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

Projective Codes

Spherical codes: Tammes’s problem (1930) n =1 in R D that maximize Problem: Find N unit vectors { x n } N n 1 � = n 2 � x n 1 − x n 2 � . min Equivalently, since � x n 1 − x n 2 � 2 = 2(1 − min n 1 � = n 2 � x n 1 , x n 2 � ), minimize max n � = n ′ � x n , x n ′ � . Example: D = 3, N = 2 , 3 , 4 , 5 , 6: 1/24

Rankin’s bounds (1955) Theorem: When N ≤ D + 1, { optimal spherical codes } = { regular simplices } . Proof: For any unit vectors { x n } N n =1 in R D , N N N 2 � � � � � � � 0 ≤ x n = � x n 1 , x n 2 � ≤ N + N ( N − 1) max n 1 � = n 2 � x n 1 , x n 2 � . � � � � n =1 n 1 =1 n 2 =1 Theorem: max n 1 � = n 2 � x n 1 , x n 2 � ≥ 0 when N ≥ D + 2. It’s achievable when N ≤ 2 D . n =1 are mutually obtuse in R D then projecting Proof: By induction: if { x n } N { x n } N − 1 N yields N − 1 mutually obtuse vectors in R D − 1 . n =1 onto x ⊥ To achieve equality when N ≤ 2 D , choose { x n } N n =1 as a subset of the orthoplex . 2/24

Projective spaces Let F be either R or C . The projective space of F D is the set of its one-dimensional subspaces. We identify each with its rank-one projection: P ( F D ) := { ϕϕ ∗ : ϕ ∈ F D , � ϕ � = 1 } . The Frobenius inner product of two such projections is � ϕ 1 ϕ ∗ 1 , ϕ 2 ϕ ∗ 2 � Fro = Tr( ϕ 1 ϕ ∗ 1 ϕ 2 ϕ ∗ 2 ) = Tr( ϕ ∗ 2 ϕ 1 ϕ ∗ 1 ϕ 2 ) = |� ϕ 1 , ϕ 2 �| 2 The squared Frobenius distance between two such projections is thus 2 � 2 2 � Fro ) = 2(1 − |� ϕ 1 , ϕ 2 �| 2 ) . � ϕ 1 ϕ ∗ 1 − ϕ 2 ϕ ∗ Fro = 2(1 − � ϕ 1 ϕ ∗ 1 , ϕ 2 ϕ ∗ 3/24

Projective codes n =1 in F D that maximize Problem: Design N unit vectors { ϕ n } N n 2 � 2 = min n 1 � = n 2 � ϕ n 1 ϕ ∗ n 1 − ϕ n 2 ϕ ∗ n 1 � = n 2 2(1 − |� ϕ n 1 , ϕ n 2 �| 2 ) min or equivalently have minimal coherence : coh( { ϕ n } N n =1 ) := max n � = n ′ |� ϕ n , ϕ n ′ �| . � � That is, design a D × N matrix Φ = ϕ 1 · · · ϕ N so that its Gram matrix     ϕ ∗ � ϕ 1 , ϕ 1 � · · · � ϕ 1 , ϕ N � 1 . . . ... Φ ∗ Φ = . � � . . ϕ 1 · · · ϕ N =  .   . .      ϕ ∗ � ϕ N , ϕ 1 � · · · � ϕ 1 , ϕ N � N has off-diagonal entries of minimal ∞ -norm, subject to having 1’s in its diagonal. 4/24

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 ∗ n A ϕ n } N n =1 = � ϕ n ϕ ∗ n , A � Fro ◮ Design of experiments ( ± 1-valued real vectors in 1 ⊥ ) 5/24

