8 Boolean Atoms Spanning the 256-Dimensional Entanglement-Probability Three-Set Algebra of the Two-Qutrit Hiesmayr-Loffler Magic Simplex of Bell States
88 Boolean Atoms Spanning the 256-DimensionalEntanglement-Probability Three-Set Algebra of the Two-QutritHiesmayr-L¨offler Magic Simplex of Bell States
Paul B. Slater ∗ Kavli Institute for Theoretical Physics,University of California,Santa Barbara, CA 93106-4030 (Dated: July 1, 2020)
Abstract
We obtain formulas (bot. p. 12)–including and ( √ π − ) –for the eight atoms (Fig. 11),summing to 1, which span a 256-dimensional three-set (P, S, PPT) entanglement-probability booleanalgebra for the two-qutrit Hiesmayr-L¨offler states. PPT denotes positive partial transpose, while Pand S provide the Li-Qiao necessary and sufficient conditions for entanglement. The constraintsensuring entanglement are s > ≈ . p > · · ≈ . · − . Here, s is thesquare of the sum (Ky Fan norm) of the eight singular values of the 8 × p , the square of the product of the singular values. In the two- ququart Hiesmayr-L¨offler case, one constraint is s > ≈ .
25, while ≈ . · − is an upperbound on the appropriate p value, with an entanglement probability ≈ . S constraints,in both cases, prove equivalent to the well-known CCNR/realignment criteria. Further, we detectand verify–using software of A. Mandilara–pseudo-one-copy undistillable (POCU) negative partialtransposed two-qutrit states distributed over the surface of the separable states. Additionally, westudy the best separable approximation problem within this two-qutrit setting, and obtain explicitdecompositions of separable states into the sum of eleven product states. Numerous quantities ofinterest–including the eight atoms–were, first, estimated using a quasirandom procedure. PACS numbers: Valid PACS 03.67.Mn, 02.50.Cw, 02.40.Ft, 02.10.Yn, 03.65.-w ∗ Electronic address: [email protected] a r X i v : . [ qu a n t - ph ] J un . INTRODUCTION In our recent preprint “Jagged Islands of Bound Entanglement and Witness-ParameterizedProbabilities” [1], we reported a PPT (positive partial transpose) Hilbert-Schmidt probabilityof π √ ≈ . + log ( −√ ) √ ≈ . W (+) witness test [3, 4], obtaininga total entanglement (that is, bound plus “non-bound”/“free”) probability for each test of ≈ . ≈ . ≈ . − + π √ + log(3)6 ≈ . (cid:0)
27 + √ (cid:0)
97 + 56 √ (cid:1)(cid:1) ≈ . (cid:0) √ π − (cid:1) ≈ . S constraint [7, 8]–but ignoring their [product] P constraint–equals (1 − π √ )+ (cid:0) √ π − (cid:1) = ≈ . as the entire entan-glement probability–but now must revise it to 1 − = ≈ . − ( + log ( −√ ) √ )) + 0 . ≈ . and sufficient conditions recently put forth by Li and Qiao [7, 8] (cf. [11]) for thethree-parameter qubit-ququart model, ρ (1) AB = 12 · ⊗ + 14 ( t σ ⊗ λ + t σ ⊗ λ + t σ ⊗ λ ) , (1)where t µ (cid:54) = 0, t µ ∈ R , and σ i and λ ν are SU(2) (Pauli matrix) and SU(4) generators,respectively (cf. [12]). We also examined there, certain three-parameter two-ququart andtwo-qutrit scenarios. 2 et Probability Numerical Value——- 1 1.PPT π √ ∧ MUB − + π √ + log(3)6 ∧ Choi − + π √ + log(3)6 ∧ Choi ∨ Choi ¬ MUB ∧ Choi ∧ ¬
Choi ∧ ¬
MUB (cid:0)
72 + 8 √ π −
27 log(3) (cid:1) ∧ ¬
Choi (cid:0)
72 + 8 √ π −
27 log(3) (cid:1) ∧ MUB ∧ Choi 0 0PPT ∧ (MUB ∨ Choi) − + π √ + log(3)3 ¬ PPT ∧ MUB + √ + √ π − π √ − log(3)8 ¬ PPT ∧ Choi + √ + √ π − π √ − log(3)8 ¬ PPT ∧ ¬
MUB (cid:0) − √ π (cid:1) ¬ PPT ∧ ¬
Choi (cid:0) − √ π (cid:1) ¬ PPT ∧ ¬
MUB ∧ ¬
Choi (3 log(3) −
1) 0.255092985PPT ∧ ¬
MUB ∧ ¬
Choi (8 − ∨ (MUB ∧ Choi) (cid:0) √ π (cid:1) d = 3 two-qutrit model.Notationally, ¬ is the negation logic operator (NOT); ∧ is the conjunction logic operator (AND);and ∨ is the disjunction logic operator (OR). The mutually unbiased and Choi witness tests areindicated. I. LI-QIAO HIESMAYR-L ¨OFFLER TWO-QUTRIT ANALYSES
