# A method to determine which quantum operations can be realized with linear optics with a constructive implementation recipe

## TL;DR

This paper introduces a method to determine if a given quantum operation can be realized with linear optical devices and provides a constructive recipe for implementation, advancing the design of optical quantum systems.

## Contribution

It presents a novel approach to decide realizability of quantum operations with linear optics and offers a constructive implementation method based on Lie algebra analysis.

## Key findings

- Method to verify if a unitary can be implemented with linear optics.
- Inverse transformation for implementable unitaries.
- Framework to design optical systems for specific quantum operations.

## Abstract

The evolution of quantum light through linear optical devices can be described by the scattering matrix $S$ of the system. For linear optical systems with $m$ possible modes, the evolution of $n$ input photons is given by a unitary matrix $U=\varphi_{m,M}(S)$ given by a known homomorphism, $\varphi_{m,M}$, which depends on the size of the resulting Hilbert space of the possible photon states, $M$. We present a method to decide whether a given unitary evolution $U$ for $n$ photons in $m$ modes can be achieved with linear optics or not and the inverse transformation $\varphi_{m,M}^{-1}$ when the transformation can be implemented. Together with previous results, the method can be used to find a simple optical system which implements any quantum operation within the reach of linear optics. The results come from studying the adjoint map bewtween the Lie algebras corresponding to the Lie groups of the relevant unitary matrices.