12.02.04 · quantum / formalism

Measurement postulates: Born rule, wavefunction collapse, and the projection postulate

shipped3 tiersLean: nonepending prereqs

Anchor (Master): von Neumann, Mathematical Foundations of Quantum Mechanics (1932), Ch. V; Wigner, Group Theory (1959), Ch. 26

Intuition Beginner

Units 12.02.01 and 12.02.02 set up the mathematical stage: states are vectors in a Hilbert space, observables are self-adjoint operators, and the spectral theorem decomposes every observable into eigenvalues and eigenstates. This unit answers the question those setups raised but did not resolve: what actually happens when you measure?

Three rules govern every quantum measurement. Together they are called the measurement postulates.

The first rule is the Born rule. When the system sits in state $∣ ψ ⟩$ and you measure an observable $A$ whose eigenstates are $∣ a_{1} ⟩, ∣ a_{2} ⟩, \dots$ with eigenvalues $a_{1}, a_{2}, \dots$ , the number that appears on the instrument dial is one of the $a_{n}$ . Which one is not determined in advance — it is random. The probability of seeing $a_{n}$ equals the squared overlap $∣ ⟨ a_{n} ∣ ψ ⟩ ∣^{2}$ . Born proposed this rule in 1926 and it has survived every experimental test since.

The second rule is wavefunction collapse. After the measurement returns $a_{n}$ , the state is no longer $∣ ψ ⟩$ . It has become the eigenstate $∣ a_{n} ⟩$ . A second measurement of the same observable, made immediately, will return $a_{n}$ again with certainty. The first measurement destroyed the old state and replaced it with a new one.

The third rule is the projection postulate (Luders rule). Collapse is mathematically a projection. The operator $P_{n} = ∣ a_{n} ⟩ ⟨ a_{n} ∣$ projects any state onto the eigenstate $∣ a_{n} ⟩$ . After measuring $a_{n}$ , the new state is $P_{n} ∣ ψ ⟩$ normalised to unit length. For a degenerate eigenvalue shared by several eigenstates, the projector is the sum of the individual projectors, and the post-measurement state is the normalised projection onto the whole degenerate subspace.

These three rules are the operational heart of quantum mechanics. Every prediction the theory makes about laboratory data passes through them. The rest of this unit works through the consequences: what happens when two observables share eigenstates (compatible observables), what happens when they do not (incompatible observables), and two striking effects that follow directly from the projection postulate — the quantum Zeno effect and the no-cloning theorem.

Visual Beginner

The state vector $∣ ψ ⟩$ sits in a space whose axes are the eigenstates of the observable being measured. It casts a shadow on each axis. The squared length of each shadow is the probability of the corresponding outcome. When the dice land on one outcome, the vector snaps to that axis — that is collapse.

For a degenerate eigenvalue, several axes share the same label. The projection is then onto the subspace spanned by all of them, not onto any single axis. The direction within that subspace is preserved; only the component outside the subspace is erased.

Worked example Beginner

A spin-1/2 particle is in the state $∣ ψ ⟩ = \frac{1}{3} ∣ ↑ ⟩ + \frac{2}{3} ∣ ↓ ⟩$ . A Stern-Gerlach apparatus measures spin along the $z$ -axis.

Born rule. The eigenstates of $S_{z}$ are $∣ ↑ ⟩$ (eigenvalue $+ ℏ/2$ ) and $∣ ↓ ⟩$ (eigenvalue $- ℏ/2$ ). The probability of spin-up is $∣ ⟨ ↑ ∣ ψ ⟩ ∣^{2} = ∣1/ 3 ∣^{2} = 1/3$ . The probability of spin-down is $∣ ⟨ ↓ ∣ ψ ⟩ ∣^{2} = ∣ 2/3 ∣^{2} = 2/3$ . The probabilities sum to one.

Collapse. Suppose the apparatus reads spin-up. The state after measurement is $∣ ↑ ⟩$ . If you measure $S_{z}$ again immediately, the probability of spin-up is $∣ ⟨ ↑ ∣ ↑ ⟩ ∣^{2} = 1$ — certainty. The first measurement converted a probabilistic state into a definite one.

Projection postulate. The projector for spin-up is $P_{↑} = ∣ ↑ ⟩ ⟨ ↑ ∣$ . Applying it: $P_{↑} ∣ ψ ⟩ = ∣ ↑ ⟩ ⟨ ↑ ∣ ψ ⟩ = \frac{1}{3} ∣ ↑ ⟩$ . Normalise: the new state is $∣ ↑ ⟩$ . The projector extracted the spin-up component and discarded the spin-down component.

Now measure $S_{x}$ on the same state $∣ ψ ⟩$ . The eigenstates of $S_{x}$ are $∣ + ⟩_{x} = \frac{1}{2} (∣ ↑ ⟩ + ∣ ↓ ⟩)$ and $∣ - ⟩_{x} = \frac{1}{2} (∣ ↑ ⟩ - ∣ ↓ ⟩)$ , both with eigenvalue magnitudes $ℏ/2$ . Compute the overlaps: $⟨ + ∣_{x} ∣ ψ ⟩ = \frac{1}{2} (\frac{1}{3} + \frac{2}{3}) = \frac{1 + 2}{6}$ . The probability of $+ x$ is $\frac{( 1 + 2 ) ^{2}}{6} = \frac{3 + 2 2}{6} \approx 0.971$ . The $S_{x}$ measurement is almost certain to return $+ x$ because the original state is heavily weighted toward spin-down, which is close to $∣ + ⟩_{x}$ in this basis. After the measurement the state collapses to $∣ + ⟩_{x}$ , destroying all information about the prior $S_{z}$ content.

