On spherical designs, a survey Eiichi Bannai Shanghai Jiao Tong - - PowerPoint PPT Presentation

On spherical designs, a survey

Eiichi Bannai Shanghai Jiao Tong University 2016 April 21 at SJTU

The same talk was given at Kinki University, Feb 15, 2016 Osaka, Japan

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Main References

• 1. P. Delsarte, J.M. Goethals and J. J. Seidel: Spherical codes and

designs, Geom. Dedicata 6 (1977), 363–388.

• 2. 坂内英一・坂内悦子、球面上の代数的組合せ理論、シュプリンガー

東京、１９９９.

• 3. Eiichi Bannai and Etsuko Bannail: A survey on spherical designs

and algebraic combinatorics on spheres, Europ. J. Combinatorics, 30 (2009), 1392–1425.

• 4. H. Cohn and A. Kumar: Universally optimal distribution of points
• n spheres, J. Amer. Math. Soc. 20 (2007), 99–148.
• 5. H. Cohn: Order and disorder in energy minimization, Proceedings
• f ICM, Hyderbad, India, 2010, Volume IV, pages 2416–2443.

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Sn−1 = {(x1, x2, · · · , xn) ∈ Rn | x2

1 + x2 2 + · · · + x2 n = 1}.

X ⊂ Sn−1, |X| < ∞.

What X are good?

Answer may not be unique. There are several different viewpoints (or criterions) about this question. What criterions are ”good” is a part of the question.

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Definition (Optimal code). X(⊂ Sn−1) is called an optimal code if the minimum distance Min{d(x, y) | x, y ∈ X, x ̸= y} is max- imum among all the subsets (on Sn−1) of the same cardinality. Optimal codes do exist for each given pair of n and |X|. (They may not be unique.) Classifications of optimal codes on S2 are known for |X| ≤ 12 and |X| = 24. (See the book, Ericson-Zinoviev: Codes on Euclidean Spheres, 2001.) Optimal codes on Sn−1, n ≥ 4 are deter- mined only for very special cases. (Many candidates are given by computer simulations.)

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Recently, Musin-Tarazov solved the case of n = 3 and |X| = 13. (Strong thirteen spheres problem. Discrete & Computational Geome- try, 48:1 (2012), 128-141) More recently, Musin-Tarazov announced the solution of the case of n = 3 and |X| = 14. (The Tammes problem for N = 14, arXiv: 1410:2536.) Finding optimal codes is also called Tammes’ problem, named after the Dutch botanist who posed the problem in connection with the study of pores in spherical pollen grains (1930).

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Coulomb-Thomson problem.

Find X ⊂ Sn−1 with the property that ∑

x,y∈X,x̸=y

1 ||x − y|| is minimum among all the subsets on Sn−1 of the same cardinality as |X|, where ||x − y|| = √ (x1 − y1)2 + (x2 − y2)2 + · · · + (xn − yn)2. We can also consider energy minimizing sets X on Sn−1 for other potential functions, for example, ∑

x,y∈X,x̸=y

1 ||x − y||k, (k ∈ Z>0) is minimum among all the subsets on Sn−1 of the same cardinality as |X|.

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Energy minimizing subset (w.r.t. f). Let

f : (0, 4] − → R≥0 be a decreasing function. Then X is called an energy minimizing set (on Sn−1) w.r.t. f, if ∑

x,y∈X,x̸=y

f(||x − y||2) is minimum, among all the subsets on Sn−1 of the same cardinality as |X|. It is Coulomb-Thomson problem, if we take f(r) = r− 1

2.

(Of course, for different functions f and g, X may be an energy min- imizing set for f, but not an energy minimizing set for g.)

Are there X ⊂ Sn−1 which are energy minimizing sets for all reasonably good classes of functions f : (0, 4] − → R≥0?

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Definition (Universally optimal codes).

(Cohn-Kumar, 2007). X ⊂ Sn−1 is called a universally optimal code (on Sn−1), if X is an energy minimizing set, among all the subsets (on Sn−1) of the same cardinality as X, w.r.t. all completely monotonic (decreasing) functions f : (0, 4] − → R≥0, where f is called completely monotonic, if f ∈ C∞ and (−1)kf (k)(r) ≥ 0 for all k = 0, 1, 2, · · · .

