Almost Hadamard & Complex Hadamard Matrices Karol Zyczkowskichaos.if.uj.edu.pl/~karol/pdf2/kz_Banff14.pdf · & Complex Hadamard Matrices Karol Zyczkowski_ in collaboration with

Almost Hadamard& Complex Hadamard Matrices

Karol Zyczkowskiin collaboration with

Teo Banica (Cergy–Pontoise) and Ion Nechita (Toulouse)

Jagiellonian University, Cracow, Polandand

Polish Academy of Science, Warsaw

Banff Center, July 13, 2014KZ (IF UJ/CFT PAN ) Almost Hadamard Matrices July 13, 2014 1 / 38

(standard) Hadamard matrices

Definition (Sylvester 1867)

mutually orthogonal row and columns,

HH∗ = N1 , Hij = ±1. (1)

simplest example: N = 2 Hadamard matrix

H2 =

[1 11 −1

]Hadamard conjecture

(real) Hadamard matrices do exist for N = 2 and 4k for any k = 1, 2, ...After a discovery of N = 428 Hadamard matrix (Kharaghani andTayfeh-Razaie, 2005) this conjecture is known to hold up to N = 664

KZ (IF UJ/CFT PAN ) Almost Hadamard Matrices July 13, 2014 2 / 38

Generalized Hadamard matrices

Standard Hadamard matrices (real matrix, unitary, unimodular entries)do not exist for all dimensions, (interesting for quantum theory)e.g. N = 3, 5, 6, 7, 9...

possible solution:use ’generalized Hadamard matrices’ by relaxing some constraints:

a) keeping matrix real but relaxing the orthogonality

Quasi Hadamard matrices: (real, unimodular)

b) keeping matrix real but relaxing the unimodularity

Almost Hadamard matrices: (real, orthogonal)

c) keeping matrix unitary but allowing it to be complex

Complex Hadamard matrices: (unitary, unimodular, complex)


Generalized Hadamard matrices

Standard Hadamard matrices (real matrix, unitary, unimodular entries)do not exist for all dimensions, (interesting for quantum theory)e.g. N = 3, 5, 6, 7, 9...

possible solution:use ’generalized Hadamard matrices’ by relaxing some constraints:

a) keeping matrix real but relaxing the orthogonality

Quasi Hadamard matrices: (real, unimodular)

b) keeping matrix real but relaxing the unimodularity

Almost Hadamard matrices: (real, orthogonal)

c) keeping matrix unitary but allowing it to be complex

Complex Hadamard matrices: (unitary, unimodular, complex)



b) Almost-Hadamard matrices for N 6= 2, N 6= 4k

Real orthogonal matrix A with ’almost equal’ moduli

Formally: Orthogonal matrix with maximal 1–norm, ||A||1 =∑

ij |Aij |.For N = 2, 4, 8, 12, 16, ... any (rescaled) Hadamard matrix is also AlmostHadamard with one–norm equal to N

√N.

N = 3, A3 = 13

−1 2 22 −1 22 2 −1

for which ||A3||1 = 5 < 3√

3 ≈ 5.196

N = 5, A5 = 15

−3 2 2 2 22 −3 2 2 22 2 −3 2 22 2 2 −3 22 2 2 2 −3

T. Banica, I. Nechita, K. Z.,Almost Hadamard matrices: general theory and examples,

Open Systems & Information Dynamics 19, 1250024 (2012).


A chess problem on a N × N chessboard

Take NM chess rooks with M ≤ N/2 and fix an integer K ≤ M,

Try to place all the rooks on the board such that:a) there are exactly M rooks on each row and each column of the board,

andb) for any pair of rows or columns,

there are exactly K pairs of mutually attacking rooks.

Example 1: Arbitrary N and M = 1 and K = 0:Identity or a permutation matrix does the job (no attacking roocks !)

N = 4,

X . . .. X . .. . X .. . . X

,

. X . .. . X .. . . XX . . .


Example 2. Let K = 1, N = 7 and M = 3

X X X . . . .X . . X X . .X . . . . X X. X . X . . X. X . . X X .. . X . X . X. . X X . X .

‘Symmetric balanced incompleteblock designs’ (BIBD)Colbourn & Dinitz,Handbook of Combinatorial Designs(2007)

finite plane of Fano



X X X . . . .X . . X X . .X . . . . X X. X . X . . X. X . . X X .. . X . X . X. . X X . X .

