= Quantum theory
Quantum theory supplies a common language for <quantum mechanics>, <measurement in quantum mechanics>, and <quantum information>: prepare a <quantum state>, transform it, and ask which experimental outcomes occur. In a finite-dimensional idealization, a <pure state> is a normalized <vector> of <complex numbers>. Its entries are <probability amplitudes>; the <Born rule> turns their squared magnitudes into <probabilities>. Between measurements, an isolated system evolves by a <unitary operator>, preserving its <norm> and <inner products>. A <quantum circuit> makes this evolution explicit as a sequence of <quantum gates>.
The extra information carried by <quantum phases> is essential. Relative <quantum phases> affect how <probability amplitudes> add when different contributions reach the same output. Their <interference> can enhance one outcome and cancel another. A <global phase>, multiplying the entire <quantum state> by the same complex factor of magnitude one, changes neither these relative phases nor observable predictions. The <Hadamard gate> below provides a small experiment in which this distinction becomes a deterministic change of output.
<Quantum coherence> describes this capacity to interfere relative to a specified <orthonormal basis>. In a <density matrix>, it appears in off-diagonal entries; a diagonal <mixed state> retains the populations but lacks that basis coherence. Thus matching the <probabilities> of one measurement need not match the predictions of a later experiment. Baumgratz, Cramer, and Plenio made this basis-dependent viewpoint precise in their resource framework for <quantum coherence>. https://arxiv.org/abs/1311.0275[Baumgratz, Cramer, and Plenio, Quantifying Coherence, 2014, pp. 1–2]{ref}.
This makes <quantum information> an operational question: which distinctions can a preparation, an available <unitary operator>, and <measurement in quantum mechanics> reveal? <Amplitude amplification> illustrates how carefully arranged <interference> converts a weakly represented answer into a likely observable result. Preserving the relevant <quantum coherence> throughout the computation is part of the physical task. https://quantum.cloud.ibm.com/learning/en/courses/basics-of-quantum-information/single-systems/introduction[Watrous, IBM Quantum Learning, introduction to single systems]{ref}.
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