Check your understanding Beginner

Exercise (easy, multiple choice).

The projection postulate says that after measuring $A$ and obtaining $a_{n}$ , the new state is:

A. $∣ ψ ⟩$ unchanged B. $∣ a_{n} ⟩ ⟨ a_{n} ∣ ψ ⟩$ (unnormalised) C. $∣ a_{n} ⟩$ always D. $P_{n} ∣ ψ ⟩ /∥ P_{n} ∣ ψ ⟩ ∥$ , where $P_{n}$ projects onto the $a_{n}$ -eigenspace

Hint

For a non-degenerate eigenvalue there is a unique eigenstate and option C works. For a degenerate eigenvalue, multiple eigenstates share the same label. Which option covers both cases?

Answer

D. $P_{n} ∣ ψ ⟩ /∥ P_{n} ∣ ψ ⟩ ∥$ .

For a non-degenerate eigenvalue, $P_{n} = ∣ a_{n} ⟩ ⟨ a_{n} ∣$ and the normalised projection is just $∣ a_{n} ⟩$ . For a degenerate eigenvalue with a multi-dimensional eigenspace, the projection preserves the component of $∣ ψ ⟩$ lying inside that subspace. Option B is the unnormalised projection (missing the normalisation). Option C fails for degenerate spectra.

Exercise (medium, numeric).

A quantum system has three orthonormal states $∣1 ⟩, ∣2 ⟩, ∣3 ⟩$ and an observable $A$ with eigenvalues $a_{1} = 1, a_{2} = 2, a_{3} = 2$ (the eigenvalue 2 is degenerate). The state is $∣ ψ ⟩ = \frac{1}{6} ∣1 ⟩ + \frac{1}{3} ∣2 ⟩ + \frac{1}{2} ∣3 ⟩$ . What is the probability of measuring $a_{2} = 2$ ? If the outcome is 2, what is the post-measurement state?

Hint

For a degenerate eigenvalue, sum the probabilities over all eigenstates in that eigenspace. The projector for eigenvalue 2 is $P_{2} = ∣2 ⟩ ⟨ 2∣ + ∣3 ⟩ ⟨ 3∣$ .

Answer

Probability of $a_{2}$ : $∣ ⟨ 2∣ ψ ⟩ ∣^{2} + ∣ ⟨ 3∣ ψ ⟩ ∣^{2} = 1/3 + 1/2 = 5/6$ .

The unnormalised post-measurement state is $P_{2} ∣ ψ ⟩ = \frac{1}{3} ∣2 ⟩ + \frac{1}{2} ∣3 ⟩$ . Its norm is $1/3 + 1/2 = 5/6$ . Normalising:

∣ ψ^{'} ⟩ = \frac{6}{5} (\frac{1}{3} ∣2 ⟩ + \frac{1}{2} ∣3 ⟩) = \frac{2}{5} ∣2 ⟩ + \frac{3}{5} ∣3 ⟩ .

The relative weights of $∣2 ⟩$ and $∣3 ⟩$ are preserved — only the component along $∣1 ⟩$ is erased. This is the Luders rule in action.

Formal definition Intermediate+

Born rule (general form). Let $A$ be a self-adjoint operator on a Hilbert space $H$ with projection-valued measure $E_{A}$ . For a system in state $∣ ψ ⟩$ with $∥∣ ψ ⟩ ∥ = 1$ , the probability that a measurement of $A$ yields a value in the Borel set $Ω \subset R$ is

Prob (A \in Ω ∣ ψ) = ⟨ ψ ∣ E_{A} (Ω) ∣ ψ ⟩ = ∥ E_{A} (Ω) ∣ ψ ⟩ ∥^{2} .

For pure point spectrum with distinct eigenvalues $a_{n}$ and eigenprojectors $P_{n}$ , this reduces to $Prob (a_{n} ∣ ψ) = ⟨ ψ ∣ P_{n} ∣ ψ ⟩ = ∣ ⟨ a_{n} ∣ ψ ⟩ ∣^{2}$ when $P_{n} = ∣ a_{n} ⟩ ⟨ a_{n} ∣$ .

Expectation value. The mean of the measurement outcomes is

⟨ A ⟩_{ψ} = ⟨ ψ ∣ A ∣ ψ ⟩ = n \sum a_{n} Prob (a_{n} ∣ ψ) .

This follows by inserting the spectral decomposition $A = \sum_{n} a_{n} P_{n}$ and the Born rule.

Projection postulate (Luders rule). After a measurement of $A$ yields outcome $a_{n}$ (possibly degenerate), the post-measurement state is

∣ ψ^{'} ⟩ = \frac{P _{n} ∣ ψ ⟩}{∥ P _{n} ∣ ψ ⟩ ∥},

where $P_{n} = E_{A} ({a_{n}})$ is the projector onto the full eigenspace of $a_{n}$ . For a mixed state $ρ$ , the post-measurement state is

ρ^{'} = \frac{P _{n} ρ P _{n}}{tr ( P _{n} ρ )} .

