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Quantum Measurement

Quantum measurement is the process by which quantum states are observed, causing the probabilistic collapse of superposition states into definite classical outcomes.

Mathematical Framework

Born Rule

The probability of measuring state \(|i\rangle\) from quantum state \(|\psi\rangle\) is given by the Born rule:

\[P(|i\rangle) = |\langle i|\psi\rangle|^2\]

For a general state \(|\psi\rangle = \sum_i \alpha_i |i\rangle\):

\[P(|i\rangle) = |\alpha_i|^2\]

Normalization

Probabilities must sum to 1:

\[\sum_i P(|i\rangle) = \sum_i |\alpha_i|^2 = 1\]

This is ensured by the normalization condition on quantum states.

Measurement Operators

Projective Measurement

A projective measurement is described by projection operators \(P_i\):

\[P_i = |i\rangle\langle i|\]

Properties:

  • \(P_i^2 = P_i\) (idempotent)
  • \(P_i P_j = 0\) for \(i \neq j\) (orthogonal)
  • \(\sum_i P_i = I\) (completeness)

General Measurements (POVM)

Positive Operator-Valued Measures generalize projective measurements:

\[\sum_i E_i = I, \quad E_i \geq 0\]

Where \(E_i\) are positive operators (not necessarily projections).

State Collapse

Post-Measurement State

If measurement outcome \(i\) is observed, the state collapses to:

\[|\psi\rangle \rightarrow \frac{P_i|\psi\rangle}{\sqrt{\langle\psi|P_i|\psi\rangle}}\]

For computational basis measurement:

\[|\psi\rangle = \alpha|0\rangle + \beta|1\rangle \xrightarrow{\text{measure}} \begin{cases} |0\rangle & \text{probability } |\alpha|^2 \\ |1\rangle & \text{probability } |\beta|^2 \end{cases}\]

Irreversibility

Quantum measurement is irreversible:

  • Information about the original superposition is lost
  • Cannot reconstruct \(|\psi\rangle\) from measurement outcome
  • Multiple measurements on identical states give statistics

Types of Measurements

Computational Basis Measurement

Standard measurement in the \(\{|0\rangle, |1\rangle\}\) basis:

\[M_0 = |0\rangle\langle 0| = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}\]
\[M_1 = |1\rangle\langle 1| = \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}\]

Pauli Measurements

X-basis measurement in the \(\{|+\rangle, |-\rangle\}\) basis:

\[|+\rangle = \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle), \quad |-\rangle = \frac{1}{\sqrt{2}}(|0\rangle - |1\rangle)\]

Y-basis measurement in the \(\{|+i\rangle, |-i\rangle\}\) basis:

\[|+i\rangle = \frac{1}{\sqrt{2}}(|0\rangle + i|1\rangle), \quad |-i\rangle = \frac{1}{\sqrt{2}}(|0\rangle - i|1\rangle)\]

Multi-Qubit Measurements

Joint Measurement

Measuring all qubits simultaneously in computational basis:

For \(|\psi\rangle = \sum_{i_1,i_2,\ldots,i_n} \alpha_{i_1 i_2 \cdots i_n} |i_1 i_2 \cdots i_n\rangle\)

\[P(|i_1 i_2 \cdots i_n\rangle) = |\alpha_{i_1 i_2 \cdots i_n}|^2\]

Partial Measurement

Measuring only a subset of qubits causes partial collapse.

For two qubits \(|\psi\rangle = \sum_{ij} \alpha_{ij}|ij\rangle\), measuring first qubit:

\[P(0) = \sum_j |\alpha_{0j}|^2, \quad P(1) = \sum_j |\alpha_{1j}|^2\]

Post-measurement state (if outcome 0):

\[|\psi'\rangle = \frac{1}{\sqrt{P(0)}} \sum_j \alpha_{0j}|0j\rangle\]

Expectation Values

Observable Measurement

For Hermitian operator \(A\) representing an observable:

\[\langle A \rangle = \langle\psi|A|\psi\rangle\]

This gives the expected value of measuring observable \(A\).

Variance

The uncertainty in measurement is:

\[(\Delta A)^2 = \langle A^2 \rangle - \langle A \rangle^2\]

Examples with Bell States

Measuring Bell State \(|\Phi^+\rangle\)

For \(|\Phi^+\rangle = \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)\):

Joint measurement probabilities:

  • \(P(00) = \frac{1}{2}\)
  • \(P(01) = 0\)
  • \(P(10) = 0\)
  • \(P(11) = \frac{1}{2}\)

Individual qubit probabilities:

  • \(P(\text{qubit 0} = 0) = \frac{1}{2}\)
  • \(P(\text{qubit 0} = 1) = \frac{1}{2}\)
  • \(P(\text{qubit 1} = 0) = \frac{1}{2}\)
  • \(P(\text{qubit 1} = 1) = \frac{1}{2}\)

Correlations: If qubit 0 is measured as 0, qubit 1 is guaranteed to be 0.

Implementation in the Simulator

Single Qubit Measurement

from quantum_simulator import QuantumSimulator
from quantum_simulator.gates import H_GATE

# Create superposition
sim = QuantumSimulator(1)
H_GATE.apply(sim.state_vector, [0])

# Measure multiple times to see statistics
results = []
for _ in range(1000):
    sim_copy = QuantumSimulator(1)
    sim_copy.state_vector = sim.state_vector.copy()
    result = sim_copy.measure(0)
    results.append(result)

print(f"Fraction of 0s: {results.count(0)/1000}")  # ≈ 0.5
print(f"Fraction of 1s: {results.count(1)/1000}")  # ≈ 0.5

Bell State Measurement

from quantum_simulator.gates import CNOT_GATE

# Create Bell state
sim = QuantumSimulator(2)
H_GATE.apply(sim.state_vector, [0])
CNOT_GATE.apply(sim.state_vector, [0, 1])

# Measure both qubits
result_0 = sim.measure(0)  
result_1 = sim.measure(1)

print(f"Results: ({result_0}, {result_1})")  # Will be (0,0) or (1,1)

Quantum Non-Demolition Measurements

Special measurements that can be repeated without disturbing the state:

  • Measure eigenstate of the measurement operator
  • Subsequent measurements give same result
  • Used in quantum error correction

Deferred Measurement

In quantum circuits, measurements can be deferred to the end:

  • Replace mid-circuit measurements with conditional operations
  • Equivalent to measuring at the end
  • Useful for theoretical analysis

Measurement is the bridge between the quantum and classical worlds, converting quantum superposition into definite classical information.