Here, we seek–in two different manners–to extend these procedures developed by Li andQiao to the Hiesmayr-L¨offler two-qutrit magic simplex of Bell states [2], earlier studied byus in [1]. To do so, constitutes a substantial challenge, since now the associated correlationmatrix of the Bloch representation of the bipartite state ρ HL = 19 ⊗ + 14 (cid:16) t λ ⊗ λ + t λ ⊗ λ + Σ i =1 t i λ i ⊗ λ i ) (cid:17) . (2)is 8 ×
8, rather than 2 × × SU (3), theCartan subalgebra is the set of linear combinations [with real coefficients] of the two matrices λ and λ , which commute with each other.) In the simplifying parameterization of theHiesmayr–L¨offler states introduced in [1, sec. II.A], ρ HL = γ γ γ Q γ γ γ γ γ Q Q γ γ γ γ , (3)where γ = ( Q + 2 Q ) , γ = ( Q − Q ), and γ = ( − Q − Q − Q + 1), we have t = t = t = ( Q − Q ) , t = t = − ( Q − Q ) , t = t = − ( + Q + 2 Q ). Further, t = Q +6 Q +2 Q − √ and t = − t .The requirement that ρ HL be a nonnegative definite density matrix–ensured by requiringthat its nine leading nested minors all be nonnegative [13]–takes the form [1, eqs. (29)], Q > ∧ Q > ∧ Q > ∧ Q + 3 Q + 2 Q < . (4)Additionally, the constraint that the partial transpose of the 9 × Q > ∧ Q > ∧ Q + 3 Q + 2 Q < ∧ Q + 3 Q Q + (3 Q + Q ) < Q + 2 Q Q . (5)4urther, the Hiesmayr-L¨offler mutually-unbiased-bases (MUB) criterion for bound entangle-ment, I = Σ k =1 C A k ,B k >
2, where C A k ,B k are correlation functions for observables A k , B k [14, Fig. 1] is Q > Q + 4 Q , (6)In the Hiesmayr-L¨offler d = 3 two-qutrit density-matrix setting, the Choi-witness entangle-ment requirement that Tr[ W ρ HL ] < Q + 1 − Q − Q < . (7)The realignment constraint that, if satisfied, ensures entanglement is (cid:113) − Q − Q + 3 ( Q + (3 Q + 4 Q − Q + 9 Q + 4 Q + 6 Q Q ) + 1+ (8)+3 | Q − Q | > . A. Singular values of the × Hiesmayr–L¨offler two-qutrit correlation matrix
The pair (
P, S ) of entanglement constraints in the Li-Qiao framework, for which we seekthe appropriate bounds, would be based on the eight singular values of the 8 × λ ⊗ λ and λ ⊗ λ . The coefficientsof these terms in the indicated reparameterization being Q +6 Q +2 Q − √ and − Q +6 Q +2 Q − √ ,respectively, as noted earlier.)Entanglement is achieved if either the square ( p ) of the product of the eight singularvalues exceeds a certain threshold, or the square ( s ) of the sum (the Ky Fan norm) of thesingular values exceeds a corresponding threshold. Our research here is first focused ondetermining the appropriate thresholds to employ. (The set of two-qutrit states satisfyingthe first [product-form] of these two constraints we denote P and the second [sum-form], S .)To so proceed, we found that six of the eight singular values of thecorrelation matrix of (2) are (cid:112) ( Q − Q ) and the remaining two are (cid:112) − Q − Q + 3 ( Q + (3 Q + 4 Q − Q + 9 Q + 4 Q + 6 Q Q ) + 1. The square ofthe product of the eight values is, then,65536 ( Q − Q ) (3 Q + 3 (3 Q + 4 Q − Q + 27 Q + 9 Q (2 Q −
1) + 6 Q (2 Q −
1) + 1) (cid:18) √ ζ (cid:112) ( Q − Q ) (cid:19) (10)where (cf. (8)) ζ = − Q − Q + 3 (cid:0) Q + (3 Q + 4 Q − Q + 9 Q + 4 Q + 6 Q Q (cid:1) + 1 . (11)These are the two quantities–in the Li-Qiao framework–for which we must find suitable lowerbounds. If a particular Hiesmayr-L¨offler state exceeds either bound it is necessarily entangled.We, preliminarily, found that the maxima–over the entire magic simplex (of both entangledand separable states)–for P is = ( ) ≈ . S , ≈ . not bound-entangled according to the realignment test. Then,our numerics indicated that the maxima are = · · ≈ . · − [16] for P and ≈ . S (at Q = , Q = 0 , Q = ). (This last maximum canalso be achieved at Q = , Q = (3 − √ , Q = 0–which in the original Hiesmayr-L¨offlercoordinates, converts to q = (cid:0) √ − (cid:1) , q = − − √ , q = − √ . If, on the other hand, wesimply search for the maxima over the Hiesmayr–L¨offler two-qutri states with positive partialtranspose–within which all the separable states must lie, but now do not omit those statesthat are bound-entangled based on the realignment test, we obtain the larger values for S , s = ≈ . p = · ≈ . · − for P (at Q = , Q = , Q = 0).)Enforcement of the constraint S ≡ s > proves, interestingly (algebraically demon-strable), fully equivalent (at least for d = 3) to the application of both the realign-ment (CCNR) and SIC POVMs tests [5, 6], in yielding a total entanglement probabil-ity of (cid:0)
27 + √ (cid:0)
97 + 56 √ (cid:1)(cid:1) ≈ . (cid:0) √ π − (cid:1) ≈ . (10 − ≈ . s ≤ is one of the knownresults for separability, using the Bloch representation [8, eq. (48)].)We have also, interestingly, found that of this bound-entanglement probability of (cid:0) √ π − (cid:1) ≈ . ≈ . P ≡ p > = · · ≈ . · − constraint.6 . Graphic representations Now, in a series of figures, let us attempt to gain insight into the specific relations betweenthe constraints and the geometric structure of entanglement. To begin, in Fig. 1 we showa sampling of just those entangled Hiesmayr–L¨offler two-qutrit states that do satisfy the P ≡ p > · · constraint, but do not satisfy the S ≡ s > constraint. (The sampling isbased on use of the Mathematica FindInstance command to generate points satisfying thebasic feasible density matrix constraint (4), which points are, then, employed to test furtherconstraints. We so proceed, although we are not aware of any particular measure [Hilbert-Schmidt, Bures, . . . ] underlying this command.) The bound-entangled states correspond tothe green points, and the free-entangled states to the red. There appear to be two islands ofentanglement.In Fig. 2, we reverse the role of the two constraints.Now, in Fig. 3, we present a sampling of those states which satisfy neither of theentanglement constraints. The (predominantly) green points are separable in nature, whilethe red ones appear to be pseudo-one-copy undistillable (POCU) negative partial transposedstates [17]. (“Our results are disclosing that for the two-qutrit system the BE [bound-entangled] states have negligible volume and that these form tiny islands sporadicallydistributed over the surface of the polytope of separable states. The detected families of BEstates are found to be located under a layer of pseudo-one-copy undistillable negative partialtransposed states with the latter covering the vast majority of the surface of the separablepolytope” [17]. The term “pseudo” is used to emphasize that although a single copy of thestate is undistillable, a collection of more than one might be.) A Mathematica program isavailable for testing for the POCU property [18]. (One instance of such a point to be sotested is Q = , Q = , Q = , while another is Q = , Q = , Q = .) Infact, employing the indicated program on a sample of ten candidate POCU states, we wereable to confirm that they all possess this property. (Also, all ten 9 × + π √ − √ − cosh − (97)54 √ ≈ . IG. 1: A sampling of just those entangled Hiesmayr–L¨offler two-qutrit states that do satisfy the p > · · constraint, but do not satisfy the s > constraint. The bound-entangled statescorrespond to the green points, and the free-entangled states to the red. There appear to be two islands of entanglement. value that s can attain is 0.47742 (at Q = , Q = , Q = ). In Figs. 4, 5 and 6,we show plots based on additional Boolean combinations of the two constraints. (Note thatthere are some differences in scaling among the several figures in the paper.)
