= Single-qubit process tomography
{title2=$R_{ij}=\tfrac12\operatorname{Tr}(\sigma_iV\sigma_jV^\dagger)$}
To identify the physical channel $\rho\mapsto V\rho V^\dagger$ of a single-<qubit> <unitary gate>, prepare each of the three positive <Pauli matrix> <eigenstates> and estimate each output Pauli <expectation>. Their nine means are
$$
R_{ij}=\frac12\operatorname{Tr}(\sigma_iV\sigma_jV^\dagger),\qquad i,j\in\{x,y,z\}.
$$
They determine the <Bloch vector> <rotation> $\boldsymbol r\mapsto R\boldsymbol r$, hence the channel on all states. If two gates have the same channel, their relative unitary commutes with all three <Pauli matrices>: commuting with $Z$ forces it diagonal and commuting with $X$ forces equal diagonal entries. Thus they differ by a scalar <global phase>, proving uniqueness up to that phase. This procedure needs no controlled gate or <eigenstate> of the unknown gate. Estimating its bounded measurement means to error $\epsilon$ with fixed confidence uses order $\epsilon^{-2}$ repetitions. It determines the relative <eigenphases> but cannot determine an overall <eigenvalue> phase from uncontrolled gate access alone.
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