More on symmetry resolved operator entanglement

Abstract

The `operator entanglement' of a quantum operator $O$ is a useful indicator of its complexity, and, in one-dimension, of its approximability by matrix product operators. Here we focus on spin chains with a global $U(1)$ conservation law, and on operators $O$ with a well-defined $U(1)$ charge, for which it is possible to resolve the operator entanglement of $O$ according to the $U(1)$ symmetry. We employ the notion of symmetry resolved operator entanglement (SROE) introduced in [PRX Quantum 4, 010318 (2023)] and extend the results of the latter paper in several directions. Using a combination of conformal field theory and of exact analytical and numerical calculations in critical free fermionic chains, we study the SROE of the thermal density matrix ${\rho }_{\beta }=e^{-\beta H}$ and of charged local operators evolving in Heisenberg picture $O=e^{itH}O{e}^{-itH}$. Our main results are: i) the SROE of ${\rho }_{\beta }$ obeys the operator area law; ii) for free fermions, local operators in Heisenberg picture can have a SROE that grows logarithmically in time or saturates to a constant value; iii) there is equipartition of the entanglement among all the charge sectors except for a pair of fermionic creation and annihilation operators.

arXiv comment26 pages, 6 figures

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• On symmetry-resolved generalized entropies
• Total and Symmetry resolved Entanglement spectra in some Fermionic CFTs from the BCFT approach
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• Symmetry resolved out-of-time-order correlators of Heisenberg spin chains using projected matrix product operators
• Simulation of the Dissipative Dynamics of Strongly Interacting NV Centers with Tensor Networks