Macroscopicity of quantum superpositions on a one-parameter unitary path in Hilbert space

We analyze quantum states formed as superpositions of an initial pure product state and its image under local unitary evolution, using two measurement-based measures of superposition size: one based on the optimal quantum binary distinguishability of the branches of the superposition and another based on the ratio of the maximal quantum Fisher information of the superposition to that of its branches, i.e., the relative metrological usefulness of the superposition. A general formula for the effective sizes of these states according to the branch-distinguishability measure is obtained and applied to superposition states of $N$ quantum harmonic oscillators composed of Gaussian branches. Considering optimal distinguishability of pure states on a time-evolution path leads naturally to a notion of distinguishability time that generalizes the well-known orthogonalization times of Mandelstam and Tamm and Margolus and Levitin. We further show that the distinguishability time provides a compact operational expression for the superposition size measure based on the relative quantum Fisher information. By restricting the maximization procedure in the definition of this measure to an appropriate algebra of observables, we show that the superposition size of, e.g., NOON states and hierarchical cat states, can scale linearly with the number of elementary particles comprising the superposition state, implying precision scaling inversely with the total number of photons when these states are employed as probes in quantum parameter estimation of a 1-local Hamiltonian in this algebra.

Figure 1

Qualitative phase space representations of the variances of $x$ and $p$ quadratures in the vacuum branches (blue) and displaced branches (red) of the states $|{\Psi }_0\rangle$, $|{\Psi }_1\rangle$, $|{\Psi }_{2,\pm }\rangle$ written in the form $\propto (\mathbb{I}+U)|0\rangle$ for $N=1$ (axes are not to scale). The semimajor and semiminor axes of the ellipses in (a), (b), and (d) are equal to $e^{4\xi }$ and $e^{-4\xi }$, respectively, corresponding to the variances of the respective quadratures. (a) $U=S\left(2\xi \right)D(-\alpha (1+e^{2\xi }))$, (b) $U=S(-2\xi )D(-\alpha (1+e^{2\xi }))$, (c) $U=D(-2\alpha e^{\xi })$, (d) $U=S(-2\xi )D(-2\alpha e^{-\xi })$ with $\xi$ taken to be large enough that the squeezed branch quadrature ellipse overlaps that of the vacuum branch.