Complex Hadamard matrices and some of their applications

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1 Complex Hadamard matrices and some of their applications Karol Życzkowski in collaboration with W. Bruzda and W. S lomczyński (Cracow) W. Tadej (Warsaw) and I. Bengtsson (Stockholm) Institute of Physics, Jagiellonian University, Cracow, Poland and Center for Theoretical Physics, Polish Academy of Sciences Bȩdlewo, August 21, 2009 KŻ (IF UJ/CFT PAN ) Complex Hadamard matrices August 21, / 26

2 Hadamard matrices (real) definition (Sylvester 1867) mutually orthogonal row and columns, HH = ½, H ij = ±1. (1) simplest example of order n = 2 [ ] 1 1 H 2 = 1 1. (2) 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 and Tayfeh-Razaie, 2005) this conjecture is known to hold up to n = 664 KŻ (IF UJ/CFT PAN ) Complex Hadamard matrices August 21, / 26

3 Generalized Hadamard matrices Hadamard matrices of the Butson type composed off roots of unity; H H(q, n) iff Butson, 1962 HH = ½, (H ij ) q = 1 for i, j = 1,...n (3) special case: q = 4 H H(4, n) iff HH = ½ and H ij = ±1, ±i (also called complex Hadamard matrices, Turyn, 1970) Complex Hadamard matrices (general case) HH = ½ and H ij = 1, hence H ij = exp(iφ ij ) with an arbitrary complex phase. KŻ (IF UJ/CFT PAN ) Complex Hadamard matrices August 21, / 26

4 Complex Hadamard matrices do exist for any n! example: the Fourier matrix special case : n = 4 (F n ) jk := exp(ijk2π/n) with j, k = 0, 1,...,n 1. (4) F 4 = i 1 i i 1 i H(4, 4) (5) The Fourier matrices are constructed of n th root of unity, so they are of the Butson type, F n H(n, n). KŻ (IF UJ/CFT PAN ) Complex Hadamard matrices August 21, / 26

5 Equivalent Hadamard matrices H 1 H 2 iff there exist permutation matrices P 1 and P 2 and diagonal unitary matrices D 1 and D 2 such that Dephased form of a Hadamard matrix H 1 = D 1 P 1 H 2 P 2 D 2. (6) H 1,j = H j,1 = 1 for j = 1,...,n. (7) Any complex Hadamard matrix can be brought to the dephased form by an equivalence relation. example for N = 3, here α [0, 2π) while w = exp(i 2π/3), so w 3 = 1 w 1 w F 3 = e iα w w 2 =: F 3, (8) w 2 1 w 1 w 2 w KŻ (IF UJ/CFT PAN ) Complex Hadamard matrices August 21, / 26

6 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 order n reads n = E = Fang and Ge For n = 32 and n = 36 this number is not smaller than 3, 578, 006 and 4, 745, 357 respectively (Orrick 2005), but the problem of ennumerating all equivalence classes of Hadamard matrices remains open. KŻ (IF UJ/CFT PAN ) Complex Hadamard matrices August 21, / 26

7 Classification of Complex Hadamard matrices I n = 2 all complex Hadamard matrices are equivalent to the real Hadamard (Fourier) matrix [ ] 1 1 H 2 = F 2 =. (9) 1 1 n = 3 all complex Hadamard matrices are equivalent to the Fourier matrix F 3 = 1 w w 2, w = e 2πi/3. (10) 1 w 2 w U. Haagerup, Orthogonal maximal abelian -subalgebras of the n n matrices and cyclic n rots, in Operator Algebras and Quantum Field Theory, KŻ (IF UJ/CFT PAN ) Complex Hadamard matrices August 21, / 26

8 Classification of Complex Hadamard matrices II n = 4 Lemma (Haagerup). For n = 4 all complex Hadamard matrices are equivalent to one of the matrices from the following 1 d orbit, w = i F (1) 4 (a) = w 1 exp(i a) w 2 w 3 exp(i a) 1 w 2 1 w 2 1 w 3 exp(i a) w 2 w 1 exp(i a) n = 5 All n = 5 complex Hadamard matrices are equivalent to the Fourier matrix F 5 (Haagerup 1996)., a [0, /pi]. n 6 Several orbits of Complex Hadamard matrices are known, but the problem of their complete classification remains open! KŻ (IF UJ/CFT PAN ) Complex Hadamard matrices August 21, / 26

