1.1.3 Hermitian Operators#

Prompts

Why must observables be represented by Hermitian operators rather than arbitrary operators? What special property of Hermitian operators makes them suitable for describing measurements?
Find the eigenvalues and eigenvectors of \(\hat{X}\) and \(\hat{Z}\) by solving the characteristic equation. Why are all Pauli eigenvalues \(\pm 1\)?
For a general qubit state \(\vert\psi\rangle = \alpha\vert 0\rangle + \beta\vert 1\rangle\), write down the expectation values \(\langle\hat{X}\rangle\), \(\langle\hat{Y}\rangle\), \(\langle\hat{Z}\rangle\) in terms of \(\alpha\) and \(\beta\). What do these three numbers represent geometrically?
Prove that eigenvectors of a Hermitian operator with different eigenvalues are orthogonal. Does this result depend on the eigenvectors being normalized?
Interpret the Bloch vector: for a qubit state parameterized by angles \((\theta, \varphi)\), explain how the expectation values of the Pauli operators encode position on the Bloch sphere.

Lecture Notes#

Overview#

In quantum mechanics, observables (measurable quantities) are represented by Hermitian operators. This is a fundamental postulate: for measurement outcomes to be real numbers, the operators must satisfy special properties. This section establishes eigenvalues, spectral decomposition, and the Pauli matrices—the mathematical foundation that makes quantum predictions testable.

Why Hermitian?#

A quantum state is described by complex numbers (superposition coefficients). Yet every measurement outcome is a real number. How do we extract reality from complexity?

For a complex number \(z = x + \mathrm{i}y\), the real part is \(x = \frac{z + z^*}{2}\). The key: \(x = x^*\) (real part equals its complex conjugate). The same principle applies to operators: for an observable to yield real outcomes, we require:

(9)#\[ \hat{O}^\dagger = \hat{O} \]

This is the definition of a Hermitian (self-adjoint) operator.

Definition: Hermitian Conjugate

For an operator \(\hat{O}\), the Hermitian conjugate \(\hat{O}^\dagger\) is defined by:

\[ \langle \psi \vert \hat{O} \phi \rangle = \langle \hat{O}^\dagger \psi \vert \phi \rangle \]

for all states \(\vert \psi\rangle, \vert \phi\rangle\).

Operators as Matrices#

Quantum operators act on the Hilbert space (spanned by basis states \(\vert 0\rangle, \vert 1\rangle\)). Represent them as matrices using:

(10)#\[ O_{ij} = \langle i \vert \hat{O} \vert j \rangle \]

A matrix is Hermitian if all diagonal elements are real and off-diagonal elements are conjugate pairs: \(O_{ij} = O_{ji}^*\).

Example: matrix elements of \(\hat{X}\)

For \(\hat{X}\) on a qubit (basis \(\vert 0\rangle, \vert 1\rangle\)):

\[\begin{split} \hat{X}\vert 0\rangle = \vert 1\rangle \quad \Rightarrow \text{column 0} = \begin{pmatrix} 0 \\ 1 \end{pmatrix}, \quad \hat{X}\vert 1\rangle = \vert 0\rangle \quad \Rightarrow \text{column 1} = \begin{pmatrix} 1 \\ 0 \end{pmatrix} \end{split}\]

Thus \(\hat{X} = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}\).

Eigenvalues and Eigenvectors#

Some states \(\vert \psi\rangle\) are only rescaled by an operator:

(11)#\[ \hat{O}\vert \psi\rangle = \lambda \vert \psi\rangle \]

Here \(\vert \psi\rangle\) is an eigenvector and \(\lambda\) is its eigenvalue.

Theorem: Hermitian Operators Have Real Eigenvalues

If \(\hat{O}\) is Hermitian and \(\vert \psi\rangle\) is an eigenvector with eigenvalue \(\lambda\), then \(\lambda \in \mathbb{R}\).

Proof: real eigenvalues of Hermitian operators

Take \(\langle \psi\vert \) times the eigenvalue equation:

\[ \langle \psi\vert \hat{O}\vert \psi\rangle = \lambda \langle\psi\vert \psi\rangle = \lambda \]

Complex conjugate both sides:

\[ \langle \psi\vert \hat{O}\vert \psi\rangle^* = \lambda^* \]

But by Hermiticity:

\[ \langle \psi\vert \hat{O}\vert \psi\rangle^* = \langle \psi\vert \hat{O}^\dagger\vert \psi\rangle = \langle \psi\vert \hat{O}\vert \psi\rangle = \lambda \]

Thus \(\lambda = \lambda^*\), so \(\lambda\) is real.

