We show that the isotropic lines in the lattice Z_{d}^{2} are the Lagrangian
submodules of that lattice and we give their number together with the number of
them through a given point of the lattice. The set of isotropic lines decompose
into orbits under the action of SL(2,Z_d). We give an explicit description of
those orbits as well as their number and their respective cardinalities. We
also develop two group actions on the group \Sigma_{D}(M) related to the topic.Comment: 10 page

Albouy, Olivier

English

arXiv

10 pagesWe show that the isotropic lines in the lattice Z_{d}^{2} are the Lagrangian submodules of that lattice and we give their number together with the number of them through a given point of the lattice. The set of isotropic lines decompose into orbits under the action of SL(2,Z_d). We give an explicit description of those orbits as well as their number and their respective cardinalities. We also develop two group actions on the group \Sigma_{D}(M) related to the topic

Albouy, O.

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The isotropic lines of Z_{d}^{2}

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The isotropic lines of Z2dOlivier Albouy1,2,31 Universite´ de Lyon, F-69622, Lyon, France2 Universite´ Lyon 1, Villeurbanne, France3 CNRS/IN2P3, UMR5822, Institut de Physique Nucle´aire de Lyon, FranceE-mail: o.albouy@ipnl.in2p3.frAbstractWe show that the isotropic lines in the lattice Z2d are the Lagrangian submodulesof that lattice and we give their number together with the number of them througha given point of the lattice. The set of isotropic lines decompose into orbits underthe action of SL(2,Zd). We give an explicit description of those orbits as well astheir number and their respective cardinalities. We also develop two group actionson the group ΣD(M) related to the topic.PACS numbers: 03.65.Fd, 02.10.Ox, 02.10.Ud, 03.67.-aKeywords: discrete Wigner distributions - isotropic lines - Lagrangian submod-ulesIntroductionWigner distributions are a major tool of quantum mechanics. They offer a useful,alternative way besides density matrices of representing pure and mixed states ofa quantum system. But whereas in the continuous phase space those distributionsare well-defined [1][2], there is still a need for a sound mathematical definitionover a discrete phase space. In particular, the structure of such a phase space is ofsome importance. In 1974, Buot introduced a Wigner distribution over an d × dphase space with d an odd integer [3]. In 1980, Hannay and Berry followed anotherapproach to build a Wigner distribution over a 2d× 2d lattice [4]. Still in anotherway, in 2004, Gibbons et al. constructed Wigner distributions over a finite fieldparametrised lattice [5].More recently, in their way to set up discrete Wigner distributions on the dis-crete phase space Z2d, with Zd the set of integers modulo d, Chaturvedi et al [6]encountered undetermined signs S(q, p), one at each point (q, p) of the lattice. Anatural question then arises: To what extent can these signs be fixed by demand-ing that averages of Wigner distributions over isotropic lines in the lattice yield1probabilities, where an isotropic line is a set of d points on the lattice such thatthe symplectic product of any two of them is 0 (modulo d). In order to answerthis and related questions one needs a detailed knowledge of the structure of theisotropic lines in Z2d. In particular, it would be useful to know their number asa whole or with special conditions and also how they arrange in orbits under theaction of the symplectic group SL(2,Zd).This communication is only concerned with the mathematical properties of theisotropic lines in Z2d. In Section 1, we derive the number of isotropic lines in Z2d andthen in Section 2 the number of them through a given point of the lattice. Thisshould be compared with the results obtained by Havlicek and