Generalized Cat States via Daubechies Wavelet Transform

We introduce a quantum version of Daubechies-transformed states and explore its applications to quantum analogues of classical problems for which Daubechies wavelets are useful. Our states are defined in terms of a quantum Daubechies transform applied to the overcomplete coherent-state basis, and we calculate the corresponding Wigner functions, which are then used to analyze the properties of these states. We investigate recursive application of Daubechies transforms and compare these states to the Gottesman-Kitaev-Preskill comb state. The Daubechies transform is constructed via a weighted sum of Glauber displacement operators followed by a squeezing operator, thereby connecting our Daubechies-transformed states to generalized cat states. The Wigner function patterns are complex and we have developed useful methods for understanding these phase-space patterns. In addition, we identify the quantum limit to how many times the Daubechies transform can be recursively applied. Our work is a foray into a representation that exploits Daubechies transform advantages in the quantum domain.