Phase Damping Channel

Of all the ways a qubit can go wrong, phase damping is the most quietly quantum. It takes no energy, moves no population, flips nothing you could catch with a voltmeter — and yet it is usually the process that kills a computation first. A qubit prepared in the equal superposition $(|0\rangle+|1\rangle)/\sqrt2$ carries its information not in which level is occupied but in the relative phase between them, and that phase is exactly what phase damping erases. In the computational basis the populations $\rho_{00}$ and $\rho_{11}$ are conserved to all orders while the off-diagonal coherence $\rho_{01}$ is multiplied by a shrink factor $\sqrt{1-\lambda}\le 1$ . Nothing relaxes toward the ground state; the qubit simply forgets how to interfere. The mechanism is decoherence in its purest form. Whenever a stray environmental degree of freedom — a scattered photon, a fluctuating charge, a wandering flux — couples to the qubit's $Z$ operator, it becomes entangled with the qubit and quietly records which $Z$ -eigenstate is present. Trace that environment away and the reduced dynamics is a non-unitary channel in which the environment has performed a weak, incomplete measurement of $Z$ . This is why the $Z$ -eigenstates $\{|0\rangle,|1\rangle\}$ are the robust "pointer states" that survive, while their superpositions decay: the environment is measuring in that basis whether we like it or not. Structurally, phase damping is a completely-positive trace-preserving map with a two-element Kraus representation, and a striking fact organizes everything that follows — it is unitarily equivalent to the phase-flip channel. A partial dephasing is nothing but applying $Z$ with some probability $p$ . This article builds the Kraus operators, derives the Bloch-sphere pancake, proves the phase-flip equivalence, dilates the channel to a system–environment unitary, and connects the discrete map to its Lindblad generator and the measured $T_2$ .

Key takeaways

Phase damping is a CPTP channel that shrinks a qubit's off-diagonal coherences by $\sqrt{1-\lambda}$ while leaving the diagonal populations exactly fixed — pure dephasing, with no energy exchanged.
Its two Kraus operators are $E_0=\mathrm{diag}(1,\sqrt{1-\lambda})$ and $E_1=\mathrm{diag}(0,\sqrt{\lambda})$ , and they satisfy the completeness relation $E_0^\dagger E_0+E_1^\dagger E_1=I$ for every $\lambda\in[0,1]$ .
It is unitarily equivalent to the phase-flip channel: $\mathcal{E}(\rho)=(1-p)\rho+p\,Z\rho Z$ with $1-2p=\sqrt{1-\lambda}$ , so partial dephasing is just a random $Z$ .
On the Bloch ball it contracts the equatorial coordinates $r_x,r_y$ by $\sqrt{1-\lambda}$ and fixes $r_z$ , flattening the ball into a pancake and leaving the whole $z$ -axis as a line of fixed points.
Its continuous-time Markovian limit is a Lindblad equation with the single Hermitian jump operator $L=\sqrt{\gamma_\phi/2}\,Z$ , giving exponential coherence decay $\rho_{01}(t)=e^{-\gamma_\phi t}\rho_{01}(0)$ and a pure-dephasing time $T_\phi=1/\gamma_\phi$ that combines with $T_1$ to set the measured $T_2$ .
Key idea: phase damping kills coherences but never moves population — $\rho_{01}\mapsto\sqrt{1-\lambda}\,\rho_{01}$ with $\rho_{00},\rho_{11}$ fixed. It is exactly the phase-flip channel in disguise, driven by an environment that learns which $Z$ -eigenstate the qubit is in without taking any energy.

The core insight

Phase damping kills coherences but never moves population: $\rho_{01}\mapsto\sqrt{1-\lambda}\,\rho_{01}$ with $\rho_{00},\rho_{11}$ fixed. It is exactly the phase-flip channel in disguise, and its environment is one that learns which $Z$ -eigenstate the qubit is in without taking any energy.

Where this lives today

Phase damping is not a textbook idealization — it is usually the dominant error on real quantum hardware, and characterizing it is a daily task. On superconducting transmons and semiconductor spin qubits, low-frequency flux, charge, and magnetic-field noise modulate the qubit's transition frequency and dephase it without inducing transitions; the resulting $T_\phi$ is extracted from the decay envelope of Ramsey fringes and combined with $1/2T_1$ to predict the measured $T_2$ . Because the noise is dominated by slow $1/f$ fluctuations, it can be partly refocused: a Hahn echo inserts a $\pi$ -pulse to cancel static frequency offsets, and CPMG or more general dynamical-decoupling sequences push the effective coherence time from $T_2^\ast$ up toward $T_2$ by filtering out the low-frequency components — the whole art of dynamical decoupling is engineering against phase damping. Designers also chase "sweet spots" in parameter space where the transition frequency is first-order insensitive to a noise source, directly minimizing $\gamma_\phi$ . Downstream, because phase damping equals a phase-flip channel, dephasing-biased hardware is matched to biased-noise codes — the XZZX surface code and bosonic cat qubits deliberately trade weaker $X$ protection for stronger $Z$ protection, an allocation that only pays off once the channel model tells you $Z$ errors dominate. The open frontier is the noise this Markovian channel misses: temporally correlated, non-Markovian dephasing from structured environments, where a single rate $\gamma_\phi$ is no longer enough.

