6.3.1 Projective Measurement

6.3.1 Projective Measurement#

Prompts

  • Explain the measurement postulate in density matrix language. How do you compute outcome probabilities and post-measurement states using projectors?

  • What happens when you measure the same observable twice in succession? Prove that the outcome is deterministic.

  • What is the quantum Zeno effect? How does frequent measurement suppress time evolution, and what is the scaling of survival probability with measurement interval?

  • When can two observables be measured simultaneously without disturbing each other? What mathematical condition must they satisfy?

Lecture Notes#

Overview#

Measurement in quantum mechanics is not a passive observation—it is a projective act that forces the system into one of several possible states. For any Hermitian observable, the measurement outcome is an eigenvalue, the probability depends on the initial state’s alignment with the corresponding eigenspace (Born rule), and the post-measurement state is the normalized projection onto that eigenspace. This section develops the mathematical framework and explores key consequences: repeated measurements give the same outcome with certainty, frequent measurements can suppress evolution (quantum Zeno effect), and commuting observables can be measured simultaneously.

Spectral Decomposition and Hermitian Operators#

Definition: Spectral Decomposition

For a Hermitian operator \(\hat{A}\) (observable), the spectral theorem states:

(130)#\[A = \sum_i a_i P_i\]

where:

  • \(a_i\) = eigenvalues (real, since \(\hat{A}\) is Hermitian)

  • \(P_i\) = projector onto eigenspace of \(a_i\)

  • \(P_i\) are orthogonal: \(P_i P_j = \delta_{ij} P_i\)

  • \(P_i\) are complete: \(\sum_i P_i = I\) (sum to identity)

  • \(P_i^2 = P_i\) (idempotent)

If eigenvalue \(a_i\) is degenerate (multiplicity \(> 1\)), projector \(P_i\) projects onto a multi-dimensional subspace.

Example: Qubit Spin Observable

For \(\hat{\sigma}^z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}\) (Pauli-z operator):

  • Eigenvalues: \(\lambda_1 = +1, \lambda_2 = -1\).

  • Eigenvectors: \(\vert 0\rangle = \begin{pmatrix} 1 \\ 0 \end{pmatrix}, \vert 1\rangle = \begin{pmatrix} 0 \\ 1 \end{pmatrix}\).

  • Projectors:

\[\begin{split}P_+ = \vert 0\rangle\langle 0\vert = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}, \quad P_- = \vert 1\rangle\langle 1\vert = \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}\end{split}\]

  • Verification: \(P_+ + P_- = I\) ✓, \(P_+^2 = P_+\) ✓, \(P_+ P_- = 0\) ✓.

  • Spectral form: \(\hat{\sigma}^z = (+1) P_+ + (-1) P_-\).

The Measurement Postulate#

Recall the measurement postulate from §1.2.1: For a system in state \(\hat{\rho}\) measured for observable \(A = \sum_i a_i P_i\), the outcome is eigenvalue \(a_i\) with probability \(p_i = \text{Tr}(P_i \hat{\rho})\), and the state collapses to \(\hat{\rho}' = P_i \hat{\rho} P_i / \text{Tr}(P_i \hat{\rho})\).

Here we explore the mathematical structure more carefully, generalizations, and physical consequences.

Measurement Postulate: Probability and Collapse

Outcome: Measurement yields eigenvalue \(a_i\) with probability:

(131)#\[p_i = \text{Tr}(P_i \hat{\rho})\]

Post-measurement state (state collapse):

(132)#\[\hat{\rho}' = \frac{P_i \hat{\rho} P_i}{\text{Tr}(P_i \hat{\rho})} = \frac{P_i \hat{\rho} P_i}{p_i}\]

The state is projected onto the eigenspace corresponding to the measured eigenvalue.

Normalization Check#

\[\text{Tr}(\hat{\rho}') = \frac{\text{Tr}(P_i \hat{\rho} P_i)}{p_i} = \frac{p_i}{p_i} = 1 \quad ✓\]

since \(P_i^2 = P_i\) and \(P_i\) is Hermitian.

Born Rule: Expectation Value

The average outcome of repeated measurements is:

(133)#\[\langle A \rangle = \sum_i a_i p_i = \sum_i a_i \text{Tr}(P_i \hat{\rho}) = \text{Tr}(A \hat{\rho})\]

This is the Born rule in density matrix form.

Measurement of Pure States

For a pure state \(\hat{\rho} = \vert\psi\rangle\langle\psi\vert\):

Probability:

(134)#\[p_i = \text{Tr}(P_i \vert\psi\rangle\langle\psi\vert) = \langle\psi\vert P_i \vert\psi\rangle\]

If \(\vert\psi\rangle = \sum_j c_j \vert u_j^{(i)}\rangle\) where \(\vert u_j^{(i)}\rangle\) are eigenstates of eigenvalue \(a_i\) (with multiplicity \(d_i\)):

\[p_i = \sum_j \vert c_j\vert^2\]

Post-measurement state:

(135)#\[\vert\psi'\rangle = \frac{P_i \vert\psi\rangle}{\sqrt{p_i}}\]

The state is normalized projection onto eigenspace \(i\).