Saniga in [7] and[8] about the number of projective points in the lattice and the number of themunder the same condition. In Section 3, we give a full description of the orbitsof isotropic lines under the action of SL(2,Zd) with the help of some parameters.All that is achieved on the basis of a work by the author on symplectic reductionof matrices and Lagrangian submodules [9]. In a fourth section, we develop twogroup actions on the group ΣD (M) relevant to the understanding of that lattergroup. For any result appearing in this communication without proof, the readeris referred to [9].To end this introduction, we note two features of the results presented here. Onthe one hand, they do not depend on the parity of d, contrary to what happenedin [3] and [4]. On the other hand, they have direct relevance to the commutingsubgroups of the Pauli group for a general d. That latter feature has been of greatimportance in the building up of both Wigner ditrbutions and mutually unbiasedbases [5][10].1 The number of isotropic linesLet ω denote the symplectic product of two vectors of Z2d. With matrices, itconsists in computing a determinant:ω((α, β), (γ, δ)) =∣∣∣∣ α γβ δ∣∣∣∣ = αδ − βγ. (1)The orthogonal of a submodule M of Z2d will be denoted Mω:Mω = {x ∈ Z2d; ∀y ∈M,ω(x, y) = 0}. (2)Isotropic submodules are defined to satisfy the set inclusion M ⊂ Mω. Lagrangiansubmodules are the maximal isotropic submodules for inclusion, what is equivalentto M =Mω.In a first time, we are going to find the number of isotropic lines in Z2d for d apower of a prime, say d = ps, s ≥ 1. The way we derive this number is a strictapplication of Theorem 7 in [9]. This brings about a hint for Section 4, but weshall also see that there exists a shortcut. We then address the case of a generald.21.1 Special case: d a power of a primeLet s˜ = ⌊s/2⌋, the floor part of s/2. As shown in [9], for any Lagrangian submoduleM , there exist S ∈ SL(2,Zps) and k ∈ {0, . . . , s˜} such that M is linearly generatedby the column vectors ofS ×(pk 00 ps−k). (3)In other words, with S1 and S2 the two column vectors of S, (S1, S2) is a symplecticcomputational basis of (Zps)2 and M is the set of all linear combinations of pkS1and ps−kS2 with coefficients in Zps. As a converse, any submodule thus generatedis Lagrangian. In fact, the number k is a property of M , that is to say for anyconvenient pair (S, k′) in order to generate M as in (3), we have k′ = k. We willdenote Ok(ps) the set of all Lagrangian submodules thus obtained for a given kand S varying. The cardinality of any M ∈ Ok(ps) isp(s−1)−(k−1)p(s−1)−(s−k−1) = ps. (4)Let ℓ be an isotropic line and 〈ℓ〉 the submodule it generates, the set of all finitelinear combinations of vectors of ℓ. Any two vectors in 〈ℓ〉 are orthogonal andhence 〈ℓ〉 is an isotropic submodule containing at least ps vectors. Thus isotropiclines and Lagrangian submodules are the same.The number of free vectors x in Zps is p2s − p2(s−1). The number of vectors ysuch that for a given free x we have ω(x, y) = 1 is ps. The number of pairs (x, y)such that ω(x, y) = 1 is the product of the two previous ones:nω = |SL(2,Zps)| = p3s − p3s−2. (5)Several symplectic matrices S may give rise to the same submodule in Ok(ps)according to the form (3). Let k ∈ {0, . . . , s˜} and M ∈ Ok(ps). Let ΣD(M) be thematrix group of the changes of computational basis such that if P ∈ ΣD(M) andif M is generated by the column vectors of the matrix given in (3), then M is alsogenerated by the column vectors of the matrixSP ×(pk 00 ps−k), (6)where SP need not be symplectic. In fact, we derived in [9] that the group ΣD(M)is completely determined by the value of k. So, the number of symplectic matricesthat give rise to a given M ∈ Ok(ps) isnD(k) = |ΣD(M) ∩ SL(2,Zps)| (7)and hence|Ok(ps)| =nωnD(k). (8)Let us suppose that 2k < s. In ΣD (M), the number of matrices with determinant1 is the same