aa r X i v : . [ qu a n t - ph ] A p r Characterizing Incomparability in Quantum Information
Liwen Hu ∗ School of Physics, Beijing Institute of Technology, Beijing 100081, China (Dated: September 9, 2018)The theory of majorization has seen substantial application in quantum information. Its frame-work predicates on the comparability between real vectors. We explore the antithesis of this premise,namely, incomparability . Specifically, we provide ways to measure the incomparability between apair of spectra. We show that distinct spectra isoentropic by generalized entropies are mutually strongly incomparable. The inversion rank is proposed to classify incomparability. Majorizationrelations are advanced to probe the scale of incomparability, referred to as inconvertibility . Introduction. —The theory of majorization [1] tracesits roots to the characterization of inequalities and hasdeepseated foundation in matrix theory. In its incipi-ent stages of application, majorization has become in-strumental in the explanation of many crucial aspectsof quantum physics. Examples include the deterministicconversion of quantum states [2–5], the characterizationof mixing and measurement [6], the detection of separa-bility [7], the formulation of uncertainty relations [8] andthe framework of quantum relative Lorenz curves [9], etc.Majorization permits only a preorder on real vectors, itsutility in large relies on their comparability while casesproving otherwise are deemed undesirable.The notion of incomparability is a prevalent theme inorder theory and an ubiquitous phenomenon in nature.First educed by Nielsen [6], majorization incomparabilitywas used to account for the restrictions placed on entan-glement transformation. Work has been done to uncoverits underlying nature [10–13], however such understand-ing is far from exhaustion. Here, we examine majoriza-tion incomparability under a quantum setting in hopesof facilitating such discussions.As a preliminary, let us outline the basic principlesand provide some necessary notations. For d -dimensionaldensity matrix ρ , its spectrum λ [ ρ ] occupies a ( d − d − which can be parti-tioned into d ! minor simplices called Weyl chambers , eachcontaining the complete set of spectra but with differentorder [10]. We arrange λ [ ρ ] in nonincreasing order as avector r = λ ↓ [ ρ ] ∈ W d − such that 1 ≥ r i ≥ r i +1 ≥ , i ∈ { , . . . , d − } , where W d − is the one with non-increasing order, which we henceforth merely refer to asthe Weyl chamber. It pertains to the scope of our anal-ysis. For such spectral vectors r and s , if they satisfy A j ( r ) ≡ P ji =1 r i ≥ P ji =1 s i ≡ A j ( s ) , j ∈ { , . . . , d } , withequality when j = d , then r majorizes s , i.e., r ≻ s or ρ ≻ σ . Alternatively, we write A ′ j ( r ) = 1 − A j − ( r ) forsummation in nondecreasing order. We say the majoriza-tion is exact if there exists a coincidence A j ( r ) = A j ( s )where j < d , i.e., r (cid:23) s . We let M − d − ( r ) ≡ { r ′ ∈ W d − : r ≻ r ′ } be the set of nonincreasing spectra r majorizesin a d -system and let M + d − ( r ) ≡ { r ′ ∈ W d − : r ′ ≻ r } be the set of those majorizing r . There are instanceswhen r ⊁ s and s ⊁ r meaning neither party majorizes the other, nonetheless, if the final equality holds then wesay that r and s are incomparable, i.e., r ≁ s . We let I d − ≡ { r ′ ∈ W d − : r ′ ≁ r } be the set incomparableto r . There are times when r does not necessarily ma-jorize s , but do so by appending a catalyst c [14], i.e., r ⊗ c ≻ s ⊗ c . We say that r trumps s , i.e., r ≻ T s . When r ⊁ T s and s ⊁ T r they are called strongly incompara-ble [15, 16], i.e., r ≁ T s .We adopt the viewpoint of deterministic state conver-sion in quantum resource theories of which we highlightthree. Regarding entanglement, the bipartite pure statetransformation | ψ i → | φ i eventuates via local operationand classical communication iff the Schmidt vector of | φ i majorizes that of | ψ i [2]. Similarly for coherence, thepure state transformation | ψ i → | φ i eventuates via inco-herent operation iff the dephasing of | φ i majorizes thatof | ψ i [3, 4]. As for the resource theory of nonuniformity,the transformation ρ → σ eventuates via unital operationiff ρ ≻ σ [5]. The majorizing party holds less resource forthe first two theories, but more resourceful for the last.When state conversion cannot occur with certainty,especially in the event of strong incomparability, whereeven assistance does not guarantee success, it is reason-able to inquire the modifications a source or end statehas to undergo for assured transformation. Althoughthis may constitute costly operation, our interest divergesfrom that of resource theories and is not consigned to freeoperations. If the amount of resource a state possesses,when in excess, is of no immediate consequence to the re-alization of a task, the lack thereof from a bare minimumis then of more import. Incomparability is a condition tobe extricated from, in some sense it implicates a notionof “anti-resource” that needs to be expended.Aside from quantifying incomparability, the stronglyincomparable nature of distinct isoentropic spectra is an-alyzed, the concept of incomparability is refined throughcategorization by inversion rank, representative spectralfamilies are developed to ultimately survey the scale ofincomparability, referred to as inconvertibility . Measuring incomparability. —To gauge the incompara-bility of a pair of spectra we need to determine the expen-diture necessary to render them comparable by a particu-lar means with optimality. By this logic, we demonstratecharacteristic ways to quantify incomparability, includ-ing approaches ranging operational, distansal and alge-braic perspectives. They are designed to be nonnegative,vanishing for comparable pairs and symmetric.Quantum states tend toward entropy increase underquantum operations. We capture this through measure-ments that enable majorization.
