Uhlmann fidelities from tensor networks

Abstract

Given two states $|\psi ⟩$ and $|\varphi ⟩$ of a quantum many-body system, one may use the overlap or fidelity $|⟨\psi |\varphi ⟩|$ to quantify how similar they are. To further resolve the similarity of $|\psi ⟩$ and $|\varphi ⟩$ in space, one can consider their reduced density matrices $\rho$ and $\sigma$ on various regions of the system, and compute the Uhlmann fidelity $F(\rho ,\sigma )=\mathrm{Tr}\sqrt{\sqrt{\rho }\sigma \sqrt{\rho }}$. In this paper, we show how computing such subsystem fidelities can be done efficiently in many cases when the two states are represented as tensor networks. Formulated using Uhlmann's theorem, such subsystem fidelities appear as natural quantities to extract for certain subsystems for Matrix Product States and Tree Tensor Networks, and evaluating them is algorithmically simple and computationally affordable. We demonstrate the usefulness of evaluating subsystem fidelities with three example applications: studying local quenches, comparing critical and non-critical states, and quantifying convergence in tensor network simulations.

arXiv comment9+3 pages, many figures; v2 matches the journal version, with some new content

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• Biorthogonal quantum criticality in non-Hermitian many-body systems
• Dynamic process and Uhlmann process: Incompatibility and dynamic phase of mixed quantum states
• Benchmarking a novel efficient numerical method for localized 1D Fermi-Hubbard systems on a quantum simulator
• Uhlmann quench and geometric dynamic quantum phase transition of mixed states
• Accuracy vs Memory Advantage in the Quantum Simulation of Stochastic Processes
• A tensor network approach to sensing quantum light-matter interactions
• Yang-Lee edge singularity and quantum criticality in non-Hermitian PXP model
• Quantum Fisher information from tensor network integration of Lyapunov equation