The Luders rule (1951) ^{[Luders 1951]} is the correct generalisation to degenerate spectra. The earlier von Neumann rule selected a single eigenstate from the degenerate subspace, introducing an arbitrary choice; Luders showed that the subspace projection is uniquely determined by the requirement that the post-measurement state depend only on the projector $P_{n}$ and not on the choice of basis within the eigenspace.

Compatible observables. Two observables $A$ and $B$ are compatible if their spectral measures commute: $E_{A} (Ω) E_{B} (Ω^{'}) = E_{B} (Ω^{'}) E_{A} (Ω)$ for all Borel sets $Ω, Ω^{'}$ . For operators with pure point spectrum this is equivalent to $[A, B] = 0$ . Compatible observables possess a complete set of simultaneous eigenstates, and measuring one does not disturb the statistics of the other.

Incompatible observables. Two observables are incompatible if their spectral measures do not commute. Measuring one observable collapses the state into an eigenstate of that observable, which is in general a superposition of eigenstates of the other. The post-measurement state has indeterminate value for the second observable, and the measurement statistics change. The uncertainty relation $Δ A Δ B \geq \frac{1}{2} ∣ ⟨[A, B]⟩ ∣$ is the quantitative expression of this incompatibility.

Counterexamples to common slips

Collapse is not physical propagation. The projection postulate describes how the state assignment updates after a measurement result is recorded. It is not a dynamical process in space-time with a speed or a spatial profile. The formalism does not say how the wavefunction collapses — only that the correct pre-measurement state for predicting future results is the projected state. The measurement problem (addressed in the Master tier and in unit 20.03.01) is precisely the question of whether and how to give a dynamical account of this update.
Degenerate eigenvalues require the full projector. If $A$ has eigenvalue $a$ with eigenspace spanned by ${∣ e_{1} ⟩, ∣ e_{2} ⟩}$ , the post-measurement state is $P_{a} ∣ ψ ⟩ /∥ P_{a} ∣ ψ ⟩ ∥$ where $P_{a} = ∣ e_{1} ⟩ ⟨ e_{1} ∣ + ∣ e_{2} ⟩ ⟨ e_{2} ∣$ , not a random selection of $∣ e_{1} ⟩$ or $∣ e_{2} ⟩$ . The relative weights within the degenerate subspace are preserved by the measurement.
The Born rule is not $⟨ ψ ∣ A ∣ ψ ⟩$ . The expectation value is the probability-weighted average of outcomes, not a probability itself. The probability of each individual outcome is $⟨ ψ ∣ P_{n} ∣ ψ ⟩$ . The expectation value can be a number that is not itself an eigenvalue (e.g., $⟨ σ_{z} ⟩ = 0$ for $∣ + ⟩_{x}$ , even though the only possible outcomes are $\pm 1$ ).

Key theorem with proof Intermediate+

Theorem (no-cloning theorem). There exists no unitary operator $U$ on a Hilbert space $H$ such that for all states $∣ ψ ⟩$ and some fixed "blank" state $∣0 ⟩$ ,

U (∣ ψ ⟩ \otimes ∣0 ⟩) = ∣ ψ ⟩ \otimes ∣ ψ ⟩ .

Proof. Suppose such a $U$ exists. Then for two distinct states $∣ ψ ⟩$ and $∣ ϕ ⟩$ :

U (∣ ψ ⟩ \otimes ∣0 ⟩) = ∣ ψ ⟩ \otimes ∣ ψ ⟩, U (∣ ϕ ⟩ \otimes ∣0 ⟩) = ∣ ϕ ⟩ \otimes ∣ ϕ ⟩ .

Take the inner product of the left-hand sides. Since $U$ is unitary it preserves inner products:

⟨ ψ ∣ ϕ ⟩ \cdot ⟨ 0∣0 ⟩ = ⟨ ψ ∣ ϕ ⟩ .

Now take the inner product of the right-hand sides:

⟨ ψ ∣ ϕ ⟩ \cdot ⟨ ψ ∣ ϕ ⟩ = ⟨ ψ ∣ ϕ ⟩^{2} .

Unitarity forces $⟨ ψ ∣ ϕ ⟩ = ⟨ ψ ∣ ϕ ⟩^{2}$ , which gives $⟨ ψ ∣ ϕ ⟩ (⟨ ψ ∣ ϕ ⟩ - 1) = 0$ . This requires either $⟨ ψ ∣ ϕ ⟩ = 0$ (orthogonal) or $⟨ ψ ∣ ϕ ⟩ = 1$ (identical). For two distinct, non-orthogonal states this is a contradiction. $□$

The no-cloning theorem was proved independently by Wootters and Zurek (1982) and Dieks (1982) ^{[Wootters-Zurek 1982; Dieks 1982]}. Its physical content: quantum information cannot be copied. This is a direct consequence of the linearity of quantum mechanics — unitary operators are linear, and the cloning map $∣ ψ ⟩ \mapsto ∣ ψ ⟩ \otimes ∣ ψ ⟩$ is nonlinear. The theorem underpins the security of quantum key distribution and the impossibility of amplifying an unknown quantum signal.

Theorem (quantum Zeno effect). Let $P$ be a projector and $∣ ψ ⟩$ a state with $P ∣ ψ ⟩ = ∣ ψ ⟩$ (an eigenstate of $P$ with eigenvalue 1). Perform $N$ sequential measurements of the observable $P$ in a total time $T$ , with spacing $τ = T / N$ . As $N \to \infty$ , the probability that every measurement returns 1 approaches certainty.

