[Math] Another $2 \times 2$ matrix question

at.algebraic-topologylinear algebramatrices

This question is similar to this previous one but I think it is harder.

Let $X$, $Y$, $Z$, and $W$ be $2\times 2$ Hermitian matrices. Can we always find $\theta,\phi \in [0,\pi/2]$ and $2\times 2$ unitaries $U$ and $V$ such that
$$SUXU^*S + CVYV^*C$$ and $$SUZU^*S + CVWV^*C,$$
are both scalar multiples of the identity matrix,
where $S = {\rm diag}(\sin \theta, \sin \phi)$ and $C = {\rm diag}(\cos \theta, \cos\phi)$?

I'm guessing this is true. Maybe there is a homotopy reason as in the previous question. I think an answer in either direction would likely lead to a solution to this other previous question.

Edit: another way to ask this is, given $X$, $Y$, $Z$, and $W$, can we find $2\times 2$ matrices $A$ and $B$ such that $$AA^* + BB^*$$ $$AXA^* + BYB^*$$ $$AZA^* + BWB^*$$ are all scalar multiples of the identity (and the first one is not zero).

Edit 2: another reformulation. Let $\alpha$ and $\beta$ be the column vectors of $A^*$ and $\gamma$ and $\delta$ the column vectors of $B^*$ in the first edit. Then by looking at matrix entries of the expressions given there we see that the conditions become $$\langle X\alpha, \alpha\rangle + \langle Y\gamma, \gamma\rangle = \langle X\beta, \beta\rangle + \langle Y\delta, \delta\rangle$$
and $$\langle X\alpha,\beta\rangle + \langle Y\gamma,\delta\rangle = 0,$$
plus similar conditions with $Z, W$ or $I_2,I_2$ in place of $X,Y$. The problem is to find $\alpha,\beta,\gamma,\delta \in \mathbb{C}^2$, not all zero, satisfying these equations.

Best Answer

Too long for a comment, maybe someone with a respectable CAS and the skill to use it can try. MATLAB's symbolic toolbox returns "Warning: Cannot find explicit solution" from the obvious/naive attempt

%% Parameters
syms x0 x1 x2 y0 y1 y2 z0 z1 z2 w0 w1 w2;   % all assumed complex
X = [x1,x0;conj(x0),x2];    % Hermitian
Y = [y1,y0;conj(y0),y2];    % Hermitian
Z = [z1,z0;conj(z0),z2];    % Hermitian
W = [w1,w0;conj(w0),w2];    % Hermitian
%% Variables
syms a11 a12 a21 a22 b11 b12 b21 b22 L M;   % all assumed complex
A = [a11,a12;a21,a22];
B = [b11,b12;b21,b22];
%% Equations
C = A*A'+B*B'-eye(2);
D = A*X*A'+B*Y*B'-L*eye(2);
E = A*Z*A'+B*W*B'-M*eye(2);
eqns = [C(1,1) == 0, C(1,2) == 0, C(2,1) == 0, C(2,2) == 0,...
    D(1,1) == 0, D(1,2) == 0, D(2,1) == 0, D(2,2) == 0,...
    E(1,1) == 0, E(1,2) == 0, E(2,1) == 0, E(2,2) == 0];
%% Solution
S = solve(eqns,[a11 a12 a21 a22 b11 b12 b21 b22 L M])
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