# Special Operators: Hermitian, Unitary, Projection, Positive
> Four families of operators carry all of quantum mechanics. **Hermitian** operators are
> observables (and Hamiltonians). **Unitary** operators are the reversible evolutions and quantum
> gates. **Projection** operators are measurements. **Positive** operators are density matrices and
> POVM elements. Each is a special case of "normal," so the spectral theorem from
> [0.1.5](05-eigen-spectral-theorem.md) describes them all — we just read off which eigenvalues each
> allows.
## Learning Objectives
After this lesson you will be able to:
1. Characterize **Hermitian** operators (real spectrum) and connect them to observables.
2. Prove that **unitary** operators preserve inner products and have unit-modulus eigenvalues.
3. Manipulate **orthogonal projection** operators and use them for measurement.
4. Define **positive (semidefinite)** operators and recognize density matrices.
5. Build **functions of operators**, especially $e^{-iHt}$, linking Hermitian generators to unitary gates.
---
## Intuition
Diagonalize a normal operator (spectral theorem): it's "multiply each eigen-direction by its
eigenvalue $\lambda_k$." The *type* of operator is then dictated entirely by **where the eigenvalues
live**:
| Operator | Constraint | Eigenvalues live on… | Role in QM |
|---|---|---|---|
| Hermitian | $A^\dagger = A$ | the real line $\mathbb{R}$ | observables, Hamiltonians |
| Unitary | $U^\dagger U = I$ | the unit circle $\{e^{i\theta}\}$ | gates, time evolution |
| Projection | $P^2 = P = P^\dagger$ | $\{0,1\}$ | measurement onto a subspace |
| Positive | $A^\dagger=A,\ \langle\psi|A|\psi\rangle\ge0$ | $[0,\infty)$ | density matrices, POVM elements |
Everything below is making this table precise.
---
## Theory
### Hermitian operators
$A$ is **Hermitian (self-adjoint)** if $A^\dagger = A$. By the spectral theorem (Hermitian ⇒ normal)
it has an orthonormal eigenbasis, and (proved in [0.1.5 E3](05-eigen-spectral-theorem.md)) **its
eigenvalues are real**. Conversely, a normal operator with real spectrum is Hermitian. So:
$$
A = A^\dagger \iff A = \sum_k \lambda_k|v_k\rangle\langle v_k| \text{ with } \lambda_k\in\mathbb{R}.
$$
This is the **measurement postulate's** backbone (Term 1.3): an observable is a Hermitian operator,
its eigenvalues are the possible measured values, and its eigenvectors are the states of definite
value. The expectation value $\langle A\rangle = \langle\psi|A|\psi\rangle = \sum_k \lambda_k
|\langle v_k|\psi\rangle|^2$ is a real, probability-weighted average of eigenvalues.
### Unitary operators
$U$ is **unitary** if $U^\dagger U = U U^\dagger = I$, i.e. $U^{-1} = U^\dagger$.
> **Theorem.** The following are equivalent: (i) $U$ is unitary; (ii) $U$ preserves inner products,
> $\langle U\phi|U\psi\rangle = \langle\phi|\psi\rangle$; (iii) $U$ maps an ONB to an ONB; (iv) $U$ is
> normal with all eigenvalues of modulus $1$.
**Proof of (i)⇔(ii).** $\langle U\phi|U\psi\rangle = \langle\phi|U^\dagger U|\psi\rangle$, which equals
$\langle\phi|\psi\rangle$ for all $\phi,\psi$ iff $U^\dagger U = I$. **(iv):** $U$ normal gives an
orthonormal eigenbasis with $U|v_k\rangle = \mu_k|v_k\rangle$; then $\langle v_k|U^\dagger U|v_k\rangle =
|\mu_k|^2 = 1$, so $\mu_k = e^{i\theta_k}$. ∎
Preserving inner products means **preserving probabilities and normalization** — which is precisely
why closed-system quantum evolution and every quantum gate must be unitary. Spectral form:
$$
U = \sum_k e^{i\theta_k}\,|v_k\rangle\langle v_k|, \qquad \theta_k\in\mathbb{R}.
$$
### Projection operators
$P$ is an **orthogonal projector** if $P^2 = P$ (idempotent) and $P^\dagger = P$ (Hermitian). Its
eigenvalues are $0$ or $1$ ([0.1.5 E2](05-eigen-spectral-theorem.md)); it projects onto the subspace
$\mathcal{S} = \operatorname{range}(P)$ along $\mathcal{S}^\perp$. For an orthonormal set
$\{|u_j\rangle\}$ spanning $\mathcal{S}$,
$$
P = \sum_j |u_j\rangle\langle u_j|, \qquad I - P = \text{projector onto } \mathcal{S}^\perp .
$$
A rank-1 projector $|\psi\rangle\langle\psi|$ (with $\|\psi\|=1$) projects onto the line through
$|\psi\rangle$. Projectors implement **projective measurement** (Term 1.3): the spectral projectors
$\{P_k\}$ of an observable give outcome $k$ with probability $\langle\psi|P_k|\psi\rangle$.
