Quantifying the imaginarity via different distance measures

Bo Li; Meng-Li Guo; Shao-Ming Fei; Si-Yin Huang

REVIEW 4 major objections 7 minor 51 references

Quantifying the imaginarity via different distance measures

T0 review · 4 major / 7 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read Three entropy-based measures are proven valid for quantum imaginarity.

desk verdict The Tsallis imaginarity measure's closed-form formulas are invalid because the minimization is over all real states, not just pure ones; this undercuts the decay and ordering results, though the resource-theoretic axioms may still hold. read the letter →

arxiv 2501.07775 v1 pith:H5W34TJA submitted 2025-01-14 quant-ph

Meng-Li Guo , Si-Yin Huang , Bo Li , Shao-Ming Fei This is my paper · ORCID

classification quant-ph MSC 81P45 PACS 03.67.-a

keywords imaginarityresourcetheoryTsallisrelativealpha-entropysandwichedRényientropyoperatorquantumchannelsstateorderingbit-flipchannel

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper introduces three new quantitative measures of imaginarity: the Tsallis relative α-entropy of imaginarity, the sandwiched Rényi relative entropy of imaginarity, and the Tsallis relative operator entropy of imaginarity, defined for α in [1/2,1). It proves that each one satisfies the four axioms required of a legitimate resource measure, so each can be used to rank how much of a quantum state is irreducible to real amplitudes. For single-qubit states the measures all become decreasing functions of the same real parameter, meaning they impose the same ordering on states, and the paper shows that the bit-flip channel leaves that ordering intact. In comparing the decay of the measures under bit-flip, phase-damping, and amplitude-damping channels at parameter value 3/4, the body of the paper finds the Tsallis relative α-entropy measure to be the most stable under noise, a conclusion opposite to the abstract's wording that it has a higher decay rate.

What carries the argument

The argument is carried by the resource-theoretic frame of imaginarity: real states (density matrices whose entries are all real) are free, real operations (Kraus maps with real matrix entries) are free operations, and a valid measure must obey axioms (M1)-(M4), equivalently (M1), (M2), and additivity (M5). For the measures to be bona fide, the paper imports the monotonicity of the Tsallis relative α-entropy, the sandwiched Rényi relative entropy, and the Tsallis relative operator entropy under completely positive trace-preserving maps, together with Hölder's inequality to enforce additivity. For the ordering results, the load-bearing device is the canonical form of a qubit state $\rho = \begin{pmatrix} 1/2 & x-iy \\ x+iy & 1/2 \end{pmatrix}$ with $x^2+y^2=1/4$ for pure states; every one of the three measures is a decreasing function of $x$, and the bit-flip channel preserves this monotonicity.

What would settle it

On a dense grid of qubit states and channel strengths, compare the three closed-form expressions in Proposition 3 with a direct numerical minimization over real states; the first mismatch, or any point where the stated derivative signs reverse, would falsify the ordering-invariance claim.

Watch

Extended reading notes

Core claim

The central claim of the paper is Theorem 1: for every $\alpha\in[1/2,1)$, the quantities $M_{T,\alpha}$, $M_{S,\alpha}$, and $M_{O,\alpha}$, each defined as a minimization over the set $\mathcal{F}$ of real states of an entropy-based distance, are bona fide imaginarity measures. The proof verifies the standard axioms: faithfulness (nonzero exactly for states with non-real density-matrix entries), monotonicity under real operations, strong monotonicity, and convexity, with the convexity step handled via Hölder's inequality. For qubit pure states, all three measures are functions only of $|\langle\psi^*|\psi\rangle|$, and for the bit-flip channel the paper supplies closed-form expressions whose derivatives are all non-positive, establishing that state ordering is preserved. The authors also compare the decay of the three measures under three common noise channels at $\alpha=3/4$ and conclude, from their figures and discussion, that the Tsallis relative α-entropy measure is the most stable of the three.

Load-bearing premise

The ordering result for the bit-flip channel depends on closed-form formulas for the three measures after the channel, which the paper states as 'easy to derive' without showing the derivation; if any formula is incorrect, the ordering claim loses its support.

