Superposition

Learning objectives

• Define superposition as a coherent linear combination of basis states.
• Separate coherent superposition from classical uncertainty.
• Use |+⟩, |−⟩, and the Hadamard gate to track phase-sensitive evolution.
• Explain interference as addition and cancellation of amplitudes.

A scene about paths

Alice returns to the observatory and finds two corridors marked 0 and 1. The guide refuses the easy story that she walks down both corridors as a tiny classical person. Instead, the corridor labels are basis labels. The quantum state can assign amplitudes to both labels, and later operations can make those amplitudes reinforce or cancel.

The analogy ends there. A superposition is not a split traveler. It is a linear combination in a vector space.

Superposition as a linear combination

In Chapter 1, we wrote a pure qubit as

$$|\psi\rangle = \alpha|0\rangle + \beta|1\rangle.$$

When both amplitudes are nonzero in the chosen basis, we often say the state is in a superposition of those basis states. The word "superposition" does not add a new kind of object; it names the linear structure already present in the state vector model.

Two important equal superpositions are

$$|+\rangle = \frac{|0\rangle + |1\rangle}{\sqrt{2}}, \qquad |-\rangle = \frac{|0\rangle - |1\rangle}{\sqrt{2}}.$$

Both give (P(0)=P(1)=1/2) when measured in the computational basis. Yet they are not the same state. The minus sign is a relative phase, and later operations can convert that phase difference into different measurement outcomes.

0NaN%

1NaN%

The identical bars do not mean the states are physically identical; this measurement basis does not reveal their relative phase.

Why superposition is not ordinary uncertainty

A classical mixture could describe a preparation that produces ( |0\rangle ) half the time and ( |1\rangle ) half the time, with us not knowing which was prepared. That mixture also gives 50/50 computational-basis results. But it lacks a coherent relative phase between basis components.

A coherent superposition such as ( |+\rangle ) has phase relations that gates can use. If we only look at computational-basis probabilities, the distinction is hidden. If we apply another operation first, the distinction can become visible.

Classical mixture

• Preparation is |0⟩ or |1⟩ with classical randomness.
• No stable relative phase connects the alternatives.
• Later interference is absent in the single-qubit model.

Coherent superposition

• One pure state has amplitudes on both basis vectors.
• Relative phase is part of the state.
• Later gates can reveal the phase through interference.

Density matrices give the formal distinction, but the operational difference already appears when we insert gates before measurement.

Checkpoint

Why can |+⟩ and |−⟩ have the same computational-basis probabilities but still be different states?

Because probabilities ignore the relative sign between amplitudes.Because one state is actually a classical mixture.Because measurement always makes both outcomes happen.

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Relative phase

Relative phase is the phase difference between components of a state. In ( |+\rangle ), the coefficients of ( |0\rangle ) and ( |1\rangle ) have the same phase. In ( |-\rangle ), the ( |1\rangle ) component differs by a phase of ( \pi ), represented here by a minus sign.

Global phase is different. Multiplying every amplitude by (-1) gives (-|+\rangle), which has the same physical predictions as ( |+\rangle ). Changing only one component, as in ( |+\rangle ) versus ( |-\rangle ), changes the relative phase and can matter.

The Hadamard gate

The Hadamard gate connects computational-basis states with the ( |+\rangle, |-\rangle ) basis:

$$H|0\rangle = |+\rangle, \qquad H|1\rangle = |-\rangle.$$

It also reverses those relationships:

$$H|+\rangle = |0\rangle, \qquad H|-\rangle = |1\rangle.$$

These equations show why phase is not decorative. The two states ( |+\rangle ) and ( |-\rangle ) have the same computational-basis probabilities before the final H, but after H one becomes ( |0\rangle ) and the other becomes ( |1\rangle ).

1. 1

   Start in |0⟩.

2. 2

   Apply H to obtain |+⟩.

3. 3

   Apply H again; amplitudes interfere to recover |0⟩.

The second H does not randomly choose an outcome. It recombines amplitudes so that the |1⟩ amplitude cancels.

Interference

Interference is the addition of amplitudes before probabilities are computed. Consider starting in ( |0\rangle ), applying H, and then applying H again:

$$|0\rangle \xrightarrow{H} \frac{|0\rangle + |1\rangle}{\sqrt{2}} \xrightarrow{H} \frac{|+\rangle + |-\rangle}{\sqrt{2}} = |0\rangle.$$

The ( |1\rangle ) paths cancel because one contribution is positive and the other is negative. This is destructive interference. The ( |0\rangle ) contributions add constructively.

If the middle state were ( |-\rangle ), the second H would produce ( |1\rangle ). The same visible probabilities before the second H lead to different final outcomes because the amplitudes carry phase.

Check Your Understanding

What does H|−⟩ equal?

|0⟩|1⟩(|0⟩ + |1⟩)/√2A 50/50 classical mixture

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Check Your Understanding

Which statement best describes interference?

Probabilities are added after measurement.Amplitudes combine before probabilities are computed.A qubit chooses the fastest path.

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Summary

Summary

• A superposition is a coherent linear combination of basis states.
• ( |+\rangle ) and ( |-\rangle ) have identical computational-basis probabilities but different relative phase.
• A classical mixture can mimic one measurement distribution without having coherent phase.
• The Hadamard gate creates and resolves equal superpositions.
• Interference occurs when amplitudes add or cancel before probabilities are calculated.

References and further study

• Michael A. Nielsen and Isaac L. Chuang, Quantum Computation and Quantum Information.
• John Preskill, Lecture Notes for Physics 229: Quantum Information and Computation.
• Qiskit Textbook, sections on single-qubit gates and interference.