Entropic uncertainty of random quantum states and their application to data encryption

Random quantum state, as column or row vectors of a random unitary operator, play a key role in random circuit sampling to demonstrate quantum supremacy, quantum chaotic systems, randomized benchmarking, etc. Here, we present one interesting property of random quantum states and their application to the encryption of data. First, we find that the Shannon entropy of random quantum states are invariant under the quantum Fourier transform. Thus, random quantum states have the equal entropic uncertainty in a computational basis and a quantum Fourier transformed basis. This is analogus to a coherent state that has the equal Heisenberg-Robertson uncertainty in the position and the momentum, or generally between two conjugate operators. If a random quantum state is noisy, its Shannon entropy is higher in a computational basis, but less in the quantum Fourier transformed basis. We show random quantum states generated by the Google Sycamore processor were nosy and do not satisfy the equal entropic uncertainty in a computational basis and a quantum Fourier transformed basis. The equal entropic uncertainty for ideal random quantum states could be used in analyzing noise in random circuits. Second, we show that random circuit could be used to encrypt classical data or a quantum state.