The physical picture

Picture a qubit poised in the superposition $(|0\rangle+|1\rangle)/\sqrt2$ . A passing environmental particle scatters a little differently depending on whether the qubit sits in $|0\rangle$ or $|1\rangle$ , and in doing so it becomes entangled with the qubit — it now holds a partial copy of "which state". Crucially, no quantum jumps: the particle carries away no energy, the two qubit levels keep their occupation, and nothing about the populations changes. But the moment you average over (trace out) that particle, the relative phase that would have shown up as interference is gone. Only the correlation across the $|0\rangle/|1\rangle$ divide was leaked, and leaking it is enough to destroy the superposition. The first figure follows one such event and shows the bookkeeping: the coherence $\rho_{01}$ is multiplied by $\sqrt{1-\lambda}$ , here $\sqrt{0.64}=0.80$ , while the energy-level diagram is untouched.

Figure

A qubit superposition entangles with an environment that records which Z-eigenstate is present, without exchanging energy: the level spacing is unchanged so populations are fixed, but tracing out the environment multiplies the coherence ρ₀₁ by √(1−λ) = 0.80 for λ = 0.36

On the Bloch sphere the same story becomes a shape. The equatorial plane — the states of maximal coherence, where $r_x^2+r_y^2$ is largest — is squeezed uniformly toward the polar axis by the factor $\sqrt{1-\lambda}$ , while the north–south axis of definite $Z$ never moves. The ball flattens into a pancake, and every diagonal state along the $z$ -axis is a fixed point, in sharp contrast to amplitude damping, which drags the whole ball toward a single pole. Full dephasing, $\lambda=1$ , collapses the ball entirely onto the $z$ -axis segment: a superposition becomes the classical mixture $\rho_{00}|0\rangle\langle0|+\rho_{11}|1\rangle\langle1|$ , exactly what an averaged $Z$ -measurement would leave behind. The second figure shows the pancake with concrete numbers.

Figure

The Bloch ball under phase damping: equatorial coordinates rₓ and r_y contract by √(1−λ), here 0.80 for λ = 0.36, while r_z is untouched, so the unit ball flattens into a pancake and the entire z-axis is a line of fixed points; full dephasing λ = 1 collapses it onto the z-axis segment

The core equations

Every quantum channel begins with the operator-sum form. The map acts by conjugating the input state with a set of Kraus operators and summing them, and the completeness relation on those operators is precisely what keeps the trace — the total probability — equal to one. This single line packages complete positivity and trace preservation together.

\mathcal{E}(\rho)=E_0\,\rho\,E_0^{\dagger}+E_1\,\rho\,E_1^{\dagger},\qquad E_0^{\dagger}E_0+E_1^{\dagger}E_1=I \qquad (1)

where: $\mathcal{E}$ — the phase-damping CPTP map on a single qubit; $\rho$ — input density matrix, $2\times2$ , $\mathrm{Tr}\,\rho=1$ ; $E_0,E_1$ — the two Kraus operators; $I$ — the $2\times2$ identity. Reading: $E_0$ is the "no phase event" branch and $E_1$ the "phase event registered" branch, and the sum averages coherently over the branch the environment took but we never read. The specific Kraus operators of phase damping are diagonal in the computational basis. That diagonality is the whole story: operators diagonal in $\{|0\rangle,|1\rangle\}$ commute with the populations, so they can never move probability between levels — they can only rescale the coherence.

E_0=\begin{pmatrix}1&0\\0&\sqrt{1-\lambda}\end{pmatrix},\qquad E_1=\begin{pmatrix}0&0\\0&\sqrt{\lambda}\end{pmatrix} \qquad (2)

where: $\lambda$ — the dephasing parameter $\in[0,1]$ , the probability that a phase-scattering event is registered by the environment. Reading: at $\lambda=0$ both operators leave the state alone (the identity channel); at $\lambda=1$ the environment fully resolves the $|1\rangle$ branch and coherence is destroyed. Completeness holds for every $\lambda$ : $E_0^\dagger E_0+E_1^\dagger E_1=\mathrm{diag}(1,1-\lambda)+\mathrm{diag}(0,\lambda)=I$ . Applying equation (2) to a general density matrix exposes the channel's signature. The diagonal entries pass through untouched while both off-diagonal entries are multiplied by the same factor $\sqrt{1-\lambda}$ .

