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1Faculty of Mathematics and Information Science, Warsaw University of Technology, Plac Politechniki 1, 00-661 Warsaw, Poland,E-mail: [email protected]

^{2}Faculty of Mathematics and Natural Sciences, Cardinal Stefan Wyszynski University,
ul. Dewajtis 5, 01-815 Warsaw, Poland,
E-mail: [email protected]

- *Corresponding Author:
- Alicja SMOKTUNOWICZ

Faculty of Mathematics and Information Science, Warsaw University of Technology,

Plac Politechniki 1, 00-661 Warsaw, Poland

E-mail: [email protected]

**Received date:** December 16, 2007; **Revised date:** March 07, 2008

**Visit for more related articles at** Journal of Generalized Lie Theory and Applications

The problem of reconstructing the phases of a unitary matrix with prescribed moduli is of a broad interest to people working in many applications, e.g in the circuit theory, phase shift analysis, multichannel scattering, computer science (e.g in the theory of error correcting codes, design theory). We propose efficient algorithms for computing Hermitian unitary matrices for given symmetric bistochastic matrices A(n × n) for n = 3 and n = 4. We mention also some results for matrices of arbitrary size n.

We will study the set of unistochastic matrices which is a subset of the set of bistochastic matrices. We say that a matrix is bistochastic (doubly stochastic) if all its entries are nonnegative real numbers and all its row sums and column sums are equal to 1. A unistochastic matrix is a bistochastic matrix whose entries are the squares of the absolute values of the entries of some unitary matrix U. We recall that a matrix is Hermitian if B = B*, i.e. for i, j = 1, . . . , n. A matrix is unitary if A matrixis orthogonal if We say that a symmetric bistochastic matrix is H-unistochastic if there exists a Hermitian unitary matrix such thatIf U is real (so U is orthogonal) then A is called H-orthostochastic.

Perhaps the van der Waerden matrix (W_{n}) is the most famous unistochastic matrix. Its
elements are all equal to . For example, for n = 4 there exists an orthogonal preimage U (2U
is called an Hadamard matrix):

so W_{4} is even orthostochastic (and also H- orthostochastic). However, it is easy to verify that
W_{3} is not H-unistochastic!

Hadamard’s Conjecture (still open!) says that for n > 2 the Hadamard matrices exist when n = 4k and only for such n, see W.Tadej et al. [5] for explicit examples of the Hadamard matrices.

We consider the following research problems.

• I. Given a bistochastic matrix check if there exists a unitary matrix such that (so A is unistochastic).

• II. Given a symmetric bistochastic matrix check if there exists a Hermitian unitary matrix such that (so A is H-unistochastic).

For n = 2 every bistochastic matrix A is symmetric and is orthostochastic (U can be chosen to be orthogonal). We have

Given a 3×3 bistochastic matrix A it is easy to check whether it is unistochastic or not (see e.g [2]). We get

Therefore the problem is to form a triangle from 3 line segments of given lengths Then A is unistochastic if and only if the chain-link conditions are fulfilled: In this case

Some methods for constructing unitary preimages to 3×3 bistochastic matrices are discussed in [3].

However, for a given 4×4 bistochastic matrix A it is not easy to check whether it is unistochastic or not! There are only partial results, so it is reasonable to try to develop efficient algorithms to check whether a given bistochastic matrix A(n × n) is unistochastic or not. We focus our attention only on symmetric bistochastic matrices and their Hermitian unitary preimages.

For simplicity define a matrix

Notice that without loss of generality we can seek a Hermitian unitary matrix U(n × n) in the dephased form, i.e. such that the first row and the first column of U are the same as the first row and the first column of M. It is obvious because if U is not dephased, we can find a unitary diagonal matrix D such that the matrix satisfies these conditions and it is still Hermitian.

Given a 3×3 symmetric bistochastic matrix A it is easy to check whether it is H-unistochastic or not.

Notice that we can assume that the diagonal elements of A are ordered in such a way that for some permutation {p1, p2, p3} of {1, 2, 3}, then we can permute rows and columns of A. Define a permutation matrix where I = [e1, e2, e3]. Then has the desired property. Note also that A is H-unistochastic iff is H-unistochastic. That is, U is a unitary preimage for A iff PTUP is a unitary preimage for .