Definition 1.
Let ρ, σ ∈ B ( H d ), the majorization cost of ρ with respect to σ can be expressed as M ( ρ | σ ) = inf { S (Λ π [ ρ ]) − S ( ρ ) : σ ≻ Λ π [ ρ ] } , (1)where S ( ρ ) is the von Neumann entropy, infimum is takenover measurements Λ π [ ρ ] in the bases (cid:8) π = { Π j } di =1 (cid:9) .Their operational incomparability is the minimum oftheir majorization costs: I O ( ρ, σ ) = min { M ( ρ | σ ) , M ( σ | ρ ) } . (2) M ( ρ | σ ) is simply the minimal entropy cost of measur-ing ρ for σ to majorize it. We know from the Schur-Horntheorem [1] that M − d − ( r ) is completely contained withinthe diagonal entries of possible density matrices for r ,which are obtained by Λ π [ ρ ]. Thus, deriving M ( ρ | σ )and M ( σ | ρ ) translates to finding their optimal posteriorspectra r ⋆ = s ⋆ ∈ M − d − ( r ) ∩ M − d − ( s ) which coincide.Wherefore, I O ( ρ, σ ) = S ( r ⋆ ) − max { S ( ρ ) , S ( σ ) } . Hence,by Uhlmann’s theorem [1] supplanting Λ π [ ρ ] with generalbistochastic operations yields the same result.We present a general framework for distance mea-sures, unlike before the party under optimization maynow move up and down the majorization hierarchy. Definition 2.
Let r, s ∈ W d − , the ascending and de-scending majorization distances of r with respect to s can be expressed as D ← ( r | s ) = inf {k r ′ − r k : r ′ ≻ s } , (3) D → ( r | s ) = inf {k r ′ − r k : s ≻ r ′ } , (4)where “ ← ” and “ → ” each stands for r moving up anddown the majorization order. Their incomparability dis-tance is the minimum of their majorization distances: I D ( r, s ) = min { D α ( r | s ) , D α ( s | r ) } , α = ← , → . (5)When D is taken to be the trace distance, D ← ( s | r )corresponds to the maximal fidelity achievable in a faith-ful entanglement transformation with known form [17].Here, we care not for which one majorizes which, all suchdistances are found and their infimum serves as the in-comparability of a spectral pair.For the algebraic method we intend to ”stretch” theLorenz curve [1, 9] of r upwards to majorize s andvice versa, whichever gives the least distortion is usedto represent incomparability. We know from proba-bilistic entanglement conversion, the optimal probabil-ity for | ψ i → | φ i , with respective Schmidt vectors r and s , is specified by Q ( s | r ) ≡ P ( | ψ i → | φ i ) =min j ∈{ ,...,d } A ′ j ( r ) /A ′ j ( s ) [18]. It suits our motive to de-termine the least failure rate for interconversion. r ss' r' j A j ( a ) χ χ j A j ( b ) FIG. 1. (a) Lorenz curves of r = (0 . , . , . , . s = (0 . , . , . , .
14) (solid black), r ′ =(0 . , . , . , .
