Classical proxy of single-particle entanglement in quantum walks with classical randomness

Quantum walks utilize quantum coin operator and translation operators to generate superposition of spread wavefunctions. When classical randomness is introduced to the coin operator, a transition to localization occurs. The single-particle entanglement between the internal and positional degrees of freedom can be characterized by the entanglement entropy of the internal space, and we show that the entanglement survives in the presence of time-dependent and spatially dependent randomness in the localized regimes. We propose a classical proxy called the overlap, which literally measures the overlap of the probability distributions of the two internal states in real space. The overlap corresponds to the off-diagonal elements of the reduced density matrix and increases with its the purity. Therefore, the overlap negatively indicates entanglement in quantum walks and serves as an experimentally accessible proxy for the entanglement entropy. Our simulations of various types of quantum walks with different forms of classical randomness confirm the validity of the overlap, except in a special case with high population imbalance which blinds the overlap.