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No information transmission through quantum channels above capacity
Hao-Chung Cheng, Marco Tomamichel
TL;DR
Sharp strong converses for unassisted communication over general finite-dimensional memoryless quantum channels have been elusive. Combining the Arimoto strategy with Rényi information representations and continuity bounds, the paper proves exponential decay of quantum and classical communication performance above their respective capacities.
Problem
Establishing the same sharp coding transition for finite-dimensional memoryless quantum channels has been elusive, and their capacity formulas have intricate structure.
Source: §I · p. 1Method
The proof combines Arimoto converse bounds with integral representations and asymptotic continuity of Rényi capacities at order one.
Source: §VI · p. 21Results
The strong-converse exponents are strictly positive at every rate above the respective quantum and classical capacities.
Source: D. Continuity of the Rényi capacities… · p. 7, §VI · p. 21Takeaways & Limitations
For quantum communication, the exponential strong-converse property also holds for standard unassisted entanglement-transmission and worst-case fidelity definitions.
Source: A. The Arimoto route to a strong… · p. 4
Abstract
from arXiv · showhide
We show that the capacity of a quantum channel demarcates a phase transition: while reliable transmission below capacity is always possible, any attempt to transmit information above it fails catastrophically. Specifically, we prove exponential strong converse theorems for unassisted quantum and classical communication over arbitrary finite-dimensional memoryless quantum channels. At rates beyond the respective capacity, the entanglement-generation fidelity and the success probability for classical communication decay exponentially with the number of channel uses. This rules out transmission above capacity even when one tolerates arbitrarily large errors. Our proof follows the classical Arimoto strategy, augmented by a crucial new ingredient: integral representations of R\'enyi information measures that lead to asymptotic continuity bounds for R\'enyi capacities.
I. INTRODUCTION
For finite-dimensional memoryless quantum channels, establishing a sharp coding transition has been difficult because the capacity formulas are intricate and regularized. The paper supplies the continuity ingredient needed to combine Rényi information bounds with the Arimoto strategy and prove exponential strong converses for unassisted quantum and classical communication.
I. INTRODUCTION
- Both unassisted capacity formulas are regularized, with inputs optimized jointly across channel copies; this is necessary because coherent information and the Holevo quantity can be superadditive.Entangled inputs can provide a strict advantage for the Holevo quantity, and coherent information can be zero for every fixed tensor power even when the channel capacity is positive.Source: §I · p. 2
- A uniform comparison with ordinary information quantities is needed because fixed-block-length continuity may require Rényi orders approaching one as block length grows.Without a uniform order, the resulting exponential rate need not stay positive.Source: §I · p. 3
- The comparison for Rényi coherent information and the Holevo quantity yields exponential strong converses for unassisted entanglement generation and classical communication over arbitrary finite-dimensional memoryless channels.The regularized Rényi quantities converge to the respective capacities.Source: §I · p. 3
- One-shot integral representations, local support estimates, and sequential restoration yield a correction linear in block length that vanishes per use as the Rényi order approaches one.The proof adapts the Leung–Smith telescoping argument [40] and bounds entropy changes using conditional-entropy continuity.Source: §I · p. 3
A. The Arimoto route to a strong converse
- For every rate above quantum capacity, the Arimoto route would make entanglement-generation fidelity decay exponentially once the Rényi-capacity continuity statement is established.The quantum statement uses squared fidelity, and the exponential strong-converse property extends to standard unassisted definitions.Source: A. The Arimoto route to a strong… · p. 4
- For classical communication, the Arimoto converse bounds success probability, and the missing continuity statement makes its exponent positive at every rate above classical capacity.The converse applies to every Rényi order α > 1; the continuity result supplies the needed above-capacity exponent.Source: A. The Arimoto route to a strong… · p. 5
B. One-shot integral representations
The section builds uniform continuity bounds for Rényi capacities from integral representations and local support restoration, then uses them to establish exponential strong converses above capacity. Its proof handles tilted states that need not be valid channel outputs by controlling their local support violations.
