---
title: Damping-Dephasing Channel in Quantum Noise
url: https://www.emergentmind.com/topics/damping-dephasing-channel
type: topic
---

# Damping-Dephasing Channel in Quantum Noise

Searching arXiv for recent and relevant papers on damping-dephasing channels and closely related noise models.
A damping-dephasing channel is a quantum noise model that combines amplitude damping and dephasing within a single completely positive trace-preserving map. In the single-qubit formulation introduced for entanglement sharing, the channel $\Lambda_{p,\gamma}$ is parameterized by a dephasing probability $p\in[0,\tfrac12]$ and an amplitude-damping probability $\gamma\in[0,1]$, and it captures realistic noise in settings where both $T_1$-type energy relaxation and $T_2$-type phase relaxation are present rather than idealized Pauli noise alone. In this form, the channel is used to study both practical one-way distillation and asymptotic entanglement-sharing rates, including coherent-information and reverse-coherent-information lower bounds, as well as two-letter non-additivity phenomena [2405.06231].

## 1. Definition and Kraus structure

The joint damping-dephasing channel $\Lambda_{p,\gamma}$ acts on an input qubit system $A$ and produces an output $D$ through three Kraus operators,
\[
K_0=\sqrt{1-p}\,\bigl(|0\rangle\langle 0|+\sqrt{1-\gamma}\,|1\rangle\langle 1|\bigr),
\qquad
K_1=\sqrt{\gamma}\,|0\rangle\langle 1|,
\]
\[
K_2=\sqrt{p}\,\bigl(|0\rangle\langle 0|-\sqrt{1-\gamma}\,|1\rangle\langle 1|\bigr),
\]
so that
\[
\Lambda_{p,\gamma}(\rho)=\sum_{i=0}^2 K_i\,\rho\,K_i^\dagger.
\]
This representation is equivalent to first dephasing with probability $p$ and then amplitude damping with probability $\gamma$, or vice versa, since the two commute [2405.06231].

This commuting structure is significant because it separates a non-Pauli component, amplitude damping, from a dephasing component without imposing an ordering ambiguity. A common simplification is to analyze pure dephasing or pure damping in isolation; the joint model instead formalizes the case in which both mechanisms act simultaneously and are operationally relevant for communication and modular quantum-computing platforms. In the notation of the channel, $p=0$ reduces to pure damping, while $\gamma=0$ removes the damping component.

## 2. Relation to physical relaxation times

In many architectures, noise is modeled by the time evolution
\[
\rho(0)\mapsto
\begin{pmatrix}
1-\rho_{11}e^{-t/T_1} & \rho_{01}e^{-t/T_2}\\
\rho_{01}^*e^{-t/T_2} & \rho_{11}e^{-t/T_1}
\end{pmatrix}.
\]
For the damping-dephasing channel, the parameters are identified as
\[
\gamma = 1-e^{-t/T_1},
\qquad
p=\tfrac12\bigl[1-e^{-t\,(1/(2T_1)-1/T_2)}\bigr].
\]
This gives the constraint $2T_1\ge T_2$, and in practice $T_1\neq 2T_2$ yields $p>0$ and $\gamma>0$ [2405.06231].

This parametrization makes explicit that the channel is not merely a formal interpolation between two textbook noise models. It is designed to encode the experimentally relevant regime in which realistic decoherence times induce simultaneous phase and energy relaxation. The condition $T_1\neq 2T_2$ is especially important because it excludes the special case in which the dephasing parameter vanishes. In that sense, the damping-dephasing channel is a direct model of realistic $T_1,T_2$ noise rather than a purely idealized construction.

