# Minimal sufficient positive-operator valued measure on a separable Hilbert space

## Abstract

We introduce a concept of a minimal sufficient positive-operator valued measure (POVM), which is the least redundant POVM among the POVMs that have the equivalent information about the measured quantum system. Assuming the system Hilbert space to be separable, we show that for a given POVM, a sufficient statistic called a Lehmann-Scheffé-Bahadur statistic induces a minimal sufficient POVM. We also show that every POVM has an equivalent minimal sufficient POVM and that such a minimal sufficient POVM is unique up to relabeling neglecting null sets. We apply these results to discrete POVMs and information conservation conditions proposed by the author.

- Authors:

- Department of Nuclear Engineering, Kyoto University, 6158540 Kyoto (Japan)

- Publication Date:

- OSTI Identifier:
- 22479542

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Mathematical Physics; Journal Volume: 56; Journal Issue: 10; Other Information: (c) 2015 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; FUZZY LOGIC; HILBERT SPACE; QUANTUM SYSTEMS; STATISTICS

### Citation Formats

```
Kuramochi, Yui, E-mail: kuramochi.yui.22c@st.kyoto-u.ac.jp.
```*Minimal sufficient positive-operator valued measure on a separable Hilbert space*. United States: N. p., 2015.
Web. doi:10.1063/1.4934235.

```
Kuramochi, Yui, E-mail: kuramochi.yui.22c@st.kyoto-u.ac.jp.
```*Minimal sufficient positive-operator valued measure on a separable Hilbert space*. United States. doi:10.1063/1.4934235.

```
Kuramochi, Yui, E-mail: kuramochi.yui.22c@st.kyoto-u.ac.jp. Thu .
"Minimal sufficient positive-operator valued measure on a separable Hilbert space". United States. doi:10.1063/1.4934235.
```

```
@article{osti_22479542,
```

title = {Minimal sufficient positive-operator valued measure on a separable Hilbert space},

author = {Kuramochi, Yui, E-mail: kuramochi.yui.22c@st.kyoto-u.ac.jp},

abstractNote = {We introduce a concept of a minimal sufficient positive-operator valued measure (POVM), which is the least redundant POVM among the POVMs that have the equivalent information about the measured quantum system. Assuming the system Hilbert space to be separable, we show that for a given POVM, a sufficient statistic called a Lehmann-Scheffé-Bahadur statistic induces a minimal sufficient POVM. We also show that every POVM has an equivalent minimal sufficient POVM and that such a minimal sufficient POVM is unique up to relabeling neglecting null sets. We apply these results to discrete POVMs and information conservation conditions proposed by the author.},

doi = {10.1063/1.4934235},

journal = {Journal of Mathematical Physics},

number = 10,

volume = 56,

place = {United States},

year = {Thu Oct 15 00:00:00 EDT 2015},

month = {Thu Oct 15 00:00:00 EDT 2015}

}

DOI: 10.1063/1.4934235

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