That is, ∑

x,y∈X,x̸=y

f(||x − y||2) ≤ ∑

x,y∈Y,x̸=y

f(||x − y||2) for all Y ⊂ Sn−1 with |Y | = |X| and for all complete monotonic function f : (0, 4] − → R≥0.

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Another equivalent definition of universally opti- mal codes. X ⊂ Sn−1 becomes a universally optimal code (on Sn−1), if X satisfy ∑

x,y∈X,x̸=y

f(x · y) ≤ ∑

x,y∈Y,x̸=y

f(x · y) for all Y ⊂ Sn−1 with |Y | = |X| and for all absolutely

monotonic (increasing) function f : [−1, 1) −

→ R≥0. Here x·y is the usual Euclidean inner product, and f is called absolutely monotonic, if f ∈ C∞ and f (k)(t) ≥ 0 for all k = 0, 1, 2, · · · .

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Remarks and Examples.

(i) f(r) = r−s, s > 0 are complete monotonic functions. f(r) = (4 − r)k, k = 1, 2, . . . are complete monotonic functions. (ii) Universally optimal codes are optimal codes. (Consider the function f(r) = r−s, with s → ∞.) (iii) For n = 2, i.e., for S1 ⊂ R2, the set of vertices of a regular polygon is a universally optimal code. (These are the only universally optimal codes on S1. (iv) For n = 3, the sets of vertices of regular tetrahedron, octahedron, and icosahedron, are universally optimal codes, (4, 6, 12 vertices, re- spectively.) Together with the trivial cases of up to 3 points, they are the only universally optimal codes on S2. Also, note that the sets of vertices of cube and dodecahedron (8, 20 vertices, respectively) are not universally optimal. They are not even optimal.

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(v) The set of 240 roots of type E8 (240 min. vectors of E8-lattice) in R8 gives a universally optimal code. (vi) The set of 196560 min. vectors of Leech lattice in R24 gives a universally optimal code. (vii) There are some more examples of universally optimal codes, but they are fairly rare. Cohn-Kumar (2007) conjectures that there are

• nly finitely many universally optimal codes for each n ≥ 4. (The

classifications are open for any dimension n ≥ 4.) The next table gives known universally optimal codes (after Levenshtein and Cohn-Kumar)

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n |X| t tight inner product Name 2 N N − 1 yes cos(2πj/N) (1 ≤ j ≤ N/2) regular N-gon n N ≤ n 1 no −1/(N − 1) regular simplex n n + 1 2 yes −1/n regular simplex n 2n 3 yes −1, 0 regular cross polytope 3 12 5 yes −1, ±1/ √ 5 regular icosahedron 4 120 11 no −1, ±1/2, 0, (±1 ± √ 5)/4 regular 600-cell 5 16 3 no −3/5, 1/5 Clebsch graph(=hemicube) 6 27 4 yes −1/2, 1/4 Schl¨ afli graph 7 56 5 yes −1, ±1/3 28 equangular lines 8 240 7 yes −1, ±1/2, 0 E8 root system 21 112 3 no −1/3, 1/9 U4(3) isotropic subspaces 21 162 3 no −2/7, 1/7 U4(3)/L4(3), strong regular graph 22 100 3 no −4/11, 1/11 Higman-Sims graph 22 275 4 yes −1/4, 1/6 MacLaughlin graph 22 891 5 no −1/2, −1/8, 1/4 generalized hexagon 23 552 5 yes −1, ±1/5 276 equangular lines 23 4600 7 yes −1, ±1/3, 0 iterated kissing configuration 24 196560 11 yes −1, ±1/2, ±1/4, 0

• min. vectors of Leech lattice

q q3+1

q+1

(q + 1)(q3 + 1) 3∗ no∗ −1/q, 1/q2 U3(q) isotropic subspaces ( q is a prime power. The cases q = 2 (t = 4 and tight) and q = 3 are already in the list.)

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Definition (Spherical t-designs).