‘Symmetric balanced incompleteblock designs’ (BIBD)Colbourn & Dinitz,Handbook of Combinatorial Designs(2007)

finite plane of Fano


Almost Hadamard matrix for N = 7

is based on incidence matrix of the Fano plane

Let x = 2− 4√

2 and y = 2 + 3√

2.

Then Almost Hadamard matrix A7 of size N = 7 reads

A7 = 12√

7

x x x y y y yx y y x x y yx y y y y x xy x y x y y xy x y y x x yy y x y x y xy y x x y x y



. X . X X X . . . X .

. . X . X X X . . . XX . . X . X X X . . .. X . . X . X X X . .. . X . . X . X X X .. . . X . . X . X X XX . . . X . . X . X XX X . . . X . . X . XX X X . . . X . . X .. X X X . . . X . . XX . X X X . . . X . .

the Paley biplane



. X . X X X . . . X .

. . X . X X X . . . XX . . X . X X X . . .. X . . X . X X X . .. . X . . X . X X X .. . . X . . X . X X XX . . . X . . X . X XX X . . . X . . X . XX X X . . . X . . X .. X X X . . . X . . XX . X X X . . . X . .

the Paley biplane


Almost Hadamard matrix for N = 11

is based onincidence matrix of the Paley biplane

Let x = 6− 12√

3, y = 6 + 10√

3

Then Almost Hadamard matrix A11 of size N = 11 reads :

A11 = 16√

11

y x y x x x y y y x yy y x y x x x y y y xx y y x y x x x y y yy x y y x y x x x y yy y x y y x y x x x yy y y x y y x y x x xx y y y x y y x y x xx x y y y x y y x y xx x x y y y x y y x yy x x x y y y x y y xx y x x x y y y x y y



Almost Hadamard matrices II

Other options:

a) optimize other p–norm of a matrix A, Banica, Nechita 2013

b) minimize (a certain) distance to the unimodular matrix, |Mij |2 = 1/N

c) maximize average entropy S or Renyi entropy Sαof the corresponding orthostochastic matrix Bij = |Aij |2,as due to orthogonality of A relation

∑i Bij =

∑j Bij = 1 holds,

S(A) := − 1

N

∑ij

|A|2ij ln |A|2ij ,

(as used by Erickson, 2014) and

Sα(A) :=1

N

∑i

1

1− αln∑

j

|A|2αij .

Average entropy of random Haar unitary/orthogonal matrix of order Nreads 〈S(A)〉 = ln N − C , K.Z, Kus, S lomczynski, Sommers, 2003



other possibility: Complex Hadamard matrices

Hadamard matrices of the Butson type

composed off roots of unity; H ∈ H(N, q) iff

HH∗ = N 1 , (Hij)q = 1 for i , j = 1, . . .N (2)

Butson, 1962

special case: q = 4H ∈ H(N, 4) iff HH∗ = N 1 and Hij = ±1,±i

(also called complex Hadamard matrices, Turyn, 1970)

Complex Hadamard matrices (general case)

HH∗ = N 1 and |Hij | = 1,hence Hij = exp(iφij) with an arbitrary complex phase.


Complex Hadamard matrices do exist for any N !

example: the Fourier matrix

(FN)jk := exp(ijk2π/N) with j , k = 0, 1, . . . ,N − 1. (3)

special case : N = 4

F4 =

1 1 1 11 i −1 −i1 −1 1 −11 −i −1 i

∈ H(4, 4) (4)

The Fourier matrices are constructed of N–th root of unity, so they areof the Butson type,

FN ∈ H(N,N).


Equivalent Hadamard matrices

H ′ ∼ Hiff there exist permutation matrices P1 and P2 anddiagonal unitary matrices D1 and D2 such that

H ′ = D1P1 H P2D2 . (5)

Dephased form of a Hadamard matrix

H1,j = Hj ,1 = 1 for j = 1, . . . ,N. (6)

Any complex Hadamard matrix can be brought to the dephased formby an equivalence relation.

example for N = 3, here α ∈ [0, 2π) while w = exp(i · 2π/3), so w3 = 1

F ′3 = e iα

w 1 w2

1 1 1w2 1 w

∼ 1 1 1

1 w w2

1 w2 w

=: F3 , (7)


Classification of (real) Hadamard matrices

n ≤ 12

For n = 2, 4, 8, 12 all (real) Hadamard matrices are equivalent

higher dimensions

The number E of equivalence classes of real Hadamard matrices of ordern reads

n = 2 4 8 12 16 20 24 28 32E = 1 1 1 1 5 3 60 487 13 710 027

Fang and Ge 2004, Orrick 2005, Tayfeh-Razaie 2014

For n = 36 and n = 40 this number is not smaller than 1.8× 107 and3× 1011 respectively (Orrick, Tonchev), but the problem ofennumerating all equivalence classes of Hadamard matrices remains open.



