Squeezed Generalized Amplitude Damping Channel

• The SGAD channel is a quantum noise model that extends generalized amplitude damping by incorporating squeezed thermal environments to capture both energy relaxation and dephasing.
• It employs a detailed Lindblad master equation and Kraus operator formalism to encode dissipative dynamics, with parameters that depend on temperature, squeezing, and bath memory effects.
• The model provides practical insights into phenomena like decoherence, quantum speed limits, and error mitigation, making it key for advanced quantum information processing and metrology.

The squeezed generalized amplitude damping (SGAD) channel is a quantum noise model that extends the well-studied generalized amplitude damping (GAD) channel by incorporating the effects of a squeezed thermal environment. It characterizes dissipative dynamics of open quantum systems in contact with nonzero-temperature and nonzero-squeezing bosonic baths, capturing both energy relaxation and dephasing processes with explicit dependency on squeezing parameters and memory correlations. This model provides a physically motivated framework for evaluating decoherence, information loss, and error mechanisms in quantum information processing, quantum walks, quantum metrology, and the paper of multipartite entanglement in both finite and infinite-dimensional Hilbert spaces.

1. Mathematical Structure and Dynamics of the SGAD Channel

The SGAD channel is described, in the interaction picture, by the Lindblad-type master equation:

$$\frac{d\rho^s}{dt} = \gamma_0 (N+1) [\sigma_- \rho^s \sigma_+ - \tfrac{1}{2}\{\sigma_+\sigma_-, \rho^s\}] + \gamma_0 N [\sigma_+ \rho^s \sigma_- - \tfrac{1}{2}\{\sigma_-\sigma_+, \rho^s\}] - \gamma_0 M \sigma_+ \rho^s \sigma_+ - \gamma_0 M^* \sigma_- \rho^s \sigma_-$$

where $\gamma_0$ is the spontaneous emission rate, $\sigma_+$ and $\sigma_-$ are raising and lowering operators, and parameters $N$ and $M$ encode both thermal ($N_\text{th}$) and squeezing ($r$, $\Phi$) effects: \begin{align*} N &= N_\mathrm{th} (\cosh^2 r + \sinh^2 r) + \sinh^2 r,\ M &= -\tfrac{1}{2} \sinh 2r\, e^{i\Phi} (2N_\mathrm{th} + 1),\ N_\mathrm{th} &= \frac{1}{e^{\hbar\omega_0 / k_BT} - 1}. \end{align*} For fixed time $t = \Delta$, the noise map is given by the Kraus operator set $\{E_0, E_1, E_2, E_3\}$, with entries parameterized by $\alpha(\Delta)$, $\mu(\Delta)$, $\nu(\Delta)$ and weights $p_1$, $p_2$ (with $p_1 + p_2 = 1$). These parameters are functions of the bath characteristics (e.g., $N$, $M$, $\gamma_0$, $\Delta$), capturing decay and squeezing-induced noise.

The full Kraus operator structure is as follows (suppressing identity elements): $$\begin{aligned} E_0 &= \sqrt{p_1} \begin{bmatrix} \sqrt{1-\alpha(\Delta)} & 0 \ 0 & 1 \end{bmatrix}, \quad E_1 = \sqrt{p_1} \begin{bmatrix} 0 & 0 \ \sqrt{\alpha(\Delta)} & 0 \end{bmatrix},\ E_2 &= \sqrt{p_2} \begin{bmatrix} \sqrt{1-\mu(\Delta)} & 0 \ 0 & \sqrt{1-\nu(\Delta)} \end{bmatrix}, \quad E_3 = \sqrt{p_2} \begin{bmatrix} 0 & \sqrt{\nu(\Delta)} \ \sqrt{\mu(\Delta)} e^{-i\theta(\Delta)} & 0 \end{bmatrix} \end{aligned}$$ where the time-dependent parameters encode both dissipative and nonclassical bath properties.

In the context of field modes, the SGAD (and its bosonic generalization) can also be formulated using phase-space quasi-distributions, expressing the evolution as a convolution of the initial Wigner function with a normalized, time-dependent Gaussian kernel determined by the bath's temperature and squeezing parameters. This convolution represents both dissipation and phase-sensitive diffusion, signifying a dynamic reordering of the Gaussian quasi-distribution (Bazrafkan et al., 2014).