Delsarte-Goethals-Seidel (1977) Let t be a positive integer. X ⊂ Sn−1, |X| < ∞, is called a spherical t-design on Sn−1 , if the following condition is satisfied. 1 |Sn−1| ∫

Sn−1 f(x)dσ(x) =

1 |X| ∑

x∈X

f(x) for any polynomials f(x) = f(x1, x2, . . . , xn) of degrees up to

• t. Here, |Sn−1| denotes the area of the sphere Sn−1, and the

integral is the surface integral on Sn−1. We are interested in the t-designs with the cardi- nality |X| as small as possible.

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Theorem (Delsarte-Goethals-Seidel, 1977).

X = 2e-design = ⇒ |X| ≥ (n−1+e

e

) + (n−1+e−1

e−1

) . X = (2e + 1)-design = ⇒ |X| ≥ 2 (n−1+e

e

) . (These inequalities are called Fisher type inequalities.) X is called a tight t-design if equality ” = ” holds in one of the above two inequalities.

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Classifications of tight t-designs.

(i) Tight t-designs on S1 are regular (t + 1)-gons. (ii) If a tight t-design on Sn−1 exists for n ≥ 3, then t ∈ {1, 2, 3, 4, 5, 7, 11} (Bannai-Damerell (1979/80), Bannai-Sloane (1981). (iii) tight t-designs are completely classified for t ≤ 3 and t = 11. (iv) There are some developments of the classification of tight t-designs for t = 4, 5, 7. See Bannai-Munemasa-Venkov, The nonexistence of certain tight spher- ical designs, Algebra i Analiz(2004). See also,

• G. Nebe and B. Venkov, On tight spherical designs, Algebra i Analiz

(2012). The complete classifications are still open for t = 4, 5, 7.

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• Remarks. (i) Let X ⊂ Sn−1. X is called an s-distance set if

|{d(x, y) | x, y ∈ X, x ̸= y}| = s. (ii) Suppose X is an s-distance set and t-design, then we have t ≤ 2s. Moreover, we have t = 2s ⇐ ⇒ X= tight 2s-design. and X is antipodal, and t = 2s − 1 ⇐ ⇒ X= tight (2s − 1)-design.

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Theorem (Cohn-Kumar (2007)).

If X is an s-distance set and t-design on Sn−1, and satisfy t ≥ 2s − 1, then X become a universally optimal code on Sn−1. (In particular, all tight t-designs are universally optimal.) Theorem (Cohn-Kumar (2007)). If X is the set of 120 vertices of a regular polytope 600-cell on S3 ⊂ R4, then X becomes a universally optimal code on S3. (In this example, we have t = 11 and s = 8, and so t ≥ 2s − 1 is not satisfied.)

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• Observation. All so far known examples of universally op-

timal codes on Sn−1 satisfy the condition t ≥ 2s − 1, except for the last example of n = 4 and |X| = 120. Remarks.

(i) It is known (Delsarte-Goethals-Seidel, 1977) that if X is an s- distance set and t-design on Sn−1, and satisfy t ≥ 2s − 2, then X has the strucuture of a Q-polynomial association scheme. (It is a very strong condition.) (ii) We are interested in the classification of X ⊂ Sn−1 with t close to 2s. (How about the classification of those with t ≥ 2s − 1 with t large? ) Also, it would be very interesting to try to classify universally optimal codes on Sn−1.

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Further search for universally optimal codes

Cohn and others (see Ballinger et al., Experiment. Math. 18 (2009), 257–283) systematically searched for universally optimal codes by com- puter. Cohn (2004) found the following two new candidates: (i) n = 10, and |X| = 40. (ii) n = 14, and |X| = 64. (Ballinger et al. says that these are the only possible new examples in the range of n ≤ 32 and |X| ≤ 100, besides those already known.) They become association schemes of classes 4 and 3 (respectively), with A(X) = {−1

2, −1 3, 0, 1 6} and

A(X) = {−3

7, −1 7, 1 7} (respectively).

(They do not satisfy the condition t ≥ 2s − 1.)

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With the request of Cohn, Bannai-Bannai-Bannai (Europ. J.

• Comb. (2008)) proved that these two association schemes are

uniquely characterized by their parameters pk

i,js.

But it is still unknown whether these two examples are actually universally optimal or not. Ballinger et al. found two more candidates by expanding the range of search. n = 7, and |X| = 182 (E7 ∪ E∗

7 (normalized)), and

n = 15, and |X| = 128.

