6.4.1 Decoherence#

Prompts

Explain decoherence: how does coupling to an environment turn a coherent superposition into a classical mixture? Walk through the partial trace calculation step by step.
What are pointer states, and why does the environment “select” them (einselection)? Give a specific example with a coupling Hamiltonian.
Why do macroscopic objects decohere almost instantaneously while small quantum systems maintain coherence? What determines the timescale \(T_2\)?
How is decoherence related to the dephasing quantum channel? What does “measurement + forgetting” mean physically when you trace out the environment?
What is a decoherence-free subspace, and why is the two-qubit singlet \(\vert\Psi^-\rangle\) immune to common-mode dephasing? Identify the symmetry of the collective coupling \(\hat{Z}_1+\hat{Z}_2\) that protects it, and why such subspaces underpin quantum error correction.

Lecture Notes#

Overview#

When a qubit couples to its environment—a thermal bath, stray photons, or surrounding atoms—the environment becomes entangled with the system and effectively “measures” it. The off-diagonal elements of the density matrix decay, turning a coherent superposition into a classical mixture. This is decoherence: the mechanism behind the quantum-to-classical transition.

Decoherence can be understood in two steps: the environment entangles with the system (like a measurement), then we trace out the environment (like forgetting the outcome). No new postulate is needed — decoherence is simply the dephasing quantum channel (§6.3.3) applied by the environment. Pointer states are the robust states that survive this process; all other superpositions decohere on a timescale \(T_2 = 1/\gamma\).

Decoherence Process#

Consider a qubit \(S\) initially in a superposition, coupled to environment \(E\) (bath of many modes). After interaction, the total state evolves to:

(255)#\[ \alpha\vert 0\rangle_S \vert E_0\rangle_E + \beta\vert 1\rangle_S \vert E_1\rangle_E \;\longrightarrow\; \alpha\vert 0\rangle_S \vert E_0(t)\rangle_E + \beta\vert 1\rangle_S \vert E_1(t)\rangle_E \]

where \(\vert E_0(t)\rangle\) and \(\vert E_1(t)\rangle\) are (nearly) orthogonal environmental states. The environment has become a which-path detector: it encodes information about which branch the system is in.

Pointer States and Einselection

Pointer states are the eigenstates of the system-environment interaction Hamiltonian \(\hat{H}_\text{int}\). They are the states robust under decoherence: the environment correlates with them without creating superpositions between them.

Einselection (Zurek): The environment selects a preferred pointer basis. Superpositions of pointer states decohere rapidly into mixtures; individual pointer states remain stable.

Example: For \(\hat{H}_\text{int} = \hat{Z} \otimes \hat{B}_E\), the pointer basis is \(\{\vert 0\rangle, \vert 1\rangle\}\) (eigenstates of \(\hat{Z}\)). The superposition \(\vert+\rangle = \frac{1}{\sqrt{2}}(\vert 0\rangle + \vert 1\rangle)\) decoheres rapidly; \(\vert 0\rangle\) and \(\vert 1\rangle\) do not.

Reduced Density Matrix#

Tracing out the environment gives the reduced density matrix:

(256)#\[ \hat{\rho}_S(t) = \operatorname{Tr}_E\!\left(\vert\Psi(t)\rangle\langle\Psi(t)\vert\right) \]

Substituting the entangled state (255) and carrying out the partial trace gives:

(257)#\[ \hat{\rho}_S(t) = \vert\alpha\vert^2 \vert 0\rangle\langle 0\vert + \vert\beta\vert^2 \vert 1\rangle\langle 1\vert + \alpha\beta^*\,\langle E_1(t)\vert E_0(t)\rangle\,\vert 0\rangle\langle 1\vert + \beta\alpha^*\,\langle E_0(t)\vert E_1(t)\rangle\,\vert 1\rangle\langle 0\vert \]

Derivation: Reduced Density Matrix from the Partial Trace

Write \(\vert E_0\rangle \equiv \vert E_0(t)\rangle\) and \(\vert E_1\rangle \equiv \vert E_1(t)\rangle\) for brevity. The partial trace over the environment acts only on the environmental factor of a product operator, and on a pair of environmental states it returns their overlap:

\[ \operatorname{Tr}_E\!\left(\vert s\rangle\langle s'\vert \otimes \vert E_i\rangle\langle E_j\vert\right) = \vert s\rangle\langle s'\vert \sum_k \langle k\vert E_i\rangle\langle E_j\vert k\rangle = \langle E_j\vert E_i\rangle\;\vert s\rangle\langle s'\vert \]

where \(\{\vert k\rangle\}\) is any orthonormal environment basis and \(\sum_k \vert k\rangle\langle k\vert = \hat{I}_E\).

