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

Juan Carlos Garcia-Escartin, Vicent Gimeno, Julio José Moyano-Fernández

TL;DR

The paper tackles the problem of determining which quantum operations on $n$ photons in $m$ modes can be realized with linear optics and provides a constructive recipe to obtain the corresponding interferometer. It reframes realizability as a Lie-algebraic problem using the adjoint representation of the photonic homomorphism $\varphi_{m,M}:U(m)\to U(M)$, turning the question into a linear decomposition of the desired Hamiltonian in the image subalgebra. The main contribution is a necessary-and-sufficient criterion: $U$ is optically realizable iff $\mathrm{Ad}_U|_{\mathfrak{d}}$ is an automorphism, together with an explicit procedure to recover $S$ when realizable. This framework enables exact construction of optical implementations (via beam splitters and phase shifters) for realizable operations and clarifies the role of matrix logarithms, with potential impact on linear-optics quantum computing and boson sampling.

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.

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

TL;DR

The paper tackles the problem of determining which quantum operations on photons in modes can be realized with linear optics and provides a constructive recipe to obtain the corresponding interferometer. It reframes realizability as a Lie-algebraic problem using the adjoint representation of the photonic homomorphism , turning the question into a linear decomposition of the desired Hamiltonian in the image subalgebra. The main contribution is a necessary-and-sufficient criterion: is optically realizable iff is an automorphism, together with an explicit procedure to recover when realizable. This framework enables exact construction of optical implementations (via beam splitters and phase shifters) for realizable operations and clarifies the role of matrix logarithms, with potential impact on linear-optics quantum computing and boson sampling.

Abstract

The evolution of quantum light through linear optical devices can be described by the scattering matrix of the system. For linear optical systems with possible modes, the evolution of input photons is given by a unitary matrix given by a known homomorphism, , which depends on the size of the resulting Hilbert space of the possible photon states, . We present a method to decide whether a given unitary evolution for photons in modes can be achieved with linear optics or not and the inverse transformation 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.

Paper Structure

This paper contains 23 sections, 4 theorems, 61 equations.

Table of Contents

  1. Introduction
  2. Organization of the paper
  3. Outline of the results and definitions
  4. Optical realizations and their implementation
  5. Linear optical evolution from the unitary (group) and Hermitian (algebra) matrices point of view
  6. Summary of the results
  7. The adjoint representation
  8. Bases for the $\mathfrak{u}(m)$ algebra and the image subalgebra
  9. Example space: 5 photons in 2 modes
  10. Existence of a unitary evolution via linear optics
  11. Example 1: An impossible operation
  12. Example 2: An optical realization
  13. Implementation of the possible operations using linear optics
  14. Implementation recipe
  15. Implementation example
  16. ...and 8 more sections

Key Result

Theorem 1

$U \in \mathrm{im} \varphi_{m,M} \Longleftrightarrow \mathrm{Ad}_U\mid_{\mathfrak{d}}$ is an automorphism.

Theorems & Definitions (5)

  • Definition 1
  • Theorem 1
  • Theorem 2
  • Lemma 1
  • Lemma 2