Proof sketch. Under unitary evolution with Hamiltonian $H$ for time $τ$ , the state evolves to $∣ ψ (τ)⟩ = e^{- i H τ /ℏ} ∣ ψ ⟩$ . For small $τ$ :

⟨ ψ ∣ e^{i H τ /ℏ} P e^{- i H τ /ℏ} ∣ ψ ⟩ = 1 - \frac{τ ^{2}}{ℏ ^{2}} (⟨ H^{2} ⟩ - ⟨ H ⟩^{2}) + O (τ^{3}) = 1 - \frac{τ ^{2}}{ℏ ^{2}} (Δ H)^{2} + O (τ^{3}) .

The probability of surviving one measurement is $1 - (τ^{2} / ℏ^{2}) (Δ H)^{2} + O (τ^{3})$ . After $N = T / τ$ measurements, the survival probability is

P_{surv} \approx (1 - \frac{τ ^{2}}{ℏ ^{2}} (Δ H)^{2})^{T / τ} = (1 - \frac{T ^{2}}{N ^{2} ℏ ^{2}} (Δ H)^{2})^{N} .

As $N \to \infty$ , $(1 - c / N^{2})^{N} \to 1$ for any constant $c$ , so $P_{surv} \to 1$ . Frequent enough measurement freezes the state. $□$

The quantum Zeno effect was analysed by Misra and Sudarshan (1977) ^{[Misra-Sudarshan 1977]}, though the underlying mathematics is already in von Neumann's 1932 monograph. The name alludes to Zeno's arrow paradox: if you observe continuously, the system never changes. The effect has been confirmed experimentally in trapped-ion systems and polarised-neutron beams.

Exercises Intermediate+

Exercise 5 (medium, symbolic).

An observable $A$ on a three-dimensional Hilbert space has eigenvalues $a_{1} = 1$ (non-degenerate, eigenstate $∣1 ⟩$ ) and $a_{2} = 3$ (doubly degenerate, eigenspace spanned by $∣2 ⟩, ∣3 ⟩$ ). The state is $∣ ψ ⟩ = \frac{1}{2} ∣1 ⟩ + \frac{1}{2} ∣2 ⟩ + \frac{1}{2} ∣3 ⟩$ . Verify normalisation, compute the probability of each measurement outcome, and find the post-measurement state if $a_{2} = 3$ is observed.

Hint

Check $∣ \frac{1}{2} ∣^{2} + ∣ \frac{1}{2} ∣^{2} + ∣ \frac{1}{2} ∣^{2} = 1$ . For the degenerate eigenvalue, sum the probabilities over the eigenspace and use the Luders rule.

Answer

Normalisation: $1/4 + 1/4 + 1/2 = 1$ . Correct.

$Prob (a_{1}) = ∣ ⟨ 1∣ ψ ⟩ ∣^{2} = 1/4$ .

$Prob (a_{2}) = ∣ ⟨ 2∣ ψ ⟩ ∣^{2} + ∣ ⟨ 3∣ ψ ⟩ ∣^{2} = 1/4 + 1/2 = 3/4$ .

Post-measurement state after $a_{2}$ : $P_{2} ∣ ψ ⟩ = \frac{1}{2} ∣2 ⟩ + \frac{1}{2} ∣3 ⟩$ , norm $= 3/4$ . Normalising:

∣ ψ^{'} ⟩ = \frac{1}{3/4} (\frac{1}{2} ∣2 ⟩ + \frac{1}{2} ∣3 ⟩) = \frac{1}{3} ∣2 ⟩ + \frac{2}{3} ∣3 ⟩ .

The relative weights $1/2 : 1/ 2 = 1 : 2$ within the degenerate subspace are preserved, while the $∣1 ⟩$ component is completely removed.

Exercise 6 (medium, symbolic).

A spin-1/2 particle in state $∣ ψ ⟩ = \frac{1}{2} (∣ ↑ ⟩ + ∣ ↓ ⟩)$ is sent through a Stern-Gerlach measuring $S_{z}$ , then through a second Stern-Gerlach measuring $S_{x}$ . Compute the probability of each path: (a) first spin-up, then $+ x$ ; (b) first spin-up, then $- x$ .

Hint

After the first measurement the state collapses. Use the new state to compute the Born-rule probabilities for the second measurement.

Answer

First measurement ( $S_{z}$ ): $Prob (↑) = 1/2$ . After collapse, the state is $∣ ↑ ⟩$ .

Second measurement ( $S_{x}$ ) on $∣ ↑ ⟩$ : $⟨ + ∣_{x} ∣ ↑ ⟩ = 1/ 2$ , so $Prob (+ x) = 1/2$ and $Prob (- x) = 1/2$ .

Path probabilities: (a) $1/2 \times 1/2 = 1/4$ . (b) $1/2 \times 1/2 = 1/4$ .

The total probability of spin-up at the first apparatus is $1/2$ , split equally between $+ x$ and $- x$ at the second. The first measurement destroyed the coherence that made $∣ ψ ⟩$ an eigenstate of $S_{x}$ , so the $S_{x}$ statistics become fifty-fifty regardless of the original state.

Exercise 7 (hard, proof).

Prove the no-cloning theorem using linearity directly (without assuming unitarity). Show that no linear map $C : H \to H \otimes H$ can satisfy $C ∣ ψ ⟩ = ∣ ψ ⟩ \otimes ∣ ψ ⟩$ for all $∣ ψ ⟩ \in H$ .

