Quantifying the Performance of Bidirectional Quantum Teleportation*

Bidirectional teleportation is a fundamental protocol for exchanging quantum
information between two parties by means of a shared resource state and local
operations and classical communication (LOCC). In this work, we develop two
seemingly different ways of quantifying the simulation error of unideal
bidirectional teleportation by means of the normalized diamond distance and the
channel infidelity, and we prove that they are equivalent. By relaxing the set
of operations allowed from LOCC to those that completely preserve the
positivity of the partial transpose, we obtain semi-definite programming lower
bounds on the simulation error of unideal bidirectional teleportation. We then evaluate the
performance of some schemes for bidirectional teleportation due to [Kiktenko et
al., Phys. Rev. A 93, 062305 (2016)] and find that they are suboptimal and do
not go beyond the aforementioned classical limit for bidirectional
teleportation. We offer a scheme alternative to theirs that is provably
optimal. Finally, we establish semi-definite programming lower bounds on the simulation error for
this task.