Editorial extensions

If this is right

Any task that needs to quantify imaginarity can now choose among three entropy-based measures that all satisfy the standard resource-theory axioms for $\alpha\in[1/2,1)$.
For single-qubit states the three measures induce the same ordering, so the distinction between measures is irrelevant for ranking qubits by imaginarity.
Bit-flip noise does not reorder single-qubit states under any of the three measures in the stated range, so resource rankings are stable under that channel.
At $\alpha=3/4$, the Tsallis relative α-entropy measure loses the least imaginarity under bit-flip, phase-damping, and amplitude-damping channels, making it the preferred measure in noisy settings.
The proof pattern, checking axioms (M1), (M2), and (M5), provides a template for validating future distance-based imaginarity measures.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The ordering-invariance result rests on closed-form formulas the paper states without derivation; a careful user should numerically verify those formulas before relying on Proposition 3.
The same construction could be applied to other relative-entropy families, such as (α,z)-Rényi relative entropies; whether they yield valid imaginarity measures would depend on the same CPTP monotonicity used here.
The stability of the Tsallis measure is a property of the formula alone; turning that into an operational advantage in a specific quantum protocol requires separate work.
Because all three measures collapse to monotone functions of one parameter for qubits, experimental comparison of imaginarity in photonic polarisation qubits could be reduced to estimating that single parameter.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Let me give you the short version: this is a resource-theory paper that defines three imaginarity measures and proves they satisfy the standard axioms. That part is plausibly correct and is a legitimate, if conventional, extension of known distance-based coherence and entanglement measures. The trouble is that the paper's quantitative results—the closed-form formulas for pure states, the decay-rate comparison, and the ordering under bit-flip—are built on a minimization error for the Tsallis measure MT,α.

Theorem 1 defines MT,α as a minimum over all real density matrices σ. In Theorem 2, the proof restricts the minimization to pure real states |v⟩. For a concave function of σ—and σ↦tr(ρ^α σ^{1−α}) is concave for α∈[1/2,1)—the maximum over a convex set can sit at a mixed state, not a pure one. Concretely, take |ψ⟩=(|0⟩+i|1⟩)/√2 and α=3/4. The claimed formula gives 1−2^{−4/3}≈0.603, but choosing σ=I/2 (a real state) gives 1−2^{−1/3}≈0.206. Since MT is a minimum, the claimed value is not the value of the measure. The same pure-state restriction reappears in the Section III decay formulas and in the MT part of Proposition 3, so those results are unsupported as stated. This is not a small gap: it is the main quantitative content of the paper.

There is also an internal contradiction the authors did not catch: the abstract says the Tsallis measure has a higher decay rate, while Section III concludes it is more stable. Proposition 1's proof of MO≥MT relies on a figure instead of an analytic argument, which is insufficient for a published claim. And the 'it is easy to derive' formulas for the bit-flip channel output in Section IV are exactly where the pure-state restriction bites; for MT they do not match the definition.

What is worth keeping: Theorem 1's axiomatic proof (M1, M2, M5) looks mostly right, and the definitions are new even if they are direct analogues of existing entanglement measures. The paper engages the literature honestly, and the two cited supporting inequalities from the same group are published, so self-citation is not the issue.

My take: this deserves peer review rather than desk rejection, because the definitions and the axiom proofs are a legitimate contribution and the minimization error is identifiable and fixable. But the authors need to either prove the pure-state reduction (which is false for MT) or abandon the closed-form formulas and the decay/ordering claims built on them. For a reading group, I'd skip it; the error is instructive, but the paper as it stands would mislead.

Referee Report

4 major / 7 minor

Summary. This paper proposes three quantitative measures of quantum imaginarity for α∈[1/2,1): MT,α based on Tsallis relative α-entropy, MS,α based on sandwiched Rényi relative entropy, and MO,α based on Tsallis relative operator entropy, each defined as a minimization over the set F of real states. Theorem 1 claims that these quantities satisfy the resource-theoretic axioms (M1)–(M5), with proofs in the Appendix that rely on published data-processing inequalities. Theorem 2 gives closed-form expressions for pure qubit states in terms of A = |⟨ψ*|ψ⟩|, and Proposition 1 compares the three measures on pure states. Section III reports decay formulas under bit-flip, phase-damping, and amplitude-damping channels at α = 3/4 and concludes that the Tsallis measure is the most stable under these channels. Section IV examines ordering of single-qubit states, claiming order invariance under the bit-flip channel in Proposition 3.

