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Orbit dimensions in linear and Gaussian quantum optics

Eliott Z. Mamon

Abstract

We study the dimension of the manifold of quantum states (called orbit) that a given quantum state of light can reach under the dynamics of linear or Gaussian quantum optics. That is, we investigate how many directions in the Hilbert space a given state can explore under these sub-universal regimes. We find that these orbit dimensions reveal fundamental insights into the structure of attainable state spaces (e.g. boson bunching does not increase the number of accessible directions) with multi-faceted consequences. By showcasing a simple way to compute this topological quantity, we reveal how it can alone yield no-go results for some transformations. Our framework is proven to hold in both discrete and continuous-variable settings, and can be used with Fock as well as phase-space representations such as the Wigner or stellar representations. We study genericity and robustness properties of orbit dimensions, and propose strategies to probe them using homodyne/heterodyne measurements on pure states, or photon counters on two copies of general states. We also highlight how under Gaussian unitaries, orbit dimensions witness non-Gaussianity of quantum states. Lastly, we rigorously establish implications of orbit dimensions for the number of directions that a bosonic variational circuit can explore in state space. Our approach sheds new light on the structure of reachable states in quantum optics, which can help practitioners understand limitations and sources of expressivity or non-Gaussianity in bosonic quantum information protocols such as quantum machine learning.

Orbit dimensions in linear and Gaussian quantum optics

Abstract

We study the dimension of the manifold of quantum states (called orbit) that a given quantum state of light can reach under the dynamics of linear or Gaussian quantum optics. That is, we investigate how many directions in the Hilbert space a given state can explore under these sub-universal regimes. We find that these orbit dimensions reveal fundamental insights into the structure of attainable state spaces (e.g. boson bunching does not increase the number of accessible directions) with multi-faceted consequences. By showcasing a simple way to compute this topological quantity, we reveal how it can alone yield no-go results for some transformations. Our framework is proven to hold in both discrete and continuous-variable settings, and can be used with Fock as well as phase-space representations such as the Wigner or stellar representations. We study genericity and robustness properties of orbit dimensions, and propose strategies to probe them using homodyne/heterodyne measurements on pure states, or photon counters on two copies of general states. We also highlight how under Gaussian unitaries, orbit dimensions witness non-Gaussianity of quantum states. Lastly, we rigorously establish implications of orbit dimensions for the number of directions that a bosonic variational circuit can explore in state space. Our approach sheds new light on the structure of reachable states in quantum optics, which can help practitioners understand limitations and sources of expressivity or non-Gaussianity in bosonic quantum information protocols such as quantum machine learning.

Paper Structure

This paper contains 56 sections, 24 theorems, 227 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Structure of orbits, and orbit dimensions
  3. States of quantum optics
  4. Unitary groups and Lie algebras of quantum optics
  5. Orbits of quantum optics
  6. Properties of orbit dimensions
  7. Invariance property
  8. Genericity on finite-dimensional shells
  9. Robustness from below
  10. Implications for number of directions explored in state space by bosonic variational quantum circuits
  11. Evaluating orbit dimensions
  12. Calculations in the Fock representation
  13. Calculations in phase-space representations
  14. Experimental estimation of orbit dimensions
  15. Relation with quantum non-Gaussianity
  16. ...and 41 more sections

Key Result

Theorem 1

Let $G$ and $\mathcal{B}_\mathfrak{g}=i\{H_1,\dots,H_d\}$ be one of the $m$-mode quantum optical unitary groups and associated Lie algebra bases considered in tab:Lie-algebra-bases. For any physical state $\ket{\psi}$ and physical density operator $\rho$, the orbits $\operatorname{Orb}_{G}(\ket{\psi

Figures (1)

  • Figure 1: Illustration of an orbit of a state $\ket{\psi}$ under group $G$ (red). The tangent space to the orbit at point $\ket{\psi}$ (yellow) encodes the possible movement directions (blue) under small evolutions from the group $G$. The vectors $H_I \ket{\psi}$ ($H_I \in \mathcal{B}_{\mathfrak{g}}$) are translations of a basis of these directions to the origin of the Hilbert space. The dimension of the orbit, equal to the tangent space dimension, is found by counting the number of linearly independent (w.r.t. real linear combinations) vectors $H_I \ket{\psi}$ (\ref{['eq:concrete-rank-formula-ket']}).

Theorems & Definitions (45)

  • Theorem 1
  • Remark 1
  • Proposition 1
  • Proposition 2
  • Theorem 2
  • Proposition 3
  • Conjecture 1
  • Lemma S1
  • proof
  • Theorem S1
  • ...and 35 more