Randomness Expansion Secured by Quantum Contextuality

The output randomness from a random number generator can be certified by observing the violation of quantum contextuality inequalities based on the Kochen-Specker theorem. Contextuality can be tested in a single quantum system, which significantly simplifies the experimental requirements to observe the violation comparing to the ones based on nonlocality tests. However, it is not yet resolved as to how to ensure compatibilities for sequential measurements that is required in contextuality tests. Here, we employ a modified Klyachko-Can-Binicioğlu-Shumovsky contextuality inequality, which can ease the strict compatibility requirement on measurements. On a trapped single ${}^{138}{\mathrm{Ba}}^{+}$ ion system, we experimentally demonstrate violation of the contextuality inequality and realize quantum random number expansion by closing detection loopholes. We perform $1.29\times {10}^8$ trials of experiments and extract a randomness of $5.28\times {10}^5$ bits with a speed of $270\mathrm{bits}{\mathrm{s}}^{-1}$. Our demonstration paves the way for practical high-speed spot-checking quantum random number expansion and other secure information processing applications.

Figure 1

KCBS pentagram and experimental procedure. (a) Initial state and five axes that form a pentagram in $d=3$ space. The five observables $A_1,A_2,\dots ,A_5$ are the projectors on the five axes, respectively. The connected axes $|v_i⟩$ and $|v_{i+1}⟩$ are orthogonal, representing compatibility of the corresponding observables $A_i$ and $A_{i+1}$. (b) Initially, we prepare $|3⟩$ state, then perform two sequential measurements of $A_i$ and $A_j$. Each sequential measurement contains a unitary rotation $U_i$, projective measurement, and an inverse unitary rotation $U_i^{†}$. Each unitary rotation $U_i$ is comprised of first $R_2\left({\theta }_{2i},{\varphi }_{2i}\right)$ then $R_1\left({\theta }_{1i},{\varphi }_{1i}\right)$. In projective measurement, we assign $a_i=1(-1)$ if fluorescence is (is not) detected.

Figure 2

Experimental setup of the ¹³⁸Ba⁺ ion system. (a) The energy level diagram of a ¹³⁸Ba⁺ ion for a qutrit system, which is represented by two Zeeman sublevels ($|m_D=+1/2⟩\equiv |1⟩$, $|m_D=+3/2⟩\equiv |2⟩$) in the ${}^5{D}_{5/2}$ manifold and one sublevel ($|m_S=+1/2⟩\equiv |3⟩$) in the ${}^6{S}_{1/2}$ manifold. The quadrupole transitions between ${}^6{S}_{1/2}$ and ${}^5{D}_{5/2}$ are coherently manipulated using a narrow-line 1762-nm laser that is stabilized to a high-finesse cavity. Lasers of 493 and 650 nm are used for Doppler cooling, EIT cooling, optical pumping, and detection. A 614-nm laser is used for depopulation of the ${}^5{D}_{5/2}$ level to the ${}^6{S}_{1/2}$ level. (b) The experimental setup of a trapped ¹³⁸Ba⁺ ion for testing the KCBS inequality and for spot checking random number expansion. One of 11 measurement configurations $\left\{A_i,A_j\right\}$ is randomly selected. Their pulse sequences are independently generated by their own direct digital synthesizer and amplifiers, sent to an AOM through independent paths, and finally applied to the ion at different time order. Fluorescence is observed by a photomultiplier tube at different time order and the values of the observables are assigned accordingly.

Figure 3

(a),(b) For the Miller-Shi bound the relation of the score of KCBS game $g_{\mathrm{KCBS}}$, number of total rounds $N$, test probability $q$, and randomness expansion rate with smoothing parameter $\delta ={10}^{-2}$ in Eq. (A4) of Appendix pp1. (a) The minimum number of rounds to have net randomness depending on the score $g_{\mathrm{KCBS}}$. The minimum $N$ decreases as $g_{\mathrm{KCBS}}$ increases. We can get net randomness only within the shaded area. Our experimental $g_{\mathrm{KCBS}}=0.790$ and $N_{\exp }=1.29\times {10}^8$ are shown as the green circle. (b) Randomness expansion rate at different $g_{\mathrm{KCBS}}$ and $q$ for our $N_{\exp }$. Only with the combination of large enough $g_{\mathrm{KCBS}}$ and appropriate $q$ can we obtain net randomness. Our experimental $g_{\mathrm{KCBS}}=0.790$ and $q_{\exp }=0.0001$ are shown as the red circle, the resulting expansion rate being $2.6\times {10}^{-3}$ per bit. (c),(d) For the Huang-Shi bound the relation of the score of KCBS game $g_{\mathrm{KCBS}}$, number of total rounds $N$, test probability $q$, and randomness expansion rate with smoothing parameter $\delta ={10}^{-4}$ in Eq. (A4) of Appendix pp1. (c) The minimum number of rounds to have net randomness depending on the score $g_{\mathrm{KCBS}}$. Our experimental condition is shown as the green circle. (d) Randomness expansion rate at different $g_{\mathrm{KCBS}}$ and $q$ for our $N_{\exp }$. Our experimental $g_{\mathrm{KCBS}}=0.790$ and $q_{\exp }=0.0001$ are shown as the red circle, the resulting expansion rate being $2.3\times {10}^{-3}$ per bit, although our $q_{\exp }$ is not optimal for this case.

Figure 4

The results for random tests [53] of the outputs of the first measurement $a_i$ and the second measurement $a_j$, and both measurement $a_i{a}_j$. Outputs of $a_i^N$ and $a_j^N$ pass the listed tests since all $p$ values exceed the threshold of 0.01, while the outputs of ${(a_i{a}_j)}^N$ fail to pass the first test of “Cumulative Sums (CuSm).”