Significance. The construction is a natural and potentially useful extension of the imaginarity resource theory: the measures contain no fitted parameters (only the parameter α), the axiom proofs invoke established monotonicity inequalities from the literature rather than ad hoc assumptions, and the paper attempts concrete falsifiable predictions in the form of closed forms for pure states and explicit decay functions. If Theorem 1 is taken alone, the contribution is a legitimate, if modest, addition to the catalogue of imaginarity measures. However, the advertised quantitative content — the decay comparison, the measure ordering of Proposition 1, and the bit-flip order invariance of Proposition 3 — is currently unreliable because the closed-form evaluations supporting it are invalidated by the pure-state restriction in the proof of Theorem 2 (see major comment 1), and the headline decay claim is internally contradictory (see major comment 2). The significance of the work therefore hinges on whether the underlying minimizations can be corrected and the quantitative claims re-derived.

major comments (4)

[§II (Theorem 2 and its proof)] The closed-form expressions in Theorem 2 are not the values of the measures defined in Theorem 1, because the proof minimizes over pure real states only, whereas the definitions in Theorem 1 range over all real density matrices σ ∈ F, including mixed ones. Concretely, for |ψ⟩ = (|0⟩+i|1⟩)/√2, which has A = |⟨ψ*|ψ⟩| = 0, and α = 3/4, taking σ = I/2 ∈ F yields MT,α(|ψ⟩) ≤ 1 − [tr(ρ^{3/4}(I/2)^{1/4})]^{4/3} = 1 − 2^{−1/3} ≈ 0.206, while Theorem 2 claims MT,α(|ψ⟩) = 1 − ((1+A)/2)^{1/α} = 1 − 2^{−4/3} ≈ 0.603. Since MT,α is a minimum over F, the claimed value cannot be correct; in fact, for this state tr(ρ^α σ^{1−α}) = tr(σ^{1−α})/2 for every real σ, and tr(σ^{1−α}) is maximized at σ = I/2, so the true value is exactly 0.206. The same unjustified pure-state restriction underlies the evaluations of MS,α and MO,α, the Section III decay formulas (which are applied to mixed channel outputs), and the Section IV bit-flip formulas; consequently the decay-rate comparison, Proposition 1, and Proposition 3 are unsupported as stated, even though Theorem 1 may remain true as an abstract existence statement.
[Abstract vs. Sections III and V] The abstract states that "the Tsallis relative α-entropy of imaginarity exhibits higher decay rate under quantum channels compared to other measures," whereas Section III concludes that this same measure "exhibits a smaller attenuation difference, indicating that it retains more information during transmission and exhibits higher stability," and Section V repeats that it "exhibits greater stability compared to the other two measures." A higher decay rate and a smaller attenuation are mutually incompatible statements, and since this comparison is the paper's advertised main finding, the contradiction must be resolved and the surviving claim re-derived from correct decay formulas.
[§II (Proposition 1)] The proof of ΔM2 ≥ 0 (i.e., MO,α ≥ MT,α) is not analytic: it concludes with "As can be seen from Fig. 1," which is a graphical assertion rather than a proof. Moreover, the displayed closed form for ΔM2 is numerically inconsistent: at A = 0 and α = 3/4, the preceding expression evaluates to approximately 0.222, while the simplified expression displayed after it evaluates to approximately 8.56, so the algebraic simplification as printed is wrong or missing a factor. Independently of this algebra, the proposition inherits the invalid Theorem 2 evaluation of MT,α, so the claimed ordering is not established.
[§IV (Proposition 3)] The formulas for MT,α(ε_BF(ρ)), MS,α(ε_BF(ρ)), and MO,α(ε_BF(ρ)) are introduced with the sentence "It is easy to derive that" and no derivation is supplied, yet ε_BF(ρ) in Eq. (8) is a mixed state for m ∉ {0,1} (its purity is 1/2 + 2x² + 2(2m−1)²y², which is less than 1 whenever y ≠ 0 and m ∉ {0,1}). The pure-state closed forms of Theorem 2 therefore do not apply to this output state, and no mixed-state evaluation is provided. The monotonicity argument ∂x M(ε_BF(ρ)) ≤ 0 and the resulting order-invariance conclusion rest on these unverified expressions, so Proposition 3 is not proven.

minor comments (7)

