Experimental test of Born's rule by inspecting third-order quantum interference on a single spin in solids

As a fundamental postulate of quantum mechanics, Born's rule assigns probabilities to the measurement outcomes of quantum systems and excludes multiorder quantum interference. Here we report an experiment on a single spin in diamond to test Born's rule by inspecting the third-order quantum interference. The ratio of the third-order quantum interference to the second order in our experiment is bounded to the scale of $1\times {10}^{-3}$, which provides a stringent constraint on the potential breakdown of Born's rule.

Figure 1

Experimental processes for measuring the third-order quantum interference. (a) A qutrit with a set of orthonormal basis $|0\rangle , |+1\rangle$, and $|-1\rangle$, with $|0\rangle \leftrightarrow |+1\rangle$ and $|0\rangle \leftrightarrow |-1\rangle$ being allowed transitions. (b)–(h) Preparation of the seven states {$|{\psi }_i\rangle$} ($i=1,2,...,7$) by different operator $U_i$. Measurement operator $M$ acts on $|{\psi }_i\rangle$ in each of the seven experiments. The third-order quantum interference term can be obtained from the seven experiments. (i) Representation of the state vectors {$|{\psi }_i\rangle$} in the case where $a=1/\sqrt{3}$ and $b=c=-1/\sqrt{3}$ in real three-dimensional space. The states {$|{\psi }_i\rangle$}, in turn, correspond to the prepared states in (b)–(h).

Figure 2

Experimental system and pulse sequences. (a) An NV center in diamond is formed by a substitutional nitrogen atom and an adjacent vacancy. Spin projections $m_s=0,\pm 1$ are defined with respect to the NV symmetry axis. Electronic spin polarization and readout is performed by optical excitation and red fluorescence detection. The microwave pulses ${\mathrm{MW}}_1$ and ${\mathrm{MW}}_2$ drive the selective transitions $|0\rangle \leftrightarrow |-1\rangle$ and $|0\rangle \leftrightarrow |+1\rangle$, respectively. (b) Each experiment includes two steps: preparation of the state $|{\psi }_i\rangle$ by $U_i$, and implementation of the measurement operator $M_1$. The rotation operations $R_1\left({\theta }_1\right)$ and $R_2\left({\theta }_2\right)$ are realized by ${\mathrm{MW}}_1$ and ${\mathrm{MW}}_2$, respectively.

Figure 3

Experimental data of the normalized variant $\kappa$. (a) The data associated with the measurement operator $M_1$ result in $\kappa =0.0017\pm 0.0045$. (b) The data associated with the measurement operator $M_2$ result in $\kappa =-0.0017\pm 0.0042$. The left-hand figures are the values of $\kappa$ with error bars representing standard deviations, and each data point is averaged by $2\times {10}^6$ times. The right-hand figures are the distribution of $\kappa$ corresponding to the left. In the left-hand figures, the horizontal color lines represent the mean values of $\kappa$, and the blue shaded regions represent a band of one standard deviation of the distribution of $\kappa$ values around the mean.