as the number of matrices with any other (invertible) determinant.Indeed, if u ∈ U(Zps) and P = (P1|P2) ∈ ΣD(M) ∩ SL(2,Zps), with P1 and P2the first and second columns of P respectively, then (uP1|P2) ∈ ΣD(M) but withdeterminant u. This transformation is injective so that the number of matrices in3ΣD(M) with determinant u is greater than or equal to the number of matrices inΣD(M) with determinant 1. The converse inequality may be shown the same way.So we havenD(k) =|ΣD(M)||U(Zps)|=(ps − ps−1)2 · p(s−1)−(s−2k−1) · psps − ps−1= (ps − ps−1)ps+2k(9a)= p2s(p2k − p2k−1) (9b)and so|Ok(ps)| =ps − ps−2p2k − p2k−1= ps−2k−1(p+ 1). (10)If 2k = s (what supposes that s is even), then ΣD(M) ∩ SL(2,Zps) = SL(2,Zps)and so ∣∣Os/2(ps)∣∣ = nωnω= 1. (11)For s odd, then 2s˜ = s− 1,es∑k=0p−2k =1− p−2(es+1)1− p−2=p2(es+1) − 1p2(es+1) − p2es=ps+1 − 1ps+1 − ps−1(12)and hence the number of isotropic lines isnL(ps) =es∑k=0|Ok(ps)| = ps−1(p+ 1)ps+1 − 1ps+1 − ps−1=ps+1 − 1p− 1. (13)If s is even, then 2s˜ = s,es−1∑k=0p−2k =1− p−2es1− p−2=p2es − 1p2es − p2es−2=ps − 1ps − ps−2(14)and hence the number of isotropic lines is againnL(ps) =es−1∑k=0|Ok(ps)|+ 1 = ps−1(p+ 1)ps − 1ps − ps−2+ 1= pps − 1p− 1+ 1 =ps+1 − p+ p− 1p− 1=ps+1 − 1p− 1. (15)1.2 General case: d any integer ≥ 2Now let d be any integer greater than or equal to 2 andd =∏i∈Ipsii (16)be the prime factor decomposition of d. Due to the Chinese remainder theorem, wecan study the structure of an isotropic line ℓ in each of the Chinese factor (Zpsii)2.For every i ∈ I, let ℓi = πpi(ℓ) be the i-th Chinese projection of ℓ. As a subgroup4of (Zpsii)2, 〈ℓi〉 has cardinality a power of pi, say ptii . As an isotropic submodule of(Zpsii)2, 〈ℓi〉 is included in a Lagrangian submodule and then ti ≤ si. Sod = |ℓ| ≤∏i∈I|ℓi| ≤∏i∈Iptii ≤ d, (17)what proves that ti = si. Moreover, if ℓi ( 〈ℓi〉 for some i, the second inequalityjust above would be strict, what is impossible and so ℓi = 〈ℓi〉 is a Lagrangiansubmodule of (Zpsii)2. As to the converse, for all i ∈ I, let ℓ′i be a Lagrangiansubmodule of (Zpsii)2. The set ℓ′ of all vectors x ∈ Z2d such that for all i, πpi(x) ∈ ℓ′i,is an isotropic set with cardinality d, namely an isotropic line. The reader maycheck that the maps ℓ 7→ (ℓi)i∈I and (ℓ′i)i∈I 7→ ℓ′ thus defined are reciprocal of oneanother.So, isotropic lines and Lagrangian submodules are the same sets of Z2d and thenumber of isotropic lines of Z2d isnL(d) =∏i∈InL (psii ) =∏i∈Ipsi+1i − 1pi − 1. (18)Remark 1 In (3), the left-hand-side factor was a symplectic matrix. But in fact,any invertible matrix would be convenient since we are to consider all the linearcombinations of the columns in the product. Thus we could have calculated thecardinality of an orbit asnω |U(Zps)||ΣD(M)|instead ofnω(|ΣD(M)| / |U(Zps)|)(19)and the argument between (8) and (9) could have been avoided.Remark 2 Let us assume that s is even. It should be noticed that the formula forthe cardinality of Ok(ps) given in (10) is not valid for k = s/2. Indeed, equation(10) gives 1 + 1/p for that particular value of k, what is even not an integer.Equivalently, nD(k) and |ΣD(M)| have no unique expression for all values of k.This must be traced back to the behaviour of ΣD(M) when k is ranging up to s/2(see [9]).2 The number of lines through a given pointWe now give the number of isotropic lines through a given point of the lattice.We suppose that d = ps is a power of a prime. Let x ∈ Z2d and let t = vp(x) bethe p-valuation of x. Since all the vectors in an isotropic line ℓ ∈ Ok(ps) havep-valuation at least k, the vector x cannot be in ℓ unless k ≤ t. Let us assume thatk is such that s−k ≤ t, what implies that k ≤ t. Then for any computational basis(f1, f2), symplectic or not, x is a linear combination of pkf1 and ps−kf2. Hence∀k ∈ {0, . . . , ⌊s/2⌋}, ∀ℓ ∈ Ok(ps), (k ≥ s− t =⇒ x ∈ ℓ). (20)That case can occur only if t ≥ ⌈s/2⌉, the ceiling part of s/2. Now, let