9 Complex Hadamard matrices: some applications 1 subalgebras of finite dimensional von Neuman algebras, Haagerup, Popa, de la Harpe & Jones, Munemasa & Watatani, 2 Bi-unimodular sequences and cyclic n roots, Björk, Fröberg, Saffari 3 Equiangular lines, Godsil & Roy 4 quantum optics: symmetric multiports, Jex, Stenholm & Zeilinger 5 quantum designs, Zauner quantum information: bases of unitary operators, Werner KŻ (IF UJ/CFT PAN ) Complex Hadamard matrices August 21, / 26

10 Complex Hadamards & Quantum Information Werner lemma (2001) A complex Hadamard matrix H allows one to construct a) Basis of unitary operators the set of mutually orthogonal unitary operators, {U k } n2 U k U(n) and TrUk U l = nδ kl for k, l = 1,...,n 2, k=1 such that b) Basis of maximally entangled states the set { ψ k } n2 k=1 such that each ψ k belongs to a composed Hilbert space H = H A H B with the partial trace (marginal) Tr n ( ψ k ψ k ) = ½/n, and they are mutually orthogonal, ψ k ψ l = δ kl, c) Unitary depolarisers the set {U k } n2 k=1 such that for any bounded linear operator A the following property holds: n 2 k=1 U k AU k = n(tra) ½. KŻ (IF UJ/CFT PAN ) Complex Hadamard matrices August 21, / 26

11 Werner construction: shift and multiply basis Basis of unitaries in U(n) Take a set of n arbitrary complex Hadamard matrices {H (j) } n j= and a Latin square λ of size n, e.g. λ 3 = Then the basis in the set of unitary matrices is given by of order n U i,j k := H (j) λ(j, k) / n, for i, j, k = 1,...n. (11) Basis of entangled states in H = H A H B Define a maximally entangled state φ + := 1 n n i=1 i i. Then the maximally entangled basis is given by the set of n 2 states ψ m := (U i,j ½) φ +, m = i + n(j 1) = 1,..n 2 (12) KŻ (IF UJ/CFT PAN ) Complex Hadamard matrices August 21, / 26

12 How does the Werner construction work? Example for n = 2 Set n = 2 and H (1) = H (2) = H 2 = [ ] and λ 2 = [ ]. Unitary basis in U(2) reads: [ ] 1 0 U 1 = U 1,1 = / [ ] 2 = ½ ; U 2 = U 1,2 = 0 1 [ ] 0 1 U 3 = U 2,1 = / [ 2 = 0 1 σx ; U 4 = U 2,2 = 1 0 / 2 = σz 2 ; ] / 2 = iσy 2 ; Bell basis consisting of entangled states in H 4 = H 2 H 2 φ 1 = Ψ + = 1 2 [ 0, 0 + 1, 1 ]; φ 2 = Ψ = 1 2 [ 0, 0 1, 1 ] φ 3 = Φ + = 1 2 [ 0, 1 + 1, 0 ]; φ 4 = Φ = 1 2 [ 0, 1 + 1, 0 ]. KŻ (IF UJ/CFT PAN ) Complex Hadamard matrices August 21, / 26

13 If you would like to know more about quantum entanglement please have a look at I. Bengtsson and K. Życzkowski, Geometry of Quantum States: an Introduction to Quantum Entanglement (Cambridge 2006, 2008) KŻ (IF UJ/CFT PAN ) Complex Hadamard matrices August 21, / 26

14 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 continuous family of inequivalent complex Hadamard matrices. Examples: F 2, F 3, F 5 b) non-isolated Hadamard matrices, which belong to a k-dimensional continuous family of inequivalent complex Hadamard matrices. Examples: F 4 F (1) 4 (a) and F 6 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 are known: KŻ (IF UJ/CFT PAN ) Complex Hadamard matrices August 21, / 26