[§IV] There are typos in this section: "singe-qubit" appears twice and "imagimarity" appears once; these should be corrected.
[§III] The sentence "for phase flip and amplitude damping channels, these quantities do not attain their maximum values at m = 0.5" uses the bit-flip parameter m for channels whose noise parameters are n (phase damping) and p (amplitude damping), so the sentence should refer to n and p; also "phase flip" and "phase damping" are used inconsistently to name the same channel.
[Figure 2 caption] The caption refers to undefined quantities "ΔMgl" and "ΔMg" and does not identify which colored surface corresponds to which measure in each of the panels (a)–(i); the caption should be rewritten to match the panels it describes.
[Appendix, proof of (i) and (iii)] In the proof of (M5), "p2 1 + p2 2 = 1" and the analogous expression for q should read p1 + p2 = 1 and q1 + q2 = 1; near the end of the (i) proof, "MF,α" should be "MT,α"; and in the (iii) proof the line "max σ∈F (...) = min σ∈F (...)" contains a typo, since both sides should be "max."
[Appendix, proof of (iii)] The faithfulness condition (M1) for MO,α is asserted with the sentence "It is easy to find that MO,α(ρ) = 0 iff ρ is a real state" rather than proved; since (M1) is part of the defining property of an imaginarity measure, a proof or an explicit citation is required.
[References] Reference [18] is cited in arXiv form (arXiv:1309.6586) and reference [36] has an incomplete article identifier ("Sci. China-Phys. Mech. Astron. 67, 1 (2025)"); both should be updated to their published versions.
[§IV, after Eq. (5)] The sentence "if ρ is a pure state, then x² + y² = 1/4. Therefore, we focus exclusively on the pure state ρ" does not explain why mixed states can be excluded from the ordering analysis; the restriction to pure states should be stated explicitly as an assumption rather than implied.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the imaginarity measures are defined by minimization over real states and proved using external, parameter-free inequalities; self-citations are supporting lemmas, not load-bearing circular reductions.

full rationale

The paper's central definitions (Theorem 1) are minimization-based quantifiers built from standard relative entropies, with no fitted parameters and no quantity being predicted from itself. The proofs of the measure axioms rely on external data-processing inequalities: for the Tsallis and sandwiched Rényi terms, citations [42] and [45] are independent published results, and for the operator-entropy term the monotonicity inequality is cited to the authors' prior work [46]. That cited inequality is a parameter-free mathematical statement about CPTP maps and does not assume or contain the new imaginarity measures, so it is independent evidence rather than a circular premise. The later quantitative formulas for pure states and channel outputs are asserted with little derivation and may be mathematically deficient (notably, Theorem 2 minimizes only over pure real states while the definition allows mixed real states), but that is a correctness issue, not a circularity of the kind defined here: no equation is shown to equal its own input by construction, and no fitted value is renamed as a prediction. The figure-based comparison in Proposition 1 is an omitted analytic justification, not a circular step. Overall, the claimed derivation chain is self-contained with respect to its external assumptions, and the self-citations do not carry the central claim by definition.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The paper contributes three new functionals but no new physical entities. It rests on standard resource-theory axioms and external data-processing inequalities; the only hand-chosen numeric value is α=3/4 for the decay plots.

free parameters (1)

α (measure parameter) = 3/4 for decay analysis; range [1/2,1) in definitions
The imaginarity measures are defined for any α in [1/2,1), but the decay-rate comparison and all figures are computed only at α=3/4. Conclusions about relative stability are not established for other α values.

assumptions (4)

domain assumption Equivalence of (M1)-(M4) with (M1), (M2), (M5) for imaginarity measures (from [35])
Used in the proof of Theorem 1 to establish that proving nonnegativity, monotonicity under real operations, and additivity is sufficient for a bona fide measure. The paper does not re-derive this equivalence.
domain assumption Data processing inequalities for Tsallis relative entropy, sandwiched Rényi relative entropy, and Tsallis relative operator entropy (from [42] and [46])
Monotonicity (M2) in Theorem 1 relies on these external inequalities; [46] is a prior paper by the same group, but it is a published result.
standard math Any pure state can be brought to the canonical form (3) via a real orthogonal transformation that preserves imaginarity (from [33])
Used in Theorem 2 to reduce the minimization over real states to an optimization over b0,b1.
domain assumption For α ∈ [1/2,1), Dα(ρ||σ) ≥ 0 with equality iff ρ=σ, and the trace inequality leading to (10) (from [45])
Used to prove (M1) and the monotonicity direction for MT,α.

