bibkey: wuwu2025unifiedimaginarity
authors: Chuanfa Wu; Zhaoqi Wu
year: 2025
title: "Two imaginarity monotones induced by unified (α,β)-relative entropy"
doi: 10.1088/1572-9494/adbdbd
url: https://arxiv.org/abs/2506.09799v1
claim: "The paper defines the imaginarity quantifiers M^H, the unified (α,β)-relative entropy between a state and its complex conjugate, and M^E, its minimum over real states, proves that both are imaginarity monotones for α in (0,1) and β in (0,1], verifies M^H >= M^E on a qubit family, and conjectures M^H >= M^E for every quantum state."
strata_touched:
  - D5/S3/Quantum/Divergence/UnifiedImaginarityMonotoneComparison
license: citation-only
triage: anchor

Two imaginarity monotones induced by unified (α,β)-relative entropy

C. Wu and Z. Wu, arXiv:2506.09799v1 [quant-ph]; Commun. Theor. Phys. 77 (2025). Quotations are from the v1 TeX source.

Real states:

$\rho$ is a free state or real state, if $\rho ={\sum }_{jk}{\rho }_{jk}|j⟩⟨k|$, where ${\rho }_{jk}\in \mathbb{R}$ for arbitrary $j$,$k\in \{0,1,\dots ,d-1\}$. We denote the set of all real states by $\mathcal{F}$.

The unified relative entropy (Eq. (7)):

$H_{\alpha }^{\beta }(\rho ||\sigma )=\frac{1}{(\alpha -1)\beta }[(\mathrm{tr}({\rho }^{\alpha }{\sigma }^{1-\alpha }){)}^{\beta }-1]$,

The two quantifiers (Eqs. (13) and (20)):

$M_{\alpha ,\beta }^H(\rho )=\frac{1}{(\alpha -1)\beta }[(\mathrm{tr}({\rho }^{\alpha }({\rho }^{\ast }{)}^{1-\alpha }){)}^{\beta }-1]$, where $\alpha \in (0,1),\beta \in (0,1].$

$M_{\alpha ,\beta }^E(\rho )={\min }_{\sigma \in \mathcal{F}}\frac{1}{(\alpha -1)\beta }[(\mathrm{tr}({\rho }^{\alpha }{\sigma }^{1-\alpha }){)}^{\beta }-1]$, where $\alpha \in (0,1)$ and $\beta \in (0,1]$.

The conjecture (Example 1):

We have not found a proof of $M_{\alpha ,\beta }^H(\rho )\ge M_{\alpha ,\beta }^E(\rho )$ for any $\alpha \in (0,1)$, $\beta \in (0,1]$ and any qubit state $\rho$, however, numerical experiments shows that it might be true, and it seems difficult to find a counterexample. It is also conjectured that $M_{\alpha ,\beta }^H(\rho )\ge M_{\alpha ,\beta }^E(\rho )$ for any $\alpha \in (0,1)$, $\beta \in (0,1]$ and any quantum state $\rho$.

Verified locator

Source: https://arxiv.org/abs/2506.09799v1

• URL: https://arxiv.org/abs/2506.09799v1 (source 20250301WuCF-imaginarity_monotone_R2.tex, md5 eb104c03d3638b12044677758c200bae):
  • real states, l. 120–124;
  • the unified relative entropy, l. 164–166;
  • $M^H$, l. 203–205;
  • $M^E$, l. 347–349;
  • the conjecture, l. 573–579.
• DOI: 10.1088/1572-9494/adbdbd (https://doi.org/10.1088/1572-9494/adbdbd; Crossref lists Communications in Theoretical Physics, volume 77, issued 30 May 2025; arXiv lists only v1 of 11 June 2025).