Quantum teleportation is one of the earliest and most widely used primitives in Quantum Information Science which performs an arbitrary quantum state transfer between two spatially separated systems. It involves pre-sharing an entangled resource state and consists of three simple stages. The first stage involves a joint measurement of the teleported subsystem together with the share of the resource state on the sender’s side. In the second step, a classical measurement outcome is communicated to the receiver. The last step consists of applying a requisite correction operation which recovers the transmitted quantum state.

Port-based teleportation (PBT) is a unique set of teleportation protocols in that they do not require unitary correction. We study PBT protocols and fully characterize their performance for an arbitrary dimensions and number of ports. We find optimal probability of success and the fidelity of teleportation for all probabilistic and deterministic PBT schemes. In the latter case, surprisingly, the answer depends only on a largest eigenvalue of a certain easy to construct matrix which encodes the relationship between a set of Young diagrams and emerges as the the optimal solution to the relevant semidefinite program. To derive our results, we develop new mathematical tools to study the symmetries of the operators that arise in PBT protocols and belong to the algebra of partially transposed permutation operators. These tools can be used to characterize quantum systems with partial symmetries. Quantum states occurring in the PBT protocol are one such example. Systems with partial symmetries are widespread but in contrast to their permutational-invariant counterparts very little is known about how to efficiently estimate their properties.