Mark Wilde

Gaussian Quantum Information—PHYS 7895

Time and Location: Tuesday and Thursday 10:30am-11:50am, Room: Nicholson 262

This course introduces the subject of Gaussian quantum information. Due to the experimental “ease” with which bosonic Gaussian states can be prepared and manipulated in the laboratory, and the theoretical elegance and striking simplicity of the bosonic Gaussian mathematical formalism, the topic of Gaussian quantum information seems to penetrate nearly every research area of interest in modern quantum information, including computing, communication, metrology, cryptography, etc. As such, it is thus essential to have a systematic introduction to and presentation of this fundamental topic, and the objective of this course is to provide such an introduction. In particular, we will cover far-ranging topics within Gaussian quantum information, as listed in the syllabus.

Required Textbook:

[1] Quantum Continuous Variables: A Primer of Theoretical Methods by Alessio Serafini

Other references:

[4] Lecture notes of Stephane Attal

Course grade: Pass / fail. Based on homework assignments and a final presentation.

Office Hours: TBA

Homeworks

Homework 2 due Friday, March 1, by 4pm in Nicholson 447

Homework 1 due Tuesday, February 5 in class

Lectures

All scribe notes as a single PDF

Lecture 24: heterodyne detection, general-dyne detection, and conditional dynamics
PDF

Lecture 23: experimental implementation of homodyne detection and its strong convergence to ideal homodyne detection
PDF

Lecture 22: graphical depiction of single-mode phase-insensitive bosonic Gaussian channels, channel transformations, combinations, partial transposition
PDF

Lecture 21: Holevo classification of single-mode bosonic Gaussian channels
PDF

Lecture 20: additive-noise channels, thermal channels, amplifier channels, phase-insensitive channels
PDF (some material taken from Ref. [1])

Lecture 19: Gaussian Stinespring dilation theorem (any Gaussian channel can be realized by Gaussian unitary acting on input + Gaussian environment state, followed by partial trace), action of adjoint of Gaussian channel on displacement operators
PDF (material taken from Ref. [1])

Lecture 18: two-mode squeezed vacuum state, Gaussian quantum channels, Choi state and channel uncertainty relation, Wigner function of a quantum channel
PDF (some material taken from Ref. [1])

Lecture 17: physical implementation of Gaussian unitary as linear-optical interferometer, array of single-mode squeezers, and linear-optical interferometer, along with Clements et al. decomposition of linear-optical interferometer
PDF (some material taken from Ref. [1] and Clements et al.)

Lecture 16: Gaussian unitaries, symplectic singular value decomposition of a symplectic matrix
PDF (some material taken from Ref. [1]) (Scribe notes: PDF | LaTeX)

Lecture 15: Characteristic and Wigner functions of Gaussian states, overlap formula for Gaussian states, Gaussian quadratic evolutions
PDF (some material taken from Ref. [1])

Lecture 14: Properties of characteristic function, quasi-probability distribution (Wigner function), phase-space point operators and their properties
PDF (some material taken from Ref. [1])

Lecture 13: characteristic functions and quasi-probability distributions, Hilbert-Schmidt orthogonality of displacement operators, expansion of density operator with characteristic function and displacement operators
PDF (some material taken from Ref. [1])
(Scribe notes: PDF | LaTeX)

Lecture 12: overlap formulas for quantum Gaussian states, including Holevo fidelity, Uhlmann fidelity, Petz-Renyi relative entropy, and sandwiched Renyi relative entropy
PDF
(Scribe notes: PDF | LaTeX)

Lecture 11: quantum relative entropy of faithful Gaussian states, Renyi entropies and powers of Gaussian states, purity and entropy of Gaussian states
PDF (some material taken from Ref. [1])
(Scribe notes: PDF | LaTeX)

Lecture 10: constraints on symplectic eigenvalues related to uncertainty principle, purification of Gaussian states, purity and entropy of Gaussian states
PDF (some material taken from Ref. [1])
(Scribe notes: PDF | LaTeX)

Lecture 9: Faithful Gaussian states as thermal states of quadratic Hamiltonians (cont'd.), Williamson theorem, relation between Hamiltonian matrix and covariance matrix for faithful Gaussian states
PDF (some material taken from Ref. [1])
(Scribe notes: PDF | LaTeX)

Lecture 8: Faithful Gaussian states as thermal states of quadratic Hamiltonians
PDF (some material taken from Ref. [1])
(Scribe notes: PDF | LaTeX)

Lecture 7 Addendum: Hamiltonian operators and symplectic matrices
PDF

Lecture 7: displacement operator and its properties, quadratic Hamiltonians and symplectic matrices
PDF (material taken from Ref. [1])
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Lecture 6: mean vector, covariance matrix, finite energy iff finite covariance matrix is finite, uncertainty principle for covariance matrix, covariance matrix is positive definite
PDF (material taken from Ref. [1])
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Lecture 5: creation, annihilation, position, and momentum operators, commutators, multiple modes, symplectic form
PDF (material taken from Ref. [1])
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Lecture 4: norm topology, weak operator topology, sepctral and singular value decompositions for compact operators, duality of trace-class and bounded operators, effects, partial trace, quantum channels, Stinespring
PDF (material taken from Ref. [3])
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Lecture 3: continuous functional calculus, polar decomposition, unitary operators, exponential map, trace-class operators, trace norm, Hilbert--Schmidt operators
PDF (material taken from Refs. [3] and [4] above)
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Lecture 2: bounded operators, operator norm, C*-algebra, spectrum, self-adjoint and positive operators
PDF (material taken from Ref. [3] above) (Scribe notes: PDF | LaTeX)

Lecture 1: introduction to course, background on separable Hilbert spaces
PDF (material taken from Ref. [3] above) (Scribe notes: PDF | LaTeX)

Last modified: January 01, 2023.