A quantum depolarizing channel is a model for quantum noise in quantum systems. The d {\displaystyle d} -dimensional depolarizing channel can be viewed as a completely positive trace-preserving map Δ λ {\displaystyle \Delta _{\lambda }} , depending on one parameter λ {\displaystyle \lambda } , which maps a state ρ {\displaystyle \rho } onto a linear combination of itself and the maximally mixed state,
Δ λ ( ρ ) = ( 1 − λ ) ρ + λ d ( t r ( ρ ) ) I = ( 1 − λ ) ρ + λ d I . {\displaystyle \Delta _{\lambda }(\rho )=(1-\lambda )\rho +{\frac {\lambda }{d}}(\mathrm {tr} (\rho ))I=(1-\lambda )\rho +{\frac {\lambda }{d}}I.}
The condition of complete positivity requires λ {\displaystyle \lambda } to satisfy the bounds
0 ≤ λ ≤ 1 + 1 d 2 − 1 . {\displaystyle 0\leq \lambda \leq 1+{\frac {1}{d^{2}-1}}.}
Qubit channel The single qubit depolarizing channel has operator-sum representation on a density matrix ρ {\displaystyle \rho } given by
Δ λ ( ρ ) = ∑ i = 0 3 K i ρ K i † , {\displaystyle \Delta _{\lambda }(\rho )=\sum _{i=0}^{3}K_{i}\rho K_{i}^{\dagger },}
where K i {\displaystyle K_{i}} are the Kraus operators given by
K 0 = 1 − 3 λ 4 I , K 1 = λ 4 X , K 2 = λ 4 Y , K 3 = λ 4 Z {\displaystyle K_{0}={\sqrt {1-{\frac {3\lambda }{4}}}}I,K_{1}={\sqrt {\frac {\lambda }{4}}}X,K_{2}={\sqrt {\frac {\lambda }{4}}}Y,K_{3}={\sqrt {\frac {\lambda }{4}}}Z}
and { I , X , Y , Z } {\displaystyle \{I,X,Y,Z\}} are the Pauli matrices. The trace preserving condition is satisfied by the fact that ∑ i K i † K i = I . {\displaystyle \sum _{i}K_{i}^{\dagger }K_{i}=I.}
Geometrically the depolarizing channel Δ λ {\displaystyle \Delta _{\lambda }} can be interpreted as a uniform contraction of the Bloch sphere, parameterized by λ {\displaystyle \lambda } . In the case where λ = 1 {\displaystyle \lambda =1} the channel returns the maximally-mixed state for any input state ρ {\displaystyle \rho } , which corresponds of the complete contraction of the Bloch-sphere down to the single-point I 2 {\displaystyle {\frac {I}{2}}} given by the origin.
Classical capacity The HSW theorem states that the classical capacity of a quantum channel Ψ {\displaystyle \Psi } can be characterized as its regularized Holevo information:
lim n → ∞ 1 n χ ( Ψ ⊗ n ) . {\displaystyle \lim _{n\to \infty }{\frac {1}{n}}\chi (\Psi ^{\otimes n}).}
This quantity is difficult to compute and this reflects our ignorance on quantum channels. However, if the Holevo information is additive for a channel Ψ {\displaystyle \Psi } , i.e.,
χ ( Ψ ⊗ Ψ ) = χ ( Ψ ) + χ ( Ψ ) . {\displaystyle \chi (\Psi \otimes \Psi )=\chi (\Psi )+\chi (\Psi ).}
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