Mixed States, Open Systems, and the Density Operator

Gilbert Reinisch

Gilbert Reinisch

Yuchen Guo, Shuo Yang

Abstract Open quantum system simulations are essential for exploring novel quantum phenomena and evaluating noisy quantum circuits. In this Letter, we investigate whether mixed states generated from noisy quantum circuits can be efficiently represented by locally purified density operators (LPDOs). We map an LPDO of N qubits to a pure state of size 2 × N defined on a ladder and introduce a unified method for managing virtual and Kraus bonds. We numerically simulate noisy random quantum circuits with depths of up to d = 40 using fidelity and entanglement entropy as accuracy measures. The LPDO representation is effective in describing mixed states in both the quantum and classical regions; however, it encounters significant challenges at the quantum-classical critical point, restricting its applicability to the quantum region. In contrast, matrix product operators (MPO) successfully characterize the entanglement trend throughout the simulation, while the truncation in MPOs breaks the positivity condition required for a physical density matrix. This work advances our understanding of efficient mixed-state representations in open quantum systems and provides insights into the entanglement structure of noisy quantum circuits.

Faisal A. A. El-Orany

The mixed states are important in quantum optics since they frequently appear in the decoherence problems. When one of the components of the system is prepared in the mixed state and the evolution operator of this system is not available, one cannot deduce the density matrix. We present analytical approach to accurately solve this problem. The approach can be applied on the condition that the Schrödinger's equation of the system is solvable with any arbitrary initial state. In deriving the solution we exploit the fact that any mixed state can be expressed in terms of a phase state. The approach is illustrated by deriving the density matrix of a single-mode heat environment interacting asymmetrically with two qubits. Our results are in good agreement with the available results in the literature. This approach opens new perspectives for treating complicated systems and may impact other applications in the quantum theory.