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Abdukhalikov-Bannai-Suda (J. Comb. Theory A (2009)) obtained further candidates of universally optimal codes, i.e., the higher dimen- sional analogues of those with (n, |X|) = (14, 64) and (15, 128). We proved that if there exists a maximal real MUB (mutually unbiased bases), then the following set X on Sn−1 are constructed. Then, these are the candidates of universally optimal codes on Sn−1. (n, |X|) = (N, N 2 + 2N), (N − 1, N 2 2 ), (N − 2, N 2 4 ). Here, N = 4 or N = 16·(a square of an integer).) (It is known that for any power of 4, such maximal real MUB exist.) The last one with (n, |X|) = (N − 2, N 2

4 ) is the most likely to be

universally optimal.) All these X form association schemes.

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What are MUB? Two orthonormal bases B and B′ of RN are said to be mutually unbiased, if |x · y| = 1 √ N , for all x ∈ B, y ∈ B′. There are at most N

2 + 1 number of mutually unbiased bases.

So, there are 2N(N 2 + 1) = N 2 + 2N number of points on SN−1.

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A brief survey on spherical t-designs

• The set of vertices of a regular polyhedron: tetrahedron,

cube, octahedron, dodecahedron, icosahedron, is a 2-, 3-, 3-, 5-, 5-design, respectively.

• Many examples of t-designs are obtained as orbits of a finite

group G ⊂ O(n). However, for n ≥ 3, all these examples are t ≤ 19. It is an interesting open question whether we can construct t-designs as orbits of a finite group G ⊂ O(n) for larger t.

• Many examples of t-designs are obtained as shells of a lattice

L ⊂ Rn. However, for n ≥ 2, all these examples are t ≤ 11. It is an interesting open question whether we can construct t-designs as shells of a lattice L ⊂ Rn for larger t.

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Theorem (Seymour-Zaslavsky, 1984). Spherical t- designs on Sn−1 exist for any n and t. However, this is an existence theorem, and the explicit con- structions are very difficult. (Generally open if both n and t are large.) Theorem (Bondarenko-Radchenko-Viazovska, 2013, 2015) For each N ≥ cdtd there exists a spherical t-design in the sphere Sd consisting of N points, where cd is a constant depending only on d.

(Cf. Optimal asymptotic bounds for spherical designs, Ann. of Math. (2013) and Well separated spherical designs, Constr. Approx. (2015))

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The case of S2. Theorem (Chen-Womersley-Lang, 2011) The existence of spherical t-designs on S2 of size (t + 1)2 was proved for all t ≤ 100.

Computational existence proofs for spherical t-designs, Nemerische

• Math. (2011).

(Open for bigger t.)

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Connection with number theory Let L be the E8-lattice in R8. Let L2m = {x ∈ L | ∥x∥2 = 2m}. (Then it is known that X = (1/ √ 2m)L2m is a spherical 7-design on S7 for any m.) Is there any m for which X = (1/ √ 2m)L2m be- comes an 8-design? Theorem (Venkov, 2005 or earlier). X = (1/ √ 2m)L2m is an 8-design if and only if τ(m) = 0, where τ(m) is the Ramanujan τ function.) (Proof uses the theory of modular forms.)

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Ramanujan’s τ function τ(m).

q = eπiz, z ∈ H, (0.1) η(z) = q

1 12 ∏

m≥1

(1 − q2m) = q

1 12(1 − q2 − q4 + q10 + · · ·),

∆24 = η(z)24= q2 ∏

m≥1

(1 − q2m)24 = q2 − 24q4 + 252q6 − 1472q8 + 4830q10 −6048q12 − 16744q14 + · · · = ∑

m≥1

τ(m)q2m.

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Conjecture (D. H. Lehmer, 1947). τ(m) ̸= 0 for any positive integer. So, Lehmer’s conjecture is interpreted in terms of spherical designs. However, Lehmer’s conjecture is still difficult to prove, and it is still an open prob- lem. We can prove an analogue of D. H. Lehmer’s con- jecture for some 2-dimensional lattices. Bannai-Miezaki, Toy models of D.H. Lehmer’s conjecturte, I, II (2010, 2013) and Bannai-Miezaki-Yudin (Izv. Math. 2013).