Form the joint projector from \(\vert\Psi(t)\rangle = \alpha\vert 0\rangle\vert E_0\rangle + \beta\vert 1\rangle\vert E_1\rangle\):

\[\begin{split} \begin{split} \vert\Psi\rangle\langle\Psi\vert = {}& \vert\alpha\vert^2\,\vert 0\rangle\langle 0\vert\otimes\vert E_0\rangle\langle E_0\vert + \alpha\beta^*\,\vert 0\rangle\langle 1\vert\otimes\vert E_0\rangle\langle E_1\vert \\ & + \beta\alpha^*\,\vert 1\rangle\langle 0\vert\otimes\vert E_1\rangle\langle E_0\vert + \vert\beta\vert^2\,\vert 1\rangle\langle 1\vert\otimes\vert E_1\rangle\langle E_1\vert \end{split} \end{split}\]

Apply \(\operatorname{Tr}_E\) to each term with the overlap rule above:

\[\begin{split} \begin{split} \hat{\rho}_S(t) = {}& \vert\alpha\vert^2\,\langle E_0\vert E_0\rangle\;\vert 0\rangle\langle 0\vert + \alpha\beta^*\,\langle E_1\vert E_0\rangle\;\vert 0\rangle\langle 1\vert \\ & + \beta\alpha^*\,\langle E_0\vert E_1\rangle\;\vert 1\rangle\langle 0\vert + \vert\beta\vert^2\,\langle E_1\vert E_1\rangle\;\vert 1\rangle\langle 1\vert \end{split} \end{split}\]

The environmental states are normalized, \(\langle E_0\vert E_0\rangle = \langle E_1\vert E_1\rangle = 1\), which gives (257). The \(\vert 0\rangle\langle 1\vert\) coefficient is \(\alpha\beta^*\) — the ket \(\vert 0\rangle\) carries the amplitude \(\alpha\) while the bra \(\langle 1\vert\) carries \(\beta^*\) — and the \(\vert 1\rangle\langle 0\vert\) entry is its complex conjugate, so \(\hat{\rho}_S\) is Hermitian.

As the environmental states become orthogonal (\(\langle E_0\vert E_1\rangle \to 0\)), the off-diagonal coherences vanish. The system transitions from a pure state to a mixed state — not from any measurement postulate, but purely from entanglement with the environment.

Decoherence Timescale#

In the weak coupling limit, the environmental overlap decays exponentially:

(258)#\[ \langle E_0(t) \vert E_1(t)\rangle \approx \mathrm{e}^{-t/T_2} \]

where \(T_2 = 1/\gamma\) is the dephasing time. The microscopic origin of this decay:

Microscopic Origin: The Spin-Boson Decoherence Function

The exponential form just quoted is the weak-coupling, high-temperature limit of a more general result. Model the environment as a bath of harmonic oscillators (bosonic modes) coupled to the qubit through

\[ \hat{H}_\text{int} = \hat{Z} \otimes \sum_k g_k\left(\hat{a}_k\,\mathrm{e}^{-\mathrm{i}\omega_k t} + \hat{a}_k^\dagger\,\mathrm{e}^{\mathrm{i}\omega_k t}\right) \]

with \(g_k\) the coupling energy of mode \(k\). This is the spin-boson model in its pure-dephasing form: \(\hat{H}_\text{int}\) commutes with \(\hat{Z}\), so the qubit never leaves its pointer basis \(\{\vert 0\rangle, \vert 1\rangle\}\) — the bath only records which branch the qubit occupies.

