Decoherence

What decoherence is

Decoherence is the loss of quantum coherence — the suppression of off-diagonal elements of the system's density matrix in a preferred basis — caused by the system becoming entangled with an environment it cannot track. It is the mechanism by which quantum superpositions stop behaving quantum-mechanically and start looking like classical statistical mixtures, without any collapse postulate beyond ordinary unitary evolution of $S+E$ .

The canonical model: a which-path measurement by the environment

Let a system qubit start in a superposition $|\psi\rangle_S = \alpha|0\rangle + \beta|1\rangle$ and an environment start in $|e_0\rangle$ . Suppose the interaction copies the system's state into the environment, like a measurement:

\bigl(\alpha|0\rangle + \beta|1\rangle\bigr)\otimes|e_0\rangle \;\longrightarrow\; \alpha\,|0\rangle\otimes|e_0\rangle + \beta\,|1\rangle\otimes|e_1\rangle .

Tracing out the environment, the system's reduced density matrix is

\rho_S = \begin{pmatrix} |\alpha|^2 & \alpha\beta^{*}\,\langle e_1|e_0\rangle \\[4pt] \alpha^{*}\beta\,\langle e_0|e_1\rangle & |\beta|^2 \end{pmatrix}.

The diagonal (the populations) is untouched. The off-diagonal coherences are multiplied by the overlap $\langle e_0|e_1\rangle$ of the environment states the two branches push the bath into.

The decoherence factor

Define the decoherence factor $D = \langle e_0|e_1\rangle$ . When the environment cannot distinguish the branches ( $|e_0\rangle = |e_1\rangle$ , $D=1$ ) nothing happens: the qubit stays coherent. When the branches drive the environment to orthogonal states ( $\langle e_0|e_1\rangle = 0$ , $D=0$ ) the coherences vanish entirely:

\rho_S \;\longrightarrow\; \begin{pmatrix} |\alpha|^2 & 0 \\ 0 & |\beta|^2 \end{pmatrix}.

The pure superposition has become an incoherent classical mixture of $|0\rangle$ and $|1\rangle$ . For a macroscopic environment, $|D(t)|$ typically decays toward zero on a timescale far shorter than any other dynamical scale — which is why large superpositions are essentially never seen.

Einselection and the pointer basis

The interaction $H_{SB}$ singles out a preferred set of system states — the pointer states — that are least disturbed by the coupling, because the environment records them without disturbing them back. Superpositions of pointer states decohere rapidly, while the pointer states themselves remain robust. This environment-induced superselection (einselection) explains why we observe definite positions, definite currents, definite spins — the pointer basis of the relevant interaction — rather than arbitrary superpositions. For a dephasing coupling $\propto \sigma_z$ , the pointer states are $|0\rangle$ and $|1\rangle$ , and coherences in that basis are exactly what decay.

Why it matters for quantum computing

Every qubit technology fights decoherence. A logical superposition entangling with stray photons, two-level defects, or nuclear spins loses its phase, corrupting the computation. The coherence time sets how many gates can run before the information is lost to the bath. The rest of this module turns this qualitative picture into an equation of motion — the Lindblad master equation — whose dephasing and relaxation terms quantify exactly how fast $\rho_S$ decoheres.