Classification of Complex Hadamard matrices I

N = 2

all complex Hadamard matrices are equivalent to the real Hadamard(Fourier) matrix

H2 = F2 =

[1 11 −1

]. (8)

N = 3

all complex Hadamard matrices are equivalent to the Fourier matrix

F3 =

1 1 11 w w2

1 w2 w

, w = e2πi/3. (9)

U. Haagerup, Orthogonal maximal abelian ∗-subalgebras of the N × Nmatrices and cyclic N–rots,in Operator Algebras and Quantum Field Theory, 1996.


Classification of Complex Hadamard matrices II

N = 4

Lemma (Haagerup). For N = 4 all complex Hadamard matrices areequivalent to one of the matrices from the following 1–d orbit, w = i

F(1)4 (a) =

1 1 1 11 w1 · exp(i · a) w2 w3 · exp(i · a)1 w2 1 w2

1 w3 · exp(i · a) w2 w1 · exp(i · a)

, a ∈ [0, π].

N = 5

All N = 5 complex Hadamard matrices are equivalent tothe Fourier matrix F5 (Haagerup 1996).

N ≥ 6

Several orbits of Complex Hadamard matrices are known, but the problemof their classification remains open!


Our aim:

Classify all Complex Hadamard matrices for ’small’ matrix size n.

One distinguishes:

a) isolated Hadamard matrices: if they do not belong to any continuousfamily of inequivalent complex Hadamard matrices.Examples: F2,F3,F5

b) non-isolated Hadamard matrices, which belong to a k-dimensionalcontinuous family of inequivalent complex Hadamard matrices.

Examples: F4 ∈ F(1)4 (a) and F6 ∈ F

(2)6 (a, b)

For n ≥ 6 such a full classification is still missing...

For n = 6 only partial results are available and several special cases areknown:


Zoo of n = 6 complex Hadamard matrices

S(0)6 isolated ’spectral set’ matrix, Tao 2004, Moorhouse 2001

S(0)6 =

1 1 1 1 1 11 1 w w w2 w2

1 w 1 w2 w2 w1 w w2 1 w w2

1 w2 w2 w 1 w1 w2 w w2 w 1

; w = exp(2πi/3) .

Alternative notation with entry-by-entry exponentiation EXP

S(0)6 = EXP[i R]; R =

2π

3

0 0 0 0 0 00 0 1 1 2 20 1 0 2 2 10 1 2 0 1 20 2 2 1 0 10 2 1 2 1 0

∈ H(6, 3).

Butson–like design...KZ (IF UJ/CFT PAN ) Almost Hadamard Matrices July 13, 2014 22 / 38

C ′6: Dephased circulant matrix of Bjork ’1995

C ′6 =

1 i d −d −i −d−1 id−1

id−1 1 i d −d −i −d−1

−d−1 id−1 1 i d −d −i−i −d−1 id−1 1 i d −d−d −i −d−1 id−1 1 i di d −d −i −d−1 id−1 1

where a unimodular complex number d = 1−

√3

2 + i ·√√

32

is a solution of the equation x2 − (1−√

3)x + 1 = 0.

C ′6 =

1 1 1 1 1 11 −1 −d −d2 d2 d1 −d−1 1 d2 −d3 d2

1 −d−2 d−2 −1 d2 −d2

1 d−2 −d−3 d−2 1 −d1 d−1 d−2 −d−2 −d−1 −1


D(1)6 : 1-d affine family of Dita ’2001

D(1)6 (c) = D6 ◦ EXP[i · R

D(1)6

(c)]; a Hadamard product of

D6 =

1 1 1 1 1 11 −1 i −i −i i1 i −1 i −i −i1 −i i −1 i −i1 −i −i i −1 i1 i −i −i i −1

∈ H(4, 6)

and a single parameter matrix of phases, R(c), in which (•) = 0,

RD

(1)6

(c) =

• • • • • •• • • • • •• • • c c •• • −c • • −c• • −c • • −c• • • c c •

(′enphasing′)