Because \(\hat{Z}\) is conserved, each branch drives the bath under its own Hamiltonian: branch \(\vert 0\rangle\) (\(\hat{Z} = +1\)) and branch \(\vert 1\rangle\) (\(\hat{Z} = -1\)) push every oscillator with equal and opposite force. The conditional bath states \(\vert E_0(t)\rangle\) and \(\vert E_1(t)\rangle\) are therefore displaced from equilibrium in opposite directions, so the relative coupling that sets their overlap is \(2g_k\).

Evaluating this overlap for a bath in thermal equilibrium at temperature \(T\) is a standard result of the independent-boson model:

\[ \langle E_0(t)\vert E_1(t)\rangle = \exp\!\left(-\frac{4}{\hbar^2}\int_0^\infty \frac{J(\omega)}{\omega^2}\,(1 - \cos\omega t)\,\coth\frac{\hbar\omega}{2k_BT}\,\mathrm{d}\omega\right) \]

where \(J(\omega) = \sum_k g_k^2\,\delta(\omega - \omega_k)\) is the bath spectral density. The prefactor \(4 = 2^2\) is the square of the relative coupling \(2g_k\) between the two branches; the \(\coth(\hbar\omega/2k_BT)\) weight counts both thermal and zero-point bath fluctuations. Writing the exponent as \(\Gamma(t)\), the coherence decays as \(\langle E_0(t)\vert E_1(t)\rangle = \mathrm{e}^{-\Gamma(t)}\).

For an ohmic bath at high temperature (\(k_BT \gg \hbar\omega_c\), with \(\omega_c\) the bath cutoff frequency), \(\Gamma(t)\) grows linearly in time, \(\Gamma(t) \approx \gamma t\), reproducing the exponential decay \(\mathrm{e}^{-\gamma t}\) of (258). The dephasing rate is fixed by the low-frequency bath modes and grows with temperature, \(\gamma \propto k_BT\) — hotter environments decohere the qubit faster.

Macroscopic objects: The decoherence rate scales extensively with system size \(N\): \(\Gamma \sim N\gamma_\text{atom}\). For a macroscopic object (\(N \sim 10^{26}\)), \(T_2 \sim 10^{-26}\,\text{s}\) — decoherence is essentially instantaneous. This is why we never observe macroscopic superpositions.

Poll: Decoherence and superposition suppression

In the spin-environment model, a two-level spin couples to many environmental modes. Starting from a superposition \(\vert \psi\rangle = (\vert 0\rangle + \vert 1\rangle)/\sqrt{2}\), the density matrix of the spin acquires off-diagonal terms (coherences) that decay: \(\rho_{01}(t) \propto \exp(-\gamma t)\). What is the physical origin of this suppression?

(A) The spin loses energy to the environment; dissipation destroys coherence.

(B) Entanglement with the environment causes the spin’s state to become mixed; off-diagonal terms depend on environmental correlations that are unobservable.

(D) Quantum mechanics is fundamentally non-unitary on open systems.

Discussion: Bound Between \(T_1\) and \(T_2\)

For a qubit coupled to a thermal bath, the decoherence time satisfies \(T_2 \leq 2T_1\). This bound is saturated by pure dephasing. Express \(T_2\) in terms of the relaxation rate \(\Gamma_1 = 1/T_1\) and pure dephasing rate \(\Gamma_\varphi = 1/T_\varphi\): the relation is \(\frac{1}{T_2} = \frac{1}{2T_1} + \frac{1}{T_\varphi}\). Why does energy relaxation contribute to dephasing even without random phase kicks?

Decoherence as a Quantum Channel#

Decoherence is not a new phenomenon — it is the dephasing channel (§6.3.3) derived from a microscopic model:

Entanglement (environment measures): \(\vert \psi\rangle \otimes \vert E_0\rangle \to \alpha\vert 0\rangle\vert E_0(t)\rangle + \beta\vert 1\rangle\vert E_1(t)\rangle\)
Partial trace (forgetting): \(\operatorname{Tr}_E(\cdot)\) discards the measurement outcome.

