Strong converse for quantum capacity
For every finite-dimensional memoryless channel , does unassisted entanglement transmission at a fixed rate above force entanglement fidelity to tend to zero for all code sequences? Prove the statement or construct a counterexample.
Problem statement
For every finite-dimensional memoryless channel , does unassisted entanglement transmission at a fixed rate above force entanglement fidelity to tend to zero for all code sequences? Prove the statement or construct a counterexample.
Lean 4
No Lean statement or proof is attached yet. Have a formalization? Share its source to add it to this problem.
Background
A strong converse makes quantum capacity a sharp threshold: above it, entanglement fidelity tends to zero as blocklength grows. The weak converse only prevents asymptotically perfect transmission. Exponential decay is a stronger quantitative conclusion and should be distinguished from the basic strong-converse question.
Earlier work established quantum strong converses for selected channel classes and strong-converse upper bounds using the Rains quantity. The general degradable-channel baseline was a “pretty strong” converse, not an all-code full strong converse. Strong converses for classical communication through entanglement-breaking channels do not prove the corresponding quantum-capacity statement.
Preprint progress (July 2026): Tomamichel claims an exponential strong converse for full-joint-eigenspace stabilizer codes over Pauli channels, with a threshold determined by the corresponding coherent-information optimization. This restricted-code result is not a theorem for every possible encoder.
Further preprint progress (August 2026): Kondra–Brinster–Kampermann–Bruß–Wyderka claim an all-code exponential strong converse for finite-dimensional degradable and antidegradable channels, including the first all-code exponential erasure-channel result over its full parameter range. They also give general SDP strong-converse bounds and a result for a multilevel amplitude-damping family. These are newly proposed proofs; this catalogue does not independently certify them.
Neither manuscript resolves the unrestricted finite-dimensional channel question stated here. Code restrictions, assistance and fidelity conventions remain part of each result.
Source disclosures
Tomamichel reports author-directed Claude proof development, writing and literature work, plus ChatGPT adversarial review, and states that he checked the statements, proofs and references. Source
Kondra et al. disclose ChatGPT 5.6 Sol assistance on technical proof steps and later manuscript work with Claude Opus 4.8; they state that the initial draft was handwritten and all assisted material was reviewed. Source
Progress & source assessment
Source catalogue review: 4 September 2026. Improved indicates partial progress, including qualified preprint claims.
preprint: Strong converse restricted to full-joint-eigenspace stabilizer codes over Pauli channels. Evidence
preprint: Claimed all-code exponential strong converse for finite-dimensional degradable and antidegradable channels; general channels remain open. Evidence
- [1]Tomamichel, Wilde, Winter, 'Strong converse rates for quantum communication', IEEE Trans. Inf. Theory 63 (2017) link
- [2]Wilde, Winter, Yang, 'Strong converse for the classical capacity of entanglement-breaking and Hadamard channels', Comm. Math. Phys. 331 (2014) link
- [3]Tomamichel, 'A strong converse for stabilizer codes over Pauli channels via the blowing-up lemma' (2026 preprint) link
- [4]Kondra et al., 'Sharp Quantum Capacity Thresholds: Exponential Strong Converses for Degradable and Antidegradable Channels' (2026 preprint) link
Recorded progress
Add yours →Nothing recorded yet. A pointer to a paper the entry is missing counts.
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Discussion
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