Lindbladian

System described by a Master equation of the form $\frac{d}{dt}{\rho }_t=\mathbb{L}[\rho ]=-\frac{i}{\hslash }[\hat{H},\hat{\rho }]+{\sum }_{\alpha =1}^{N^2-1}{L}_{\alpha }({\hat{K}}_{\alpha }\rho {\hat{K}}_{\alpha }^{†}-\frac{1}{2}\{{\hat{K}}_{\alpha }^{†}{\hat{K}}_{\alpha },\hat{\rho }\})$, where

• $\mathbb{L}$ is the Quantum Dynamical Semigroup generator
• $L_{\alpha }:=\sqrt{C_{\alpha }}{\hat{K}}_{\alpha }$ are the Lindblad Operators.
• $C_{\alpha }$ are positive operators. This follows from the Quantum Dynamical Semigroup generated by $\mathbb{L}$ being Completely Positive. I need to write down this proof. Help me ❓

The Lindbladian generalises the Schrodinger Equation to an open quantum system $S$. In open quantum systems the noise introduced by the environment changes the dynamics of the system, requiring the addition of “adjustment” terms. These adjustments make Unitary Operators unfit to describe a open quantum system, requiring us to use more general Operators.

Deriving the Master Equation

We start with a generic Stochastic Differential Equation of the form $d|\psi ⟩=\left[-\frac{i}{\hslash }Hdt+i\sqrt{\lambda }\hat{q}d{W}_t-\frac{1}{2}\lambda q^{w}dt\right]|{\psi }_t⟩$. To show unitarity, we apply the Converting between Ito vs Stratonovich Calculus methodology to get the equivalent Stratonovich Calculus integral $d|{\psi }_t⟩=-\frac{1}{\hslash }Hdt+i\sqrt{\lambda }\hat{q}\circ d{W}_t$, which is a Unitary Operator. This is useful to show that the result is unitary, however most calculations are more practical under the Ito Calculus framework. This is due to independence of $W_t$ and $|{\psi }_t⟩$: Ito Calculus allows us to have $\mathbb{E}[W_t|{\psi }_t⟩]=\mathbb{E}[W_t]\mathbb{E}[|{\psi }_t⟩]$, simplifying a significant number of calculations. The Stratonovich Calculus does not allow us to to assume this independence.