Mutually unbiased bases, orthogonal Latin squares, and hidden-variable models

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1 Mutually unbiased bases, othogonal Latin squaes, and hidden-vaiable odels Toasz Pateek, 1 Boivoje Dakić, 1,2 and Časlav Bukne 1,2 1 Institute fo Quantu Optics and Quantu Infoation, Austian Acadey of Sciences, Boltzanngasse 3, A-1090 Vienna, Austia 2 Faculty of Physics, Univesity of Vienna, Boltzanngasse 5, A-1090 Vienna, Austia Received 14 Apil 2008; evised anuscipt eceived 27 Octobe 2008; published 16 Januay 2009 Mutually unbiased bases encapsulate the concept of copleentaity the ipossibility of siultaneous knowledge of cetain obsevables in the foalis of quantu theoy. Although this concept is at the heat of quantu echanics, the nube of these bases is unknown except fo systes of diension being a powe of a pie. We develop the elation between this physical poble and the atheatical poble of finding the nube of utually othogonal Latin squaes. We deive in a siple way all known esults about the unbiased bases, find thei lowe nube, and dispove the existence of cetain fos of the bases in diensions diffeent than powe of a pie. Using the Latin squaes, we constuct hidden-vaiable odels which efficiently siulate esults of copleentay quantu easueents. DOI: /PhysRevA PACS nube s : Ta, Ox I. INTRODUCTION Copleentaity is a fundaental pinciple of quantu physics which fobids siultaneous knowledge of cetain obsevables. It is anifested aleady fo the siplest quantu echanical syste spin If the syste is in a definite state of, say, spin along x, the spin along y o z is copletely unknown, i.e., the outcoes spin-up and spin-down occu with the sae pobability. The eigenbases of ˆ x, ˆ y, and ˆ z Pauli opeatos fo so-called utually unbiased bases MUBs : Evey vecto fo one basis has equal ovelap with all the vectos fo othe bases. MUBs encapsulate the concept of copleentaity in the quantu foalis. Although copleentaity is at the heat of quantu physics, the question about the nube of MUBs eains unansweed. Apat fo being of foundational inteest, MUBs find applications in quantu state toogaphy 1, quantukey distibution 2, and the ean King poble 3. A d-level quantu syste can have at ost d+1 MUBs, and such a set is efeed to as the coplete set of MUBs. In 1981 Ivanović poved by constuction that thee ae indeed d+1 copleentay easueents fo d being a pie nube 4. This esult was genealized by Woottes and Fields to cove powes of pies 1. Fo othe diensions the nube of MUBs is unknown, the siplest case being diension six. A consideable aount of wok was done towads undestanding this poble. New poofs of pevious esults wee established 5 8 and the poble was linked with othe unsolved pobles 9,10. It was also noticed that it is siila in spiit to cetain pobles in cobinatoics and finite geoety 14. Hee, we build upon these elations. We descibe the poble of the nube of othogonal Latin squaes OLSs, which was initiated by Eule 15 and still attacts a lot of attention in atheatics. Although this poble is not solved yet in full geneality, oe is known about it than about the nube of MUBs. Using a black box which physically encodes infoation contained in a Latin squae, we link evey OLS of ode being a powe of a pie with a MUB. Fo diension six, ou ethod gives thee MUBs, which is the axial nube found by the nueical eseach 10,11. Utilizing known esults fo OLSs we deive a inial nube of MUBs, and dispove the existence of cetain fos of MUBs fo abitay d. Finally, using OLSs we constuct hidden-vaiable odels that efficiently siulate copleentay quantu easueents. II. ORTHOGONAL LATIN SQUARES A Latin squae of ode d is an aay of nubes 0,...,d 1 whee evey ow and evey colun contains each nube exactly once. Two Latin squaes, A= A ij and B= B ij, ae othogonal if all odeed pais A ij,b ij ae distinct. Thee ae at ost d 1 OLSs and this set is called coplete. The existence of L OLSs is equivalent to the existence of a