1. States on the boundary of separability
The points in the next two figures (Figs. 7 and 8) all saturate the S entanglementconstraint, i. e., s = . The points in the former lie, in general, within the PPT states,while in the latter, they lie on the boundary of the PPT states. Efforts of ours to produce a8 IG. 2: A sampling of just those entangled Hiesmayr–L¨offler two-qutrit states that do not satisfythe p > · · constraint, but do satisfy the s > constraint. The bound-entangled statescorrespond to the green points, and the free-entangled states to the red. There appear to be multipleislands of entanglement. companion pair of figures to these last two, in which instead of the S entanglement constraintbeing saturated, the P constraint would be, proved much more computationally challenging.However, we were able to obtain a fewer-point analogue of Fig. 8, that is, Fig. 9. In Fig. 10we jointly plot the two curves (Fig. 8 and Fig. 9), showing the intersection of the PPTboundary with points saturating the S and P constraints, respectively.In Table II, we summarize several of our analyses.9 IG. 3: A sampling of those Hiesmayr–L¨offler two-qutrit states which satisfy neither of theentanglement constraints (
P, S ). The green points are separable in nature, while the red onesare pseudo-one-copy undistillable (POCU) negative partial transposed states. Numerical analysesindicated that for these POCU states, an upper bound on the lowest value that s can attain is0.47742. C. Boolean-analysis-based derivation of the formulas in Table II
The formulas in this table were derived making use of the decomposition into eight “atoms”of the 256-dimensional algebra associated with the three sets
P P T, P, S . We now presentthe final answer to [23]–omitting the already-presented Table II–discussing the underlyinganalysis (in terms of the notation in [23], A ≡ P, B ≡ P, C ≡ P P T ):“We determine–making strong use of the Mathematica code given by user250938 in the10
IG. 4: A sampling of those Hiesmayr–L¨offler two-qutrit states which do not satisfy at least one ofthe entanglement constraints (
P, S ). The green points are separable in nature, while the red onesappear to be pseudo-one-copy undistillable negative partial transposed states . The highest value of s for the red points in this plot is 3.11447. answer to this question–the eight atoms of our 256-dimensional Boolean algebra on threesets. Then, we are able to present a table of imposed constraints and their (now partiallyrevised) associated probabilities fully consistent with this framework.(The several integer denominators [in Table II] all have prime factorizations with primesno greater than 13–but certainly not the numerators. The prime 97 plays a conspicuousrole.)To obtain these results, we began by estimating the values of the eight atoms–in the11 IG. 5: A sampling of those Hiesmayr–L¨offler two-qutrit states which satisfy at least one of theentanglement constraints. The bound-entangled states correspond to the green points, and