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Strong converse for the classical capacity of entanglement-breaking and Hadamard channels via a sandwiched Renyi relative entropy
Mark M. Wilde, Andreas Winter, Dong Yang
TL;DR
The paper asks whether the classical capacity has a strong converse for broad classes of quantum channels, rather than only limited settings. It uses a sandwiched Renyi relative entropy to bound success probability and proves the resulting quantity is subadditive, establishing the theorem for all entanglement-breaking and Hadamard channels.
Problem
When the Holevo formula is not additive, the classical capacity requires regularization, motivating sharper strong-converse results for channels with additive HSW formulas.
Method
The proof bounds success probability using a sandwiched Renyi relative entropy, relates its Holevo-like quantity to an alpha-information radius, and proves subadditivity for the relevant channels.
Results
A strong converse holds for the classical capacity of all entanglement-breaking channels and all Hadamard channels, with success probability decaying exponentially when R > χ(NEB).
Takeaways & Limitations
The result sharpens the interpretation of classical capacity for entanglement-breaking and Hadamard channels and establishes monotonicity of the sandwiched Renyi relative entropy.
Takeaways & Limitations
A possible route using tools from Refs. [47] [46] remains an open question.
Abstract
from arXiv · showhide
A strong converse theorem for the classical capacity of a quantum channel states that the probability of correctly decoding a classical message converges exponentially fast to zero in the limit of many channel uses if the rate of communication exceeds the classical capacity of the channel. Along with a corresponding achievability statement for rates below the capacity, such a strong converse theorem enhances our understanding of the capacity as a very sharp dividing line between achievable and unachievable rates of communication. Here, we show that such a strong converse theorem holds for the classical capacity of all entanglement-breaking channels and all Hadamard channels (the complementary channels of the former). These results follow by bounding the success probability in terms of a "sandwiched" Renyi relative entropy, by showing that this quantity is subadditive for all entanglement-breaking and Hadamard channels, and by relating this quantity to the Holevo capacity. Prior results regarding strong converse theorems for particular covariant channels emerge as a special case of our results.
1 Introduction
The introduction frames classical capacity as the maximum reliable communication rate and motivates strong converses as a sharper boundary between achievable and unachievable rates. It highlights the challenge of characterizing capacity when the HSW formula is not additive and reviews prior quantum strong-converse results.
- Classical capacity is the maximum communication rate at which error probability decreases to zero over many independent channel uses.
- The HSW theorem characterizes a quantum channel’s classical capacity through a Holevo-information formula.
- When Holevo information is not additive, the best general capacity characterization is a regularized formula, and entangled codewords can increase capacity for some channels.
- The achievability theorem gives exponentially vanishing error below capacity, whereas the ordinary converse only bounds error away from zero above capacity.
- A strong converse removes the rate–error trade-off by making error converge to one whenever communication exceeds capacity.
- For quantum channels, prior strong converses covered classical-input quantum-output channels and particular covariant channels using Renyi-based or combinatorial approaches.
2 Summary of results
The paper proves strong converses for all entanglement-breaking channels and their complementary Hadamard channels. Its strategy combines generalized-divergence success bounds, a sandwiched Renyi quantity, subadditivity, and continuity arguments.
- The main theorem establishes a strong converse for the classical capacity of all entanglement-breaking channels and their complementary Hadamard channels.
- An entanglement-breaking channel measures its input and prepares an output state conditioned on the measurement result, destroying entanglement with any retained system.
- Hadamard channels are complementary channels of entanglement-breaking channels and include generalized dephasing, cloning, and Unruh channels.
- The conclusion identifies possible applications in noisy bounded storage cryptography and independent interest for the new information quantity.
- The proof first bounds success probability using a data-processing generalized divergence, then introduces a sandwiched Renyi relative entropy and its associated Holevo-like quantity.
- The sandwiched Renyi divergence is monotone for α ∈(1, 2], approaches von Neumann relative entropy as α →1, and yields exponentially vanishing success probability when R > χ(NEB).
- For entanglement-breaking channels, the Holevo-like quantity is related to an alpha-information radius and shown subadditive using multiplicativity of maximum output alpha-norms.