cobinatoial design called a net with L+2 ows 16. The net design has a fo of a table in which evey ow contains d 2 distinct nubes. They ae split into d cells of d nubes each, in such a way that the nubes of any cell in a given ow ae distibuted aong all cells of any othe ow. The additional two ows of the net coespond to othogonal but not Latin squaes, with the enties A ij = j and A ij =i. The following algoith allows us to constuct the net fo a set of OLSs: i Wite the squaes in the standad fo in which the nubes of the fist colun ae in ascending ode by peuting the enties, it is always possible to wite the set of OLSs in the standad fo without copoising Latiness and othogonality. ii Augent the set of OLSs by the two othogonal non- Latin squaes A ij = j and A ij =i. iii Wite the ows of the squaes as cells in a single ow of the table. The nube of the table s ows is now equal to the nube of squaes in the augented set. iv In the ow of the table which coesponds to the squae A ij = j, efeed to as the coodinate ow, eplace the nube A ij in the ith cell by A ij =id+ j, whee d is the ode of the squae /2009/79 1 / The Aeican Physical Society

2 PATEREK, DAKIĆ, AND BRUKNER v In evey cell of the othe ows eplace nube B ij on position j by the intege associated to the nube B ij of the jth cell in the coodinate ow, i.e., B ij B ij = jd+b ij. We shall pove that the table geneated by this pocedue is indeed a net design. We use anothe popety defining the design: Two nubes in one cell do not epeat in any othe cell. This aleady includes that any two cells of two diffeent ows shae exactly one coon nube, as if thee wee no coon nubes shaed by these cells, thee would have to be at least two coon nubes shaed by othe cells. Due to the definitions of A ij and B ij and the fact that the coluns of B ij contain all distinct nubes 0,...,d 1, evey ow of the table contains d 2 distinct nubes 0,...,d 2 1. By constuction, the nubes of any cell of the coodinate ow ae distibuted aong all the cells of all the othe ows. Theefoe, it is sufficient to pove the popety of the net fo the eaining ows. Assue to the contay, that two nubes epeat in two cells of diffeent ows, jd+b ij, j d +B ij = ld+c kl,l d+c kl. Since j, j,l,l,b ij,c ij 0,...,d 1 the equality can only hold if B ij =C kj and B ij =C kj, i.e., thee ae ows of the squaes B and C which contain the sae nubes, in the coluns defined by j and j. This, howeve, cannot be because one can always peute the enties of, say, squae C such that its kth ow becoes the ith ow without copoising othogonality and the two squaes would not be othogonal. III. QUBIT Conside the squaes fo d = 2. We link the with the copleentay easueents of a qubit. The augented set of othogonal squaes eads as The ight-hand side squae is Latin, the left and iddle squaes ae othogonal to each othe and to the Latin squae. These thee squaes lead to the following net design on the left-hand side, in which the nubes ae epesented by pais n in odulo-two decoposition: b =0 b = = b? n = b? n = b? On the ight-hand side, we wite the copleentay questions associated with each ow. They ae answeed by pais n in the left- and ight-hand coluns of the net design left colun answe 0, ight colun answe 1. In this way, the questions ae linked to the othogonal squaes. The copleentay questions can be answeed in quantu expeients involving MUBs. Conside a device encoding paaetes and n via application of the unitay 1 2 Û= ˆ x ˆ zn. When it acts on z states, they get a phase dependent on n and ae flipped ties. Thus, knowing the initial state, a final easueent in the ˆ z eigenbasis eveals, giving the answe to the fist copleentay question. Siilaly, taking x and y as initial states, the esults of x and y easueent answe the second and the thid copleentay question, espectively. IV. PRIME DIMENSIONS Fo pie d the net has d+1 ows. The enties of the ows coesponding to the OLSs ae geneated fo the following foula: n = a + b, 3 whee the intege