Theorem: Orthogonality of Eigenvectors

If \(\hat{O}\) is Hermitian and \(\vert \psi_1\rangle, \vert \psi_2\rangle\) are eigenvectors with different eigenvalues \(\lambda_1 \neq \lambda_2\), then \(\langle\psi_1\vert \psi_2\rangle = 0\).

Proof: orthogonality of eigenvectors

From the eigenvalue equation:

\[ \langle \psi_1\vert \hat{O}\vert \psi_2\rangle = \lambda_2 \langle\psi_1\vert \psi_2\rangle \]

Also:

\[ \langle \psi_1\vert \hat{O}\vert \psi_2\rangle = \lambda_1 \langle\psi_1\vert \psi_2\rangle \]

(using Hermiticity). Equating gives:

\[ (\lambda_2 - \lambda_1)\langle\psi_1\vert \psi_2\rangle = 0 \quad \Rightarrow \quad \langle\psi_1\vert \psi_2\rangle = 0 \]

since \(\lambda_2 \neq \lambda_1\).

Spectral Decomposition and Pauli Operators#

Any Hermitian operator can be decomposed as:

(12)#\[ \hat{O} = \sum_i \lambda_i \vert \psi_i\rangle\langle \psi_i\vert \]

where \(\lambda_i\) are eigenvalues and \(\vert \psi_i\rangle\langle \psi_i\vert \) is the projector onto eigenstate \(\vert \psi_i\rangle\).

A general \(2\times 2\) Hermitian matrix has 4 real degrees of freedom—thus 4 linearly independent Hermitian \(2\times 2\) matrices: the identity \(\hat{I}\) and the three Pauli operators \(\hat{X}, \hat{Y}, \hat{Z}\). Every Hermitian \(2\times 2\) operator is uniquely \(a_0\hat{I} + a_x\hat{X} + a_y\hat{Y} + a_z\hat{Z}\) with \(a_0, a_x, a_y, a_z\in\mathbb{R}\).

Table: Pauli Operators and Spectral Decompositions

Operator	Matrix	Eigenstate (\(+1\))	Projector \(\hat{P}_{+1}\)
\(\hat{X}\)	\(\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}\)	\(\vert +\rangle = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ 1 \end{pmatrix}\)	\(\frac{1}{2}(\hat{I} + \hat{X})\)
\(\hat{Y}\)	\(\begin{pmatrix} 0 & -\mathrm{i} \\ \mathrm{i} & 0 \end{pmatrix}\)	\(\vert \mathrm{i}\rangle = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ \mathrm{i} \end{pmatrix}\)	\(\frac{1}{2}(\hat{I} + \hat{Y})\)
\(\hat{Z}\)	\(\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}\)	\(\vert 0\rangle = \begin{pmatrix} 1 \\ 0 \end{pmatrix}\)	\(\frac{1}{2}(\hat{I} + \hat{Z})\)

Pauli Algebra#

The Pauli matrices satisfy elegant algebraic relations:

(13)#\[ \hat{\sigma}^i \hat{\sigma}^j = \delta_{ij}\hat{I} + \mathrm{i}\epsilon_{ijk}\hat{\sigma}^k \]

Commutation relations:

(14)#\[ [\hat{\sigma}^i, \hat{\sigma}^j] = 2\mathrm{i}\epsilon_{ijk}\hat{\sigma}^k \]

For example: \([\hat{X}, \hat{Y}] = 2\mathrm{i}\hat{Z}\) and cyclic permutations.

Pauli operators do not commute

The Pauli operators do not commute. Measuring along one axis disturbs the result of measuring along another—a manifestation of the uncertainty principle.

Anti-commutation relations:

\[ \{\hat{\sigma}^i, \hat{\sigma}^j\} = 2\delta_{ij}\hat{I} \]

where \(\{\hat{A}, \hat{B}\} = \hat{A}\hat{B} + \hat{B}\hat{A}\) is the anti-commutator.

Discussion: must observables be Hermitian?

Are all physical observables necessarily Hermitian?

We require Hermiticity for real measurement outcomes. Yet creation operators \(\hat{a}^\dagger\) are non-Hermitian but physically meaningful. Recent work on non-Hermitian PT-symmetric Hamiltonians shows certain non-Hermitian operators can have entirely real spectra. What is the minimal mathematical requirement for an observable?

Expectation Values and the Bloch Sphere#

The expectation value of a Hermitian observable is:

(15)#\[ \langle\hat{O}\rangle = \langle\psi\vert\hat{O}\vert\psi\rangle \]

This is the average outcome over many measurements on identically prepared copies of \(\vert\psi\rangle\).

