Iterating a quantum channel related function and producing a unique fixed point.

Let E be a completely positive superoperator mapping d dimensional complex matrices to d dimensional complex matrices.
Let f denote the mapping defined by f(X)=(E(X))^2. The goal is to find a non-zero complex matrix X where X=c f(X). In this visualization, we find the projective complex matrix X iteratively. More specifically, we begin with a random matrix X, and we set X_0=X/tr(X) and we set X_{n+1}=f(X_n)/tr(f(X_n)) for all n.
In Frame n of the animation, we show the spectrum of X_n. To compare how the iteration behaves under different initializations, we run the iteration twice and show two spectra overlayed on top of each other; one spectrum is colored red and the other one is blue. We observe that the function f is Hermitian preserving in the sense that f(adjoint(X))=adjoint(f(X)) for all X, so the iteration of f under two initializations could converge to a matrix Z along with its adjoint, and in this case, the spectra will be mirror images of each other (reflected over the real number line), so in this visualization, I had to reflect one of the spectra over the real number line.
We observe that for this particular completely positive superoperator E, the sequence (X_n)_n does not converge to a positive semidefinite operator or even a normal operator, but rather to an operator that does not seem to follow the circular law at all. For other completely positive superooperators, the sequence (X_n)_n would have easily converged to a positive semidefinite matrix, and the matrices X_n somewhat resemble positive semidefinite matrices for small n, but for larger n, the matrices X_n no longer resemble positive semidefinite matrices.
The following result guarantees the existence of a fixed point. For the quaternionic case, we define the trace to be the real part of the sum of all diagonal entries in the matrix (the quaternionic trace is only well-behaved if we take its real part).
Proposition: Let K denote the field of real numbers, complex numbers, or quaternions. Suppose that E is a superoperator from the set of d by d matrices over K to d by d matrices over K. Suppose that E(P) is positive semidefinite whenever P is positive semidefinite and E(P) is non-zero for each non-zero positive semidefinite P. Then the mapping F defined by F(X)=E(X)^2/Tr(E(X)^2) has a positive semidefinite fixed point.
Proof: Let C denote the collection of all positive semidefinite d by d matrices over X with trace 1. Then C is a convex set, and F maps C to C, so by Brouwer's fixed point theorem, the function F has a fixed point on C. Q.E.D.
The above result does not give us an algorithm for constructing such a fixed point.
Quantum channels always increase the similarity between quantum states for several notions of similarity including the quantum relative entropy and the fidelity between quantum states, and unital quantum channels always increase the amount that a state is mixed under the majorization ordering (see the textbook Quantum Information Theory by John Watrous for proofs). On the other hand, the squaring function tends to magnify differences between positive semidefinite operators and purifies states. Since the function f is the composition of both of these operations, the function f does not tend to purify or mix quantum states, so if everything goes right the function f tends to have a fixed point that is neither overly mixed nor overly pure.
The function f is my own, but this function is just a simple adaptation of the notion of a quantum channel, so others probably have thought of it before. The notion of taking the fixed point of f is also my own. A fixed point X of the function F may be useful in machine learning especially if X is unique (or canonical in some sense) and if X is positive semidefinite. If X is positive semidefinite, then one should interpret X as a 'cluster of dimensions' in d dimensional complex Euclidean space. The uniqueness of X means that the matrix X is likely to be quite interpretable.=
While the function f maps complex matrices to complex matrices, one can also define f to map real matrices to real matrices or even quaternionic matrices to quaterionic matrices.
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I have not met too many complex matrices whose spectrum looks like Oumuamua especially when they are so far from being Hermitian. The circular law states that such spectra should look like disks.