Mutually Unbiased Equiangular Tight Frames Matthew Fickus Benjamin - - PowerPoint PPT Presentation

Mutually Unbiased Equiangular Tight Frames

Matthew Fickus Benjamin R. Mayo

Department of Mathematics and Statistics Air Force Institute of Technology Wright-Patterson Air Force Base, Ohio

July 28, 2020

The views expressed in this talk are those of the speaker and do not reflect the official policy

• r position of the United States Air Force, Department of Defense, or the U.S. Government.

Projective Codes

Spherical codes: Tammes’s problem (1930)

Problem: Find N unit vectors {xn}N

n=1 in RD that maximize

min

n1=n2 xn1 − xn2.

Equivalently, since xn1 − xn22 = 2(1 − min

n1=n2xn1, xn2), minimize

max

n=n′xn, xn′.

Example: D = 3, N = 2, 3, 4, 5, 6:

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Rankin’s bounds (1955)

Theorem: When N ≤ D + 1, {optimal spherical codes} = {regular simplices}. Proof: For any unit vectors {xn}N

n=1 in RD,

0 ≤

• N
• n=1

xn

• 2

=

N

• n1=1

N

• n2=1

xn1, xn2 ≤ N + N(N − 1) max

n1=n2xn1, xn2.

Theorem: max

n1=n2xn1, xn2 ≥ 0 when N ≥ D + 2. It’s achievable when N ≤ 2D.

Proof: By induction: if {xn}N

n=1 are mutually obtuse in RD then projecting

{xn}N−1

n=1 onto x⊥ N yields N − 1 mutually obtuse vectors in RD−1.

To achieve equality when N ≤ 2D, choose {xn}N

n=1 as a subset of the orthoplex.

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Projective spaces

Let F be either R or C. The projective space of FD is the set of its

• ne-dimensional subspaces. We identify each with its rank-one projection:

P(FD) := {ϕϕ∗ : ϕ ∈ FD, ϕ = 1}. The Frobenius inner product of two such projections is ϕ1ϕ∗

1, ϕ2ϕ∗ 2Fro = Tr(ϕ1ϕ∗ 1ϕ2ϕ∗ 2) = Tr(ϕ∗ 2ϕ1ϕ∗ 1ϕ2) = |ϕ1, ϕ2|2

The squared Frobenius distance between two such projections is thus ϕ1ϕ∗

1 − ϕ2ϕ∗ 22 Fro = 2(1 − ϕ1ϕ∗ 1, ϕ2ϕ∗ 2Fro) = 2(1 − |ϕ1, ϕ2|2).

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Projective codes

Problem: Design N unit vectors {ϕn}N

n=1 in FD that maximize

min

n1=n2 ϕn1ϕ∗ n1 − ϕn2ϕ∗ n22 = min n1=n2 2(1 − |ϕn1, ϕn2|2)

• r equivalently have minimal coherence:

coh({ϕn}N

n=1) := max n=n′ |ϕn, ϕn′|.

That is, design a D × N matrix Φ =

• ϕ1 · · · ϕN
• so that its Gram matrix

Φ∗Φ =    ϕ∗

1

. . . ϕ∗

N

  

• ϕ1 · · · ϕN
• =

   ϕ1, ϕ1 · · · ϕ1, ϕN . . . ... . . . ϕN, ϕ1 · · · ϕ1, ϕN    has off-diagonal entries of minimal ∞-norm, subject to having 1’s in its diagonal.

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Motivating applications

Spherical codes: ◮ Error-detecting and -correcting binary codes (±1-valued real vectors) Projective codes: ◮ Waveform design for communication, radar, sonar, etc.:

◮ autocorrelation function gives coherence with own translates ◮ ambiguity function gives coherence with own translates and modulates ◮ pings, chirps, OFDM, CDMA, etc.

◮ Compressed sensing ◮ Quantum tomography: recover a D × D self-adjoint matrix A from {ϕ∗

nAϕn}N n=1 = ϕnϕ∗ n, AFro

◮ Design of experiments (±1-valued real vectors in 1⊥)

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Embedding projective spaces into Euclidean spaces

Idea: Since ϕ1ϕ∗

1, ϕ2ϕ∗ 2Fro = |ϕ1, ϕ2|2, consider the mapping

ϕ ∈ FD − → x := ϕϕ∗ ∈ {A ∈ FD×D : A = A∗} ∼ =

• R

D(D+1) 2

, F = R, RD2, F = C. In particular, {ϕn}N

n=1 has minimal coherence if {xn}N n=1, xn := ϕnϕ∗ n is an

• ptimal spherical code in {A ∈ FD×D : A = A∗}. Unfortunately, it never will be.