The result is exactly the dephasing channel:

(259)#\[\begin{split} \mathcal{E}(\hat{\rho}) = \begin{pmatrix} \rho_{00} & \rho_{01}\,\mathrm{e}^{-t/T_2} \\ \rho_{10}\,\mathrm{e}^{-t/T_2} & \rho_{11} \end{pmatrix} \end{split}\]

This identification — decoherence = quantum channel — means the full machinery of §6.3.3 applies to open-system evolution without any new framework.

Decoherence Is Not Measurement Collapse

Decoherence explains why off-diagonal elements of \(\hat{\rho}_S\) decay, making interference unobservable. But the full system+environment state remains pure — decoherence does not select a single outcome. The measurement problem (why we observe definite results) is a separate philosophical question.

Quantum-to-Classical Transition#

Under decoherence, a superposition of pointer states rapidly becomes an effective classical mixture:

\[\begin{split} \hat{\rho}_S(t) \approx \begin{pmatrix} 1/2 & \mathrm{e}^{-t/T_2}/2 \\ \mathrm{e}^{-t/T_2}/2 & 1/2 \end{pmatrix} \xrightarrow{t \gg T_2} \frac{\hat{I}}{2} \end{split}\]

For macroscopic objects, this happens in \(\sim 10^{-26}\) s, far below any observable timescale. This is why classical physics works: quantum coherence is perpetually destroyed by environmental coupling before it can manifest at macroscopic scales.

Decoherence-Free Subspaces#

Some states are immune to specific decoherence channels due to symmetry. For two qubits coupling identically to the environment (\(\hat{H}_\text{int} = (\hat{Z}_1 + \hat{Z}_2)\otimes \hat{B}_E\)), the noise couples only through \(\hat{Z}_1 + \hat{Z}_2\). Any state annihilated by \(\hat{Z}_1 + \hat{Z}_2\)—equivalently, of zero total \(z\)-spin—is therefore decoherence-free, and both Bell states \(\vert\Psi^\pm\rangle\) qualify:

\[\begin{split} \begin{split} (\hat{Z}_1 + \hat{Z}_2)\,\vert\Psi^\pm\rangle &= 0 \\ \vert\Psi^\pm\rangle &= \frac{1}{\sqrt{2}}(\vert 01\rangle \pm \vert 10\rangle) \end{split} \end{split}\]

The singlet \(\vert\Psi^-\rangle\) and the \(S^z = 0\) triplet \(\vert\Psi^+\rangle\) are both left unchanged by the common-mode dephasing, so the decoherence-free subspace is the entire \(S^z = 0\) subspace, spanned by \(\{\vert\Psi^+\rangle, \vert\Psi^-\rangle\}\). Such symmetry-protected subspaces are a foundation of quantum error correction (§6.4.3).

Summary#

Decoherence mechanism: Entanglement between system and environment creates entangled state; tracing out environment (since inaccessible) leaves reduced density matrix; off-diagonal (coherence) terms decay.
Pointer basis: Environment couples to specific system observable (e.g., position); only that basis is robust—superpositions in the pointer basis decohere exponentially (\(T_2 \sim 1/\gamma\)).
Decoherence time: \(T_2\propto 1/(N\gamma)\) where \(N\) is the environment size and \(\gamma\) is the per-particle coupling rate (so the total rate \(\Gamma\sim N\gamma\)). Macroscopic objects decohere in femtoseconds; microscopic quantum systems survive much longer.
Decoherence as a quantum channel: The two-step process—environment entangles with the system, then the environment is traced out—reproduces exactly the dephasing channel of §6.3.3, so open-system evolution needs no new framework beyond the channel formalism. Decoherence makes interference unobservable, but the joint system+environment state stays pure: it does not by itself collapse the state or resolve the measurement problem.
Pure to mixed: Initial pure state \(\vert\psi\rangle\) evolves to mixed state \(\hat{\rho}(t)\) with vanishing off-diagonal terms in pointer basis; superposition becomes incoherent mixture.
Arrow of time: Decoherence is irreversible in practice (environment has huge dimension); explains classical behavior of macroscopic objects and the emergence of a classical world from quantum mechanics.
Decoherence-free subspaces: States with the right symmetry escape a given decoherence channel—any two-qubit state of zero total \(z\)-spin (the subspace spanned by \(\vert\Psi^+\rangle\) and \(\vert\Psi^-\rangle\)) is untouched by common-mode dephasing. Such symmetry-protected subspaces are a foundation of quantum error correction.

