# A Fisher Information-Based Incompatibility Criterion for Quantum Channels

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## Abstract

**:**

## 1. Introduction

## 2. Classical and Quantum Fisher Information

## 3. Compatibility of Quantum Channels

- complete positivity: for any dimension $k\ge 1$, the linear map ${\mathrm{id}}_{k}\otimes \Phi :\mathcal{L}({\mathbb{C}}^{k}\otimes {H}_{d})\to \mathcal{L}({\mathbb{C}}^{k}\otimes {H}_{D})$ is a positive operator;
- trace-preservation: for all operators $X\in \mathcal{L}\left({H}_{d}\right)$, $Tr\Phi \left(X\right)=TrX$.

**Definition**

**1.**

**Definition**

**2.**

**Φ**as

**Proposition**

**1.**

## 4. Channel Incompatibility via POVM Incompatibility

- positivity: the operators ${A}_{1},\dots ,{A}_{k}\in \mathcal{L}\left({H}_{d}\right)$ are positive semidefinite;
- normalization: ${\sum}_{i=1}^{k}{A}_{i}={I}_{d}$.

**Definition**

**3.**

**Proposition**

**2.**

**Remark**

**1.**

**Remark**

**2.**

**Lemma**

**1.**

**Proof.**

**Conjecture**

**1.**

**Theorem**

**1.**

**Proof.**

**Remark**

**3.**

- channel compatibility: the joint channel has a Choi matrix of size ${d}^{N+1}$
- incompatibility criterion from Theorem 1: the variable H has size ${d}^{2}$.

**Proposition**

**3.**

**Proof.**

**Lemma**

**2.**

**Lemma**

**3.**

## 5. Incompatibility of Two Schur Channels

**Lemma**

**4.**

**Proof.**

**Lemma**

**5.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Remark**

**4.**

## 6. Channel Assemblages

**Definition**

**4.**

**Φ**is called:

- $(N,K)$-compatible if all K-subsets of
**Φ**are compatible. - $(N,K)$-incompatible if at least one K-subset of
**Φ**is incompatible. - $(N,K)$-strong incompatible if all K-subsets of
**Φ**are incompatible. - $(N,K+1)$-genuinely incompatible if it is $(N,K)$-compatible and $(N,K+1)$-incompatible.
- $(N,K+1)$-genuinely strong incompatible if it is $(N,K)$-compatible and $(N,K+1)$-strong incompatible.

**Theorem**

**3.**

**Φ**is $(N,K)$-incompatible. Moreover, if for all K-subsets $S\subseteq \left[N\right]$, there exists a K-tuple of bases ${\mathbf{e}}_{S}$ such that $\mathrm{val}(\mathsf{\Phi},S,{\mathbf{e}}_{S})>d$, the assemblage Φ is $(N,K)$-strong incompatible.

## 7. Assemblages of Depolarizing Channels

**Proposition**

**4.**

**Proof.**

**Corollary**

**1.**

**Corollary**

**2.**

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**The Fisher information-based incompatibility criterion for Schur channels. In the left panel, we consider two noisy copies of the Schur channel corresponding to $B=\left[\begin{array}{cc}1& 1/2\\ 1/2& 1\end{array}\right]$. In the right panel, we consider noisy versions of ${\Sigma}_{B}$ and ${\Sigma}_{C}$, where $C=\left[\begin{array}{cc}1& \sqrt{3/4}\\ \sqrt{3/4}& 1\end{array}\right]$. Shaded regions correspond to the conditions from (16), while the red dots correspond to the maximally compatible channels in the respective directions.

**Figure 2.**Comparing the incompatibility criterion from Proposition 4 (filled region) with the incompatibility thresholds from Equation (5) (dashed curves) for different values of d: $d=2$ (red curve), $d=5$ (brown curve), $d=20$ (black curve).

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**MDPI and ACS Style**

Zhang, Q.-H.; Nechita, I.
A Fisher Information-Based Incompatibility Criterion for Quantum Channels. *Entropy* **2022**, *24*, 805.
https://doi.org/10.3390/e24060805

**AMA Style**

Zhang Q-H, Nechita I.
A Fisher Information-Based Incompatibility Criterion for Quantum Channels. *Entropy*. 2022; 24(6):805.
https://doi.org/10.3390/e24060805

**Chicago/Turabian Style**

Zhang, Qing-Hua, and Ion Nechita.
2022. "A Fisher Information-Based Incompatibility Criterion for Quantum Channels" *Entropy* 24, no. 6: 805.
https://doi.org/10.3390/e24060805