Nevertheless, this gives the (proto)-Gerzon bound: rank(X) = rank(X∗X) = rank(|Φ∗Φ|2) ≤

• D(D+1)

2

, F = R, D2, F = C, where equality holds ⇔ any self-adjoint A can be recovered from {ϕ∗

nAϕn}N n=1.

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Embedding projective spaces into Euclidean spaces (redux)

Idea: [Conway, Hardin, Sloane 96] Instead map any ϕ ∈ FD to x := (

D D−1)

1 2(ϕϕ∗ − 1

DI)

in {A ∈ FD×D : A = A∗, Tr(A) = 0} ∼ =

• R

D(D+1) 2

−1, F = R,

RD2−1, F = C. For any unit vectors ϕ1, ϕ2 ∈ FD, x1, x2Fro =

D D−1 Tr[(ϕ1ϕ∗ 1 − 1 DI)(ϕ2ϕ∗ 2 − 1 DI)] = D D−1(|ϕ1, ϕ2|2 − 1).

Problem: Design {ϕn}N

n=1 so that {xn}N n=1 is an optimal spherical code.

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Equiangular Tight Frames (ETFs) and Mutually Unbiased Bases (MUBs)

The Welch bound and equiangular tight frames (ETFs)

Theorem: [Rankin 56, Welch 74, Conway Hardin Sloane 96]: Let N > 1, N ≥ D ≥ 1. For any N unit vectors {ϕn}N

n=1 in FD,

max

n1=n2 |ϕn1, ϕn2| ≥

• N−D

D(N−1)

1

2,

where equality holds ⇔ {ϕn}N

n=1 is an ETF(D, N) for FD, namely

◮ ∃A > 0 such that ΦΦ∗ =

N

• n=1

ϕnϕ∗

n = AI,

◮ ∃C ≥ 0 such that |(Φ∗Φ)(n1, n2)| = |ϕn1, ϕn2| = C for all n1 = n2. Gerzon’s bound: If ∃ ETF(D, N) then N ≤

• D(D+1)

2

, F = R, D2, F = C.

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Example: RETF(3, 4) (vertices of tetrahedron)

Φ = 1 √ 3   + − + − + + − − + − − +   , ΦΦ∗ = 1 3   4 0 0 0 4 0 0 0 4   , Φ∗Φ = 1 3     3 − − − − 3 − − − − 3 − − − − 3     .

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Example: RETF(3, 6) (antipodes of icosahedron)

ΦΦ∗ =   2 0 0 0 2 0 0 0 2   , Φ∗Φ = I + 1 √ 5         0 + + + + + + 0 + − − + + + 0 + − − + − + 0 + − + − − + 0 + + + − − + 0        

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Example: CETF(3, 7) (quadratic residues modulo 7)

Φ = 1 √ 3   1 ω ω2 ω3 ω4 ω5 ω6 1 ω2 ω4 ω6 ω1 ω3 ω5 1 ω4 ω ω5 ω2 ω6 ω3   ΦΦ∗ = 1 3   7 0 0 0 7 0 0 0 7   Φ∗Φ = 1 3           1 ζ ζ ζ ζ ζ ζ ζ 1 ζ ζ ζ ζ ζ ζ ζ 1 ζ ζ ζ ζ ζ ζ ζ 1 ζ ζ ζ ζ ζ ζ ζ 1 ζ ζ ζ ζ ζ ζ ζ 1 ζ ζ ζ ζ ζ ζ ζ 1           ω = exp( 2πi

7 )

ζ = 1 + ω + ω4

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Example: CETF(3, 9) (SIC-POVM)