## 3. Coherent and reverse-coherent information

Let $\Lambda^c_{p,\gamma}$ denote the complementary channel to $\Lambda_{p,\gamma}$. For any input density operator $\rho$,
\[
I_c\bigl(\Lambda_{p,\gamma},\rho\bigr)
= S\bigl(\Lambda_{p,\gamma}(\rho)\bigr)-S\bigl(\Lambda^c_{p,\gamma}(\rho)\bigr),
\]
\[
I_r\bigl(\Lambda_{p,\gamma},\rho\bigr)
= S(\rho)-S\bigl(\Lambda^c_{p,\gamma}(\rho)\bigr).
\]
Both optimizations can be restricted by symmetry to Bloch states
\[
\rho_z=\tfrac12(I+zZ),\qquad z\in[-1,1].
\]
Writing the output Bloch vector as
\[
r_d=\bigl((1-2p)\sqrt{1-\gamma}\,x,\,(1-2p)\sqrt{1-\gamma}\,y,\,(1-\gamma)z+\gamma\bigr),
\]
one obtains
\[
I_c(\Lambda_{p,\gamma})
=\max_{z\in[-1,1]}
\Bigl[
h\!\bigl(\tfrac{1+|(1-\gamma)z+\gamma|}{2}\bigr)
-
h\!\bigl(\tfrac{1+|\gamma(z-1)|}{2}\bigr)
\Bigr],
\]
and
\[
I_r(\Lambda_{p,\gamma})
=\max_{z\in[-1,1]}
\Bigl[
h\!\bigl(\tfrac{1+z}{2}\bigr)
-
h\!\bigl(\tfrac{1+|\gamma(z-1)|}{2}\bigr)
\Bigr],
\]
where $h(x)=-x\log_2 x-(1-x)\log_2(1-x)$. Numerically, the maximum of $I_c$ lies on the $z$-axis. Closed-form asymptotics for small $1-\gamma$ show
\[
I_r\sim (1-\gamma)(1-q)\,(q(1-\gamma))^{q/(1-q)},
\qquad q=4p(1-p)
\]
[2405.06231].

These expressions place the damping-dephasing channel in the standard information-theoretic framework for quantum communication. They also show that the relevant single-letter optimizations are unusually tractable: the optimization reduces to a concave optimization over $z\in[-1,1]$. At the same time, later results on non-additivity show that this single-letter tractability does not imply that one-letter formulas are capacity formulas in general.

## 4. One-way distillation and asymptotic lower bounds

A practical one-way distillation protocol with backward classical communication is organized in two stages. In Stage 1, Alice encodes half a Bell pair into two qubits $(a_1,a_2)$ and sends them through two copies of $\Lambda_{p,\gamma}$. Bob performs $\mathrm{CNOT}_{d_1\to d_2}$ and measures the target in the $Z$ basis. Outcome $+1$ is kept and outcome $-1$ is discarded. This post-selection removes all amplitude-damping events, while dephasing survives as flipped-phase pairs. In Stage 2, Bob performs a standard hashing protocol on the surviving pairs, using one-way classical feedback [2405.06231].

The probability to pass Stage 1 is
\[
p_s=1-\gamma,
\]
and the post-selected two-qubit state is
\[
\rho_{\mathrm{post}}
=(1-q/2)\,\phi+\tfrac q2\,Z\phi Z,
\qquad
q:=4p(1-p),
\qquad
\phi=|\Phi^+\rangle\langle\Phi^+|.
\]
Hashing on $\rho_{\mathrm{post}}$ yields the entanglement yield
\[
Y=\tfrac12\,p_s\,[1-h(q/2)]
=\tfrac{1-\gamma}{2}\,[1-h(2p(1-p))].
\]

In the asymptotic setting, the channel satisfies the lower bounds
\[
Q(\Lambda_{p,\gamma})\ge I_c(\Lambda_{p,\gamma})
\]
for forward-only communication, and
\[
L_{\leftarrow}(\Lambda_{p,\gamma})\ge \max\{I_r(\Lambda_{p,\gamma}),Y\}
\]
for backward-only communication. Comparison with pure damping shows that pure-damping $I_r>I_c$, but for $p>0$ the yield $Y$ can exceed $I_r$ in realistic-noise regimes with $T_1\neq 2T_2$. This directly distinguishes the joint channel from the pure amplitude-damping case: a backward-only protocol that isolates damping errors can outperform the reverse-coherent-information strategy that is best known for pure damping.

## 5. Two-letter non-additivity and the limits of single-letter formulas

The forward-communication setting motivates the comparison
\[
\tfrac12\,I_c\bigl(\Lambda_{p,\gamma}^{\otimes 2}\bigr)\stackrel{?}{>}I_c(\Lambda_{p,\gamma}).
\]
To test this, a two-letter ansatz is used,
\[
|\Psi\rangle=\sum_{i=1}^4 \alpha_i |\psi_i\rangle,
\qquad
\{\psi_i\}\in\{|00\rangle,|\Phi^+\rangle,Z_1|\Phi^+\rangle,|11\rangle\},
\]
and $I_c(\Lambda^{\otimes 2},|\Psi\rangle\langle\Psi|)$ is numerically maximized. The result is a swath of $(p,\gamma)$ values for which the channel is non-additive at the two-letter level, with gaps
\[
\delta:=\tfrac12\,I_c(\Lambda^{\otimes 2})-I_c(\Lambda)=O(10^{-3})
\]
[2405.06231].