thefree-entangled states to the red. indicated order P ∧ S ∧ P P T, ¬ P ∧ S ∧ P P T, P ∧ ¬ S ∧ P P T, P ∧ S ∧ ¬ P P T, ¬ P ∧ ¬ S ∧ P P T, (12) ¬ P ∧ S ∧ ¬ P P T, P ∧ ¬ S ∧ ¬ P P T, ¬ P ∧ ¬ S ∧ ¬ P P T as– { , , , , , , , } (13) ≈ { . , . , . , . , . , . , . , . } . The estimation procedure employed was the ”quasirandom” (”generalized golden ratio”)one of Martin Roberts https://math.stackexchange.com/questions/2231391/how-can-one-generate-an-open-ended-sequence-of-low-discrepancy-points-in-3d . It was used to generate12
IG. 6: A sampling of those states which satisfy both of the entanglement constraints. Thebound-entangled states correspond to the green points, and the free-entangled states to the red. six-and-a half billion points (triplets in [0 , ), only approximately one-thirty-sixth of them–those yielding feasible density matrices–being further utilized.These eight estimated values (summing to 1) are well fitted, we find (using the MathematicaSolve command), by , ( √ π − ) , + π √ − √ − cosh − (97)54 √ , − π √ − √ + cosh − (97)54 √ , − π √ + √ + cosh − (97)54 √ , − − π √ + √ + cosh − (97)54 √ , − π √ + √ − cosh − (97)54 √ , + π √ − √ − cosh − (97)54 √ . ≈ . , . , . , . , . . , . , . π √ , (cid:0)
27 + √ (cid:0)
97 + 56 √ (cid:1)(cid:1) , (cid:0) √ π − (cid:1) –havingearlier been obtained [1] through symbolic integration. Then, having strong confidence inthe previously (tabulated) used values of , and π √ − expressions, we incorporatedthem too. 13 IG. 7: Hiesmayr–L¨offler two-qutrit states on the boundary of the separable states for which the S entanglement constraint is saturated, i. e. s = . Since these six values were not fully sufficient for Solve, we additionally employed theWolframAlpha site–searching over the 256 BooleanFunction results to find simple well-fitting formulas, using the above-given numerically estimated values of the eight atoms.For instance, for BooleanFunction[133,P,S,PPT]=( P ∧ P P T ∧ S ) ∨ ( ¬ P ∧ ¬ P P T ), the sitesuggested , fitting the estimated corresponding value to a ratio of 1.00000006615. Also,for BooleanFunction[62,P,S,PPT]= ¬ ( P ∧ S ) ∧ ( P ∨ P P T ∨ S ), the suggestion was √ ,having an analogous ratio of 0.999999807781.Incorporating as well, these last two results, as well as the previously tabulated for ¬ P P T ∨ S , proved sufficient to obtain the eight “atomic” formulas.The close-to-1 ratios of these formulas to the estimated values,given above, are { . , . , . , . . , . , . , . } .”In Fig. 11, we now, additionally, display the eight atoms spanning the entanglement-probability boolean algebra for the Hiesmayr–L¨offler two-qutrit model. (We have alsoinvestigated–as a supplement to Fig. 11–the potential use of [planar] Venn diagrams to repre-14 IG. 8: Hiesmayr–L¨offler two-qutrit states on the boundaries of both the separable states and PPTstates for which the S entanglement constraint is saturated, i. e. s = . sent the various entanglement-related probabilities associated with the boolean combinationsof P, S and
P P T [24].)