Φ = 1 √ 2   1 ω ω2 0 1 ω2 ω 1 ω2 ω 1 ω ω2 0 1 ω2 ω 1 ω ω2   ΦΦ∗ =   3 0 0 0 3 0 0 0 3   Φ∗Φ = 1 2               2 − − 1 ω ω2 1 ω2 ω − 2 − ω ω2 1 ω2 ω 1 − − 2 ω2 1 ω ω 1 ω2 1 ω2 ω 2 − − 1 ω ω2 ω2 ω 1 − 2 − ω ω2 1 ω 1 ω2 − − 2 ω2 1 ω 1 ω ω2 1 ω2 ω 2 − − ω ω2 1 ω2 ω 1 − 2 − ω2 1 ω ω 1 ω2 − − 2              

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The orthoplex bound

Theorem: [Conway Hardin Sloane 96]: Let N >

• D(D+1)

2

, F = R, D2, F = C. Then maxn1=n2 |ϕn1, ϕn2| ≥

1 √ D for any N unit vectors {ϕn}N n=1 in FD.

If {ϕn}N

n=1 achieves equality, it’s an orthopletic Grassmannian frame (OGF).

Examples: (from the infinite families of OGFs of [Bodmann Hass 16]): ∃ ROGF(3, 7) : Φ = 1 √ 3   + − + − √ 3 0 + + − − 0 √ 3 0 + − − + 0 √ 3   ∃ COGF(3, 10) : Φ = 1 √ 3   1 ω ω2 ω3 ω4 ω5 ω6 √ 3 0 1 ω2 ω4 ω6 ω1 ω3 ω5 √ 3 0 1 ω4 ω ω5 ω2 ω6 ω3 √ 3  

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Mutually unbiased bases (MUBs)

Definition: A sequence of M orthonormal bases {ϕm,d}M

m=1, D d=1 for FD are

mutually unbiased if |ϕm1,d1, ϕm2,d2| =

1 √ D for all m1, m2, d1, d2, m1 = m2.

Example: ∃ RMUB(2, 2) : Φ = 1 √ 2 + + √ 2 0 + − 0 √ 2

• Example: ∃CMUB(3, 4) : Φ = 1

√ 3   1 1 1 1 1 1 1 1 1 √ 3 0 1 ω ω2 ω ω2 1 ω2 1 ω √ 3 0 1 ω2 ω ω 1 ω2 ω2 ω 1 √ 3   Corollary of Proto-Gerzon: If ∃ FMUB(D, M) then M ≤ D

2 + 1, F = R,

D + 1, F = C. Moreover, if equality is achieved then the MUB is an OGF and any self-adjoint A can be recovered from {ϕ∗

m,dAϕm,d}M m=1, D d=1.

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Some open problems about ETFs and MUB existence

◮ When does an ETF(D, N) exist?

◮ Zauner’s conjecture: Does a CETF(D, D2) (SIC-POVM) exist for all D? ◮ Necessary integrality conditions on CETF(D, N)?

◮ [Sz¨

• ll˝
• si 14] ∄ CETF(3, 8)

◮ For all known ETF(D, N), one member of {D, N − D, N − 1} divides the product of the other two.

◮ When does an MUB(D, M) exist?

◮ ∃ CMUB(Q, Q + 1) for prime power Q and ∃ RMUB(Q, Q

2 + 1) for Q = 2, 4j

◮ if ∃ FMUB(D1, M) and ∃ FMUB(D2, M) then ∃ FMUB(D1D2, M) ◮ [Wocjan Beth 05] MUBs from MOLS ◮ Does a CMUB(6, M) exist for M = 3, 4, 5, 6, 7?

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Mutually Unbiased Equiangular Tight Frames (MUETFs)

MUETFs defined

Definition: A sequence {ψm,n}M

m=1 N n=1 of unit vectors in FD is an

MUETF(D, N, M) for FD if |ψm1,n1, ψm2,n2|2 =

• N−D

D(N−1), m1 = m2, n1 = n2, 1 D,

m1 = m2. Theorem: [F Mayo 20] (Proto-Gerzon corollary) If ∃ FMUETF(D, N, M) then M ≤

• D2−1

N−1 ,

F = C,

(D−1)(D+2) 2(N−1)

, F = R. ◮ Reduces to Gerzon bound when M = 1 ◮ Reduces to MUB bound when N = D

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Example: CMUETF(4, 5, 3) (F Schmitt 20)

