Consider a system of \(N\) \(q\)-ary variables \(x=x_1,\ldots,x_N\) evolving according to a stochastic process governed by a master equation \(\begin{equation} \frac{d}{dt}p(x,t)=\sum_{x'}\left[p(x',t)w(x'\to x) - p(x,t)w(x\to x')\right] \end{equation}\) where \(w(x\to x')\) is the probability of transitioning from state \(x\) to state \(x'\).
As explained in
Now, the time evolution for \(\ket{p(t)}\), is given by \(\begin{align} \frac{d}{dt}\ket{p(t)} &= \sum_x \frac{d}{dt}p(x,t)\ket{x}\\ &= \sum_{x,x'} \left[p(x',t)w(x'\to x) - p(x,t)w(x\to x')\right]\ket{x}\\ &= \sum_{x,x'}\left[w(x'\to x)\braket{x'|p(t)} - w(x\to x')\braket{x|p(t)}\right]\ket{x}\\ &= \sum_{x,x'}\left[w(x'\to x)\ket{x}\bra{x'} - w(x\to x')\ket{x}\bra{x}\right]\ket{p(t)}\\ &= \sum_{x,x'}\left[w(x\to x')\ket{x'}\bra{x} - w(x\to x')\ket{x}\bra{x}\right]\ket{p(t)}\\ &= \sum_{x,x'}w(x\to x')\left(\ket{x'}\bra{x} - \ket{x}\bra{x}\right)\ket{p(t)}\\ \end{align}\)
where to go from \((8)\) to \((9)\) we renamed \(x\to x'\), \(x'\to x\) in the first term. Finally \(\begin{align} \frac{d}{dt}\ket{p(t)} &= -\mathcal{H} \ket{p(t)} \end{align}\) where we defined \(\begin{equation} \mathcal{H} = \sum_{x,x'}w(x\to x')\left(\ket{x}\bra{x}-\ket{x}\bra{x'}\right) \end{equation}\)
Now \((11)\) is an eigenvalue problem, formally solved by \(\begin{equation} \ket{p(t)} =e^{-t\mathcal H}\ket{p(0)} \end{equation}\) but intractable in general since the size is exponential in \(N\).
At this point, we notice that \((11)\) somehow resembles a Schrodinger equation for the so-called “stochastic Hamiltonian” \(\mathcal{H}\). There are some differences with a Schrodinger equation, namely \(\mathcal H\) is not Hermitian in general, \(p(t)\) is a probability density instead of a wavefunction, there is no imaginary unit. However, it turns out that for a variety of stochastic processes, there exists an equivalent quantum system whose Hamiltonian is precisely \(\mathcal H\). In such cases, any technique for the diagonalization of the quantum Hamiltonian can be transferred directly to the time evolution of $p$. For example, the steady state of the stochastic process coincides with the eigenvector relative to the smallest eigenvalue of \(\mathcal H\). This means that if there exists an equivalent quantum system whose Hamiltonian is \(\mathcal H\), then knowlegde of the ground state of the quantum system directly implies knowledge of the steady state of \(p(x,t)\). Neat!
Even when exact results are not available for such quantum problems, efficient approximation techniques such as the Density Matrix Renormalization Group can be employed.
Consider a system of \(N\) sites and a bunch a particles hopping from site to site. Variable \(x_i\) is \(1\) if site \(i\) is occupied, \(0\) otherwise. The term “exclusion” refers to the impossibility of having more than one particle occupying a site. The only allowed transition is: a particle can hop from an occupied site \(i\) to an empty site \(j\) with probability \(w_{ij}\). More formally, the transitions are \(w(x\to x') = \sum_{i=1}^N w_i(x\to x')\) with \(\begin{align} w_i(x\to x') = \sum_{j=i}^N w_{ij}\delta(x_i,1)\delta(x_j,0)\delta(x'_i,0)\delta(x'_j,1) \prod_{k\neq i,j} \delta(x_k,x'_k) \end{align}\) For this choice of transitions, the Hamiltonian \((10)\) reads \(\begin{align} \mathcal H &= \sum_{i\neq j}\sum_{\substack{x_i,x_j\\x'_i,x'_j}}w_{ij}\delta(x_i,1)\delta(x_j,0)\delta(x'_i,0)\delta(x'_j,1) \left(\ket{x_i x_j}\bra{x_i x_j}-\ket{x_i x_j}\bra{x'_i x'_j}\right)\\ &= \sum_{i\neq j} w_{ij} \left(\ket{1_i0_j}\bra{1_i0_j}-\ket{0_i1_j}\bra{1_i0_j}\right)\\ &= \sum_{i<j} w_{ij} \left(\ket{1_i0_j}\bra{1_i0_j}-\ket{0_i1_j}\bra{1_i0_j}\right) +w_{ji} \left(\ket{1_j0_i}\bra{1_j0_i}-\ket{0_j1_i}\bra{1_j0_i}\right) \\ \end{align}\) If transition rates are symmetric \(w_{ij}=w_{ji}\), \(\begin{align} \mathcal H &= \sum_{i<j} w_{ij} \left(\ket{1_i0_j}\bra{1_i0_j}-\ket{0_i1_j}\bra{1_i0_j}+\ket{1_j0_i}\bra{1_j0_i}-\ket{0_j1_i}\bra{1_j0_i}\right) \\ &= \sum_{i<j} w_{ij} \mathcal T^{(ij)} \end{align}\) On the standard \(4\)-dimensional basis \(B_i\otimes B_j\), with \(B_\alpha=\left\{\begin{pmatrix}1\\0\end{pmatrix}=\ket{1_\alpha}, \begin{pmatrix}0\\1\end{pmatrix}=\ket{0_\alpha}\right\}\), the action of \(\mathcal T^{(ij)}\) is \(\begin{align} \mathcal T^{(ij)} &= \begin{pmatrix} 0 & 0 & 0 & 0\\ 0 & -1 & 1 & 0 \\ 0 & 1 & -1 & 0 \\ 0 & 0 & 0 & 0\\ \end{pmatrix} \end{align}\) An educated eye (not mine!) will notice that \(\begin{align} \mathcal T^{(ij)} &=\sigma^x_i\sigma^x_j+\sigma^y_i\sigma^y_j+\sigma^z_i\sigma^z_j-\mathbb{1}\\ &= \vec{\sigma}_i\cdot\vec{\sigma}_j-\mathbb{1} \end{align}\) where \(\vec{\sigma}_\alpha=(\sigma_\alpha^x, \sigma_\alpha^y, \sigma_\alpha^z)\) are the Pauli matrices . The notation \(\sigma^x_i\sigma^x_j\) means \(\sigma^x_i\) acting on \(\ket{x_i}\), \(\sigma^x_j\) acting on \(\ket{x_j}\). Finally we get \(\begin{align} \mathcal H &= \sum_{i<j} w_{ij} \left(\vec{\sigma}_i\cdot\vec{\sigma}_j-\mathbb{1}\right) \end{align}\) which is the Hamiltonian of a fully-connected, anti-ferromagnetic quantum Heisenberg model.