Local Certification of Many-Body Steady States

TL;DR

The paper addresses the challenge of characterizing steady states in open many-body quantum systems by shifting from the full density operator to local reduced density matrices and enforcing consistency with a global steady state. It develops a relaxation-based semidefinite programming framework that bounds local observable expectations for Lindblad dynamics, with bounds tightening as the local patch size $k$ increases and remaining valid in the thermodynamic limit. The method is demonstrated on one- and two-dimensional Ising and Dicke-type models, achieving fast convergence and, in several cases, two-decimal accuracy, while resolving degeneracies that hinder other approaches. This approach provides rigorous, scalable benchmarks for steady-state properties and can serve as a tool to certify quantum devices and guide theory, with potential extensions to non-TI systems, higher dimensions, and near phase transitions.

Abstract

We present a relaxation-based method to bound expectation values on the steady state of dissipative many-body quantum systems described by master equations of the Lindblad form. Instead of targeting to represent the entire state, we promote the reduced density matrices to our variables and enforce the constraints that are imposed on them by consistency with a global steady state. The resulting constraints have the form of a semidefinite program, which allows us to efficiently bound the values a given expectation value can take in the steady state. Our results show fast convergence of the bounds with the size of the reduced density matrices, giving very competitive predictions for the steady state of several one- and two-dimensional models for an arbitrary number of particles.

Figures (4)

• Figure 1: Diagrammatic depiction of the setup considered. (a) We target the steady state of an open system, here shown for the one-dimensional case. The dynamics of the system are generated by a Lindbladian which is a sum of local terms. (b) All reduced states of $k$ contiguous particles are described by the same density matrix, which has to satisfy the local translation invariance condition Eq. (\ref{['eq:lti']}). (c) Due to the strict locality of the Lindbladian, the growth of the support of an operator $X$ when acted upon by the adjoint Lindbladian is bounded. For a Lindbladian with two-body terms, an operator with support on $k'$ sites extends to an operator of size $k'+2$ after the action of the adjoint Lindbladian.
• Figure 2: Diagram of the implementation of our method for two-dimensional open systems. (a) Consider an operator $X$ supported on a region of the lattice. The action of the adjoint Lindbladian on the operator, $\mathcal{L}^\dagger \left( X \right)$, is an operator supported on a larger region. The particular growth of the operator is determined by the connectivity of the Lindbladian. If it acts only on nearest-neighbours, the support of the operator grows by one unit in the directions in which the Lindbladian acts non-trivially. (b) The operator $\mathcal{L}^\dagger \left( X \right)$, however, can be written as a sum of terms, $\left( \mathcal{L}^\dagger \left( X \right)\right)_n$, where each term acts non-trivially on the initial support of $X$ and one extra qudit. If the support of the original operator $X$ is symmetric under rotation and reflection symmetries, the expectation value of some of the terms $\left( \mathcal{L}^\dagger \left( X \right) \right)_n$ can be related to others.
• Figure 3: Bounds obtained with our method for the local expectation values of two different one-dimensional models. In all the main figures, markers indicate the center of the allowed region for each operator, while the error bars represent is full extent. (a)-(c) Bounds on the expectation value of (a) $\langle \sigma^x\rangle$, (b) $\langle \sigma^y\rangle$, and (c) $\langle \sigma^z\rangle$ for the one-dimensional dissipative Ising model as a function of $k$, the size of the reduced density matrix $\rho^{(k)}$ in the SDP \ref{['eq:sdp_rdm']}. The insets show a zoom-in of the results for larger values of $k$, as the error bars become much smaller than the markers. (d) Results obtained with $k = 8$ for the one-dimensional short-range Dicke model as a function of $g/\gamma$. Except for $\langle \sigma^z \rangle$ at $g/\gamma = 0$, the error bars are smaller than the markers. For reference, the inset displays the size of the allowed region for the same expectation values.
• Figure 4: Bounds on the local expectation values (a) $\langle \sigma^y\rangle$ and (b) $\langle \sigma^z\rangle$ for the phase diagram of the steady state of the considered two-dimensional short-range Dicke model. Each error bar shows different sizes of the reduced density matrix used to impose the stationarity of the local expectation values. In both figures, the dashed gray line shows the result of a mean-field computation, whose validity is ruled out by our method. In the inset of (b), we show the size of the error bars of $\langle \sigma^z\rangle$ for the different regions we use as a function of $g/\gamma$. The horizontal lines show, as an indicator, an error bar size of $10^{-3}$, $10^{-2}$, and $10^{-1}$, respectively. We see that with the largest reduced density matrix, we can get results accurate to two decimal digits through practically the entire phase diagram.