High-dimensional multi-input quantum random access codes and mutually unbiased bases

Quantum random access codes (QRACs) provide a basic tool for demonstrating the advantages of quantum resources and protocols, which have a wide range of applications in quantum information processing tasks. However, the investigation and application of high-dimensional (d) multi-input (n) n^(d)→1 QRACs are still lacking. Here, we present a general method to find the maximum success probability of n^(d)→1 QRACs. In particular, we give the analytical solution for maximum success probability of 3^(d)→1 QRACs when measurement bases are mutually unbiased bases (MUBs). Based on the analytical solution, we show the relationship between MUBs and n^(d)→1 QRACs. First, we provide a systematic method of searching for the operational inequivalence of MUBs (OI-MUBs) when the dimension d is a prime power. Second, we theoretically prove that, surprisingly, the commonly used Galois MUBs are not the optimal measurement bases to obtain the maximum success probability of n^(d)→1 QRACs, which indicates a breakthrough according to the traditional conjecture regarding the optimal measurement bases. Furthermore, based on high-fidelity high-dimensional quantum states of orbital angular momentum, we experimentally achieve two-input and three-input QRACs up to dimension 11. We experimentally confirm the OI-MUBs when d=5. Our results open alternative avenues for investigating the foundational properties of quantum mechanics and quantum network coding.