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arXiv:1411.6881 · Annales mathématiques Blaise Pascal

Additivity rates and PPT property for random quantum channels

Motohisa Fukuda, Ion Nechita

Published Jan 1, 2015• 7 citations
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Abstract

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 .

DOI
10.5802/ambp.345

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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 .
We find an upper bound for the minimum output entropy of a unital quantum channel, and obtain an exact formula for general qubit channels. Our techniques incorporate the Rényi entropies, particularly, with Rényi parameter α = 2. Moreover, since our upper bound is additive under tensor product, we get as a corollary an upper bound for the classical capacity of unital quantum channels. Interestingly, our upper bound for the classical capacity depends only on the operator norm of matrix representations of channels on the space of traceless Hermitian operators, and is tight in the sense that it gives the precise quantity of classical capacity of the Werner-Holevo channel. As an example, we study quantum channels with operator sum representation that is made of the discrete Weyl operators (generalized Pauli operators), and explain how our formula works in this case. Finally, we find new examples for which the minimum output Rényi 2-entropy is additive.
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.
Additivity violation of minimum output entropy, which shows non-classical properties in quantum communication, had been proved in most cases for random quantum channels defined by Haar-distributed unitary matrices. In this paper, we investigate random completely positive maps made of Gaussian Unitary Ensembles and Ginibre Ensembles regarding this matter. Using semi-circular systems and circular systems of free probability, we not only show the multiplicativity violation of maximum output norms in the asymptotic regimes but also prove the additivity violation via Haagerup inequality for a new class of random quantum channels constructed by rectifying the above completely positive maps based on strong convergence.
ABSTRACT. The free contraction norm (or the (t)-norm) was introduced in [BCN12] as a tool to compute the typical location of the collection of singular values associated to a random subspace of the tensor product of two Hilbert spaces. In turn, it was used in [BCN13] in order to obtain sharp bounds for the violation of the additivity of the minimum output entropy for random quantum channels with Bell states. This free con-traction norm, however, is difficult to compute explicitly. The purpose of this note is to give a good estimate for this norm. Our technique is based on results of super convergence in the context of free probability theory. As an application, we give a new, simple and conceptual proof of the violation of the additivity of the minimum output entropy. 1.
We investigate non-random quantum channels generated from the representation the- ory of orthogonal quantum groups to perform numerical computations for estimates of their minimum output entropies. In doing so, we address the violation of the additiv- ity conjecture for minimum output entropies, previously established through probabilistic proofs. This violation is relevant because given two quantum channels whose minimum output entropies sum to a value greater than the minimum output entropy of their tensor product channel it can be shown that more information can be sent with greater fidelity via the two channels in parallel than by using each separately. This is directly related to the relevance of quantum entanglement in quantum information theory. \nWe numerically compute minimum output entropies for single quantum channels and tensor products of quantum channels with respect to both Von Neumann and Renyi en- tropies. We then proceed to compute entropy estimates for nonplanar channels directly with Temperley-Lieb Category Theory. We conclude by discussing which of our choices for channels are optimal for entangled inputs.
The free contraction norm (or the (t)-norm) was introduced by Belinschi, Collins and Nechita as a tool to compute the typical location of the collection of singular values associated to a random subspace of the tensor product of two Hilbert spaces. In turn, it was used again by them in order to obtain sharp bounds for the violation of the additivity of the minimum output entropy (MOE) for random quantum channels with Bell states. This free contraction norm, however, is difficult to compute explicitly. The purpose of this note is to give a good estimate for this norm. Our technique is based on results of super convergence in the context of free probability theory. As an application, we give a new, simple and conceptual proof of the violation of the additivity of the MOE.