Embedding projective spaces into Euclidean spaces Idea: Since � ϕ 1 ϕ ∗ 1 , ϕ 2 ϕ ∗ 2 � Fro = |� ϕ 1 , ϕ 2 �| 2 , consider the mapping � D ( D +1) R , F = R , ϕ ∈ F D �− → x := ϕϕ ∗ ∈ { A ∈ F D × D : A = A ∗ } ∼ 2 = R D 2 , F = C . In particular, { ϕ n } N n =1 has minimal coherence if { x n } N n =1 , x n := ϕ n ϕ ∗ n is an optimal spherical code in { A ∈ F D × D : A = A ∗ } . Unfortunately, it never will be. Nevertheless, this gives the (proto)-Gerzon bound: � D ( D +1) , F = R , rank( X ) = rank( X ∗ X ) = rank( | Φ ∗ Φ | 2 ) ≤ 2 D 2 , F = C , where equality holds ⇔ any self-adjoint A can be recovered from { ϕ ∗ n A ϕ n } N n =1 . 6/24

Embedding projective spaces into Euclidean spaces (redux) Idea: [Conway, Hardin, Sloane 96] Instead map any ϕ ∈ F D to 2 ( ϕϕ ∗ − 1 1 D x := ( D − 1 ) D I ) � D ( D +1) R − 1 , F = R , in { A ∈ F D × D : A = A ∗ , Tr( A ) = 0 } ∼ 2 = R D 2 − 1 , F = C . For any unit vectors ϕ 1 , ϕ 2 ∈ F D , D − 1 ( |� ϕ 1 , ϕ 2 �| 2 − 1) . D − 1 Tr[( ϕ 1 ϕ ∗ D 1 − 1 D I )( ϕ 2 ϕ ∗ 2 − 1 D � x 1 , x 2 � Fro = D I )] = Problem: Design { ϕ n } N n =1 so that { x n } N n =1 is an optimal spherical code. 7/24

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 { ϕ n } N n =1 in F D , � 1 � N − D n 1 � = n 2 |� ϕ n 1 , ϕ n 2 �| ≥ max 2 , D ( N − 1) where equality holds ⇔ { ϕ n } N n =1 is an ETF( D , N ) for F D , namely N ◮ ∃ A > 0 such that ΦΦ ∗ = � ϕ n ϕ ∗ n = A I , n =1 ◮ ∃ C ≥ 0 such that | ( Φ ∗ Φ )( n 1 , n 2 ) | = |� ϕ n 1 , ϕ n 2 �| = C for all n 1 � = n 2 . � D ( D +1) , F = R , 2 Gerzon’s bound: If ∃ ETF( D , N ) then N ≤ D 2 , F = C . 8/24

Example: R ETF(3 , 4) (vertices of tetrahedron)   3 − − −     + − + − 4 0 0 Φ = 1  , ΦΦ ∗ = 1  , Φ ∗ Φ = 1 − 3 − −   √ + + − − 0 4 0  .     − − 3 − 3 3 3  + − − + 0 0 4 − − − 3 9/24

Example: R ETF(3 , 6) (antipodes of icosahedron)   0 + + + + + + 0 + − − +     2 0 0   Φ ∗ Φ = I + 1 + + 0 + − − ΦΦ ∗ =  ,   0 2 0 √    + − + 0 + − 5   0 0 2   + − − + 0 +   + + − − + 0 10/24

Example: C ETF(3 , 7) (quadratic residues modulo 7) ω 2 ω 3 ω 4 ω 5 ω 6     1 ω 7 0 0 Φ = 1 ΦΦ ∗ = 1 1 ω 2 ω 4 ω 6 ω 1 ω 3 ω 5 √ 0 7 0     3 3 ω 5 ω 2 ω 6 ω 3 1 ω 4 ω 0 0 7 1 ζ ζ ζ ζ ζ ζ   ζ 1 ζ ζ ζ ζ ζ     ω = exp( 2 π i ζ ζ 1 ζ ζ ζ ζ 7 )   Φ ∗ Φ = 1   ζ ζ ζ 1 ζ ζ ζ   3   ζ = 1 + ω + ω 4 ζ ζ ζ ζ 1 ζ ζ     ζ ζ ζ ζ ζ 1 ζ   ζ ζ ζ ζ ζ ζ 1 11/24