2. Noise-Induced Decoherence, Classicalization, and Quantum Correlations

When applied to qubit systems, the SGAD channel drives decoherence in both the population and coherence sectors of the density matrix, resulting in loss of quantumness under increasing environmental noise:

• In discrete-time quantum walks (DTQWs), when the coin is subjected to SGAD noise at each step, quantum correlations between coin and position degrees of freedom, as measured by the quantum mutual information and measurement-induced disturbance (QMID), decay monotonically with increased squeezing parameter $r$ or temperature $T$ (Srikanth et al., 2010). This classicalization is witnessed by the transition to Gaussian-like position distributions for walks on a line, or diminished interference effects on an $n$-cycle.

For a generic bipartite state $\rho$, the quantumness of correlations is quantified as

$Q(\rho) = I(\rho) - I[\Pi(\rho)],$

where $I(\rho)$ is the mutual information and $\Pi(\rho)$ denotes the state post-local projective measurement. $Q(\rho)$ decays as SGAD parameters $r, T$ grow, indicating the systematic loss of quantum resource strength (Srikanth et al., 2010). In cyclic quantum walks, enhanced self-interference due to topological periodicity leads to more rapid classicalization compared to linear geometries.

The SGAD channel also provides mechanisms for noise-activated generation of quantum correlations: for example, local action of a (generalized or squeezed) amplitude damping channel can create nonzero quantum discord from initially classical (discord-free) two-qubit states, provided the environmental noise parameters are appropriately asymmetric (Xu, 2011). High environmental temperature typically suppresses this activation, whereas squeezing offers further tunability.

3. Multipartite Entanglement, Memory Effects, and Long-Time Behavior

When multiple qubits traverse an SGAD channel—especially in the presence of correlated (non-Markovian) noise—the interplay between dissipation, squeezing, and memory results in rich dynamics for multipartite entanglement:

• For three-qubit Greenberger-Horne-Zeilinger (GHZ) or W states, each mixed with white noise, memory parameter $\mu$ quantifies correlation between successive channel uses, interpolating between fully uncorrelated ($\mu=0$) and maximally correlated ($\mu=1$) regimes (Ali, 29 Jan 2024).
• The evolution map for $n$-qubits with memory assumes the form $$\rho(t) = \mu K_\text{correlated}(\rho) + (1-\mu) K_\text{uncorr}(\rho)$$.

A key finding is that in the asymptotic ($t \to \infty$) limit, genuine multipartite entanglement in these systems can only persist for a narrow regime of high memory ($\mu \gtrsim 0.97$, depending on initial state and thermal parameter $n$). For small $n$, even initially biseparable states can become genuinely entangled via the channel, but for sufficiently large $n$ (thermal limit), all asymptotic states become biseparable. Importantly, in the $t\to\infty$ limit, residual squeezing ($m$) has no effect on the asymptotic entanglement, showing that squeezing-induced coherence is washed out by environmental equilibrium (Ali, 29 Jan 2024).

4. Quantum Speed Limit, Channel Memory, and Evolution Rate

The presence of memory (correlation) in the SGAD channel reduces the quantum speed limit (QSL) time with increasing correlation strength $\mu$, as given by

$$\tau_\text{QSL} = \max \left\{ \frac{1}{\langle \sum_n \Lambda_n \beta_n \rangle}, \frac{1}{\langle \sqrt{\sum_n \Lambda_n^2} \rangle} \right\} \times |f(\tau+\tau_D) - 1| \,\text{tr}(\rho_\tau^2)$$

where $\Lambda_n$ and $\beta_n$ are singular values of the generator and state, and $f(\cdot)$ is the relative purity (Awasthi et al., 2019). For SGAD channels, increasing $\mu$ (correlation strength) leads to a monotonic decrease in $\tau_\text{QSL}$ (faster evolution), a result that contrasts with certain unital channels (such as colored dephasing), where memory may slow evolution. This property has significance in optimizing quantum control protocols and in the trade-off between speed and decoherence.

