Lindbladian

System described by a Master equation of the form $\frac{d}{dt}{\rho }_t=\mathbb{L}[\rho ]=-\frac{i}{\hslash }[\hat{H},\rho ]+{\sum }_{i=1,j=1}^d{C}_{ij}{F}_i{\rho }_t{F}_j^{†}-\frac{1}{2}\{{\sum }_{i=1,j=1}^d{C}_{ij}{F}_j^{†}{F}_i,{\rho }_t\}$, where

• $\mathbb{L}[{\rho }_t]:=\frac{d}{dt}{\rho }_t$ is the Quantum Dynamical Semigroup generator
• $C_{ij}\in \mathbb{C}$ forms an Hermitian Operator, i.e. $C^{†}=C$, called the Kossohowski Matrix.
• $C_{dd}=0$
• $F_i\in M_d\mathbb{C}$ are Linear Operators.
• $\mathrm{Tr}[F_j^{†}{F}_i]={\delta }_{ij}$, i.e. $\{F_i{\}}_i$ satisfy the Hilbert Schmidt orthogonality condition.
• $F_d=\frac{1}{\sqrt{d}}{1}_d$, where $1_d$ is the identity map on a $d$-dimensional space.

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.