Estimating the Trace Norm of Matrix Product Operators

The trace norm is a widely used quantity in quantum information, crucial for diagnosing entanglement and quantifying distance between quantum states. We propose an efficient tensor-network method to estimate the trace norm ||A||_1 of matrix product operators (MPOs) A, which compactly represent physically relevant operators in quantum many-body settings, such as systems at thermal equilibrium or out of equilibrium. Using our tensor-network method, we compute the entanglement negativity of various quantum many-body states, including critical spin chains at finite temperature. We further demonstrate that our method enables estimation of the quantum fidelity between MPOs.