5. Information-Theoretic Properties, Channel Capacities, and Error Correction

The SGAD channel's impact on information transmission is intimately linked with its entanglement-breaking and anti-degradable regimes. For the closely related GAD channel, the channel is entanglement-breaking for sufficiently large damping and thermal noise, and anti-degradable for $\gamma \geq 1/2$ (for all $N$) (Khatri et al., 2019). The addition of squeezing (in SGAD) modifies these boundaries, with practical implications for quantum communication—parameter regimes with high squeezing or high thermal excitation reduce quantum/private capacity.

Upper bounds on channel capacities, such as those derived via SDP programs for the max-Rains information,

$$E_\text{max}(\mathcal{A}_{\gamma,N}) = \log_2 \left[1-\gamma/2 + \frac{1}{2}\sqrt{[\gamma(2N-1)]^2 + 4(1-\gamma)}\right],$$

must generally be recalibrated in the presence of squeezing (Khatri et al., 2019).

For error correction, generic results for the GAD channel indicate superior performance for self-complementary nonadditive codes and certain degenerate stabilizer codes, which achieve lower leading-order infidelity compared to standard stabilizer codes. The analytic structure of these codes and the recovery maps are independent of specific details of the damping, suggesting direct applicability to SGAD channels after explicit update of code parameters for squeezing-induced noise (Cafaro et al., 2013).

6. Quantum Metrology, Fisher Information, and Squeezing as Error Mitigation

The action of the SGAD channel on entangled states used in quantum metrology, such as three-qubit entangled Dirac states in curved spacetime, demonstrates the role of squeezing as an error-mitigating resource (Iyen et al., 23 Jul 2025). For strong squeezing ($r=1$), the quantum Fisher information (QFI) of the entanglement weight parameter $\theta$ is rendered insensitive to Hawking temperature ($T_H$, gravitational decoherence), although it continues to decay under increased channel temperature ($T_C$). The QFI decay is substantially slower for $r=1$ compared to $r=0$. A transient spike in the QFI for the phase parameter $\phi$ at $T_C=2$ (possibly a thermal resonance or non-monotonic decoherence) is observed, but is also largely independent of $T_H$. These findings indicate that engineering the SGAD channel's squeezing can offer practical routes to robustness in quantum parameter estimation, even under strong environmental and relativistic noise.

7. Related Physical Models and Extensions

In continuous-variable (bosonic) settings, the SGAD channel generalizes to a convolution operation in phase space, with the Wigner function evolution

$$W(\alpha, t) = e^{2Kt} [ g(\alpha, t) * W(\alpha,0) ],$$

where $g(\alpha,t)$ is a normalized Gaussian whose width, orientation, and anisotropy are set by the bath's damping, thermal, and squeezing parameters (Bazrafkan et al., 2014). This formalism makes explicit the SGAD channel's encoding of both dissipative and phase-sensitive (nonclassical) fluctuations, offering conceptual clarity for the inclusion of squeezing in dissipative quantum channels.

Summary Table: SGAD Channel Parameterization and Effects

Effect/Quantity	SGAD Parameter Dependence	Key Consequence
Decoherence rate	$\gamma_0, N, r, \Phi$	Coherence loss; depends on squeezing
Memory/memorylessness	$\mu$	Correlates noise over qubits
Entanglement persistence	$n, m, \mu, t$	High $\mu$ and low $n$ preserve $E$
Capacity (quantum/private)	$\gamma, N, r$	Decreases with noise and squeezing
QSL (evolution time)	$\mu$	Decreases for high $\mu$
Quantum Fisher Information	$r, T_H, T_C$	High $r$ mitigates $T_H$-induced loss

In summary, the squeezed generalized amplitude damping channel constitutes a paradigmatic model for quantifying and controlling decoherence, quantum-to-classical transition, and the resilience of quantum resources in realistic noisy environments, with explicit handles from both temperature and squeezing parameters as well as inter-use channel memory. Its comprehensive mathematical structure, operational relevance, and technical adaptability render it fundamental in contemporary quantum information theory and experimental device modeling.