### Abstract

In the problem of quantum state discrimination, one has to determine by measurements the state of a quantum system, based on the a priori side information that the true state is one of the two given completely known states, ρ or σ. In general, it is not possible to decide the identity of the true state with certaintythe optimal measurement strategy depends on whether the two possible errors (mistaking ρ for σ, or the other way around) are treated as of equal importance or not. Results on the quantum Chernoff Hoeffding bounds the quantum Stein's lemma show that, if several copies of the system are available then the optimal error probabilities decay exponentially in the number of copiesthe decay rate is given by a certain statistical distance between ρ σ (the Chernoff distance, the Hoeffding distancesthe relative entropy, respectively). While these results provide a complete solution to the asymptotic problem, they are not completely satisfying from a practical point of view. Indeed, in realistic scenarios one has access only to finitely many copies of a systemtherefore it is desirable to have bounds on the error probabilities for finite sample size. In this paper we provide finite-size bounds on the so-called Stein errors, the Chernoff errors, the Hoeffding errorsthe mixed error probabilities related to the Chernoff the Hoeffding errors.

Original language | English |
---|---|

Article number | 122205 |

Journal | Journal of Mathematical Physics |

Volume | 53 |

Issue number | 12 |

DOIs | |

Publication status | Published - Dec 19 2012 |

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### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Journal of Mathematical Physics*,

*53*(12), [122205]. https://doi.org/10.1063/1.4768252

**Quantum state discrimination bounds for finite sample size.** / Audenaert, Koenraad M R; Mosonyi, M.; Verstraete, Frank.

Research output: Contribution to journal › Article

*Journal of Mathematical Physics*, vol. 53, no. 12, 122205. https://doi.org/10.1063/1.4768252

}

TY - JOUR

T1 - Quantum state discrimination bounds for finite sample size

AU - Audenaert, Koenraad M R

AU - Mosonyi, M.

AU - Verstraete, Frank

PY - 2012/12/19

Y1 - 2012/12/19

N2 - In the problem of quantum state discrimination, one has to determine by measurements the state of a quantum system, based on the a priori side information that the true state is one of the two given completely known states, ρ or σ. In general, it is not possible to decide the identity of the true state with certaintythe optimal measurement strategy depends on whether the two possible errors (mistaking ρ for σ, or the other way around) are treated as of equal importance or not. Results on the quantum Chernoff Hoeffding bounds the quantum Stein's lemma show that, if several copies of the system are available then the optimal error probabilities decay exponentially in the number of copiesthe decay rate is given by a certain statistical distance between ρ σ (the Chernoff distance, the Hoeffding distancesthe relative entropy, respectively). While these results provide a complete solution to the asymptotic problem, they are not completely satisfying from a practical point of view. Indeed, in realistic scenarios one has access only to finitely many copies of a systemtherefore it is desirable to have bounds on the error probabilities for finite sample size. In this paper we provide finite-size bounds on the so-called Stein errors, the Chernoff errors, the Hoeffding errorsthe mixed error probabilities related to the Chernoff the Hoeffding errors.

AB - In the problem of quantum state discrimination, one has to determine by measurements the state of a quantum system, based on the a priori side information that the true state is one of the two given completely known states, ρ or σ. In general, it is not possible to decide the identity of the true state with certaintythe optimal measurement strategy depends on whether the two possible errors (mistaking ρ for σ, or the other way around) are treated as of equal importance or not. Results on the quantum Chernoff Hoeffding bounds the quantum Stein's lemma show that, if several copies of the system are available then the optimal error probabilities decay exponentially in the number of copiesthe decay rate is given by a certain statistical distance between ρ σ (the Chernoff distance, the Hoeffding distancesthe relative entropy, respectively). While these results provide a complete solution to the asymptotic problem, they are not completely satisfying from a practical point of view. Indeed, in realistic scenarios one has access only to finitely many copies of a systemtherefore it is desirable to have bounds on the error probabilities for finite sample size. In this paper we provide finite-size bounds on the so-called Stein errors, the Chernoff errors, the Hoeffding errorsthe mixed error probabilities related to the Chernoff the Hoeffding errors.

UR - http://www.scopus.com/inward/record.url?scp=84871858487&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84871858487&partnerID=8YFLogxK

U2 - 10.1063/1.4768252

DO - 10.1063/1.4768252

M3 - Article

AN - SCOPUS:84871858487

VL - 53

JO - Journal of Mathematical Physics

JF - Journal of Mathematical Physics

SN - 0022-2488

IS - 12

M1 - 122205

ER -