Hint

Apply $C$ to a superposition $∣ ψ ⟩ + ∣ ϕ ⟩$ and compare the two ways of computing the result.

Answer

By linearity, $C (∣ ψ ⟩ + ∣ ϕ ⟩) = C ∣ ψ ⟩ + C ∣ ϕ ⟩ = ∣ ψ ⟩ \otimes ∣ ψ ⟩ + ∣ ϕ ⟩ \otimes ∣ ϕ ⟩$ . But if $C$ clones every state, it must also clone $∣ ψ ⟩ + ∣ ϕ ⟩$ :

C (∣ ψ ⟩ + ∣ ϕ ⟩) = (∣ ψ ⟩ + ∣ ϕ ⟩) \otimes (∣ ψ ⟩ + ∣ ϕ ⟩) = ∣ ψ ⟩ \otimes ∣ ψ ⟩ + ∣ ψ ⟩ \otimes ∣ ϕ ⟩ + ∣ ϕ ⟩ \otimes ∣ ψ ⟩ + ∣ ϕ ⟩ \otimes ∣ ϕ ⟩ .

The two expressions differ by the cross terms $∣ ψ ⟩ \otimes ∣ ϕ ⟩ + ∣ ϕ ⟩ \otimes ∣ ψ ⟩$ , which vanish only when $∣ ψ ⟩$ and $∣ ϕ ⟩$ are proportional. The cloning map is nonlinear; no linear map can implement it. Since quantum-mechanical time evolution is linear (unitary), cloning is impossible within quantum mechanics.

Exercise 8 (hard, proof).

Let $A$ and $B$ be compatible observables ( $[A, B] = 0$ ) on a finite-dimensional Hilbert space, and let ${P_{n}}$ and ${Q_{m}}$ be their respective spectral projectors. Show that $P_{n} Q_{m} = Q_{m} P_{n}$ for all $n, m$ , and that the joint probability of obtaining eigenvalue $a_{n}$ for $A$ and $b_{m}$ for $B$ is $⟨ ψ ∣ P_{n} Q_{m} ∣ ψ ⟩$ , independent of the order of measurement.

Hint

Commuting self-adjoint operators share a simultaneous eigenbasis. Use the spectral theorem for both operators simultaneously.

Answer

Since $[A, B] = 0$ , the spectral theorem for commuting self-adjoint operators guarantees a basis of simultaneous eigenvectors ${∣ n, m ⟩}$ with $A ∣ n, m ⟩ = a_{n} ∣ n, m ⟩$ and $B ∣ n, m ⟩ = b_{m} ∣ n, m ⟩$ . The projectors are $P_{n} = \sum_{m} ∣ n, m ⟩ ⟨ n, m ∣$ and $Q_{m} = \sum_{n} ∣ n, m ⟩ ⟨ n, m ∣$ .

Then $P_{n} Q_{m} = ∣ n, m ⟩ ⟨ n, m ∣ = Q_{m} P_{n}$ (the projector onto the simultaneous eigenspace). The joint probability is

Prob (a_{n}, b_{m} ∣ ψ) = ⟨ ψ ∣ P_{n} Q_{m} ∣ ψ ⟩ .

If $A$ is measured first and yields $a_{n}$ , the state collapses to $P_{n} ∣ ψ ⟩ /∥ P_{n} ∣ ψ ⟩ ∥$ . Then measuring $B$ : $Prob (b_{m}) = ⟨ ψ ∣ P_{n} Q_{m} P_{n} ∣ ψ ⟩ / ⟨ ψ ∣ P_{n} ∣ ψ ⟩ = ⟨ ψ ∣ P_{n} Q_{m} ∣ ψ ⟩ / ⟨ ψ ∣ P_{n} ∣ ψ ⟩$ (using $P_{n}^{2} = P_{n}$ and $P_{n} Q_{m} = Q_{m} P_{n}$ ). Multiplying by $Prob (a_{n}) = ⟨ ψ ∣ P_{n} ∣ ψ ⟩$ gives $⟨ ψ ∣ P_{n} Q_{m} ∣ ψ ⟩$ . Measuring $B$ first gives the same expression with $Q_{m}$ and $P_{n}$ swapped, which is identical by commutativity. The joint probability is order-independent for compatible observables.

Exercise 9 (hard, symbolic).

A spin-1/2 particle is initially in state $∣ ↑ ⟩$ . The Hamiltonian is $H = (ℏ ω /2) σ_{x}$ , which drives rotation from $z$ -up toward $z$ -down. After time $τ$ , an $S_{z}$ measurement is performed. Compute the probability of finding spin-up. Now suppose $N$ such measurements are performed in total time $T = N τ$ , with the system evolving freely between measurements. Show that the probability of spin-up at every measurement approaches 1 as $N \to \infty$ .

Hint

The evolution operator is $e^{- i H t /ℏ} = e^{- iω t σ_{x} /2}$ . Use the formula for the survival probability and take $N \to \infty$ .

Answer

The evolution operator is $U (τ) = cos (ω τ /2) 1 - i sin (ω τ /2) σ_{x}$ . After time $τ$ , the state is $U (τ) ∣ ↑ ⟩ = cos (ω τ /2) ∣ ↑ ⟩ - i sin (ω τ /2) ∣ ↓ ⟩$ . The survival probability is $∣ ⟨ ↑ ∣ U (τ) ∣ ↑ ⟩ ∣^{2} = cos^{2} (ω τ /2)$ .

