We study the robustness of quantum and classical information to perturbations implemented by local operator insertions. We do this by computing multipartite entanglement measures in the Hilbert space of local operators in the Heisenberg picture. The sensitivity to initial conditions that we explore is an illuminating manifestation of the butterfly effect in quantum many-body systems. We present a ``membrane theory'' in Haar random unitary circuits to compute the mutual information, logarithmic negativity, and reflected entropy in the local operator state by mapping to a classical statistical mechanics problem and find that any local operator insertion delocalizes information as fast as is allowed by causality after taking the large local Hilbert space dimension limit. Identical behavior is found for conformal field theories admitting holographic duals where the bulk geometry is described by the eternal black hole with a local object situated at the horizon. In contrast to these maximal scramblers, only an $O(1)$ amount of information is found to be delocalized by local operators in free fermionic systems and random Clifford circuits.
We study the robustness of quantum and classical information to perturbations implemented by local operator insertions. We do this by computing multipartite entanglement measures in the Hilbert space of local operators in the Heisenberg picture. The sensitivity to initial conditions that we explore is an illuminating manifestation of the butterfly effect in quantum many-body systems. We present a ``membrane theory'' in Haar random unitary circuits to compute the mutual information, logarithmic negativity, and reflected entropy in the local operator state by mapping to a classical statistical mechanics problem and find that any local operator insertion delocalizes information as fast as is allowed by causality after taking the large local Hilbert space dimension limit. Identical behavior is found for conformal field theories admitting holographic duals where the bulk geometry is described by the eternal black hole with a local object situated at the horizon. In contrast to these maximal scramblers, only an O(1) amount of information is found to be delocalized by local operators in free fermionic systems and random Clifford circuits.
A bstract We consider the logarithmic negativity and related quantities of time evolution operators. We study free fermion, compact boson, and holographic conformal field theories (CFTs) as well as numerical simulations of random unitary circuits and integrable and chaotic spin chains. The holographic behavior strongly deviates from known non- holographic CFT results and displays clear signatures of maximal scrambling. Intriguingly, the random unitary circuits display nearly identical behavior to the holographic channels. Generically, we find the “line-tension picture” to effectively capture the entanglement dynamics for chaotic systems and the “quasi-particle picture” for integrable systems. With this motivation, we propose an effective “line-tension” that captures the dynamics of the logarithmic negativity in chaotic systems in the spacetime scaling limit. We compare the negativity and mutual information leading us to find distinct dynamics of quantum and classical information. The “spurious entanglement” we observe may have implications on the “simulatability” of quantum systems on classical computers. Finally, we elucidate the connection between the operation of partially transposing a density matrix in conformal field theory and the entanglement wedge cross section in Anti-de Sitter space using geodesic Witten diagrams.
Bipartite entanglement entropy is one of the most useful characterizations of universal properties in a many-body quantum system. Far from equilibrium, there exist two highly effective theories describing its dynamics -- the quasiparticle and membrane pictures. In this work we investigate entanglement dynamics, and these two complementary approaches, in a quantum circuit model perturbed by an impurity. In particular, we consider a dual unitary quantum circuit containing a spatially fixed, non-dual-unitary impurity gate, allowing for differing local Hilbert space dimensions to either side. We compute the entanglement entropy for both a semi-infinite and a finite subsystem within a finite distance of the impurity, comparing exact results to predictions of the effective theories. We find that for a semi-infinite subsystem, both theories agree with each other and the exact calculation. For a finite subsystem, however, both theories qualitatively differ, with the quasiparticle picture predicting a non-monotonic growth in contrast to the membrane picture. We show that such non-monotonic behavior can arise even in random chaotic circuits, shedding light on the range of validity of the membrane picture in such systems.
Abstract We study operator entanglement measures of the unitary evolution operators of (1 + 1)-dimensional conformal field theories (CFTs), aiming to uncover their scrambling and chaotic behaviors. In particular, we compute the bi-partite and tri-partite mutual information for various configurations of input and output subsystems, and as a function of time. We contrast three different CFTs: the free fermion theory, compactified free boson theory at various radii, and CFTs with holographic dual. We find that the bi-partite mutual information exhibits distinct behaviors for these different CFTs, reflecting the different information scrambling capabilities of these unitary operators; while a quasi-particle picture can describe the case of the free fermion and free boson CFTs well, it completely fails for the case of holographic CFTs. Similarly, the tri-partite mutual information also distinguishes the unitary evolution operators of different CFTs. In particular, its late-time behaviors, when the output subsystems are semi-infinite, are quite distinct for these theories. We speculate that for holographic theories the late-time saturation value of the tri-partite mutual information takes the largest possible negative value and saturates the lower bound among quantum field theories.
We study entanglement in thermofield double states of strongly coupled conformal field theories by analyzing two-sided Reissner-Nordstr\"om solutions in anti--de Sitter space. The central object of study is the mutual information between a pair of regions, one on each asymptotic boundary of the black hole. For large regions the mutual information is positive and for small ones it vanishes; we compute the critical length scale, which goes to infinity for extremal black holes, of the transition. We also generalize the butterfly effect of Shenker and Stanford [J. High Energy Phys. 03 (2014) 067] to a wide class of charged black holes, showing that mutual information is disrupted upon perturbing the system and waiting for a time of order $\mathrm{log}E/\ensuremath{\delta}E$ in units of the temperature. We conjecture that the parametric form of this time scale is universal.
We study the growth of genuine multipartite entanglement in random quantum circuit models, which include random unitary circuit models and the random Clifford circuit. We find that for the random Clifford circuit, the growth of multipartite entanglement remains slower in comparison to the random unitary case. However, the final saturation value of multipartite entanglement is almost the same in both cases. The behavior is then compared to the genuine multipartite entanglement obtained in random matrix product states with a moderately high bond dimension. We then relate the behavior of multipartite entanglement to other global properties of the system, viz., the delocalization of the many-body wave functions in Hilbert space. Along with this, we analyze the robustness of such highly entangled quantum states obtained through random unitary dynamics under weak measurements.
The delocalization or scrambling of quantum information has emerged as a central ingredient in the understanding of thermalization in isolated quantum many-body systems. Recently, significant progress has been made analytically by modeling non-integrable systems as periodically driven systems, lacking a Hamiltonian picture, while honest Hamiltonian dynamics are frequently limited to small system sizes due to computational constraints. In this paper, we address this by investigating the role of conservation laws (including energy conservation) in the thermalization process from an information-theoretic perspective. For general non-integrable models, we use the equilibrium approximation to show that the maximal amount of information is scrambled (as measured by the tripartite mutual information of the time-evolution operator) at late times even when a system conserves energy. In contrast, we explicate how when a system has additional symmetries that lead to degeneracies in the spectrum, the amount of information scrambled must decrease. This general theory is exemplified in case studies of holographic conformal field theories (CFTs) and the Sachdev-Ye-Kitaev (SYK) model. Due to the large Virasoro symmetry in 1+1D CFTs, we argue that, in a sense, these holographic theories are not maximally chaotic, which is explicitly seen by the non-saturation of the second Rényi tripartite mutual information. The roles of particle-hole and U(1) symmetries in the SYK model are milder due to the degeneracies being only two-fold, which we confirm explicitly at both large- and small-N. We reinterpret the operator entanglement in terms of the growth of local operators, connecting our results with the information scrambling described by out-of-time-ordered correlators, identifying the mechanism for suppressed scrambling from the Heisenberg perspective.