07) (dotdashed red), s ′ = (0 . , . , . , . r ≁ s . r ′ (cid:23) s , s ′ (cid:23) r . (b) Lorenz curvesof pure depolarization bounds. χ = (0 . , . , . , . (cid:23) (0 . , . , . , . (cid:23) (0 . , . , . , .
13) = χ . Definition 3.
Let r, s ∈ W d − , their algebraic incompa-rability is defined as I A ( r, s ) = 1 − max { Q ( r | s ) , Q ( s | r ) } . (6)Ref. [19] details an intuitive procedure: r ′ = (1 − µ + µr , µr , . . . , µr d ) , µ ∈ [0 , r is enhanced while r i> are attenuated. r ′ (cid:23) s for µ = Q ( r | s ) [FIG. 2(a)].A comparison of the measures is made in FIG. 2(b).Operationally, besides I O , a local variant I lO (assuming a2 × M − ( η ) ∩ M − ( ξ ). Thelatter enlists parameterizing local measurements withBloch polar angles. Note, η l ⋆ = ξ l ⋆ . The Euclideandistance is used for I D while I A is straightforwardly cal-culated. I O ( ξ, η ) ≤ I lO ( ξ, η ), indicating limited local ac-cess. I O ( ˜ ξ, η ) is non-smooth as S ( ˜ ξ ) = S ( η ) where theoptimal party is reversed. Also, I O ( ˜ ξ, η ) ≈ I lO ( ˜ ξ, η ). I D and I A are linear with respect to k ξ − η k . Isoentropic spectra. —It was shown that isoentropicstates are either unitarily connected or incomparable [13,20]. This was also proven for unified entropies [21]. Here,we solidify and advance these claims to the extent ofstrong incomparability for generalized entropies.Trumpability is equivalent to a series of inequalities fora family of R´enyi entropy based functions [22, 23]. Let f ν ( r ) = ν (1 − ν ) log P i r νi ν = 0 , , d P i log r i ν = 0 , − P i r i log r i ν = 1 , (7)where f ( r ) is the Burg entropy and f ( r ) is the Shan-non entropy. Let r = s ∈ W d − , suppose extra d − max { rank( r ) , rank( s ) } zeros are omitted from both, r ≻ T s iff F ( ν, r, s ) ≡ f ν ( s ) − f ν ( r ) > ν ∈ ( −∞ , ∞ ). This can also be expressed in terms of powermajorization [24].Markedly, all distinct spectra on an isoentropy surface f ν ( r ) are mutually strongly incomparable. Since assisted-comparability is dictated by whether the sign of F ( ν, r, s )is consistent with respect to ν . For r = s , F ( ν, r, s )cannot vanish for all ν . Any discrepancy would include F ( ν, r, s ) = 0, leading to strong incomparability. I O L L L L - - - I D - - I A - - ( b ) FIG. 2. (a) Weyl chamber W . η = (0 . , . , . , . I ( η ): n I ( η ) = 1 (purple), n I ( η ) = 2noncatalyzable (red), n I ( η ) = 2 catalyzable (orange). Redand orange regions are demarcated by f ( η ) and f ( η ) for ei-ther branch. (b) Comparison of I (l)O ( ξ, η ), I D ( ξ, η ), I A ( ξ, η ). ξ ∈ L L [green (blue) lines], horizontal axes reflect theirnatural length. I lO (dashed lines). ˜ ξ (blank point). Furthermore, any pair of strongly incomparable spec-tra is on some isoentropy surface f ν ( r ). When rank( r ) =rank( s ), F ( ν, r, s ) is ν -continuous [22], any change insgn[ F ( ν, r, s )] implies F ( ν, r, s ) = 0. When rank( r ) < rank( s ), F ( ν ≤ , r, s ) are nonexistent, similarly, r ≁ T s are isoentropic by f ν> ( r ).Hence, the aim of catalysis is to catalyze incomparablespectra that are not isoentropic by generalized entropies.We find that a rigorous measure of a resource con-cerned with majorization should have it that equivaluestates with distinct spectra are incomparable. If this isnot fulfilled then such states are not truly equals as con-versions between them are possible. Inversion rank. —We introduce the following notion.
Definition 4.