\mathcal{E}\!\left(\begin{pmatrix}\rho_{00}&\rho_{01}\\\rho_{10}&\rho_{11}\end{pmatrix}\right)=\begin{pmatrix}\rho_{00}&\sqrt{1-\lambda}\,\rho_{01}\\[2pt]\sqrt{1-\lambda}\,\rho_{10}&\rho_{11}\end{pmatrix} \qquad (3)

where: $\rho_{00},\rho_{11}$ — populations (diagonal), invariant under the channel; $\rho_{01},\rho_{10}$ — coherences (off-diagonal), scaled by $\sqrt{1-\lambda}$ . Reading: this is decoherence with no relaxation. Because $\rho_{00}$ and $\rho_{11}$ never change, no state ever relaxes toward a thermal or ground state; only the ability to interfere is degraded. The same channel has a second face in the Pauli family. Writing it as a probabilistic mixture of the identity and a $Z$ rotation makes the connection to error correction immediate — phase damping is the phase-flip channel.

\mathcal{E}(\rho)=(1-p)\,\rho+p\,Z\rho Z,\qquad E_0'=\sqrt{1-p}\,I,\quad E_1'=\sqrt{p}\,Z,\qquad 1-2p=\sqrt{1-\lambda} \qquad (4)

where: $p$ — the probability of applying a $Z$ phase flip, $p=\tfrac12\!\left(1-\sqrt{1-\lambda}\right)\in[0,\tfrac12]$ ; $Z=\mathrm{diag}(1,-1)$ — the Pauli- $Z$ operator. Reading: since $Z\rho Z$ flips the sign of the off-diagonals, the mixture multiplies coherence by $1-2p$ ; matching to (3) fixes $1-2p=\sqrt{1-\lambda}$ . The two Kraus sets are related by a $2\times2$ unitary on the environment index, so they describe the identical map. Geometrically the action is an affine contraction of the Bloch vector. Two of the three coordinates shrink by the coherence factor; the third is frozen.

(r_x,\,r_y,\,r_z)\;\longmapsto\;\bigl(\sqrt{1-\lambda}\,r_x,\;\sqrt{1-\lambda}\,r_y,\;r_z\bigr) \qquad (5)

where: $r_x,r_y,r_z$ — Bloch coordinates in $\rho=\tfrac12(I+r_xX+r_yY+r_zZ)$ ; $\sqrt{1-\lambda}$ — the equatorial contraction factor. Reading: the unit ball becomes a pancake of polar radius $1$ and equatorial radius $\sqrt{1-\lambda}$ ; the entire $z$ -axis ( $r_x=r_y=0$ ) is a line of fixed points. Finally, the continuous-time version. In the Markovian limit the discrete channel is generated by a Lindblad master equation with one Hermitian jump operator, and integrating it recovers the exponential coherence decay that experiments actually measure as $T_2$ .

\frac{\mathrm{d}\rho}{\mathrm{d}t}=\frac{\gamma_\phi}{2}\bigl(Z\rho Z-\rho\bigr),\qquad L=\sqrt{\tfrac{\gamma_\phi}{2}}\,Z,\qquad \frac{1}{T_2}=\frac{1}{2T_1}+\frac{1}{T_\phi} \qquad (6)

where: $\gamma_\phi=1/T_\phi$ — the pure-dephasing rate; $L$ — the single (Hermitian) Lindblad jump operator, so no energy is dissipated; $T_1$ — longitudinal relaxation time (from amplitude damping); $T_2$ — the measured transverse coherence time. Reading: the off-diagonal obeys $\dot\rho_{01}=-\gamma_\phi\rho_{01}$ , giving $\rho_{01}(t)=e^{-\gamma_\phi t}\rho_{01}(0)$ , so the discrete parameter is $\lambda=1-e^{-2\gamma_\phi t}$ and pure dephasing adds the rate $1/T_\phi$ to the coherence decay.

Derivation ABuilding the channel from a which-path environment

The goal is to derive the phase-damping Kraus operators from first principles, starting not from equation (2) but from an honest microscopic story: a qubit couples to a fresh environment through a unitary that lets the environment record whether the qubit is excited, and we then trace the environment out. The payoff is that the abstract map $\mathcal{E}$ acquires a mechanism — the "loss" of coherence is revealed to be information that has walked next door into modes we cannot follow, with no energy changing hands anywhere in the qubit.

Step 1. Initialize the environment in a pure state $|e_0\rangle$ and choose a joint unitary that leaves the ground branch alone but partially excites the environment on the $|1\rangle$ branch: $U|0\rangle|e_0\rangle=|0\rangle|e_0\rangle$ and $U|1\rangle|e_0\rangle=|1\rangle\bigl(\sqrt{1-\lambda}\,|e_0\rangle+\sqrt{\lambda}\,|e_1\rangle\bigr)$ . Why: the environment now carries a partial record of the qubit's $Z$ -value, with distinguishability set by $\sqrt{\lambda}$ , and — crucially — the qubit label $|0\rangle$ or $|1\rangle$ is never flipped, so the qubit's energy is conserved.

