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Counterexamples to the maximal p-norm multiplicativity conjecture for all p > 1
Patrick Hayden, Andreas Winter
TL;DR
The paper addresses whether maximal output p-norms of quantum channels are multiplicative, a route previously used to study minimum output entropy additivity. By combining and strengthening two counterexample constructions, it shows that multiplicativity and minimum output Rényi-entropy additivity fail for every p > 1, while the p = 1 case remains unresolved.
Problem
The paper tests whether channel-independent maximal p-norm multiplicativity can support the minimum output entropy additivity conjecture.
Method
The authors combine approximate-encryption-based random unitary channels with random generic quantum channels and analyze their output norms and Rényi entropies.
Results
Maximal p-norm multiplicativity and minimum output Rényi-entropy additivity fail for all p > 1, with product-channel entropy not significantly exceeding the individual factors’ entropy.
Takeaways & Limitations
The previously favored proof strategy through maximal p-norm multiplicativity cannot establish the minimum output von Neumann entropy conjecture in general.
Takeaways & Limitations
The counterexamples are nonconstructive, and the paper reports no evidence of a p = 1 violation for its channels.
Abstract
from arXiv · showhide
For all p > 1, we demonstrate the existence of quantum channels with non-multiplicative maximal output p-norms. Equivalently, for all p >1, the minimum output Renyi entropy of order p of a quantum channel is not additive. The violations found are large; in all cases, the minimum output Renyi entropy of order p for a product channel need not be significantly greater than the minimum output entropy of its individual factors. Since p=1 corresponds to the von Neumann entropy, these counterexamples demonstrate that if the additivity conjecture of quantum information theory is true, it cannot be proved as a consequence of any channel-independent guarantee of maximal p-norm multiplicativity. We also show that a class of channels previously studied in the context of approximate encryption lead to counterexamples for all p > 2.
I. INTRODUCTION
The paper places maximal p-norm multiplicativity within the broader additivity problem for quantum-channel capacities and minimum output entropy. It then introduces counterexamples that extend beyond earlier restricted results, including channels violating the conjecture for every p > 1.
- I. INTRODUCTION: The Holevo-capacity additivity conjecture would imply that entangled codewords do not increase a quantum channel’s classical capacity.This conjecture is equivalent, through established connections, to the minimum output entropy conjecture.
- I. INTRODUCTION: Earlier work established many special cases and found counterexamples only in restricted p-regimes, including p > 4.79 and all p > 2.Before this paper, multiplicativity was still conjectured for 1 ≤ p ≤ 2 or at least near p = 1.
- I. INTRODUCTION: The minimum output entropy conjecture asserts additivity of the minimum output entropy across product channels, while maximal p-norm multiplicativity is a natural strengthening of it.The paper reformulates the p-norm conjecture using minimum output Rényi entropies of order p.
- I. INTRODUCTION: The paper merges and slightly strengthens earlier counterexamples, presenting simpler approximate-encryption examples and improved generic-channel examples that work for all p > 1.Section II treats Winter’s counterexamples for p > 2, while Section III extends Hayden’s examples beyond 1 < p < 2.
- I. INTRODUCTION: For every p > 1, the authors find channels whose product-channel minimum output entropy need not be significantly larger than those of the individual factors.The counterexamples are described as essentially strongest possible up to a constant additive term, with p-dependence absorbed into asymptotic notation.
- I. INTRODUCTION: The examples provide no evidence against additivity at p = 1, so whether entangled codewords increase classical capacity remains open.The paper therefore separates the failure of Rényi-entropy additivity for p > 1 from the unresolved von Neumann-entropy case.
II. RANDOM UNITARY CHANNELS: p > 2
For p > 2, approximate-encryption-inspired random unitary channels provide counterexamples to maximal output p-norm multiplicativity, with violations for sufficiently large dimensions.
- II. RANDOM UNITARY CHANNELS: p > 2: The construction uses ε-randomizing random unitary channels, a class previously studied for approximate encryption.Such channels are straightforward counterexamples for p > 2, although stronger constructions appear later.
- II. RANDOM UNITARY CHANNELS: p > 2: The counterexample compares a channel with its complex conjugate and uses a maximally entangled test state to obtain the violating product-channel behavior.The proof relies on lemmas bounding the individual and paired output p-norms.
- II. RANDOM UNITARY CHANNELS: p > 2: For an ε-randomizing channel, every output eigenvalue is bounded above by (1+ε)/d and the eigenvalues sum to one.These constraints enable the p-norm bound used in the argument.
- II. RANDOM UNITARY CHANNELS: p > 2: For p > 2 and sufficiently large d, these channels have a strictly supermultiplicative maximum output p-norm.Theorem II.3 assumes an ε-randomizing map with n > 134 d ln d/ε^2.
- II. RANDOM UNITARY CHANNELS: p > 2: This class reaches its validity limit at p = 2 and applies only in rather large dimension d.The paper notes that n ≥ d for every ε-randomizing map, while explicit constructions are discussed separately.
