Probabilistic Metrology Attains Macroscopic Cloning of Quantum Clocks

It has recently been shown that probabilistic protocols based on postselection boost the performances of the replication of quantum clocks and phase estimation. Here we demonstrate that the improvements in these two tasks have to match exactly in the macroscopic limit where the number of clones grows to infinity, preserving the equivalence between asymptotic cloning and state estimation for arbitrary values of the success probability. Remarkably, the cloning fidelity depends critically on the number of rationally independent eigenvalues of the clock Hamiltonian. We also prove that probabilistic metrology can simulate cloning in the macroscopic limit for arbitrary sets of states when the performance of the simulation is measured by testing small groups of clones.

Figure 1

Probability mass functions $p_{E,m}$ (a) and $p_{\tilde{\mathbf{m}},m}$ (b) for $m=30$ quantum clocks with energies $\mathbf{e}=(0,-1,1+\pi )$ and initial state given by $\mathbf{p}=(\frac{1}{3},\frac{1}{3},\frac{1}{3}{)}^t$. In (b), ${\tilde{m}}_1=m_3$, ${\tilde{m}}_2=m_3-m_2$, which corresponds to the choice of energy units $\mathit{ϵ}=(\pi ,1)$. Note that $p_{E,m}$ in (a) resembles neither a Gaussian (only its outline does) nor any continuous function. Instead, it is the projection of the bell-shaped distribution of points in (b), which approach a smooth surface, onto the gray plane.