Step 2. Apply $U$ to a general superposition $\alpha|0\rangle+\beta|1\rangle$ to get $\alpha|0\rangle|e_0\rangle+\beta|1\rangle\bigl(\sqrt{1-\lambda}\,|e_0\rangle+\sqrt{\lambda}\,|e_1\rangle\bigr)$ . Why: the qubit and environment are now entangled; the cross term that would produce interference is spread across two orthogonal environment states, and orthogonality is what makes it vanish on averaging.

Step 3. Trace out the environment by projecting onto the orthonormal basis $\{|e_0\rangle,|e_1\rangle\}$ , which defines the Kraus operators $E_k=\langle e_k|U|e_0\rangle$ . Sandwiching $U$ gives $E_0=\langle e_0|U|e_0\rangle=\mathrm{diag}(1,\sqrt{1-\lambda})$ and $E_1=\langle e_1|U|e_0\rangle=\mathrm{diag}(0,\sqrt{\lambda})$ . Why: these are exactly the operators of equation (2); the "no-click" and "click" environment outcomes become the two Kraus branches, and $E_0^\dagger E_0+E_1^\dagger E_1=I$ falls out of $U$ being unitary.

Step 4. Read off the reduced state $\mathcal{E}(\rho)=E_0\rho E_0^\dagger+E_1\rho E_1^\dagger$ and confirm the diagonal is untouched while the coherence is multiplied by $\sqrt{1-\lambda}$ . Why: this closes the loop — a purely unitary, energy-conserving interaction, once the environment is discarded, is identical to the phenomenological phase-damping channel, so decoherence needs no new law of physics beyond "we cannot track the environment".

Go deeper · Derivation B · 1 minPhase damping is the phase-flip channel in disguise

Derivation BPhase damping is the phase-flip channel in disguise

The aim is to prove rigorously that the phase-damping family and the phase-flip (Pauli- $Z$ ) family are the very same set of channels, related only by a relabelling of the environment, and to pin down the exact dictionary between their parameters $\lambda$ and $p$ . This is more than a curiosity: it means every threshold, every code overhead, and every simulation of dephasing can be phrased in terms of a single probability $p$ of applying $Z$ , which is why fault-tolerance theory treats pure dephasing as a stochastic Pauli channel.

Step 1. Characterize the phase-damping map by its action on matrix elements from equation (3): $\mathcal{E}(\rho)_{00}=\rho_{00}$ , $\mathcal{E}(\rho)_{11}=\rho_{11}$ , and $\mathcal{E}(\rho)_{01}=\sqrt{1-\lambda}\,\rho_{01}$ . Why: a single-qubit channel is completely fixed once its action on this complete set of matrix elements is known, so if a phase-flip channel reproduces all three relations it must be the identical map.

Step 2. Compute the phase-flip action on the same elements. Since $Z\rho Z$ negates the off-diagonals and fixes the diagonals, $\bigl[(1-p)\rho+p\,Z\rho Z\bigr]_{01}=(1-p)\rho_{01}-p\,\rho_{01}=(1-2p)\rho_{01}$ , with $\rho_{00},\rho_{11}$ unchanged. Why: this is the same structural signature — populations fixed, coherence rescaled — differing from phase damping only in the numerical shrink factor, which we now match.

Step 3. Equate the two coherence factors, $1-2p=\sqrt{1-\lambda}$ , and solve to get $p=\tfrac12\bigl(1-\sqrt{1-\lambda}\bigr)$ , which runs over $[0,\tfrac12]$ as $\lambda$ runs over $[0,1]$ . Why: because both maps now agree on every matrix element, they are equal as channels; a partial dephasing of strength $\lambda$ is operationally indistinguishable from flipping the phase with probability $p$ .

Step 4. Exhibit the Kraus freedom explicitly: the two operator sets are related by a $2\times2$ unitary $W$ acting on the environment index, $\{E_0',E_1'\}=W\{E_0,E_1\}$ . Why: two Kraus representations describe the same channel if and only if they are related by such an isometry, so producing $W$ is the formal proof that $\{\mathrm{diag}(1,\sqrt{1-\lambda}),\mathrm{diag}(0,\sqrt{\lambda})\}$ and $\{\sqrt{1-p}\,I,\sqrt{p}\,Z\}$ are two unravelings of one map. *As a quick check: $p=\tfrac12(1-\sqrt{1-\lambda})$ gives $p=0.05$ at $\lambda=0.19$ and $p=\tfrac12$ at $\lambda=1$ , confirming the dictionary runs over the full range.*

Go deeper · Derivation C · 1 minFrom the Lindblad generator to exponential coherence decay

Derivation CFrom the Lindblad generator to exponential coherence decay

The goal is to connect the discrete channel to its continuous-time origin: to write the Lindblad (GKSL) master equation whose single Hermitian jump operator is proportional to $Z$ , integrate it, and show that the time- $t$ propagator is exactly the phase-damping channel with $\lambda=1-e^{-2\gamma_\phi t}$ . This is what licenses using a constant "dephasing rate" on hardware and cleanly separates the pure-dephasing contribution $1/T_\phi$ from the relaxation contribution $1/2T_1$ inside the measured $T_2$ .