For $N$ measurements in time $T$ : $τ = T / N$ , and the single-measurement survival probability is $cos^{2} (ω T /2 N) \approx 1 - (ω T /2 N)^{2}$ for large $N$ . After $N$ measurements:

P_{surv} \approx (1 - \frac{ω ^{2} T ^{2}}{4 N ^{2}})^{N} .

As $N \to \infty$ : $ln P_{surv} \approx N \cdot (- ω^{2} T^{2} /4 N^{2}) = - ω^{2} T^{2} /4 N \to 0$ , so $P_{surv} \to 1$ . The system is frozen by frequent measurement — the quantum Zeno effect.

Exercise 10 (hard, conceptual).

Explain why the no-cloning theorem does not prevent a quantum computer from preparing a known state $∣ ψ ⟩$ in multiple qubits. What is the difference between cloning an unknown state and preparing known copies?

Hint

Preparing a known state uses a fixed unitary that depends on the description of the state. Cloning an unknown state would require a single unitary that works for all inputs.

Answer

If you know the state $∣ ψ ⟩ = α ∣0 ⟩ + β ∣1 ⟩$ — that is, you know the numbers $α$ and $β$ — you can design a circuit that prepares this state from $∣0 ⟩$ on as many qubits as you like. The circuit depends on $α$ and $β$ , and is different for each state.

The no-cloning theorem forbids something else: a single unitary $U$ that takes as input a qubit in an unknown state $∣ ψ ⟩$ (which could be anything) plus a blank $∣0 ⟩$ , and outputs $∣ ψ ⟩ \otimes ∣ ψ ⟩$ . Such a unitary would need to work for all $∣ ψ ⟩$ simultaneously, which linearity forbids.

The distinction is between having a classical description (which lets you prepare arbitrarily many copies) and having only a single quantum sample (which does not). This is why quantum state tomography — reconstructing $α$ and $β$ from measurements on many copies — requires many identically prepared systems, not one system measured repeatedly.

Advanced content Master

The measurement problem

The formalism contains two dynamical laws that cannot both be universal. Unitary evolution (postulate P4) preserves superpositions: $U (\sum_{n} c_{n} ∣ n ⟩) = \sum_{n} c_{n} e^{- i E_{n} t /ℏ} ∣ n ⟩$ . The projection postulate (P3) destroys them: measuring in the ${∣ n ⟩}$ basis collapses $\sum_{n} c_{n} ∣ n ⟩$ to a single $∣ n_{0} ⟩$ . The formalism gives no precise criterion for when each law applies.

The boundary is sometimes called the Heisenberg cut: systems below the cut evolve unitarily, and the measurement apparatus above the cut triggers collapse. The cut's location is not specified by the theory. Moving it does not change any prediction (a fact sometimes called the "movability of the cut"), but its existence is a structural feature that every interpretation must address.

Copenhagen treats the cut as fundamental: classical apparatus, quantum system. Measurement is a primitive notion not further analysable.
Everett / many-worlds abolishes the cut. The global state always evolves unitarily. The appearance of collapse is an emergent phenomenon: after a unitary interaction with a measuring device, the composite state is $\sum_{n} c_{n} ∣ n ⟩ \otimes ∣ M_{n} ⟩$ , and decoherence ensures the branches $∣ n ⟩ \otimes ∣ M_{n} ⟩$ do not interfere. Each branch experiences an effective collapse.
Objective-collapse theories (GRW, continuous spontaneous localisation) modify the Schrödinger equation by adding a stochastic, nonlinear term that causes collapse with a rate proportional to system size. Microscopic systems are nearly unaffected; macroscopic superpositions collapse rapidly.
Bohmian mechanics adds particle positions as hidden variables. The wavefunction always evolves unitarily (it never collapses in the universal dynamics). Collapse is an effective description of the conditional wavefunction for a subsystem, after conditioning on the actual particle positions.

The formal content of this unit is neutral among these interpretations. All agree on the Born rule, the projection postulate, and every prediction derived from them.

POVMs and Naimark dilation

The projective (von Neumann) measurement described by the spectral measure of $A$ is a special case. The most general quantum measurement is described by a positive operator-valued measure (POVM): a set of positive operators ${E_{k}}$ with $\sum_{k} E_{k} = 1$ , where outcome $k$ occurs with probability $Prob (k) = ⟨ ψ ∣ E_{k} ∣ ψ ⟩$ . The post-measurement state is not uniquely determined by the POVM alone — it depends on the choice of Kraus operators ${M_{k}}$ with $E_{k} = M_{k}^{†} M_{k}$ , giving $ρ^{'} = M_{k} ρ M_{k}^{†} / tr (M_{k}^{†} M_{k} ρ)$ .

Naimark's dilation theorem states that every POVM on a system $H$ can be realised as a projective measurement on an enlarged space $H \otimes K$ , followed by tracing out the ancilla. Projective measurements are thus the fundamental case, and POVMs are the derived, operational generalisation. The quantum-information unit 12.17.01 develops the POVM framework fully.

