Minimax quantum state estimation under Bregman divergence
We investigate minimax estimators for quantum state tomography under general Bregman divergences. First, generalizing the work of Koyama et al. [Entropy 19, 618 (2017)] for relative entropy, we find that given any estimator for a quantum state, there always exists a sequence of Bayes estimators that...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften
2019-03-01
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| Series: | Quantum |
| Online Access: | https://quantum-journal.org/papers/q-2019-03-04-126/pdf/ |
| _version_ | 1819087332045750272 |
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| author | Maria Quadeer Marco Tomamichel Christopher Ferrie |
| author_facet | Maria Quadeer Marco Tomamichel Christopher Ferrie |
| author_sort | Maria Quadeer |
| collection | DOAJ |
| description | We investigate minimax estimators for quantum state tomography under general Bregman divergences. First, generalizing the work of Koyama et al. [Entropy 19, 618 (2017)] for relative entropy, we find that given any estimator for a quantum state, there always exists a sequence of Bayes estimators that asymptotically perform at least as well as the given estimator, on any state. Second, we show that there always exists a sequence of priors for which the corresponding sequence of Bayes estimators is asymptotically minimax (i.e. it minimizes the worst-case risk). Third, by re-formulating Holevo's theorem for the covariant state estimation problem in terms of estimators, we find that there exists a covariant measurement that is, in fact, minimax (i.e. it minimizes the worst-case risk). Moreover, we find that a measurement that is covariant only under a unitary 2-design is also minimax. Lastly, in an attempt to understand the problem of finding minimax measurements for general state estimation, we study the qubit case in detail and find that every spherical 2-design is a minimax measurement. |
| first_indexed | 2024-12-21T21:34:27Z |
| format | Article |
| id | doaj.art-891daab3608f4c4e918927dcd4c7c4bf |
| institution | Directory Open Access Journal |
| issn | 2521-327X |
| language | English |
| last_indexed | 2024-12-21T21:34:27Z |
| publishDate | 2019-03-01 |
| publisher | Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften |
| record_format | Article |
| series | Quantum |
| spelling | doaj.art-891daab3608f4c4e918927dcd4c7c4bf2022-12-21T18:49:33ZengVerein zur Förderung des Open Access Publizierens in den QuantenwissenschaftenQuantum2521-327X2019-03-01312610.22331/q-2019-03-04-12610.22331/q-2019-03-04-126Minimax quantum state estimation under Bregman divergenceMaria QuadeerMarco TomamichelChristopher FerrieWe investigate minimax estimators for quantum state tomography under general Bregman divergences. First, generalizing the work of Koyama et al. [Entropy 19, 618 (2017)] for relative entropy, we find that given any estimator for a quantum state, there always exists a sequence of Bayes estimators that asymptotically perform at least as well as the given estimator, on any state. Second, we show that there always exists a sequence of priors for which the corresponding sequence of Bayes estimators is asymptotically minimax (i.e. it minimizes the worst-case risk). Third, by re-formulating Holevo's theorem for the covariant state estimation problem in terms of estimators, we find that there exists a covariant measurement that is, in fact, minimax (i.e. it minimizes the worst-case risk). Moreover, we find that a measurement that is covariant only under a unitary 2-design is also minimax. Lastly, in an attempt to understand the problem of finding minimax measurements for general state estimation, we study the qubit case in detail and find that every spherical 2-design is a minimax measurement.https://quantum-journal.org/papers/q-2019-03-04-126/pdf/ |
| spellingShingle | Maria Quadeer Marco Tomamichel Christopher Ferrie Minimax quantum state estimation under Bregman divergence Quantum |
| title | Minimax quantum state estimation under Bregman divergence |
| title_full | Minimax quantum state estimation under Bregman divergence |
| title_fullStr | Minimax quantum state estimation under Bregman divergence |
| title_full_unstemmed | Minimax quantum state estimation under Bregman divergence |
| title_short | Minimax quantum state estimation under Bregman divergence |
| title_sort | minimax quantum state estimation under bregman divergence |
| url | https://quantum-journal.org/papers/q-2019-03-04-126/pdf/ |
| work_keys_str_mv | AT mariaquadeer minimaxquantumstateestimationunderbregmandivergence AT marcotomamichel minimaxquantumstateestimationunderbregmandivergence AT christopherferrie minimaxquantumstateestimationunderbregmandivergence |