Step 1. Take the Lindblad dissipator for the single jump operator $L=\sqrt{\gamma_\phi/2}\,Z$ : since $L$ is Hermitian and $Z^2=I$ , one finds $L\rho L^\dagger-\tfrac12\{L^\dagger L,\rho\}=\tfrac{\gamma_\phi}{2}\bigl(Z\rho Z-\rho\bigr)$ , which also equals $-\tfrac{\gamma_\phi}{4}[Z,[Z,\rho]]$ . Why: a Hermitian jump operator has no anti-Hermitian part, so it cannot dissipate energy — the generator only scrambles phase, exactly the feature that distinguishes pure dephasing from amplitude damping.

Step 2. Read the equation componentwise. The diagonal entries obey $\dot\rho_{00}=\dot\rho_{11}=0$ , while the off-diagonal obeys $\dot\rho_{01}=\tfrac{\gamma_\phi}{2}\bigl((Z\rho Z)_{01}-\rho_{01}\bigr)=\tfrac{\gamma_\phi}{2}(-\rho_{01}-\rho_{01})=-\gamma_\phi\rho_{01}$ . Why: $(Z\rho Z)_{01}=-\rho_{01}$ , so the two terms add rather than cancel; the populations sit in the kernel of the generator, confirming that no relaxation occurs.

Step 3. Integrate to get $\rho_{01}(t)=e^{-\gamma_\phi t}\rho_{01}(0)$ with $\rho_{00},\rho_{11}$ constant. Why: coherence decays as a simple exponential with rate $\gamma_\phi=1/T_\phi$ , the defining behaviour of a Ramsey fringe envelope under low-frequency dephasing.

Step 4. Match to the discrete channel by identifying the coherence factor: $\sqrt{1-\lambda}=e^{-\gamma_\phi t}$ , hence $\lambda=1-e^{-2\gamma_\phi t}\approx 2\gamma_\phi t$ for short times. Combining with the amplitude-damping coherence factor $e^{-t/2T_1}$ gives $1/T_2=1/2T_1+1/T_\phi$ . Why: this makes the phase-damping channel the exact time- $t$ solution of the master equation and slots pure dephasing into the standard $T_1$ – $T_2$ budget, with $T_2\le 2T_1$ always. *Note: with $L=\sqrt{\gamma_\phi/2}\,Z$ the dissipator equals $\tfrac{\gamma_\phi}{2}(Z\rho Z-\rho)$ , giving $\rho_{01}(t)=e^{-\gamma_\phi t}\rho_{01}(0)$ and $\lambda=1-e^{-2\gamma_\phi t}$ , and the standard relation $1/T_2=1/2T_1+1/T_\phi$ follows directly.*

Limiting cases

The single parameter $\lambda\in[0,1]$ — equivalently the phase-flip probability $p$ with $1-2p=\sqrt{1-\lambda}$ , or the elapsed time through $\lambda=1-e^{-2\gamma_\phi t}$ — interpolates from no noise to complete dephasing.

Case	Result	Meaning
No dephasing	$\lambda=0,\ p=0$	The channel is the identity; every coherence and population is preserved and the Bloch ball is unchanged.
Full dephasing	$\lambda=1,\ p=\tfrac12$	Off-diagonals vanish entirely, $\rho\mapsto\rho_{00}\|0\rangle\langle0\|+\rho_{11}\|1\rangle\langle1\|$ — an averaged $Z$ -measurement; the ball collapses to the $z$ -axis segment.
Weak / continuous limit	$\lambda=1-e^{-2\gamma_\phi t}\approx2\gamma_\phi t$	Coherence decays as $e^{-\gamma_\phi t}$ , recovering the Lindblad equation with pure-dephasing time $T_\phi=1/\gamma_\phi$ .
Any diagonal input	$\rho=\mathrm{diag}(\rho_{00},\rho_{11})\mapsto\rho$	Every diagonal state is a fixed point — the whole $z$ -axis is invariant, unlike amplitude damping's single fixed point at the pole.

Common trapPhase damping relaxes the qubit to a fixed point like amplitude damping

It does not, and the confusion causes real modeling errors. Phase damping conserves the diagonal populations exactly, so it has an entire line of fixed points — every state on the $z$ -axis is left alone — whereas amplitude damping is non-unital with a single fixed point, the ground state $|0\rangle$ at the north pole, toward which it drags the whole ball while losing energy. Phase damping loses no energy at all; its jump operator is Hermitian. A second, sharper trap is the two parametrizations: the Kraus parameter $\lambda$ and the phase-flip probability $p$ are related by $1-2p=\sqrt{1-\lambda}$ , not by $p=\lambda/2$ or $p=\lambda$ . Confusing them corrupts every downstream rate — for instance $\lambda=0.36$ is $p=0.10$ , not $p=0.18$ — so a coherence factor $\sqrt{1-\lambda}=0.80$ gets mis-reported. Always convert through the square root.