This establishes that the damping-dephasing channel can exhibit non-additivity with magnitudes comparable to those found in more idealized noise channels. It also corrects a common misconception that physically realistic noise necessarily behaves more regularly than idealized models. Here, the one-letter coherent information is computable, but it is not in general additive. A plausible implication is that joint damping-dephasing noise may provide a useful intermediate testbed between analytically simple Pauli channels and more structured open-system models when studying super-additivity and capacity lower bounds.

## 6. Correlated and multi-qubit damping-dephasing variants

In two coupled nuclear spins, a broader damping-dephasing model has been studied in which the non-unitary evolution is the sum of a correlated phase-damping channel acting jointly on both spins and two independent generalized amplitude-damping channels, one on each spin. The dynamics is written as a Lindblad master equation,
\[
\frac{d\rho}{dt}
=
\bigl[
\gamma_1 D[\sigma_z\otimes\mathbbm1]
+
\gamma_2 D[\mathbbm1\otimes\sigma_z]
+
\gamma_3 D[\sigma_z\otimes\sigma_z]
\bigr]
+
\sum_{i=1}^2 \frac{\Gamma_i}{2}
\Bigl(D[\sigma_+^{(i)}]+D[\sigma_-^{(i)}]\Bigr),
\]
with independent dephasing rates $\gamma_1,\gamma_2$, correlated dephasing rate $\gamma_3$, and lattice-relaxation rates $\Gamma_i=1/T_1^{(i)}$. In this model, experimental data fit well to a correlated phase damping channel acting simultaneously on both spins together with a generalized amplitude damping channel acting independently on both spins [2007.12972].

The model also yields differential decay laws for multiple-quantum coherences. A single-quantum element on spin $i$ decays as
\[
\exp\bigl[-t(\gamma_i+\tfrac12\Gamma_i)\bigr],
\]
a zero-quantum element decays as
\[
\exp\bigl[-t(\gamma_1+\gamma_2-\gamma_3+\tfrac{\Gamma_1+\Gamma_2}{2})\bigr],
\]
and a double-quantum element decays as
\[
\exp\bigl[-t(\gamma_1+\gamma_2+\gamma_3+\tfrac{\Gamma_1+\Gamma_2}{2})\bigr].
\]
Because of the sign flip in front of $\gamma_3$, zero-quantum coherences decay more slowly than double-quantum coherences when $\gamma_3>0$. This shows that “damping-dephasing channel” can denote a family of models rather than a single normal form: the single-qubit $\Lambda_{p,\gamma}$ is one instance, while correlated multi-spin channels incorporate joint dephasing generators and independent generalized amplitude damping in a Lindbladian setting.

## 7. Alternative representations and methodological connections

For two-qubit amplitude-damping dynamics, the operator sum-difference formalism provides an extended Kraus representation by decomposing the Choi matrix into simpler Hermitian blocks. The resulting extended Kraus operators come in “positive” and “negative” families and satisfy
\[
\sum_p (A_p^+)^\dagger A_p^+ - \sum_n (A_n^-)^\dagger A_n^- = I.
\]
In the two-qubit amplitude-damping channel, the diagonal block yields nine positive Kraus operators that produce a “maximally dephasing” evolution of populations, while the ten off-diagonal rank-2 blocks yield phase-damping effects through pairs $K_x^+,K_x^-$. The decomposition separates a purely amplitude-damping component from dephasing components, and the special channels derived from this separation have distinct entanglement-breaking and broadcasting properties: the maximally dephasing component is entanglement-breaking, while the purely dephasing component is not entanglement-breaking for finite $t$ but becomes entanglement-breaking in the $t\to\infty$ limit [1212.2780].

A distinct methodological connection comes from dynamic quantum tomography for phase-damping channels. There, the channel is written in Hadamard form,
\[
\Phi_t(\rho)=C(t)\circ \rho,
\]
with $C(t)\succeq 0$, $C_{ii}(t)=1$, and $C_{ij}(0)=1$, and the same observables are measured at different times so that the known time dependence of the channel provides additional linear constraints for reconstructing the initial state [1509.09318]. This tomography result is established for phase-damping channels rather than for the full joint damping-dephasing channel. A plausible implication is that whenever the dephasing sector of a damping-dephasing model admits a fixed-basis description with analytically controlled coefficients, analogous time-resolved measurement strategies may be useful, although that extension is not established by the cited tomography analysis itself.

Source: https://www.emergentmind.com/topics/damping-dephasing-channel