Quantum Decoherence Scaling with Bath Size: Importance of Dynamics, Connectivity, and Randomness

TL;DR

This paper derives and verifies a scaling law for quantum decoherence in system-environment setups, showing that decoherence strength decreases with environment size and is influenced by dynamics, connectivity, and randomness.

Contribution

It introduces a universal scaling relationship for quantum decoherence with environment size and demonstrates how environment complexity and randomness affect this scaling.

Findings

Decoherence sum scales as 1/√D_E for large environment Hilbert space.

Simulations confirm the scaling law under certain dynamical conditions.

Increased environment connectivity or randomness facilitates the observed scaling behavior.

Abstract

The decoherence of a quantum system $S$ coupled to a quantum environment $E$ is considered. For states chosen uniformly at random from the unit hypersphere in the Hilbert space of the closed system $S+E$ we derive a scaling relationship for the sum of the off-diagonal elements of the reduced density matrix of $S$ as a function of the size $D_E$ of the Hilbert space of $E$. This sum decreases as $1/\sqrt{D_E}$ as long as $D_E\gg 1$. This scaling prediction is tested by performing large-scale simulations which solve the time-dependent Schr{\"o}dinger equation for a ring of spin-1/2 particles, four of them belonging to $S$ and the others to $E$. Provided that the time evolution drives the whole system from the initial state toward a state which has similar properties as states belonging to the class of quantum states for which we derived the scaling relationship, the scaling prediction holds. For systems which do not exhibit this feature, it is shown that increasing the complexity (in terms of connections) of the environment or introducing a small amount of randomness in the interactions in the environment suffices to observe the predicted scaling behavior.