Realization of a Quantum Random Generator Certified with the Kochen-Specker Theorem

Random numbers are required for a variety of applications from secure communications to Monte Carlo simulation. Yet randomness is an asymptotic property, and no output string generated by a physical device can be strictly proven to be random. We report an experimental realization of a quantum random number generator (QRNG) with randomness certified by quantum contextuality and the Kochen-Specker theorem. The certification is not performed in a device-independent way but through a rigorous theoretical proof of each outcome being value indefinite even in the presence of experimental imperfections. The analysis of the generated data confirms the incomputable nature of our QRNG.

Figure 1

The theoretical protocol of the QRNG certified by the strong Kochen-Specker theorem proposed in Ref. [12]. The protocol is formulated for a spin-1 particle and consists of two sequential measurements. The first measurement is used to initialize the particle in the $S_z=0$ eigenstate of the spin operator $S_z$. The second measurement is performed in the eigenbasis of the $S_x$ operator with the two outcomes $S_x=\pm 1$ realized randomly as proven by the Kochen-Specker theorem. The outcome $S_x=0$ is never realized in the ideal case but can be used to monitor the quality of the protocol implementation.

Figure 2

(a) Simplified diagram of the measurement setup. A transmon-type multilevel quantum system is incorporated into a 3D microwave copper cavity attached to the cold stage of a dilution cryostat. A magnetically tunable Josephson junction (SQUID) is used to control the transition frequency of the qutrit by a superconducting coil attached to the cavity. Amplitude-controlled and phase-controlled microwave pulses are applied to the input port of the cavity by a quadrature IF (IQ) mixer driven by a local oscillator (LO) and sideband modulated by an arbitrary waveform generator (AWG). The measurement signals transmitted through the cavity are amplified by a quantum Josephson parametric amplifier (JPA), a high-electron-mobility transistor (HEMT) amplifier at 4 K, and a chain of room temperature (RT) amplifiers. The sample at 20 mK is isolated from the higher-temperature stages by three circulators ($C$) in series. The amplified transmission signal is down-converted to an intermediate frequency of 25 MHz in an IQ mixer driven by a dedicated LO and is digitized by an analog-to-digital converter (ADC) for data analysis. (b) The energy-level diagram of a qutrit coupled to a microwave cavity. The transition frequencies of the qutrit and cavity are in gigahertz, while the anharmonicity of the qutrit is $\sim 300\mathrm{MHz}$. When the coupling $g$ between the transmon and the cavity is much smaller than their mutual detuning, the system is in the dispersive regime used for measurement of the qutrit.

Figure 3

(a) Hexbin histogram plot of single-shot three-level readout of different qutrit states. Red, ground qutrit state; green, excited state ($|1⟩$); blue, second excited state ($|2⟩$). The intensity of the color represents the number of measurement outcomes falling in each bin. (b) Hexbin plot of the output of the protocol. Shown are the logical encoding of the resulting states and the correspondence to the spin-1 protocol. Note that the $S_x=0$ state is almost ($<0.1\%$) never realized. The black lines sketch the boundaries of the classification regions.