a=1,...,d 1 enueates the ows of the table, while the intege b=0,...,d 1 enueates diffeent coluns, and the su is odulo d. Additional two ows coespond to the questions about and n, espectively. The table fo the ows coesponding to the OLSs is built in the following way: i Choose a ow, a, and the colun, b. ii Vay =0,...,d 1 and copute n using 3. iii Wite pais n in the cell. Fo exaple, fo d=3, one has b =0 b =1 b = = b? n = b? n = + b? n =2 + b? 4 The copleentay questions ae given on the ight-hand side. Diffeent values of b enueate possible answes. We shall see, again, that the copleentay questions can be answeed using MUBs. Conside encoding of paaetes and n via application of Û=Xˆ Ẑ n, whee the Weyl- Schwinge opeatos Xˆ Ẑ n span a unitay opeato basis. In the basis of Ẑ, denoted as, the two eleentay opeatos satisfy Ẑ = d, Xˆ = +1, 5 whee d =exp i2 /d is a coplex dth oot of unity. Fo the sae easons as fo a qubit, the fist two questions ae answeed by applying Û on the eigenstates of Ẑ and Xˆ opeatos, and then by easuing the eeging state in these bases. In all othe cases the action of the device is Û=Xˆ Ẑ a+b =Xˆ Ẑ a Ẑ b. The eleentay opeatos do not coute, instead one has ẐXˆ = d Xˆ Ẑ, and it follows that Xˆ Ẑ a = 1/2 a 1 d Xˆ Ẑ a. Finally, the action of the device is, up to the global phase, given by Û Xˆ Ẑ a Ẑ b. The eigenstates of the Xˆ Ẑ a opeato, expessed in the Ẑ basis, ae given by j a = 1/ d 1 j as d =0 d, whee s = + + d 1 5, and the Ẑ opeato shifts the, Ẑ j a = j 1 a. Afte the de

3 MUTUALLY UNBIASED BASES, ORTHOGONAL LATIN vice, j a is shifted exactly b ties and subsequent easueent in this basis eveals the answe to the ath question. On the othe hand, the eigenbases of Xˆ Ẑ a fo a=1,...,d 1 and eigenbases of Xˆ and Ẑ ae known to fo a coplete set of MUBs 5. Not only the nube of MUBs is the sae as the nube of OLSs, but they ae indexed by the sae vaiable, a. This allows to associate MUB to evey OLS fo pie d. V. POWERS OF PRIMES If d is a powe of a pie, a coplete set of OLSs is obtained using opeations in the finite field of d eleents, and one expects that a coplete set of MUBs also follows fo the existence of the field. Indeed, explicit foulas fo MUBs in tes of the field opeations wee pesented in 1,7,8. Hee, we pove this esult in a siple way elated to 17, using the theoe of Bandyopadhyay et al. 5,19 : If thee is a set of othogonal unitay atices, which can be patitioned into M subsets of d couting opeatos, then thee ae at least M MUBs. They ae the joint eigenbases of the d couting opeatos. To illustate the idea, conside again pie d. Take the othogonal unitay opeatos Ŝ n =Xˆ Ẑ n with thei powes n taken fo the fist colun of the net. The cell of the fist and second ow coesponds to the eigenbases of Ẑ and Xˆ, espectively, wheeas the othe ows ae defined by b=0, i.e., n=a. Accoding to the coutation ule of the eleentay opeatos Xˆ and Ẑ, Ŝ n and Ŝ n coute if and only if n n=0 od d. Thus, fo a fixed ow, i.e., fixed a, the set of d opeatos Ŝ n coute, because a a =0, and, due to the entioned theoe, thee is a set of d +1 MUBs. Fo d= p being a powe of a pie, the OLSs and the net ae geneated by the foula n = a b, 6 whee and denote ultiplication and addition in the field, a,b,,n F d ae field eleents, and a 0. The fist two ows of the table ae defined by =b and n=b. Inthe case of d=4, the fou eleents 0,1,, +1 of the field F 4 is the oot of x 2 +x+1 17, when indexed with the nubes 0, 1, 2, 3, lead to the following net design: We use the concept of a basis in the finite field F d.it consists of eleents e i, with i=1,...,. Evey basis has a unique dual basis, ē j, such that t e i ē j = ij, whee the tace in the field, t x, aps eleents of F d into the eleents of the pie field F p. We use lowecase t fo the tace in the field in ode to distinguish it fo the usual tace ove an opeato, which we denote by T. It has the following useful popeties: t x y =t x +t y, and t a x =a t x, whee opeations on the ight-hand side ae odulo p and a is in the pie field. We decopose in the basis e i, = 1 e 1 e, whee i =t ē i, and n in the dual basis, n=n 1 ē 1 n ē, with n i =t n e i. Due to the popeties of the tace in the field and the dual basis t n = i n i = n, i=1 whee = 1,..., and n = n 1,...,n have coponents in the pie field, i.e., nubes 0,...,p 1. 