Recovering full coherence in a qubit by measuring half of its environment

TL;DR

This paper investigates how measuring a portion of a qubit's environment can restore its quantum coherence, revealing a sharp transition in recoverability when accessing about half of the environment, with implications for quantum information processing.

Contribution

It quantifies the maximum coherence recoverable by measuring part of the environment and uncovers a phase transition at half the environment's degrees of freedom.

Findings

Recovering full coherence requires access to roughly half of the environment.

A sharp transition in recoverable coherence occurs around half the environment.

Recoverable coherence becomes typical in large ensembles.

Abstract

When quantum systems interact with the environment they lose their quantum properties, such as coherence. Quantum erasure makes it possible to restore coherence in a system by measuring its environment, but accessing the whole of it may be prohibitive: realistically one might have to concentrate only on an accessible subspace and neglect the rest. If that is the case, how good is quantum erasure? In this work we compute the largest coherence $\langle \mathcal C\rangle$ that we can expect to recover in a qubit, as a function of the dimension of the accessible and of the inaccessible subspaces of its environment. We then imagine the following game: we are given a uniformly random pure state of $n+1$ qubits and we are asked to compute the largest coherence that we can retrieve on one of them by optimally measuring a certain number $0\leq a\leq n$ of the others. We find a surprising effect around the value $a\approx n/2$: the recoverable coherence sharply transitions between 0 and 1, indicating that in order to restore full coherence on a qubit we need access to only half of its physical environment (or in terms of degrees of freedom to just the square root of them). Moreover, we find that the recoverable coherence becomes a typical property of the whole ensemble as $n$ grows.