B. One-shot integral representations
- The one-shot integral representations express Rényi information quantities through ordinary entropy differences or mutual information evaluated on tilted states.In the classical case, the input-family probabilities are optimized separately at each integration parameter.Source: B. One-shot integral representations · p. 5
- Tilting can move states outside the channel’s Stinespring image, so a direct comparison with optimization over admissible channel inputs does not generally work.The proof instead quantifies and controls the off-image component near Rényi order one.Source: B. One-shot integral representations · p. 5
C. One channel image at a time
- Each tilted state has a local support violation of order O(u^2), with bounds independent of other channel dimensions and, classically, ensemble size.The proof restores the local Stinespring constraints one channel use at a time, applying conditional-entropy continuity bounds with local dimension factors.Source: C. One channel image at a time · p. 6
D. Continuity of the Rényi capacities and the exponential strong converse
- As the Rényi order decreases to one, the regularized Rényi coherent-information and Holevo capacities converge to their respective capacities.The finite-block comparison has an explicit correction that vanishes with the order and depends only on local channel dimensions.Source: D. Continuity of the Rényi capacities… · p. 7
- The strong-converse exponents are strictly positive above the respective quantum and classical capacities.The resulting exponent formulas are exact, with matching achievability bounds.Source: D. Continuity of the Rényi capacities… · p. 7
- Evaluating the regularized exponent formulas for concrete nonadditive channels remains an important open problem.The paper’s novelty claim is the strict positivity of both exponents above capacity, not the form of the exponent formulas.Source: D. Continuity of the Rényi capacities… · p. 7
III. INTEGRAL REPRESENTATIONS
The section develops integral representations by relating optimized Rényi quantities to derivatives evaluated on tilted states. These identities turn the Rényi quantities into ordinary entropy and information expressions to which continuity arguments can apply.
A. Rényi divergence preliminaries
- The sandwiched Rényi divergence is infinite unless the state’s support is contained in the comparison operator’s support, and the paper collects its properties for later use.The order-one quantity is the Umegaki relative entropy.Source: A. Rényi divergence preliminaries · p. 8
B. Quantum coherent information
- For quantum coherent information, a tilted state converts the derivative of a Rényi quantity into an ordinary entropy difference.The exact marginals of the tilted state convert logarithms of optimizing states into ordinary entropies.Source: B. Quantum coherent information · pp. 9–10
III. INTEGRAL REPRESENTATIONS
- The optimized Rényi conditional-entropy identities generalize fixed-state differential and integral representations to optimized quantities.The paper identifies the corresponding lemmas as generalizations of fixed-state identities for optimized sandwiched Rényi conditional entropy and Holevo information.Source: B. Quantum coherent information · pp. 10–11
C. Classical Holevo information
For classical communication, the proof represents the Rényi Holevo quantity through tilted ensembles and controls its dependence on Rényi order. The optimizing probabilities may vary with that order, so the derivative identity is used almost everywhere.
C. Classical Holevo information
- The classical integral representation optimizes the probability distribution defining the tilted ensemble separately at each integration parameter.The representation is stated for finite families of pure input states.Source: C. Classical Holevo information · p. 11
- The tilted-state construction uses a saddle pair and comparison states restricted to the support of the output marginal.The paper notes that this support restriction does not change the infimum.Source: C. Classical Holevo information · pp. 11–12
- The optimizing probabilities may change with the Rényi order, but an almost-everywhere derivative identity suffices for integration.For each fixed finite output family, the resulting function is Lipschitz on [0, 1).Source: C. Classical Holevo information · p. 13
D. Proof of Theorem 3
The proof obtains the integral identities by integrating derivative formulas and then taking the relevant input suprema. For classical communication, refining mixed-input ensembles to pure states preserves the supremum.
