Quantum decoherence

Quantum decoherence is the suppression of observable phase coherence when a system becomes entangled with unobserved environmental degrees of freedom. In a preferred basis, tracing out the environment damps off-diagonal entries of the system's reduced density matrix while leaving the corresponding populations unchanged.

A device qubit couples diagonally to a finite bath of spins. Conditional bath states acquire opposite phase rotations, and their conditional environment overlap multiplies the device coherence. An initially aligned bath leaves the overlap's modulus at one. Initially equatorial spins yield $z_N(t)={\prod }_k\cos (2g_{k}t)$. Global unitary time evolution remains reversible, and a finite spin-bath coherence recurrence prevents a strict zero long-time limit at finite bath size.

Independent random couplings suppress typical coherence as the bath size increases. This conclusion concerns an ensemble or a specified large-bath limit, and must be distinguished from a pointwise long-time limit for one finite realization. Ensemble spin-bath coherence supplies exact mean and mean-square formulas for a uniform coupling distribution, while short-time Gaussian spin-bath decoherence gives the initial decay scale.

For initially equatorial spins and uniform couplings on $[0,g_{\ast }]$, expand $\log \cos (2g_{k}t)=-2g_k^2{t}^2+O(g_k^4{t}^4)$. The strong law of large numbers then gives $\sum g_k^2\simeq N{g}_{\ast }^2/3$. On the scale $t=O((g_{\ast }\sqrt{N}{)}^{-1})$, the total fourth-order remainder tends to zero, giving the Gaussian coherence envelope. This approximation concerns early times and does not remove finite spin-bath coherence recurrence.

For independent uniform couplings on $[0,g_{\ast }]$ and equatorial initial bath spins,At fixed finite $N$, the mean tends to zero and the mean square tends to $2^{-N}$ as time grows. At any fixed nonzero time, the mean-square bracket is strictly below one; the Markov inequality proves coherence tends to zero in probability as $N$ grows. Individual finite realizations still exhibit finite spin-bath coherence recurrence.

For any finite set of couplings in $z_N(t)={\prod }_k\cos (2g_{k}t)$, the simultaneous Dirichlet approximation theorem supplies arbitrarily late near-alignment of all phases, or an exact common period. Thus finite bath coherence does not tend to zero as time tends to infinity. This remains true for almost every realization of continuous random couplings; an ensemble mean can decay while each realization has recurrences.

If a joint pure state is $\alpha |0⟩|E_0⟩+\beta |1⟩|E_1⟩$, the off-diagonal entry of the system's reduced density matrix is multiplied by the environmental overlap $⟨E_1|E_0⟩$. Independent repeated interactions multiply these overlaps.

For a pure state $a|0⟩|E_0⟩+b|1⟩|E_1⟩$ with normalized environment states, the upper-right device reduced density matrix entry is $a{b}^{\ast }⟨E_1|E_0⟩$. Distinguishability of the conditional environment states suppresses device coherence. States differing only by a global phase have unit-modulus overlap and do not cause quantum decoherence.

Prepare a qubit in $|+⟩$, apply a phase gate $P(\varphi )=\mathrm{diag}(1,e^{i\varphi })$, and correlate its computational basis states with normalized environment states $|e_0⟩,|e_1⟩$. Write $⟨e_0|e_1⟩=v{e}^{i\alpha }$. After a final Hadamard gate, the environment amplitude accompanying output zero is $(|e_0⟩+e^{i\varphi }|e_1⟩)/2$. Its squared norm is $P_0=[1+v\cos (\varphi +\alpha )]/2$. Thus $v$ is the interference visibility, while $\alpha$ shifts the fringe. This quantum decoherence map commutes with the phase gate, so placing it immediately before that gate gives the same probabilities.

Quantum recoherence is a later restoration of phase coherence. It can occur when a finite environment revisits a state having substantial overlap with its initial conditional state.

The quantum measurement problem asks how a unitary superposition of system and apparatus outcomes gives rise to one definite observed outcome. Quantum decoherence explains the suppression of interference between outcome branches, but by itself does not select one branch.