Three earlier chapters have all been quietly pointing at this one. Ch. 1 told you a gate matrix has to satisfy "one extra rule" beyond being any old 2×2 matrix. Ch. 2 gave you the dagger (†) notation for a vector's conjugate transpose. Ch. 4 generalizes † to matrices and uses it to state, precisely, the two rules that separate "a matrix that could be a real quantum gate" from "a matrix that's just numbers in a grid," and the rule that separates "something you could actually measure" from "something you can't."
The conjugate transpose of a matrix
You already met this for vectors in Ch. 2: the bra ⟨ψ| is the ket |ψ⟩ transposed into a row, with every entry conjugated. The exact same operation on a matrix is written M† ("M dagger"): transpose it, then conjugate every entry.
Unitary: the rule every quantum gate obeys
A matrix U is unitary if U†U = I. Here's why that specific condition is the right one: Ch. 3 required every valid qubit state to satisfy ⟨ψ|ψ⟩ = 1, and a gate has to take valid states to valid states. If U preserves every state's norm — ⟨Uψ|Uψ⟩ = ⟨ψ|ψ⟩ for every ψ — then writing that out as ψ†U†Uψ = ψ†ψ forces U†U = I directly. Unitarity isn't an arbitrary rule bolted onto quantum mechanics; it's just "doesn't break normalization," translated into one matrix equation.
This is exactly the "extra rule" Ch. 1 mentioned: the Shear preset in Fig. 1.2 fails it (Shear†·Shear ≠ I), which is precisely why no real quantum gate looks like a shear.
Hermitian: the rule every observable obeys
A matrix M is Hermitian if it equals its own dagger: M = M†. Concretely, that forces the diagonal entries to be real (a number that equals its own conjugate has to be real) and the two off-diagonal entries to be complex conjugates of each other.
Hermitian matrices have one property that makes them indispensable in quantum mechanics, and it's the reason this chapter exists right after Ch. 1's eigenvalue section: every eigenvalue of a Hermitian matrix is guaranteed to be a real number. Since a measurement outcome (Ch. 7) is defined as an eigenvalue of the observable being measured, and a measured quantity has to be a real number, observables are required to be represented by Hermitian matrices — there's no other option that would even make sense.
M†
M†M (check against I)
Both at once: Pauli matrices
Try I, X, Y, and Z in Fig. 4.1 — every one of them is both unitary and Hermitian simultaneously, with real eigenvalues of exactly +1 and −1. That's not a coincidence: it's exactly why the Pauli matrices do double duty throughout this book, as gates (Ch. 3, where unitary is the requirement) and as observables (Ch. 7, where Hermitian is the requirement). Most matrices are one, the other, neither, or — like the Pauli matrices — happily both.
- Video3Blue1Brown — Eigenvectors and eigenvaluesUseful refresher before the spectral-theorem claim in this chapter (Hermitian ⇒ real eigenvalues).
- TextbookNielsen & Chuang, §2.1.7–2.1.9The full spectral theorem and proof that Hermitian operators have real eigenvalues and orthogonal eigenvectors.
- ToolNumPy docs — linalg.eigvalsReference for the function used in Fig. 4.2's code cell.