8 Conside opeatos defined by the decoposition of and n, Ŝ n =Xˆ 1 n p Ẑ 1 p Xˆ n p Ẑ p, whee, e.g., Xˆ i p is the unitay opeato acting on the ith p-diensional subspace of the global d-diensional space. Opeatos Ŝ n fo an othogonal basis. They coute, if and only if n n = 0 od p. Take the opeatos coesponding to a fixed ow of the fist colun of the net, i.e., a is fixed, b=0 and theefoe n=a. Fo Eq. 8, all these d opeatos coute if t a =t a, which is satisfied due to associativity and coutativity of ultiplication in the field. Theefoe, thei eigenbases define MUBs. Again, each ow of the table is linked with the MUB. To ake an illustation, conside again the exaple of d =4. Choose e 1,e 2 =,1 as a basis in the field, such that the nubes ae decoposed into pais 1 2 in the usual way: 0 00, 1 01, 2 10, The dual basis eads as ē 1,ē 2 = 1, +1, which iplies that the nubes n ae decoposed into pais n n 1 n 2 as follows: 0 00, 1 10, 2 11, Each pai of nubes of table 7 is now witten vetically as a cobination of two pais of nubes:

4 PATEREK, DAKIĆ, AND BRUKNER MUBs ae foed by the eigenbases of opeatos ˆ 1 n x ˆ 1 z ˆ 2 n x ˆ 2 z, whee the powes ae taken fo the fist colun of this table. The esult is in ageeent with othe ethods 5,6. The copleentay questions answeed by the states of these MUBs ae foulated in tes of individual bits 1, 2, n 1, n 2, which ae encoded by Û= ˆ 1 n x ˆ 1 z ˆ 2 n x ˆ 2 z. Fo exaple, the question of the last ow is about the values of 1 +n 1 and 2 +n 2. An inteesting featue stengthening the link between MUBs and OLSs is the existence of the set of OLSs and MUBs which cannot be copleted. Fo exaple, the following net design cannot have oe ows. The MUBs elated to this table ae the eigenbases of Xˆ, Ẑ, and Xˆ Ẑ fo d=4. Coespondingly, thee ae no othe bases which ae utually unbiased with espect to these thee 18. VI. GENERAL DIMENSION Tay was the fist to pove that no two OLSs of ode six exist 20, i.e., the net fo d=6 has only thee ows. The opeatos Xˆ Ẑ n coute fo nubes and n fo the fist cell of these ows and the coesponding MUBs ae the eigenbases of Xˆ, Ẑ, and Xˆ Ẑ. Siilaly to the case of d=4 no othe MUB with espect to these thee exists 19. Of couse, the question whethe diffeent thee MUBs can be augented with additional MUBs eains open. A. MacNeish s bound Moe geneally, the lowe bound on the nube of OLSs was given by MacNeish 21. If two squaes of ode a ae othogonal, A B, and two squaes of ode b ae othogonal, C D, then the squaes obtained by a diect poduct, of ode ab, ae also othogonal, A C B D. This iplies that the nube of OLSs, L, of the ode d= p 1 1 p n n, with p i being pie factos of d, is at least L in i p i i 1, whee p i i 1 is the nube of OLSs of ode p i i. A paallel esult holds fo MUBs 7,19. If a and b ae the states of two MUBs in diension d 1, and c and d ae the states of MUBs in diension d 2, then the tenso poduct bases a c and b d fo MUBs in diension d 1 d 2. Thus, fo d = p 1 1 p n n thee ae at least in i p i i +1 MUBs. B. Latin opeato basis In geneal, we know oe about the nube of OLSs than about the nube of MUBs 16. We use this knowledge to deive conditions which estict the fo of MUBs. Conside the opeatos Bˆ n 0 n d = 1ˆ + =0 =1 n d Ŝ, 11 whee n =0,...,d 1 and Ŝ = j=0 j d j j have a coplete set of MUBs as eigenbases, =0,...,d. We show that existence of such a set and othogonality of d 2 opeatos Bˆ n 0,...,n d iplies copleteness of the set of OLSs. The tace d d