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Pith. "Pith review of Quantifying the imaginarity via different distance measures." pith.science (2026). https://pith.science/paper/H5W34TJA

@misc{pith2026250107775,
  author       = {Pith},
  title        = {Pith review of: Quantifying the imaginarity via different distance measures},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/H5W34TJA}},
  note         = {Machine review of arXiv:2501.07775}
}

abstract

The recently introduced resource theory of imaginarity facilitates a systematic investigation into the role of complex numbers in quantum mechanics and quantum information theory. In this work, we propose well-defined measures of imaginarity using various distance metrics, drawing inspiration from recent advancements in quantum entanglement and coherence. Specifically, we focus on quantitatively evaluating imaginarity through measures such as Tsallis relative $\alpha$-entropy, Sandwiched R\'{e}nyi relative entropy, and Tsallis relative operator entropy. Additionally, we analyze the decay rates of these measures. Our findings reveal that the Tsallis relative $\alpha$-entropy of imaginarity exhibits higher decay rate under quantum channels compared to other measures. Finally, we examine the ordering of single-qubit states under these imaginarity measures, demonstrating that the order remains invariant under the bit-flip channel for specific parameter ranges. This study enhances our understanding of imaginarity as a quantum resource and its potential applications in quantum information theory.

Figures

Figures reproduced from arXiv: 2501.07775 by the authors.

**Figure 2.** FIG. 2. The decay of [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

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Works this paper leans on

51 extracted references · 45 canonical work pages

[1]

tr h ε(ρ) 1 2 ε(ρ)− 1 2 ε(σ∗)ε(ρ)− 1 2 1−α ε(ρ) 1 2 i# 1 α − 1 ) ≥ min σ∈F 1 α − 1 (

+ qα 2 (p1−α 2 t ′ 2) ≤ h (qα 1 ) 1 α + (qα 2 ) 1 α iαh (pα 1 t ′ 1) 1 1−α + (pα 2 t ′ 2) 1 1−α i1−α = h p1(t ′ 1) 1 1−α + p2(t ′ 2) 1 1−α i1−α , the equality holds when q1 q2 = p1(t ′ 1) 1 1−α p2(t′ 2) 1 1−α . Consequently, max σ∈F nh tr (ρ 1−α 2α σρ 1−α 2α )α i 1 1−α o = n max σ∈F tr h (ρ 1−α 2α σρ 1−α 2α )α io 1 1−α = p1(t ′ 1) 1 1−α + p2(t ′ 2) 1 1−α ...

work page
[2]

Therefore, we focus exclusively on the pure state ρ. Using the definitions of the measures MT,α, MS,α, and MO,α for qubit pure states, we can calculate: MT,α (ρ) = 1 − x + 1 2 1 α , MS,α(ρ) = x + 1 2 α 1−α − 1 α − 1 , MO,α(ρ) = x + 1 2 1 α −1 − 1 α − 1 . From the derivations of the imaginarity measures with respect to x, ∂xMT,α (ρ) = − x + 1 2 1 α −1 α , ...

work page
[3]

Consequently, we conclude that ∂xMT,α (ρ) ≤ 0, ∂xMS,α(ρ) ≤ 0, and ∂xMO,α(ρ) ≤

work page
[4]

It is important to note that if ρ is a pure state, then x2 + y2 = 1

ρ′ and ρ have the same imaginarity. It is important to note that if ρ is a pure state, then x2 + y2 = 1

work page
[5]

6 Based on the analysis above, we observe thatMT,α, MS,α, and MO,α exhibit the same monotonicity for single-qubit states

This indicates that MT,α (ρ), MS,α(ρ), and MO,α(ρ) are all decreasing functions. 6 Based on the analysis above, we observe thatMT,α, MS,α, and MO,α exhibit the same monotonicity for single-qubit states. Therefore, we can draw the following conclusions. Proposition 2 For any two single-qubit statesρ1 and ρ2 in the form given by formula (5), the imaginarity...

work page
[6]

W. K. Wootters, Found. Phys. 42, 19 (2012)

work page 2012
[7]

Hardy, W

L. Hardy, W. K. Wootters, Found. Phys. 42, 454 (2012)

work page 2012
[8]

Aleksandrova, V

A. Aleksandrova, V . Borish, W. K. Wootters, Phys. Rev. A 87, 052106 (2013)

work page 2013

Show all 51 references

[9]

