The bosonic birthday paradox

TL;DR

This paper establishes a quantum version of the birthday paradox for bosons, showing that fewer bosons are needed to find two in the same state compared to distinguishable particles, with implications for quantum and classical probability.

Contribution

It proves a bosonic birthday paradox, generalizes the hypothesis, and links quantum boson statistics to classical probability theorems.

Findings

Bosons require approximately √n particles for a collision, half as many as distinguishable particles.

Bosons in a uniform mixed state exhibit birthday statistics similar to classical multinomial distributions.

Provides a quantum proof of a classical probability theorem and a classical proof for comparison.

Abstract

We motivate and prove a version of the birthday paradox for $k$ identical bosons in $n$ possible modes. If the bosons are in the uniform mixed state, also called the maximally mixed quantum state, then we need $k \sim \sqrt{n}$ bosons to expect two in the same state, which is smaller by a factor of $\sqrt{2}$ than in the case of distinguishable objects (boltzmannons). While the core result is elementary, we generalize the hypothesis and strengthen the conclusion in several ways. One side result is that boltzmannons with a randomly chosen multinomial distribution have the same birthday statistics as bosons. This last result is interesting as a quantum proof of a classical probability theorem; we also give a classical proof.