### Positive (semidefinite) operators
$A$ is **positive semidefinite** ($A \succeq 0$) if it is Hermitian and $\langle\psi|A|\psi\rangle\ge 0$
for all $|\psi\rangle$; equivalently, all eigenvalues are $\ge 0$. It is **positive definite**
($A\succ 0$) if strictly $>0$. Facts: any $A = B^\dagger B$ is positive; a positive operator has a
unique positive **square root** $\sqrt{A}$ (take $\sqrt{\lambda_k}$ in the spectral form). A **density
matrix** $\rho$ is exactly a positive operator with $\operatorname{Tr}(\rho)=1$ (Term 1.5), and POVM
elements are positive operators summing to $I$ (Term 1.3).
### Functions of operators and the exponential map
For normal $A = \sum_k\lambda_k P_k$ and any $f:\mathbb{C}\to\mathbb{C}$, define $f(A) = \sum_k
f(\lambda_k)P_k$ (consistent with power series where they converge). The case that powers the whole
field:
> **Hermitian generates unitary.** If $H = H^\dagger$ (Hermitian), then for real $t$,
> $$ U(t) = e^{-iHt} = \sum_k e^{-i\lambda_k t}\,|v_k\rangle\langle v_k| $$
> is **unitary**, because each eigenvalue $e^{-i\lambda_k t}$ has modulus $1$.
**Proof of unitarity.** $U^\dagger = \sum_k e^{+i\lambda_k t}|v_k\rangle\langle v_k|$ (real $\lambda_k$),
and $U^\dagger U = \sum_k e^{-i\lambda_k t}e^{+i\lambda_k t}|v_k\rangle\langle v_k| = \sum_k P_k = I$. ∎
This is the bridge from physics (Hamiltonians) to computation (gates): every gate $U = e^{-iHt}$ for
some Hermitian $H$ and "time" $t$. We use it constantly from Term 1.6 onward, and computationally it's
`scipy.linalg.expm(-1j*H*t)` (Appendix B). Conversely, by **Stone's theorem / the matrix log**, every
unitary is $e^{-iH}$ for some Hermitian $H$.
> **Caution.** $e^{A}e^{B} = e^{A+B}$ **only if** $[A,B]=0$. In general use Baker–Campbell–Hausdorff
> ([Appendix E](../../appendix/E-math-identities.md)); the failure of this identity is exactly why
> Trotterization is needed for Hamiltonian simulation (Term 3.4).
### Polar and singular-value decompositions (brief)
Every operator factors as $A = U|A|$ with $|A| = \sqrt{A^\dagger A}\succeq 0$ and $U$ unitary (**polar
decomposition**), and as $A = U\Sigma V^\dagger$ with $\Sigma\succeq0$ diagonal (**SVD**). We will need
the SVD for the **Schmidt decomposition** of entangled states (Term 1.4) and operator norms; flagged
here, developed when needed.
---
## Worked Examples
### Example 1 — $Z$ is Hermitian and unitary; build $R_z(\theta) = e^{-i\theta Z/2}$
$Z = \operatorname{diag}(1,-1)$ has real eigenvalues $\pm1$ (Hermitian) and $Z^\dagger Z = I$ (unitary)
— an involution, as in [0.1.4](04-linear-operators-matrices.md). Using the spectral form $Z =
(+1)|0\rangle\langle0| + (-1)|1\rangle\langle1|$:
$$
e^{-i\theta Z/2} = e^{-i\theta/2}|0\rangle\langle0| + e^{+i\theta/2}|1\rangle\langle1|
= \begin{pmatrix} e^{-i\theta/2} & 0 \\ 0 & e^{i\theta/2}\end{pmatrix} = R_z(\theta),
$$
the $z$-rotation gate ([Appendix E](../../appendix/E-math-identities.md)). The Hermitian $Z$
*generates* the unitary $R_z$ — a concrete instance of the bridge.
### Example 2 — A rank-1 projector and an expectation value
For $|\psi\rangle = \tfrac1{\sqrt2}(|0\rangle+|1\rangle) = |+\rangle$, the projector onto $|0\rangle$ is
$P_0 = |0\rangle\langle0| = \operatorname{diag}(1,0)$. Probability of outcome "0" measuring in the
computational basis:
$$
\langle\psi|P_0|\psi\rangle = \tfrac12\langle 0+1|\,|0\rangle\langle0|\,|0+1\rangle = \tfrac12\cdot 1 = \tfrac12 .
$$
And $\langle Z\rangle = \langle+|Z|+\rangle = \tfrac12(1 - 1) = 0$: the $\pm1$ outcomes are equally
likely, averaging to $0$. Consistent with the Bloch picture of $|+\rangle$ lying on the equator
(Term 1.2).