15 Zoo of n = 6 complex Hadamard matrices S (0) 6 isolated spectral set matrix, Tao 2004, Moorhouse w w w 2 w 2 S (0) 6 = 1 w 1 w 2 w 2 w 1 w w 2 1 w w 2 ; w = exp(2πi/3). 1 w 2 w 2 w 1 w 1 w 2 w w 2 w 1 Alternative notation with entry-by-entry exponentiation EXP S (0) 6 = EXP[i R]; R = 2π H(3, 6). KŻ (IF UJ/CFT PAN ) Complex Hadamard matrices August 21, / 26

16 C 6 : Dephased circulant matrix of Björk 1995 C 6 = 1 id d i d 1 id 1 id 1 1 id d i d 1 d 1 id 1 1 id d i i d 1 id 1 1 id d d i d 1 id 1 1 id id d i d 1 id 1 1 where a unimodular complex number d = i is a solution of the equation x 2 (1 3)x + 1 = 0. C 6 = d d 2 d 2 d 1 d 1 1 d 2 d 3 d 2 1 d 2 d 2 1 d 2 d 2 1 d 2 d 3 d 2 1 d 1 d 1 d 2 d 2 d 1 1 KŻ (IF UJ/CFT PAN ) Complex Hadamard matrices August 21, / 26 2

17 D (1) 6 : 1-d affine family of Diţă (c) = D 6 EXP[i R (1) D (c)]; a Hadamard product of i i i i D 6 = 1 i 1 i i i 1 i i 1 i i H(4, 6) 1 i i i 1 i 1 i i i i 1 D (1) and a single parameter matrix of phases, R(c), in which ( ) = 0, R (1) D (c) = 6 c c c c c c c c ( enphasing ) KŻ (IF UJ/CFT PAN ) Complex Hadamard matrices August 21, / 26

18 F (2) 6 (a, b): affine 2-d family of Haagerup (1996) stemming from the Fourier matrix F 6 6 (a, b) = F 6 EXP[i R (2) F (a, b)]; a Hadamard product of w 1 w 2 w 3 w 4 w 5 F 6 = 1 w 2 w 4 1 w 2 w 4 1 w 3 1 w 3 1 w 3 ; w = exp(2πi/6). 1 w 4 w 2 1 w 4 w 2 1 w 5 w 4 w 3 w 2 w 1 F (2) and a two-parameter matrix of phases, R (2) F (a, b) = 6 a b a b a b a b a b a b ; a, b [0, 2 π) KŻ (IF UJ/CFT PAN ) Complex Hadamard matrices August 21, / 26

19 B (1) 6 (y): non-affine 1-d family of Beauchamp & Nicoara, April 2006 B (1) 6 (y) = /x y y 1/x 1 x 1 y 1/z 1/t 1 1/y 1/y 1 1/t 1/t 1 1/y z t 1 1/x 1 x t t x 1 where y = exp(i s) is a free parameter and x(y) = 1 + 2y + y2 ± y + 2y 3 + y y y 2 z(y) = 1 + 2y y2 y( 1 + 2y + y 2 ) ; t(y) = xyz W. Bruzda discovered his family independenty in May 2006 KŻ (IF UJ/CFT PAN ) Complex Hadamard matrices August 21, / 26

20 n = 6, state of the art, July 2008 New 1-d non-affine orbit M 6 (x) found by Matolcsi and Szöllősi Sketch of the set of n = 6 inequivalent complex Hadamards KŻ (IF UJ/CFT PAN ) Complex Hadamard matrices August 21, / 26

21 X (2) 6 (a, b): non-affine 2-d family Szöllősi, Oct Bi-circulant n = 6 complex Hadamard matrix [ ] A B X = B A, (13) where A and B are 3 by 3 circulant matrices a b c A = c a b, B = b c a d e f f d e e 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): KŻ (IF UJ/CFT PAN ) Complex Hadamard matrices August 21, / 26