us assumethat k is such that k ≤ t < s − k. Thus 2k < s and we search for the symplectic5computational bases (f1, f2) such that x is a linear combination of pkf1 and ps−kf2.Let (f1, f2) be a symplectic computational basis and x = af1 + bf2. Sincevp(ω(x, f2)) = vp(a) ≥ t ≥ k, (21)we have no extra conditions on the choice of f2. But we must havevp(ω(x, f1)) = vp(b) ≥ s− k, (22)what shows that in a symplectic basis where x = (pt, 0), f1 must be of the formf1 = (α, βps−k−t), (23)with α, β ∈ Zd. The number of suitable vectors f1 is(ps − ps−1) · p(s−1)−(s−k−t−1) = (ps − ps−1)pk+t. (24)The number of suitable vectors f2 for a given f1 is ps. Then the number of suitable,symplectic computational bases (f1, f2) is (ps − ps−1)ps+k+t. Moreover, if f is aconvenient basis and 〈pkf1, ps−kf2〉=〈pkf ′1, ps−kf ′2〉, (25)then f ′ is convenient too. With (9a), we deduce that the number of isotropic linesin Ok(ps) containing x is(ps − ps−1)ps+k+t(ps − ps−1)ps+2k= pt−k. (26)Thus, if t < ⌈s/2⌉, the number of isotropic lines containing x ist∑k=0pt−k = pt ·1− p−(t+1)1− p−1=pt+1 − 1p− 1. (27)If t ≥ ⌈s/2⌉ and s˜ = ⌊s/2⌋, the number of isotropic lines containing x iss−t−1∑k=0pt−k +es∑k=s−t|Ok(ps)| . (28)The first term is equal topt ·1− p−(s−t)1− p−1=pt+1 − p2t−s+1p− 1. (29)For s odd, then 2s˜ = s− 1,es∑k=s−tp−2k = p−2(s−t) ·1− p−2(es−s+t+1)1− p−2=p2t−s+1 − 1ps−1(p2 − 1), (30)and the second term in (28) is equal tops−1(p+ 1)p2t−s−1 − 1ps−1(p2 − 1)=p2t−s+1 − 1p− 1. (31)6For s even, then 2s˜ = s,es−1∑k=s−tp−2k = p−2(s−t) ·1− p−2(es−1−s+t+1)1− p−2=p2t−s+1 − pps−1(p2 − 1), (32)and the second term in (28) is againps−1(p+ 1)p2t−s+1 − pps−1(p2 − 1)+ 1 =p2t−s+1 − 1p− 1. (33)Hence, in any case, the number of isotropic lines containing some given vector xwith p-valuation t isnL(ps; x) = nL(ps; t) =pt+1 − 1p− 1. (34)In particular,nL(ps; t = 0) = 1 and nL(ps; t = s) = nL(ps). (35)That is to say the sole isotropic line containing a free vector is the submodule itgenerates and every isotropic line goes through the null vector.If d is not necessarily a power of a prime, then with (16) and for all i, ti = vpi(x),we obtain that the number of isotropic lines containing x isnL(d; x) = nL(d; (ti)i∈I) =∏i∈Ipti+1i − 1pi − 1. (36)3 Orbits under the action of SL(2,Zd)As in Section 1, we first suppose that d is a power of a prime, say d = ps, s ≥ 1.Then it is obvious from (3) that the orbits of the left-action of SL(2,Zps) amongthe isotropic lines are the Ok(ps). Their number is ⌊s/2⌋+ 1 and we have alreadyseen what their cardinalities are in (10) and (11).Now if d is a composite integer as in (16), then the set of the orbits is parametrisedbyk = (ki)i∈I ∈∏i∈I{0, . . . , ⌊si/2⌋} (37)and the orbit with index k isOk(d) ={ℓ ⊂ Z2d; |ℓ| = N, πpi(ℓ) ∈ Oki(psii )}. (38)The number of orbits is ∏i∈I(⌊si/2⌋+ 1) , (39)and the cardinality of one of them is|Ok(d)| =∏i∈I|Oki(psii )| . (40)7Example Let us suppose that d contains no square factor, that is to say in(16), for all i ∈ I, si = 1. According to (3), with k necessarily equal to 0, theisotropic lines are the submodules that can be generated by a single free vector.These submodules are called the projective points of Z2d. With (18), we find thatthe number of isotropic lines isnL(d) =∏i∈I(pi + 1). (41)They all belong to the sole orbit under the action of SL(2,Zd) corresponding toki = 0 for all i.4 Some group actions on ΣD(M)In this section, we assume that d = ps is a power of a prime. In order to establishequation (9), we showed that the number of matrices in ΣD(M) with determinant1 is the same as the number of matrices in the same set with any other (invertible)determinant. The simple reasoning we used was enough in the frame of Section 1.But we are going to introduce here two other group actions that are linked to thatpoint and to Remarks 1 and 2. Let ρ0 be the action of U(Zps) on ΣD (M) definedby∀u ∈ U(Zps), ∀P = (P1|P2) ∈ ΣD (M), ρ0(u) · P = (uP1|u−1P2) (42)and ρ1 the action of U(Zps)2 on ΣD(M) defined by∀(u1, u2) ∈ U(Zps)2, ∀P = (P1|P2) ∈ ΣD(M), ρ1(u1, u2) · P = (u1P1|u2P2). (43)All the orbits of ρ0 (resp. ρ1) have the same cardinality, namely |U(Zps)| = ps−ps−1(resp. |U(Zps)|2). In a given orbit of ρ0, every matrix has the same determinant.Since Zps is a commutative ring, those two actions ”commute”:ρ1(u1, u2) · (ρ0(u) · P ) = ρ0(u) · (ρ1(u1, u2) · P ). (44)Let (u1, u2), (v1, v2) ∈ U(Zps)2 such that u1u2 = v1v2, that is to say∀P ∈ ΣD(M), det(ρ1(u1, u2) · P ) = det(ρ1(v1, v2) · P ). (45)With λ = u2v−12 = u−11 v1 ∈ U(Zps), we have(v1, v2) = (λu1, λ−1u2). (46)Thus we have a kind of a discrete Hopf fibration. It is given by the action h ofU(Zps) on U(Zps)2 defined by∀λ ∈ U(Zps), ∀(u1, u2) ∈ U(Zps)2, h(λ) · (u1, u2) = (λu1, λ−1u2). (47)Moreover, the action ρ = ρ1/(h, ρ0) of U(Zps)2/h on ΣD (M)/ρ0 is well-defined.For any u ∈ U(Zps), letDu = {P ∈ ΣD(M); detP = u} . (48)8Every orbit of ρ is transversal to Du/ρ0. Indeed, let P be in some orbit O of ρ1with some determinant v. Then (uv−1P1|P2) is in O with determinant u so thatthere is at least one orbit of ρ0 in Du ∩O. Then if P and Q = (u1P1|u2P2) are inO and have the same determinant, then u2 = u−11 and thus P and Q are in thesame orbit of ρ0.As a conclusion, we have a partition E = {Eij} of ΣD (M): The Eij ’s are theorbits of ρ0, i ∈ U(Zps) is the determinant of every matrix in Eij and j stands foran orbit of ρ1 (or equivalently of ρ). The number of different values that j canassume isnρ =|ΣD(M)||U(Zps)|2 . (49)If 2k < s, then nρ = ps+2k according to (9a). But if k = s/2, thennρ =|GL(2,Zps)||U(Zps)|2 =(p2s − p2(s−1)) · (ps − ps−1) · ps(ps − ps−1)2= p2s + p2s−1 > p2s. (50)Let P ∈ Ei1j1 and Q ∈ Ei2j2 . On the one hand, detP = i1 and detQ =i2. On the other hand, j1 = j2 iff Q1 and Q2 are proportional to P1 and P2respectively. In passing, we find again that the number of matrices in ΣD(M) withsome determinant u is the same as the number of matrices in ΣD(M) with anyother (invertible) determinant v.AcknowledgmentsThe author wishes to thank Subhash Chaturvedi for calling his attention to allthe features about isotropic lines addressed in this communication and to theirinvolvement in the setting-up of discrete Wigner distributions.This work is part of the Ph.D. thesis by the author. He is indebted to hisadvisor for useful comments and also to Michel Planat and Metod Saniga forintroducing him to finite geometry in the frame of quantum physics.References[1] E. P. Wigner. On the Quantum Correction For Thermodynamic Equilibrium.Physical Review, 40:749–759, 1932.[2] S. de Groot. La transformation de Weyl et la fonction de Wigner: une formealternative de la mcanique quantique. Les Presses Universitaires de Montral,1975.[3] F. A. Buot. Method for calculating TrHn in solid-state theory. PhysicalReview B, 10:3700–3705, 1974.[4] J. H. Hannay and M. V. Berry. On the Quantum Correction For Thermody-namic Equilibrium. Physica D, 1:267, 1980.[5] K. S. Gibbons, F. J. Hoffman, and W. K. Wootters. Discrete phase spacebased on finite fields. Physical Review A, 70:062101, 2004.9[6] S. Chaturvedi, E. Ercolessi, G. Marmo, G. Morandi, N. Mukunda, and R. Si-mon. Wigner-Weyl correspondence in quantum mechanics for continuous anddiscrete systems - a Dirac-inspired view. Journal of Physics A, 39:1405–1423,2006.[7] H. Havlicek and M. Saniga. Projective Ring Line of a Specific Qudit. Journalof Physics A, 40:F943–F952, 2007.[8] H. Havlicek and M. Saniga. Projective Ring Line of an Arbitrary Single Qudit.Journal of Physics A, 41:015302, 2008.[9] Olivier Albouy. Matrix reduction and Lagrangian submodules.arXiv:0809.1059v1 [quant-ph].[10] S. Bandyopadhyay, P. O. Boykin, V. Roychowdhury, and F. Vatan. A newproof for the existence of mutually unbiased bases. Algorithmica, 34:512–528,2002.10

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