Linear optics only allows every possible quantum operation for one photon or one port

TL;DR

This paper demonstrates that linear optical systems can only implement arbitrary quantum operations in single-photon or single-mode scenarios, highlighting fundamental limitations for multi-photon and multi-mode quantum evolutions.

Contribution

It proves that universal quantum operations are impossible with linear optics unless involving only one photon or one mode, clarifying the scope of linear optical quantum computing.

Findings

Single-photon evolution can realize any unitary operation.

Single-mode evolution corresponds to trivial phase shifts.

Multi-photon and multi-mode evolutions cannot be fully implemented with linear optics.

Abstract

We study the evolution of the quantum state of $n$ photons in $m$ different modes when they go through a lossless linear optical system. We show that there are quantum evolution operators $U$ that cannot be built with linear optics alone unless the number of photons or the number of modes is equal to one. The evolution for single photons can be controlled with the known realization of any unitary proved by Reck, Zeilinger, Bernstein and Bertani. The evolution for a single mode corresponds to the trivial evolution in a phase shifter. We analyze these two cases and prove that any other combination of the number of photons and modes produces a Hilbert state too big for the linear optics system to give any desired evolution.