Fig. 3: Bell state tomography.

a Adiabatic inversion probability of both qubits as a function of detuned microwave frequency Δf_MW, where the carrier frequency is chosen to be the single-qubit operation frequency for Q2 and J gate voltage ΔV_J, with qubits initialized in the \(\left|\downarrow \downarrow \right\rangle\) state. Horizontal dashed lines represent J gate voltages applied for various single-qubit and two-qubit gates. Yellow dotted lines are a guide indicating the other resonance frequencies that would be observed at ΔV_J > 100 mV if the spins were initialized randomly. b Schematic of an example microwave and voltage pulse sequence for state tomography. It initializes the qubits as \(\left|\uparrow \downarrow \right\rangle\) by performing two \(\frac{\pi }{2}\) ×1 pulses (all calibration is performed for \(\frac{\pi }{2}\) pulses, such that a high-fidelity π pulse is obtained by composing it out of two \(\frac{\pi }{2}\) gates, each starting and finishing at a common voltage ΔV_J = −70 mV, which is shown as a blue dashed line in a), then perform IZ projection operation, by converting the parity readout into single-qubit readout via a CNOT gate⁴. Horizontal lines align with ΔV_J from a. c Example qubit states and operations required to obtain projections along the indicated axes. The first, two columns of Bloch spheres represent the eigenstates of Q1 (red) and Q2 (blue) before state tomography, whereas the rest illustrates the logic gate operations required for state tomography, before parity readout. For IX and IZ, all possible initial eigenstates are displayed, with parity results shown on the last column. d–g Quantum-state tomography of Bell states d \({{{\Phi }}}^{+}=\frac{\left|\uparrow \uparrow \right\rangle +\left|\downarrow \downarrow \right\rangle }{\sqrt{2}}\), e \({{{\Phi }}}^{-}=\frac{\left|\uparrow \uparrow \right\rangle -\left|\downarrow \downarrow \right\rangle }{\sqrt{2}}\), f \({{{\Psi }}}^{+}=\frac{\left|\uparrow \downarrow \right\rangle +\left|\downarrow \uparrow \right\rangle }{\sqrt{2}}\), g \({{{\Psi }}}^{-}=\frac{\left|\uparrow \downarrow \right\rangle -\left|\downarrow \uparrow \right\rangle }{\sqrt{2}}\). The height of the bars represents the absolute value of density matrix elements, whereas complex phase information is encoded in the colour map. Inset: bar graph of the ideal density matrix of the corresponding Bell state. The measured fidelities of each Bell state are (87.1 ± 2.8)%, (90.3 ± 3.0)%, (90.3 ± 2.4)%, and (90.2 ± 2.9)%, from d to g, respectively.