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No-go theorems for photon state transformations in quantum linear optics

Pablo V. Parellada, Vicent Gimeno i Garcia, Julio-José Moyano-Fernández, Juan Carlos Garcia-Escartin

TL;DR

This work derives a necessary condition for transforming photon-number states in lossless linear optical networks that preserve the total photon number, by identifying conserved tangent and perpendicular invariants under the adjoint action $ ext{Ad}_U$ of optical evolutions. The authors formulate a closed, computable tangent invariant $I_t$ (and a complementary invariant $I_p$) for states and density matrices, and show how these invariants constrain possible state-to-state transformations, yielding concrete no-go results such as the impossibility of deterministic redistribution of photons among modes, deterministic Bell-state generation from separable inputs with any ancilla, and exact transformations between GHZ and W states. They provide explicit invariant formulas, reduced criteria, and illustrative examples (including a two-photon, two-mode scenario) and release software for computing these invariants ($ ext{QOptCraft}$). The results offer a practical screening tool for optical-state engineering and clarify fundamental limits of linear-optical state preparation. The work connects linear optics with Lie-algebraic structures, illustrating how algebraic invariants govern physically feasible evolutions and guiding future resource-efficient designs in photonic quantum information.

Abstract

We give a necessary condition for photon state transformations in linear optical setups preserving the total number of photons. From an analysis of the algebra describing the quantum evolution, we find a conserved quantity that appears in all allowed optical transformations. We comment some examples and numerical applications, with example code, and give three general no-go results. These include (i) the impossibility of deterministic transformations which redistribute the photons from one to two different modes, (ii) a proof that it is impossible to generate a perfect Bell state in heralded schemes with a separable input for any number of ancillary photons and modes and a fixed herald and (iii) a restriction for the conversion between different types of entanglement (converting GHZ to W states).

No-go theorems for photon state transformations in quantum linear optics

TL;DR

This work derives a necessary condition for transforming photon-number states in lossless linear optical networks that preserve the total photon number, by identifying conserved tangent and perpendicular invariants under the adjoint action of optical evolutions. The authors formulate a closed, computable tangent invariant (and a complementary invariant ) for states and density matrices, and show how these invariants constrain possible state-to-state transformations, yielding concrete no-go results such as the impossibility of deterministic redistribution of photons among modes, deterministic Bell-state generation from separable inputs with any ancilla, and exact transformations between GHZ and W states. They provide explicit invariant formulas, reduced criteria, and illustrative examples (including a two-photon, two-mode scenario) and release software for computing these invariants (). The results offer a practical screening tool for optical-state engineering and clarify fundamental limits of linear-optical state preparation. The work connects linear optics with Lie-algebraic structures, illustrating how algebraic invariants govern physically feasible evolutions and guiding future resource-efficient designs in photonic quantum information.

Abstract

We give a necessary condition for photon state transformations in linear optical setups preserving the total number of photons. From an analysis of the algebra describing the quantum evolution, we find a conserved quantity that appears in all allowed optical transformations. We comment some examples and numerical applications, with example code, and give three general no-go results. These include (i) the impossibility of deterministic transformations which redistribute the photons from one to two different modes, (ii) a proof that it is impossible to generate a perfect Bell state in heralded schemes with a separable input for any number of ancillary photons and modes and a fixed herald and (iii) a restriction for the conversion between different types of entanglement (converting GHZ to W states).
Paper Structure (29 sections, 11 theorems, 68 equations)

This paper contains 29 sections, 11 theorems, 68 equations.

Table of Contents

  1. Overview and summary of the results
  2. Quantum evolution in linear optical systems
  3. Preliminaries: The image algebra
  4. Optical realizations
  5. The image algebra
  6. Orthonormalization. A simple example.
  7. The adjoint map: invariants in linear optical transformations.
  8. Conserved quantities
  9. Proof:
  10. Proof:
  11. Allowed and forbidden transformations
  12. Density matrix description
  13. Example for two photons and two modes
  14. The invariant conservation condition is necessary but not sufficient. Counterexample: NOON states.
  15. Explicit formulae for the tangent invariant of a pure state and consequences
  16. ...and 14 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 (12)

  • Definition 1
  • Theorem 1
  • Proposition 1
  • Corollary 1
  • Corollary 2
  • Theorem 1
  • Theorem 2
  • Corollary 3
  • Corollary 4
  • Corollary 5
  • ...and 2 more