Course notes, last updated March 4, 2024. (To be finished spring 2026!)
Syllabus, spring 2024 version:
Overview: This course is intended as an introduction to the field of quantum measurements. The primary goal is understanding of the basics of measurements in quantum mechanics, and in particular how the formal theoretical description can be implemented in a wide range of physical architectures. The role of quantum properties in the single-body (non-commuting observables) and many-body (entanglement) settings will be emphasized. Examples covered will include systems of interest in both quantum computing (qubits), metrology (atomic clocks, interferometers), as well as fundamental physics (gravitational wave detection).
Requirements: Students are expected to have understanding of quantum mechanics at the level of a standard introductory graduate course (e.g., based on Sakurai’s textbook). In particular, students should be comfortable with calculations in bra-ket notation, simple systems like the harmonic oscillator and spin-½ particle, and basic atomic physics concepts like electron energy levels. Familiarity with basic statistics will also be assumed (hypothesis testing, variance and standard deviation, etc.).
Readings/references: We will not follow a particular textbook, but these all have valuable materials and I will draw from them periodically:
A. Peres, Quantum Theory: Concepts and Methods
G. Milburn and H. Wiseman, Quantum Measurement and Control
K. Jacobs, Quantum Measurement Theory and its Applications
A. Jordan and I. Siddiqi, Quantum Measurement: Theory and Practice
J. Preskill, Lecture Notes on Quantum Computation, http://theory.caltech.edu/~preskill/ph229/
Office hours: Tuesdays 4-5pm, Phys North 395. Please note that I will not respond to technical questions sent via email! If you want to talk physics, show up to office hours or catch me after class.
Evaluation: Weekly homeworks, assigned fridays, due the following friday. Mid term and final exams will be 24-hour take home exams. Midterm tentatively given March 8, final is May 9 as set by the registrar. Grades will be 50% homework, 50% exams.
Attendance policy: This is a graduate elective course, so I assume you are an adult and will attend as you see the need. Note however that lecture notes and class recordings will be posted sporadically, so if you miss a lecture you may be on your own to figure out how to get the materials.
Weekly topics:
1 What is a measurement?
Born’s rule and when it does/does not apply
General measurement rules (POVMs and state updates)
What the measurement problem is and why we are not going to discuss it further
2 Qubit measurements
Projective and weak qubit measurements
Projective measurements as the limit of a sequence of weak measurements
Examples: Stern-Gerlach, hyperfine qubits
3 Continuous variable measurements
Projective and weak CV measurements
Measurement disturbance/backaction
Examples: free particle, electromagnetic field mode
4-5 Quantum state tomography
Definition of local measurements
Statement of “axiom of local tomography”
Explicit algorithm on N qubits: Pauli tomography
Homodyne tomography
6-7 Time evolution and noise
Quantum channels, operations, and their Stinespring dilations
Reversibility and unitarity
Decoherence, dephasing
Markov approximations, Lindblad equation
8-9 Continuous weak measurements
Continuous measurement, input-output formalism
Standard quantum limits
Examples: superconducting qubits, LIGO
10 Classical vs quantum states, single degree of freedom
Glauber’s theory of coherence
Squeezed light and its detection
11 Classical vs quantum states, multipartite
Bell locality
Aspect/CHSH experiments
Entanglement witness theory
12-13 Advantages of quantum states
Quantum sensing of external parameters
Fisher information, Cramer-Rao bounds
Heisenberg scaling
Sub-SQL continuous measurements
Applications: spin squeezing, GW detection with squeezed light