Quantum superposition of high-dimensional states enables both computational speed-up and security in cryptographic protocols. However, the exponential complexity of tomographic processes makes certification of these properties a challenging task. In this work, we experimentally certify coherence witnesses tailored for quantum systems of increasing dimension using pairwise overlap measurements enabled by a six-mode universal photonic processor fabricated with a femtosecond laser writing technology. In particular, we show the effectiveness of the proposed coherence and dimension witnesses for qudits of dimensions up to 5. We also demonstrate advantage in a quantum interrogation task and show it is fueled by quantum contextuality. Our experimental results testify to the efficiency of this approach for the certification of quantum properties in programmable integrated photonic platforms.
A dimension witness provides a device-independent certification of the minimal dimension required to reproduce the observed data without imposing assumptions on the functioning of the devices used to generate the experimental statistics. In this paper, we provide a family of Bell expressions where Alice and Bob perform ${2}^{n\ensuremath{-}1}$ and
n number of dichotomic measurements, respectively, which serve as the device-independent dimension witnesses of Hilbert space of
2m dimensions with $m=1,2,...,{2}^{\ensuremath{\lfloor}n/2\ensuremath{\rfloor}}$. The family of Bell expressions considered here determines the success probability of a communication game known as the
n-bit parity-oblivious random access code. The parity obliviousness constraint is equivalent to the preparation noncontextuality assumption in an ontological model of an operational theory. For any given $n\ensuremath{\ge}3$, if such a constraint is imposed on the encoding scheme of the random access code, then the local bound of the Bell expression reduces to the preparation noncontextual bound. We provide explicit examples for the
n=4and5 case to demonstrate that the relevant Bell expressions certify the qubit and two-qubit system, and for the
n=6 case to demonstrate that the relevant Bell expression certifies the qubit, two-qubit, and three-qubit systems. We further demonstrate the sharing of quantum preparation contextuality by multiple Bobs sequentially to examine whether the number of Bobs sharing the preparation contextuality is dependent on the dimension of the system. We provide explicit examples of
n=5 and 6 to demonstrate that the number of Bobs sequentially sharing the contextuality remains the same for any of the
2m-dimensional systems.