Worked examples

1 · Ramsey coherence decay — the pancake with numbers (fully worked)

A qubit is prepared in $|+\rangle=(|0\rangle+|1\rangle)/\sqrt2$ and left idle for $t=10\ \mu\mathrm{s}$ in a device with pure-dephasing time $T_\phi=20\ \mu\mathrm{s}$ (relaxation negligible on this timescale). Find the dephasing parameter $\lambda$ , the output state, the Ramsey-fringe visibility, and the purity.

Step 1. Convert the elapsed time to the dephasing parameter. The dimensionless decay is $\gamma_\phi t=t/T_\phi=10/20=0.5$ , so the coherence factor is $\sqrt{1-\lambda}=e^{-\gamma_\phi t}=e^{-0.5}=\mathbf{0.6065}$ , and therefore $\lambda=1-e^{-2\gamma_\phi t}=1-e^{-1}=1-0.3679=\mathbf{0.6321}$ .

Step 2. Write the input. $|+\rangle\langle+|=\tfrac12\begin{pmatrix}1&1\\1&1\end{pmatrix}$ , so $\rho_{00}=\rho_{11}=\rho_{01}=\rho_{10}=\tfrac12$ . Apply the channel action (3): the diagonal is untouched, and each coherence is scaled by $0.6065$ , giving $\rho_{01}'=0.6065\times0.5=\mathbf{0.3033}$ . The output is $\rho'=\begin{pmatrix}0.5 & 0.3033\\ 0.3033 & 0.5\end{pmatrix}$ .

Step 3. Read the fringe visibility. Ramsey contrast is proportional to the surviving coherence, so the visibility has dropped to $V=\sqrt{1-\lambda}=\mathbf{0.61}$ (61% of its initial value). On the Bloch sphere the vector went from $(1,0,0)$ to $(0.6065,0,0)$ — pulled toward the polar axis but staying in the equatorial plane, since $r_z=0$ is fixed.

Step 4. Compute the purity. $\mathrm{Tr}(\rho'^2)=\rho_{00}^2+\rho_{11}^2+2\rho_{01}'\rho_{10}'=0.25+0.25+2(0.3033)^2=0.5+0.1840=\mathbf{0.684}$ . The purity has fallen from $1$ toward the maximally mixed value $0.5$ , quantifying exactly how much "quantumness" the idle wait cost — all of it phase, none of it energy.

2 · Parametrizations and the $T_2$ budget (fully worked)

(a) A noise model quotes a per-gate phase-flip probability $p=0.05$ ; find the equivalent Kraus $\lambda$ and Bloch contraction. (b) A device has $T_1=80\ \mu\mathrm{s}$ and $T_\phi=40\ \mu\mathrm{s}$ ; find the measured $T_2$ .

Step 1. Convert $p$ to $\lambda$ with the dictionary $1-2p=\sqrt{1-\lambda}$ . Here $1-2(0.05)=0.90=\sqrt{1-\lambda}$ , so $1-\lambda=0.81$ and $\lambda=\mathbf{0.19}$ . The equatorial Bloch coordinates contract by $\sqrt{1-\lambda}=\mathbf{0.90}$ — a 10% loss of coherence per gate, which is why phase noise accumulates so quickly across a deep circuit.

Step 2. Assemble the $T_2$ budget from equation (6): $\dfrac{1}{T_2}=\dfrac{1}{2T_1}+\dfrac{1}{T_\phi}=\dfrac{1}{160}+\dfrac{1}{40}=\dfrac{1}{160}+\dfrac{4}{160}=\dfrac{5}{160}=\dfrac{1}{32}\ \mu\mathrm{s}^{-1}$ , so $T_2=\mathbf{32\ \mu s}$ .

Step 3. Interpret. The relaxation ceiling here would be $2T_1=160\ \mu\mathrm{s}$ , but pure dephasing has dragged the coherence time down to $32\ \mu\mathrm{s}$ — a factor of five below the ceiling. The gap between the measured $T_2$ and $2T_1$ is a direct readout of the phase-damping rate: pure dephasing dominates this qubit's decoherence, contributing $1/T_\phi=0.025\ \mu\mathrm{s}^{-1}$ against relaxation's $1/2T_1=0.00625\ \mu\mathrm{s}^{-1}$ .

Physics of the portrait

Phase damping is the mathematical skeleton of decoherence, and decoherence as a physical program — the idea that the classical world is selected by the environment rather than assumed — has a clear founder.