22 Hot news: August 2009 (during the conference) Mail from I. Bengtsson (Stockholm), August 19, 2009 B. Karlsson (Uppsala) anounced a 2-d non-affine family of complex Hadamard matrices,k 6 (x, y). Apparently a new one i.e. not equivalent to the previously known families :) An attempt for a conclusion Many new families discoverd recently suggest that we know only a small fraction (perhaps of measure zero) of the landscape of all inequivalent n = 6 complex Hadamard matrices! Conjecture: Bengtsson et al. J.Math-Phys All known n = 6 complex Hadamard matrices (besides the isolated matrix S 6 of Tao) belong to an (yet unknown) 4-diemesional family of complex Hadamard matrices. KŻ (IF UJ/CFT PAN ) Complex Hadamard matrices August 21, / 26

23 Defect of a unitary matrix d(u) a) - Take matrix U to the dephased form b) - Multiply matrix elements by arbitrary phase factors, and expand to the first orders in the angles: U ij U ij e iφ ij U ij (1 + iφ ij ), 2 i, j n. c) - Solve the unitarity equations to first order in the angles φ ij. The number of free parameters in the solution of this linear equations is called defect of the matrix U (Tadej & Życzkowski 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 have d(h) = 4 with a single exception [d(s 6 ) = 0, since Tao matrix is isolated]. KŻ (IF UJ/CFT PAN ) Complex Hadamard matrices August 21, / 26

24 The Defect for the Fourier matrix F n Theorem: (Tadej, S lomczyński, Życzkowski, 2008) Factorize into power of primes the number n = m j=1 p j k j where p 1 > p 2 >... > p m. Then the defect of the Fourier matrix F n of size n reads m ( d(f n ) = n 1 + kj k j ) (14) p j Special cases j=1 a) n = p 1 is prime, so m = 1, k 1 = 1, so d(f p ) = p(2 1/p 2) + 1 = 0 hence F p is isolated. b) product of primes, n = pq, so d = 2(p 1)(q 1), [e.g. d(f 6 ) = 4 gives the upper bound of the dimension of the orbit stemming from F 6.] c) power of primes, n = p k, so d(f p k) = p k 1 [(p 1)k p] + 1. In this case we know the family of (generalized) Fourier matrices of this dimension! KŻ (IF UJ/CFT PAN ) Complex Hadamard matrices August 21, / 26

25 A more general set-up Unistochastic matrices Let B be a bistochastic matrix of order n, so that i B ij = j B ij = 1 and B ij 0. B is called unistochastic if there exist a unitary U such that B ij = U ij 2 what 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 = 3 Au-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. Set B ij = 1/n. Then the problem of classification of all preimages of B reduces to the search for all complex Hadamard matrices of size n. KŻ (IF UJ/CFT PAN ) Complex Hadamard matrices August 21, / 26

26 Concluding remarks 1 Complex Hadamard matrices (CHM) form a natural generalization of (real) Hadamard matrices. They are useful in theoretical physics and are related to several problems in pure mathematics. 2 CHM do exist for any matrix size n, (Fourier matrix F n ). However their classification is completed only for n = 2, 3, 4, 5. 3 For n = 6 three 2-dim orbits of CHM are known. It is conjectured that they are embedded into a single 4-d orbit but this problem remains open. 4 Several CHM for n = are listed in a Catalog of CHM, see Tadej & Życzkowski OSID, 13, (2006) and its updated online version at karol/hadamard 5 The defect of a Hadamard (unitary) matrix H gives the upper bound of the dimensionality of the orbit stemming from H. Entire orbit stemming form F n is known only if n = p k. KŻ (IF UJ/CFT PAN ) Complex Hadamard matrices August 21, / 26

Structured Hadamard matrices and quantum information

Structured Hadamard matrices and quantum information Karol Życzkowski in collaboration with Adam G asiorowski, Grzegorz Rajchel (Warsaw) Dardo Goyeneche (Concepcion/ Cracow/ Gdansk) Daniel Alsina, José