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L3-05 · Eigenvectors and observables

Find an eigenvalue and connect it to a possible measurement result.

Level 3 icon Level 3 · High-school bridge
Objective. Find an eigenvalue and connect it to a possible measurement result.
Before this lesson. L3-03
Hands-on lab. Qubits and the Bloch sphere

Learn

An eigenvector keeps its direction under a matrix transformation. The matrix can multiply the vector by a number called its eigenvalue. The vector must be nonzero. Solving an eigenvalue problem reveals directions that the transformation treats in a particularly simple way.

A Hermitian matrix equals its conjugate transpose. Its eigenvalues are real, and an orthonormal eigenbasis can be chosen in finite dimensions. Quantum observables use Hermitian operators. Their eigenvalues name possible outcomes in an ideal projective measurement, while the state determines the probabilities.

An expectation value is a weighted average of measurement outcomes. It need not be one of the possible outcomes. Measure many independently prepared copies to estimate it. A single measurement generally does not reveal the full state or its expectation value exactly.

Practice

Solution

Multiply each outcome by its probability and add. The expectation is 1(0.8) + (-1)(0.2) = 0.6. A single ideal measurement still returns either 1 or -1.

Check your understanding

Can the expectation of a two-outcome Z measurement equal 0.6?

Show the answer

A. Yes, as a weighted average

An average can lie between the possible outcomes. Individual ideal Z outcomes remain 1 and -1 in this convention.

Sources

This lesson draws on Watrous, 2018Preskill, n.d..

Next: L3-06 · Derivatives and gradients

References
  1. Watrous, J. (2018). The Theory of Quantum Information. https://cs.uwaterloo.ca/~watrous/TQI/
  2. Preskill, J. (n.d.). Course Information for Physics 219/Computer Science 219 Quantum Computation (Formerly Physics 229). California Institute of Technology course page with lecture notes. https://www.preskill.caltech.edu/ph219/