Stabilizers may be poor bounds for fidelities

TL;DR

This paper demonstrates that stabilizer expectation values only upper bound the fidelity to ideal GKP states, indicating that high stabilizer values do not necessarily imply high fidelity.

Contribution

It reveals that stabilizer measurements are poor proxies for GKP state fidelity, challenging common assumptions in quantum error correction.

Findings

Stabilizer expectation values upper bound GKP fidelity

High stabilizer values do not guarantee high fidelity

Fidelity assessment requires more than stabilizer measurements

Abstract

The defining feature of ideal Gottesman-Kitaev-Preskill (GKP) states is that they are unchanged by stabilizers, which allow them to detect and correct for common errors without destroying the quantum information encoded in the states. Given this property, can one use the amount to which a state is unchanged by the stabilizers as a proxy for the quality of a GKP state? This is shown to hold in the opposite manner to which it is routinely assumed, because in fact the fidelity a state has to an ideal GKP state is only upper bounded by the stabilizer expectation values. This means that, for qubits encoded in harmonic oscillators via the GKP code, a good stabilizer expectation value does not guarantee proximity to an ideal GKP state in terms of any distance based on fidelity.