For r ≁ s , examining their inequalitiesin sequence disregarding intermediate equalities, we calleach index l where the order reverses an inversion index.The totality of such is named the inversion rank n I ( r, s ).We note n I ( r, s ) ≤ d − i = 1 , d ); whencomparable, n I ( r, s ) = 0.Incomparability can be classified by inversion rankand further distinguished by the position and directionof the inversion. We denote for spectral sets incom-parable to r by the signs of majorization inequalitiessgn { A j ( r ′ ) − A j ( r ) } written as region vectors. For ex-ample in FIG. 2(a) we have: n I ( η ) = 0 comprises M − ( η ) = ( − , − , − , η , M +3 ( η ) = (0+ , , , η ; n I ( η ) = 1 comprises ( − , ˙0+ , ˙0+ , η , ( − ˙0 , − ˙0 , + , η ,(+ , − ˙0 , − ˙0 , η , ( ˙0+ , ˙0+ , − , η ; n I ( η ) = 2 comprises( − , + , − , η , (+ , − , + , η . Note “ −
0” (“0+”) signifiesinclusion of a boundary, “ ˙0” indicates mutually exclusiveboundaries, ending “0” upholds final equality.We present a condition for strongly incomparability us-ing inversion rank contrapositioning a result in Ref. [14].
Remark 1. If n I ( r, s ) is an odd number then r ≁ T s . proof. Odd n I ( r, s ) means an odd traversal of the Lorenz curves giving r > s and r d > s d . For any d ′ -catalyst c , also r c > s c and r d c d ′ > s d c d ′ . Thus A ( r ⊗ c ) >A ( s ⊗ c ) and A dd ′ − ( r ⊗ c ) < A dd ′ − ( s ⊗ c ), r ≁ T s . (cid:3) The following knowledge is geometrically relevant.
Remark 2.
Any set indicated by a region vector is convex. proof.
Let t = (1 − p ) r + pr ′ , if A j ( r ) − A j ( s ) > A j ( r ′ ) − A j ( s ) > A j ( t ) − A j ( s ) = (1 − p )[ A j ( r ) − A j ( s )] + p [ A j ( r ′ ) − A j ( s )] >
0. Likewise for all cases. (cid:3)
Majorization relations. —In this segment we developsome convenient majorization relations.We raise a family of spectra with nonfixed dimensions,which has been used as a standard for nonuniformity [5].
Proposition 1.
Let u ( k ) be a k -uniform spectrum, u i ( k ) = k γ ki , k ∈ Z + , (8)where γ ki = 1 ( i ≤ k ), 0 ( i > k ). Given r ∈ W d − , fora specified k the following hold: (1) r ≻ u iff rank( r ) ≤ rank( u ); (2) u ≻ r iff r ≤ u ; (3) r ≁ u iff rank( r ) > rank( u ) and r > u . proof. (1) More plainly, u ( k ) ∈ W d − are of the formand order: (1 , , . . . , ≻ ( , , , . . . , ≻ . . . ≻ ( d − , . . . , d − , ≻ ( d , . . . , d ). Clearly, they correspondto maximally mixed states for their respective embeddeddimensions, the claim becomes apparent. (2) u i ≤ k ( k ) areuniform while r i are nonincreasing. (3) When the priorconditions are unmet, r ≁ u . (cid:3) We see W d − is the convex hull of the extreme points { u ( k ) } d [FIG. 2(a)]. As any convex sum of { u ( k ) } d isnonincreasing, and any r ∈ W d − can be decomposed as: r = P dn =1 p k u ( k ), where d p d = r d , k p k = r k − r k +1 , k ∈{ , . . . , d − } , ( p k ) ∈ ∆ d − . Additionally, not only is u ( k ) useless as a catalyst [14], but also noncatalyzable inrelated conversions since for r ≁ u , provably n I ( r, u ) = 1.We now detail a spectral family constructed from u ( k ). Proposition 2.
