Majorization relations and entanglement generation in a beam splitter

TL;DR

This paper demonstrates that a beam splitter's ability to generate quantum entanglement is governed by majorization relations, which depend on photon number and transmittance, revealing conditions for monotonic entanglement growth and cases requiring catalysis.

Contribution

It establishes majorization relations for beam splitter outputs, showing how entanglement varies with photon number and transmittance, and introduces catalysis to recover majorization in incomparable cases.

Findings

Entanglement increases monotonically with photon number k.

Majorization holds within certain transmittance regions, indicating monotonic entanglement growth.

Catalysis can restore majorization when output states are incomparable.

Abstract

We prove that a beam splitter, one of the most common optical components, fulfills several classes of majorization relations, which govern the amount of quantum entanglement that it can generate. First, we show that the state resulting from k photons impinging on a beam splitter majorizes the corresponding state with any larger photon number k'>k, implying that the entanglement monotonically grows with k. Then, we examine parametric infinitesimal majorization relations as a function of the beam-splitter transmittance, and find that there exists a parameter region where majorization is again fulfilled, implying a monotonic increase of entanglement by moving towards a balanced beam splitter. We also identify regions with a majorization default, where the output states become incomparable. In this latter situation, we find examples where catalysis may nevertheless be used in order to recover majorization. The catalyst states can be as simple as a path-entangled single-photon state or a two-mode vacuum squeezed state.