# Observables & the Measurement Postulate > Postulate 2 connects the abstract state vector to the laboratory. It tells us what is *measurable* > (the eigenvalues of Hermitian operators), with what *probability* (the Born rule), and what happens > to the state *afterward* (collapse). This is the postulate that makes quantum mechanics > probabilistic and irreversible-on-measurement — and it is built entirely from the spectral theorem > you proved in Term 0. Course 1.3 expands every piece; here we state the axiom and its core > consequences. ## Learning Objectives After this lesson you will be able to: 1. State **Postulate 2** using the spectral decomposition of a Hermitian observable. 2. Compute outcome probabilities via the **Born rule** and the **expectation value** $\langle A\rangle$. 3. Determine the **post-measurement state** (collapse) and explain repeatability. 4. Connect "observable = Hermitian operator" to the spectral theorem and real eigenvalues. 5. Simulate projective measurements in NumPy. --- ## Intuition Classically, you read off a property and the system is unchanged. Quantum measurement is different in three ways. First, the *possible results* of measuring a quantity are constrained — they are exactly the eigenvalues of the operator representing that quantity. Second, the *result is random*, with probabilities set by how much the state "overlaps" each eigenspace (the Born rule). Third, the act of measuring *changes the state*: it collapses onto the eigenspace of the result you got. Measure again immediately and you get the same answer — measurement is repeatable, but the first one already disturbed the system. --- ## Theory ### Postulate 2 (Observables and measurement) > **Postulate 2.** An **observable** is a Hermitian operator $A = A^\dagger$ on the state space, with > spectral decomposition (from [0.1.5](../../term-0-foundations/c01-linear-algebra/05-eigen-spectral-theorem.md)) > $$ A = \sum_k \lambda_k P_k, \qquad P_k P_l = \delta_{kl}P_k,\quad \sum_k P_k = I, $$ > where $\lambda_k$ are the (real, distinct) eigenvalues and $P_k$ the orthogonal projectors onto their > eigenspaces. Measuring $A$ on state $\lvert\psi\rangle$ yields outcome $\lambda_k$ with probability > $$ \boxed{\,p(\lambda_k) = \langle\psi| P_k |\psi\rangle = \lVert P_k\lvert\psi\rangle\rVert^2\,} \quad(\textbf{Born rule}), $$ > and immediately after a result $\lambda_k$, the state **collapses** to the normalized projection > $$ \lvert\psi'\rangle = \frac{P_k\lvert\psi\rangle}{\sqrt{p(\lambda_k)}} . $$ Why Hermitian? Because measured values are real numbers, and Hermitian operators are exactly those with **real eigenvalues** and an **orthonormal eigenbasis** ([0.1.6](../../term-0-foundations/c01-linear-algebra/06-special-operators.md)). The orthonormal eigenbasis guarantees the outcomes are perfectly distinguishable and the projectors resolve the identity. ### The Born rule is a probability distribution The numbers $p(\lambda_k)$ are a valid distribution: each $p(\lambda_k) = \langle\psi|P_k|\psi\rangle = \langle\psi|P_k^\dagger P_k|\psi\rangle = \lVert P_k\lvert\psi\rangle\rVert^2 \ge 0$ (projectors are Hermitian idempotents), and they sum to one, $$ \sum_k p(\lambda_k) = \sum_k \langle\psi|P_k|\psi\rangle = \langle\psi|\Big(\sum_k P_k\Big)|\psi\rangle = \langle\psi|I|\psi\rangle = \langle\psi|\psi\rangle = 1, $$ using completeness $\sum_k P_k = I$ and normalization. This is the physical meaning of the [Parseval identity](../../term-0-foundations/c01-linear-algebra/02-inner-products-norms.md) and of normalization from [1.1.1](01-state-postulate.md). For a **non-degenerate** observable with eigenvectors $\lvert v_k\rangle$, $P_k = \lvert v_k\rangle\langle v_k|$ and the Born rule takes its familiar form $p(\lambda_k) = |\langle v_k|\psi\rangle|^2$. ### Expectation value The **expectation value** (mean outcome over many measurements on identically prepared states) is $$ \langle A\rangle_\psi := \sum_k \lambda_k\, p(\lambda_k) = \sum_k \lambda_k \langle\psi|P_k|\psi\rangle = \langle\psi|\Big(\sum_k\lambda_k P_k\Big)|\psi\rangle = \langle\psi|A|\psi\rangle . $$ So the operator does double duty: its