Under certain conditions the evolution of a quantum system interacting with its environment can be described by a quantum dynamical semigroup and shown to satisfy a Lindblad master equation

\[\begin{equation}\label{eq:lindblad}\tag{1} \dot{\rho}(t)=-i[H,\rho]+\cL_D\rho(t) \end{equation}\]

where $\rho(t)$ is positive unit-trace operator on the system’s Hilbert space $\cH$, and \(\cL_D\rho(t)=\sum_d \cD[V_d]\rho(t)\), where $V_d$ are operators on $\cH$ and

\[\cD[V_d]\rho(t):=V_d\rho V_d^\dagger -\frac{1}{2}\{V_d^\dagger V_d,\rho\}.\]

We consider $\cH\simeq \CC^N$.

Bloch Representation

From a mathematical point of view the Lindblad master equation is a complex matrix differential equation. To use dynamical systems tools to study its stationary solutions and the stability, it is desirable to find a real representation for $\ref{eq:lindblad}$ by choosing an orthonormal basis $\mathbb \sigma=\{\sigma_k\}_{k=1}^{N^2}$ for all Hermitian matrices on $\cH$.

For $k=r+(s-1)N$ and $1\leq r<s\leq N$, the generalized Pauli matrices is defined by $\sigma_k=\lambda_{rs}$ where

\[\begin{align} & \lambda_{rs}=\frac{1}{\sqrt{2}}(\ket{r}\bra{s}+\ket{s}\bra{r})\\ & \lambda_{sr}=\frac{1}{\sqrt{2}}(-i\ket{r}\bra{s}+i\ket{s}\bra{r})\\ & \lambda_{rr}=\frac{1}{\sqrt{r^2+r}}(\sum_{k=1}^r \ket{k}\bra{k}-r\ket{r+1}\bra{r+1}) \end{align}\]

and $\sigma_{N^2}=\frac{1}{\sqrt{N}}\II$.

The state of the systen $\rho$ can then be represented as a real vector $\br=(r_k)\in\RR^{N^2}$ of coordinates with respect to this basis \(\{\sigma_k\}\),

\[\rho=\sum_{k=1}^{N^2}r_k\sigma_k,\]

where $r_k$ can be calculated by $\tr(\rho\sigma_k)$.

Thus the Lindblad dynamics $\ref{eq:lindblad}$ can be rewritten as a real systems of differential equations:

\[\dot{\br}=(\bL+\sum_d \bD^{(d)})\br,\]

where $\bL,\bD^{(d)}$ are real $N^2\times N^2$ matrices with entries

\[\begin{align} &L_{mn}=\tr(iH[\sigma_m,\sigma_n])\\ &D^{(d)}_{mn}=\tr(V_d^\dagger\sigma_m V_d\sigma_n)-\frac{1}{2}\tr(V_d^\dagger V_d\{\sigma_m,\sigma_n\}) \end{align}\]

Characterization of the Stationary States

Let

\[\mathfrak{E}_{ss}=\{\rho|\dot{\rho}=\cL(\rho)=0\}\]

be the set of steady states for the dynamic given by Equation $\ref{eq:lindblad}$. It is convex as Equation $\ref{eq:lindblad}$ is linear.

We can show that $\mathfrak{E}_{ss}$ is always non-empty using Brouwer’s Fixed Point Theorem.

As any convex set is the convex hull of its extremal points, next we want to characterize the extremal points of $\mathfrak{E}_{ss}$.

We record the following lemma first: if $\rho_s=s\rho_0+(1-s)\rho_1$ is a convex combination of the positive operators $\rho_0,\rho_1$ with $0<s<1$, then the support of $\rho_0$ and $\rho_1$ is contained in the support of $\rho_s$. Further, $\text{rank}(\rho_s)$ is constant for all $0<s<1$.

  • The support of $\rho$ is defined by $$\text{supp}(\rho)^\perp:=\{v\in\cH v^\dagger\rho v=0\}\(. As $\rho$ is positive, we have\)\{v v^\dagger\rho v=0\}=\{v\in\cH \rho v=0\}=\ker \rho$$, so $\text{supp}(\rho)=\text{im}\rho$.
  • For $v\in \text{supp}(\rho_s)^\perp$, $0=v^\dagger\rho_s v = s v^\dagger\rho_0 v + (1-s)v^\dagger\rho_1 v$. Thus, $v^\dagger\rho_0 v=v^\dagger\rho_1 v=0$, so $\text{supp}(\rho_0)\subset\text{supp}(\rho_s)$ and $\text{supp}(\rho_1)\subset\text{supp}(\rho_s)$.
  • For $s\neq t$ and $s,t\in (0,1)$, $\rho_s$ can be written as a convex combination of $\rho_t$ and either $\rho_0$ or $\rho_1$, and vice versa. So $\text{supp}(\rho_s)=\text{supp}(\rho_t)$. Notice that $\text{rank}(\rho_s)=\dim\text{im}\rho$, we yield the desired result.

From this, we can see that the steady state of $\mathfrak{E}_{ss}$ is extremal if and only if it is the unique steady state in its support.

  • Let $\rho_s$ be the unique steady state in its support. Suppose it is not an extremal point, which means that there exist $\rho_1$ and $\rho_2$ such that $\rho_s=s\rho_1+(1-s)\rho_2$ with $0<1<s$. From the above lemma, $\rho_1$ and $\rho_2$ also lie in $\supp

Uniqueness Implies Asymptotic Stability

A steady state of ? is attractive if and only if it is unique.

Reference: Stabilizing open quantum systems by Markovian reservoir engineering