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On estimation of the entropy infimum for the output of the Weyl channels being covariant with respect to the maximum commutative group of unitaries

Grigori Gennadievich Amosov

Published May 19, 2006• 1 citation
OpenAlex

Abstract

We estimate the von Neumann entropy infimum for the output of the Weyl channels being covariant with respect to the maximum commutative group. In qubit case, such the class includes the quantum depolarizing channel and the "two-Pauli" channel as well. For the dimesion $d=2$ our approach allows to prove the additivity conjecture for the Holevo-Schumacher-Westmoreland bound. Our method is based upon using the decreasing property of the relative entropy. Earlier we had shown that the method gives the proof of the additivity conjecture for the quantum depolarizing channel in any dimension.

Related papers

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

Grigori Gennadievich Amosov

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.

arXiv:quant-ph/0605177•Journal of Mathematical Physics•Jan 1, 2007•29 citations

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

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Some new examples of quantum channels for which the infimum of the output entropy is additive under taking a tensor product of channels are given.

arXiv:quant-ph/0606040•Jun 6, 2006•0 citations

The additivity conjecture for the Weyl channels in the dimension d=3

Grigori Gennadievich Amosov

Following the method introduced in quant-ph/0408004 and quant-ph/0605177 we prove that the additivity conjecture for the Holevo-Schumacher-Westmoreland bound of the output of a quantum channel holds for the channels of the form $\\Phi (x)=\\sum \\limits _{m=0}^{2}(r_{m}W_{m,0}xW_{m,0}^{*}+p_{1}W_{m,1}xW_{m,1}^{*}+p_{2}W_{m,2}x W_{m,2}^{*})$. Here $x$ runs the set of states $\\sigma (H)$ in the Hilbert space $H, dimH=3,$ $W_{m,n}$ are the Weyl operators satisfying the relation $W_{m,n}W_{m',n'}=e^{\\frac {2\\pi i(m'n-mn')}{3}}W_{m+m',n+n'}, 0\\le m,n\\le 2,$ and $0\\le r_{m},p_{1},p_{2}\\le 1, \\sum \\limits _{m=0}^{2}r_{m}+3(p_{1}+p_{2})=1$. This class includes the quantum depolarizing channel as well as a number of the other channels being covariant with respect to the maximum commutative group of unitaries.

•Jun 6, 2006•0 citations

ENTROPIC CHARACTERIZATION OF QUANTUM OPERATIONS

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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.

International Journal of Quantum Information•Jun 1, 2011•27 citations

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We investigate decoherence induced by a quantum channel in terms of minimal output entropy and of map entropy. The latter is the von Neumann entropy of the Jamiolkowski 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 Φ1 acting on a finite dimensional system there exists a class of channels Φ2 sufficiently close to a unitary map such that additivity of minimal output entropy for Ψ1 x Ψ2 holds.

arXiv:1101.4105•Jan 21, 2011•1 citation

Remark on the additivity conjecture for the quantum depolarizing channel

Grigori Gennadievich Amosov

We consider bistochastic quantum channels generated by unitary representations of the discret group. The proof of the additivity conjecture for the quantum depolarizing channel $Φ$ based on the decreasing property of the relative entropy is given. We show that the additivity conjecture is true for the channel $Ξ=Ψ\circ Φ$, where $Ψ$ is the phase damping.

arXiv:quant-ph/0408004•Aug 1, 2004•4 citations
We estimate the von Neumann entropy infimum for the output of the Weyl channels being covariant with respect to the maximum commutative group. In qubit case, such the class includes the quantum depolarizing channel and the "two-Pauli" channel as well. For the dimesion d=2d=2d=2 our approach allows to prove the additivity conjecture for the Holevo-Schumacher-Westmoreland bound. Our method is based upon using the decreasing property of the relative entropy. Earlier we had shown that the method gives the proof of the additivity conjecture for the quantum depolarizing channel in any dimension.
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.
Following the method introduced in quant-ph/0408004 and quant-ph/0605177 we prove that the additivity conjecture for the Holevo-Schumacher-Westmoreland bound of the output of a quantum channel holds for the channels of the form Phi(x)=sumlimitsm=02(rmWm,0xWm,0∗+p1Wm,1xWm,1∗+p2Wm,2xWm,2∗)\\Phi (x)=\\sum \\limits _{m=0}^{2}(r_{m}W_{m,0}xW_{m,0}^{*}+p_{1}W_{m,1}xW_{m,1}^{*}+p_{2}W_{m,2}x W_{m,2}^{*})Phi(x)=sumlimitsm=02​(rm​Wm,0​xWm,0∗​+p1​Wm,1​xWm,1∗​+p2​Wm,2​xWm,2∗​). Here xxx runs the set of states sigma(H)\\sigma (H)sigma(H) in the Hilbert space H,dimH=3,H, dimH=3,H,dimH=3, Wm,nW_{m,n}Wm,n​ are the Weyl operators satisfying the relation Wm,nWm′,n′=efrac2pii(m′n−mn′)3Wm+m′,n+n′,0lem,nle2,W_{m,n}W_{m',n'}=e^{\\frac {2\\pi i(m'n-mn')}{3}}W_{m+m',n+n'}, 0\\le m,n\\le 2,Wm,n​Wm′,n′​=efrac2pii(m′n−mn′)3Wm+m′,n+n′​,0lem,nle2, and 0lerm,p1,p2le1,sumlimitsm=02rm+3(p1+p2)=10\\le r_{m},p_{1},p_{2}\\le 1, \\sum \\limits _{m=0}^{2}r_{m}+3(p_{1}+p_{2})=10lerm​,p1​,p2​le1,sumlimitsm=02​rm​+3(p1​+p2​)=1. This class includes the quantum depolarizing channel as well as a number of the other channels being covariant with respect to the maximum commutative group of unitaries.
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.
We investigate decoherence induced by a quantum channel in terms of minimal output entropy and of map entropy. The latter is the von Neumann entropy of the Jamiolkowski 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 Φ1 acting on a finite dimensional system there exists a class of channels Φ2 sufficiently close to a unitary map such that additivity of minimal output entropy for Ψ1 x Ψ2 holds.
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.