Example: Measuring \(\hat{\sigma}^z\)

Initial state: \(\vert\psi\rangle = \frac{1}{\sqrt{2}}(\vert 0\rangle + \vert 1\rangle)\) (superposition).

Measure \(\hat{\sigma}^z\):

  • Outcome \(+1\) (\(\vert 0\rangle\)):

    • Probability: \(p_+ = \vert\frac{1}{\sqrt{2}}\vert^2 = 1/2\).

    • Post-measurement: \(\vert\psi'\rangle = \vert 0\rangle\) (definite \(\vert 0\rangle\)).

  • Outcome \(-1\) (\(\vert 1\rangle\)):

    • Probability: \(p_- = \vert\frac{1}{\sqrt{2}}\vert^2 = 1/2\).

    • Post-measurement: \(\vert\psi'\rangle = \vert 1\rangle\) (definite \(\vert 1\rangle\)).

After measurement, the superposition is destroyed; the state is definite.

Repeated Measurements#

Theorem: If you measure observable \(A\) and obtain outcome \(a_i\), then immediately measure \(A\) again, you get the same outcome \(a_i\) with certainty (probability 1).

Proof: After first measurement, state is \(\hat{\rho}' = P_i \hat{\rho} P_i / p_i\), which is supported on eigenspace of \(a_i\).

Second measurement probability:

\[p_i^{(2)} = \text{Tr}(P_i \hat{\rho}') = \text{Tr}\left(P_i \frac{P_i \hat{\rho} P_i}{p_i}\right) = \frac{\text{Tr}(P_i P_i \hat{\rho} P_i)}{p_i} = \frac{\text{Tr}(P_i \hat{\rho} P_i)}{p_i} = \frac{p_i}{p_i} = 1\]

Consequence: Measurement outcome is perfectly deterministic on repeated measurement; the result doesn’t change.

Discussion

The quantum Zeno effect says frequent measurements suppress transitions. For a two-level system with measurement interval \(\tau\), compute the survival probability \(P(T)\) as \(\tau\to 0\) and \(\tau\to\infty\).

The Quantum Zeno Effect#

Setup: System starts in state \(\vert\psi_0\rangle\) that would evolve to \(\vert\psi(t)\rangle\) under unitary evolution (e.g., decay).

Frequent projective measurements: Measure at times \(0, \Delta t, 2\Delta t, \ldots\) whether the system is still in \(\vert\psi_0\rangle\) (projector \(P = \vert\psi_0\rangle\langle\psi_0\vert\)).

Result (Zeno’s paradox): As \(\Delta t \to 0\) (infinite measurement rate), the system remains in \(\vert\psi_0\rangle\); evolution is frozen.

Mechanism: Each measurement collapses the state back to the measured outcome. If measurements are frequent enough, evolution is prevented before the system can deviate significantly.

Consider the effect quantitatively:

Derivation: Quantum Zeno Effect

After time \(\Delta t\) without measurement, state evolves to \(\vert\psi(\Delta t)\rangle \approx \vert\psi_0\rangle - \mathrm{i} H \vert\psi_0\rangle \Delta t/\hbar\) (first order).

Overlap: \(\vert\langle\psi_0\vert\psi(\Delta t)\rangle\vert^2 \approx 1 - O((\Delta t)^2)\).

After measurement: State either stays in \(\vert\psi_0\rangle\) (probability \(\approx 1 - O((\Delta t)^2)\)) or exits (probability \(\propto (\Delta t)^2\)).

With \(n = T/\Delta t\) measurements over time \(T\):

\[P(\text{remain in } \vert\psi_0\rangle) \approx (1 - c(\Delta t)^2)^{T/\Delta t} \to 1 \quad \text{as } \Delta t \to 0\]

Degenerate Eigenvalues#

If eigenvalue \(a_i\) is degenerate (multiple eigenstates), the projector projects onto the entire eigenspace:

\[P_i = \sum_{j=1}^{d_i} \vert u_j^{(i)}\rangle\langle u_j^{(i)}\vert\]

where \(d_i\) is the multiplicity (degeneracy).

Post-measurement state:

\[\hat{\rho}' = \frac{P_i \hat{\rho} P_i}{p_i}\]

is supported on the entire eigenspace (possibly mixed, if initial state was a superposition of degenerate eigenstates).

Example: Spin vs. Angular Momentum

  • \(\hat{\sigma}^z\) measurement: Non-degenerate eigenvalues (+1, -1); outcome fully specifies the state.

  • \(L^2\) measurement (orbital angular momentum squared): Degenerate eigenvalues (\(l(l+1)\hbar^2\) with degeneracy \(2l+1\)); outcome specifies angular momentum but not magnetic quantum number.

