Superposition

Superposition#

A quantum system can exist in multiple states simultaneously. Mathematical formalism, physical intuition, and the Stern-Gerlach experiment.

Superposition is the principle that a quantum system, rather than being confined to a single definite state, can exist in a linear combination of states. This is not a statement about our ignorance of the system’s state. The system genuinely occupies multiple states at once, with complex probability amplitudes governing the likelihood of each measurement outcome.

Mathematical Formalism#

Given a basis \(\{|0\rangle, |1\rangle\}\) for a qubit, the most general state is:

\[|\psi\rangle = \alpha|0\rangle + \beta|1\rangle\]

where \(\alpha, \beta \in \mathbb{C}\) satisfy the normalization condition:

\[|\alpha|^2 + |\beta|^2 = 1\]

The coefficients \(\alpha\) and \(\beta\) are probability amplitudes. They are not probabilities: amplitudes are complex numbers that can interfere. Probabilities are the squared moduli of amplitudes:

\[P(0) = |\alpha|^2, \quad P(1) = |\beta|^2\]

Superposition is Basis-Dependent#

A state that appears to be a superposition in one basis may be a definite state in another. The state \(|+\rangle = \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)\) is a superposition in the computational (\(Z\)) basis but a definite eigenstate in the Hadamard (\(X\)) basis.

This is analogous to how a vector \((1,1)/\sqrt{2}\) is a superposition of the \(x\) and \(y\) unit vectors but points in a definite direction along the diagonal. The physics does not depend on the choice of basis; only measurement outcomes depend on which observable is measured.

General Superposition#

For a \(d\)-dimensional system with orthonormal basis \(\{|k\rangle\}_{k=0}^{d-1}\):

\[|\psi\rangle = \sum_{k=0}^{d-1} c_k |k\rangle, \quad \sum_{k=0}^{d-1} |c_k|^2 = 1\]

The number of independent real parameters needed to specify the state is \(2d - 2\) (accounting for normalization and global phase).

The Stern-Gerlach Experiment#

The Stern-Gerlach experiment (1922) provided the first direct evidence of quantized angular momentum and remains the clearest physical demonstration of superposition and measurement.

Setup#

A beam of silver atoms passes through an inhomogeneous magnetic field. The field exerts a force proportional to the component of the atom’s magnetic moment along the field direction.

Classical Prediction#

The magnetic moment should be oriented in all directions, producing a continuous distribution on the detector screen.

Quantum Result#

The beam splits into exactly two spots, corresponding to spin-up (\(|{\uparrow}\rangle\)) and spin-down (\(|{\downarrow}\rangle\)) along the field axis. The angular momentum component is quantized: \(S_z = \pm\hbar/2\).

Sequential Measurements#

The power of superposition becomes clear with sequential Stern-Gerlach devices:

Z then Z: A beam filtered to \(|{\uparrow}_z\rangle\) and passed through a second \(Z\)-oriented device always emerges as \(|{\uparrow}_z\rangle\). No splitting occurs.
Z then X: A beam filtered to \(|{\uparrow}_z\rangle\) and passed through an \(X\)-oriented device splits 50/50. The \(Z\)-eigenstate is a superposition of \(X\)-eigenstates:

\[|{\uparrow}_z\rangle = \frac{1}{\sqrt{2}}(|{\uparrow}_x\rangle + |{\downarrow}_x\rangle)\]

Z then X then Z: After filtering \(|{\uparrow}_z\rangle\), then filtering \(|{\uparrow}_x\rangle\), the beam splits 50/50 in a final \(Z\)-measurement. The \(X\)-measurement has destroyed the \(Z\)-information, because \(|{\uparrow}_x\rangle\) is a superposition in the \(Z\)-basis.

Key Insight

The sequential Stern-Gerlach experiment demonstrates that quantum measurement is not passive observation. Measuring spin along \(X\) does not simply reveal information; it actively resets the spin state, erasing any prior information about the \(Z\)-component.

Superposition vs. Classical Mixtures#

Superposition is fundamentally different from classical uncertainty (not knowing which state the system is in):

Property	Superposition	Classical mixture
Description	\(\alpha\|0\rangle + \beta\|1\rangle\)	“50% chance of \(\|0\rangle\), 50% chance of \(\|1\rangle\)”
Density matrix	\(\|\psi\rangle\langle\psi\|\) (pure, rank 1)	\(\frac{1}{2}\|0\rangle\langle 0\| + \frac{1}{2}\|1\rangle\langle 1\|\) (mixed)
Interference	Yes	No
Off-diagonal elements	Non-zero	Zero
Basis dependence	Changes form	Remains mixed

The off-diagonal elements of the density matrix (coherences) are the signature of superposition. Decoherence, the loss of these off-diagonal elements through interaction with the environment, is what converts a quantum superposition into a classical mixture.

Superposition in Quantum Computing#

In quantum computing, superposition has a specific operational meaning: a Hadamard gate \(H\) maps a computational basis state to an equal superposition:

\[H|0\rangle = |+\rangle = \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)\]

Applying \(H\) to each of \(n\) qubits initialized to \(|0\rangle\) produces:

\[H^{\otimes n}|0\rangle^{\otimes n} = \frac{1}{\sqrt{2^n}} \sum_{x=0}^{2^n - 1} |x\rangle\]

This is an equal superposition over all \(2^n\) computational basis states. The subsequent algorithm uses interference to concentrate probability amplitude on desired outcomes.

Superposition is Necessary but Not Sufficient

Creating a superposition is the easy part. The hard part is arranging for the amplitudes to interfere in a way that amplifies correct answers and suppresses wrong ones. Superposition without interference gives no computational advantage over random guessing.