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arXiv:quant-ph/0403075 · Physical Review A

Additivity properties of a Gaussian channel

Vittorio Giovannetti, Seth Lloyd

Published Jun 7, 2004• 62 citations
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Abstract

The Amosov-Holevo-Werner conjecture implies the additivity of the minimum R\'enyi entropies at the output of a channel. The conjecture is proven true for all R\'enyi entropies of integer order greater than two in a class of Gaussian bosonic channel where the input signal is randomly displaced or where it is coupled linearly to an external environment.

DOI
10.1103/physreva.69.062307

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The Amosov-Holevo-Werner conjecture implies the additivity of the minimum R'enyi entropies at the output of a channel. The conjecture is proven true for all R'enyi entropies of integer order greater than two in a class of Gaussian bosonic channel where the input signal is randomly displaced or where it is coupled linearly to an external environment.
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.
Inspired by Montanaro’s work, we introduce the concept of additivity rates of a quantum channel L , which give the first order (linear) term of the minimum output p -Rényi entropies of L ⊗ r as functions of r . We lower bound the additivity rates of arbitrary quantum channels using the operator norms of several interesting matrices including partially transposed Choi matrices. As a direct consequence, we obtain upper bounds for the classical capacity of the channels. We study these matrices for random quantum channels defined by random subspaces of a bipartite tensor product space. A detailed spectral analysis of the relevant random matrix models is performed, and strong convergence towards free probabilistic limits is shown. As a corollary, we compute the threshold for random quantum channels to have the positive partial transpose (PPT) property. We then show that a class of random PPT channels violate generically additivity of the p -Rényi entropy for all p ≥ 30 . 95 .