On Hastings' Counterexamples to the Minimum Output Entropy Additivity Conjecture

2009/07/19 by Fernando G. S. L. Brandão, Fernando G. S. L. Brandao, Michał Horodecki +1 · 60 citations
Computer Science · Mathematics · Physics and Astronomy · #Additive function #Combinatorics #Conjecture #Counterexample #Dimension (graph theory) #Discrete mathematics #Entropy (arrow of time) #Markov Chains and Monte Carlo Methods #Mathematical analysis #Mathematics #Physics #Pure mathematics #Quantum Computing Algorithms and Architecture #Stochastic Gradient Optimization Techniques #quant-ph

paper · pdf · doi:10.1142/s1230161210000047

published in Open Systems & Information Dynamics 17(01), 31-52 (World Scientific) · 17 pages + 1 line

arxiv created 2009/07/19 · openalex publication_date 2010/03/01 · arxiv updated 2010/07/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

Hastings recently reported a randomized construction of channels violating the minimum output entropy additivity conjecture. Here we revisit his argument, presenting a simplified proof. In particular, we do not resort to the exact probability distribution of the Schmidt coefficients of a random bipartite pure state, as in the original proof, but rather derive the necessary large deviation bounds by a concentration of measure argument. Furthermore, we prove non-additivity for the overwhelming majority of channels consisting of a Haar random isometry followed by partial trace over the environment, for an environment dimension much bigger than the output dimension. This makes Hastings' original reasoning clearer and extends the class of channels for which additivity can be shown to be violated.

Citations

• Isotropic quantum spin channels and additivity questions
• Strong superadditivity of the entanglement of formation follows from its additivity
• Superdense Coding of Quantum States
• Multiparty data hiding of quantum information
• Simplifying additivity problems using direct sum constructions
• Random Quantum Channels I: Graphical Calculus and the Bell State Phenomenon
• Classical, quantum and total correlations
• Classical information capacity of a class of quantum channels

Cited by

• Trade-off capacities of the quantum Hadamard channels
• On the convergence of output sets of quantum channels

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