From the Bloch sphere to phase space representations with the Gottesman-Kitaev-Preskill encoding

TL;DR

This paper investigates the phase-space representation of GKP-encoded qubit states, linking Wigner function negativity to quantum computational resources and identifying the extremal states in negativity.

Contribution

It establishes a connection between Wigner negativity and non-stabilizer states in GKP-encoded qubits, revealing how negativity correlates with quantum computational resources.

Findings

Stabilizer states have the lowest Wigner negativity in GKP encoding.

Non-stabilizer states like H-type and T-type exhibit maximum negativity.

Negativity serves as a resource indicator for quantum advantage in CV architectures.

Abstract

In this work, we study the Wigner phase-space representation of qubit states encoded in continuous variables (CV) by using the Gottesman-Kitaev-Preskill (GKP) mapping. We explore a possible connection between resources for universal quantum computation in discrete-variable (DV) systems, i.e. non-stabilizer states, and negativity of the Wigner function in CV architectures, which is a necessary requirement for quantum advantage. In particular, we show that the lowest Wigner logarithmic negativity of qubit states encoded in CV with the GKP mapping corresponds to encoded stabilizer states, while the maximum negativity is associated with the most non-stabilizer states, H-type and T-type quantum states.