TY - JOUR

T1 - Discrimination of Quantum States Under Locality Constraints in the Many-Copy Setting

AU - Cheng, Hao Chung

AU - Winter, Andreas

AU - Yu, Nengkun

N1 - Publisher Copyright:
© 2023, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.

PY - 2023/11

Y1 - 2023/11

N2 - We study quantum hypothesis testing between orthogonal states under restricted local measurements in the many-copy scenario. For testing arbitrary multipartite entangled pure state against its orthogonal complement state via the local operation and classical communication (LOCC) operation, we prove that the optimal average error probability always decays exponentially in the number of copies. Second, we provide a sufficient condition for the LOCC operations to achieve the same performance as the positive-partial-transpose (PPT) operations. We further show that testing a maximally entangled state against its orthogonal complement and testing extremal Werner states both fulfill the above-mentioned condition. Hence, we determine the explicit expressions for the optimal average error probability, the optimal trade-off between the type-I and type-II errors, and the associated Chernoff, Stein, Hoeffding, and strong converse exponents. Then, we show an infinite asymptotic separation between the separable (SEP) and PPT operations by providing a pair of states constructed from an unextendible product basis (UPB). The quantum states can be distinguished perfectly by PPT operations, while the optimal error probability, with SEP operations, admits an exponential lower bound. On the technical side, we prove this result by providing a quantitative version of the well-known statement that the tensor product of UPBs is a UPB.

AB - We study quantum hypothesis testing between orthogonal states under restricted local measurements in the many-copy scenario. For testing arbitrary multipartite entangled pure state against its orthogonal complement state via the local operation and classical communication (LOCC) operation, we prove that the optimal average error probability always decays exponentially in the number of copies. Second, we provide a sufficient condition for the LOCC operations to achieve the same performance as the positive-partial-transpose (PPT) operations. We further show that testing a maximally entangled state against its orthogonal complement and testing extremal Werner states both fulfill the above-mentioned condition. Hence, we determine the explicit expressions for the optimal average error probability, the optimal trade-off between the type-I and type-II errors, and the associated Chernoff, Stein, Hoeffding, and strong converse exponents. Then, we show an infinite asymptotic separation between the separable (SEP) and PPT operations by providing a pair of states constructed from an unextendible product basis (UPB). The quantum states can be distinguished perfectly by PPT operations, while the optimal error probability, with SEP operations, admits an exponential lower bound. On the technical side, we prove this result by providing a quantitative version of the well-known statement that the tensor product of UPBs is a UPB.

UR - http://www.scopus.com/inward/record.url?scp=85171261338&partnerID=8YFLogxK

U2 - 10.1007/s00220-023-04836-0

DO - 10.1007/s00220-023-04836-0

M3 - Article

AN - SCOPUS:85171261338

SN - 0010-3616

VL - 404

SP - 151

EP - 183

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

IS - 1

ER -