Reconstructing a (unitary) matrix from the determinant of its sub-matrices

Question

I want to find the unitary $N \times N$ matrix U from the following data. Let $M$ be an integer $(1< M<N-1)$ and let $\mathcal S$ be the space of all the possible subsets of $\{1,2,\dots, N\}$ with exactly $M$ elements.

Let $U(s,s')$ be the $M\times M$ submatrix of U whose rows are specified by the elements of $s\in\mathcal S$ and whose columns are specified by the elements of $s'\in\mathcal S$.

My data is a set of $I(s,s')\in\mathbb Q$ such that $$ \det U(s,s') = e^{i \pi I(s,s')}.$$

Can I find the full matrix $U$? I would be interested also in understand why this problem might not have a solution. However, if the solution exists (for certain values of $M,N$), than I would like to find the algorithm to obtain $U$.

Answer 1

$\newcommand{\bC}{\mathbb{C}}$ View the unitary matrix as a unitary operator $U:\bC^N\to\bC^N$. For any $1\leq M\leq N$ it canonically induces a unitary map

$$\Lambda^MU: \Lambda^M\bC^N\to\Lambda^M\bC^N, $$

where $\Lambda^M$ denotes the $M$-th exterior power of a vector space. Your problem can be rephrased as follows: how much of $U$ can one recover given that we have complete knowledge of $\Lambda^M U$.

Immediately one ses that we can recover quite a bit. If $\lambda_1,\dotsc,\lambda_N$ are the eigenvalues of $U$ then the eigenvalues of $\Lambda^M U$ are

$$ \lambda(S)=\prod_{k\in S}\lambda_k, $$

where $S$ runs through the subsets of $\{1,\dotsc, N\}$ of cardinality $M$. Observe that if $M< N$, then the collection of eigenvalues

$$\{ \lambda(S);\;\;|S|=M\} $$

determines the collection of ratios

$$\frac{\lambda_i}{\lambda_j},\;\; 1\leq i<j\leq N. $$

With this extra knowledge we can compute the powers $\lambda_1^M,\dotsc, \lambda_N^M$, and thus the eigenvalues of $U$ up to an overall multiplicative ambiguity $\rho$ which is an $M$-th root of $1$. You cannot do better than this because

$$\Lambda^M U=\Lambda^M(\rho U). $$