On classical capacity of Weyl channels
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On classical capacity of Weyl channels Grigori Amosov1 Received: 1 July 2020 / Accepted: 20 October 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We study an old problem concerning the resolution of the question of whether the supremum of the Holevo upper bound for the output of a quantum channel coincides with the classical capacity of this channel. It is shown that this property takes place for one special case. The additivity of minimal output entropy is proved for the Weyl channel obtained by the deformation of a q-c Weyl channel. The classical capacity of channel is calculated. Keywords Quantum Weyl channel · Classical capacity of a channel · Holevo upper bound · Additivity conjecture Mathematics Subject Classification 81P45 · 81P47 · 94A40
1 Introduction The quantum coding theorem proved independently by A.S. Holevo [1] and B. Schumacher, M.D. Westmoreland [2] posed the task of calculating the Holevo upper bound C(Φ ⊗N ) for a tensor product of N copies of quantum channel Φ because a classical capacity of Φ is given by the formula C(Φ) =
C(Φ ⊗N ) . N →+∞ N lim
The additivity conjecture asks whether the equality C(Φ ⊗ Ω) = C(Φ) + C(Ω)
(1)
This work is supported by the Russian Science Foundation under Grant N 19-11-00086.
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Grigori Amosov [email protected] Steklov Mathematical Institute of Russian Academy of Sciences, 8 Gubkina St., Moscow, Russia 119991 0123456789().: V,-vol
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holds true for the fixed channel Φ and an arbitrary channel Ω. If the additivity property (1) takes place for Φ, the classical capacity can be calculated as follows: C(Φ) = C(Φ).
(2)
The same level of interest has the additivity conjecture in the weak form asking whether C(Φ ⊗N ) = N C(Φ) takes place for a fixed channel Φ. The validity of this statement also leads to (2). The additivity conjecture for C is closely related to the additivity conjecture for the minimal output entropy of a channel and the multiplicativity conjectures for trace norms of a channel [3]. At the moment, the additivity is proved for many significant cases [4–8] including the solution to the famous problem of Gaussian optimizers [9,10]. On the other hand, there are channels for which the additivity conjecture does not hold true [11]. Recently, the method of majorization was introduced to estimate the Holevo upper bound for Weyl channels [12]. In the present paper, we prove the additivity conjecture for one subclass of Weyl channels that are “deformations” of q-c channels of [1]. Our method is based upon [12]. Throughout this paper, we denote S(H ) the set of positive unit-trace operators (quantum states) in a Hilbert space H , I H is the identity operator in H and S(ρ) = −T r (ρ log ρ) is the von Neumann entropy of ρ ∈ S(H ). Quantum channel Φ : S(H ) → S(K ) is a completely positive trace preserving map between the algebras of all bounded operators B(H ) and B(K ) in Hilbert spaces H and K , respectively. Given two ρ, σ ∈ S(H ) for which suppρ ⊂ suppσ the quantum relative entropy is
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