Example: C ETF(3 , 9) (SIC-POVM) 1 ω ω 2 0 1 ω 2 ω     0 0 3 0 0 Φ = 1 ΦΦ ∗ = 1 ω 2 ω 1 ω ω 2 0 √ 0 0 0 3 0     2 1 ω 2 ω 1 ω ω 2 0 0 0 0 0 3 ω ω 2 1 ω 2 ω  2 − − 1  − 2 − ω ω 2 1 ω 2 ω 1   − − 2 ω 2 1  1 ω 2  ω ω   1 ω 2 ω  ω ω 2  2 − − 1 Φ ∗ Φ = 1   ω 2 ω 1 − 2 − ω ω 2 1     2 1 ω 2 − − 2 ω 2 1   ω ω   ω ω 2 1 ω 2 ω   1 2 − −   ω ω 2 1 ω 2 ω   1 − 2 −   ω 2 1 1 ω 2 − − 2 ω ω 12/24

The orthoplex bound � D ( D +1) , F = R , 2 Theorem: [Conway Hardin Sloane 96]: Let N > D 2 , F = C . 1 D for any N unit vectors { ϕ n } N n =1 in F D . Then max n 1 � = n 2 |� ϕ n 1 , ϕ n 2 �| ≥ √ If { ϕ n } N n =1 achieves equality, it’s an orthopletic Grassmannian frame (OGF) . Examples: (from the infinite families of OGFs of [Bodmann Hass 16]): √   + − + − 3 0 0 √ Φ = 1 ∃ R OGF(3 , 7) : √ + + − − 0 3 0 √   3 + − − + 0 0 3 1 ω ω 2 ω 3 ω 4 ω 5 ω 6 √  3 0 0  √ Φ = 1 1 ω 2 ω 4 ω 6 ω 1 ω 3 ω 5 ∃ C OGF(3 , 10) : √ 0 3 0  √  3 1 ω 4 ω ω 5 ω 2 ω 6 ω 3 0 0 3 13/24

Mutually unbiased bases (MUBs) d =1 for F D are Definition: A sequence of M orthonormal bases { ϕ m , d } M D m =1 , 1 mutually unbiased if |� ϕ m 1 , d 1 , ϕ m 2 , d 2 �| = D for all m 1 , m 2 , d 1 , d 2 , m 1 � = m 2 . √ � + + √ Example: ∃ R MUB(2 , 2) : Φ = 1 � 2 0 √ √ + − 0 2 2 √  1 1 1 1 1 1 1 1 1 3 0 0  √ Example: ∃ C MUB(3 , 4) : Φ = 1 1 ω ω 2 ω ω 2 1 ω 2 1 ω √ 0 3 0  √  3 1 ω 2 ω ω 1 ω 2 ω 2 ω 1 0 0 3 � D 2 + 1 , F = R , Corollary of Proto-Gerzon: If ∃ F MUB( D , M ) then M ≤ 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 , d A ϕ m , d } M D d =1 . m =1 , 14/24

Some open problems about ETFs and MUB existence ◮ When does an ETF( D , N ) exist? ◮ Zauner’s conjecture: Does a C ETF( D , D 2 ) (SIC-POVM) exist for all D ? ◮ Necessary integrality conditions on C ETF( D , N )? ◮ [Sz¨ osi 14] ∄ C ETF(3 , 8) oll˝ ◮ 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? ◮ ∃ C MUB( Q , Q + 1) for prime power Q and ∃ R MUB( Q , Q 2 + 1) for Q = 2 , 4 j ◮ if ∃ F MUB( D 1 , M ) and ∃ F MUB( D 2 , M ) then ∃ F MUB( D 1 D 2 , M ) ◮ [Wocjan Beth 05] MUBs from MOLS ◮ Does a C MUB(6 , M ) exist for M = 3 , 4 , 5 , 6 , 7? 15/24

Mutually Unbiased Equiangular Tight Frames (MUETFs)