eigenvalues are the outcomes, and the sandwich $\langle\psi|A|\psi\rangle$ is their average. (This is real because $A$ is Hermitian.) The variance and the uncertainty relation are built from this in [1.3.2](../c03-measurement/02-expectation-uncertainty.md). ### Collapse and repeatability After outcome $\lambda_k$, the state is $P_k\lvert\psi\rangle/\sqrt{p(\lambda_k)}$, which lies entirely in the $\lambda_k$-eigenspace. Measure $A$ again: now $P_k\lvert\psi'\rangle = \lvert\psi'\rangle$ so $p(\lambda_k) = 1$ — the **same** result with certainty. Measurement is **repeatable**, but the first measurement has irreversibly disturbed any prior superposition. This irreversibility is unique to measurement; ordinary evolution (Postulate 3) is reversible. > **Compatibility (preview).** Two observables can be measured simultaneously with definite values iff > they **commute** ($[A,B]=0$), i.e. share an eigenbasis ([0.1.5](../../term-0-foundations/c01-linear-algebra/05-eigen-spectral-theorem.md)). > Incompatible observables (like $X$ and $Z$) obey an uncertainty relation — [1.3.2](../c03-measurement/02-expectation-uncertainty.md). --- ## Worked Examples ### Example 1 — Measuring $Z$ on $\lvert+\rangle$ The observable $Z = (+1)\lvert0\rangle\langle0| + (-1)\lvert1\rangle\langle1|$ has projectors $P_{+1} = \lvert0\rangle\langle0|$, $P_{-1} = \lvert1\rangle\langle1|$. On $\lvert+\rangle = \tfrac1{\sqrt2}(\lvert0\rangle+\lvert1\rangle)$: $$ p(+1) = \langle+|P_{+1}|+\rangle = |\langle0|+\rangle|^2 = \tfrac12, \qquad p(-1) = |\langle1|+\rangle|^2 = \tfrac12 . $$ Expectation $\langle Z\rangle = (+1)\tfrac12 + (-1)\tfrac12 = 0$. If the result is $+1$, the state collapses to $P_{+1}\lvert+\rangle/\sqrt{1/2} = \lvert0\rangle$. ### Example 2 — Same state, different observable ($X$ on $\lvert+\rangle$) Measure $X = (+1)\lvert+\rangle\langle+| + (-1)\lvert-\rangle\langle-|$ on $\lvert+\rangle$. Since $\lvert+\rangle$ *is* the $+1$ eigenvector, $p(+1) = |\langle+|+\rangle|^2 = 1$ and $p(-1) = 0$: a **certain** outcome, $\langle X\rangle = 1$, and no collapse. The same state is "random" for $Z$ but "definite" for $X$ — measurement outcomes are observable-relative, the seed of complementarity. --- ## Hands-on (Python) ```python import numpy as np ket0 = np.array([1, 0], dtype=complex) ket1 = np.array([0, 1], dtype=complex) plus = (ket0 + ket1) / np.sqrt(2) def projective_measure(psi, projectors, eigenvalues, rng=None): """Sample one outcome via the Born rule and return (eigenvalue, collapsed state).""" rng = rng or np.random.default_rng() probs = [np.real(psi.conj() @ (P @ psi)) for P in projectors] # <ψ|P_k|ψ> k = rng.choice(len(projectors), p=probs) collapsed = (projectors[k] @ psi) / np.sqrt(probs[k]) return eigenvalues[k], collapsed # Z-measurement on |+>: P0 = np.outer(ket0, ket0.conj()) # projector for eigenvalue +1 P1 = np.outer(ket1, ket1.conj()) # projector for eigenvalue -1 val, post = projective_measure(plus, [P0, P1], [+1, -1], rng=np.random.default_rng(0)) print("outcome:", val, " collapsed to:", np.round(post, 3)) # +1 -> |0> (rng-dependent) # Expectation = <ψ|Z|ψ> directly: Z = np.array([[1, 0], [0, -1]], dtype=complex) print("〈Z〉 =", np.real(plus.conj() @ Z @ plus)) # 0.0 ``` ```python # Empirical Born statistics over many shots match the predicted 50/50: from collections import Counter rng = np.random.default_rng(1) counts = Counter(projective_measure(plus, [P0, P1], [+1, -1], rng)[0] for _ in range(2000)) print(counts) # ~ Counter({1: ~1000, -1: ~1000}) ``` > **On Braket.** A real device or simulator only ever returns *samples* (shots), exactly like > `projective_measure` above — never the amplitudes. You estimate probabilities and expectation values > from finite shots, with the $1/\sqrt{M}$ error you derived in > [0.2.2](../../term-0-foundations/c02-probability-information/02-random-variables-expectation.md). You'll > measure $\lvert+\rangle$ on the local simulator in [1.2.3](../c02-qubits-bloch/03-bases-and-expectation.md). --- ## Exercises **E1 (easy).** For $\lvert\psi\rangle = \tfrac1{\sqrt5}(\lvert0\rangle + 2\lvert1\rangle)$, give the $Z$-measurement probabilities and $\langle Z\rangle$.