Let χ ( q ) be the depolarized pure spectrumof a d -system, χ ( q ) = u (1) u ( d ) ≡ (1 − q ) u (1) + qu ( d ) , (9)where q ∈ [0 , r ∈ W d − , for a specified q thefollowing hold: (1) r ≻ χ iff r ≥ χ ; (2) χ ≻ r iff r d ≥ χ d ; (3) r ≁ χ iff r < χ and r d < χ d . proof. (1) Necessity is self-evident, we give proof of suf-ficiency. Let r ≥ χ , assuming r ≁ χ , then there ex-ists an inversion index l < d where A j ( r ) ≥ A j ( χ ) , j ∈{ , . . . , l − } and A l ( r ) < A l ( χ ) or A ′ l +1 ( r ) > A ′ l +1 ( χ ).The inversion requires r l < χ l , and since r i are non-increasing while χ i> are strictly uniform, A ′ l +1 ( r ) χ it must be r ≻ χ . When r = χ ,if χ ≻ r then χ ≥ r , due to uniformity of χ i> theonly possibility is r = χ where reflectively r ≻ χ . (2) For χ ≻ r , equivalently A ′ j ( r ) ≥ A ′ j ( χ ) , j ∈ { , . . . , d } withequality for j = 1. Necessarily r d ≥ χ d whence the otherinequalities are implied by uniformity of χ i> . (3) Whenthe prior conditions are unmet, r ≁ χ . (cid:3) Also, χ ( q ) is noncatalyzable since for r ≁ χ , again n I ( r, χ ) = 1. We now derive a majorization criterionimplying a general range for incomparability. Corollary 1.
Let r, s ∈ W d − , if r ≥ χ [ s ] = 1+(1 − d ) s d then r ≻ s , if r d ≥ χ d [ s ] = − s d − then s ≻ r . proof. Through Proposition 2 we can find for r its upperand lower pure depolarization bound χ [ s ] ≡ χ ( q = ds d )and χ [ s ] ≡ χ ( q = d − s d − ) respectively such that χ [ s ] (cid:23) s (cid:23) χ [ s ]. When q ∈ ( q, q ), χ ( q ) ≁ s [Fig. 1(b)]. Thecorollary ensues after a repetition of Proposition 2 byidentifying spectrum r satisfying r ≻ χ [ s ] or χ [ s ] ≻ r . (cid:3) Inconvertibility. —Nielsen conjectured that the proba-bility of picking at random a pair of incomparable spectrain a d -system tends to 1 as d → ∞ [2]. It was establishedthat densely many of these are in fact strongly incompa-rable [11]. Our objective is to find for a single spectrumits ratio of incomparable spectra in the Weyl chamber.We refer to this as inconvertibility. The higher its valuethe less likely it is to engage in majorization. Definition 5.
Let r ∈ W d − , its inconvertibility C d ( r ) isdefined as its incomparable spectral volume V [ I d − ( r )]divided by the volume of the entire chamber V [ W d − ]: C d ( r ) = V [ I d − ( r )] V [ W d − ] . (10)Technically, for V [ I d − ( r )] we calculate the volumes of M ∓ d − ( r ) by means of computational geometry and sub-tract them from the whole, i.e., V [ I d − ( r )] = V [ W d − ] − V [ M − d − ( r )] − V [ M + d − ( r )]. For rank( r ) < d , M + d − ( r )lies on the boundaries of W d − and is negligible. u ( k )and χ ( q ) can be directly integrated.We find C d [ u ( d − − ( d − (1 − d ) . We assess u ( d −
1) maximizes C d since its value tends to 1 as d → ∞ .Hence, maximal inconvertibility may occur very close tocomplete disorder which is most convertible.We choose χ ( q ) as a probe of inconvertibility since itspans the entire entropy range and has manifest ma-jorization relations [FIG. 3(a)]. The figure agrees withthe intuition of statistical complexity [26], affirming theuse of C d as an indicator of complexity [25].We believe V [ M ∓ d − ( r )] are natural reflections of dis-order and disequilibrium [26]. They can be supplemen-tal to related measures by imposing a finer hierarchy,e.g., for isoentropic spectra, greater V [ M + d − ( r )] meansmore parties majorizing r , which signifies greater disor-der and more utility for say its associated entangled (co-herent) pure state. To inquire further, the complemen-tary relation C ( r ) = 1 − E ( r ) − H ( r ) where E ( r ) = V [ M − ( r )] /V [ W ] and H ( r ) = V [ M +3 ( r )] /V [ W ] isshown by random points in W [FIG. 3(b)]. We surmisethat the central region of C d would compress upwards forgreater d with χ ( q ) touching the upperbound. Conclusion. —Throughout this treatise we have stud-ied the circumstance of incomparability in majorizationcomprehensively under a quantum setting. The rudi-ments of our theory can be accommodated to more spe- d = = = = = q C d [ χ ( q ) ] ( a ) FIG. 3. (a) Inconvertibility of χ ( q ) for d ∈ { , . . . , } . C d [ χ ( q )] tapers to zero at the extremities and vanishes for d =2. It increases with d , as C m d [ χ ( q m → . →