D. Analyses employing Li-Qiao variables α i , β i As a matter of analytical interest, we had initially concentrated upon attempting toconstruct the proper entanglement bounds–now reported above–for P and S applicable to theHiesmayr–L¨offler two-qutrit model, but strictly within the Li-Qiao framework. In so doing,we follow [10], in which we employed the well-known necessary and sufficient conditions fornonnegative-semidefiniteness that all leading minors be nonnegative [13]. There are twenty-two sets of such minors of 3 × ρ HL into eleven separable two-qutrit states. (We were able to obtain this explicitexpansion, lending us confidence in our further analyses. In the Li-Qiao setup, we initiallyhave twenty parameters, ten α i and ten β i , with t i = α i β i . Then, the solution yielding thecorrect expansion was expressible as β i = Q − Q )3 α i , i = 1 , , β i = − Q − Q )3 α i , i = 2 , , IG. 9: Hiesmayr–L¨offler two-qutrit states on the boundaries of both the separable states andPPT states for which the P entanglement constraint is saturated, i. e. p = = · · ≈ . · − . There are three curves, one much smaller than the other two. and β i = − Q +6 Q α i , i = 3 ,
8, and β i = − Q +6 Q +2 Q α i , i = 9 , thirteen -dimensional setting ( Q , Q , Q and theten α i ’s), our highest estimate of the (multiplicative) bound for P was p = 8 . ∗ − ,and of the (additive) bound for S was s = 0 . p > . ∗ − ,yields an entanglement probability estimate of 0.764984, and enforcing s > . π √ ≈ . (cid:0) √ π − (cid:1) ≈ . s and p for theHiesmayr-L¨offler two-qutrit magic simplex of Bell states, this was only achievable in the firstof our two lines of two-qutrit analysis, employing simply the trivariate ( Q , Q , Q ) set ofconstraints ((4)-(8)). The second line of 13-variable ( Q , Q , Q and ten Li-Qiao parameters16 IG. 10: A joint plot of the two curves (Fig. 8 and Fig. 9), showing the intersection of the PPTboundary of the Hiesmayr–L¨offler two-qutrit states with points for which s = and p = · · ,respectively. α i , i = 1 , · · · ,
10) analyses, conducted within the Li-Qiao framework, had not similarlysucceeded.
1. Explicit decompositions of separable states
However, further analyses allowed us to construct multiple (about twenty, presently)sets of Li-Qiao parameters α i , β i , i = 1 , . . . ,
10 (cf. [8, eqs. (63)-(66)]) each yielding aseparable expansion of length eleven (each component product density matrix being equallyweighted by ) for specific Hiesmayr-L¨offler two-qutrit states. For example, the ten α parameters (cid:8) , , , , , − , − , , , − (cid:9) , together with the ten β parame-ters (cid:8) − , − , , − , , − , , , , − (cid:9) gave us a separable decomposition for thestate with Q , = , Q = , Q = . Somewhat disappointingly however, allthe twenty-or-so examples so far generated had Q = Q , so the multiplicative norm (9)simply reduced to zero. The greatest value for the additive norm (10) so far generated is17 ABLE II: Exact formulas and underlying quasirandom estimates [19–22] of various Hilbert-Schmidtprobabilities for the Hiesmayr-L¨offler d = 3 two-qutrit model. More than four hundred million pointswere employed–and results appear accurate to six-seven decimal places. S denotes the set satisfyingthe constraint s > ≈ . P , the constraint p > = · · ≈ . · − . Notationally, ¬ is the negation logic operator (NOT); ∧ is the conjunction logicoperator (AND); and ∨ is the disjunction logic operator (OR). Alternative integration procedures toquasirandom estimation were used for the last two entries. (Somewhat interesting observations withregard to the entries of the revised table are that cosh − (97) = log (cid:0)
97 + 56 √ (cid:1) = sinh − (cid:0) √ (cid:1) ,so that √ Set P robability QuasirandomEstimate . π √ . ¬ P ∧ ¬ S . P − π √ − √ − cosh − (97)54 √ . S (cid:0)
27 + √ (cid:0)
97 + 56 √ (cid:1)(cid:1) . P ∧ S − π √ − √ + cosh − (97)54 √ . P ∨ S . ¬ P ∨ ¬ S − π √ + √ + cosh − (97)54 √ . ∧ ¬ P ∧ ¬ S − π √ + √ + cosh − (97)54 √ . ∧ P + π √ − √ − cosh − (97)54 √ . ∧ S (cid:0) √ π − (cid:1) . ∧ P ∧ S . ∧ ( P ∨ S ) − + π √ − √ − cosh − (97)54 √ . ∧ ( ¬ P ∨ ¬ S ) π √ − . ∧ S ∧ ¬ P ( √ π − ) . ¬ PPT ∨ S . ≈ . < ≈ . IG. 11: Decomposition of the Hiesmayr–L¨offler two-qutrit states into its mutually exclusive eightatoms. The PPT states are in the interior of the body and the entangled states at the extremities.Exclamation signs in the legend denote set negation, and the double ampersand, set intersection.