3 Preliminaries
The preliminaries define the operator, map, channel, complementary-channel, Hadamard, and entanglement-breaking frameworks used in the proof. A key structural property is that positive conjugation preserves entanglement-breaking maps.
- The paper works with bounded operators and finite-dimensional Hilbert spaces, defining positive operators, density operators, tensor products, partial traces, and positive maps.
- An entanglement-breaking map admits a positive operator representation and equivalent characterizations through rank-one Kraus operators or separable outputs.
- Positive conjugation preserves the entanglement-breaking property, which is central to transferring maximum-output-norm arguments into the proof.
- For trace-preserving entanglement-breaking maps, the operator representation has a POVM and output density operators, while applying the map to part of an entangled state yields a separable state.
- For a channel, a Stinespring isometry produces both the original channel and its complementary channel by tracing out the environment or output system.
- A channel is Hadamard precisely when it is complementary to an entanglement-breaking channel.
4 Bounding the success probability with a generalized divergence
This section develops a generalized-divergence converse bound for classical communication codes. Data processing converts distinguishability through decoding and equality testing into a channel-level Holevo-like quantity that can bound success probability.
- A generalized divergence measures distinguishability and is useful here when it is monotone under quantum operations.
- Monotonicity implies invariance under appending an independent state and under unitary transformations.
- The divergence induces a generalized Holevo information by optimizing distinguishability between an ensemble’s classical–quantum state and states σB.
- For an (n, R, ε) code, the sender transmits uniformly selected quantum codewords through n channel uses and the receiver decodes with a POVM.
- Data processing under decoding and an equality test bounds the code’s success probability in terms of the divergence from a product reference state.
- Maximizing over input ensembles removes code dependence and produces the channel’s generalized Holevo quantity as the converse bound.
- The paper argues that the chosen sandwiched divergence gives tighter success-probability bounds than earlier approaches based on other divergences.
5 The sandwiched quantum R´enyi relative entropy
The paper introduces the sandwiched quantum Rényi relative entropy and establishes its main information-theoretic properties for α ∈ (1, 2]. It uses these properties to relate sandwiched Holevo information to an α-information radius.
- Definition and basic properties: The sandwiched Rényi relative entropy is introduced as an information measure parameterized by α, with the main range α ∈ (1, 2].The quantity is defined from a sandwiched quasi-relative entropy on positive operators.
- Comparison with traditional Rényi divergence: For α > 1, the sandwiched quantity is no larger than the traditional quantum Rényi relative entropy, while the two coincide for commuting operators and reduce to the classical Rényi relative entropy.This comparison explains why the sandwiched form gives tighter upper bounds on classical communication success probabilities.
- Definition and basic properties: For α ∈ (1, 2], the sandwiched quasi-relative entropy is jointly convex, which yields monotonicity of both sandwiched quantities under quantum operations.The proof combines unitary invariance, tensor-product multiplicativity, tensoring invariance, and joint convexity.
- Definition and basic properties: The sandwiched Rényi relative entropy is non-negative and equals zero exactly when the two density operators are identical.These properties are established using monotonicity, classical Rényi divergence, and informationally complete measurements.
- Definition and basic properties: As α approaches one, the sandwiched Rényi relative entropy converges to the von Neumann relative entropy, assuming the first operator has unit trace.The argument assumes supp(A) is contained in supp(B); otherwise both quantities are infinite.
- Holevo-like information and information radius: The α-Holevo information equals the α-information radius for α ∈ (1, 2], improving a prior relation that provided only two inequalities.The equality is stated in Lemma 14 and the section identifies it as an improvement over Lemma I.3 of Ref. [41].
6 Bounding the success probability with the sandwiched relative R´enyi entropy
The paper bounds the success probability of classical communication schemes using the sandwiched Rényi relative entropy. This reduces the strong converse proof to establishing subadditivity of the sandwiched Holevo-like quantity over multiple channel uses.
- Success-probability bound: For any rate-R scheme using n copies of a quantum channel, the success probability is bounded using the sandwiched Rényi relative entropy for α ∈ (1, 2].The bound follows by combining the generalized-divergence success-probability bound with the sandwiched divergence’s properties.
- Reduction to subadditivity: The strong converse proof is reduced to proving subadditivity of the sandwiched Holevo-like quantity eχα(N ⊗n).The reduction follows after applying the induced classical divergence inequality.