Φ = 1 2     1 ω3 ω6 ω9 ω12 ω ω4 ω7 ω10 ω13 ω2 ω5 ω8 ω11 ω14 1 ω6 ω12 ω3 ω9 ω2 ω8 ω14 ω5 ω11 ω4 ω10 ω ω7 ω13 1 ω9 ω3 ω12 ω6 ω8 ω2 ω11 ω5 ω14 ω ω10 ω4 ω13 ω7 1 ω12 ω9 ω6 ω3 ω4 ω ω13 ω10 ω7 ω8 ω5 ω2 ω14 ω11     ω = exp( 2πi

15 )

Φ∗Φ = 1 4                             4 −1 −1 −1 −1 ζ ζ ζ −2 ζ ζ −2 ζ ζ ζ −1 4 −1 −1 −1 ζ ζ ζ ζ −2 ζ ζ −2 ζ ζ −1 −1 4 −1 −1 −2 ζ ζ ζ ζ ζ ζ ζ −2 ζ −1 −1 −1 4 −1 ζ −2 ζ ζ ζ ζ ζ ζ ζ −2 −1 −1 −1 −1 4 ζ ζ −2 ζ ζ −2 ζ ζ ζ ζ ζ ζ −2 ζ ζ 4 −1 −1 −1 −1 ζ ζ ζ −2 ζ ζ ζ ζ −2 ζ −1 4 −1 −1 −1 ζ ζ ζ ζ −2 ζ ζ ζ ζ −2 −1 −1 4 −1 −1 −2 ζ ζ ζ ζ −2 ζ ζ ζ ζ −1 −1 −1 4 −1 ζ −2 ζ ζ ζ ζ −2 ζ ζ ζ −1 −1 −1 −1 4 ζ ζ −2 ζ ζ ζ ζ ζ ζ −2 ζ ζ −2 ζ ζ 4 −1 −1 −1 −1 −2 ζ ζ ζ ζ ζ ζ ζ −2 ζ −1 4 −1 −1 −1 ζ −2 ζ ζ ζ ζ ζ ζ ζ −2 −1 −1 4 −1 −1 ζ ζ −2 ζ ζ −2 ζ ζ ζ ζ −1 −1 −1 4 −1 ζ ζ ζ −2 ζ ζ −2 ζ ζ ζ −1 −1 −1 −1 4                             ζ = 1

2 (1 +

√ 15i)

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Why MUETFs matter

Theorem: [F Mayo 20] If {ϕn1}N1

n1=1 is an ETF(D1, N1) for FD1 and

{ψn1,n2}N1

n1=1, N2 n2=1 is an MUETF(D2, N2, N1) for FD2, and N1−D1 D1(N1−1) = N2−D2 N2−1 ,

then {ϕn1 ⊗ ψn1,n2}N1

n1=1, N2 n2=1 is an ETF(D1D2, N1N2) for FD1D2.

Example: Let Φ and Ψ be an ETF(2, 3) and an MUETF(4, 5, 3): Φ =

• ϕ1 ϕ2 ϕ3
• ∈ C2×3,

Ψ =

• Ψ1 Ψ2 Ψ3
• ∈ C4×12.

Since

3−2 2(3−1) = 1 4 = 5−4 5−1,

• ϕ1 ⊗ Ψ1 ϕ2 ⊗ Ψ2 ϕ3 ⊗ Ψ3
• is a CETF(8, 15).

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Towards an MUETF construction: harmonic frames

Definition: Let Γ be the G × ˆ G character table of a finite abelian group G. Restricting the characters of G to D and normalizing gives the harmonic frame {ϕγ}γ∈ ˆ

G,

ϕγ(d) =

1 √ Dγ(d),

i.e., √ DΦ is the D × ˆ G submatrix of Γ. Facts: ◮ Harmonic frames are tight: (ΦΦ∗)(d1, d2) = 1

D(ΓΓ∗)(d1, d2) = N DI(d1, d2).

◮ Harmonic frames are unit norm with |ϕγ1, ϕγ2|2 = 1

D|(Γ∗χD)(γ1γ−1 2 )|2 = 1 DΓ∗(χD ∗ χ−D)(γ1γ−1 2 ).

◮ Here, χD ∗ χ−D is the autocorrelation of the characteristic function of D: (χD ∗ χ−D)(g) = #[D ∩ (g + D)] = {(d1, d2) ∈ D × D : g = d1 − d2}.