F(2)6 (a, b): affine 2-d family of Haagerup (1996)

stemming from the Fourier matrix F6, 3× 2 = 6

F(2)6 (a, b) = F6 ◦ EXP[i · R

F(2)6

(a, b)]; a Hadamard product of

F6 =

1 1 1 1 1 11 w1 w2 w3 w4 w5

1 w2 w4 1 w2 w4

1 w3 1 w3 1 w3

1 w4 w2 1 w4 w2

1 w5 w4 w3 w2 w1

; w = exp(2πi/6).

and a two-parameter matrix of phases,

RF

(2)6

(a, b) =

• • • • • •• a b • a b• • • • • •• a b • a b• • • • • •• a b • a b

; a, b ∈ [0, 2 π)


B(1)6 (y): non-affine 1-d family of Beauchamp &

Nicoara, April 2006

B(1)6 (y) =

1 1 1 1 1 11 −1 −1/x −y y 1/x1 −x 1 y 1/z −1/t1 −1/y 1/y −1 −1/t 1/t1 1/y z −t 1 −1/x1 x −t t −x −1

where y = exp(i s) is a free parameter and

x(y) =1 + 2y + y2 ±

√2√

1 + 2y + 2y3 + y4

1 + 2y − y2

z(y) =1 + 2y − y2

y(−1 + 2y + y2); t(y) = xyz

W. Bruzda discovered his family independenty in May 2006KZ (IF UJ/CFT PAN ) Almost Hadamard Matrices July 13, 2014 26 / 38

X(2)6 (a, b): non-affine 2-d family Szollosi, Oct. 2008

Bi-circulant n = 6 complex Hadamard matrix

X =

[A BB† −A†

], (10)

where A and B are 3 by 3 circulant matrices

A =

a b cc a bb c a

, B =

d e ff d ee f d

, |a| = |b| = · · · = |f | = 1 .

X is an Hadamard matrix if AA∗ + BB∗ = 61.These conditions lead to a 2-parameter family X

(2)6 (a, b):


Hadamard family from Sweden

In 2009 B. Karlsson (Uppsala) constructed a 2-d non-affine family ofcomplex Hadamard matrices,K6(x , y).

A 4-dimensional orbit

Semi-analitic 4–dim family of F. Szollosi 2011(Which arguably contains all previously known continuous families!)

A (Swedish-Polish) Conjecture

Bengtsson, Bruzda, Ericsson, Larsson, Tadej, K. Z, 2007

All n = 6 complex Hadamard matrices(besides the isolated matrix S6 of Tao)

belong to a single, 4-dimesional family of complex Hadamard matrices.Numerical confirmation: Skinner, Newell, Sanchez 2008, Szollosi 2011.

Why 4 dimensional ?



Defect of a unitary matrix d(U)

a) - Take matrix U to the dephased formb) - Multiply matrix elements by arbitrary phase factors, and expand tothe first orders in the angles:

Uij → Uijeiφij ' Uij(1 + iφij) , 2 ≤ i , j ≤ n .

c) - Solve the unitarity equations to first order in the angles φij . Thenumber of free parameters in the solution of this linear equations is calleddefect of the matrix U (Tadej & Zyczkowski 2006, 2008),and can be determined be calculating the rank of a differential matrix.

defect d(U) gives an upper bound on the dimension

of the family of inequivalent Hadamard matrices containing√

nU.If d(U) = 0 then H =

√nU is isolated.

Observation: for all known n = 6 complex Hadamard matrix H we haved(H) = 4 with a single exception [d(S6) = 0, since Tao matrix is isolated].


The Defect for the Fourier matrix Fn

Theorem: (Tadej, S lomczynski, Zyczkowski, 2008)

Factorize into power of primes the number n =∏m

j=1 pjkj where

p1 > p2 > . . . > pm. Then the defect of the Fourier matrix Fn of size nreads

d(Fn) = n ·

m∏j=1

(1 + kj −

kj

pj

)− 2

+ 1 . (11)

Special cases

a) n = p1 is prime, so m = 1, k1 = 1, so d(Fp) = p(2− 1/p − 2) + 1 = 0hence Fp is isolated.b) power of primes, n = pk , so d(Fpk ) = pk−1[(p − 1)k − p] + 1.In this case the family of (generalized) Fourier matrices

of this dimension is known, Tadej 2009.c) product of primes, n = pq, so d = 2(p − 1)(q − 1), [e.g. d(F6) = 4gives the upper bound of the dimension of the orbit stemming from F6.]


A more general set-up

Unistochastic matrices

Let B be a bistochastic matrix of order n, so that∑

i Bij =∑

j Bij = 1and Bij ≥ 0 (also called doubly stochastic).B is called unistochastic if there exist a unitary U such that Bij = |Uij |2what implies B = f (U)

Existence problem: Which B is unistochastic?

