Seminarium med Nicolas Fabre: Gottesman--Kitaev--Preskill states on a double cylinder phase space: Wigner negativity and stellar zeros

Abstract:

Cylinder phase spaces, built from an angle and an integer angular-momentum-like variable, provide a natural representation for twisted photons, Josephson junctions and bosonic codes. Periodic states, such as the grid states of the Gottesman--Kitaev--Preskill (GKP) code, are naturally represented on a double cylinder, built from modular position and momentum variables. Using covariant integral quantization, a technique that generates whole families of phase-space representations, we construct the Wigner and Husimi distributions on the double cylinder [1, 2, 3]. We show that the Wigner distribution is positive for angular-momentum and phase eigenstates, which include ideal GKP logical states, and negative for their superpositions [1]. The zeros of the stellar function associated with the Husimi distribution define a hierarchy of states tied to their usefulness for quantum computation [2,3]. We identify the operations that change the number of zeros and show that this hierarchy does not govern the localization of the Husimi distribution. We find that phase eigenstates have no zero, whereas the wrapped coherent state, a natural coherent state of the cylinder, has infinitely many and is therefore the most quantum state: a coherent state need not signal classicality. Finally, we show that physical GKP states, which reduce to wrapped coherent states for a broad envelope, acquire zeros on the lattice of the phase space, as does the vacuum, although it is Gaussian in the plane. This echoes their role as resources in GKP-based quantum computing architectures.