Solution $p(+1) = |1/\sqrt5|^2 = 1/5$, $p(-1) = |2/\sqrt5|^2 = 4/5$. $\langle Z\rangle = (1)(1/5)+(-1)(4/5) = -3/5$.
**E2 (easy).** Show the Born probabilities sum to $1$ for any normalized state and any observable, in one line.
Solution $\sum_k\langle\psi|P_k|\psi\rangle = \langle\psi|\sum_k P_k|\psi\rangle = \langle\psi|I|\psi\rangle = 1$ by completeness and normalization. ∎
**E3 (medium).** Prove $\langle A\rangle = \langle\psi|A|\psi\rangle$ is real for any Hermitian $A$ (without invoking the eigenvalue formula).
Solution $\overline{\langle\psi|A|\psi\rangle} = \langle\psi|A^\dagger|\psi\rangle = \langle\psi|A|\psi\rangle$ using $A=A^\dagger$ (and $\overline{\langle\psi|M|\psi\rangle} = \langle\psi|M^\dagger|\psi\rangle$). A number equal to its own conjugate is real. ∎
**E4 (medium).** A qubit is measured with the observable $A = 3\lvert+\rangle\langle+| - 1\lvert-\rangle\langle-|$ in state $\lvert0\rangle$. Find the outcome probabilities, $\langle A\rangle$, and the post-measurement states.
Solution $\lvert0\rangle = \tfrac1{\sqrt2}(\lvert+\rangle+\lvert-\rangle)$, so $p(3) = |\langle+|0\rangle|^2 = 1/2$ and $p(-1) = |\langle-|0\rangle|^2 = 1/2$. $\langle A\rangle = 3(\tfrac12) + (-1)(\tfrac12) = 1$. Post-measurement: outcome $3 \Rightarrow \lvert+\rangle$; outcome $-1 \Rightarrow \lvert-\rangle$.
**E5 (hard).** Show that if $\lvert\psi\rangle$ is an eigenvector of observable $A$, the measurement of $A$ is deterministic and leaves the state unchanged. Conversely, argue that zero variance implies the state is an eigenvector.
Solution If $A\lvert\psi\rangle = \lambda_j\lvert\psi\rangle$ then $P_k\lvert\psi\rangle = \delta_{kj}\lvert\psi\rangle$, so $p(\lambda_j) = 1$ and all others $0$; the collapse $P_j\lvert\psi\rangle/\sqrt1 = \lvert\psi\rangle$ is unchanged. Conversely the variance $(\Delta A)^2 = \langle A^2\rangle - \langle A\rangle^2 = \sum_k(\lambda_k-\langle A\rangle)^2 p(\lambda_k)$; being a sum of nonnegative terms, it is $0$ iff all probability sits on a single eigenvalue, i.e. $\lvert\psi\rangle$ lies in one eigenspace — an eigenvector. ∎ (Formalized in [1.3.2](../c03-measurement/02-expectation-uncertainty.md).)
--- ## Checkpoint 1. State Postulate 2, including the role of the spectral decomposition. 2. Write the Born rule and prove the probabilities sum to $1$. 3. Give two equivalent formulas for the expectation value $\langle A\rangle$. 4. What is the post-measurement state, and what does repeatability mean? 5. Why must observables be Hermitian?
Answers 1. Observables are Hermitian $A=\sum_k\lambda_k P_k$; measuring gives outcome $\lambda_k$ w.p. $\langle\psi|P_k|\psi\rangle$ and collapses the state to $P_k\lvert\psi\rangle/\sqrt{p_k}$. 2. $p(\lambda_k)=\langle\psi|P_k|\psi\rangle$; $\sum_k p_k = \langle\psi|\sum_k P_k|\psi\rangle = \langle\psi|\psi\rangle = 1$. 3. $\langle A\rangle = \sum_k\lambda_k p(\lambda_k) = \langle\psi|A|\psi\rangle$. 4. $P_k\lvert\psi\rangle/\sqrt{p_k}$, lying in the $\lambda_k$-eigenspace; an immediate re-measurement returns $\lambda_k$ with certainty. 5. Measured values are real and outcomes must be perfectly distinguishable; Hermitian operators have real eigenvalues and an orthonormal eigenbasis.
--- ## Further Reading - **[NC]** Nielsen & Chuang, §2.2.3 (Postulate 3, measurement) and §2.2.5 (projective measurements). - **[Sak]** Sakurai & Napolitano, §1.4 — measurements, observables, expectation values. - **[Pre]** Preskill, *Ph219*, Ch. 2 — the measurement axiom. --- ← Prev: [The State Postulate](01-state-postulate.md) · Up: [Term 1](../README.md) · Next: [The Evolution Postulate](03-evolution-postulate.md) →