5 MUTUALLY UNBIASED BASES, ORTHOGONAL LATIN scala poduct T Bˆ n 0,...,n Bˆ d n 0,...,n d is given by d2 k 1, whee k denotes the su of Konecke deltas, k n0 n nd n d. Opeatos Bˆ n 0,...,n d and Bˆ n 0,...,n d ae othogonal if and only if k=1, i.e., n =n fo exactly one. This condition applied to d 2 othogonal opeatos, defines a coplete set of othogonal squaes. To see this, take d 2 othogonal opeatos Bˆ n 0 b,...,n d b with b=1,...,d 2 and conside d+1 squaes defined by thei indices n b fo a fixed. If the squaes wee not othogonal, one could find at least two identical pais, (n b,n b )=(n b,n b ), iplying that opeatos 11 ae not othogonal k 1. Theefoe, e.g., fo d=6, thee is no coplete set of MUBs fo which opeatos Bˆ n 0,...,n d ae othogonal because thee is no coplete set of OLSs in this case. C. Othogonal functions The second condition is obtained by noting that a net defines othogonal functions, F a,n, which give the colun of the ath ow whee the pai n is enteed. The othogonality eans that fo the pais n fo which the function F a,n has a fixed value, the function F a,n acquies all its values. We show that if d 2 unitaies, Û n, shift up to a phase the states of diffeent bases in accodance with the net Û n j a j + F a,n a, 12 then these bases ae MUBs. Fo the poof, note that d 1 i =0 a i i a 2 =1. Fo othogonality of the functions, this su can be witten as S a j+f a,n j +F a,n a 2 =1, whee S is the set of pais n fo which F a,n has a fixed value. By 12, the last is S a j Û n Û n j a 2, which due to unitaity, Û n Û n =1ˆ, is the su of d identical tes a j j a 2. Theefoe, a j j a 2 =1/d. Futhe, given d 2 unitaies with popety 12, one ecoves the table in the following expeient: Pepae 0 a, act on it with Û n, easue in the sae basis, and wite the pai n in the ath ow and the colun coesponding to the esult. Thus, in diension six, thee cannot be 36 unitaies satisfying 12, with the othogonal functions, fo oe than thee bases, because othewise one could constuct oe than thee othogonal squaes of ode six, which is ipossible. VII. HIDDEN-VARIABLE SIMULATION OF MUBS The net designs can be used to constuct hidden-vaiable odels which siulate esults of copleentay easueents on cetain states. Recently, Spekkens showed that only fou ontic states hidden vaiables ae sufficient to siulate copleentay easueents of a qubit pepaed in a state of a MUB 22. In his odel, quantu states of MUBs coespond to the episteic states satisfying the knowledge balance pinciple: The aount of knowledge one possesses about the ontic state is equal to the aount of knowledge one lacks 22. This pinciple lies behind the net design. Left-hand table of 2 coesponds to the oiginal Spekkens odel: The nubes enueate ontic states, cells coespond to the episteic states and ows to the copleentay easueents. All othe tables genealize the odel. To identify the ontic state one needs two dits of infoation thee ae d 2 ontic states, wheeas the episteic state is defined by a single dit, leaving the othe one unknown. The quantu states descibed by these odels equie a classical ixtue of only two dits to odel d outcoes of d+1 quantu copleentay easueents. Ou appoach allows us to ask the question how any episteic states satisfying the knowledge balance pinciple, i.e., having d undelying ontic states, coespond to quantu states. Fo exaple, in the case of a two-level syste thee ae fou ontic states, and six possible episteic states see the net design of 2. All six coespond to quantu eigenstates of copleentay obsevables. In geneal, any episteic state is epesented by a cell of d nubes i 1 i 2 i d. Since each nube takes on one of d 2 values, the nubes cannot epeat and thei ode is not ipotant, thee ae E d = D D+1 D+d 1 i1 =1 i2 =i 1 +1 id =i d 1 +1 possible episteic states, with D=d 2 d+1. Fo d being a powe of a pie the quantu states coesponding to the cells of the net design ae basis vectos of a coplete set of MUBs. They can be used to uniquely decopose abitay Heitian opeato Ô = T Ô 1ˆ + =0 d d 1 j=0 p j j j, 13 whee p j = j Ô j and j is the jth state of the th MUB. Fo the poof, note that the coplete set of MUBs can be used