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 found a wide range of applications in quantum information processing tasks. However, the investigation and application of high-dimensional multi-input QRACs are still lacking. Here, we focus on $n$-dit string input QRACs with a $d$-dimensional system ($n^{(d)}\rightarrow1$ QRACs) and present a general method to find the maximum success probability of $n^{(d)}\rightarrow1$ QRACs. In particular, we give the analytical solution for maximum success probability of $3^{(d)}\rightarrow1$ QRACs under the limitation of mutually unbiased bases (MUBs). Based on the analytical solution, we show the relationship between MUBs and $n^{(d)}\rightarrow1$ 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, which means that the choice of the subset of MUBs will affect the final results of quantum information tasks. Second, we theoretically prove that MUBs are not the optimal measurement bases to obtain the maximum success probability of $n^{(d)}\rightarrow1$ 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 3-input QRACs up to dimension 11. Finally, for the first time, we experimentally confirm the OI-MUBs when $d=5$. Our results open new avenues for investigating the foundational properties of quantum mechanics and quantum network coding.