Origins · 1 minH. Dieter Zeh and how this was discovered

H. Dieter Zeh

1932–2018 · the founding of decoherence theory

Heinz-Dieter Zeh, a theoretical physicist at Heidelberg, is the person who first took seriously the idea that a quantum system is never isolated and that this changes everything. In a 1970 paper — met at the time with considerable skepticism — he argued that a macroscopic object is inevitably entangled with its environment, and that tracing out those uncontrolled degrees of freedom suppresses interference between "pointer" configurations without any collapse postulate. That is precisely the mechanism of the phase-damping channel: an environment that reads which $Z$ -eigenstate the system occupies, kills the coherence between them, and takes no energy in the process. Zeh's insight reframed the measurement problem as a question about where the information goes rather than a mysterious discontinuity, and it laid the groundwork for the operator-sum and master-equation machinery that Kraus, Lindblad, and later Wojciech Zurek would sharpen into calculable channels. Zurek's einselection and pointer-state program of the 1980s is a direct descendant of Zeh's proposal, and today every quoted $T_\phi$ on a superconducting or spin qubit is a number Zeh's picture predicts: coherence leaking, irreversibly, into modes we chose not to follow. He spent the rest of his career on the arrow of time and the foundations of quantum theory, insisting to the end that decoherence is unitary quantum mechanics on a Hilbert space too large to track.

History

1970

— Zeh proposes that entanglement with an inaccessible environment destroys interference

1971

— Kraus casts general open-system dynamics in the operator-sum representation

1976

— Lindblad, and Gorini, Kossakowski and Sudarshan, derive the Markovian master equation

1981

— Zurek introduces pointer states and environment-induced superselection

2003

— Zurek's review codifies einselection and the pointer basis as the origin of the classical

Going further

Texts. Nielsen and Chuang, Quantum Computation and Quantum Information, section 8.3.5 (phase damping) and section 8.3 (quantum operations) · Preskill, Lecture Notes for Physics 219, Chapter 3 (quantum channels and decoherence) · Wilde, Quantum Information Theory, Chapter 4 (quantum channels) · Watrous, The Theory of Quantum Information, Chapter 2 (channels and operations) · Kitaev, Shen and Vyalyi, Classical and Quantum Computation, Chapter 11 · Zurek, Reviews of Modern Physics 75, 715 (2003), "Decoherence, einselection, and the quantum origins of the classical". The key relations to keep in mind: the completeness relation $E_0^\dagger E_0+E_1^\dagger E_1=I$ ; the channel action showing diagonal invariance and off-diagonal scaling by $\sqrt{1-\lambda}$ ; the phase-flip equivalence $1-2p=\sqrt{1-\lambda}$ , hence $p=\tfrac12(1-\sqrt{1-\lambda})$ ; the Bloch map $(r_x,r_y,r_z)\mapsto(\sqrt{1-\lambda}r_x,\sqrt{1-\lambda}r_y,r_z)$ ; the Lindblad generator $\dot\rho=\tfrac{\gamma_\phi}{2}(Z\rho Z-\rho)$ with $L=\sqrt{\gamma_\phi/2}\,Z$ , yielding $\rho_{01}(t)=e^{-\gamma_\phi t}\rho_{01}(0)$ and $\lambda=1-e^{-2\gamma_\phi t}$ ; the worked numbers $\lambda=0.6321$ , $\sqrt{1-\lambda}=0.6065$ , purity $0.684$ at $t/T_\phi=0.5$ , and $T_2=32\ \mu\mathrm{s}$ for $T_1=80,\ T_\phi=40\ \mu\mathrm{s}$ ; and the standard relation $1/T_2=1/2T_1+1/T_\phi$ . The Lindblad prefactor $\tfrac{\gamma_\phi}{2}$ is chosen so that the coherence factor $\sqrt{1-\lambda}=e^{-\gamma_\phi t}$ and $\gamma_\phi=1/T_\phi$ are mutually consistent.

Prerequisites. Kraus Representation of Quantum Channels · CPTP Maps and Quantum Operations · Open Quantum Systems and Reduced Dynamics · Pauli, Bit-Flip and Phase-Flip Channels. Leads to. Decoherence · Relaxation and Dephasing: T1 and T2 · Lindblad / GKSL Master Equation · Depolarizing Channel.

Ask the tutor: Ask me to run any qubit state through phase damping and watch its coherence, Bloch pancake, purity, or Ramsey visibility change step by step.

Why is phase damping called the most "quantum" kind of noise?

Because it attacks the one resource that has no classical counterpart: relative phase. A classical bit has only a population — which value it holds — and phase damping leaves populations completely untouched, so a classical bit would not even notice it. What phase damping destroys is the coherence between $|0\rangle$ and $|1\rangle$ , the very thing that lets a qubit sit in a superposition and interfere with itself. Amplitude damping, by contrast, has a classical shadow (an excited state decaying to the ground state looks like ordinary energy relaxation). Phase damping erases quantumness while conserving every classical observable, which is exactly why it is the textbook model of decoherence and the reason a qubit can "look fine" in its populations while having lost all its computational power.

