# Equivalence of quantum state diffusion and heterodyne trajectory

According to Breuer-Petruccione, the SDE quantum trajectory evolution for heterodyne detection

$$d\psi=-iH\psi dt-\frac{\gamma}{2}\left(C^\dagger C-\langle C^\dagger \rangle_{\psi} C+\frac{1}{2}\langle C\rangle_{\psi} \langle C^\dagger\rangle_{\psi} \right)\psi dt +\sqrt{\gamma}(C-\langle C\rangle_\psi)\psi dW(t) +\frac{\sqrt{\gamma}}{2}(\langle C\rangle_\psi dW(t)-\langle C^\dagger\rangle_\psi dW^*(t))\psi$$

is equivalent to the SDE of quantum state diffusion (stochastic collapse)

$$d\psi=-iH\psi dt-\gamma\left(\frac{1}{2} C^\dagger C-\langle C^\dagger \rangle_{\psi} C+\frac{1}{2}\langle C\rangle_{\psi} \langle C^\dagger\rangle_{\psi} \right)\psi dt +\sqrt{\gamma}(C-\langle C\rangle_\psi)\psi dW(t)$$

as can be obtained trough a phase transformation $\psi(t)\rightarrow e^{i\phi(t)}\psi(t)$.

My question is: how deep can this correspondence been interpreted:

• Is the correspondence only on the level of the density matrix (i.e. separate unraveling) or can individual QSD samples 'physically' be interpreted as a time- evolution under heterodyne measurement?
• Does the global phase of the ensemble remain the same when time evolves?