For a qubit \(\vert\psi\rangle = \alpha\vert 0\rangle + \beta\vert 1\rangle\) (with \(|\alpha|^2 + |\beta|^2 = 1\)):

\[ \langle\hat{X}\rangle = 2\operatorname{Re}(\alpha^*\beta), \quad \langle\hat{Y}\rangle = 2\operatorname{Im}(\alpha^*\beta), \quad \langle\hat{Z}\rangle = \vert\alpha|^2 - |\beta|^2 \]

In the Bloch parametrization \(\vert\psi\rangle = \cos(\theta/2)\vert 0\rangle + \mathrm{e}^{\mathrm{i}\varphi}\sin(\theta/2)\vert 1\rangle\):

(16)#\[ \langle\hat{X}\rangle = \sin\theta\cos\varphi, \quad \langle\hat{Y}\rangle = \sin\theta\sin\varphi, \quad \langle\hat{Z}\rangle = \cos\theta \]

The Bloch vector \(\boldsymbol{n} = (\sin\theta\cos\varphi,\,\sin\theta\sin\varphi,\,\cos\theta)\) lies on the unit sphere and completely specifies the qubit state.

Derivation: Pauli Expectation Values

From \((\alpha,\beta)\) to \((\theta,\varphi)\):

Substitute \(\alpha = \cos(\theta/2)\), \(\beta = \mathrm{e}^{\mathrm{i}\varphi}\sin(\theta/2)\):

\[ \alpha^*\beta = \cos(\theta/2)\,\mathrm{e}^{\mathrm{i}\varphi}\sin(\theta/2) = \tfrac{1}{2}\mathrm{e}^{\mathrm{i}\varphi}\sin\theta \]

Thus:

\[ \langle\hat{X}\rangle = 2\operatorname{Re}\!\left(\tfrac{1}{2}\mathrm{e}^{\mathrm{i}\varphi}\sin\theta\right) = \sin\theta\cos\varphi \quad \checkmark \]

\[ \langle\hat{Z}\rangle = \cos^2(\theta/2) - \sin^2(\theta/2) = \cos\theta \quad \checkmark \]

Poll: Measurement outcomes are eigenvalues

For a qubit in state \(\vert\psi\rangle = \frac{3}{5}\vert 0\rangle + \frac{4}{5}\vert 1\rangle\), you measure \(\hat{Z} = \vert 0\rangle\langle 0\vert - \vert 1\rangle\langle 1\vert\). What outcome(s) are possible?

(A) Always \(-7/25\) (the expectation value).

(B) Either \(+1\) or \(-1\) (the eigenvalues of \(\hat{Z}\)).

(D) Either \(3/5\) or \(4/5\) (the amplitudes of the state).

Summary#

Hermitian operators have real eigenvalues and orthogonal eigenvectors—essential for physical observables.
The Pauli matrices (\(\hat{X}, \hat{Y}, \hat{Z}\)) represent spin measurements along three orthogonal directions.
Spectral decomposition expresses any observable as a sum of projectors onto eigenstates.
Non-commuting observables cannot be simultaneously diagonal—measuring one disturbs the other.
Pauli expectation values (\(\langle\hat{X}\rangle, \langle\hat{Y}\rangle, \langle\hat{Z}\rangle\)) encode the qubit state as a point on the Bloch sphere.

Homework#

1. Spectral decomposition of a two-level Hamiltonian. Consider the Hermitian operator \(\hat{H} = \omega\,\hat{X} + \Delta\,\hat{Z}\), with \(\omega,\Delta\in\mathbb{R}\). This represents a generic two-level system in which \(\hat Z\) encodes an energy splitting \(2\Delta\) between \(\vert 0\rangle\) and \(\vert 1\rangle\) and \(\hat X\) provides a coupling of strength \(\omega\) between them.

(a) Verify that \(\hat H\) is Hermitian, and write its matrix in the \(\{\vert 0\rangle,\vert 1\rangle\}\) basis.

(b) Find the eigenvalues \(E_\pm\) and the corresponding eigenstates \(\vert E_\pm\rangle\). Express the result using the mixing angle \(\theta_0\) defined by \(\tan\theta_0 = \omega/\Delta\) (with \(\theta_0\in[0,\pi]\)).

(c) Write the spectral decomposition \(\hat H = E_+\vert E_+\rangle\langle E_+\vert + E_-\vert E_-\rangle\langle E_-\vert\) and use it to compute \(\hat H^2\) without doing any matrix multiplication. Show that \(\hat H^2 = (\omega^2+\Delta^2)\,\hat I\).

