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arXiv:1312.2200

Potential Examples for Non-Additivity of the Minimal Output Entropy

Muneerah Al Nuwairan

Published Dec 8, 2013• 1 citation
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Abstract

In this paper, we study the minimal output entropy of EPOSIC channels. We determine the cases where their minimal output entropy is zero, and obtain some partial results on the fulfillment of their entanglement breaking property.Our results show that these channels provide potential examples for non-additivity of the minimal output entropy.

DOI
10.48550/arxiv.1312.2200

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In this paper, we study the minimal output entropy of EPOSIC channels. We determine the cases where their minimal output entropy is zero, and obtain some partial results on the fulfillment of their entanglement breaking property.Our results show that these channels provide potential examples for non-additivity of the minimal output entropy.
In this thesis, we introduce EPOSIC channels, a class of SU(2) -covariant quantum channels. For each of them, we give a Stinespring representation, a Kraus representation, its Choi matrix, a complementary channel, and its dual map. We show that these channels are the extreme points of all SU(2) -irreducibly covariant channels. As an application of these channels to the theory of quantum information, we study the minimal output entropy of EPOSIC channels, and show that a large class of these channels is a potential example of violating the well-known problem, the additivity problem. We determine the cases where their minimal output entropy is not zero, and obtain some partial results on the fulfillment of their entanglement breaking property. We find a bound of the minimal output entropy of the tensor product of two SU(2) -irreducibly covariant channels. We also get an example of a positive map that is not completely positive.
It is known that the minimal output entropy is additive for any product of entanglement breaking (EB) channels. The same is true for the Renyi entropy, where additivity is equivalent to multiplicativity of the \norm{1}{q} norm for all q≥1q \ge 1q≥1. In this paper we consider the related question of multiplicativity of the \norm{2}{q} norm for q>2q > 2q>2 for entanglement breaking channels. We prove that multiplicativity holds in this case for certain classes of EB channels, including both the CQ and QC channels.
We investigate decoherence induced by a quantum channel in terms of minimal output entropy and map entropy. The latter is the von Neumann entropy of the Jamiołkowski state of the channel. Both quantities admit q-Renyi versions. We prove additivity of the map entropy for all q. For the case q = 2, we show that the depolarizing channel has the smallest map entropy among all channels with a given minimal output Renyi entropy of order two. This allows us to characterize pairs of channels such that the output entropy of their tensor product acting on a maximally entangled input state is larger than the sum of the minimal output entropies of the individual channels. We conjecture that for any channel Φ1acting on a finite dimensional system, there exists a class of channels Φ2sufficiently close to a unitary map such that additivity of minimal output entropy for Ψ1⊗ Ψ2holds.