K. D. Wu, T. V . Kondra, S. Rana, C. M. Scandolo, G. Y . Xiang, C. F. Li, G. C. Guo, A. Streltsov, Phys. Rev. Lett. 126, 090401 (2021)

2021
[10]

M. O. Renou, D. Trillo, M. Weilenmann, T. P. Le, A. Tavakoli, N. Gisin, A. Ac´ın, M. Navascu´es, Nature 600, 625 (2021)

2021
[11]

M. C. Chen, C. Wang, F. M. Liu, et al, Phys. Rev. Lett. 128, 040403 (2022)

2022
[12]

Z. D. Li, Y . L. Mao, M. Weilenmann, et ai, Phys, Rev. Lett.128, 040402 (2022)

2022
[13]

Horodecki, P

R. Horodecki, P. Horodecki, M. Horodecki, K. Horodecki, Rev. Mod. Phys. 81, 865 (2009)

2009
[14]

Baumgratz, M

T. Baumgratz, M. Cramer, and M. B. Plenio, Phys. Rev. Lett. 113, 140401 (2014)

2014
[15]

Chitambar and G

E. Chitambar and G. Gour, Phys. Rev. Lett.117, 030401 (2016)

2016
[16]

Marvian and R

I. Marvian and R. W. Spekkens, Phys. Rev. A94, 052324 (2016)

2016
[17]

Chitambar and M

E. Chitambar and M. H. Hsieh, Phys. Rev. Lett. 117, 020402 (2016)

2016
[18]

Winter and D

A. Winter and D. Yang, Phys. Rev. Lett. 116, 120404 (2016)

2016
[19]

Napoli, T

C. Napoli, T. R. Bromley, M. Cianciaruso, M. Piani, N. John- ston, and Gerardo Adesso, Phys. Rev. Lett.116, 150502 (2016)

2016
[20]

F. G.S.L. Brand ˜ao, M. Horodecki, J. Oppenheim, J. M. Renes, and R. W. Spekkens, Phys. Rev. Lett.111, 250404 (2013)

2013
[21]

Horodecki and J

M. Horodecki and J. Oppenheim, Nature Communications 4, 2059 (2013)

2013
[22]

Lostaglio, D

M. Lostaglio, D. Jennings, and T. Rudolph, Nature Communi- cations 6, 6383 (2015)

2015
[23]

G. Gour, M. P. M ¨uller, V . Narasimhachar, R. W. Spekkens, and N. Y . Halpern, arXiv:1309.6586 (2014)

2014 arXiv
[24]

Narasimhachar and G

V . Narasimhachar and G. Gour, Nature Communications 6, 7689 (2015)

2015
[25]

Gour and R

G. Gour and R. W. Spekkens, New J. Phys. 10 (2008) 033023

2008
[26]

G. Gour, I. Marvian, and R. W. Spekkens, Phys. Rev. A 80, 012307 (2009)

2009
[27]

Skotiniotis and G

M. Skotiniotis and G. Gour, New J. Phys. 14, 073022 (2012)

2012
[28]

Marvian and R

I. Marvian and R. W. Spekkens, New J. Phys. 15, 033001 (2013)

2013
[29]

del Rio, L

L. del Rio, L. Kraemer, and R. Renner, arXiv:1511.08818 (2015)

2015 arXiv
[30]

Veitch, S

V . Veitch, S. A. H. Mousavian, D. Gottesman, and J. Emerson, New J. Phys. 16, 013009 (2014)

2014
[31]

Howard and E

M. Howard and E. T. Campbell, Phys. Rev. Lett. 118, 090501 (2017)

2017
[32]

Ahmadi, H

M. Ahmadi, H. B. Dang, G. Gour, and B. C. Sanders, arXiv:1706.03828 (2017)

2017 arXiv
[33]

Theurer, N

T. Theurer, N. Killoran, D. Egloff, and M. B. Plenio, arXiv:1703.10943 (2017)

2017 arXiv
[34]

Gallego and L

R. Gallego and L. Aolita, Phys. Rev. X 5, 041008 (2015)

2015
[35]

J. I. de Vicente, J. Phys. A 47, 424017 (2014)

2014
[36]

Amaral, A

B. Amaral, A. Cabello, M. T. Cunha, and L. Aolita, arXiv:1705.07911 (2017)

2017 arXiv
[37]

Hickey and G

A. Hickey and G. Gour, J. Phys. A: Math. Theor. 51, 414009 (2018)

2018
[38]

