Entanglement in the full state vector of boson sampling

TL;DR

This paper analyzes the entanglement properties of the full state vector in boson sampling, revealing how entanglement scales with system size and particle number, and providing insights into the structure of quantum states in this context.

Contribution

It introduces an exact method to compute the full state vector in boson sampling using generalized coherent states, enabling entanglement analysis for large systems.

Findings

Renyi entanglement entropies follow symmetric Page curves with maximum at equal partition.

Maximum entropy saturates as mode number increases in the collision-free subspace.

Entanglement builds up rapidly, reaching maximum before full mode distribution occurs.

Abstract

The full state vector of boson sampling is generated by passing S single photons through beam splitters of M modes. The initial Fock state is expressed withgeneralized coherent states, and an exact application of the unitary evolution becomes possible. Due to the favorable polynomial scaling in M , we can investigate Renyi entanglement entropies for moderate particle and huge mode numbers. We find (almost) Renyi index independent symmetric Page curves with maximum entropy at equal partition. Furthermore, the maximum entropy as a function of mode index saturates as a function of M in the collision-free subspace case. The asymptotic value of the entropy increases linearly with S. Furthermore, we show that the build-up of the entanglement leads to a cusp at subsystem size equal to S in the asymmetric entanglement curve. The maximum entanglement is reached surprisingly early before the mode population is distributed over the whole system.