Registration Deadline: January 24, 2027

Instructor: Professor Jason Crann, Carleton University

Course Dates: January 7 - April 8, 2027
Mid-Semester Break: February 15-19, 2027
Lecture Time: Tuesdays & Thursdays | 2:35 PM - 3:55 PM (ET)
Office Hours: Thursdays | 1:30 PM - 2:30 PM (ET)

Registration Fee:

• Students from our Principal Sponsoring & Affiliate Universities: Free
• Other Students: CAD$500

Capacity Limit: 30 students (Auditing is allowed; please indicate if you are auditing in your registration)

Format: Online via Zoom

Course Description

Continuous variable quantum information aims to encode and process information using continuous degrees of freedom, such as the quadrature amplitudes of quantum harmonic oscillators or electromagnetic elds. Its increasing experimental realization within quantum optics makes it a desirable platform for quantum computation and communication. All the while, it has a rich mathematical foundation, incorporating group theory, representation theory, symplectic liner algebra, Fourier and functional analysis. The course will introduce students to the mathematical formalism of continuous variable quantum information together with important applications such as Gaussian states and channels, homodyne detection, classical communication over Gaussian channels and continuous variable quantum key distribution. Successful students will have the knowledge base to read current literature in continuous variable quantum information theory.

Approximate List of Topics

Part I: Gaussian States and Channels

• Operator theoretic preliminaries: position and momentum, canonical commutation relations. Schwartz space, Weyl operators, Stone-von Neumann theorem.
• Phase space: Wigner and characteristic functions, equivalence with density operators.
• Gaussian states: quadratic Hamiltonians, covariance matrices, the symplectic group, normal mode decompositions. Coherent states. Separability and partial transposition.
• Measurements: rigorous formulation of homodyne and heterodyne detection.
• Gaussian channels: completely positive trace preserving maps in innite dimensions.
• Action of Gaussian channels on covariance matrices. Energy-constrained diamond norm, application to robustness of Gaussian states.

Part II: Information Capacities and Quantum Key Distribution

• Information and entanglement measures: von Neumann entropy, relative entropy, monotonicity, mutual information, entanglement entropy.
• Classical capacity over Gaussian channels.
• Teleportation stretching: innite-dimensional state channel duality. Continuous variable quantum teleportation, and teleportation stretching.
• Capacities for pure loss channels.
• Quantum key distribution with coherent states, proof of security.

Prerequisites: High-level of mathematical maturity. Linear algebra and functional analysis. Knowledge of elementary quantum mechanics and/or quantum computing is an asset, but not necessary.

Evaluation: Final project (50%) consisting of a 5-10 page written summary of a topic related to the course, an oral examination (40%) consisting of a one-on-one 30-minute (virtual) meeting with the instructor, and attendance/participation in lecture (10%). Details to be provided at a later date.

Textbook & Resources

A. Serani, Quantum Continuous Variables: A Primer of Theoretical Methods, CRP Press, 2017.

Other sources:

• F. Grosshans and P. Grangier, Continuous Variable Quantum Cryptography Using Coherent States, Phys. Rev. Lett. 88, 057902.
• A. S. Holevo, Quantum Systems, Channels, Information. A Mathematical Introduction, De Gruyter Studies in Mathematical Physics, 16. De Gruyter, Berlin, 2012.
• S. Pirandola, R. Laurenza, C. Ottaviani et al. Fundamental limits of repeaterless quantum communications, Nat Commun 8, 15043 (2017).
• C. Weedbrook et. al., Gaussian quantum information, Reviews of Modern Physics, 84 (2) (2021), 621-669.