Motivated by the problem of constructing the \(S\)-matrix for a scattering process from experimental data, the authors examine the following matrix question: If an \(n\times n\) matrix is known to be unitary, to what extent is it determined by the values of the moduli of its elements?
One can begin by doing a parameter count: An \(n\times n\) matrix depends on \(n^ 2\) real parameters, and there are \(n^ 2\) moduli. The latter are, however, not independent. In fact, the \(n\) rows and the \(n\) columns all have unit length. These \(2n\) relations can be reduced to \(2n-1\) independent relations by noting that the sum of all row lengths equals the sum of all column lengths.
Thus, in the original question, there appears to be a shortfall of \(2n-1\) real parameters. Fortunately this is not a genuine difficulty. For, taking out the \(n\) phases of the first row and the remaining \(n-1\) phases of the first column, the parameter counts balance and there remains a well-posed problem.
In particular, one might expect that the reconstruction problem, modulo the \(2n-1\) trivial phase ambiguities, has only a finite number of solutions. This is known to be true for \(n=2\) and \(n=3\). Using contraction mapping techniques, these results are extended to general \(n\) and for symmetric matrices which are not too far from the unit matrix.
However, for non-symmetric matrices and \(n>3\), a continuous set of solutions appear for certain configurations of the moduli. This implies, in particular, that the unitary group \(U(n)\) with \(n>3\) cannot be fully parametrized, even locally, by the set of \((n-1)^ 2\) independent moduli and the \(2n-1\) trivial phases.
The case \(n=4\), with one non-trivial free phase, is examined exhaustively, and there are further comments on the nearly-symmetric case.