Source-linked AI summary
Strong converse rates for quantum communication
Marco Tomamichel, Mark M. Wilde, Andreas Winter
TL;DR
The paper asks whether quantum information can be sent above channel capacity when decoding error need not vanish, a question important to the interpretation of capacity. It proves that Rains information is a strong-converse rate, with fidelity decaying exponentially even under classical pre- and post-processing. For generalized dephasing channels, Rains information is achievable as well, establishing the strong-converse property for that class.
Problem
Whether rates above quantum capacity force decoding error toward one when non-vanishing error is allowed remains an open question for quantum communication.
Source: §I · p. 1Method
The proof uses generalized divergences to compare channel outputs with PPT′ operators and establishes weak subadditivity for a Rényi-like Rains information quantity.
Source: A. Overview of Results and Outline · p. 2Results
Above Rains information, fidelity decays exponentially with channel uses even with classical pre- and post-processing; for generalized dephasing channels, Rains information equals coherent information.
Source: A. Overview of Results and Outline · p. 2, §VI · p. 9Takeaways & Limitations
Rains information is a strong-converse rate for any quantum channel, and the strong-converse property holds for generalized dephasing channels.
Source: §V · p. 8, §VI · p. 9, §VIII · p. 10
Abstract
from arXiv · showhide
We revisit a fundamental open problem in quantum information theory, namely whether it is possible to transmit quantum information at a rate exceeding the channel capacity if we allow for a non-vanishing probability of decoding error. Here we establish that the Rains information of any quantum channel is a strong converse rate for quantum communication: For any sequence of codes with rate exceeding the Rains information of the channel, we show that the fidelity vanishes exponentially fast as the number of channel uses increases. This remains true even if we consider codes that perform classical post-processing on the transmitted quantum data. As an application of this result, for generalized dephasing channels we show that the Rains information is also achievable, and thereby establish the strong converse property for quantum communication over such channels. Thus we conclusively settle the strong converse question for a class of quantum channels that have a non-trivial quantum capacity.
I. INTRODUCTION
Quantum capacity is the asymptotically reliable communication rate, but whether rates above it force error to one remained open for quantum communication. The paper proves that Rains information is a strong-converse rate, then applies the result to generalized dephasing channels.
I. INTRODUCTION
- Quantum capacity is the maximum rate in qubits per channel use achievable with fidelity approaching one over many memoryless channel uses.The coherent information is an achievable rate, and its regularization is also an upper bound on quantum capacity.Source: §I · p. 1
- The strong-converse question asks whether rates above quantum capacity force error toward one; this matters because practical codes use finite block lengths.The paper frames its goal as sharpening the interpretation of quantum capacity.Source: §I · p. 1
A. Overview of Results and Outline
- Above the Rains information, fidelity decays exponentially with channel uses, even with classical pre- and post-processing.The result gives an operational proof of I_c(N) ≤ R(N).Source: A. Overview of Results and Outline · p. 2
- The proof uses generalized divergences to compare channel outputs with PPT′ operators, which represent channels incapable of quantum data transmission.A Rains-like Rényi information quantity obeys weak subadditivity, enabling the strong-converse proof.Source: A. Overview of Results and Outline · p. 2
- For generalized dephasing channels, Rains information equals coherent information, establishing the strong-converse property for this class.The channel class includes maps defined by orthonormal input and output bases and arbitrary pure environment states; the qubit dephasing channel is an example.Source: A. Overview of Results and Outline · p. 2
B. Related Work
Earlier work made partial progress on strong converses for quantum capacity, but had not established the property for a nontrivial class of channels. This section also sets notation for states, channels, and PPT′ operators used in the paper.
B. Related Work
- Prior work addressed quantum-capacity strong converses through entanglement-assisted rates, degradable channels, and random erasure codes, but none established a strong converse for any nontrivial class of channels.For degradable channels, the result was a pretty strong converse: quantum error jumps from zero to at least 1/2 as soon as the communication rate exceeds quantum capacity. Separately, randomly selected codes above the quantum capacity of the quantum erasure channel have fidelity that decreases exponentially with channel uses.Source: B. Related Work · pp. 2–3
- The paper strengthens a prior claim that the Rains bound gives a weak-converse upper bound, identifying missing support for Rains-information weak subadditivity.The authors establish the corresponding property for Rényi Rains information.Source: B. Related Work · pp. 2–3
A. States and Channels
- The paper uses finite-dimensional Hilbert spaces; states are positive operators of trace one, and quantum channels are completely positive, trace-preserving maps.Bipartite states are separable when expressible as mixtures of product states; PPT requires positivity after partial transpose.Source: A. States and Channels · p. 3
- PPT′ enlarges the PPT set to include positive operators whose partial-transpose trace norm is at most one, including strictly subnormalized operators.Every PPT state belongs to PPT′, but PPT′ also contains operators that are not PPT.Source: A. States and Channels · p. 3
B. Codes, Rates and Capacity
The paper studies entanglement-generation codes over many parallel uses of a memoryless channel. It defines classical pre- and post-processing assistance, achievable rates, and the strong-converse criterion through fidelity decay above a rate threshold.
