3. Postulates: Measurement
Postulate 3: Measurement (The Born Rule and State Collapse)
The Rule:
Quantum measurements are governed by observables, which are described by Hermitian operators ($M$).
The Outcome (Eigenvalues): The only possible results of a measurement are the eigenvalues ($\lambda_i$) of the observable $M$. (Since measurement results must be real, $M$ must be Hermitian, as all Hermitian operators have real eigenvalues).
The Probability (Born Rule): The probability of observing a specific eigenvalue $\lambda_i$ is given by the squared magnitude of the projection of the state vector $|\psi\rangle$ onto the corresponding eigenvector $|e_i\rangle$:
$$P(\lambda_i) = |\langle e_i | \psi \rangle|^2$$The State Collapse: Immediately after the measurement yields the result $\lambda_i$, the state of the system instantaneously collapses to the corresponding eigenvector $|e_i\rangle$.
The Trap: Non-Determinism
Before measurement, the qubit exists in a superposition of all possible outcomes. The state is perfectly known ($\alpha$ and $\beta$ are known). However, the result is fundamentally non-deterministic. We only know the probabilities. After measurement, the amplitude information is gone; the system is forced into one definite classical state ($|0\rangle$ or $|1\rangle$).
Your Task: Calculating Collapse
The standard measurement in quantum computing is performed with the Pauli-Z Observable ($M=Z$).
Z-Observable: $Z = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$
Eigenvectors/Outcomes:
- $|e_0\rangle = |0\rangle$ (Eigenvalue $\lambda_0 = +1$)
- $|e_1\rangle = |1\rangle$ (Eigenvalue $\lambda_1 = -1$)
Suppose you prepare the qubit in the balanced superposition state $|+\rangle$:
$$|+\rangle = \frac{1}{\sqrt{2}}|0\rangle + \frac{1}{\sqrt{2}}|1\rangle$$Calculate the Probability: What is the probability $P(+1)$ of measuring the eigenvalue $+1$ (i.e., collapsing to $|0\rangle$)?
$$P(+1) = |\langle 0 | + \rangle|^2$$State After Measurement: If you perform the measurement and the result is $+1$, what is the state of the qubit immediately after the collapse?