Measurement- and comparison-based sizes of Schrödinger cat states of light

We extend several measurement-based definitions of effective superposition size to coherent state superpositions with branches composed of either single coherent states or tensor products of coherent states. These measures of superposition size depend on determining the maximal quantum distinguishability of certain states associated with the superposition state: e.g., in one measure, the maximal distinguishability of the branches of the superposition is considered as in quantum binary decision theory; in another measure, the maximal distinguishability of the initial superposition and its image after a one-parameter evolution generated by a local Hermitian operator is of interest. The scaling of the size of the superposition with the number of modes and mode intensity (i.e., photon number) is compared to the scaling of certain geometric properties of the Wigner function of the superposition and also to the superposition size estimated experimentally from decoherence. We also apply earlier comparison-based methods for determining macroscopic superposition size that require a reference Greenberger-Horne-Zeilinger (GHZ) state. The case of a hierarchical Schrödinger cat state with branches composed of smaller superpositions is also analyzed from a measurement-based perspective.

Figure 1

Graphs of the Wigner function $W^{{\text{HCS}}_2\left(\alpha \right)}({\gamma }_1,{\gamma }_2)$ [$({\gamma }_1,{\gamma }_2)\in \mathbb{C}\times \mathbb{C}$] of $|{\text{HCS}}_2\left(\alpha \right)\rangle$ (with $z={\gamma }_1$ and $\left|\alpha \right|=3$) for (a) ${\gamma }_2=0$, (b) ${\gamma }_2=3$, and (c) ${\gamma }_2=-3$. See Eq. (C1) in Appendix pp3 for the explicit form of this function on $\mathbb{C}\times \mathbb{C}$.