where 𝜌 ⃗ is the vector representation of 𝜌 >, and 𝛼 and 𝛽 are the expectation values of the basis operators in 𝜌 > " and 𝜌 > $ , respectively (i.e. the elements of 𝜌 ⃗ " and 𝜌 ⃗ $ ). Thus two directly opposing quantum states, with equal and opposite observables, will have a pseudo-fidelity of 0.25, whereas two identical quantum states will have a pseudo-fidelity of 1.00. By this metric, the pseudo-fidelity of the fitted state (Figure and 8b, green) to the desired state (black) is 0.788. Note that the fitted experimental state was normalised by the magnitude of the calculated expectation values because the experimental starting state is unknown. This means that the measurement is insensitive to an overall loss in magnetisation, and therefore this measurement of this pseudo-fidelity is a normalised pseudo-fidelity and an upper-bound (see further discussion later in the 'Relation to experiment' section). Factors leading the loss of fidelity are examined further in Section 4, however while discussing the fitted expectation values it is pertinent to note that due to the relatively low value found for the filter pulse flip probability, the majority of the echo intensity in all traces arises from 'leakage' of unfiltered magnetisation. If the actual filter pulse efficiency is greater than the value used in the fit, then the expectation values of observables which are not single-quantum coherences will be artificially inflated as these operators cannot be detected without being affected by a filter pulse. This will be especially problematic for experiments with filter pulses at both frequencies, in which less than 20% of the fitted signal arises from the desired coherence pathway. For example, there is no pathway by which 2𝑆 0 % 𝐼 0 % or 2𝑆 0 ' 𝐼 0 ' magnetisation can be generated by the DFHE entangling gate, and yet these are the largest expectation values in the fitted density matrix (Figure ). This distortion of the fitted quantum state would lead to errors in the readout and thus a loss of fidelity.