7 Subadditivity of the α-information radius for entanglement-breaking channels
The section establishes subadditivity of the α-information radius for entanglement-breaking channels by reducing the argument to multiplicativity of maximum output α-norms.
- The main result states eχ_α(N^⊗n) ≤ n eχ_α(N) for every entanglement-breaking channel N.
- The proof uses multiplicativity of the maximum output α-norm for an entanglement-breaking completely positive map paired with an arbitrary completely positive map.
- For α ∈ (1, 2], the α-information radius is subadditive when an entanglement-breaking channel is combined with any other channel.
- An inductive argument extends the two-channel subadditivity relation to arbitrary tensor powers.
8 Final steps for the strong converse for entanglement-breaking channels
The subadditivity result yields an exponentially vanishing success probability whenever the communication rate exceeds the Holevo capacity of an entanglement-breaking channel, establishing its strong converse.
- The success-probability bound is obtained from the subadditivity relation and an α-information-radius argument around an optimal output state σ*.
- Choosing α sufficiently close to one connects the α-information-radius quantity to χ(N_EB), allowing the exponent to remain positive above capacity.
- Optimizing over α ∈ (1, 2] gives a stronger decay bound than the alternative proof approach.
- For every rate R > χ(N_EB), the success probability converges exponentially fast to zero.
- Prior results on particular covariant channels follow as a special case: The same framework recovers a strong converse rate of log d − H_min(N) when a channel’s minimum output Rényi entropy is additive for all α ≥ 1.
9 Strong converse for the classical capacity of Hadamard channels
The section extends the strong converse to Hadamard channels by exploiting their complementary-channel relationship with entanglement-breaking channels and a continuity argument at the boundary.
- The strong converse for Hadamard channels follows from a theorem on the multiplicativity of maximum output α-norms for complementary maps.
- For complements in the interior of the entanglement-breaking set, sandwiching preserves the relevant Hadamard structure for α sufficiently close to one.
- This preservation yields subadditivity of the α-information radius and an upper bound on success probability.
- The resulting success probability converges exponentially fast to zero when the communication rate exceeds capacity.
- For boundary cases, depolarizing the environment produces interior entanglement-breaking complements, while continuity of Holevo information gives χ(M_p) → χ(N_H).
- Every code for N_H is also a code for M_p with the same rate and error parameters, transferring the strong converse from M_p to N_H.
10 Conclusion
The conclusion states that strong converses hold for entanglement-breaking and Hadamard channels through sandwiched Rényi methods, while identifying open questions and broader relevance.
- The paper proves strong converse theorems for the classical capacity of all entanglement-breaking channels and their complementary Hadamard channels.
- The proofs bound success probability using sandwiched Rényi relative entropy, establish subadditivity, and relate the quantity to Holevo capacity.
- The superadditivity of eχ_α(N₁ ⊗ N₂) was identified as an open question in the paper, with a subsequent solution attributed to Beigi.
- The authors note that other tools might prove strong converse theorems, but this possibility remained open in the paper.
- The sandwiched Rényi relative entropy is presented as independently interesting because it is monotone under quantum operations.
A Appendix
The appendix reproduces operator-convexity and monotonicity results for multivariate matrix functionals, including conditions ensuring joint operator convexity and a tensor-product monotonicity principle.
- A Appendix: A general functional constructed from jointly operator-concave and positive g, semidefinite h, and a function f satisfies stated convexity conditions.Joint operator convexity holds when h is jointly operator concave with operator anti-monotone f, or when h is affine with operator-convex f.
- A Appendix: The map (L, R) → L^x ⊗ R^y is jointly operator concave for x, y ≥ 0 with x + y ≤ 1.The statement applies to positive operators.
- A Appendix: A dimension-independent functional that is jointly convex, unitarily invariant, and invariant under tensoring each argument with a density operator is monotone under all CPTP maps.The monotonicity applies to the stated domain and completely positive trace-preserving maps.
- A Appendix: The displayed inequality states that applying the same transformation T to all arguments does not increase the functional F.It summarizes the monotonicity relation F(T(A1), …, T(An)) ≤ F(A1, …, An).