Hybrid ququart-encoded quantum cryptography protected by Kochen-Specker contextuality

Quantum cryptographic protocols based on complementarity are not secure against attacks in which complementarity is imitated with classical resources. The Kochen-Specker (KS) theorem provides protection against these attacks, without requiring entanglement or spatially separated composite systems. We analyze the maximum tolerated noise to guarantee the security of a KS-protected cryptographic scheme against these attacks and describe a photonic realization of this scheme using hybrid ququarts defined by the polarization and orbital angular momentum of single photons.

Figure 1

The protocol is based on a KS set of 18 states which can be grouped in 9 bases represented by 9 colors. Every state belongs to two different bases. No set of 18 balls can have all the properties required to imitate the KS set; at least two balls must have different symbols. Therefore, the imitation can be detected.

Figure 2

(a) Setup for the generation of ququart states: One of the two photons emitted by SPDC acts as a trigger, while the other one is sent to a polarizing beam splitter (PBS), wave plates, and a quantum transferrer based on the $q$ plate in order to generate the desired ququart. (b) Setup for the analysis of bases (I-III-IV-IX): The setup in the dotted rectangle analyzes the four states of basis I; basis III can be measured by inserting a half-wave plate (HWP) at $\pi /8$ before the PBS. A polarizing Sagnac interferometer (PSI) and a quarter-wave plate are needed to analyze bases IV and IX (adding a HWP at $\pi /8$ before the PSI). (c) Setup for the analysis of bases (II-V-VI-VII-VIII): The part in the dotted rectangle is suitable to sort the four states of all the bases (the gray wave plate can be a HWP or a QWP depending on the particular basis as shown in Table ); this part is sufficient to analyze basis VIII. Basis II can be analyzed by adding a mode converter (MC). Using a PSI before the MC makes it possible to analyze bases VI (adding a HWP at $\pi /8$) and VII. Finally, the states of basis V can be sorted by an additional PSI and QWP. The pictures in the three boxes on the right represent the Sagnac interferometer, the LG mode sorter, and the cylindrical lens mode converter, respectively. The detection stage consists of a $q$ plate, a single-mode fiber, and a detector.