B. Codes, Rates and Capacity
- Classical pre- and post-processing assisted codes allow LOCC preparation and decoding, including a separable shared resource, but exclude interactive communication between channel uses.Unassisted codes restrict the preparation and decoding operations to local operations.Source: B. Codes, Rates and Capacity · p. 4
- An achievable rate requires a sequence of codes whose fidelity approaches one as channel uses grow, and capacity is the supremum of achievable rates.The assisted capacity is defined analogously using assisted achievable rates.Source: B. Codes, Rates and Capacity · p. 4
- A strong-converse rate forces fidelity to zero for every code sequence whose rate exceeds that threshold; the strong-converse capacity is the infimum of such rates.The paper defines the strong-converse property when the ordinary and strong-converse capacities coincide.Source: B. Codes, Rates and Capacity · p. 4
III. GENERALIZED DIVERGENCE FRAMEWORK
The generalized-divergence framework supplies monotonicity tools for the paper’s converse argument. The section defines Rains relative entropy and channel information using optimization against PPT′ operators, whose maximally entangled overlap is bounded.
III. GENERALIZED DIVERGENCE FRAMEWORK
- A generalized divergence is monotone under quantum channels and is invariant under isometries and tensoring with another quantum state.These invariances follow from monotonicity and the availability of a channel reversing an isometry.Source: §III · p. 4
A. Rains Relative Entropy and Rains Information
- Generalized Rains relative entropy optimizes a divergence against PPT′ operators; the channel Rains information applies it to channel outputs generated from reference-input states.The paper introduces these quantities as extensions of the Rains bound on distillable entanglement.Source: A. Rains Relative Entropy and Rains… · pp. 4–5
- Rains relative entropy is monotone under PPT-preserving operations and thus under LOCC, because such operations preserve PPT′.The argument uses trace-norm monotonicity under completely positive trace-preserving maps.Source: A. Rains Relative Entropy and Rains… · pp. 4–5
- A PPT′ operator has overlap at most 1/M with a maximally entangled state of Schmidt rank M.The same bound applies to PPT states because PPT is contained in PPT′.Source: A. Rains Relative Entropy and Rains… · p. 5
B. Covariance of Quantum Channels
The section defines channel covariance through compatible input and output unitary actions, then uses that symmetry to restrict the states needed to optimize generalized Rains information. It also specializes the divergence framework to sandwiched Rényi quantities, emphasizing their behavior for α > 1.
B. Covariance of Quantum Channels
- A channel is covariant when applying the input representation and then the channel matches applying the output representation after the channel.The representations act on the input and output spaces, respectively.Source: B. Covariance of Quantum Channels · p. 5
- Covariance lets the optimization of a covariant channel’s generalized Rains information be restricted to a simplified form of input state.The proof uses dephasing, channel covariance, divergence invariance, and discarding a register.Source: B. Covariance of Quantum Channels · p. 5
C. Specializing to R´enyi Divergence
- For α > 1, sandwiched Rényi relative entropy is monotone under CPTP maps, and its trace term is jointly convex in its arguments.As α approaches one, the divergence converges to the ordinary relative entropy.Source: C. Specializing to R´enyi Divergence · p. 5
- The Rényi Rains relative entropy is subadditive for bipartite product states.The text attributes this to tensor-product additivity of the sandwiched Rényi relative entropy and restriction to product states in the minimization.Source: C. Specializing to R´enyi Divergence · p. 5
RAU †
This section defines Rényi Rains information for a channel and establishes properties that support a fidelity bound for entanglement-generation codes. The resulting one-shot bound applies to CPPP-assisted codes for every α > 1.
RAU †
- The Rényi Rains relative entropy is quasi-convex in the state, and it is subadditive on product states.The quasi-convexity follows from joint convexity of the expression being minimized.Source: RAU † · p. 6
- The Rényi Rains information optimization can use pure bipartite inputs with a reference system no larger than the channel input.The reduction follows from quasi-convexity and Schmidt decomposition.Source: RAU † · p. 6
- As α approaches one from above, the channel’s Rényi Rains information converges to its Rains information.The convergence is stated for any quantum channel.Source: RAU † · p. 6
D. Relating Fidelity of an Entanglement Generation Code to the R´enyi Rains Information of a Channel
- The proof relates code fidelity to the Rényi Rains information by optimizing over channel inputs after bounding against PPT′ states.The argument removes dependence on a particular code by optimizing over inputs to the channel.Source: D. Relating Fidelity of an… · pp. 6–7
- Proposition 4 gives a one-shot fidelity bound for every CPPP-assisted entanglement-generation code and every α > 1.Its proof applies a binary test for maximally entangled output and uses monotonicity under decoding and testing.Source: D. Relating Fidelity of an… · pp. 6–7
IV. WEAK SUBADDITIVITY OF THE α-RAINS INFORMATION FOR MEMORYLESS CHANNELS
The section establishes weak subadditivity of the α-Rains information for memoryless channels. Its proof combines an operator-ordering lemma with permutation symmetry and properties of the Rényi Rains quantity.