Simultaneous measurement and joint measurability

Two observables $A$ and $B$ are jointly measurable if there exists a single POVM on $H \otimes H$ whose marginals reproduce the Born-rule distributions of $A$ and $B$ individually. For projective measurements, joint measurability is equivalent to commutativity $[A, B] = 0$ . For general POVMs the situation is richer: non-commuting observables can be jointly measurable if one allows noisy measurements. This is the content of the unsharpness programme in quantum measurement theory.

Weak measurement and the two-state formalism

A weak measurement is one where the coupling between system and apparatus is so small that the system state is negligibly disturbed. The outcome, called a weak value, is $⟨ ϕ ∣ \hat{A} ∣ ψ ⟩ / ⟨ ϕ ∣ ψ ⟩$ , where $∣ ψ ⟩$ is the pre-selected state and $∣ ϕ ⟩$ is the post-selected state (conditioned on a later measurement result). Weak values can lie outside the spectrum of $A$ and can be complex. The two-state formalism of Aharonov, Albert, and Vaidman (1988) treats the weak value as a time-symmetric quantity that depends on both boundary conditions. Weak measurements do not contradict the projection postulate — they trade information per trial for minimal disturbance, accumulating statistics over many runs.

Connections Master

Hilbert-space formalism 12.02.01. The spectral theorem supplies the projection-valued measure that the Born rule consumes. Without the spectral decomposition of self-adjoint operators, the measurement postulate has no mathematical substrate to act on.
Operators and Hermiticity 12.02.02. The eigenvalue-eigenvector structure of self-adjoint operators is exactly the structure the projection postulate operates on: eigenvalues label outcomes, eigenvectors label post-measurement states, and commutativity determines compatibility.
Density matrix and mixed states 12.02.03. The Luders rule generalises the projection postulate from pure states to density operators: $ρ \mapsto P_{n} ρ P_{n} / tr (P_{n} ρ)$ . The no-cloning theorem has a density-matrix formulation: there is no quantum channel (completely positive trace-preserving map) that clones an arbitrary unknown state.
Time evolution and the Schrodinger equation 12.03.01. The two dynamical laws — unitary evolution and projective collapse — are the twin pillars whose coexistence constitutes the measurement problem. The Schrodinger equation governs what happens between measurements; the projection postulate governs what happens at measurements.
Quantum information [12.17.01, 12.17.02]. The no-cloning theorem is foundational for quantum cryptography (BB84, E91 protocols) and quantum teleportation. Entanglement-based protocols rely on the projection postulate: measuring one half of a Bell pair collapses the other. The quantum Zeno effect is exploited in quantum-error-correction codes to suppress decoherence.
Bell inequalities 12.17.03. The Born-rule probabilities for measurements on entangled states violate Bell inequalities. The projection postulate is what makes the EPR argument work (measuring particle $A$ collapses the state of particle $B$ ), and Bell's theorem shows that the resulting correlations cannot be reproduced by any local hidden-variable theory.
Philosophy of physics 20.03.01. The measurement problem — the tension between unitary evolution and projective collapse — is the central open question in the philosophy of quantum mechanics. Every interpretation (Copenhagen, many-worlds, Bohmian, GRW, QBism) is a response to this structural feature of the formalism.

Historical & philosophical context Master

The Born rule was proposed by Max Born in June 1926 in a footnote to his paper on scattering ^{[Born 1926 Z. Phys. 37, 863]}. Born originally wrote the probability as $∣ ψ ∣^{2}$ (using the real wavefunction); in a second paper later that year [Z. Phys. 38, 803] he corrected this to $∣ ψ ∣^{2}$ or $∣ ψ^{*} ψ ∣$ for the complex case. The rule was an interpretive addition to the formalism, not derived from it, and Born himself expressed surprise at its success.

The projection postulate was introduced by John von Neumann in his 1932 monograph [Mathematische Grundlagen der Quantenmechanik, Ch. V] ^{[von Neumann 1932/1955]}. Von Neumann's original formulation selected a single eigenstate from a degenerate eigenspace, introducing an arbitrary choice. Gerhart Luders corrected this in 1951 [Ann. Phys. 8, 322] ^{[Luders 1951]} by showing that the projection onto the full eigenspace is the unique update rule consistent with the structure of the observable algebra. The corrected rule is now called the Luders rule.

The no-cloning theorem was discovered independently by William Wootters and Wojciech Zurek [Nature 299, 802 (1982)] and by Dennis Dieks [Phys. Lett. A 92, 271 (1982)] ^{[Wootters-Zurek 1982; Dieks 1982]}, both in response to a proposal by Herbert (1982) for a superluminal communication scheme based on quantum amplification. The no-cloning theorem closed that loophole and established the absolute security of quantum key distribution.

The quantum Zeno effect was analysed by Baidyanath Misra and George Sudarshan in 1977 [J. Math. Phys. 18, 756] ^{[Misra-Sudarshan 1977]}, though Alan Turing had noted the mathematical possibility in a 1954 letter. The effect was first observed experimentally by Itano, Heinzen, Bollinger, and Wineland in 1990 using trapped beryllium ions, confirming that frequent measurement suppresses transitions between atomic energy levels.

The measurement problem has been the central unresolved question in quantum mechanics since the Bohr-Einstein debates of 1927-1935. The Copenhagen interpretation, dominant through the mid-twentieth century, treats the projection postulate as a primitive. The many-worlds interpretation (Everett 1957), Bohmian mechanics (Bohm 1952), objective-collapse theories (GRW 1986), and QBism (Fuchs et al. 2010s) each offer a different resolution. None modifies the operational predictions for standard measurements; they differ only in the ontological story they tell about what happens during a measurement event.

