Optimality of the pretty good measurement for port-based teleportation

Abstract

Port-based teleportation (PBT) is a protocol in which Alice teleports an unknown quantum state to Bob using measurements on a shared entangled multipartite state called the port state and forward classical communication. In this paper, we give an explicit proof that the so-called pretty good measurement, or square-root measurement, is optimal for the PBT protocol with independent copies of maximally entangled states as the port state. We then show that the very same measurement remains optimal even when the port state is optimized to yield the best possible PBT protocol. Hence, there is one particular pretty good measurement achieving the optimal performance in both cases. The following well-known facts are key ingredients in the proofs of these results: (i) the natural symmetries of PBT, leading to a description in terms of representation-theoretic data; (ii) the operational equivalence of PBT with certain state discrimination problems, which allows us to employ duality of the associated semidefinite programs. Along the way, we rederive the representation-theoretic formulas for the performance of PBT protocols proved in [Studziński et al., 2017] and [Mozrzymas et al., 2018] using only standard techniques from the representation theory of the unitary and symmetric groups. Providing a simplified derivation of these beautiful formulas is one of the main goals of this paper.

arXiv comment19 pages, 1 figure, comments welcome! v2: accepted for publication in Letters in Mathematical Physics (main edit: corrected statement of Lemma 1 with proof in appendix)

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• Linear programming with unitary-equivariant constraints
• The Asymmetric Quantum Cloning Region
• Port-Based State Preparation and Applications
• The pretty bad measurement
• From port-based teleportation to Frobenius reciprocity theorem: partially reduced irreducible representations and their applications
• Asymptotic teleportation schemes bridging between standard and port-based teleportation
• Port-based telecloning of an unknown quantum state
• A resource theory of asynchronous quantum information processing
• On the distinguishability of geometrically uniform quantum states