State-Independent Experimental Test of Quantum Contextuality with a Single Trapped Ion

Using a single trapped ion, we have experimentally demonstrated state-independent violation of a recent version of the Kochen-Specker inequality in a three-level system (qutrit) that is intrinsically indivisible. Three ground states of the ${}^{171}\mathrm{Yb}^{+}$ ion representing a qutrit are manipulated with high fidelity through microwaves and detected with high efficiency through a two-step quantum jump technique. Qutrits constitute the most fundamental system to show quantum contextuality and our experiment represents the first one that closes the detection efficiency loophole for experimental tests of quantum contextuality in such a system.

Figure 1

Observables and compatibility relations for the state-independent KS inequality for a qutrit system. (a) The 13 observables are represented as vectors in a three-dimensional space. $|v_i{⟩}^{\prime }$s are specified as $|v_1⟩=(1,0,0)$, $|v_2⟩=(0,1,0)$, $|v_3⟩=(0,0,1)$, $|v_4⟩=(0,1,-1)$, $|v_5⟩=(1,0,-1)$, $|v_6⟩=(1,-1,0)$, $|v_7⟩=(0,1,1)$, $|v_8⟩=(1,0,1)$, $|v_9⟩=(1,1,0)$, $|v_{10}⟩=(-1,1,1)$, $|v_{11}⟩=(1,-1,1)$, $|v_{12}⟩=(1,1,-1)$, and $|v_{13}⟩=(1,1,1)$. (b) The compatibility graph between 13 observables. The nodes represent the 13 vectors $|v_i⟩$ and the 24 edges show the orthogonality or compatibility relations. The experimental realization and joint measurements of the 24 orthogonal observables are described in Table  and Fig. 2c.

Figure 2

The ${}^{171}\mathrm{Yb}^{+}$ ion system and the measurement scheme. (a) The energy diagram of ${}^{171}\mathrm{Yb}^{+}$. Qutrit states $|1⟩$, $|2⟩$, and $|3⟩$ are mapped onto $|F=1,m_F=0⟩$, $|F=1,m_F=1⟩$, and $|F=0,m_F=0⟩$ in the $S_{1/2}$ ground state manifold, respectively. The transition frequencies are ${\omega }_1=(2\pi )12642.8213\mathrm{MHz}$ and ${\omega }_2={\omega }_1+(2\pi )7.6372\mathrm{MHz}$. The quantum state projected to $|1⟩$ or $|2⟩$ generates fluorescence, while the state collapsed to $|3⟩$ does not generate photons. Therefore, we assign zero on the observable related to state $|3⟩$ when we detect photons and one when no photons are detected. (b) The matrix representations of $R_1({\theta }_1,{\varphi }_1),R_2({\theta }_2,{\varphi }_2)$ rotations, which are realized by applying microwave pulses with the frequencies of ${\omega }_1$ and ${\omega }_2$. Here ${\theta }_1$, ${\theta }_2$ and ${\varphi }_1$, ${\varphi }_2$ are controlled by the duration and the phase of the applied microwaves. (c) The sequential measurement scheme to detect the correlations $⟨V_i{V}_j⟩$, where $M_i$ and $M_j$ denote the measurement boxes associated with the observables $V_i$ and $V_j$, respectively. The detailed implementation is explained in the text.

Figure 3

State-independent test of KS inequalities $⟨{\chi }_4⟩$ and $⟨{\chi }_{13}⟩$ for a qutrit system. The inequalities (1, 2) are examined with eleven different initial states including simple product states ${\psi }_1$ to ${\psi }_3$, superposition states at the axis of the observables ${\psi }_4$ to ${\psi }_6$, outside of the observables ${\psi }_7$ to ${\psi }_9$, and mixed states ${\psi }_{10}$ to ${\psi }_{12}$. We measure the density matrix of all states by performing qutrit state tomography [19] and confirm the prepared states with an average fidelity 99.5% for the pure states ${\psi }_1$ to ${\psi }_9$. The mixed state fidelities are around 97%, shown in the figure. All raw measurements for $⟨{\chi }_{13}⟩=27.38(\pm 0.21)$ shown as blue filled squares violate the classical limit 25 by $11.2\sigma$, confirming the quantum contextuality for various states. For each state, 480 000 realizations are used to obtain the $⟨{\chi }_{13}⟩$. For the inequality (2), the average value of all measured initial states is $⟨{\chi }_4⟩=1.35(\pm 0.04)$.