D. Proof of Theorem 3
- For a fixed quantum input, integrating the derivative identity from zero to the Rényi parameter gives the entropy integral representation.The proof then takes the supremum over input states.Source: D. Proof of Theorem 3 · p. 14
- For a fixed finite pure-input family, integrating the almost-everywhere derivative identity gives the mutual-information integral representation.Choices of saddle pairs at nondifferentiability points do not affect the integral.Source: D. Proof of Theorem 3 · p. 14
- Refining a finite mixed-input ensemble to pure states does not lower the supremum, because forgetting the refinement is a channel on the classical label.Taking the supremum over pure-state families therefore proves the stated representation.Source: D. Proof of Theorem 3 · p. 14
IV. PROJECTION ONTO THE STINESPRING IMAGE
The tilted states used in both information measures can leave the channel’s Stinespring image, so the proof controls their local support violations. Local estimates and a projection-based argument keep the resulting error independent of the block length.
A. Local support estimates
- The coherent-information and Holevo tilted states can leave the Stinespring image, requiring support estimates that also accommodate singular states.The conditional matrix-power bounds do not depend on the conditioning-system dimension or on the smallest nonzero eigenvalues.Source: A. Local support estimates · p. 14
- For either tilted state, the local support error is O(s^2) when the state-weighted norm deviation is O(s).Only the local output and environment dimensions enter the bound, even when all other channel uses and the classical label are conditioning systems.Source: A. Local support estimates · pp. 16–17
IV. PROJECTION ONTO THE STINESPRING IMAGE
- The projection estimates require no product-state assumption, alphabet-size bound, or global trace-distance approximation.The coherent and classical cases use the same local matrix-power and restoration lemmas, followed by separate entropy accounting.Source: §IV · p. 14, A. Local support estimates · p. 17
B. Restoring the Stinespring support
The proof restores the Stinespring constraint one channel use at a time, replacing each locally unsupported component with an allowed state. Conditional-entropy continuity then bounds the accumulated change using local dimensions rather than the dimensions of the remaining systems.
B. Restoring the Stinespring support
- Conditional-entropy continuity charges each repair only for the repaired system’s dimension, so summing the local costs gives a bound linear in block length.The local repair preserves the joint marginal of untouched systems, enabling the telescoping argument.Source: B. Restoring the Stinespring support · pp. 17–18
- Sequential local repairs place the final state in the joint Stinespring image while preserving the classical label’s marginal.Each repair acts on one output-environment pair and preserves constraints established at earlier pairs.Source: B. Restoring the Stinespring support · pp. 17–18
- For Holevo information, each restoration changes mutual information by at most 2δ log((dim B)^2 − 1) + 2h2(δ).The restored state specifies an admissible input ensemble; the repair channels need not be implemented by a receiver.Source: B. Restoring the Stinespring support · p. 19
V. CONTINUITY OF THE RÉNYI CAPACITIES
Finite-block continuity bounds control how far the Rényi coherent information and Holevo quantity exceed their respective capacities. Their regularized versions converge to those capacities as the Rényi order decreases to one.
V. CONTINUITY OF THE RÉNYI CAPACITIES
- The Rényi coherent information and Holevo quantity exceed their respective capacities by at most εq(α) and εc(α), respectively.These finite-block bounds apply for α > 1 within the stated range and for every block length n.Source: §V · p. 19
- The proof obtains the classical bound by integrating the derivative estimate for finite pure-input families, then taking the supremum over ensembles.Finite pure-state refinements and data processing ensure restricting to pure input families does not lower the supremum.Source: §V · p. 20
- The correction terms vanish as α decreases to one, proving convergence of the regularized Rényi quantities to the quantum and classical capacities.The proof also covers the excluded cases dim B = 1 and dim E = 1 directly.Source: §V · pp. 19–20
VI. CONCLUSION
The paper proves exponential strong converses for unassisted quantum and classical communication over finite-dimensional memoryless quantum channels. The bounds extend to certain product-channel settings, with the exact quantum exponent applying to entanglement generation.
