[Paper Review] Note on multiple additivity of minimal Renyi entropy output of the Werner-Holevo channels

Robert Alicki, M. Fannes|ArXiv.org|Jul 5, 2004

Mathematical Dynamics and Fractals5 references6 citations

TL;DR

This paper provides a self-contained proof that the minimal Renyi entropy output of products of Werner-Holevo channels is additive for all $ p \in [1,2] $, establishing the multiplicativity of maximal $ p $-norms for these channels. The result confirms the additivity conjecture for this class of quantum channels using elementary trace inequalities and symmetry-based decomposition of tensor products.

ABSTRACT

We give an elementary self-contained proof that the minimal entropy output of arbitrary products of channels $ρ\mapsto \frac{1}{d-1}(1-ρ^T)$ is additive.

Motivation & Objective

• To establish the additivity of minimal Renyi entropy output for multiple copies of Werner-Holevo channels.
• To prove the multiplicativity of the maximal $ p $-norm for these channels, which is equivalent to the additivity conjecture.
• To provide a self-contained, elementary proof using only trace identities and norm inequalities.
• To extend previous results on $ p=1 $ and $ d=3 $ to general $ d $ and $ p \in [1,2] $.
• To clarify the role of symmetry and transpose structure in the channel's behavior under tensor products.

Proposed method

• The channel $ \Gamma_d(\rho) = \frac{1}{d-1}(\mathbb{I} - \rho^T) $ is analyzed for pure state inputs, exploiting the transpose operation and complex conjugation symmetry.
• The action of the tensor product channel $ \otimes_{j=1}^N \Gamma_{d_j} $ on a pure state is expanded using inclusion-exclusion over subsets of indices.
• The $ p=2 $ Renyi entropy is computed via the trace of the square of the output state, leading to an upper bound on $ \mathrm{Tr}(X_N(\Omega))^2 $.
• Trace identities and the fact that $ \mathrm{Tr}(\rho_\Lambda)^2 \leq 1 $ are used to bound the total trace expression.
• The inequality $ \mathrm{S}_2(\rho) \leq \mathrm{S}_p(\rho) \leq \mathrm{S}(\rho) $ for $ 1 \leq p \leq 2 $ is applied to relate $ p $-Renyi entropy to the 2-Renyi case.
• The final result is derived by combining the trace bound with the duality between Renyi entropy and $ p $-norms via $ \mathrm{S}_p(\rho) = -\frac{p}{p-1} \log \|\rho\|_p $.

Experimental results

Research questions

• RQ1 Does the minimal Renyi entropy output of the Werner-Holevo channel remain additive under arbitrary tensor products for $ p \in [1,2] $?
• RQ2 Is the maximal $ p $-norm of the Werner-Holevo channel multiplicative under tensor product for $ 1 < p \leq 2 $?
• RQ3 Can the additivity of minimal entropy output be proven using only elementary trace inequalities and symmetry?
• RQ4 How does the structure of the transpose map affect the output state of the tensor product channel?
• RQ5 What is the minimal Renyi entropy output for the product channel $ \otimes_{j=1}^N \Gamma_{d_j} $, and does it equal the sum of individual outputs?

Key findings

• The minimal Renyi entropy output of $ \otimes_{j=1}^N \Gamma_{d_j} $ is additive for all $ p \in [1,2] $, i.e., $ \mathrm{MEO}_p[\otimes_{j=1}^N \Gamma_{d_j}] = \sum_{j=1}^N \mathrm{MEO}_p[\Gamma_{d_j}] $.
• For any $ d $, the minimal $ p $-Renyi entropy output of a single Werner-Holevo channel is $ \mathrm{MEO}_p[\Gamma_d] = \log(d-1) $, independent of $ p \in [1,2] $.
• The maximal $ p $-norm of the channel satisfies $ \nu_p[\otimes_{j=1}^N \Gamma_{d_j}] = \prod_{j=1}^N \nu_p[\Gamma_{d_j}] $, proving multiplicativity for $ 1 < p \leq 2 $.
• The trace of the square of the output state of the product channel is bounded above by $ \prod_{j=1}^N \frac{1}{d_j - 1} $, which is key to the entropy bound.
• The inequality $ \mathrm{S}_2(\rho) \leq \mathrm{S}_p(\rho) \leq \mathrm{S}(\rho) $ for $ 1 \leq p \leq 2 $ allows the extension of the $ p=2 $ result to all $ p \in [1,2] $.
• The proof is elementary and self-contained, relying only on trace identities, norm inequalities, and symmetry of the transpose map.