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Gaussian states in continuous variable quantum information

Alessandro Ferraro, Stefano Olivares, Matteo G. A. Paris

TL;DR

The notes present a comprehensive framework for continuous-variable quantum information with a central focus on Gaussian states, whose phase-space description is fully captured by covariance matrices and symplectic transformations. They survey how Gaussian states are generated, manipulated, and characterized (via Wigner/characteristic functions and Williamson diagonalisations), and how entanglement, separability, and nonlocality are analyzed in bipartite and multipartite Gaussian systems, including dynamics in noisy channels. The text also covers quantum measurements and state engineering in CV systems, detailing homodyne/tomography techniques and non-Gaussian de-Gaussification protocols that extend beyond Gaussian resources. Together, these methods underpin CV quantum-information tasks such as teleportation, cloning, and entanglement distribution, with a rigorous treatment of practical considerations like detector inefficiencies and environmental noise. The resulting toolkit highlights the central role of Gaussian states as both a practically accessible resource and a mathematically tractable platform for CV quantum information processing.

Abstract

These notes originated out of a set of lectures in Quantum Optics and Quantum Information given by one of us (MGAP) at the University of Napoli and the University of Milano. A quite broad set of issues are covered, ranging from elementary concepts to current research topics, and from fundamental concepts to applications. A special emphasis has been given to the phase space analysis of quantum dynamics and to the role of Gaussian states in continuous variable quantum information.

Gaussian states in continuous variable quantum information

TL;DR

The notes present a comprehensive framework for continuous-variable quantum information with a central focus on Gaussian states, whose phase-space description is fully captured by covariance matrices and symplectic transformations. They survey how Gaussian states are generated, manipulated, and characterized (via Wigner/characteristic functions and Williamson diagonalisations), and how entanglement, separability, and nonlocality are analyzed in bipartite and multipartite Gaussian systems, including dynamics in noisy channels. The text also covers quantum measurements and state engineering in CV systems, detailing homodyne/tomography techniques and non-Gaussian de-Gaussification protocols that extend beyond Gaussian resources. Together, these methods underpin CV quantum-information tasks such as teleportation, cloning, and entanglement distribution, with a rigorous treatment of practical considerations like detector inefficiencies and environmental noise. The resulting toolkit highlights the central role of Gaussian states as both a practically accessible resource and a mathematically tractable platform for CV quantum information processing.

Abstract

These notes originated out of a set of lectures in Quantum Optics and Quantum Information given by one of us (MGAP) at the University of Napoli and the University of Milano. A quite broad set of issues are covered, ranging from elementary concepts to current research topics, and from fundamental concepts to applications. A special emphasis has been given to the phase space analysis of quantum dynamics and to the role of Gaussian states in continuous variable quantum information.

Paper Structure

This paper contains 82 sections, 2 theorems, 531 equations, 19 figures, 1 table.

Table of Contents

  1. Preliminary notions
  2. Systems made of $\boldsymbol{n}$ bosons
  3. Matrix notations for bipartite systems
  4. Symplectic transformations
  5. Linear and bilinear interactions of modes
  6. Displacement operator
  7. Two-mode mixing
  8. Single-mode squeezing
  9. Two-mode squeezing
  10. Multimode interactions: ${\rm\bf SU} \boldsymbol{(p,q)}$ Hamiltonians
  11. Characteristic function and Wigner function
  12. Trace rule in the phase space
  13. A remark about parameters $\boldsymbol{\kappa}$
  14. Gaussian states
  15. Definition and general properties
  16. ...and 67 more sections

Key Result

Theorem 1

(Williamson) Given ${\boldsymbol V}\in {\rm M}(2n,{\mathbb R})$, ${\boldsymbol V}^{ T}={\boldsymbol V}$, ${\boldsymbol V}>0$ there exist ${\boldsymbol S}\in {\rm Sp}(2n,{\mathbb R})$ and ${\boldsymbol D}\in {\rm M}(n,{\mathbb R})$ diagonal and positive defined such that: Matrices ${\boldsymbol S}$ and ${\boldsymbol D}$ are unique, up to a permutation of the elements of ${\boldsymbol D}$.

Figures (19)

  • Figure 1: Schematic diagram of the balanced homodyne detector.
  • Figure 2: Schematic diagram of a heterodyne detection. Relevant modes are pointed out.
  • Figure 3: Realization of a triple coupler in terms of 50/50 beam splitters (BS) and phase shifters "$\phi$". In order to obtain a symmetric coupler the following values has to be chosen: $\phi_1=\arccos (1/3)$ and $\phi_2 = \phi_1 / 2$.
  • Figure 4: Outline of triple coupler homodyne detectors: the hexagonal box symbolizes the electronically performed Fourier transform (FT).
  • Figure 5: (a): measurement of the two-mode POVM $E(z)$ viewed as a single-mode measurement of the $\tau$-dependent POVM (\ref{['2pom:Pi1']}) on mode $a$; (b): the same for the POVM (\ref{['2pom:Pi2']}) on mode $b$.
  • ...and 14 more figures

Theorems & Definitions (2)

  • Theorem 1
  • Theorem 2