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Towards an MUETF construction: relative difference sets

Definition: A subset D of G is a difference set for G relative to a subgroup H ≤ G if χD ∗ ˜ χD = Λ(χG − χH) + Dδ0 for some Λ ≥ 0, i.e., {(d1, d2) ∈ D × D : g = d1 − d2} = 0, g ∈ H, Λ, g / ∈ H. A difference set is an RDS with H = {0}. Examples: D = {1, 2, 8, 4} is an RDS(5, 3, 4, 1) for G = Z15, H = {0, 5, 10}; D = {00, 11, 12} is an RDS(3, 3, 3, 1) for G = Z3 × Z3, H = {0} × Z3: − 1 2 8 4 1 0 14 8 12 2 1 0 9 13 8 7 6 4 4 3 2 11 0 − 00 11 21 00 00 22 12 11 11 00 01 21 21 20 00

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Harmonic MUETFs “=” relative difference sets

Theorem: [F & Mayo 20] Let H⊥ = {γ ∈ ˆ G : γ(h) = 1 ∀ h ∈ H}. Then {ψα,β}α∈ ˆ

G/H⊥, β∈H⊥ ⊆ CD,

ψα,β(d) :=

1 √ Dα(d)β(d),

is an MUETF(D, G

H , H) for CD if and only if D is an H-RDS( G H , H, D, Λ) for G.

This is a unification and generalization of: ◮ [K¨

• nig 99, Strohmer Heath 03, Xia Zhou Giannakis 05, Ding Feng 07]

ETF(D, G) = MUETF(D, G, 1) from DS(G, D, Λ) = RDS(G, 1, D, Λ). ◮ [Godsil Roy 09] MUB(D, H) = MUETF(D, D, H) from RDS( G

H , H, D, Λ) where G = HD.

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Putting it together

Theorem: [Gordon Mills Welch 62] For any prime power Q and integer J ≥ 2, ∃ RDS( QJ−1

Q−1 , Q − 1, QJ−1, QJ−2) for G = F× QJ relative to H = F× Q.

Theorem: [F Mayo 20] If an ETF(D, N) exists where D < N < 2D and Q = D(N−1)

N−D

is a prime power, then for all integers J ≥ 1, ∃ CETF(D(J), N(J)) where D(J) = DQJ−1, N(J) = N( QJ−1

Q−1 ).

Note: When applied to a harmonic ETF arising from the complement of a Singer difference set, this result yields another harmonic ETF of this same type. This theorem generalizes a difference set construction of [Gordon Mills Welch 62].

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Example: A new infinite family of ETFs from the ETF(3, 9)

The Naimark complement of an ETF(3, 9) is an ETF(6, 9) and Q = D(N−1)

N−D

= 6(9−1)

9−6

= 16 is a prime power. For any J ≥ 2, [Gordon Mills Welch 62] gives an RDS( 1

15(16J − 1), 15, 16J−1, 16J−2), i.e., an MUETF(16J−1, 1 5(16J − 1), 15).

Tensoring the ETF(6, 9) with 9 of these 15 yields an ETF(D(J), N(J)) with (D(J), N(J)) = (6(16J−1), 9

15(16J − 1)) = (96, 153), (1536, 2457), ...

Their Naimark complements have parameters (D(J), N(J)) = (57, 153), (921, 2457), ...

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Takeaways and future work

◮ ETFs and MUBs are different types of optimal projective codes. ◮ MUETFs unify and generalize ETFs and MUBs into a common framework, with a common “Gerzon” bound and “difference set” equivalence. ◮ Tensoring an ETF with a “compatible” MUETF yields another ETF. ◮ Harmonic MUETFs equate to relative difference sets, and classical examples

• f these allow us to build an infinite family of ETF from any single

ETF(D, N) where Q = D(N−1)

N−D

is a prime power. (“Most” of these are new.) ◮ When do MUETF(D, N, M) with D < N (non-MUBs) exist?

◮ Every known example of such an MUETF is complex. MUBs can be real, are not necessarily constructed harmonically. ◮ Beyond [Gordon Mills Welch 62] we only have a CMUETF(16, 21, 6).

(Thanks! Questions?)

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