≤ µ y ≤ µ x . One way to do sois to use the eigenvalues λ ′ a of ρ ′ as auxiliary variables,with µ a = 1 − λ ′ a , setting λ ′ x λ ′ y λ ′ z = sin θ ′ sin ϕ ′ sin θ ′ cos ϕ ′ cos θ ′ (22)with θ ′ ∈ ]0 , arctan(1 / cos( ϕ ′ ))] and ϕ ′ ∈ ]0 , π/ . Pointsof ∂ N corresponding to cases 2 and 3 are of measure zeroon the surface. V. SET C OF CLASSICAL STATE DENSITYMATRICES
We now characterize the set C of classical states. A nec-essary and sufficient condition for classicality of a state ρ ∈ N is Z ≥ , where Z is given by Eq. (6). Thecharacteristic polynomial of Z reads det ( x − Z ) = x − tr Zx + (tr Z ) − tr Z x − det Z. (23)Since Z is real symmetric the three roots of the charac-teristic polynomial are real. As in the previous section,Descartes’ rule of signs implies that the roots are posi-tive, i.e. ρ is a density matrix associated with a classicalstate, if and only if the three conditions tr Z ≥ , (tr Z ) ≥ tr Z and det Z ≥ (24)are fulfilled. In terms of u and W one has tr Z = 1 − | u | (25) tr Z = tr W − h u | W | u i + | u | (26) det Z = det ( W − | u ih u | ) . (27)Using (9) and (25) we see that condition tr Z ≥ is ful-filled by any density matrix. The two remaining condi-tions on Z do not depend on the basis in which u and W is expressed, thus we can write them in the basis where W is diagonal with eigenvalues µ x , µ y , µ z . Using Eqs. (25)–(26), condition (tr Z ) ≥ tr Z is equivalent to X a u a (1 − µ a ) ≤ µ x µ y + µ x µ z + µ y µ z . (28)Condition det Z ≥ becomes µ y µ z u x + µ x µ z u y + µ x µ y u z ≤ µ x µ y µ z . (29)A state ρ belongs to C if and only if it verifies Eqs. (8)–(9)and (28)–(29). A necessary condition for Z to be positiveis that its diagonal elements µ a − u a are positive, whichentails positivity of the µ a and thus µ a ∈ [0 , . If all µ a differ from 0 and 1 then (28) and (29) describe ellipsoidsin u -space, with axes lengths respectively given by r a and r ′ a with r a = µ x µ y + µ x µ z + µ y µ z − µ a , r ′ a = µ a (30)Since all µ a ∈ ]0 , , one has r a > r ′ a , thus (29) is morerestrictive than (28). It is also more restrictive than theequation of the sphere Eq. (9) since r ′ a < . If µ a = 0 or µ a = 1 for at least one value of a , one has to consider allequations again. Finally ρ ∈ C if and only if the param-eters µ a and u a correspond to the following situations:1. All µ a ∈ ]0 , and u x µ x + u y µ y + u z µ z ≤ , (31)that is, u corresponds to a point inside an ellipsoidcentered at (0 , , with half-axes of length √ µ a ; 2. Exactly one of the µ a is equal to 0, say µ z = 0 .Then from (29) one must have u z = 0 and from(28) u x µ x + u y µ y ≤ . (32)This corresponds to the situation above flattenedto 2 dimensions;3. Two µ a are zero, e. g. µ y = µ z = 0 . Then µ x = 1 and from (28) one must have u y = u z = 0 , whichleaves the condition | u z | ≤ . Again this corre-sponds to the situation (31), flattened to 1 dimen-sion.A point ρ belongs to the boundary ∂ C of C if one of theinequalities (24) becomes an equality. States with one ortwo µ a equal to 0 always verify det Z = 0 and thus lie onthe boundary ∂ C . When all µ a are in ]0 , , the conditionthat det Z = 0 is equivalent to equality in (31), whichcorresponds to points u a which lie on the surface of theellipsoid (31). Condition tr Z = 0 is equivalent to equal-ity in (9), while condition (tr Z ) = tr Z is equivalent toequality in (28). Since the ellipsoid (31) lies inside boththe sphere (9) and the