Measurement

The process of extracting classical information from a quantum system, which fundamentally disturbs the quantum state.

Measurement is how we get information out of a quantum computer. It’s also one of the most counterintuitive aspects of quantum mechanics: measuring a quantum system fundamentally changes it.

The Measurement Process

When you measure a qubit in the computational basis:

1. Before: Qubit is in state $|\psi ⟩=\alpha |0⟩+\beta |1⟩$
2. Measurement: You observe either 0 or 1
3. After: Qubit “collapses” to the observed state ($|0⟩$ or $|1⟩$)

The probabilities are given by the Born rule:

• Probability of 0: $|\alpha {|}^2$
• Probability of 1: $|\beta {|}^2$

Key Properties

Probabilistic

Even if you know the exact quantum state, you can only predict measurement probabilities, not definite outcomes (unless the state is already an eigenstate of the measurement).

Irreversible

Measurement destroys the superposition. You cannot “undo” a measurement to recover the original state.

Basis-Dependent

You can measure in different bases:

• Computational basis (Z): Measures $|0⟩$ vs $|1⟩$
• Hadamard basis (X): Measures $|+⟩$ vs $|-⟩$
• Y basis: Measures $|i⟩$ vs $|-i⟩$

The same state gives different results in different bases.

Types of Measurement

Projective Measurement

The standard quantum measurement, described by projection operators. For computational basis:

• $P_0=|0⟩⟨0|$
• $P_1=|1⟩⟨1|$

POVM (Positive Operator-Valued Measure)

A generalization that allows for more measurement outcomes than basis states. Used in quantum information theory and quantum cryptography.

Weak Measurement

Partial measurement that disturbs the state less but provides less information. Useful for studying quantum systems without fully collapsing them.

Mid-Circuit Measurement

Modern quantum computers support measuring qubits in the middle of a circuit (not just at the end). This enables:

• Classical feedforward: Conditioning later operations on measurement results
• Quantum error correction: Measuring syndromes without destroying logical information
• Repeat-until-success protocols: Retrying until desired outcome occurs

Measurement in Practice

In real quantum hardware, measurement is often the slowest and noisiest operation:

• Superconducting qubits: ~1 μs readout time
• Trapped ions: ~100 μs readout time

Measurement errors (misreading 0 as 1 or vice versa) are a significant source of noise.

Why It Matters

• Measurement is how we extract answers from quantum computations
• The measurement problem is at the heart of quantum mechanics interpretation debates
• Understanding measurement is important for designing algorithms and error correction

See also: Born Rule, Quantum State, Superposition, Quantum Circuit