Krylov Complexity of Purification.

Area of Science:

• Quantum Information Theory
• Quantum Many-Body Physics
• Quantum Gravity

Background:

• Purification in quantum systems enlarges Hilbert space, transforming mixed states to pure states and non-unitary to unitary evolution.
• Understanding the complexity of quantum states and their purification is crucial for quantum information processing and fundamental physics.

Purpose of the Study:

• Establish a connection between the complexities of mixed quantum states and their purification.
• Propose new inequalities relating these complexities.
• Investigate the implications for quantum information and potential gravity duals.

Main Methods:

• Analysis of single qubits, two-qubit Werner states, and Gaussian random unitary ensembles.
• Examination of infinite-dimensional systems.
• Application of mutual Krylov complexity and spread complexity.

Main Results:

• New inequalities among quantum state and purification complexities were established.
• The spread complexity of vacuum-to-thermal state purification equals the average Rindler particle number, adhering to the quantum speed limit.
• Mutual Krylov complexity exhibits subadditivity, contrasting with holographic volume complexity.

Conclusions:

• The study provides a framework for understanding quantum state purification complexity across diverse systems.
• Krylov mutual complexity is proposed as a diagnostic tool for potential gravity duals of Krylov complexities.
• Connections between purification, quantum speed limits, and particle physics were elucidated.