# On the Classical Capacity of General Quantum Gaussian Measurement

## Abstract

**:**

## 1. Introduction

## 2. The Measurement Channel and Its Classical Capacity

**Lemma**

**1.**

**Proof.**

## 3. Gaussian Maximizers for Multimode Bosonic Gaussian Observable

**Lemma**

**2.**

**Proof.**

**Lemma**

**3.**

**Proof.**

**Theorem**

**1.**

**Proof.**

**Remark**

**1.**

**Remark**

**2.**

**Hypothesis of Gaussian Maximizers**

**(HGM).**

## 4. Gaussian Measurements in One Mode

L: HGM open | C: HGM valid [5] | R: HGM open |

${\beta}_{q}\le {\beta}_{p};E<E\left({\beta}_{p},{\beta}_{q}\right)$ | $E\ge E\left({\beta}_{p},{\beta}_{q}\right)\vee E\left({\beta}_{q},{\beta}_{p}\right)$ | ${\beta}_{p}\le {\beta}_{q};E<E\left({\beta}_{q},{\beta}_{p}\right)$ |

$log\left(\frac{\sqrt{1+8E{\beta}_{q}+4{\beta}_{q}^{2}}-1}{2{\beta}_{q}}\right)$ | $log\left(\frac{E+\left({\beta}_{q}+{\beta}_{p}\right)/2}{\sqrt{{\beta}_{q}{\beta}_{p}}+1/2}\right)$ | $log\left(\frac{\sqrt{1+8E{\beta}_{p}+4{\beta}_{p}^{2}}-1}{2{\beta}_{p}}\right)$ |

## 5. The Dual Problem: Accessible Information

## 6. Discussion

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Case L in Tables 1 and 2

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range | L: $\frac{1}{2}\sqrt{\frac{{\beta}_{q}}{{\beta}_{p}}}<\frac{1}{4{\alpha}_{p}}$ | C:$\frac{1}{4{\alpha}_{p}}\le \frac{1}{2}\sqrt{\frac{{\beta}_{q}}{{\beta}_{p}}}\le {\alpha}_{q}$ | R: ${\alpha}_{q}<\frac{1}{2}\sqrt{\frac{{\beta}_{q}}{{\beta}_{p}}}$ |

HGM | open | valid | open |

${\delta}_{opt}$ | $1/\left(4{\alpha}_{p}\right)$ | $\frac{1}{2}\sqrt{\frac{{\beta}_{q}}{{\beta}_{p}}}$ | ${\alpha}_{q}$ |

${e}_{M}\left({\rho}_{\alpha}\right)-c$ | $\frac{1}{2}log\left[\left(\frac{1}{4{\alpha}_{p}}+{\beta}_{q}\right)\right.$ | $log\left(\sqrt{{\beta}_{q}{\beta}_{p}}+1/2\right)$ | $\frac{1}{2}log\left[\left(\frac{1}{4{\alpha}_{q}}+{\beta}_{p}\right)\right.$ |

$\times \left({\alpha}_{p}+{\beta}_{p}\right)]$ | $\times \left({\alpha}_{q}+{\beta}_{q}\right)]$ | ||

$C(M;\alpha )$ | $\frac{1}{2}log\frac{{\alpha}_{q}+{\beta}_{q}}{\frac{1}{4{\alpha}_{p}}+{\beta}_{q}}$ | $\frac{1}{2}log\frac{\left({\alpha}_{q}+{\beta}_{q}\right)\left({\alpha}_{p}+{\beta}_{p}\right)}{{\left(\sqrt{{\beta}_{q}{\beta}_{p}}+1/2\right)}^{2}}$ | $\frac{1}{2}log\frac{{\alpha}_{p}+{\beta}_{p}}{\frac{1}{4{\alpha}_{q}}+{\beta}_{p}}$ |

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Holevo, A.
On the Classical Capacity of General Quantum Gaussian Measurement. *Entropy* **2021**, *23*, 377.
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Holevo A.
On the Classical Capacity of General Quantum Gaussian Measurement. *Entropy*. 2021; 23(3):377.
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2021. "On the Classical Capacity of General Quantum Gaussian Measurement" *Entropy* 23, no. 3: 377.
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