[Paper Review] Purity and entropy evolution speed limits for open quantum systems

Raam Uzdin, Eric Lutz+1 more|arXiv (Cornell University)|Aug 6, 2014

Advanced Thermodynamics and Statistical Mechanics4 references3 citations

TL;DR

This paper derives state-independent speed limits for purity and entropy evolution in open quantum systems, using the Lindblad generator's norms to bound the rate of dephasing and thermalization. It introduces tighter bounds via Liouville space and purity deviation, proving tightness for single-qubit dephasing and enabling applications to noisy quantum control and cooling.

ABSTRACT

We derive generic upper bounds on the rate of purity change and entropy increase for open quantum systems. These bounds depend solely on the generators of the nonunitary dynamics and are independent of the particular states of the systems. They are thus perfectly suited to investigate dephasing and thermalization processes of arbitrary systems. We apply these results to single and multiple dephasing channels, to a problem of quantum control in the presence of noise, and to cooling.

Motivation & Objective

• To establish generic, state-independent upper bounds on the rate of purity change and entropy increase in open quantum systems.
• To address the challenge of characterizing environmental effects—such as dephasing and thermalization—without requiring detailed knowledge of the system's initial state.
• To improve existing quantum speed limits by leveraging Liouville space and a purity deviation formalism to achieve tighter, more physically relevant bounds.
• To apply the derived bounds to practical problems in quantum control under noise and quantum cooling processes.
• To demonstrate the tightness of the bounds for specific cases, such as single-qubit dephasing, and to link entropy evolution to thermodynamic constraints.

Proposed method

• Derive a differential purity speed limit in Hilbert space using the Hilbert-Schmidt norm of Lindblad operators, leading to an integral bound on purity change.
• Reformulate the problem in Liouville space to express the bound in terms of the spectral norm of the superoperator generator, yielding a tighter inequality.
• Introduce a purity deviation by subtracting the steady-state contribution of the time-dependent dynamics, significantly improving the bound's tightness.
• Apply the Jensen inequality to relate von Neumann entropy to purity, enabling a bound on entropy decrease during cooling processes.
• Use the spectral norm of the skew-Hermitian part of the effective Hamiltonian in Liouville space to derive a state-independent bound on the rate of purity and entropy evolution.
• Validate the bounds through analytical solutions for single-qubit dephasing and decay channels, comparing them with exact dynamics and existing bounds.

Experimental results

Research questions

• RQ1 What are the fundamental limits on the rate of purity change in open quantum systems, independent of the initial state?
• RQ2 How can the speed of entropy increase or decrease be bounded using only the generators of nonunitary dynamics?
• RQ3 Can tighter bounds on purity and entropy evolution be derived by transforming to Liouville space and subtracting steady-state contributions?
• RQ4 How do these bounds perform in practical scenarios such as noisy coherent control and quantum cooling?
• RQ5 Under what conditions is the derived bound on entropy decrease tight, and how does it compare to bounds based on purity?

Key findings

• A state-independent upper bound on the rate of purity change is derived in Hilbert space using the Hilbert-Schmidt norm of Lindblad operators.
• The bound is significantly tightened when reformulated in Liouville space using the spectral norm of the superoperator generator.
• For a single-qubit dephasing channel, the purity deviation-based bound is proven to be tight for all states and all times.
• A tighter bound on final purity in noisy coherent control is obtained by avoiding the triangle inequality in the spectral norm, yielding a 33% improvement over standard bounds.
• A state-independent bound on entropy decrease is derived using Jensen's inequality and the spectral norm of the skew-Hermitian part of the generator, improving over previous bounds.
• The bound on entropy reduction (Eq. 14) provides a minimal time requirement for cooling, linking the dynamics to thermodynamic cycle constraints in quantum heat engines.