Quantification of Macroscopic Quantum Superpositions within Phase Space

Based on phase-space structures of quantum states, we propose a novel measure to quantify macroscopic quantum superpositions. Our measure simultaneously quantifies two different kinds of essential information for a given quantum state in a harmonious manner: the degree of quantum coherence and the effective size of the physical system that involves the superposition. It enjoys remarkably good analytical and algebraic properties. It turns out to be the most general and inclusive measure ever proposed that it can be applied to any types of multipartite states and mixed states represented in phase space.

Figure 1

(a) Interference-based measure $\mathcal{I}(\rho )$ for SCSs of amplitudes $\alpha =2$ (solid line), $\alpha =4$ (dashed line), $\alpha =6$ (dotted line) and $\alpha =27.3$ (dash-dotted line) against the normalized time $r$ under the decoherence effect. The average number of particles is $⟨\widehat{n}⟩\approx {\alpha }^2$. (b) $\mathcal{I}(\rho )$ for single-mode Gaussian states of squeezing parameters $s=1.5$ (solid line), $s=2.1$ (dashed line ), $s=2.5$ (dotted line) and $s=7$ (dash-dotted line), where $⟨\widehat{n}⟩={sinh}^{2}r$. The same curve types mean (nearly) the same average particle numbers.