Semidefinite programming for understanding limitations of Lindblad equations

TL;DR

This work addresses whether a physically consistent Lindblad master equation can describe a quantum system weakly coupled to baths in a given steady state. The authors formulate the problem as semidefinite programs that minimize metrics capturing leading-order accuracy of populations and coherences, while enforcing CPTP and local conservation laws. They show rigorous no-go results for many parameter regimes, especially in non-equilibrium setups, and identify cases (notably equilibrium with all sites bath-coupled) where a CPTP LE that matches both populations and coherences can exist. The SDP framework provides tight, certificate-worthy conclusions and can output explicit candidate Lindbladians when feasible, offering a robust tool for understanding the limits of Markovian descriptions in open quantum systems.

Abstract

Lindbladian quantum master equations (LEs) are the most popular descriptions for quantum systems weakly coupled to baths. But, recent works have established that in many situations such Markovian descriptions are fundamentally limited: they cannot simultaneously capture populations and coherences even to the leading-order in system-bath couplings. This can cause violation of fundamental properties like thermalization and continuity equations associated with local conservation laws, even when such properties are expected in the actual setting. This begs the question: given a physical situation, how do we know if there exists an LE that describes it to a desired accuracy? Here we show that, for both equilibrium and non-equilibrium steady states (NESS), this question can be succinctly formulated as a semidefinite program (SDP), a convex optimization technique. If a solution to the SDP can be found to a desired accuracy, then an LE description is possible for the chosen setting. If not, no LE description is fundamentally attainable, showing that a consistent Markovian treatment is impossible even at weak system-bath coupling for that particular setting. Considering few qubit isotropic XXZ-type models coupled to multiple baths, we find that in most parameter regimes, LE description giving accurate populations and coherences to leading-order is unattainable, leading to rigorous no-go results. However, in some cases, LE description having correct populations but inaccurate coherences, and satisfying local conservation laws, is possible over some of the parameter regimes. Our work highlights the power of semidefinite programming in the analysis of physically consistent LEs, thereby, in understanding the limits of Markovian descriptions at weak system-bath couplings.

Figures (7)

• Figure 1: Schematic of an arbitrary finite-dimensional system described by Hamiltonian $H_S$ [Eq. \ref{['eq:system_ham']}], parts of which are weakly coupled to left and right thermal baths [Eq. \ref{['eq:bath_ham']}] at inverse temperatures $\beta_L$ and $\beta_R$. The Hilbert space of the system, $\mathcal{H}_S$, is divided into parts $\mathcal{H}_L$ (which directly couples to the Hilbert space of the left bath), $\mathcal{H}_R$ (which directly couples to the Hilbert space of the right bath), and the remaining part $\mathcal{H}_M$.
• Figure 2: Plots for $N_L = N_R = 1,~N_M = 2$ for the isotropic XXZ Hamiltonian [Eq. \ref{['eq:ham']}] keeping $\gamma_\ell = 1~ \forall \ell,~\omega_c=10$ [Eq. \ref{['eq:ohmic_bath']}] and $\beta_R = 10.0$ (blue diamond), $5.0$ (orange circle), $1.0$ (yellow plus), $0.5$ (purple cross). The black dashed horizontal line represents the chosen tolerance $\delta_{\rm tol} = 10^{-6}$. (a) $\tau^{\rm pop}_{\rm opt}$ [Eq. \ref{['eq:diag_optimization']}] versus $\beta_L$ with $g=0.01$. (b) $\tau^{\rm pop,coh}_{\rm opt}$ [Eq. \ref{['eq:coh_optimization']}] versus $\beta_L$ with $g=0.01$. (c) $\tau^{\rm pop}_{\rm opt}$ versus $g$ for $\beta_L = 1.0$. (d) $\tau^{\rm pop,coh}_{\rm opt}$ versus $g$ for $\beta_L = 1.0$. This figure shows that it is impossible to achieve populations correctly up to leading-order (implying incorrect coherences as well) for only single qubits attached to left and right baths as discussed in Sec. \ref{['sec:nl1']}.
• Figure 3: Plots for $N_L = N_R = 2,~N_M = 2$ (i.e. two qubits attached to left and right baths, see Sec. \ref{['sec:nl2']}) for the isotropic XXZ Hamiltonian [Eq. \ref{['eq:ham']}] keeping $\gamma_\ell = 1~ \forall \ell,~\omega_c=10$ [Eq. \ref{['eq:ohmic_bath']}] and $\beta_R = 10.0$ (blue diamond), $5.0$ (red circle), $1.0$ (yellow cross), $0.5$ (purple plus). The black dashed horizontal line represents $\delta_{\rm tol} = 10^{-6}$. (a) $\tau^{\rm pop}_{\rm opt}$ [Eq. \ref{['eq:diag_optimization']}] versus $\beta_L$ with $g=0.01$. (b) $\tau^{\rm pop,coh}_{\rm opt}$ [Eq. \ref{['eq:coh_optimization']}] versus $\beta_L$ with $g=0.01$. (c) $\tau^{\rm pop}_{\rm opt}$ versus $g$ for $\beta_L = 1.0$. (d) $\tau^{\rm pop,coh}_{\rm opt}$ versus $g$ for $\beta_L = 1.0$. This figure shows that it might be possible to obtain steady-states with correct leading-order populations for low inter-site coupling, but it is impossible to obtain both correct leading-order populations and coherences generally.
• Figure 4: Plots for (a) $\tau^{\rm pop}_{\rm opt}$ [Eq. \ref{['eq:diag_optimization']}] and (b) $\tau^{\rm pop,coh}_{\rm opt}$ [Eq. \ref{['eq:coh_optimization']}] versus $g$ with $N_L = 3$, $N_R=0$, $\beta_L = \beta = 1.0$ for isotropic XXZ Hamiltonian and four different values of $N_M =$ 0 (blue diamond), 1 (red circle), 2 (yellow cross), 3 (purple plus). The black dashed curve denotes $\delta_{\rm tol} = 10^{-6}$ with $\gamma_\ell = 1~\forall \ell,~\omega_c = 10$ [Eq. \ref{['eq:ohmic_bath']}].
• Figure 5: Plots for $\tau^{\rm pop,coh}_{\rm opt}$ [Eq. \ref{['eq:coh_optimization']}] with $N_L = 3,N_M = N_R = 0$ with $\beta_L = \beta = 1.0$, $\gamma_1 = 1,\gamma_2 = 1.5,\gamma_3=2$ (red circle) and $N_L = 1,N_M = 0,N_R = 2$ with $\beta_L = 1.0,~\beta_R = 5.0,~\gamma_\ell = 1~\forall \ell$ (blue star) versus $g$. The black dashed curve denotes $\delta_{\rm tol} = 10^{-6}$ and $\omega_c = 10$ [Eq. \ref{['eq:ohmic_bath']}].

Theorems & Definitions (3)

• Lemma 1
• Lemma 1
• proof