Observables and Hermitian Operators

Observables are represented by Hermitian operators, whose real eigenvalues are the possible measurement outcomes and whose eigenvectors are the outcome states.

Physical quantities as operators

In quantum mechanics every measurable quantity — energy, spin component, and so on — is represented by a Hermitian operator, called an observable. A matrix A is Hermitian when A = A-dagger (equal to its conjugate transpose). Hermiticity guarantees the properties a measurement needs.

Why Hermitian

• Real eigenvalues: measurement outcomes are real numbers
• Orthogonal eigenvectors: distinct outcomes are perfectly distinguishable
• A complete basis: eigenvectors span the space, so every state can be measured

The spectral picture

A Hermitian operator decomposes as A = sum_i a_i |a_i> are orthonormal eigenvectors. Measuring A returns some eigenvalue a_i with probability ||^2 (the Born rule), and the state collapses to the corresponding eigenvector |a_i>.

Qubit observables

For a qubit, the Pauli matrices X, Y, Z are the fundamental observables, each with eigenvalues +1 and -1. Measuring Z is a computational-basis measurement; measuring X is a measurement in the {|+>,|->} basis. Any qubit observable is a real combination of I, X, Y, Z.

Observables versus gates

Note the distinction: gates are unitary operators that evolve the state, while observables are Hermitian operators that describe measurements. The Paulis happen to be both Hermitian and unitary, so they play both roles, but in general the two categories are different. Understanding which role an operator plays is essential to reading quantum circuits correctly.