A two-level quantum system used as the basic unit of quantum information.

A pure qubit state is a normalized vector in a two-dimensional complex vector space. It is not a hidden classical bit and it is not simply two classical values stored at once.

One vector in a chosen coordinate system for describing quantum states.

For a single qubit, the computational-basis states are |0> and |1>. A basis is a descriptive coordinate system, not a claim that the system secretly has one of those values.

The standard basis {|0>, |1>} used for most circuit descriptions and Z-basis measurement.

Other bases are possible. Changing the basis can change the outcome probabilities even when the physical state is unchanged.

A vector representation of a pure quantum state.

The same physical state can be represented in different bases. State vectors encode amplitudes and relative phase, not just likely measurement outcomes.

A geometric representation of single-qubit pure states up to global phase.

The sphere encodes relative amplitude and relative phase for one qubit. It is useful, but it does not generalize visually to many qubits in a simple way.

A complex coefficient whose squared magnitude gives a measurement probability.

Amplitudes carry magnitude and phase. Probabilities discard phase information, which is why states with identical probabilities can later behave differently.

The requirement that the total probability over a complete measurement basis is 1.

For |psi> = alpha|0> + beta|1>, normalization requires |alpha|^2 + |beta|^2 = 1.

A coherent linear combination of basis states.

Superposition is not ordinary ignorance about a pre-existing value. Relative phase within the combination can affect later interference.

The phase difference between components of a superposition.

Relative phase is physically meaningful because later gates can turn phase differences into measurable probability differences.

A common phase factor multiplying the entire state vector.

Multiplying a whole isolated state by e^{i gamma} leaves all measurement probabilities unchanged. It differs from relative phase.

Constructive or destructive combination of probability amplitudes.

Quantum algorithms use interference to amplify amplitudes of useful outcomes and suppress others. This does not mean every answer is simply tested in parallel.

A physical operation that produces a classical outcome according to a chosen basis.

Projective measurement also changes the post-measurement state. It is not merely revealing a stored classical value.

The orthonormal basis whose alternatives define the possible measurement outcomes.

A computational-basis measurement and an X-basis measurement ask different physical questions of the same state.

The rule that maps amplitudes to outcome probabilities by squared magnitude.

For measurement in a basis, each outcome probability is obtained from the magnitude squared of the component along that basis vector.

The state assigned after a measurement outcome has occurred.

After projective measurement, repeating the same measurement immediately gives the same outcome in the ideal model.

A reversible linear operation applied to one or more qubits.

In the circuit model, gates transform amplitudes before measurement. Their reversibility is represented mathematically by unitarity.

A single-qubit gate that connects computational-basis states with |+> and |-> states.

H creates equal superpositions from |0> and |1>, and it can also resolve phase differences into deterministic outcomes.

A gate that exchanges |0> and |1> in the computational basis.

On a general superposition it acts linearly, moving each amplitude to the other basis component.

A multi-qubit state that cannot be factored into independent single-qubit states.

Entanglement produces correlations stronger than separable state descriptions, but it does not allow faster-than-light communication.