Qubit (source code)
= Qubit
{title2=$\mathcal H\cong\mathbb C^2$}
{wiki}
A <qubit>, or quantum bit, is a <quantum system> whose state space is a two-dimensional complex <Hilbert space>. Choose an <orthonormal basis> $|0\rangle,|1\rangle$, called the <computational basis>. A <pure state> has a normalized representative $|\psi\rangle=\alpha|0\rangle+\beta|1\rangle$, where $\alpha,\beta$ are complex <probability amplitudes> and $|\alpha|^2+|\beta|^2=1$. General <qubit states>, including <mixed states>, require a <density matrix>. https://quantum.cloud.ibm.com/learning/en/courses/basics-of-quantum-information/single-systems/quantum-information[Watrous, IBM Quantum Learning, Quantum information, Quantum state vectors]{ref}.
The abstract <qubit> describes the chosen degree of freedom: for <spin one-half>, $|0\rangle,|1\rangle$ can label spin up and down along a specified axis; for <photon polarization> in a fixed propagation mode, they can label horizontal and vertical polarization, $|H\rangle,|V\rangle$. These implementations share the same two-dimensional <linear algebra>, but their preparation and <measurement in quantum mechanics> use different apparatus. A <qubit> therefore need not be an entire two-level particle. <Foundations I: States and Ensembles by John Preskill>[Preskill, Physics 229 notes, sections 2.2.1–2.2.2, spin and photon polarization examples]{ref}.
The usefulness of a <qubit> lies in controlling <quantum superpositions> and their relative <quantum phases>, then converting them into outcome <probabilities> with <quantum gates> and an appropriate measurement. One ideal <computational-basis measurement> returns one label, zero or one. It does not reveal both <probability amplitudes>, their relative <quantum phase>, or all the bits of a larger register's expansion. <Quantum state tomography> instead estimates a <qubit state> using many repetitions of the preparation and suitable measurement settings. https://quantum.cloud.ibm.com/learning/en/courses/basics-of-quantum-information/single-systems/quantum-information[Watrous, IBM Quantum Learning, Quantum information, Measuring quantum states]{ref}.
An actual preparation and measurement example is Dederick and Beck's 2014 teaching-laboratory experiment. A 405-nm laser pumped two nonlinear crystals, producing signal and idler <photons> near 810 nm. In their non-entangled run the source produced $|H\rangle_s|H\rangle_i$; a quarter-wave plate in the signal arm changed its <photon polarization> to $|L\rangle=(|H\rangle+i|V\rangle)/\sqrt2$ in their convention. The plate supplies a relative <quantum phase> between polarization components. An idler detection selected the signal trials; signal detectors measured horizontal/vertical, diagonal, and <circular polarization> bases on separate runs. Before the polarization analyzer, a half-wave plate at $22.5$ degrees selected the diagonal basis, and a quarter-wave plate at $45$ degrees selected the circular basis. Coincidence counts $N_{AB},N_{AB'}$ gave the conditional positive-outcome frequency $N_{AB}/(N_{AB}+N_{AB'})$. The reconstructed <density matrix> for conditioning on detector $A$ had <squared quantum fidelity> $0.96$ with $|L\rangle$ and <purity of a density operator> $0.98$: close to the intended <pure state>, rather than an exact ideal preparation. https://people.reed.edu/~beckm/papers/Dederick_ajp.pdf[Dederick and Beck, Exploring entanglement with the help of quantum state measurement, American Journal of Physics 82, 962–971 (2014), sections II D and III A–B]{ref}.
\Image[/qubit-states-photon-preparation.png]
{title=Ideal polarization-qubit preparation and measurements in three separate settings}
{description=Original model schematic and exact Born-rule probabilities, not measured data or a reproduction of the experimental apparatus. The circular-basis sign uses the explicitly defined state $(|H\rangle+i|V\rangle)/\sqrt2$.}
The model diagram explains why those settings matter. With $|0\rangle=|H\rangle$ and $|1\rangle=|V\rangle$, the intended $|L\rangle$ gives equal <probabilities> in both the horizontal/vertical <computational basis> and diagonal <Hadamard basis>, just as the <mixed state> $I/2$ does. In the circular basis $|\pm i\rangle=(|0\rangle\pm i|1\rangle)/\sqrt2$, however, the ideal $|L\rangle=|+i\rangle$ always gives the plus outcome, whereas $I/2$ still gives equal <probabilities>. These are deductions from the <Born rule>, not the paper's measured counts. Unequal detector efficiencies and finite samples must be accounted for when interpreting experimental frequencies; the ideal comparison assumes equally efficient detection.
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