VI. CONCLUSION
- The proof establishes exponential strong converses for unassisted quantum and classical communication over every finite-dimensional memoryless quantum channel.The exact quantum exponent is for entanglement generation; other standard unassisted quantum criteria also have exponential strong converses, though their exact exponents can differ.Source: §VI · p. 21
- The bounds extend to product channels with uniformly bounded local output and Stinespring-environment dimensions, allowing arbitrary correlations among inputs.The local support estimates and sequential restoration use the tensor-product dilation and only those local dimension bounds.Source: §VI · p. 21
- For nonstationary product-channel sequences, the bounds also establish exponential strong converses when normalized coherent and Holevo information converge to the respective operational capacities.Uniformly bounded local dimensions make the per-use corrections vanish uniformly as α decreases to one.Source: §VI · p. 21
Appendix A: Quantum Arimoto converse
The appendices derive Arimoto-style converses and characterize the strong-converse exponents for quantum and classical communication. Above capacity, the relevant fidelities or success probabilities decay exponentially; the entanglement-generation exponent is matched by an upper bound.
Appendix A: Quantum Arimoto converse
- For quantum communication, data processing and a binary test bound generation fidelity using a comparison state and an optimized conditional-entropy quantity.The proof reduces mixed preparations to pure ones and uses the Schmidt-rank bound before applying the optimization.Source: Appendix A · pp. 21–22
Appendix B: Classical Arimoto converse
- For classical codes, a binary test and data processing bound success probability against a comparison state, with additivity and Fekete’s lemma yielding the regularized quantity.The proof combines the one-shot inequalities to obtain the converse at rate r.Source: Appendix B · p. 22
Appendix C: Entanglement-generation strong-converse exponent
- One-way hashing or state merging, followed by selection of a preparation branch, supports entanglement-generation achievability without forward classical assistance.The branch can be absorbed into a deterministic input preparation and the corresponding decoder.Source: Appendix C · p. 23, §2 · pp. 23–24
1. Converse bound and positivity
- At every rate above capacity, the entanglement-generation converse exponent is positive, and the bound extends to code sequences whose limiting inferior rate exceeds capacity.The rate condition is handled by choosing an intermediate rate strictly between that limiting inferior rate and capacity.Source: §1 · p. 23
2. Matching upper bound for entanglement generation
- The matching entanglement-generation upper bound uses change of measure, convexification, and universal pinching to establish equality of the exponent bounds.A finite-grid argument exchanges regularization and the order supremum.Source: §2 · pp. 23, 24, 26
- The entanglement-generation exponent’s converse bound is matched, establishing equality in (24).The upper-bound argument uses a change-of-measure bound, convexification, and universal pinching.Source: §2 · p. 25
Appendix D: Other quantum communication criteria
Exponential decay of entanglement-generation fidelity extends to several other quantum communication criteria. The exact exponent, however, need not be shared across those criteria.
Appendix D: Other quantum communication criteria
- Exponential decay of generation fidelity implies exponential decay for entanglement-transmission, minimum pure-state, and reference-assisted minimum fidelities.The passage also states exponential convergence of normalized diamond-distance error to one.Source: Appendix D · p. 26
- For a replacer channel at rate r, optimal generation and transmission fidelities are 1/M and 1/M^2, giving exponents r and 2r, respectively.Thus the exact quantum exponent in Theorem 5 is specific to entanglement generation.Source: Appendix D · p. 26
- The exact exponents for the different quantum communication criteria need not coincide.The passage relates transmission fidelity to normalized diamond-distance error through an inequality.Source: Appendix D · p. 26
Appendix E: Classical strong-converse exponent
The classical converse gives exponential decay above capacity, and a matching characterization yields the classical strong-converse exponent. The exponent is strictly positive at every rate exceeding classical capacity.
1. Converse bound and positivity
- Above classical capacity, the classical success probability decays exponentially, including for code sequences whose limiting inferior rate exceeds capacity.Choosing an intermediate rate strictly between capacity and the sequence’s limiting inferior rate gives the bound for all sufficiently large block lengths.Source: §1 · p. 27
2. Matching upper bound from Mosonyi–Ogawa
- The classical strong-converse exponent has a matching characterization, and it is strictly positive at every rate above classical capacity.The matching upper bound follows from Mosonyi and Ogawa’s coding bound for quantum channels [31].Source: §2 · pp. 27–28