The optimal bound of quantum erasure with limited means

TL;DR

This paper investigates the maximum achievable quantum coherence through erasure when only limited environmental information is accessible, deriving bounds and analytical solutions for specific cases, and analyzing the typical visibility outcomes.

Contribution

It introduces the concept of bounds on quantum erasure with limited environment access, linking sub-fidelity to the maximum coherence recoverable, and provides analytical solutions for 2D and 3D environments.

Findings

For 2D environment, the bound is given by sub-fidelity of inaccessible states.

Analytical solution provided for 3D environment case.

Random pure states can achieve up to 90% visibility after optimal erasure.

Abstract

In practical applications of quantum information science, quantum systems can have non-negligible interactions with the environment, and this generally degrades the power of quantum protocols as it introduces noise. Counteracting this by appropriately measuring the environment (and therefore projecting its state) would require access all the necessary degrees of freedom, which in practice can be far too hard to achieve. To better understand one's limitations, we calculate the upper bound of optimal quantum erasure (i.e. the highest recoverable visibility, or "coherence"), when erasure is realistically limited to an accessible subspace of the whole environment. In the particular case of a two-dimensional accessible environment, the bound is given by the sub-fidelity of two particular states of the \emph{inaccessible} environment, which opens a new window into understanding the connection between correlated systems. We also provide an analytical solution for a three-dimensional accessible environment. This result provides also an interesting operational interpretation of sub-fidelity. We end with a statistical analysis of the expected visibility of an optimally erased random state and we find that 1) if one picks a random pure state of 2 qubits, there is an optimal measurement that allows one to distill a 1-qubit state with almost 90\% visibility and 2) if one picks a random pure state of 2 qubits in an inaccessible environment, there is an optimal measurement that allows one to distill a 1-qubit state with almost twice its initial visibility.