= Quantum superposition
{title2=$|\psi\rangle=\sum_j a_j|j\rangle,\quad\sum_j|a_j|^2=1$}
{wiki}
A quantum superposition is a <linear combination> of <quantum state> vectors in a complex <Hilbert space>. In a specified <orthonormal basis>, a normalized <pure state> has the displayed expansion, where $a_j$ are <probability amplitudes>. A nonzero <linear combination> can always be normalized; an arbitrary sum of normalized vectors need not already have <norm> one. Having more than one nonzero component depends on the chosen <basis>: the <Hadamard basis> vector $|+\rangle$ is a <quantum superposition> of the two <computational-basis states>, but is a single vector of the <Hadamard basis>. https://ocw.mit.edu/courses/8-04-quantum-physics-i-spring-2016/7f930e013cef9cd7dec5aa88baa83f0a_MIT8_04S16_LecNotes1.pdf[Zwiebach, MIT 8.04, lecture 1, section 4, pp. 10–12]{ref}.
The distinction from classical ignorance is quantitative. Consider the normalized <qubit> <quantum state>
$$
|\psi_\phi\rangle=\frac{\sqrt3}{2}|0\rangle+\frac{e^{i\phi}}2|1\rangle.
$$
Its <density operator>, written in the <computational basis>, is
$$
\rho_\phi=|\psi_\phi\rangle\langle\psi_\phi|
=\begin{pmatrix}3/4&(\sqrt3/4)e^{-i\phi}\\(\sqrt3/4)e^{i\phi}&1/4\end{pmatrix}.
$$
Preparing $|0\rangle$ with <probability> $3/4$ and $|1\rangle$ with <probability> $1/4$, while discarding the preparation label, instead gives the <mixed state>
$$
\rho_{\mathrm{mix}}=\frac34|0\rangle\langle0|+\frac14|1\rangle\langle1|
=\begin{pmatrix}3/4&0\\0&1/4\end{pmatrix}.
$$
The diagonal entries agree, so both preparations give the same <computational-basis measurement> <probabilities>. The off-diagonal entries of $\rho_\phi$ retain <quantum coherence> and the relative <quantum phase>. In the <Hadamard basis>, the <Born rule> gives
$$
\langle\pm|\rho_{\mathrm{mix}}|\pm\rangle=\frac12,
\qquad
\langle\pm|\rho_\phi|\pm\rangle=\frac12\pm\frac{\sqrt3}{4}\cos\phi.
$$
The <density matrix> representation retains this distinction between a <pure state> and a <mixed state>. https://quantum.cloud.ibm.com/learning/en/courses/general-formulation-of-quantum-information/density-matrices/density-matrix-basics[Watrous, IBM Quantum Learning, Density matrix basics, Interpretation of density matrices and Connection to quantum state vectors]{ref}.
For example, at $\phi=0$, the <quantum superposition> yields the plus outcome with <probability> about $0.9330$, whereas the <mixed state> yields it with <probability> $0.5$. At $\phi=\pi/2$ these particular <probabilities> agree, although the <density matrices> still differ: equality for one measurement does not establish equality of <quantum states>. The <probability amplitude> figure shows the comparison throughout a full cycle of relative <quantum phase>.
This is why a <quantum superposition> is useful: <quantum gates> act linearly on its <probability amplitudes>, and <unitary operators> preserve the full <pure state> information, including relative <quantum phases>. A <Hadamard gate> can turn otherwise hidden <quantum coherence> into different outcome <probabilities>. The beautiful feature is that the same basis weights can support different predictions when amplitudes recombine. The expansion specifies a preparation and its measurement predictions; it does not assign every unmeasured outcome a simultaneous classical value. See <projective measurement> for the separate question of the conditional state after a measurement.
Back to article page
