Draft:Quantum Chernoff Distance
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Submission declined on 7 November 2023 by WikiOriginal-9 (talk).
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- Comment: Not enough independent, significant coverage. WikiOriginal-9 (talk) 17:51, 7 November 2023 (UTC)
The quantum Chernoff distance is a measure of similarity between quantum states, generalizing the classical Chernoff distance used for probability distributions. This stems from the fact that probability distributions can be viewed as particular instances of quantum states, specifically as quantum states with diagonal density matrices. Both metrics have a clear interpretation as the optimal decay rate of error probability in classical and quantum hypothesis testing.
Interpretation[edit]
The problem of quantum hypothesis testing consists of identifying an unknown state that belongs to a discrete set of quantum states , given copies of the state. Each hypothesis or state has a prior probability denoted by . For the case of binary quantum hypothesis testing, the optimal probability of error is obtained using the Helstrom measurement, which yields the following probability of error[1]
Finally, this formula can be used to define the Chernoff distance as
Definition[edit]
A close-form expression for the quantum Chernoff distance is given by[2] [3]
which as mention before, generalizes the classical concept for distributions,
Properties[edit]
When at least one of the states is pure, i.e., its rank is equal to 1, then the Chernoff distance becomes .
This metric is related to other commonly used metrics for quantum states[4], such as the trace distance or Ulhmann fidelity. Specifically,
where denotes the Uhlmann fidelity. Similarly,
References[edit]
- ^ Helstrom, Carl W. (1969). "Quantum detection and estimation theory". Journal of Statistical Physics. 1 (2): 231–252. Bibcode:1969JSP.....1..231H. doi:10.1007/BF01007479. hdl:2060/19690016211. S2CID 12758217.
- ^ Audenaert, K. M. R.; Calsamiglia, J.; Muñoz-Tapia, R.; Bagan, E.; Masanes, Ll.; Acin, A.; Verstraete, F. (17 April 2007). "Discriminating States: The Quantum Chernoff Bound". Physical Review Letters. 98 (16): 160501. arXiv:quant-ph/0610027. Bibcode:2007PhRvL..98p0501A. doi:10.1103/PhysRevLett.98.160501. PMID 17501405. S2CID 897578.
- ^ Nussbaum, Michael; Szkoła, Arleta (1 April 2009). "The Chernoff lower bound for symmetric quantum hypothesis testing". The Annals of Statistics. 37 (2). doi:10.1214/08-AOS593. S2CID 15418499.
- ^ Audenaert, K. M. R.; Nussbaum, M.; Szkola, A.; Verstraete, F. (2007). "Asymptotic Error Rates in Quantum Hypothesis Testing". Communications in Mathematical Physics. 279 (1): 251–283. arXiv:0708.4282. doi:10.1007/s00220-008-0417-5. S2CID 17526245.
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