Size of quantum superpositions as measured with classical detectors

We propose a criterion which defines whether a superposition of two photonic components is macroscopic. It is based on the ability to discriminate these components with a particular class of “classical” detectors, namely, a photon number measurement with a resolution coarse-grained by noise. We show how our criterion can be extended to a measure of the size of macroscopic superpositions by quantifying the amount of noise that can be tolerated and taking the distinctness of two Fock states differing by N photons as a reference. After applying our measure to several well-known examples, we demonstrate that the superpositions which meet our criterion are very sensitive to phase fluctuations. This suggests that quantifying the macroscopicity of a superposition state through the distinguishability of its components with “classical” detectors not only is a natural measure but also explains why it is difficult to observe superpositions at the macroscopic scale.

Figure 1

A simple model of the photon number measurement.

Figure 2

Size of several states for $P_g=2/3$ and increasing “particle number” $N$. (1a) $|0\rangle$ and $|\beta \rangle$ with $N={\left|\beta \right|}^2$. (1b) $|\alpha \rangle \mathrm{and}\mathcal{D}\left(\alpha \right)|\beta \rangle$ with $N={\left|\alpha \right|}^2$ and ${\left|\beta \right|}^2=4$. (2) $\mathcal{D}\left(\alpha \right)|+\rangle \mathrm{and}\mathcal{D}\left(\alpha \right)|-\rangle$ with $N={\left|\alpha \right|}^2$. (3) ${|{\varphi }_1\rangle }^{\otimes N}\mathrm{and}{|{\varphi }_2\rangle }^{\otimes N}$ with $\delta =0.3$.

Figure 3

Size of several states for a fixed “particle number” $N$ as a function of the guessing probability $P_g$. (1a) $|0\rangle$ and $|\beta \rangle$ with ${\left|\beta \right|}^2=40$. (1b) $|\alpha \rangle$ and $\mathcal{D}\left(\alpha \right)|\beta \rangle$ with ${\left|\alpha \right|}^2=400$ and ${\left|\beta \right|}^2=1$. (2) $\mathcal{D}\left(\alpha \right)|+\rangle \mathrm{and}\mathcal{D}\left(\alpha \right)|-\rangle$ with ${\left|\alpha \right|}^2=400$. (3) ${|{\varphi }_1\rangle }^{\otimes N}$ and ${|{\varphi }_2\rangle }^{\otimes N}$ with $N=500$ and $\delta =0.07$.