ellipsoid (28), the points corre-sponding to either of these cases must lie on the surfaceof the ellipsoid (31). Therefore, points on the boundary ∂ C correspond to case 1 above when (31) becomes anequality, or to cases 2 or 3.In the main case (equality in (31)) the surface is a two-parameter set of ellipsoids that can be parametrized by µ a ∈ ]0 , with P a µ a = 1 , e. g., µ = sin θ sin ϕ sin θ cos ϕ cos θ , (33)and u = sin θ sin ϕ cos θ cos ϕ sin θ cos ϕ cos θ sin ϕ cos θ sin θ . (34)Again, the parametrization requires three more anglesto take into account the orthogonal matrix required todiagonalize W . If α, β and γ are the three Euler anglesthat parametrize the orthogonal matrix O then one hasthe complete parametrization u = O ( α, β, γ ) sin θ sin ϕ cos θ cos ϕ sin θ cos ϕ cos θ sin ϕ cos θ sin θ , (35) W = O sin θ sin ϕ θ cos ϕ θ O T with O = O ( α, β, γ ) . To summarize, the number of es-sential parameters for the set C (excluding rotations ofthe coordinate system) is 5, and for the surface ∂ C it is4. In order to obtain each classical matrix ρ once andonly once we shall demand that µ x ≤ µ y ≤ µ z whichmeans that the range in (33)–(35) has to be restricted to θ ∈ ]0 , arctan(1 / cos( ϕ ))] and ϕ ∈ ]0 , π/ (see [11]). VI. SOME EXAMPLES
We first give an example of a non-classical state. Insection IV we saw that the case µ y = µ z = 1 corre-sponds to state ρ = | , ih , | . According to [7] this isthe most quantum spin–1 state. Its Majorana represen-tation corresponds to two points diametrically opposedon the Bloch sphere, e.g. north and south pole.Let us now consider the case where two of the µ a vanish(case 3 of the above section), say, µ y = µ z = 0 . Then µ x = 1 and Eq. (20) implies that u y = u z = 0 . Then ρ ∈ C if and only if u x = u ∈ [ − , . In this casethe ellipsoid is flattened to a line. The state ρ can bedecomposed as ρ = 1 − u | ψ ( − ) ih ψ ( − ) | + 1 + u | ψ (+) ih ψ (+) | , (36)with | ψ ( ± ) i = | , − i ± √ | , i + | , i . (37)The pure states | ψ ( ± ) i are eigenvectors of J x corre-sponding to the eigenvalues ± , i. e., they are coherentstates directed along or opposite to the x -axis. Since u ∈ [ − , , ρ is a classical mixture of two coherentstates. It forms a one-parameter family of classical states.Since the entire family is inside ∂ C , this represents a one-dimensional line on the surface. This indicates that thesurface ∂ C is not necessarily strictly convex. Neverthe-less, we now show that the surface ∂ C does not containfacets, that is, the surface is not locally a (7-dimensional)hyperplane.For a state ρ ∈ N and a three-dimensional real vector t with | t | = 1 , we define Q t = 2 h J t i − h J t i − . (38)As noted in [6], the classicality criterion Z ≥ is equiv-alent to Q t ≥ for all t . For fixed t , Q t = 0 definesa quadric surface S t in the eight-dimensional space ofvariables { u a , W ab } . Its equation can be rewritten as Q t = P a,b ( W ab − u a u b ) t a t b = 0 , with t a , a = x, y, z ,fixed. Each surface S t separates the space of all states N into two subsets. The subset of states with Q t < contains only genuinely quantum states, as they violatethe condition Q t ≥ for at least one t . The subset ofstates with Q t > is convex: indeed, a linear changeof variables with a new variable X = P a u a t a yields Q t = X + linear terms. Now let M be a point on ∂ C andsuppose there exists a sphere B ǫ with radius ǫ , centered δ CStt >0 t <0 Q QClass.Quant.
