Coarse Graining Makes It Hard to See Micro-Macro Entanglement

Observing quantum effects such as superpositions and entanglement in macroscopic systems requires not only a system that is well protected against environmental decoherence, but also sufficient measurement precision. Motivated by recent experiments, we study the effects of coarse graining in photon number measurements on the observability of micro-macro entanglement that is created by greatly amplifying one photon from an entangled pair. We compare the results obtained for a unitary quantum cloner, which generates micro-macro entanglement, and for a measure-and-prepare cloner, which produces a separable micro-macro state. We show that the distance between the probability distributions of results for the two cloners approaches zero for a fixed moderate amount of coarse graining. Proving the presence of micro-macro entanglement therefore becomes progressively harder as the system size increases.

Figure 1

Schematic of the experiments considered in this Letter. A source produces an entangled photon pair in a polarization singlet state. Photon $A$ is detected in an arbitrary polarization basis with the help of the phase rotator, polarizing beam splitter (PBS), and single-photon detectors (SPD). Photon $B$ is amplified, either by a unitary cloner, which preserves the initial entanglement, or by a measure-and-prepare cloner, which completely destroys the entanglement. The resulting multiphoton state is detected in a similar way, where the single-photon detectors are now replaced by photon counters which detect the number of photons in each polarization mode. The coarse-graining parameter $\sigma$ characterizes the precision with which the photon number can be measured.

Figure 2

Probabilities of measuring $j$ photons in photon counter $B1$ of Fig. 1, for a total photon number $N=100$, for both the unitary cloner defined by Eqs. (1, 2, 3, 4) and the measure-and-prepare (M&P) cloner defined by Eqs. (5, 6). The input state $|{\theta }_A,{\varphi }_A⟩$, which is prepared by the measurement in $A$, satisfies ${\theta }_A=\pi /2$, and the measurement in $B$ satisfies ${\theta }_B=\pi /2$ and ${\varphi }_B={\varphi }_A$. The probabilities for the unitary cloner have a distinctive odd-even structure, whereas those for the measure-and-prepare cloner do not. However, the inset shows that when pairs of neighboring photon numbers are put into nonoverlapping bins, the two resulting probability distribution functions are almost identical.

Figure 3

The distance between the probability distributions for the unitary and measure-and-prepare cloner, quantified by the Manhattan-norm distance $D$ of Eq. (9), as a function of the bin size $\sigma$ of Eq. (10). The distance decreases with $\sigma$ in a way that is almost independent of the photon number $N$. As a consequence, as shown in the inset, the distance decreases faster and faster as a function of the relative error $\sigma /N$, when $N$ is increased. The results shown are for ${\theta }_A=\pi /2$, ${\theta }_B=\pi /12$, and $\Delta \varphi =0$, but the behavior is generic; see text.