Quantum decoherence

Quantum decoherence is the suppression of observable quantum phase relations through interaction with unobserved degrees of freedom. A simple model starts with $(a|0⟩+b|1⟩)|E⟩$ and correlates the alternatives with environmental states:Taking the partial trace multiplies the off-diagonal entry of the reduced density matrix by $⟨E_1|E_0⟩$. If the environmental records become nearly orthogonal, interference between the two alternatives becomes locally inaccessible even though the joint state can remain pure.

Preserving diagonal populations is a property of dephasing, not a general requirement for quantum decoherence. An amplitude damping channel, for example, changes an excited-state population as ${\rho }_{11}\mapsto (1-p){\rho }_{11}$ while suppressing coherence as ${\rho }_{01}\mapsto \sqrt{1-p}{\rho }_{01}$. Thus energy relaxation and quantum decoherence can occur together, with different rates. Pure dephasing instead preserves populations in its selected pointer basis.^[ref]

A decoherence time is a characteristic timescale for loss of specified off-diagonal coherence in a density matrix. In a simple qubit dephasing model, ${\rho }_{01}(t)={\rho }_{01}(0)e^{-t/T_2}$, so $T_2$ is the time for that coherence magnitude to fall by a factor $e$.

In Ramsey interferometry, two separated pulses convert accumulated phase into an interference signal; scanning the pulse separation measures the decay of its contrast. The inferred timescale depends on the basis, environment and pulse sequence. It differs from the energy-relaxation time, and nonexponential decay may require a stated threshold or fit convention rather than a unique decay constant.

A detector qubit with $H=g{Z}_D\otimes X_E$ and environment initially $|0{⟩}_E$ generates conditional states $|E_{\pm }⟩=\cos A|0⟩\pm i\sin A|1⟩$, where $A=gt/\hslash$. Their overlap is $\cos 2A$. Tracing out the environment multiplies detector coherences by this number while preserving the populations. Orthogonal records occur at $A=\pi /4$, but coherence returns later. For equal initial detector populations the reduced purity is $(1+{\cos }^{2}2A)/2$. Thus complete reduced dephasing at one time does not imply irreversible global collapse.

A basis of apparatus records robust under the relevant environmental interaction. Its meaning is dynamical and depends on the coupling and the time scale, rather than on an arbitrary coordinate choice for the apparatus Hilbert space.

If record projectors commute with an apparatus-environment interaction, their populations are preserved while the corresponding environmental states can become distinguishable. Their conditional environment overlap then suppresses interference in the apparatus reduced density matrix. A nondegenerate record coupling can thereby select robust sectors; commutation alone does not select a unique basis in a degenerate or trivial interaction.^[ref]

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.

A thought experiment placing a friend's physical measurement record inside a larger isolated quantum laboratory. Conditioned on a record the friend has a definite outcome, while an external observer using universal unitary evolution can assign an entangled laboratory state. Its contrast with a collapsed mixture is an instance of the quantum measurement problem; it does not require giving consciousness a special dynamical law. Eugene Wigner used the observer-within-the-system construction to examine such tensions.

A thought experiment correlating a microscopic quantum alternative with macroscopically distinct records, introduced by Erwin Schrödinger to sharpen the problem of applying unitary evolution to a measuring apparatus. The resulting entangled superposition and the mixture produced by selecting an outcome are different joint density operators. Quantum decoherence can make the records' reduced density operator approximately diagonal without selecting a single global component.

The problem of explaining which basis represents stable measurement records. A maximally entangled quantum premeasurement state can have several equally weighted Schmidt decompositions. Choosing a different decomposition does not alter that pure state, but declaring a projection in a different basis generally changes the resulting mixture. Environment-induced pointer-basis selection supplies a dynamical account of robust records, while leaving the further question of a single realized outcome distinct.