IV. WEAK SUBADDITIVITY OF THE α-RAINS INFORMATION FOR MEMORYLESS CHANNELS
- If ρ ≤ γρ′ for γ ≥ 1, Lemma 5 bounds the sandwiched Rényi divergence using the same factor γ.The proof applies operator monotonicity before taking logarithms and dividing by α − 1.Source: §IV · p. 7
- The α-Rains information of a memoryless channel obeys weak subadditivity.Theorem 6 establishes the property, and Corollary 7 states the corresponding result for the Rains information.Source: §IV · pp. 7–8
- The proof exploits permutation covariance of tensor-power channels to represent a purified input by a symmetric state.It then uses quasi-convexity and product-state subadditivity to complete the bound.Source: §IV · pp. 7–8
V. THE RAINS INFORMATION IS A STRONG CONVERSE RATE FOR QUANTUM COMMUNICATION
The section proves that the Rains information gives a strong-converse rate for CPPP-assisted quantum communication. Above the stated regularized Rains threshold, fidelity vanishes exponentially with the number of channel uses.
V. THE RAINS INFORMATION IS A STRONG CONVERSE RATE FOR QUANTUM COMMUNICATION
- The authors do not know whether the stated bound on the exponential decay exponent is optimal.The exponential decay itself is established.Source: §V · p. 8
- The proof uses continuity of Rényi Rains information near α = 1 to choose an α > 1 above which the code rate exceeds the bound.For tensor-power channels, it also invokes weak subadditivity to obtain the exponential fidelity bound.Source: §V · p. 8
- Above inf over ℓ of 1/ℓ times the Rains information of N^⊗ℓ, any CPPP-assisted code sequence has fidelity that vanishes exponentially with channel uses.The theorem identifies this infimum as a strong-converse rate.Source: §V · p. 8
- The Rains information upper-bounds the strong-converse capacity for CPPP-assisted quantum communication.Theorem 8 states the capacity bound for any quantum channel.Source: §V · pp. 8–9
VI. STRONG CONVERSE PROPERTY FOR QUANTUM COMMUNICATION OVER DEPHASING CHANNELS
For generalized dephasing channels, the Rains information equals the coherent information, establishing the strong converse property for their quantum capacity. The section also shows that classical pre- and post-processing does not increase capacity for this class.
VI. STRONG CONVERSE PROPERTY FOR QUANTUM COMMUNICATION OVER DEPHASING CHANNELS
- The quantum capacity of generalized dephasing channels obeys the strong converse, and classical pre- and post-processing does not increase their capacity.Source: §VI · p. 9
- For generalized dephasing channels, the Rains information equals the coherent information.This equality collapses the hierarchy in (81).Source: §VI · p. 9
- The maximizing input has a Schmidt decomposition whose input-system basis is aligned with the channel basis.Source: §VI · p. 9
- The proof restricts the optimization to aligned pure inputs, applies relative-entropy monotonicity under ΔP, and compares with a PPT′ state.Source: §VI · p. 9
ASSISTED CAPACITY OF ERASURE CHANNELS
For erasure channels, the classical-post-processing-assisted capacities equal the rate determined by the probability of successful transmission. The conclusion states that the Rains information is a strong converse rate for quantum communication over any channel, while identifying further open questions.
ASSISTED CAPACITY OF ERASURE CHANNELS
- Qpp(N) = Q†pp(N) = (1 − p) log |A| for erasure channels.Source: ASSISTED CAPACITY OF ERASURE CHANNELS · p. 10
- Any rate below (1 − p) log |A| is achievable with CPPP assistance by transmitting entangled states and distilling those received without erasure.Source: ASSISTED CAPACITY OF ERASURE CHANNELS · p. 10
- For p ≤ 1/2, the unassisted erasure-channel capacity is Q(N) = max{(1 − 2p) log |A|, 0}, below the Rains bound, so that bound does not establish a strong converse for all degradable channels.Source: ASSISTED CAPACITY OF ERASURE CHANNELS · p. 10
VIII. CONCLUSION
- The paper establishes the Rains information as a strong converse rate for quantum communication over any quantum channel.Source: §VIII · p. 10
- The conclusion raises whether the equality with coherent information extends beyond generalized dephasing channels and whether strong converses or second-order characterizations hold for other communication tasks.Source: §VIII · pp. 10–11
APPENDIX A PROOF OF LEMMA 3
The appendices bound the limiting generalized Rains quantity by the Rains information and use a uniform bound to complete the proof of Corollary 7.
APPENDIX A PROOF OF LEMMA 3
- Monotonicity of the sandwiched relative entropy in its order establishes existence of the limit and one inequality; a regularized bound gives the reverse inequality.Source: Appendix A · p. 11
- A uniform bound over ρRB lets the proof maximize over channel output states and conclude limα→1+ eRα(N) ≤ R(N).The argument uses a PPT′-preserving mixture with the fully mixed state and a lower bound on its minimum eigenvalue.Source: Appendix A · p. 11
APPENDIX B PROOF OF COROLLARY 7
- The proof of Corollary 7 applies the uniform bound, uses Theorem 6, and takes n → ∞ to obtain the desired result.Source: Appendix B · pp. 11–12