Quantum Superposition

Quantum superposition is a fundamental principle of quantum mechanics according to which a suitable normalized linear combination of allowed state vectors is itself an allowed state. Relative to a specified set of alternatives, a system can therefore have several nonzero probability amplitudes rather than occupy just one alternative. These amplitudes can produce interference, distinguishing a coherent superposition from ordinary uncertainty about which state was prepared. The description concerns quantum states and measurement predictions, not a literal collection of simultaneously observable classical configurations. (learning.quantum.ibm.com)

Mathematical formulation

A pure quantum state is represented by a normalized vector in a Hilbert space, a complex vector space equipped with an inner product. Given an orthonormal basis $\{|i\rangle\}$, a state can be written as

The coefficients $c_i$ are complex numbers called probability amplitudes. The Born rule assigns probability $|c_i|^2$ to outcome $i$ in a measurement in that basis. Amplitudes themselves are not probabilities: their signs and complex phases affect how contributions combine. (learning.quantum.ibm.com)

For a qubit, the general pure state is

The statement that a state is a superposition is basis-dependent. A vector with multiple nonzero coefficients in one basis can be a single basis vector in another. Thus “being in superposition” is meaningful relative to specified states or measurement alternatives, rather than an absolute property independent of representation. (learning.quantum.ibm.com)

In a position representation, the same principle applies to the wave function. The Schrödinger equation is linear: if two wave functions solve the same equation, their linear combination also solves it, subject to normalization and the relevant boundary conditions. This provides the dynamical expression of the superposition principle. (galileo.phys.virginia.edu)

Phase and interference

Consider the states

Both give outcomes 0 and 1 with equal probabilities when measured in the computational basis. Nevertheless, they are different, orthogonal states. A measurement in the $\{|+\rangle,|-\rangle\}$ basis distinguishes them perfectly. Their difference is a relative phase between components, not a change in their computational-basis probabilities. (quantum.cloud.ibm.com)

Multiplying every component by the same phase factor $e^{i\theta}$ introduces only a global phase and does not change observable predictions. Relative phases, by contrast, can change interference. For alternatives contributing amplitudes $a$ and $b$ to the same outcome,

The final term produces constructive or destructive interference. It would be absent if the alternatives were combined only as classical probabilities. (quantum.cloud.ibm.com)

The double-slit experiment illustrates this distinction. An electron can yield a localized detection event, while many repeated detections reveal an interference pattern when the two paths remain coherent. If a physical interaction creates a distinguishable record of the path, interference is reduced or lost. No conscious observer is required: the relevant issue is whether path information is physically recorded. (feynmanlectures.caltech.edu)

Superpositions and statistical mixtures

A coherent superposition must be distinguished from a statistical mixture. Preparing $|+\rangle$ is not equivalent to randomly preparing $|0\rangle$ or $|1\rangle$ with equal probabilities, even though both preparations give identical computational-basis statistics. Their different behavior becomes evident in another measurement basis. (learning.quantum.ibm.com)

The density matrix makes this distinction explicit:

The off-diagonal entries of $\rho_+$ encode coherence in the computational basis. The mixture lacks those entries and gives equal probabilities for the two outcomes in the plus/minus basis; $|+\rangle$ gives the plus outcome with certainty. Off-diagonal coherence, like the identification of superposition components, depends on the chosen basis. (learning.quantum.ibm.com)

Dynamics, entanglement, and measurement

For a time-independent Hamiltonian, an energy eigenstate acquires a phase $e^{-iEt/\hbar}$. Components with different energies acquire different phases, so their relative phase evolves. This can produce time-dependent interference and changing measurement probabilities, even when the probabilities of the individual energy outcomes remain constant. (feynmanlectures.caltech.edu)

For composite systems, state spaces combine through the tensor product. Quantum entanglement occurs when a joint pure state cannot be factored into separate subsystem states. The Bell state $(|00\rangle+|11\rangle)/\sqrt2$, for example, is an entangled superposition. Superposition alone does not imply entanglement: a single qubit can be in a superposition without any second system. (learning.quantum.ibm.com)

Quantum decoherence arises when interactions correlate a system with its environment, suppressing locally accessible interference between selected alternatives. It helps explain the emergence of approximately classical behavior. However, suppression of interference does not by itself select one unique outcome from the joint quantum state; that distinction is central to the measurement problem. (arxiv.org)

Quantum information processing

A quantum gate can create, transform, or recombine superpositions. In a quantum computer, an $n$-qubit pure state has amplitudes for $2^n$ computational-basis strings. This does not mean that a measurement reveals all those strings or all associated computational results. Useful algorithms instead arrange interference so that desired information can be extracted through measurement. Preserving coherence against environmental interactions is consequently a central requirement of quantum information processing. (learning.quantum.ibm.com)