Every B of size n = 2 is unistochastic, for n = 3 it is not the case(Schur). Constructive conditions for unistochasticity are known for n = 3Au-Yeung and Poon 1979, but for n = 4 this problem remains open.

Classification problem: Assume B is unistochastic

Find all preimages U such that f −1(B) = U.Special case: Flat matrix of van der Waerden of size n, so Bij = 1/n.Then the problem of classification of all preimages of B reduces to thesearch for all complex Hadamard matrices of size n.



Mutually Unbiased Bases & Complex Hadamards

Mutually unbiased bases (MUB) in finite dimensions

Let {|αi}Ni=1 and {|βj〉}Nj=1 be two orthonormal basis HN

(e.g. given as eigenbasis of hermitian operators A and B).The bases are called maximally unbiased if all entries of the unitarytransition matrix U have the same modulus, |Uij |2 = |〈αi |βj〉|2 = 1

N ,i.e. U is a complex Hadamard matrix (CHM).

A set of d bases which satisfies this condition is called mutually unbiasedTwo CHM’s H1 and H2 are unbiased if H∗1H2/

√N is also a CHM.

Basic facts and ... prime numbers ...

∗ For any N there exists not more then d = N + 1 MUBs,∗ This upper bound is saturated for prime dimensions and for powers of

primes, N = pk (Ivanovic 1981; Wootters & Fields 1989)∗ For any N ≥ 2 there exists at least a triple of MUBs,∗ for N = 6 = 2× 3 only 3 MUBs are known

(so this is the first open case ! as the upper bound is d = 7)KZ (IF UJ/CFT PAN ) Almost Hadamard Matrices July 13, 2014 34 / 38

MUB’s and some number theory

Standard MUB solution for a prime dimension, N = p

∗ Choose Fourier matrix of order N and set H(1) = FN .

∗ Take diagonal unitary [EN ]jk := δjk exp(i · 2π

N (j − 1)2)

,

∗ Define set of N complex Hadamard matrices {H(r)}Nr=1 where

H(r) := (EN)r−1H(1) for r = 1, 2, . . . ,N

and check that they are mutually unbiasedsince Xr−s := (H(s))∗H(r)/

√N is also a complex Hadamard.

This is fine for any prime N = p.But why this construction does not work for composite dimensions ??

Analyze diagonal elements of the matrices (Xr ) for any (composite) N...


A prime distinguishing function

Consider a function of an integer argument N > 1

g(N) =1

N − 1

N−1∑r=1

1√N

∣∣∣ N∑j=1

e i · 2πN

r(j−1)2∣∣∣ − 1 .

Use the Gauss sum to show that it vanishes for any prime N = p ≥ 2,

g(p) = 1p−1

∑p−1r=1

1√p

∣∣∣∑pj=1 e i · 2π

pr(j−1)2

∣∣∣− 1 = 1p−1

∑p−1r=1

1√p

√p − 1 = 0

Rescaled prime distinguishing function g(N)T.Durt, B.Englert, I.Bengtsson, K.Z., 2010



Concluding Remarks

1 For any N one defines Almost Hadamard Matrices which are realorthogonal with ’almost equal’ absolute values of the entries: onelooks for maxima of 1–norm of an orthogonal matrix, maxO ||O||1.For N = 2, 4, 8, 12, ... Hadamard matrices maximize this norm.

2 Almost Hadamard Matrix is identified for N = 3, while theconjectured form for N = 5, 7, 9 has structure of balancedincomplete block designs (BIBD).

3 Another generalization of (real) Hadamard matrices is obtained ifwe allow for complex entries and define Complex Hadamardmatrices (CHM). They do exist for any matrix size n, (Fouriermatrix Fn), but their classification is completed only for n = 2, 3, 4, 5.

4 For n = 6 it is conjectured that all CHM are embedded into a single4-d orbit See also online Catalog at

http://chaos.if.uj.edu.pl/∼karol/hadamard5 The defect of a Hadamard (unitary) matrix H gives the upper bound

of the dimensionality of the CHM orbit stemming from H. Entireorbit stemming form Fn is known only if n = pk .


Documents

Almost Hadamard & Complex Hadamard Matrices Karol Zyczkowskichaos.if.uj.edu.pl/~karol/pdf2/kz_Banff14.pdf · & Complex Hadamard Matrices Karol Zyczkowski_ in collaboration with