TY - JOUR

T1 - Optimality of the pretty good measurement for port-based teleportation

AU - Leditzky, Felix

N1 - Funding Information:
I would like to thank Christian Majenz, Connor Paul-Paddock, and Michael Walter for valuable discussions and helpful feedback. I am also grateful to the anonymous referee for useful comments on an earlier version of this manuscript, and permission to reproduce their proof of Lemma . This research was partially funded through the Army Research Lab CDQI program.
Publisher Copyright:
© 2022, The Author(s), under exclusive licence to Springer Nature B.V.

PY - 2022/10

Y1 - 2022/10

N2 - 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. (Sci Rep 7(1):1–11, 2017) and Mozrzymas et al. (N J Phys 20(5):053006, 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.

AB - 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. (Sci Rep 7(1):1–11, 2017) and Mozrzymas et al. (N J Phys 20(5):053006, 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.

KW - Quantum information theory

KW - Quantum state discrimination

KW - Quantum teleportation

KW - Representation theory

KW - Semidefinite programming

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U2 - 10.1007/s11005-022-01592-5

DO - 10.1007/s11005-022-01592-5

M3 - Article

AN - SCOPUS:85139259522

SN - 0377-9017

VL - 112

JO - Letters in Mathematical Physics

JF - Letters in Mathematical Physics

IS - 5

M1 - 98

ER -