in order to obtain a hierarchy of semidefinite programming bounds on various regularized quantities. As applications, we give a general procedure to give efficient bounds on the regularized Umegaki channel divergence as well as the classical capacity and two-way assisted quantum capacity of quantum channels. In particular, we obtain slight improvements for the capacity of the amplitude damping channel. We also prove that for fixed input and output dimensions, the regularized sandwiched Rényi divergence between any two quantum channels can be approximated up to an

\epsilon

accuracy in time that is polynomial in

1/\epsilon

DOI: 10.1109/tit.2022.3182101

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\mathrm {D}^{\#}

\epsilon

accuracy in time that is polynomial in

1/\epsilon

\mathrm{D}^{\#}

in order to obtain a hierarchy of semidefinite programming bounds on various regularized quantities. As an application, we describe a general procedure to give efficient bounds on the regularized Umegaki channel divergence and on the classical capacity of quantum channels, and in particular we obtain slight improvements for the capacity of the amplitude damping channel.

ε

\mathrm{poly}(1/ε, \text{input dimension})

time.

in

\mathrm {poly}(1/\epsilon, \text {input dimension})

time, compared to the

\exp (1/\epsilon, \text {input dimension})$running time required for direct computation.