Extended Data Fig. 10: Resolving the population of the states.

For the detection of states |002⟩ and |200⟩, we extract their probabilities from several measurements. There are 64 states that may contribute to the oscillations, which are listed from |000⟩ to |333⟩ as an 8 × 8 square array (left). The amplitudes of these states according to our detection procedures are given by distinct colours (key at bottom right). For example, the state |002⟩ in the third column of the first row only contributes to the first observable \({A}_{|01\rangle }^{(2)}\)+\({A}_{|10\rangle }^{(2)}\) with a factor of 1, while the state |013⟩ at the end of the first row will be recorded by all these observables with the colour-denoted factors. We use seven terms to deduce the lower bound for the probabilities as \({p}_{|\ldots 002\ldots \rangle }+{p}_{|\ldots 200\ldots \rangle }\ge {A}_{|01\rangle }^{(2)}+{A}_{|10\rangle }^{(2)}+{A}_{|01\rangle }^{(1)}+{A}_{|10\rangle }^{(1)}-{\bar{n}}_{c}^{{\rm{o}}}-0.5{\bar{n}}^{{\rm{e}}}-1.5{\bar{n}}_{{\rm{c}}}^{{\rm{e}}}\). Such a relation can be captured from the chequerboard diagram.