III. GENERIC QUANTUM CHANNELS: ALL p > 1
Random subspace constructions yield quantum channels violating maximal p-norm multiplicativity for every 1 < p ≤ ∞. The violations arise from highly entangled single-copy outputs alongside an unusually large eigenvalue for a paired channel output.
- III. GENERIC QUANTUM CHANNELS: ALL p > 1: The method fixes system dimensions, selects an isometry at random, and shows that the resulting channel is likely to violate additivity.The intuition is that product-state inputs can look highly depolarizing even when the channel is not close to depolarizing in complete-bounded norm.
- III. GENERIC QUANTUM CHANNELS: ALL p > 1: A random subspace can be chosen so that all its states have high entanglement, producing high output Rényi entropy for individual channel uses.The proof combines expectation and Lipschitz bounds with concentration, nets, and a union bound.
- III. GENERIC QUANTUM CHANNELS: ALL p > 1: For the product channel N ⊗ N̄, a maximally entangled input produces an output with an eigenvalue at least |S|, yielding a large additivity gap.The paired construction exploits an approximate symmetry between a channel and its complex conjugate.
- III. GENERIC QUANTUM CHANNELS: ALL p > 1: For every 1 < p ≤ ∞, there exists a quantum channel whose inequalities contradict maximal p-norm multiplicativity.The construction therefore supplies counterexamples throughout the full p > 1 regime.
- III. GENERIC QUANTUM CHANNELS: ALL p > 1: The counterexamples are essentially optimal up to constant additive terms, because stronger violations would contradict the paper’s entropy bound.However, varying p changes the subspace dimension, so the construction gives a sequence of channels rather than one channel violating additivity for every p.
IV. THE VON NEUMANN ENTROPY CASE
The channel pair can produce a spectrum with one comparatively large eigenvalue while retaining high von Neumann entropy, so the p>1 violations do not appear to extend to p=1.
- The von Neumann entropy is less sensitive than Rényi entropies to a single large eigenvalue, so these examples do not appear to violate minimum-output-entropy additivity.The argument relies on the distinction between the spectral sensitivity of p>1 Rényi entropies and von Neumann entropy.
- The analysis assumes |A| ≤ |B| ≤ |S| when bounding the spectrum of the output for a maximally entangled input.The maximally entangled state is taken between S1 and S2, and Haar integration supplies the relevant spectral estimates.
- 2 ln |A| − O(1) lower-bounds the von Neumann entropy of the product output, keeping it near its maximum despite the dominant eigenvalue.The bound follows by separating the largest eigenvalue from the remaining normalized spectrum and applying the grouping property.
- The output spectrum has one large eigenvalue, while the remaining eigenvalues are sufficiently small to preserve high von Neumann entropy.Figure 1 illustrates the typical eigenvalue distribution, and the text explains why the small residual eigenvalues still yield high entropy.
V. DISCUSSION
The paper disproves maximal p-norm multiplicativity for every 1<p≤∞ while arguing that these counterexamples do not directly settle the von Neumann entropy conjecture. It instead removes a previously favored proof strategy and leaves explicit constructions open.
- The maximal p-norm multiplicativity and minimum-output p-Rényi entropy conjectures are false for all 1 < p ≤∞.This conclusion applies across the full range addressed by the paper.
- The counterexamples eliminate the previously favored route of proving minimum-output entropy additivity through channel-independent output p-norm multiplicativity.The paper notes that many earlier special-case proofs used this strategy before taking p toward 1.
- The examples appear consistent with the minimum von Neumann output entropy conjecture because von Neumann entropy is harder to perturb than Rényi entropies with p>1.The paper leaves open subtler channel-dependent or approximate approaches near p=1, as well as approaches using p<1.
- The counterexamples are nonconstructive, relying on probabilistic existence arguments without known explicit channel constructions.Known explicit constructions provide only bounds too weak to yield the required multiplicativity counterexamples.
APPENDIX A: PROOF OF LEMMA IV.1
The appendix proves the spectral estimate by expanding Haar integrals into diagrammatic terms and showing that one stack diagram dominates while symmetry and nonidentity permutations contribute lower-order corrections.
- Nonzero terms are represented by diagrams whose dotted connections encode a permutation between U and conjugate-U indices.The graphical representation follows from the requirement that the U-index vector be a permutation of the conjugate-U vector.
- For distinct vertex labels, each integral is evaluated using a Weingarten function, with nonidentity permutations suppressed relative to the identity contribution.The estimate uses Wg(π)=Θ(...) and Wg(e)=(|A||B|)^−4 as the leading scale.
- The dominant contribution to the Haar integral comes from the positive “stack” diagram, approximately equal to the leading term.The stack has parallel solid and dashed lines and supplies the main asymptotic contribution.
- The remaining diagram classes contribute lower-order terms because index-identification constraints sharply reduce their counts.The appendix bounds these contributions by orders such as O(|S|2|A|−4|B|−2).
- Vertex-label symmetries are handled by counting representative conjugacy-class diagrams and showing that their aggregate contribution does not alter the dominant term.The error bound is obtained after substituting |S| ≤ |A||B| and |A| ≤ |B| into the preceding estimates.