to define the opeato basis in the Hilbet-Schidt space Ŝ = d 1 j=0 j d j j. Thee ae d 2 such opeatos, because =0,...,d, the powe =0,...,d 1 and all d opeatos Ŝ 0 ae equal to the identity opeato. Since they ae noalized as T Ŝ Ŝ =d any opeato has a unique expansion Ô= 1 T Ô 1ˆ d d + =0 d 1 =1 T Ô Ŝ Ŝ. Witing Ŝ in tes of pojectos on MUBs one finds Eq. 13. If Ô is a quantu state, T Ô =1 and p j s ae pobabilities to obseve outcoes elated to suitable states of MUBs. We conside geneal episteic states, not necessaily those coesponding to the cells of the net design. Such episteic states have patial ovelap with the cells, defined as the nube of coon ontic states divided by d. Fo exaple, the episteic state has an ovelap of 2 3 and 1 3 with the fist and thid episteic state of the fist ow of table 4, espectively. To constuct opeato Ô associated with a geneal episteic state, we take these ovelaps to define the pobabilities p j. Since we would like to see how any episteic states coespond to quantu states we take opeatos Ô with a unit tace. If T Ô 2 =1 and T Ô 3 1, the opeato Ô cannot epesent a quantu state, because the fist condition excludes ixed states, and both of the exclude pue states 23. We find that fo d=3 only the episteic states of the net design coespond to the quantu

6 PATEREK, DAKIĆ, AND BRUKNER states. Thee ae Q 3 =12 such states, out of E 3 =84 diffeent episteic states. The atio of R d =Q d /E d apidly deceases with d: We checked R 3 =1/7, R 4 =8/455, and R 5 =1/1771. Thus, ost of the episteic states, constucted accoding to the knowledge balance pinciple, do not epesent a quantu-physical state. VIII. CONCLUSIONS In conclusion, we showed a one-to-one elation between OLSs and MUBs, if d is a powe of a pie. Fo geneal diensions, we deive conditions which liit the stuctue of the coplete set of MUBs and we pesented paallelis between the MacNeish s bound on the inial nube of OLSs and the inial nube of MUBs. Inteestingly, the MacNeish s bound is known not to be tight. Thee ae at least five OLSs of ode 35, whee the MacNeish s bound is fou 24. Theefoe, futhe insight into the elations between MUBs and OLSs would be gained fo studies of MUBs fo d=35. Finally, using the squaes, we constucted hidden-vaiable odels that efficiently siulate easueents of MUBs. Howeve, the ajoity of states in these odels do not have quantu-physical countepats. ACKNOWLEDGMENTS We thank Gabiele Uchida and Johannes Kofle. This wok is suppoted by the FWF Contact No. P19570-N16 and poject CoQuS Contact No. W1210-N16, and by EC Poject QAP Contact No , and the Foundational Questions Institute FQXi. 1 W. K. Wootes and B. D. Fields, Ann. Phys. N.Y. 191, N. Gisin et al., Rev. Mod. Phys. 74, L. Vaidan, Y. Ahaonov, and D. Z. Albet, Phys. Rev. Lett. 58, ; Y. Ahaonov and B.-G. Englet, Z. Natufosch., A: Phys. Sci. 56, ; P. K. Aavind, ibid. 58, ; A. Hayashi, M. Hoibe, and T. Hashioto, Phys. Rev. A 71, I. D. Ivanović, J. Phys. A 14, S. Bandyopadhyay et al., Algoithica 34, Č. Bukne and A. Zeilinge, J. Mod. Opt. 47, ; J. Lawence, Č. Bukne, and A. Zeilinge, Phys. Rev. A 65, A. Klappenecke and M. Röttele, Lect. Notes Coput. Sci. 2948, T. Dut, J. Phys. A 38, P. O. Boykin et al., e-pint axiv:quant-ph/ I. Bengtsson et al., J. Math. Phys. 48, G. Zaune, Dissetation, Univesität Wien, I. Bengtsson, epint axiv:quant-ph/ P. Wocjan and T. Beth, Quantu Inf. Coput. 5, M. Saniga, M. Planat, and H. Rosu, J. Opt. B: Quantu Seiclassical Opt. 6, L ; W. K. Wootes, Found. Phys. 36, L. Eule, Co. Aith. 2, The CRC Handbook of Cobinatoial Designs, edited by C. J. Colboun and J. H. Dinitz CRC Pess, Boca Raton, FL, K. S. Gibbons, M. J. Hoffan, and W. K. Woottes, Phys. Rev. A 70, M. Gassl pivate counication. 19 M. Gassl, epint axiv:quant-ph/ G. Tay, Coptes Rendu de l Assoc. Fancaise pou l Advanceent de Sci. Nat. 1, H. F. MacNeish, Ann. Math. 23, R. W. Spekkens, Phys. Rev. A 75, N. S. Jones and N. Linden, Phys. Rev. A 71, M. Wojtas, J. Cob. Des. 4,