Damping-Dephasing Channel in Quantum Noise

• The damping-dephasing channel is a quantum noise model that simultaneously incorporates amplitude damping and dephasing, representing realistic energy and phase relaxation.
• Its Kraus operator structure and relation to T1/T2 times provide analytical tractability for computing coherent and reverse-coherent information rates.
• The model exhibits two-letter non-additivity and supports practical one-way distillation protocols, offering a testbed for exploring capacity bounds in quantum communication.

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 (Siddhu et al., 2024).

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

$p\in[0,\tfrac12]$0

This representation is equivalent to first dephasing with probability $p\in[0,\tfrac12]$1 and then amplitude damping with probability $p\in[0,\tfrac12]$2, or vice versa, since the two commute (Siddhu et al., 2024).

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\in[0,\tfrac12]$3 reduces to pure damping, while $p\in[0,\tfrac12]$4 removes the damping component.

2. Relation to physical relaxation times

In many architectures, noise is modeled by the time evolution

$p\in[0,\tfrac12]$5

For the damping-dephasing channel, the parameters are identified as

$p\in[0,\tfrac12]$6

This gives the constraint $p\in[0,\tfrac12]$7, and in practice $p\in[0,\tfrac12]$8 yields $p\in[0,\tfrac12]$9 and $\gamma\in[0,1]$0 (Siddhu et al., 2024).

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 $\gamma\in[0,1]$1 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 $\gamma\in[0,1]$2 noise rather than a purely idealized construction.

3. Coherent and reverse-coherent information

Let $\gamma\in[0,1]$3 denote the complementary channel to $\gamma\in[0,1]$4. For any input density operator $\gamma\in[0,1]$5,

$\gamma\in[0,1]$6

$\gamma\in[0,1]$7

Both optimizations can be restricted by symmetry to Bloch states

$\gamma\in[0,1]$8

Writing the output Bloch vector as

$\gamma\in[0,1]$9

one obtains

$T_1$0

and

$T_1$1

where $T_1$2. Numerically, the maximum of $T_1$3 lies on the $T_1$4-axis. Closed-form asymptotics for small $T_1$5 show

$T_1$6

(Siddhu et al., 2024).

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 $T_1$7. 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 $T_1$8 and sends them through two copies of $T_1$9. Bob performs $T_2$0 and measures the target in the $T_2$1 basis. Outcome $T_2$2 is kept and outcome $T_2$3 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 (Siddhu et al., 2024).

The probability to pass Stage 1 is

$T_2$4

and the post-selected two-qubit state is

$T_2$5

Hashing on $T_2$6 yields the entanglement yield

$T_2$7

In the asymptotic setting, the channel satisfies the lower bounds

$T_2$8

for forward-only communication, and

$T_2$9

for backward-only communication. Comparison with pure damping shows that pure-damping $\Lambda_{p,\gamma}$0, but for $\Lambda_{p,\gamma}$1 the yield $\Lambda_{p,\gamma}$2 can exceed $\Lambda_{p,\gamma}$3 in realistic-noise regimes with $\Lambda_{p,\gamma}$4. 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

$\Lambda_{p,\gamma}$5

To test this, a two-letter ansatz is used,

$\Lambda_{p,\gamma}$6

and $\Lambda_{p,\gamma}$7 is numerically maximized. The result is a swath of $\Lambda_{p,\gamma}$8 values for which the channel is non-additive at the two-letter level, with gaps

$\Lambda_{p,\gamma}$9

(Siddhu et al., 2024).

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,

$A$0

with independent dephasing rates $A$1, correlated dephasing rate $A$2, and lattice-relaxation rates $A$3. 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 (Singh et al., 2020).

The model also yields differential decay laws for multiple-quantum coherences. A single-quantum element on spin $A$4 decays as

$A$5

a zero-quantum element decays as

$A$6

and a double-quantum element decays as

$A$7

Because of the sign flip in front of $A$8, zero-quantum coherences decay more slowly than double-quantum coherences when $A$9. This shows that “damping-dephasing channel” can denote a family of models rather than a single normal form: the single-qubit $D$0 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

$D$1

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 $D$2. 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 $D$3 but becomes entanglement-breaking in the $D$4 limit (Omkar et al., 2012).

A distinct methodological connection comes from dynamic quantum tomography for phase-damping channels. There, the channel is written in Hadamard form,

$D$5

with $D$6, $D$7, and $D$8, 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 (Czerwinski et al., 2015). 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.