Existence conjecture for quantum Schrödinger systems

Let ${\mathcal E}_{0:T}^\dagger$ be a positivity-improving Kraus map, meaning a Kraus map satisfying strict positivity, and let $\rho_0$ and $\rho_T$ be density matrices. Let ${\mathfrak H}_{++}$ denote the positive-definite observables. Existence conjecture for quantum Schrödinger systems. There exist observables $\phi_0$, $\phi_T$, $\hat{\phi}_0$, and $\hat{\phi}_T$ in ${\mathfrak H}_{++}$ such that

$${\mathcal E}_{0:T}(\phi_T)=\phi_0,\qquad {\mathcal E}_{0:T}^\dagger(\hat{\phi}_0)=\hat{\phi}_T.$$

There also exist operators ${\chi}_0$ and ${\chi}_T$ satisfying

$$\rho_0={\chi}_0\hat{\phi}_0{\chi}_0^\dagger,\qquad \rho_T={\chi}_T\hat{\phi}_T{\chi}_T^\dagger,$$

and

$$\phi_0={\chi}_0^\dagger{\chi}_0,\qquad \phi_T={\chi}_T^\dagger{\chi}_T.$$

In particular, ${\chi}_0$ and ${\chi}_T$ can be taken to be Hermitian, with ${\chi}_i=(\phi_i)^{1/2}$ for $i\in\{0,T\}$. This is the natural quantum generalization of the classical Schrödinger-system existence result; the source states that proving this assertion remains elusive. Uniqueness is discussed as part of the same unresolved existence-and-uniqueness problem.