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Problems of Information Transmission

Remark on the additivity conjecture for a quantum depolarizing channel

Grigori Gennadievich Amosov

Published Jun 1, 2006• 25 citations
Publisher

Abstract

We consider bistochastic quantum channels generated by unitary representations of a discrete group. We give a proof of the additivity conjecture for a quantum depolarizing channel Φ based on the decreasing property of the relative entropy. We show that the additivity conjecture holds for a channel Ξ = Ψ o Φ, where Ψ is a phase damping channel.

DOI
10.1134/s0032946006020013

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We consider bistochastic quantum channels generated by unitary representations of a discrete group. We give a proof of the additivity conjecture for a quantum depolarizing channel Φ based on the decreasing property of the relative entropy. We show that the additivity conjecture holds for a channel Ξ = Ψ o Φ, where Ψ is a phase damping channel.
We consider bistochastic quantum channels generated by unitary representations of the discret group. The proof of the additivity conjecture for the quantum depolarizing channel ΦΦΦ based on the decreasing property of the relative entropy is given. We show that the additivity conjecture is true for the channel Ξ=Ψ∘ΦΞ=Ψ\circ ΦΞ=Ψ∘Φ, where ΨΨΨ is the phase damping.
We investigate the Weyl channels being covariant with respect to the maximum commutative group of unitary operators. This class includes the quantum depolarizing channel and the “two-Pauli” channel as well. Then, we show that our estimation of the output entropy for a tensor product of the phase damping channel and the identity channel based upon the decreasing property of the relative entropy allows to prove the additivity conjecture for the minimal output entropy for the quantum depolarizing channel in any prime dimension and for the two-Pauli channel in the qubit case.
One of the central problems of quantum information theory [1] is the additivity conjecture for the channel capacity of quantum communication channels [2]–[4]. Recently Shor showed [5] that the validity of this conjecture for all quantum channels was equivalent to a whole series of (super)additivity properties for other important characteristics such as minimal output entropy and entanglementof formation. The present paper introduces the new property of strong additivity (additivity for channels with input constraints) with several equivalent formulations, establishes that it is satisfied for some nontrivial classes of channels and notes that the validity of the additivity conjecture for all channels implies strong additivity. Let H, H′ be finite-dimensional unitary spaces. By a quantum channel we mean a completely positive trace-preserving linear map Φ: B(H) → B(H′), where B(H) is the algebra of all operators on H. In particular, Φ generates an affine map of the convex set S(H) of states (density operators) on the space H into the set S(H′) of states on H′ [1]. The quantum analogue of Shannon’s theorem, established in [4], gives the following expression for the classical channel capacity:
We argue that a fundamental (conjectured) property of memoryless quantum channels, namely the strong superadditivity, is intimately related to the decreasing property of the quantum relative entropy. Using the latter we first give, for a wide class of input states, an estimation of the output entropy for phase damping channels and some Weyl quantum channels. Then we prove, without any input restriction, the strong superadditivity for several quantum channels, including depolarizing quantum channels, quantum-classical channels and quantum erasure channels.
We argue that a fundamental (conjectured) property of memoryless quantum channels, namely the strong superadditivity, is intimately related to the decreasing property of the quantum relative entropy. Using the latter we first give, for a wide class of input states, an estimation of the output entropy for phase damping channels and some Weyl quantum channels. Then we prove, without any input restriction, the strong superadditivity for several quantum channels, including depolarizing quantum channels, quantum-classical channels and quantum erasure 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.