How exactly do λ and the phase-flip probability p relate?

Through a square root, not a simple factor: $1-2p=\sqrt{1-\lambda}$ , equivalently $p=\tfrac12\bigl(1-\sqrt{1-\lambda}\bigr)$ and $\lambda=4p(1-p)$ . The reason is that the Kraus parameter $\lambda$ enters the amplitude level of the operators (as $\sqrt{1-\lambda}$ ), while $p$ is a probability of applying $Z$ ; the coherence shrink factor is $\sqrt{1-\lambda}$ in one language and $1-2p$ in the other, and equating those two factors gives the dictionary. A quick sanity check: $\lambda=0$ gives $p=0$ (no noise), $\lambda=1$ gives $p=\tfrac12$ (complete dephasing, maximal randomization of the sign of the coherence), and $\lambda=0.36$ gives $p=0.10$ . Mixing up $p$ and $\lambda/2$ is one of the most common numerical slips in noise modeling.

If the environment "measures" the qubit, why don't the populations change?

Because it measures the right observable. The environment couples to $Z$ , whose eigenstates are exactly $|0\rangle$ and $|1\rangle$ — the populations. A measurement of an observable never disturbs the probabilities of its own eigenstates; it only destroys coherence between them. So when the environment learns "which $Z$ -eigenstate", it is reading information the populations already encode, and it need not — and does not — move any probability between the levels. The disturbance lands entirely on the off-diagonal terms $\rho_{01}$ , which encode the relative phase the environment has now recorded elsewhere. This is why the pointer basis (the basis the environment measures) is the robust one: states in that basis are eigenstates of the monitored observable and pass through undisturbed, while their superpositions decohere.

What is the difference between T2, T2*, and Tφ?

$T_\phi$ is the pure-dephasing time, the coherence decay you would measure if phase damping were the only noise present; it is the $1/\gamma_\phi$ of the Lindblad generator. $T_2$ is the total transverse coherence time, combining pure dephasing with the coherence loss that amplitude damping unavoidably causes: $1/T_2=1/2T_1+1/T_\phi$ , so $T_2\le 2T_1$ always. $T_2^\ast$ is the coherence time actually seen in a simple Ramsey experiment, which is usually shorter than $T_2$ because it also includes quasi-static, inhomogeneous frequency offsets that a spin echo can refocus. Applying a Hahn echo cancels those static offsets and recovers a longer time approaching $T_2$ ; the gap between $T_2^\ast$ and $T_2$ measures how much of the dephasing is slow and refocusable versus genuinely irreversible.

Is phase damping the same as a Z measurement?

Only in the limit $\lambda=1$ , and even then only in an averaged sense. At full strength the channel sends $\rho$ to $\rho_{00}|0\rangle\langle0|+\rho_{11}|1\rangle\langle1|$ , which is exactly the state you get by measuring $Z$ and then forgetting the outcome — a non-selective measurement. For $\lambda<1$ it is a weak or partial measurement: the environment gains only partial which-state information, so coherence is reduced by $\sqrt{1-\lambda}$ rather than annihilated. The key difference from an ordinary projective measurement is that no one reads the result; the information is dumped into inaccessible environment modes. That is why the process is described by a trace-preserving channel (average over outcomes) rather than a state update conditioned on a click.

Can phase damping be undone or fought?

Not reversed after the fact — the information has genuinely left into the environment, and the channel is irreversible as a map. But it can be suppressed before it accumulates. Because laboratory dephasing is dominated by slow, low-frequency noise, a Hahn echo or a dynamical-decoupling sequence (CPMG, Uhrig, and relatives) inserts timed $\pi$ -pulses that flip the sign of the accumulated phase, so slow environmental drifts cancel over a cycle and the effective coherence time lengthens dramatically. At the hardware level, operating at a "sweet spot" where the frequency is first-order insensitive to a noise knob reduces $\gamma_\phi$ at the source. And at the logical level, because phase damping is a phase-flip channel, a quantum error-correcting code that detects $Z$ errors — culminating in biased-noise codes tuned for dephasing — restores coherence by measuring syndromes and applying $Z$ corrections. You cannot un-leak information, but you can arrange for very little of it to leak.

Phase Damping Channel

Where this lives today

The physical picture

The core equations

Limiting cases

Worked examples

1 · Ramsey coherence decay — the pancake with numbers (fully worked)

2 · Parametrizations and the $T_2$ budget (fully worked)

Physics of the portrait

Going further

Before this

Where this leads

Phase Damping Channel

Where this lives today

The physical picture

The core equations

Limiting cases

Worked examples

1 · Ramsey coherence decay — the pancake with numbers (fully worked)

2 · Parametrizations and the T2T_2T2​ budget (fully worked)

Physics of the portrait

Going further

Before this

Where this leads

2 · Parametrizations and the $T_2$ budget (fully worked)