  We derive several efficiently computable converse bounds for quantum
communication over quantum channels in both the one-shot and asymptotic regime.
First, we derive one-shot semidefinite programming (SDP) converse bounds on the
amount of quantum information that can be transmitted over a single use of a
quantum channel, which improve the previous bound from [Tomamichel/Berta/Renes,
Nat. Commun. 7, 2016]. As applications, we study quantum communication over
depolarizing channels and amplitude damping channels with finite resources.
Second, we find an SDP strong converse bound for the quantum capacity of an
arbitrary quantum channel, which means the fidelity of any sequence of codes
with a rate exceeding this bound will vanish exponentially fast as the number
of channel uses increases. Furthermore, we prove that the SDP strong converse
bound improves the partial transposition bound introduced by Holevo and Werner.
Third, we prove that this SDP strong converse bound is equal to the so-called
max-Rains information, which is an analog to the Rains information introduced
in [Tomamichel/Wilde/Winter, IEEE Trans. Inf. Theory 63:715, 2017]. Our SDP
strong converse bound is weaker than the Rains information, but it is
efficiently computable for general quantum channels.