1. Near q = 1,the curves cross at different points. (b) Complementary rela-tion of C ( r ) = 1 − E ( r ) − H ( r ) depicted by random points(orange). u (3) sits expectedly at the apex. The edges of W ,excepting χ ( q ) (blue) and u (2) u (4) (magenta) lie on the up-perbound (red). In contrast, normalizing the Boltzmann N -partition maximally inconvertible by entropy [25] yields thespectral analog: (1 − kN ) u (1)+ kN u ( k ) , k ∈ { , . . . , N } . Withal, u (2) u (4) is an oscillatory band expanding at midrange. Againassuming a 2 × u (1) u (2) and u (2) u (4) upperboundthe product spectral surface r (1 − r − r − r ) = r r , thelowerbound of which is close to that of the entire chamber. cific contexts. For resource theories it can be adapted toexamine the lack of resources inhibiting state conversion.Analysis can thence be conducted on the relations be-tween resource deficiency, infidelity and failure rate. Themutual strong incomparability of generalized isoentropicspectra is an intrinsic trait of entropies underlying thedifficulty of related processes. It can also be worthwhileto see what categorization by inversion rank may reveal.The spectral geometry inside a Weyl chamber entails anulterior layer of attributes pertaining to a state, by whicha bridge has been extended to complexity theory via in-convertibility. It may be of interest to know what are themost convertible states by entropy. All in all, we hope tohave contributed meaningfully to the discussion. ∗ [email protected][1] A. W. Marshall, I. Olkin, and B. Arnold, Inequalities:Theory of Majorization and Its Applications (SpringerScience & Business Media, 2010).[2] M. A. Nielsen, Phys. Rev. Lett. , 436 (1999).[3] S. Du, Z. Bai, and Y. Guo, Phys. Rev. A , 052120(2015).[4] H. Zhu, Z. Ma, Z. Cao, S.-M. Fei, and V. Vedral, Phys.Rev. A , 032316 (2017).[5] G. Gour, M. P. Mller, V. Narasimhachar, R. W.Spekkens, and N. Y. Halpern, Physics Reports , 1(2015).[6] M. A. Nielsen, Phys. Rev. A , 022114 (2001).[7] M. A. Nielsen and J. Kempe, Phys. Rev. Lett. , 5184(2001).[8] M. H. Partovi, Phys. Rev. A , 052117 (2011).[9] F. Buscemi and G. Gour, Phys. Rev. A , 012110 (2017).[10] K. ˙Zyczkowski and I. Bengtsson, Annals of Physics ,115 (2002).[11] R. Clifton, B. Hepburn, and C. W¨uthrich, Physics Let-ters A , 121 (2002).[12] I. Chattopadhyay and D. Sarkar, Quantum Information& Computation , 392 (2007).[13] I. Chattopadhyay and D. Sarkar, Phys. Rev. A ,050305 (2008).[14] D. Jonathan and M. B. Plenio, Phys. Rev. Lett. , 3566(1999).[15] S. Bandyopadhyay, V. Roychowdhury, and U. Sen, Phys.Rev. A , 052315 (2002).[16] R. Duan, Y. Feng, X. Li, and M. Ying, Phys. Rev. A , 042319 (2005).[17] G. Vidal, D. Jonathan, and M. A. Nielsen, Phys. Rev.A , 012304 (2000). [18] G. Vidal, Phys. Rev. Lett. , 1046 (1999).[19] Y. Feng, R. Duan, and M. Ying, IEEE Transactions onInformation Theory , 1090 (2005).[20] Y. Li and P. Busch, Journal of Mathematical Analysisand Applications , 384 (2013).[21] L. Liu and Y. Li, Reports on Mathematical Physics ,1 (2014).[22] M. Klimesh, “Inequalities that collectively completelycharacterize the catalytic majorization relation,” (2007),arXiv:quant-ph/0709.3680.[23] S. Turgut, Journal of Physics A: Mathematical and The-oretical , 12185 (2007).[24] D. W. Kribs, R. Pereira, and S. Plosker, Linear & Mul-tilinear Algebra , 1455 (2013).[25] W. Seitz and A. D. Kirwan, Entropy , 10 (2016).[26] R. L´opez-Ruiz, H. L. Mancini, and X. Calbet, PhysicsLetters A209