Homework#

1. Environment Entanglement. A qubit \(S\) is initially in \(\vert\psi_S\rangle = \frac{1}{\sqrt{2}}(\vert 0\rangle + \vert 1\rangle)\). After coupling to environment \(E\) (initially \(\vert 0_E\rangle\)), the joint state is \(\vert\Psi_{SE}(t)\rangle = \frac{1}{\sqrt{2}}(\vert 0\rangle_S \vert E_0(t)\rangle_E + \vert 1\rangle_S \vert E_1(t)\rangle_E)\).

(a) Compute the reduced density matrix \(\hat{\rho}_S(t) = \operatorname{Tr}_E(\vert\Psi_{SE}(t)\rangle\langle\Psi_{SE}(t)\vert)\).

(b) Show that when \(\langle E_0(t)\vert E_1(t)\rangle = 0\), the off-diagonal coherence terms vanish completely. Explain physically why orthogonality of environmental states implies loss of quantum coherence.

(c) For \(\langle E_0(t)\vert E_1(t)\rangle = \lambda(t)\) with \(\vert\lambda(t)\vert < 1\), show that \(\rho_{01}(t) = \frac{\lambda^*(t)}{2}\), which reduces to \(\lambda(t)/2\) when the overlap is real.

2. Weak Coupling Decoherence. In the weak coupling limit, \(\langle E_0(t)\vert E_1(t)\rangle = \mathrm{e}^{-t/T_2}\).

(a) Show that the coherence decays as \(\rho_{01}(t) = \rho_{01}(0)\,\mathrm{e}^{-t/T_2}\).

(b) Estimate the decoherence rate \(\gamma = 1/T_2\) for a spin-1/2 magnetic impurity in a crystal lattice, where the effective coupling to phonons is \(\lambda \sim 10^{-1}\) meV and the Debye temperature is \(\Theta_D = 300\) K. Use dimensional analysis.

(c) For a macroscopic object (\(m = 1\) mg, \(N \sim 10^{20}\) atoms) with each atom decohering at rate \(\gamma_\text{atom}\), estimate the total decoherence rate \(\Gamma = N\gamma_\text{atom}\) and the timescale \(T_2 = 1/\Gamma\). Why is macroscopic decoherence so much faster?

3. Dephasing Hamiltonian. The interaction Hamiltonian is \(\hat{H}_\text{int} = \hat{Z} \otimes \hat{B}_E\).

(a) Show that if the system is in pointer state \(\vert 0\rangle\), the system-environment evolution has the form \(\vert 0\rangle \otimes \vert E_0(t)\rangle\) — the system state remains definite, only the environment evolves.

(b) Show that if the system starts in \(\vert+\rangle = \frac{1}{\sqrt{2}}(\vert 0\rangle + \vert 1\rangle)\), the evolution creates entanglement between system and environment, leading to decoherence.

(c) Explain in words: why are \(\vert 0\rangle\) and \(\vert 1\rangle\) pointer states for this Hamiltonian but \(\vert+\rangle\) and \(\vert-\rangle\) are not?

4. Fluctuating Field Decoherence. A spin-1/2 system couples to a fluctuating field \(\hat{H}_\text{int} = \hat{Z} B_z(t)\), where \(\langle B_z(t)B_z(t')\rangle = \frac{\Delta^2}{2\tau_c}\mathrm{e}^{-\vert t-t'\vert/\tau_c}\) is colored noise.

(a) Show that \(\vert 0\rangle\) (a \(z\)-basis pointer state) commutes with \(\hat{H}_\text{int}\) and undergoes no dephasing.

(b) A spin in \(\vert+\rangle\) decoheres as \(\rho_{01}(t) = \frac{1}{2}\mathrm{e}^{-t/T_2}\) with \(T_2 = 2\tau_c/\Delta^2\). Show that at \(t \gg T_2\), the state is indistinguishable from the random mixture \(\frac{1}{2}\vert 0\rangle\langle 0\vert + \frac{1}{2}\vert 1\rangle\langle 1\vert\).

