Approximate Reduced Lindblad Dynamics via Algebraic and Adiabatic Methods

Abstract

We present an algebraic framework for approximate model reduction of Markovian open quantum dynamics that guarantees complete positivity and trace preservation by construction. First, we show that projecting a Lindblad generator on its center manifold -- the space spanned by eigenoperators with purely imaginary eigenvalue -- yields an asymptotically exact reduced quantum dynamical semigroup whose dynamics is unitary, with exponentially decaying transient error controlled by the generator's spectral gap. Second, for analytic perturbations of a Lindblad generator with a tractable center manifold, we propose a perturbative reduction that keeps the reduced space fixed at the unperturbed center manifold. The resulting generator is shown to remain a valid Lindbladian for arbitrary perturbation strengths, and explicit finite-time error bounds, that quantify leakage from the unperturbed center sector, are provided. We further clarify the connection to adiabatic elimination methods, by both showing how the algebraic reduction can be directly related to a first-order adiabatic-elimination and by providing sufficient conditions under which the latter method can be applied while preserving complete positivity. We showcase the usefulness of our techniques in dissipative many-body quantum systems exhibiting non-stationary long-time dynamics.

Figures (4)

• Figure 1: Left: Expectation values of the full and logical observables $X^{\otimes N}$ and $X_L$ versus time, for a system of $N=5$ spins. The dashed line represents the theoretical solution of Eq. \ref{['eq:theo_exp_val']}. Right: Exponential convergence of the trace-norm distance between $\rho(t)$ and $\mathcal{J}(\check{\rho}(t))$. The continuous line represents the norm computed numerically in a simulation, the dotted line represents the exponential bound given by $\Gamma e^{-\Delta_\mathcal{L} t}$, with $\Gamma =1$ and $\Delta_\mathcal{L} =0.72$ and the dashed line shows how $\Delta_\mathcal{L}$ correctly sets the asymptotic convergence rate. Here the initial condition was chosen randomly while the remaining parameters are $\omega= 1.2$, $A_{xy}=2$, $A_x = 4.6$, and $\gamma=1.2$.
• Figure 2: Left: Trace-norm error between the exact and the reduced evolution obtained through algebraic MR and AE techniques for $\mathcal{R}_{(0)}\mathcal{J}_{(1)}$ randomly chosen and randomly chosen within the commutant $\{\hat{\mathcal{L}}_{(0)}\}'$, respectively, with $\varepsilon=0.1$. In the former case, $\mathcal{R}_{(0)}\mathcal{J}_{(1)}$ was constructed as a matrix with elements sampled from a unitary distribution $\mathcal{U}([0,1])$, while in the latter case, it was constructed by picking random uniform values for the diagonal in the same base as $\hat{\mathcal{L}}_{(0)}$. The shaded areas represent the region included in one standard deviation from the average (darker line) of 30 runs of the simulation. Right: synchronization decay as a function of $\varepsilon$. The continuous line represents the evolution of the original model, i.e. $\left< X^{\otimes N}(t) \right>$ while the dotted line represents the expectation value predicted by reduced model, i.e. $\left< \mathcal{J}^\dag[X^{\otimes N}(t)] \right> = {\rm tr}\{\mathcal{J}^\dag[X^{\otimes N}] \check{\rho}(t)\}$. The curves have been shifted vertically by $\varepsilon/2$ for stylistic reasons. In both figures, $\rho_0=\left\vert + \middle>\middle< + \right\vert^{\otimes N}$, all the other parameters are as in Fig. \ref{['fig:red_xxz_crystal']}.
• Figure 3: Hilbert-Schmidt distance between the reduced model $\mathcal{J}\check{\rho}(t)$ obtained through algebraic MR and the exact evolution $\rho(t)$ (left) and the projected exact evolution $\mathcal{P}\rho(t)$ (right). The colors represent different values of $\varepsilon$ from $0.1$ to $0.9$ (from dark to light), with increments of $0.2$. For each color, the continuous line represents the Hilbert-Schmidt distance computed via simulation, whereas the dashed line represents the error bound found in Eq. \ref{['eqn:bound_1']} (left) and \ref{['eqn:bound_2']} (right). The insets show the dependence of the same distance with respect to $\varepsilon$ for different values of $\omega t$ from $0.3$ to $2.7$ (from dark to light), with increments of $0.6$. Here, $\rho_0$ is picked at random from $\mathscr{C}_0$ and all the other parameters are identical to those in Fig. \ref{['fig:gauge_choice']}. The super-operator norm $\left|\left| \mathcal{L}_{(1)} \right|\right|_\text{sop}$ used to compute the bounds is the one induced by the Hilbert-Schmidt norm $\left|\left| \cdot \right|\right|_{HS}$, i.e. the largest eigenvalue of a matrix representation of $\mathcal{L}_{(1)}$.
• Figure 4: Simulation results for the perturbed purely dissipative spin chain, with $N=5$, random initial condition $\rho_0$, and a random set of parameters $\mu_{u,j},$$u=x,+,-$. Left: Probability of measuring $\left\vert 0\dots0 \middle>\middle< 0\dots0 \right\vert$ versus time for different $\varepsilon$. The continuous line represents the full model, $\left< 0 \right|\rho(t)\left| 0 \right>$, whereas the empty dots represent the reduced model, $\left< 0 \right|{\bf p}(t)$. Right: Convergence of the trace-norm distance between $\rho(t)$ and $\mathcal{J} {\bf p}(t)$ for different $\varepsilon$. The dashed lines represent exponential convergence, according to $\Gamma e^{- \Delta_{\mathcal{L}_\varepsilon}t}$, for different $\varepsilon$.

Theorems & Definitions (36)

• Definition 1
• Lemma 1: Exponential convergence to $\mathscr{C}$
• Proposition 1: CPTP projection onto $\mathscr{C}$
• Proposition 2: Fixed-point structure
• Theorem 1: Reduced Lindbladian on algebras
• Remark 1: Beyond exact MR
• Theorem 2: Asymptotically exact reduction
• Corollary 1: Approximate Lindblad reduction
• Theorem 3: Error bounds
• Remark 2: Emerging unitarity