Write a projector as a limit of unitary matrices multiplication

Question

The idea is to write a projector, for example, $$P= \begin{pmatrix} 1 & 0 \\ 0 & 0\\ \end{pmatrix} $$ As a infinite multiplication of Unitary matrices, i.e:

$$ P= \lim_{n \to \infty} U_1 U_2 ... U_n$$

Where $U_k \in U(2)$

My question is to know if this is possible. This is of speacial interest for the quantum measurement problem, as unitary matrices describe evolution which is reversible, but projectors represent collapse (measurement) that are irreversible. The point is that measurements are done via macroscopic apparatus where there may be lots of unitary evolutions which may produce this projector.

Answer 1

This is not possible because the absolute value of the determinant of unitary matrices is $1$. On the other hand your projector (and every non-trivial projector) has $\det=0$.

The determinant also depends continuously on the matrix elements, which implies that the limit $V$ of matrices $V_n$ that have $|\det V_n|=1$ also has to satisfy $|\det V|=1$. (Note that the product of unitary matrices is also unitary.)