Fig. 3: Experimental reconstruction of the three-time QPD and marginal distribution by the ancilla-assisted measurement.
a The system qubit’s evolution is given by two successive rotations \({U}_{{t}_{1}\to {t}_{2}}={R}_{X}(\theta )\) and \({U}_{{t}_{2}\to {t}_{3}}={R}_{Y}({\theta }^{2})\) with θ = 0.74π. Note that the measurements are performed at three time points t1, t2, and t3. b The normalized observed trajectories from the measurements at times t1, t2, and t3 are represented by the 2D bar charts, where \({m}_{1},{m}_{2}\in \{\left\vert {0}_{x}\right\rangle,\left\vert {0}_{y}\right\rangle,\left\vert 0\right\rangle,\left\vert {1}_{x}\right\rangle,\left\vert {1}_{y}\right\rangle,\left\vert 1\right\rangle \}\). The bar charts on the left and right represent the observed trajectories of \({m}_{3}=\left\vert 0\right\rangle\) and \({m}_{3}=\left\vert 1\right\rangle\), respectively, which are the measurement results of time t3. c The three-time QPD p(x1, x2, x3) was reconstructed from the observed trajectories by classical processing. (i) The left blue bars indicate the real parts of the reconstructed three-time QPD, and the negativity of the real QPD is verified for p(x1 = 0, x2 = 1, x3 = 0) (in the blue-line box), which is  − 0.123(± 0.060). (ii) The right red bars indicate the imaginary parts of the reconstructed three-time QPD. d The marginal distributions for times t1, t2, and t3 under the unitary dynamics \({U}_{{t}_{1}\to {t}_{2}}\) and \({U}_{{t}_{2}\to {t}_{3}}\) show the snapshotting of the state evolution. The distributions are marginalized over all the other time points of the QPDs. For all figures, error bars indicate standard deviations (STDs).