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Connection with Physics. Anticoherent spin states Let S = (Sx, Sy, Sz) be the spin operators. So, [Sx, Sy] = √−1hSz, [Sy, Sz] = √−1hSx, [Sz, Sx] = √−1hSy. For a spin state | φ 〉 and for an operator T on states, let 〈T 〉 := 〈φ|T |φ〉.

• Def. (Zimba, 2006).

|φ〉 is anticoherent of order t if, for k = 1, 2, . . . , t, 〈(n · S)k〉 = 〈φ|(nxSx + nySy + nzSz)k|φ〉 does not depend on the choice of unit vector n = (nx, ny, nz) ∈ S2.

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Let |φ〉 = ∑s

m=−s am|m〉 be a spin s-state. Then let

M|φ〉(z) :=

s

∑

m=−s

(−1)m−sam √( 2s s + m ) zs+m (a polynomial of degree 2s in z.) Then M|φ〉 has 2s roots in C. By stereographic projection (C ∪ {∞} → S2), these 2s points are corresponding to 2s points on S2. (This is called Majorana representation of a spin state.)

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Conjecture (Crann-Pereira-Kribs, 2010) J. Phys. A:

• Math. Theor. 43, 255307.

A spin state is anticoherent to order t, if and only if its Majorana representation is a spherical t-design. True for some examples, including the set of ver- tices of any one of Platonic solids. Remark (Bannai-Tagami, 2011) J. Phys.

A: Math.

• Theor. 44, 342002.

There are counter examples in both directions. (For example, s = 5

2, |X| = 5, t = 1;

s = 7

2, |X| = 7, t = 2.)

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• It would be very interesting, if we could get more results

in positive direction.

• How about higher dimensional analogues?

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Related References.

(1) Giraud-Baguette-Martin-Bastin (2014), Multiqubit sym- metric states with maximally mixed one-qubit reductions, Phys- ical Review A90, 032314 (2) Braun-Bastin-Baguette-Martin (2014), Tensor representa- tions of spin states, PhysRevLett.114.080401 (3) de la Hoz-Leuchs-Klimov-Sanchez-Soto (2014), Unpolar- ized states and hidden polarization, Physical Review A90, 043826 (4) Baguette-Damanet-Giraud-Martin (2015), Anticoherence spin states with point-group symmetries, Physical Review A92, 052333 (5) Bjork-Grassl-de la Hoz-Leuche-Sanchez-Soto (2015), Stars

• f the quantum universe: extremal constellations on the Poincare

sphere, Phys Scr. 90 108008

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How the concept of spherical t-designs are generalized? Sphere− → Compact symmetric spaces of rank 1 (Projective spaces over R, C, H, O.) − → Compact symmetric spaces of arbitrary ranks

They were classified by E. Cartan (1930’s)), and they include Grassmannian spaces (Calderbank

• Shor-Sloane, Bachoc, Bannai, Coulangeon, Nebe,

etc.), as well as simple Lie groups such as O(n), U(n), etc. (Roy, Scott, Suda, etc.)

− → Euclidean space Rn. (Cf. Many recent papers by B-Etsuko B.)

− → (possibly) non-compact symmetric spaces. (B-B.) 34

Some other viewpoints.

(i) Rigid spherical t-designs.

Spherical t-designs which cannot be deformed keeping it as a t-design. (Bannai (1986))

(ii) Can we define the concepts of optimal or

universally optimal codes for finite subsets in real Euclidean space Rn?

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How about the following definitions?

Let X be a finite subset of Rn. We define x0 be the center of mass of X, and let m(X) = 1 |X| ∑

x∈X

||x − x0||2.

• We say X is an optimal code in Rn, if

Min{d(x, y) | x, y ∈ X, x ̸= y} ≥ Min{d(x, y) | x, y ∈ Y, x ̸= y} for any subset Y of Rn with |Y| = |X| and m(X) = m(Y).

• We say that X is a universally optimal code in Rn, if

∑

x,y∈X.x̸=y

f(||x − y||2) ≤ ∑

x,y∈Y.x̸=y

f(||x − y||2) for any subset Y of Rn with |Y | = |X| and m(Y ) = m(X), and for all complete monotonic (decreasing) functions f : (0, ∞) − → R≥0. 36

Thank You

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