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A pretty-tight converse on the classical capacity of generalized amplitude-damping channels
Kun Fang
TL;DR
The generalized amplitude-damping channel’s classical capacity remains unknown, with existing upper bounds separated from the achievable one-copy Holevo information. The paper relaxes a correlation measure in a converse bound to make it computable. The resulting bound improves on known bounds across the parameter range and nearly matches the Holevo information at positive temperature.
Problem
The generalized amplitude-damping channel’s classical capacity is unknown, and evaluating the regularized capacity expression generally requires optimization over arbitrarily many channel uses.
Source: §1 · pp. 1–2Method
Results
Takeaways & Limitations
At positive temperature, the new converse provides an almost tight estimate relative to the achievable Holevo information.
Source: §4 · p. 7
Abstract
from arXiv · showhide
The classical capacity of a quantum channel is the highest rate at which classical information can be transmitted with vanishing error. Although the capacities of many fundamental quantum channels have been established, the capacity of the generalized amplitude-damping channel remains an open problem in quantum Shannon theory. In this work, we revisit a converse bound introduced by Brand\~ao et al. [Phys. Rev. Lett. 106, 230502 (2011)], which upper-bounds the classical capacity in terms of a correlation measure. For the generalized amplitude-damping channel, we derive a relaxation of this measure that makes the converse bound efficiently computable. Numerical comparisons demonstrate that our bound is pretty tight in general and nearly coincides with the Holevo information at nonzero temperatures, significantly improving upon existing bounds across the entire parameter range.
1 Introduction
The generalized amplitude-damping channel’s classical capacity remains unknown, while existing upper bounds leave a gap to the achievable one-copy Holevo information. The paper relaxes a correlation measure in a converse bound to obtain a computable bound that improves on known bounds and is nearly tight at positive temperature.
1 Introduction
- The classical capacity is the highest rate for transmitting classical information with vanishing error, but its regularized expression generally requires optimization over arbitrarily many channel uses.Even exact, computable capacity expressions for qubit channels are known only in special cases.Source: §1 · p. 1
- The classical capacity of the generalized amplitude-damping channel remains unknown, and earlier upper bounds leave a significant gap to the achievable one-copy Holevo information.The channel describes spin relaxation in a two-level system coupled to a thermal bath; at zero temperature it reduces to amplitude damping.Source: §1 · pp. 1–2
- At positive temperature, the computable converse bound nearly matches the achievable Holevo information and improves known bounds across the parameter range.The bound is obtained by relaxing an output–environment correlation measure in the converse of Brandão, Eisert, Horodecki, and Yang [BEHY11].Source: §1 · p. 2
- The converse relaxes the correlation measure while retaining input occupation dependence, using complementary penalties from environment and output measurements.Koashi–Winter duality and two-qubit concurrence formulas [Woo98] reduce the environment-measurement estimate to scalar minimization and convexification; the fixed output-measurement penalty is already convex.Source: §1 · p. 2
2 Generalized amplitude-damping channels
The channel is parameterized by damping probability and thermal equilibrium occupation, with input and output occupation probabilities related by tγ,N(q). The section also introduces entropy and concurrence quantities used in the capacity analysis and identifies the one-copy Holevo information as an achievable benchmark.
2 Generalized amplitude-damping channels
- The input occupation probability is q, while the output occupation is tγ,N(q) := γN + (1 − γ)q.The passage also specifies q ∈ [0, 1] and |z|^2 ≤ q(1 − q).Source: §2 · p. 2
- For a qubit state, von Neumann entropy is expressed as a function of its determinant, using the binary entropy h2.The eigenvalues are given as (1 ± √(1 − 4 det ρ))/2.Source: §2 · p. 2
- The one-copy Holevo information is the achievable benchmark for the channel’s classical capacity.The passage states χ(Aγ,N) ≤ C(Aγ,N).Source: §2 · p. 2
- The section uses Wootters’ formula to express the concurrence of the channel’s normalized Choi state.For positive-part notation, [x]+ means max{x, 0}.Source: §2 · p. 2
3 A converse bound for the classical capacity
The section turns a converse based on output entropy minus output–environment correlation into a computable GADC capacity bound. It combines two measurement-based correlation penalties and reduces the optimization to the mean input occupation.
3 A converse bound for the classical capacity
- The environment-measurement penalty uses Koashi–Winter duality and two-qubit concurrence formulas, then reduces to scalar minimization and convexification.Source: §3 · pp. 3–5
- The converse separates output entropy from output–environment correlation, and bounds the correlation term by single-copy G measures without assuming product inputs.Source: §3 · p. 3
- The many-use converse reduces to optimizing the output-entropy bound minus the larger of two convex correlation penalties over mean input occupation.The resulting single-variable optimization gives U_G(γ, N).Source: §3 · pp. 3, 7
- Concavity of binary entropy and the affine output occupation reduce the output-entropy estimate to the mean input occupation.For marginal occupations q_i, the output occupations are t_γ,N(q_i).Source: §3 · p. 4
- The output-measurement penalty uses a fixed Pauli-Y measurement; symmetry and convexity show diagonal inputs minimize its correlation at fixed occupation.Source: §3 · pp. 5–6
- The correlation penalties are exact on diagonal inputs, making them the largest lower bounds on the respective G measures that depend only on occupation.Source: §3 · p. 7
4 Numerical comparison
The bound UG nearly matches the Holevo rate χ for N > 0 and improves on the compared bounds. The comparison also identifies parameter regions where capacity equals χ.
4 Numerical comparison
- The benchmarks include the KSW envelope of previously evaluated GADC bounds and the Fang–Fawzi bound UFF.The implementation uses a trace-preserving entanglement-breaking comparison channel for C^TP_EB, follows the population convention for CFil [Fil18], and evaluates UFF with geometric Rényi divergence of order 1 + 2^-10.Source: §4 · p. 7
- For N > 0, UG provides an almost tight estimate; Figure 1 compares it with χ, UKSW, and UFF.The capacity is symmetric under N ↔ 1 − N, so numerical comparisons restrict to N ≤ 1/2.Source: §4 · p. 7, Figure 1 · p. 8
- Capacity equals χ at γ = 0, γ = 1, on N = 1/2, and throughout the entanglement-breaking region.UG also recovers equality throughout the entanglement-breaking region.Source: §4 · p. 7
- Figure 2 maps the residual gap UG − χ in bits per channel use, with contours from 10^-1 to 10^-6 bits and the entanglement-breaking boundary marked.The orange curve marks the boundary defined in (44).Source: §4 · p. 7, Figure 2 · p. 8
Appendix A Optimality of the correlation penalties
The appendix establishes that the two correlation penalties are optimal among lower bounds depending only on input occupation. It also identifies measurements that attain the output-measurement penalty.
Appendix A Optimality of the correlation penalties
- The two penalties are optimal among lower bounds that depend only on input occupation q, for every q and M ∈ {E, B}.For the environment penalty, attainment uses opposite coherences and convexification over input occupation; the output penalty follows from the optimal Pauli-Y measurement.Source: Appendix A · p. 10
- The environment-penalty proof pairs opposite coherences in an ensemble and convexifies over q; mixed ensemble components are allowed.The argument lets the approximation parameter ε tend to zero to establish the claim for GE.Source: Appendix A · pp. 10–11
- For real coherence z, measuring the output in the Pauli-Y basis is optimal; for diagonal inputs, every equatorial projective measurement is optimal.Phase covariance supplies the corresponding rotated measurement for complex z.Source: Appendix A · pp. 10–11