FIG. 1: Local geometry at a point on the surface ∂ C of theset of classical states.FIG. 2: (Color online) Boundaries ∂ N of the set of physicaldensity matrices (outer ellipsoid) and ∂ C of the classical states(inner ellipsoid) for µ x = 0 . , µ y = 0 . , and µ z = 0 . interms of the dimensionless components u i of the vector u defined in eq.(4). The two ellipsoids do not touch in general,and their axes coincide. on M , such that inside B ǫ the surface ∂ C is a piece of ahyperplane (see Fig. 1). By definition of C , the sphere issplit into two equal halves, one containing only quantumstates and the other one containing only classical states.But since S t is not a flat surface, part of the states in thelatter half-sphere must lie on the subset on states with Q t < (see Fig. 1), which entails a contradiction.Another interesting example is the thermal state ρ = e − βH / tr e − βH (39)of a system with Hamiltonian H = J z and inverse tem-perature β = 1 /k B T , with k B Boltzmann’s constant. Fortemperature T = 0 , the thermal state is the ground state | , i , which is the most quantum state possible. For T → ∞ on the other hand, ρ approaches the identitymatrix and is therefore classical. The transition tem-perature to classicality can be found exactly from theboundary ∂ C . The parametrization of (39) gives u = and W = e β e β e β e β
00 0 − e β e β . (40)The inequality in (31) is always satisfied. The conditionthat µ a ∈ [0 , reduces to ≤ e β ≤ . Therefore, ρ isclassical if and only if β ≤ ln 2 .As a last example, consider a state with µ x = 0 . , µ y = 0 . , and µ z = 0 . . One can then specify theboundaries ∂ C and ∂ N solely in terms of the u a . Fig.2shows that ∂ C is indeed an ellipsoid inside the ellipsoidgiven by ∂ N . VII. CONCLUSIONS
To summarize, we have found an explicit representa-tion of the set C of classical spin–1 states, Eq. (31) de-fined as the convex hull of spin–1 SU(2) coherent states.The set C consists of a family of ellipsoids. The surfaceof the set contains straight lines, thus this allows theexistence of linear families of genuinely quantum stateswith exactly the same quantumness. Our results allowto visualize the set of classical states and to determineanalytically under what conditions a density matrix thatdepends on one or several parameters becomes genuinely“quantum”. Acknowledgments:
DB thanks Otfried Gühne for dis-cussions. This work has been supported by the Sonder- forschungsbereich TR 12 of the Deutsche Forschungsge-meinschaft and by the GDRI-471 of the CNRS.
Appendix
Let E ( κ ) be the lowest eigenvalue of the parameter-dependent Hermitian matrix H ( κ ) and ψ its correspond-ing eigenvector. At values of κ such that the groundlevel is non-degenerate the second derivative of E ( κ ) is non-positive. This follows, e.g., from the identity E ′′ = − h ψ | T | ψ i with T denoting a manifestly posi-tive operator, T = ∂H∂κ Q ( H − E ) − Q ∂H∂κ ,Q = ˆ1 − | ψ ih ψ | . (41)Assume now that at κ = 0 the ground state is non-degenerate and besides, the first derivative E ′ (0) = 0 .Then for all positive κ the ground state energy willbe monotonically decreasing (or at best non-growing)function of κ . Indeed, if we first assume that E ( κ ) is not degenerate for all k ≥ then we have E ′ ( κ ) = R κ E ′′ ( x ) dx ≤ for κ ≥ . The result remains true evenif there is level crossing at some κ = κ since then for κ >κ we can write E ′ ( κ ) = E ′ ( κ + 0 + ) + R κκ +0 + E ′′ ( x ) dx .The integral is negative from the same argument asabove, and E ′ ( κ +0 + ) is the slope immediately after thecrossing, which must be smaller than the slope immedi-ately before the crossing, which we know to be negative. [1] I. Bengtsson and K. Życzkowski, Geometry of quantumstates: an introduction to quantum entanglement (Cam-bride University Press, 2006).[2] R. F. Werner, Phys. Rev. A , 4277 (1989).[3] D. Z. Djokovic, quant-ph/0604190 (2006).[4] L. Ioannou, Phys. Rev. A , 052314 (2006).[5] O. Gühne and N. Lütkenhaus, Journal of Physics: Con-ference Series , 012004 (2007).[6] O. Giraud, P. Braun, and D. Braun, Phys. Rev. A ,042112 (2008).[7] O. Giraud, P. Braun, and D. Braun, New Journal ofPhysics , 063005 (2010).[8] R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, Rev. Mod. Phys. , 865 (2009).[9] J. A. Miszczak, Z. Puchała, P. Horodecki, A. Uhlmann,and K.Życzkowski, Quantum Information and Computa-tion , 0103 (2009).[10] J. Martin, O. Giraud, P. A. Braun, D. Braun, andT. Bastin, Phys. Rev. A , 062347 (2010).[11] M. S. Byrd and P. B. Slater, Phys. Lett. A , 152(2001).[12] M. R. Dennis, J. Opt. A: Pure Appl. Opt. , S26 (2004).[13] S. Chung and T. L. Trueman, Phys. Rev. D11