Decoherence

Prerequisites: Canonical Quantum Mechanics. Full path (17 pages)

Intuitive
Concrete
Abstract

Intuitive

Macroscopic systems are part of warm and wiggly environments, and the constant interactions scramble up quantum links between parts of the system. This scrambling—called decoherence—quickly converts quantum states to normal probability distributions, even in the absence of a measurement apparatus. Decoherence thus explains why we don’t observe superpositions of large things. The cat isn’t both dead and alive; there’s just a 50 percent probability that it’s dead. Lost in Math by Sabine Hossenfelder

Concrete

Take a two-level system $S$ in the superposition $\alpha|0\rangle + \beta|1\rangle$ and an environment $E$ in some state $|E\rangle$ . Suppose the interaction leaves the states $|0\rangle$ , $|1\rangle$ unchanged but leaves a different imprint on the environment in each case:

The total state is still pure and still a superposition. But all measurements on $S$ alone are described by the reduced density matrix, obtained by tracing out the environment, $\rho_S = \operatorname{tr}_E\,|\Psi\rangle\langle\Psi|$ :

$\rho_S = \begin{pmatrix} |\alpha|^2 & \alpha\beta^*\langle E_1|E_0\rangle \\ \alpha^*\beta\,\langle E_0|E_1\rangle & |\beta|^2 \end{pmatrix} .$

The off-diagonal elements (coherences) are multiplied by the overlap $\langle E_1|E_0\rangle$ . If the two environment states are nearly orthogonal, $\rho_S$ is nearly diagonal. Then no measurement on $S$ alone can distinguish the superposition from a classical mixture with probabilities $|\alpha|^2$ and $|\beta|^2$ , and interference between the two branches is no longer visible.

Why it is fast. Suppose the environment consists of $N$ particles, each of which scatters off the system and ends up in state $|e_0^{(k)}\rangle$ or $|e_1^{(k)}\rangle$ depending on the state of $S$ . Then

$\langle E_1|E_0\rangle = \prod_{k=1}^{N}\langle e_1^{(k)}|e_0^{(k)}\rangle .$

Even if each single overlap is close to one, the product decreases exponentially with $N$ . A dust grain in air or in sunlight is hit by an enormous number of molecules and photons each second, so superpositions of macroscopically different positions lose their coherence extremely quickly. Zurek's estimate for a particle of mass $m$ in a thermal environment at temperature $T$ compares the decoherence time of a superposition of two positions a distance $\Delta x$ apart with the relaxation time $\tau_R$ of the motion:

$\tau_D \approx \tau_R \left(\frac{\lambda_{\text{th}}}{\Delta x}\right)^2, \qquad \lambda_{\text{th}} = \frac{\hbar}{\sqrt{2mk_BT}} .$

For macroscopic $m$ and $\Delta x$ the thermal wavelength is tiny and $\tau_D$ is smaller than $\tau_R$ by many orders of magnitude.

Master equation. Often one does not follow the environment and instead writes an effective equation for $\rho_S$ . A simple example is pure dephasing of a qubit,

$\frac{d\rho}{dt} = -\frac{i}{\hbar}[H,\rho] + \gamma\left(\sigma_z\rho\,\sigma_z - \rho\right) .$

For $H \propto \sigma_z$ the diagonal elements stay constant and the coherences decay as $\rho_{01}(t) = \rho_{01}(0)\,e^{-2\gamma t}$ , times a phase from $H$ .

Which basis. The basis in which $\rho_S$ becomes diagonal is the one the environment couples to, here $\{|0\rangle,|1\rangle\}$ . For ordinary matter the interaction with the environment depends on position, which is why macroscopic objects are always found well localized and never in superpositions of different places.

Decoherence explains why interference between macroscopically different alternatives is unobservable in practice. On its own it does not explain why a single outcome occurs in each run: the total state of system and environment still contains all branches. How to interpret this is part of the debate about the measurement problem.

Abstract

Let the total Hilbert space be $\mathcal H_S\otimes\mathcal H_E$ with unitary evolution $U(t)$ . For an initial product state $\rho_S\otimes\rho_E$ the reduced dynamics

$\Phi_t(\rho_S) = \operatorname{tr}_E\!\left[U(t)\,(\rho_S\otimes\rho_E)\,U(t)^\dagger\right]$

is a completely positive trace preserving map, a quantum channel. Every such map has a Kraus representation $\Phi(\rho) = \sum_i K_i\rho K_i^\dagger$ with $\sum_i K_i^\dagger K_i = 1$ . Under the Born and Markov approximations (weak coupling, environment correlation time short compared to the system time scales), together with the secular approximation, the family $\Phi_t$ becomes a semigroup, and by the theorem of Gorini, Kossakowski, Sudarshan and Lindblad its generator has the form

$\frac{d\rho}{dt} = -\frac{i}{\hbar}[H,\rho] + \sum_k \gamma_k\left(L_k\rho L_k^\dagger - \frac12\{L_k^\dagger L_k,\rho\}\right) , \qquad \gamma_k\geq0 .$

The dephasing equation in the Concrete section is the case of a single Hermitian $L = \sigma_z$ , where $\sigma_z^2 = 1$ gives $\frac12\{L^\dagger L,\rho\} = \rho$ .

Pointer states. Consider a measurement-type interaction $H_{\text{int}} = \sum_i |i\rangle\langle i|\otimes B_i$ with Hermitian $B_i$ on $\mathcal H_E$ . If it dominates the dynamics, the states $|i\rangle\langle i|$ are left invariant, while $\langle i|\rho_S|j\rangle$ is multiplied by $\langle E_j(t)|E_i(t)\rangle$ with $|E_i(t)\rangle = e^{-iB_it/\hbar}|E\rangle$ . The preferred basis is thus selected by the system-environment coupling (Zurek's einselection): it consists of the states $|i\rangle$ , the eigenbasis of the system operators through which the environment couples. When $H_S$ does not commute with $H_{\text{int}}$ , the pointer states are those that are least entangled with the environment over time, for example coherent states for a harmonic oscillator coupled through its position.

Interpretational status. The diagonal $\rho_S$ obtained this way is an improper mixture: it describes a subsystem of an entangled pure state, and no ignorance interpretation of its eigenvalues follows from the formalism alone. In algebraic terms, decoherence makes the off-diagonal parts of the system observables between pointer subspaces dynamically inaccessible, so that the effectively observable algebra becomes commutative. This is an approximate, dynamically generated superselection rule, in contrast to exact superselection rules such as the one for electric charge.

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Authors: Jakob Schwichtenberg