---
## Hands-on (Python)
```python
import numpy as np
from scipy.linalg import expm
def dag(A): return A.conj().T
Z = np.array([[1, 0], [0, -1]], dtype=complex)
def is_hermitian(A): return np.allclose(A, dag(A))
def is_unitary(A): return np.allclose(dag(A) @ A, np.eye(A.shape[0]))
def is_projector(A): return is_hermitian(A) and np.allclose(A @ A, A)
def is_positive(A, tol=1e-12):
w = np.linalg.eigvalsh((A + dag(A)) / 2) # symmetrize for numerical safety
return np.all(w >= -tol)
print(is_hermitian(Z), is_unitary(Z)) # True True (involution)
# Hermitian generates unitary: R_z(θ) = exp(-i θ Z / 2)
theta = 0.9
Rz = expm(-1j * theta * Z / 2)
print(is_unitary(Rz)) # True
print(np.round(Rz, 3)) # diag(e^{-iθ/2}, e^{+iθ/2})
```
```python
# Projector onto |0> and a Born probability for |+>:
ket0 = np.array([1, 0], dtype=complex)
plus = np.array([1, 1], dtype=complex) / np.sqrt(2)
P0 = np.outer(ket0, ket0.conj())
print(is_projector(P0)) # True
print(np.real(plus.conj() @ P0 @ plus)) # 0.5 = Born prob of outcome "0"
# A density matrix is positive with trace 1:
rho = 0.5 * np.eye(2) # maximally mixed qubit
print(is_positive(rho), np.isclose(np.trace(rho), 1)) # True True
# Square root of a positive operator:
A = np.array([[2, 0], [0, 9]], dtype=complex)
sqrtA = expm(0.5 * np.log(2) * np.array([[1,0],[0,0]]) # illustrative; in practice:
) # prefer scipy.linalg.sqrtm
from scipy.linalg import sqrtm
print(np.round(sqrtm(A).real, 3)) # diag(√2, 3)
```
---
## Exercises
**E1 (easy).** Show that $Y = \begin{psmallmatrix}0&-i\\ i&0\end{psmallmatrix}$ is Hermitian and unitary,
and find its eigenvalues.
Solution
$Y^\dagger = \overline{Y}^T = \begin{psmallmatrix}0&-i\\ i&0\end{psmallmatrix} = Y$ (Hermitian).
$Y^2 = I$ so $Y^\dagger Y = I$ (unitary). $\det(Y-\lambda I) = \lambda^2 - 1 = 0 \Rightarrow \lambda = \pm1$.
Solution
$(UV)^\dagger(UV) = V^\dagger U^\dagger U V = V^\dagger I V = V^\dagger V = I$. ∎ (Hence circuits, which
compose gates, are unitary.)
Solution
Since $H = H^\dagger$, $U^\dagger = (e^{iH})^\dagger = e^{(iH)^\dagger} = e^{-iH^\dagger} = e^{-iH}$.
Because $iH$ and $-iH$ commute, $U^\dagger U = e^{-iH}e^{iH} = e^{0} = I$. Hermiticity is what makes the
exponent of $U^\dagger$ the negative of that of $U$ (so they cancel); for non-Hermitian $H$ this fails. ∎
Solution
$(I-P)^\dagger = I - P$ and $(I-P)^2 = I - 2P + P^2 = I - 2P + P = I - P$, a projector. $P(I-P) = P - P^2
= P - P = 0$. Interpretation: $\{P, I-P\}$ are complementary outcomes of a yes/no measurement; their
probabilities $\langle\psi|P|\psi\rangle + \langle\psi|(I-P)|\psi\rangle = 1$, and a definite outcome
lands entirely in one subspace.
Solution
Spectrally $A = \sum_k\lambda_k|v_k\rangle\langle v_k|$ with $\lambda_k\ge0$. Define $\sqrt A =
\sum_k\sqrt{\lambda_k}|v_k\rangle\langle v_k|\succeq0$; then $(\sqrt A)^2 = A$. Uniqueness among positive
roots: any positive $R$ with $R^2 = A$ commutes with $A$ (since $RA = R^3 = AR$), hence shares $A$'s
eigenbasis, and on each eigenvector must multiply by the nonnegative root $\sqrt{\lambda_k}$ — so
$R = \sqrt A$. Finally $A = (\sqrt A)^\dagger(\sqrt A) = B^\dagger B$ with $B = \sqrt A$. ∎
Answers
1. Hermitian → $\mathbb{R}$; unitary → unit circle $e^{i\theta}$; projector → $\{0,1\}$; positive →
$[0,\infty)$.
2. $\langle U\phi|U\psi\rangle = \langle\phi|U^\dagger U|\psi\rangle = \langle\phi|\psi\rangle$ iff
$U^\dagger U=I$; preserving overlaps preserves probabilities/normalization, required of valid
evolutions and gates.
3. $P = \sum_j|u_j\rangle\langle u_j|$ over an ONB of the target subspace; eigenvalues $0$ and $1$.
4. For Hermitian $H$, $e^{-iHt}$ is unitary; thus every observable/energy generates a valid gate, and
every gate is $e^{-iH}$ for some Hermitian $H$.
5. Only when $[A,B]=0$; otherwise BCH adds commutator corrections, and Hamiltonian simulation must
Trotterize (Term 3.4).