On Weyl channels being covariant with respect to the maximum commutative group of unitaries

Amosov, G. G.

doi:10.1063/1.2406054

Cited by 27 publications

(41 citation statements)

References 23 publications

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“…Therefore, King's argument does not generalize. Amosov [1] has given a new proof of additivity (33) for unital qubit channels. Because his argument is based on King's decomposition (40), it does not readily generalize to d > 2.…”

Section: Recall φmentioning

confidence: 99%

Pauli diagonal channels constant on axes

Nathanson¹,

Ruskai²

2007

J. Phys. A: Math. Theor.

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We define and study the properties of channels which are analogous to unital qubit channels in several ways. A full treatment can be given only when the dimension d = p m a prime power, in which case each of the d + 1 mutually unbiased bases (MUB) defines an axis. Along each axis the channel looks like a depolarizing channel, but the degree of depolarization depends on the axis. When d is not a prime power, some of our results still hold, particularly in the case of channels with one symmetry axis. We describe the convex structure of this class of channels and the subclass of entanglement breaking channels. We find new bound entangled states for d = 3.For these channels, we show that the multiplicativity conjecture for maximal output p-norm holds for p = 2. We also find channels with behavior not exhibited by unital qubit channels, including two pairs of orthogonal bases with equal output entropy in the absence of symmetry. This provides new numerical evidence for the additivity of minimal output entropy. * Partially supported by by the National Science Foundation under Grants DMS-0314228 and DMS-0604900 and by the National Security Agency and Advanced Research and Development Activity under Army Research Office contract number DAAD19-02-1-0065. IntroductionThe results presented here are motivated by the desire to find channels for dimension d > 2 whose properties are similar to those of the unital qubit channels, particularly with respect to optimal output purity. A channel is described by a completely positive, trace-preserving (CPT) map. The channels we construct are similar to unital qubit channels in the sense that their effect on a density matrix can be defined in terms of multipliers of components along different "axes" defined in terms of mutually unbiased bases (MUB). When all multipliers are positive, these channels are very much like unital qubit channels with positive multipliers. However, when some of the multipliers are negative the new channels can exhibit behavior not encountered for unital qubit channels.For a fixed orthonormal basis B = {|ψ k }, the quantum-classical (QC) channelprojects a density matrix ρ onto the corresponding diagonal matrix in this basis. A convex combination J t J Ψ QC J (ρ) of QC channels in a collection of orthonormal bases B J = {|ψ J k } is also a channel; in fact, it is an entanglement breaking (EB) channel. We consider channels which are a linear combination of the identity map I(ρ) = ρ and a convex combination of QC channels whose bases are mutually unbiased, i.e., satisfySuch channels can be written in the form 3The first condition ensures that Φ is trace-preserving (TP), and the pair that it is completely positive (CP), as will be shown in Section 2.It is well-known that C d can have at most d + 1 MUB and that this is always possible when d = p m is a prime power. We are primarily interested in channels of the form (3) when such a full set of d + 1 MUB exist. In that case, it is natural to generalize the Bloch sphere representation so that a density matri...

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Section: Recall φmentioning

confidence: 99%

Pauli diagonal channels constant on axes

Nathanson¹,

Ruskai²

2007

J. Phys. A: Math. Theor.

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“…The inequalities (20) and (21) hold simultaneously if and only if the following conditions are satisfied:…”

Section: Definitionmentioning

confidence: 99%

Generalizations of 2-Dimensional Diagonal Quantum Channels with Constant Frobenius Norm

Sergeev

2019

Reports on Mathematical Physics

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We introduce the set of quantum channels with constant Frobenius norm, the set of diagonal channels and the notion of equivalence of oneparameter families of channels. First, we show that all diagonal 2-dimensional channels with constant Frobenius norm are equivalent. Next, we generalize four one-parameter families of 2-dimensional diagonal channels with constant Frobenius norm to an arbitrary dimension n. Finally, we prove that the generalizations are not equivalent in any dimension n ≥ 3.

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“…for ρ, σ ∈ S(H) and for all (not only unital in general) quantum channels Φ. In [2,3] it was introduced the method based upon the property (3). Using this method the additivity in the known cases was proved without estimation of p-norms.…”

Section: Introductionmentioning

confidence: 99%

“…The same method allowed to prove the strong superadditivity for the quantum depolarizing channel [4], quantum-classical channels and quantum erasure channels [5]. Here we will improve the method introduced in [2,3]. It results in the strong estimation from below for the output entropy of a tensor product of the quantum phase-damping channel with an arbitrary quantum channel.…”

Section: Introductionmentioning

confidence: 99%