PMC

Approximation of the non-asymptotic achievable rate region of a qubit dephasing channel with γ=0.1 (see Theorem 1). All numerical results are evaluated using the binary logarithm, that is, log≡log₂. (a) Boundary of the achievable region for fixed n with different values (second order approximation). (b) Boundary of the achievable region for fixed infidelity ɛ=5% (third order approximation) in . (c) Comparison of strict bounds with third order approximation for fixed ɛ=5%.

Left and mid: Comparison of lower bounds of - and -strategy resp. in setting of Fig.  (right)—red: Theorem , black: Corollary  maximized over 1000 random choices of the -parameters, blue: (Hautphenne and Latouche , Theorem 3), and : approximation via simulation. On the right: Comparison of upper bounds—black: improved norm bound from Remark  with respect to , blue: (Hautphenne and Latouche , Theorem 2), : approximation via simulation (color figure online)

Comparison between different bounds for the approximation error. Left: The Rayleigh–Ritz bound asymptotes to infinity at long and short lag times. Center: The true approximation error stabilizes at long lag times. Right: The improved bound presented in Theorem 3.4 is asymptotically sharp at long lag times.

Difference between approximations and exact one-sided probabilities for n = 23 and a ∈ {1, 2, …, 23}. Negative values indicate the exact probability exceeds the approximation. DKWM6a corresponds to the DKWM bound with the constant e/2, discussed in Theorem 6(a). DKWM6b corresponds to the DKWM bound with the constant 2.16863/2, discussed in Theorem 6(b).