Compatibility and Commuting Observables#

Two observables \(A\) and \(B\) commute (\([A, B] = 0\)) iff they have a simultaneous eigenbasis. In this case:

  • Can measure both without disturbing the state (outcome of first doesn’t affect second).

  • Can assign definite values to both.

Non-commuting observables (e.g., \(\hat{\sigma}^x\) and \(\hat{\sigma}^z\)):

  • Measuring \(A\) disturbs \(B\) (collapses to eigenstate of \(A\), which is superposition of \(B\) eigenstates).

  • Cannot assign simultaneous definite values.

  • Measurement outcomes are incompatible.

Projective measurement is a special case

Projective (von Neumann) measurements are the textbook case where outcomes correspond to orthogonal projectors summing to identity. The more general framework (POVM) relaxes orthogonality—see §6.3.2.

Summary#

  • Projective measurement: von Neumann measurement based on spectral decomposition \(A = \sum_i a_i P_i\).

  • Measurement postulate (detailed in §1.2.1; extended here): Outcome \(a_i\) with probability \(p_i = \text{Tr}(P_i \hat{\rho})\); state collapses to \(P_i \hat{\rho} P_i / p_i\).

  • Expectation value: \(\langle A \rangle = \text{Tr}(A\hat{\rho})\) (Born rule).

  • Pure states: \(p_i = \langle\psi\vert P_i \vert\psi\rangle\); post-measurement \(\vert\psi'\rangle = P_i\vert\psi\rangle / \sqrt{p_i}\).

  • Repeated measurements: Second measurement of same observable gives same outcome with certainty (measurement result is reproducible).

  • Quantum Zeno effect: Frequent measurements freeze the state, preventing evolution.

  • Degeneracy: Multiple eigenstates for same eigenvalue; projector onto entire eigenspace.

  • Commuting observables: Share eigenbasis; can be measured simultaneously without disturbance.

  • Non-commuting: Mutually incompatible; measuring one disturbs the other.

See Also

Homework#

1. A projective measurement on a qubit is performed in the basis \(\{|+\rangle, |-\rangle\}\) where \(|\pm\rangle = \frac{1}{\sqrt{2}}(|0\rangle \pm |1\rangle)\). Write the two projection operators \(P_+\) and \(P_-\) as \(2\times 2\) matrices. Verify that \(P_+ + P_- = \mathbf{1}\) and \(P_\pm^2 = P_\pm\).

2. A qubit is prepared in state \(|\psi\rangle = \cos(\theta/2)|0\rangle + \mathrm{e}^{\mathrm{i}\phi}\sin(\theta/2)|1\rangle\). A projective measurement of \(\hat{\sigma}^z\) is performed. (a) What are the probabilities of each outcome? (b) What is the post-measurement state for each outcome? (c) What is \(\langle\hat{\sigma}^z\rangle\)?

3. Show that successive measurements of the same observable always give the same result. That is, if a projective measurement of observable \(\hat{A}\) gives outcome \(a_i\), prove that an immediate repetition of the measurement gives \(a_i\) with certainty. (Hint: use the projection postulate.)

4. Two observables \(\hat{A}\) and \(\hat{B}\) with spectral decompositions \(\hat{A} = \sum_i a_i P_i\) and \(\hat{B} = \sum_j b_j Q_j\) are said to be compatible if \([\hat{A}, \hat{B}] = 0\). Show that compatible observables can be simultaneously diagonalized, i.e., there exists a basis in which both are diagonal.

5. Consider sequential measurements: first measure \(\hat{\sigma}^z\), then measure \(\hat{\sigma}^x\). Starting from the state \(|0\rangle\): (a) What is the probability of getting \((+1, +1)\) for \(\hat{\sigma}^z\) and \(\hat{\sigma}^x\)? (b) After both measurements, what is the state? (c) Now measure \(\hat{\sigma}^z\) again. What is the probability of getting \(+1\)? Compare with the initial state.

6. Prove that for a projective measurement with projectors \(\{P_i\}\), the operation \(\hat{\rho} \mapsto \sum_i P_i \hat{\rho} P_i\) is a valid quantum channel (completely positive and trace-preserving). Also show that this map destroys all off-diagonal (coherence) terms in the measurement basis.

7. For a pure state \(|\psi\rangle\) and observable \(\hat{A} = \sum_i a_i |a_i\rangle\langle a_i|\), show that the variance \(\Delta A^2 = \langle\hat{A}^2\rangle - \langle\hat{A}\rangle^2 = 0\) if and only if \(|\psi\rangle\) is an eigenstate of \(\hat{A}\).

8. A 3-level system is in state \(|\psi\rangle = \frac{1}{\sqrt{6}}(|1\rangle + 2|2\rangle + |3\rangle)\). An observable \(\hat{A} = |1\rangle\langle 1| + 2|2\rangle\langle 2| + 3|3\rangle\langle 3|\) is measured. (a) Find the probability of each outcome. (b) Find \(\langle\hat{A}\rangle\) and \(\Delta A^2\). (c) What is the post-measurement state if outcome \(2\) is obtained?