bibkey: zhangnechita2022fisherchannelincompatibility
authors: Qing-Hua Zhang; Ion Nechita
year: 2022
title: "A Fisher information-based incompatibility criterion for quantum channels"
doi: 10.3390/e24060805
url: https://arxiv.org/abs/2204.09963v1
claim: "The paper defines compatibility of quantum channels through a joint channel with the given marginals, proves that compatible channels send every choice of orthonormal bases to compatible projective-test POVMs, conjectures the converse for two channels on the same space, and derives a Fisher information-based incompatibility criterion applied to partially depolarizing channels."
strata_touched:
  - D5/S3/Quantum/QuantumChannels/ProjectiveTestChannelCompatibilityRefutation
license: citation-only
triage: anchor

A Fisher information-based incompatibility criterion for quantum channels

Q.-H. Zhang and I. Nechita, arXiv:2204.09963v1 (only version); Entropy 24(6) (2022), 805. Quotations are from the v1 TeX source.

Channel compatibility (Definition def:channel-compatibility):

Consider two quantum channels ${\Phi }_1:\mathcal{L}(H_d)\to \mathcal{L}(H_{d_1})$ and ${\Phi }_2:\mathcal{L}(H_d)\to \mathcal{L}(H_{d_2})$ having the same input space. The pair $({\Phi }_1,{\Phi }_2)$ is said to be compatible, if there exists a joint channel $\Lambda :\mathcal{L}(H_d)\to \mathcal{L}(H_{d_1}\otimes H_{d_2})$ such that ${\Phi }_{1,2}$ are the marginals of $\Lambda$: $$\forall X\in \mathcal{L}(H_d),{\Phi }_1(X)={\mathrm{Tr}}_2\Lambda (X)\text{ and }{\Phi }_2(X)={\mathrm{Tr}}_1\Lambda (X).$$

POVM compatibility:

Two POVMs $\mathbf{A}=\{A_i{\}}_{i\in [k]}$ and $\mathbf{B}=\{B_j{\}}_{k\in [l]}$ are said to be compatible (or jointly measurable) if there exists a third POVM $\mathbf{C}=\{C_{ij}{\}}_{(i,j)\in [k]\times [l]}$, called joint measurement, such that $A_i={\sum }_{j=1}^l{C}_{ij}$ for all $i\in [k]$ and $B_j={\sum }_{i=1}^k{C}_{ij}$ for all $j\in [l]$.

The partially depolarizing channel:

${\Phi }_t=t\cdot \mathrm{id}+(1-t)\Delta ,0\le t\le 1,$ with $\mathrm{id}(A)=A$ and $\Delta (A)=\mathrm{Tr}A)I_d/d$ for any operator $A$

and

The unmatched parenthesis in $\mathrm{Tr}A)$ is the source’s.

It well known the channel ${\Phi }_t$ is self-compatible if $0\le t\le \frac{d+2}{2(d+1)}$.

Lemma lem:channel-POVM-compatible proves that compatible channels ${\Phi }_1,\dots ,{\Phi }_N$ give compatible POVMs $[{\Phi }_s^{\ast }(|e_i^{(s)}⟩⟨e_i^{(s)}|){]}_{i=1}^d$ for all orthonormal bases. The conjecture (Conjecture 4.6, PDF p. 7):

We leave open the reciprocal question, which we formulate as a conjecture (below for two channels, although the general version, for a $N$-tuple, can be easily stated).

Consider two quantum channels $\Phi ,\Psi :\mathcal{L}(H_d)\to \mathcal{L}(H_d)$ such that, for all orthonormal bases $\mathbf{e}=(e_1,\dots ,e_d)$, $\mathbf{f}=(f_1,\dots ,f_d)$ of ${\mathbb{C}}^d$, the POVMs $$[{\Phi }^{\ast }(|e_i⟩⟨e_i|){]}_{i=1}^d\text{ and }[{\Psi }^{\ast }(|f_j⟩⟨f_j|){]}_{j=1}^d$$ are compatible. Then, $\Phi$ and $\Psi$ are compatible channels.

Verified locator

Source: https://arxiv.org/abs/2204.09963v1

• URL: https://arxiv.org/abs/2204.09963v1 (the e-print is a tar archive; member article.tex, md5 614f0961ea15b65fb0b83ac83bd3fffb, 548 lines):
  • Definition def:channel-compatibility, l. 138–146;
  • the partially depolarizing channel and its self-compatibility range, l. 148–152;
  • POVM compatibility, l. 192–199;
  • Lemma lem:channel-POVM-compatible and its proof, l. 239–259;
  • the conjecture, l. 261–266.
• DOI: 10.3390/e24060805 (https://doi.org/10.3390/e24060805; Crossref lists Entropy, volume 24, issue 6, article 805, published 8 June 2022).