Generalization of Modular Spread Complexity for Non-Hermitian Density Matrices

TL;DR

This paper extends the concept of modular spread complexity to non-Hermitian density matrices, introducing pseudo-modular complexity and pseudo-capacity, with analytical and numerical results on quantum systems and topological theories.

Contribution

It generalizes modular spread complexity to non-Hermitian cases, introduces pseudo-modular complexity and pseudo-capacity, and demonstrates their applications in various quantum models and topological theories.

Findings

Pseudo-modular complexity provides richer information than pseudo-entropy.

Analytical calculations for simple quantum systems are presented.

Numerical analysis of phase transitions and topology effects using pseudo-modular complexity.

Abstract

In this work we generalize the concept of modular spread complexity to the cases where the reduced density matrix is non-Hermitian. This notion of complexity and associated Lanczos coefficients contain richer information than the pseudo-entropy, which turns out to be one of the first Lanczos coefficients. We also define the quantity pseudo-capacity which generalizes capacity of entanglement, and corresponds to the early modular-time measure of pseudo-modular complexity. We describe how pseudo-modular complexity can be calculated using a slightly modified bi-Lanczos algorithm. Alternatively, the (complex) Lanczos coefficients can also be obtained from the analytic expression of the pseudo-R\'enyi entropy, which can then be used to calculate the pseudo-modular spread complexity. We show some analytical calculations for 2-level systems and 4-qubit models and then do numerical investigations on the quantum phase transition of transverse field Ising model, from the (pseudo) modular spread complexity perspective. As the final example, we consider the $3d$ Chern-Simon gauge theory with Wilson loops to understand the role of topology on modular complexity. The concept of pseudo-modular complexity introduced here can be a useful tool for understanding phases and phase transitions in quantum many body systems, quantum field theories and holography.