As it was said above, we can assume that a Hermitian unitary U has the dephased form

**Theorem 2.1.** Let A(3 × 3) be a symmetric bistochastic matrix, for all i, j and Then A is H-unistochastic if and only if the following matrix

where s = −1 or s = 1, is orthogonal.

Now we consider the case n = 4. Assume that all the elements of a symmetric bistochastic matrix A are nonzero. We show that our problem can be reduced to the linear system of equations.

Write A(4 × 4) and U as follows

where and Here is assumed in order to avoid trivial cases.

By the orthogonality of the first column of U and the columns 2, 3, 4 we obtain

We can assume that the signs sk are prescribed, in an algorithm we have to check all the combinations of signs (±1).

Let Then can be computed as a unique solution of the following linear system of equation Bx = f, Bx = f, where x = [x1, x2, x3]^{T} and f = [f2, f3, f4]^{T} , where fk = −ak(m1 + mk(sk)) for k = 2, 3, 4. Here

Then det(B) = −2 (a2a3a4)(b3b4c4) 6= 0, so there exists a unique solution x of the linear system Bx = f. We can compute it as follows x = B−1f, where

Now it is easy to compute *y _{k}*. We should check the conditions: Then
we can compute

Finally, we have to verify the orthogonality of the computed matrix U.

We have only some partial results for arbitrary size n. Notice that all the eigenvalues of a Hermitian unitary matrix are real and equal to −1 or 1. There exists a unitary matrix such that U = QDQ*, Q*Q = I, where D = diag(1, 1, . . . , 1,−1,−1, . . . ,−1).

If we impose an additional condition on U, a Hermitian unitary preimage of A to be found, namely that U = QDQ* with Q*Q = I and D = diag(1, 1, . . . , 1,−1). Now it is not difficult to solve our problem! Notice that writing D = I − 2 diag(0, . . . , 0, 1) = I − 2 we obtainso U is a reflection (Householder transformation).

Let Then whereWe assume that a1,1 ≠ 1 (for otherwise the problem reduces to the case (n − 1) × (n − 1)). Then we can
choose z_{1} being real and positive because U does not depend on scaling of z (if z = αu with u*u = 1 and |α| = 1, then U = I − 2 uu*). Moreover, if D is a unitary diagonal matrix then
DUD* = I −2 (Du)(Du)* is also a Householder matrix, so we can search for U in the dephased
form. Then the desired Householder matrix U is real and must have the following form to have
the correct moduli in the first row of U

Notice that

so U is orthogonal The only thing to do is to compute U and check the condition the orthogonality of the columns of U we propose reorthogonalization. We apply QR decomposition to U. To compute QR decomposition we can use the Householder or Givens methods or special versions of Gram- Schmidt orthogonalization methods (see eg. [6]). Here is a code for MATLAB using the function qr (the Householder method):

[Q,R]=qr(U); M=A.^(1/2); Z=Q./abs(Q); U=M.*Z; I=eye(n); error_U=norm(I-U’*U)); error_A=norm(A-abs(U).*abs(U));

The numerical tests in MATLAB confirm the advantage of the proposed algorithms.

- AubersonG, MartinA, MennessierG (1991)On the reconstruction of a unitary matrix from its moduli. Comm Math Phys140: 523–542.
- BengtssonI, EricssonA, Ku ́sM, TadejW, ZyczkowskiK (2005)Birkhoff’spolytope and unis-tochastic matrices,N= 3 and N= 4. Comm Math Phys259: 307–324.
- Dit ̧ˇa P(2006) Separation of unistochastic matrices from the double stochastic ones. Recovery of a3×3unitary matrix from experimental data. J Math Phys47: 1–24.
- TadejW(2006) Permutation equivalence classes of Kronecker products of unitary Fourier matrices. LinearAlgebraAppl418: 719–736.
- TadejW, ZyczkowskiK (2006) A concise guide to complex Hadamard matrices. Open SystInf Dyn13: 133–177.
- SmoktunowiczA, BarlowJL, LangouJ (2006) A note on the error analysis of Classical Gram-Schmidt. Numer Math 105: 299–313.

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