E. Best separable approximation
In their pair of recent skillful papers [7, 8], Li and Qiao presented necessary and sufficientconditions for separability, the implementation of which we have investigated above. Theydid not, however, discuss the apparently related best separable approximation problem [25].To begin a study of the possible application of the Li-Qiao analytical framework to thisproblem of major interest, we sought a best separable approximation for the entangledHiesmayr-L¨offler two-qutrit density matrix (3) with its parameters having been set to Q = , Q = , Q = . Then, we obtained a value of B = 0 . B is the19arameter one seeks to minimize 0 ≤ B ≤
1, in the equation [17, eq. (2)]ˆ ρ = (1 − B ) ˆ ρ sep + B ˆ ρ ent . (14)Now, the minimum B = 0 . Q ’s for ˆ ρ is obtained if wechoose for the parameterization of ˆ ρ ent , the values Q = 1 . · − , Q = 1 . · − and Q = 0 .
5. Then, from (14), we can obtain the desired ˆ ρ sep –for which Q = 0 . , Q =0 . Q = 0 . III. TWO-QUQUART ANALYSES
For the d = 4 two-ququart Hiesmayr-L¨offler magic simplex states, ρ qqHL = κ κ κ κ Q Q κ κ κ κ κ κ Q Q Q κ κ κ κ κ Q Q Q κ κ κ κ κ , (15)where, κ = ( Q + 3 Q ) , κ = ( Q − Q ) and κ = ( − Q − Q − Q − Q + 1). ρ qqHL is not in normal form, in which “the Bloch representation of ρ AB would have a = 0and b = 0, that is, the local density matrices would be maximally mixed”. In fact, theBloch vectors of the two reduced 4 × (cid:113) ( Q + 3 Q )20ssociated with the fifteen generator of SU (4). The components associated with the fourteenother generators are all zero in both cases. (A constructive way of bringing a single copyof a quantum state into normal form under local filtering operations was presented in [26].A Matlab program for accomplishing this is given in [27],) The Li-Qiao framework requiressuch normal forms.The requirement that ρ qqHL is a nonnegative definite density matrix–or, equivalently, thatits sixteen leading nested minors are nonnegative [13]–takes the form [1, eq. (29)] Q > ∧ Q > ∧ Q > ∧ Q > ∧ Q + 4 ( Q + Q ) + 3 Q < . (16)The constraint that the partial transpose of ρ qqHL is nonnegative definite is [1, eq. (30)] Q > ∧ Q + 3 Q > ∧ Q + 4 ( Q + Q ) + 3 Q < ∧ Q + 4 Q Q + Q (17)+16 Q ( Q + Q ) + 12 Q Q < Q + 2 Q Q ∧ ( Q − Q ) < Q . With these formulas, we are able to establish that the corresponding PPT-probability is + log ( −√ ) √ ≈ . d = 3 counterpart of π √ ). In [1, sec. IIIB], we obtained free entanglement and bound-entangled probability CCNR-based estimates of 0.4509440211445637 and 0.01265489845176,respectively.Then, our 4-variable (as opposed to 32-variable [in Li-Qiao framework]) computations showthat–if we maximize over simply the PPT states–we have p = ≈ . · − (for Q = , Q = , Q = , Q = 0) and s = ≈ . s = ≈ .
25 (for Q = 0 , Q = , Q = 0 , Q = 0), while p appears to be unchanged. If we enforce the p > constraint, our estimate of the associatedentanglement probability is 0.31711552, while the s > constraint gives us 0.39717107.Unfortunately, at this point in time, we do not have an exact entanglement probability–asin the two-qutrit case studied above–to which to fit the Li-Qiao entanglement constraintbounds.Further analyses should be pursued in order to obtain the eight atoms spanning the 256-dimensional entanglement-probability three-set boolean algebra of the two- ququart Hiesmayr-L¨offler magic simplex of Bell states. The main impediment, it seems, to doing so is a lack ofprecise knowledge as to the proper lower bound for the P constraint–only knowing presently21hat ≈ . · − is greater than it, while we do know that s = ≈ .