Bibliography Master

@article{born1926quantenmechanik,
  author  = {Born, Max},
  title   = {Zur Quantenmechanik der Sto{\ss}vorg{\"a}nge},
  journal = {Zeitschrift f{\"u}r Physik},
  volume  = {37},
  pages   = {863--867},
  year    = {1926}
}

@article{born1926quantenmechanik2,
  author  = {Born, Max},
  title   = {Quantenmechanik der Sto{\ss}vorg{\"a}nge},
  journal = {Zeitschrift f{\"u}r Physik},
  volume  = {38},
  pages   = {803--827},
  year    = {1926}
}

@book{vonneumann1932grundlagen,
  author    = {von Neumann, John},
  title     = {Mathematische Grundlagen der Quantenmechanik},
  publisher = {Springer},
  year      = {1932},
  note      = {English translation: Mathematical Foundations of Quantum Mechanics, Princeton University Press, 1955, Ch. V}
}

@article{luders1951zustandsanderung,
  author  = {L{\"u}ders, Gerhart},
  title   = {Zur Zustands{\"a}nderung durch den Me{\ss}proze{\ss}},
  journal = {Annalen der Physik},
  volume  = {8},
  pages   = {322--328},
  year    = {1951}
}

@article{wootterszurek1982nocloning,
  author  = {Wootters, William K. and Zurek, Wojciech H.},
  title   = {A single quantum cannot be cloned},
  journal = {Nature},
  volume  = {299},
  pages   = {802--803},
  year    = {1982}
}

@article{dieks1982communication,
  author  = {Dieks, Dennis},
  title   = {Communication by {EPR} devices},
  journal = {Physics Letters A},
  volume  = {92},
  pages   = {271--272},
  year    = {1982}
}

@article{misrasudarshan1977zeno,
  author  = {Misra, Baidyanath and Sudarshan, E. C. George},
  title   = {The {Zeno}'s paradox in quantum theory},
  journal = {Journal of Mathematical Physics},
  volume  = {18},
  pages   = {756--763},
  year    = {1977}
}

@book{griffiths2018quantum,
  author    = {Griffiths, David J. and Schroeter, Darrell F.},
  title     = {Introduction to Quantum Mechanics},
  edition   = {3rd},
  publisher = {Cambridge University Press},
  year      = {2018}
}

@book{sakurai2021modern,
  author    = {Sakurai, Jun John and Napolitano, Jim},
  title     = {Modern Quantum Mechanics},
  edition   = {3rd},
  publisher = {Cambridge University Press},
  year      = {2021}
}

@book{nielsenchuang2010,
  author    = {Nielsen, Michael A. and Chuang, Isaac L.},
  title     = {Quantum Computation and Quantum Information},
  edition   = {10th Anniversary},
  publisher = {Cambridge University Press},
  year      = {2010}
}

@book{busch1996measurement,
  author    = {Busch, Paul and Lahti, Pekka and Mittelstaedt, Peter},
  title     = {The Quantum Theory of Measurement},
  edition   = {2nd},
  publisher = {Springer},
  year      = {1996}
}

@book{wigner1959group,
  author    = {Wigner, Eugene P.},
  title     = {Group Theory and its Application to the Quantum Mechanics of Atomic Spectra},
  publisher = {Academic Press},
  year      = {1959}
}

Prerequisites

12.02.01 pending
12.02.02 pending

Tier anchors

beginner: Griffiths & Schroeter, Introduction to Quantum Mechanics, 3e (2018), Ch. 1.2-1.4, 3.4
intermediate: Sakurai & Napolitano, Modern Quantum Mechanics, 3e (2021), §1.2-1.4
master: von Neumann, Mathematical Foundations of Quantum Mechanics (1932), Ch. V; Wigner, Group Theory (1959), Ch. 26

References

Griffiths & Schroeter — Introduction to Quantum Mechanics, 3e (Cambridge, 2018) · Ch. 1.2-1.4, 3.4
Sakurai & Napolitano — Modern Quantum Mechanics, 3e (Cambridge, 2021) · §1.2-1.4
von Neumann — Mathematical Foundations of Quantum Mechanics (Princeton, 1932/1955) · Ch. V
Wigner — Group Theory and its Application to the Quantum Mechanics of Atomic Spectra (Academic, 1959) · Ch. 26
Born — Zur Quantenmechanik der Stossvorgange · Z. Phys. 37, 863 (1926); 38, 803 (1926)
Luders — Zur Zustandsanderung durch den Messprozess · Ann. Phys. 8, 322 (1951)
Wooters & Zurek — A single quantum cannot be cloned · Nature 299, 802 (1982)
Dieks — Communication by EPR devices · Phys. Lett. A 92, 271 (1982)
Misra & Sudarshan — The Zeno's paradox in quantum theory · J. Math. Phys. 18, 756 (1977)
Nielsen & Chuang — Quantum Computation and Quantum Information, 10e (Cambridge, 2010) · §2.2 postulates of quantum mechanics, §12.1 no-cloning
Busch, Lahti & Mittelstaedt — The Quantum Theory of Measurement, 2e (Springer, 1996) · Ch. 2 POVMs and operational generalisation

Estimated time

beginner: 15m
intermediate: 35m
master: 50m