(d) Use the spectral form to write an integer power \(\hat H^n\) in closed form for any \(n\ge 1\).

2. Hermitian and anti-Hermitian decomposition. Any operator \(\hat A\) on a Hilbert space admits a unique decomposition \(\hat A = \hat H + \mathrm{i}\hat K\) with both \(\hat H\) and \(\hat K\) Hermitian.

(a) Show that \(\hat H = \tfrac{1}{2}(\hat A + \hat A^\dagger)\) and \(\hat K = \tfrac{1}{2\mathrm{i}}(\hat A - \hat A^\dagger)\) are both Hermitian, and that \(\hat A = \hat H + \mathrm{i}\hat K\).

(b) Apply to \(\hat A = \begin{pmatrix} 1 & \mathrm{i} \\ \mathrm{i} & 1\end{pmatrix}\). Compute \(\hat H\) and \(\hat K\) explicitly, and express each in terms of the Pauli operators \(\hat I, \hat X, \hat Y, \hat Z\).

(c) Draw the analogy with the complex-number decomposition \(z = x + \mathrm{i}y\): which operator plays the role of the real part of \(z\), and which the imaginary part? Explain in one sentence why this analogy makes the Hermitian operators the natural “real” subspace of the operator algebra.

3. Bloch vector from amplitudes. A qubit is in the state \(\vert\psi\rangle = \tfrac{1}{\sqrt 3}\vert 0\rangle + \sqrt{\tfrac{2}{3}}\,\vert 1\rangle\). Using the lecture’s amplitude formulas

\[ \langle\hat X\rangle = 2\operatorname{Re}(\alpha^*\beta),\quad \langle\hat Y\rangle = 2\operatorname{Im}(\alpha^*\beta),\quad \langle\hat Z\rangle = \vert\alpha\vert^2 - \vert\beta\vert^2, \]

compute the Bloch vector \(\boldsymbol n = (\langle\hat X\rangle, \langle\hat Y\rangle, \langle\hat Z\rangle)\). Verify \(\vert\boldsymbol n\vert = 1\), and explain in one sentence why this had to come out unit-length.

4. Eigenstate of spin along an arbitrary axis. Continuing from 1.1.2 Problem 7, the spin observable along a unit axis \(\boldsymbol{m}\) is \(\boldsymbol{m}\cdot\hat{\boldsymbol\sigma} = m_x\hat X + m_y\hat Y + m_z\hat Z\). Find the \(+1\) eigenstate explicitly for the axis \(\boldsymbol{m} = (\sin\theta_0, 0, \cos\theta_0)\) in the \(xz\)-plane.

(a) Write \(\boldsymbol{m}\cdot\hat{\boldsymbol\sigma}\) as an explicit \(2\times 2\) matrix.

(b) Solve the eigenvalue equation \((\boldsymbol{m}\cdot\hat{\boldsymbol\sigma})\vert\psi\rangle = +\vert\psi\rangle\). Show that the normalized \(+1\) eigenstate is \(\vert\psi\rangle = \cos(\theta_0/2)\vert 0\rangle + \sin(\theta_0/2)\vert 1\rangle\).

(c) Compute the Bloch vector of this \(\vert\psi\rangle\) from the formulas in Problem 3 and verify it equals \(\boldsymbol{m}\) itself — consistent with the result of 1.1.2 Problem 7 that the Bloch axis is the unique direction of maximal spin alignment.

5. Pauli decomposition of a 2x2 Hermitian operator. Any Hermitian operator on the qubit Hilbert space can be written uniquely as

\[ \hat O = a_0\,\hat I + a_x\,\hat X + a_y\,\hat Y + a_z\,\hat Z, \]

with \(a_0, a_x, a_y, a_z\in\mathbb{R}\) (lecture’s “\(4\) real degrees of freedom” statement).

(a) Using the lecture’s trace identities \(\operatorname{Tr}(\hat\sigma^i) = 0\), \(\operatorname{Tr}(\hat I) = 2\), and \(\operatorname{Tr}(\hat\sigma^i\hat\sigma^j) = 2\delta_{ij}\), show that the coefficients are recovered by

\[ a_0 = \tfrac{1}{2}\operatorname{Tr}(\hat O), \qquad a_i = \tfrac{1}{2}\operatorname{Tr}(\hat O\,\hat\sigma^i)\quad (i\in\{x,y,z\}). \]

(b) Apply to \(\hat O = \begin{pmatrix} 3 & 2-\mathrm{i} \\ 2+\mathrm{i} & 1\end{pmatrix}\). Compute \(a_0, a_x, a_y, a_z\) and write the result.

