Applying the Simplest Kochen-Specker Set for Quantum Information Processing

Kochen-Specker (KS) sets are key tools for proving some fundamental results in quantum theory and also have potential applications in quantum information processing. However, so far, their intrinsic complexity has prevented experimentalists from using them for any application. The KS set requiring the smallest number of contexts has been recently found. Relying on this simple KS set, here we report an input state-independent experimental technique to certify whether a set of measurements is actually accessing a preestablished quantum six-dimensional space encoded in the transverse momentum of single photons.

Figure 1

Experimental setup. The state preparation stage consists of an attenuated single-photon source and of two SLMs used to encode one of the 21 KS states on the linear transverse momentum of a single photon. These SLMs are connected to a personal computer (PC) which, for clarity’s sake, is not shown in the figure. The measurement projection stage employs two other SLMs to randomly project the incoming state onto one of the 21 KS states. An FPGA unit controls the entire experimental setup for testing the NC inequality. See main text for details.

Figure 2

Exclusivity tests performed for the 21 KS states. In these tests, the prepared state and the measured states are KS states that are supposed to be orthogonal. The KS states are numbered as they appear in Eq. (1) in Ref. [24]. The dashed line shows the parameter $\overline{ϵ}$ defined in the main text.

Figure 3

Experimental results for $\mathrm{\Sigma }$, defined in Eq. (1), using as initial states each of the KS states of KS21. The NC inequality (1) is equally violated for all the states tested. $N$-Context Limit (Ideal) indicates the noncontextual bound in the ideal case in which $\overline{ϵ}=0$. $N$-Context Limit indicates the bound when the actual value of $\overline{ϵ}$ obtained in the exclusivity tests is taking into account. Quantum Limit (Ideal) indicates the expected quantum result when measurements access the correct $d$-dimensional quantum system in the ideal case in which $\overline{ϵ}=0$. Quantum Lower (Upper) Limit indicates the limits expected when measurements access the correct $d$-dimensional quantum system and when the actual value of $\overline{ϵ}$ is taken into account.

Figure 4

Value of $\mathrm{\Sigma }$, defined in Eq. (1), as a function of the number of pulses sent, while using $|{\mathrm{KS}}_7⟩=(0,0,0,0,0,1)$ as initial state.

Figure 5

Demonstration of the quantum state independency of the violation of inequality (1) for the following states (normalization factors are omitted for simplicity): (i) $|{\varphi }_1⟩=(1,1,1,1,1,1)$, (ii) $|{\varphi }_2⟩=(1,0,0,0,1,0)$, (iii) a completely mixed state $\rho =\mathbb{1}/6$, (iv) ${\rho }_{|{\mathrm{KS}}_9⟩}^{w=30\%}=(1-w)|{\mathrm{KS}}_9⟩⟨{\mathrm{KS}}_9|+w\mathbb{1}/6$, which is a partial mixture composed of the pure state $|{\mathrm{KS}}_9⟩=(0,1,0,1,\omega ,{\omega }^2)$, where $\omega =e^{2\pi i/3}$, and 30% of added white noise, where $w$ is the amount of white noise and $\mathbb{1}$ is the identity matrix in dimension 6, and (v) $|{\mathrm{KS}}_9⟩$.