The bosonic birthday paradox

Understand

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.