# What are the requirements on conditional unitaries for overcomplete bases?

On way to describe "pure" decoherence (that is, decoherence with respect to a basis that doesn't involve transitions between basis states) between a system $\mathcal{S}$ and an environment $\mathcal{E}$ is with a unitary operator that acts conditional on the state of $\mathcal{S}$: $$U=\sum_i \vert S_i \rangle \langle S_i \vert \otimes U^\mathcal{E}_i.$$ Here, $U^\mathcal{E}_i$ describes the evolution of $\mathcal{E}$ conditional on $\mathcal{S}$ being in state $\vert S_i \rangle$. A CP map on $\mathcal{S}$ constructed by applying $U$ and tracing out $\mathcal{E}$ will leave fixed the diagonal matrix elements of the initial state $\rho_0^\mathcal{S}$ in the basis $\{\vert S_i \rangle\}$, but will in general add phases and decoherence (i.e. suppression of the norm) to off-diagonal elements. Importantly, $U$ is a valid unitary for any choice of unitary $U^\mathcal{E}_i$.

Now suppose I want to construct a unitary that acts like this on an overcomplete basis. I'll specialize to the coherent states $\vert \alpha \rangle$ where $\alpha = x + i p$ is a point in phase space, but the solution for a general basis would be interesting. If I write down $$U= \int \mathrm{d}\alpha \vert \alpha \rangle \langle \alpha \vert \otimes U^\mathcal{E}_\alpha.$$ one can check that this is not a valid unitary for arbitrary $U^\mathcal{E}_\alpha$. (This can be easily seen when $\mathcal{S}$ has only two dimensions using an overcomplete basis of 3 or 4 vectors, and choosing random conditional unitaries.)

Is there a compact way to write down the requirements on the $U^\mathcal{E}_\alpha$ for $U$ to be unitary? Obviously one can expand $U^\dagger U = I$ using the above definition, but I'm unable to transform this to something with a clear interpretation. My intuition is that $U^\mathcal{E}_\alpha$ and $U^\mathcal{E}_\beta$ should be "close" for $\vert \alpha - \beta\vert^2 \ll 1$ since they ought to be unrestricted for $\vert \alpha - \beta\vert^2 \gg 1$, but I can't formalize this in a useful way.

(Incidentally, this is closely related to my previous questions which did not generate much interest. I'm giving this one last shot.)

Edit: In the case where the unitary is taking the environment from a certain initial state $\vert E_0 \rangle$ to a conditional state $\vert E_\alpha \rangle$, one can see that the associated CP map is well behaved (i.e. that $\sum_i K_i^\dagger K_i = I$, where the $K_i$ are the Kraus operators) iff the operator $$\int \mathrm{d}\alpha\mathrm{d}\beta \vert \alpha \rangle \langle \alpha \vert \beta \rangle \langle \beta \vert \cdot f(\alpha,\beta)$$ equal the identity, where $f(\alpha,\beta) = \langle E_\alpha \vert E_\beta \rangle$ is the Gram matrix of inner products of the (normalized) conditional states. A sufficient condition for this is that $f(\alpha,\beta)$ depends only on $\alpha - \beta$ but not $\alpha + \beta$. I believe this is a necessary condition, but I am unable to prove it (or find a counter-example). This condition, along with the fact that $f(\alpha,\beta)$ is a Gram matrix, makes $f(\alpha-\beta)$ a positive-definite function, which I presume is important but don't know how to exploit it.