(c) Explain why \(\vert+\rangle\) is not a pointer state for \(B_z\) noise, but \(\vert+\rangle\) would be a pointer state for \(B_x\) noise (\(\hat{H}_\text{int} = \hat{X} B_x\)).

5. Schrödinger Cat Decoherence. Consider the Schrödinger cat state \(\vert\Psi\rangle = \frac{1}{\sqrt{2}}(\vert\text{alive}\rangle + \vert\text{dead}\rangle)\) for a macroscopic object with \(N \sim 10^{26}\) atoms. Each atom decoheres at rate \(\gamma_\text{atom} \sim 10^{12}\) s\(^{-1}\).

(a) Estimate \(\Gamma = N\gamma_\text{atom}\) and \(T_2 = 1/\Gamma\). Compare to the Planck time \(t_P \sim 10^{-44}\) s.

(b) Explain why decoherence — not a fundamental postulate — is the reason we never observe macroscopic superpositions. What would have to change for us to observe a “cat state”?

(c) Discuss: the full system+environment remains in a pure state even after decoherence. Why then does the system appear to be in a classical mixture?

6. Decoherence as Quantum Channel. A qubit \(S\) couples to an ancilla \(A\) via \(\hat{U} = \mathrm{e}^{\mathrm{i}\theta\hat{Z}_S \otimes \hat{Z}_A}\). The ancilla starts in \(\vert+\rangle_A = \frac{1}{\sqrt{2}}(\vert 0\rangle + \vert 1\rangle)\).

(a) Apply \(\hat{U}\) to \(\hat{\rho}_S \otimes \vert+\rangle\langle+\vert_A\) and trace out the ancilla. Show that the result is the dephasing channel \(\mathcal{E}(\hat{\rho}) = \hat{K}_0\hat{\rho} \hat{K}_0^\dagger + \hat{K}_1\hat{\rho} \hat{K}_1^\dagger\) with \(\hat{K}_0 = \sqrt{1-p}\,\hat{I}\), \(\hat{K}_1 = \sqrt{p}\,\hat{Z}\), where \(p = \frac{1}{2}(1 - \cos 2\theta)\).

(b) Identify this as decoherence in the \(\hat{Z}\) pointer basis. What is the role of the ancilla in this model?

(c) Explain in words why this calculation demonstrates “decoherence = measurement + forgetting”: what is measured, and what is forgotten?

7. Collective Decoherence and DFS. Two qubits couple to a common environment via \(\hat{H}_{\text{int}} = g(\hat{Z}_1 + \hat{Z}_2)\otimes\hat{B}_{E}\) (identical, correlated noise).

(a) Show that both the singlet \(\vert\Psi^{-}\rangle = (\vert 01\rangle - \vert 10\rangle)/\sqrt{2}\) and the \(S^{z}{=}0\) triplet \(\vert\Psi^{+}\rangle = (\vert 01\rangle + \vert 10\rangle)/\sqrt{2}\) are decoherence-free: both satisfy \((\hat{Z}_1 + \hat{Z}_2)\vert\psi\rangle = 0\).

(b) Show by contrast that the triplets \(\vert 00\rangle\) and \(\vert 11\rangle\) (eigenvalues \(\pm 2\) of \(\hat{Z}_1+\hat{Z}_2\)) acquire a relative phase under the noise and do decohere. Conclude that the decoherence-free subspace (DFS) under collective \(\hat{Z}\) noise is the entire \(S^{z}{=}0\) subspace, spanned by \(\{\vert\Psi^{+}\rangle, \vert\Psi^{-}\rangle\}\).

(c) Identify the symmetry responsible: the noise commutes with which collective operator, and hence preserves which conserved quantum number?

(d) Explain how to encode one logical qubit by identifying \(\vert 0\rangle_{L} = \vert\Psi^{+}\rangle\) and \(\vert 1\rangle_{L} = \vert\Psi^{-}\rangle\) — fully protected against this common-mode noise while requiring two physical qubits.