25 isthe proper bound for the S constraint. However, from the discussion in sec. III, it wouldappear to be of interest to pursue an analysis employing s = ≈ .
25 and p = .In fact, such an attempt–based on 3,645,771 quasirandom four-dimensional points–yieldedthe eight atomic estimates, { . · − , . , ., . , . , . , ., . } , (18)where the same ordering of the atoms as indicated in (12) was employed. We know alreadythrough symbolic integration that the PPT probability is + log ( −√ ) √ ≈ . ¬ P ∧ ¬ S ∧ P P T is quite close in value, that is, 0.404023.This atom corresponds to the separable states, so the estimate, in being slightly less than thePPT probability–due to the possibility of bound-entanglement–is plausible in that regard.If the lower bound for p could be found, then, it seems reasonable that the three zero ornear-zero estimates (all corresponding to atoms with P , rather than ¬ P ) would increase.Despite our lack of full knowledge as to the proper value of p to employ, we can utilize ouratomic estimates to obtain estimates free of P . For example, for the constraint S , just byitself, the derived estimate–obtained by summing the first, second, fourth and sixth atomicestimates (18)–is 0.4118991565. Further, the derived estimate of SP P T , that is, 0.0010906, isclose to ≈ . S ∨ P P T , that is, 0.815776, is close to ≈ . s ≤ , the entanglement bound for P appears to be at least as large as ≈ . · − .However, further numerical analysis suggested that the P upper bound could be lowered–from ≈ . · − –to 10 − (for Q = , Q = , Q = , Q = 0). Toso improve our knowledge of the lower bound for P , we utilized our confidence in thefull knowledge of the S constraint, to eliminate states entangled according to that singlecriterion from further consideration. (However, though doing so might prove sufficient to fullydetermine the proper P constraint–it is by no means clear that that is in fact the situation,seeing that it is not so in the two-qutrit case, as Table II indicates.)Then, we were able–by finding some computational improvements–to increase our quasiran-dom point collection to size 101,215,383, now yielding the eight Hiesmayr-L¨offler two-ququartatomic estimates of { . , . , ., . , . , . , ., . } . (19)22urther investigation revealed a two-ququart PPT state ( Q = , Q = , Q = , Q = ) with the apparently very small value p ≈ . · − , for which,nevertheless, s ≈ . > , and is, thus, entangled. (So, the entanglement of thisstate would not be revealed–by higher settings for p –as seems not inconsistent with theLi-Qiao two-constraint [ P, S ] framework. Numerical fine-tuning reduces the indicated p valuefurther still to 4 . · − .) IV. CONCLUDING REMARKS
In our analyses here, the CCNR (computable cross-norm realignment) criterion [5, 6] forentanglement proves to be equivalent to the properly enforced constraint–involving the squareof the Ky Fan norm (the sum of the singular values) [8, eq. (32)] of the correlation matrix inthe Bloch representation–on S . Whether this equivalence is true, in general, is a questionto be addressed. (In certain auxiliary analyses, we concluded that in the Hiesmayr-L¨offler d = 3 [two-qutrit] magic simplex model, the CCNR is equivalent–and not inferior, as can bethe case [6]–to the ESIC [SIC POVMs] test [6], in yielding the same sets of entangled andbound-entangled states. Efforts to similarly compare the CCNR and ESIC criteria in the d = 4 [two-ququart] version have so far proved too computationally challenging to complete.)An outstanding problem is the conversion of the two-ququart Hiesmayr-L¨offler densitymatrix (15) into normal form [28]. Although there are numerical approaches to this problem[26, 27], its symbolic character makes it still more challenging. Acknowledgments
This research was supported by the National Science Foundation under Grant No. NSFPHY-1748958. I thank A. Mandilara for providing me with the Mathematica code by whichI was able to corroborate the nature of the pseudo-one-copy undistillable states generated. [1] P. B. Slater, arXiv preprint arXiv:1905.09228 (2019).[2] B. C. Hiesmayr and W. L¨offler, Physica Scripta , 014017 (2014).[3] K.-C. Ha and S.-H. Kye, Physical Review A , 024302 (2011).
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