(c) For an observable of the form \(\hat O = a_0\,\hat I + \boldsymbol a\cdot\hat{\boldsymbol\sigma}\) with \(\boldsymbol a = (a_x, a_y, a_z)\), identify (i) what \(a_0\) represents, (ii) what \(\vert\boldsymbol a\vert\) controls, and (iii) what the direction \(\boldsymbol{e}_a = \boldsymbol a/\vert\boldsymbol a\vert\) encodes. (Hint: use the result of Problem 4 and the spectral structure of \(\boldsymbol{n}\cdot\hat{\boldsymbol\sigma}\) from Problem 7.)

6. Non-Hermitian eigenstates. The lecture proves that distinct eigenvalues of a Hermitian operator yield orthogonal eigenvectors. Investigate what happens when the operator is not Hermitian.

Consider the (non-Hermitian) operator

\[\begin{split} \hat A = \begin{pmatrix} 1 & 1 \\ 0 & 2\end{pmatrix}. \end{split}\]

(a) Verify that \(\hat A\) is not Hermitian.

(b) Find the two eigenvalues by solving the characteristic equation.

(c) Find the corresponding eigenvectors \(\vert v_1\rangle, \vert v_2\rangle\) (normalised). Compute the inner product \(\langle v_1\vert v_2\rangle\) and check whether it vanishes.

(d) Identify the step in the lecture’s proof of “orthogonal eigenvectors for distinct eigenvalues” that required Hermiticity. Why does the argument fail for \(\hat A\)?

7. Pauli square along an arbitrary axis. Use the Pauli multiplication law \(\hat\sigma^i\hat\sigma^j = \delta_{ij}\hat I + \mathrm{i}\epsilon_{ijk}\hat\sigma^k\) to prove that, for any unit vector \(\boldsymbol{n} = (n_x, n_y, n_z)\),

\[ (\boldsymbol{n}\cdot\hat{\boldsymbol\sigma})^2 = \hat I. \]

(a) Expand \((\boldsymbol{n}\cdot\hat{\boldsymbol\sigma})^2 = \sum_{i,j} n_i n_j\,\hat\sigma^i\hat\sigma^j\) using the Pauli multiplication law. Identify the two contributions (symmetric and antisymmetric in \(i,j\)).

(b) Use the symmetry of \(n_i n_j\) and the antisymmetry of \(\epsilon_{ijk}\) to argue that the \(\hat\sigma^k\) term vanishes after summation.

(c) Use the result to deduce that every eigenvalue of \(\boldsymbol{n}\cdot\hat{\boldsymbol\sigma}\) satisfies \(\lambda^2 = 1\), so \(\lambda = \pm 1\). Why does this confirm that “every spin observable along any axis has the same two-point spectrum \(\{+1, -1\}\)” — and not, say, more outcomes when \(\boldsymbol{n}\) is far from one of the coordinate axes?

8. Pauli multiplication from commutator and anti-commutator. The lecture gives two relations: the commutator \([\hat\sigma^i, \hat\sigma^j] = 2\mathrm{i}\epsilon_{ijk}\hat\sigma^k\) and the anti-commutator \(\{\hat\sigma^i, \hat\sigma^j\} = 2\delta_{ij}\hat I\). Use them to derive the product \(\hat\sigma^i\hat\sigma^j\) without computing any matrices.

(a) Write the defining identities for the operator product:

\[ \hat\sigma^i\hat\sigma^j = \tfrac{1}{2}\bigl(\{\hat\sigma^i,\hat\sigma^j\} + [\hat\sigma^i,\hat\sigma^j]\bigr). \]

Substitute the lecture’s commutator and anti-commutator to recover the Pauli multiplication law \(\hat\sigma^i\hat\sigma^j = \delta_{ij}\hat I + \mathrm{i}\epsilon_{ijk}\hat\sigma^k\).

(b) Specialise to \((i,j) = (1,2)\) to compute \(\hat X\hat Y\) and \(\hat Y\hat X\) directly. Verify \(\hat X\hat Y = \mathrm{i}\hat Z\) and \(\hat Y\hat X = -\mathrm{i}\hat Z\).

(c) The decomposition “product = (anti-commutator + commutator)/2” works for any pair of operators. Explain in one sentence why this identity is structural: every operator product splits into a symmetric (anti-commutator) and an antisymmetric (commutator) part, independent of whether the operators are Hermitian, unitary, or arbitrary.