Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1105.6122 · IEEE Transactions on Information Theory

The Minimum Entropy Output of a Quantum Channel Is Locally Additive

Gilad Gour, Shmuel Friedland

Published Aug 14, 2012• 12 citations
arXivHTMLPDF

Abstract

We show that the minimum von Neumann entropy output of a quantum channel is locally additive. Hastings' counterexample for the additivity conjecture makes this result quite surprising. In particular, it indicates that the nonadditivity of the minimum entropy output is a global effect of quantum channels.
DOI
10.1109/tit.2012.2210859

Related papers

Counterexamples to additivity of minimum output $p$-Rényi entropy of quantum channels for all $p\ge 0$

Debbie Leung, Benjamin Lovitz, Ion Nechita +1 more

The additivity of minimum output entropies is a central problem in quantum information theory. Nonadditivity is known for every Rényi order $p>1$, at the von Neumann point $p=1$, and for sufficiently small positive $p$, while much of the interval $0<p<1$ has remained open. In this work, we prove that for every $p>0$ there exist finite-dimensional quantum channels whose minimum output $p$-Rényi entropies violate additivity. Our proof combines two constructions: random projection-induced channels yield nonadditivity for $p>3/4$, and antisymmetric postprocessing extends the violation to all positive Rényi orders. Our estimates also improve the output-dimension bound obtained by Belinschi, Collins, and Nechita for additivity violation of minimum output von Neumann entropy.

quant-pharXiv:2607.15210•Jul 17, 2026•0 citations

A constructive violation of additivity of minimum output von Neumann entropy

Laura Shou, Alexey V. Gorshkov

We give an explicit non-random example of nonadditivity of minimum output von Neumann entropy. The proof uses a finite-dimensional construction which imitates free Haar unitary behavior. The same channel gives a violation of additivity for the minimum output Rényi-$p$ entropy for any $p\in[1,\infty]$. Additionally, given any $p_0>0$ and $\mathfrak g>0$, we extend the construction to obtain an explicit non-random channel which has entropy gap $\ge\mathfrak g$ uniformly over $p\in[p_0,\infty]$.

quant-pharXiv:2609.23946•Sep 22, 2026•0 citations

A counterexample to additivity of minimum output entropy

Matthew B. Hastings

We present a random construction of a pairs of channels which gives, with non-zero probability for sufficiently large dimensions, a counterexample to the minimum output entropy conjecture.As is known [4], this implies a violation of the additivity conjecture for the classical capacity of quantum channels.Thc violation of the minimum output cntropy conjecture is relatively small.

Physical Review Letters•Jan 1, 2008•21 citations

Local extrema of entropy functions under tensor products

Shmuel Friedland, Gilad Gour, Aidan Roy

We show that under a certain condition of local commutativity the minimum von-Neumann entropy output of a quantum channel is locally additive. We also show that local minima of the 2-norm entropy functions are closed under tensor products if one of the subspaces has dimension 2.

arXiv:1105.5380•May 26, 2011•0 citations

Local extrema of entropy functions under tensor products

Shmuel Friedland, Gilad Gour, Aidan Roy

We show that under a certain condition of local commutativity the minimum von-Neumann entropy output of a quantum channel is locally additive. We also show that local minima of the 2-norm entropy functions are closed under tensor products if one of the subspaces has dimension 2.

Quantum Information and Computation•Nov 1, 2011•2 citations

On Hastings' Counterexamples to the Minimum Output Entropy Additivity Conjecture

Fernando G. S. L. Brandão, Michał Horodecki

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.

arXiv:0907.3210•Open Systems & Information Dynamics•Mar 1, 2010•60 citations
Counterexamples to additivity of minimum output ppp-Rényi entropy of quantum channels for all p≥0p\ge 0p≥0
The additivity of minimum output entropies is a central problem in quantum information theory. Nonadditivity is known for every Rényi order p>1p>1p>1, at the von Neumann point p=1p=1p=1, and for sufficiently small positive ppp, while much of the interval 0<p<10<p<10<p<1 has remained open. In this work, we prove that for every p>0p>0p>0 there exist finite-dimensional quantum channels whose minimum output ppp-Rényi entropies violate additivity. Our proof combines two constructions: random projection-induced channels yield nonadditivity for p>3/4p>3/4p>3/4, and antisymmetric postprocessing extends the violation to all positive Rényi orders. Our estimates also improve the output-dimension bound obtained by Belinschi, Collins, and Nechita for additivity violation of minimum output von Neumann entropy.
We give an explicit non-random example of nonadditivity of minimum output von Neumann entropy. The proof uses a finite-dimensional construction which imitates free Haar unitary behavior. The same channel gives a violation of additivity for the minimum output Rényi-ppp entropy for any p∈[1,∞]p\in[1,\infty]p∈[1,∞]. Additionally, given any p0>0p_0>0p0​>0 and g>0\mathfrak g>0g>0, we extend the construction to obtain an explicit non-random channel which has entropy gap ≥g\ge\mathfrak g≥g uniformly over p∈[p0,∞]p\in[p_0,\infty]p∈[p0​,∞].
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.