K. D. Wu, T. V . Kondra, S. Rana, C. M. Scandolo, G. Y . Xiang, C. F. Li, G. C. Guo, A. Streltsov, Phys. Rev. A 103 032401 (2021)

2021
[39]

T. V . Kondra, C. Datta, and A. Streltsov, New Journal of Physics 25, 093043 (2023)

2023
[40]

S. Xue, J. Guo, P. Li, M. Ye, and Y . Li, Quantum Inf. Process. 20, 1 (2021)

2021
[41]

M. L. Guo, B. Li, S. M. Fei, Sci. China-Phys. Mech. Astron. 10 67, 1 (2025)

2025
[42]

J. W. Xu, Phys. Rev. A 108, 062203 (2023)

2023
[43]

Borland, A

L. Borland, A. R. Plastino, C. Tsallis, J. Math. Phys. 39, 6490 (1998)

1998
[44]

Furuichi, K

S. Furuichi, K. Yanagi, K. Kuriyama, J. Math. Phys. 45, 4868 (2004)

2004
[45]

Abe, Phys

S. Abe, Phys. Lett. A 312, 336 (2003)

2003
[46]

M. M. Wilde, A. Winter, D. Yang, Commun. Math. Phys. 331, 593 (2014)

2014
[47]

M ¨uller-Lennert, F

M. M ¨uller-Lennert, F. Dupuis, O. Szehr, S. Fehr, M. Tomamichel, J. Math. Phys. 54, 122203 (2013)

2013
[48]

Z. J. Xi, Sci. China-Phys. Mech. Astron. 45, 030302 (2015)

2015
[49]

Q. Chen, T. Gao, F. L. Yan, Sci. China Phys. Mech. & Astron. 66, 280312 (2023)

2023
[50]

A. E. Rastegin, Phys. Rev. A 93, 032136 (2016)

2016
[51]

L.Guo, Z

M. L.Guo, Z. X. Jin, B. Li, B. Hu, S. M. Fei, Quant. Inf. Pro- cess. 19, 382 (2020)

2020

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Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [1]

tr h ε(ρ) 1 2 ε(ρ)− 1 2 ε(σ∗)ε(ρ)− 1 2 1−α ε(ρ) 1 2 i# 1 α − 1 ) ≥ min σ∈F 1 α − 1 (

+ qα 2 (p1−α 2 t ′ 2) ≤ h (qα 1 ) 1 α + (qα 2 ) 1 α iαh (pα 1 t ′ 1) 1 1−α + (pα 2 t ′ 2) 1 1−α i1−α = h p1(t ′ 1) 1 1−α + p2(t ′ 2) 1 1−α i1−α , the equality holds when q1 q2 = p1(t ′ 1) 1 1−α p2(t′ 2) 1 1−α . Consequently, max σ∈F nh tr (ρ 1−α 2α σρ 1−α 2α )α i 1 1−α o = n max σ∈F tr h (ρ 1−α 2α σρ 1−α 2α )α io 1 1−α = p1(t ′ 1) 1 1−α + p2(t ′ 2) 1 1−α ...

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[2] [2]

Therefore, we focus exclusively on the pure state ρ. Using the definitions of the measures MT,α, MS,α, and MO,α for qubit pure states, we can calculate: MT,α (ρ) = 1 − x + 1 2 1 α , MS,α(ρ) = x + 1 2 α 1−α − 1 α − 1 , MO,α(ρ) = x + 1 2 1 α −1 − 1 α − 1 . From the derivations of the imaginarity measures with respect to x, ∂xMT,α (ρ) = − x + 1 2 1 α −1 α , ...

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[3] [3]

Consequently, we conclude that ∂xMT,α (ρ) ≤ 0, ∂xMS,α(ρ) ≤ 0, and ∂xMO,α(ρ) ≤

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[4] [4]

It is important to note that if ρ is a pure state, then x2 + y2 = 1

ρ′ and ρ have the same imaginarity. It is important to note that if ρ is a pure state, then x2 + y2 = 1

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[5] [5]

6 Based on the analysis above, we observe thatMT,α, MS,α, and MO,α exhibit the same monotonicity for single-qubit states

This indicates that MT,α (ρ), MS,α(ρ), and MO,α(ρ) are all decreasing functions. 6 Based on the analysis above, we observe thatMT,α, MS,α, and MO,α exhibit the same monotonicity for single-qubit states. Therefore, we can draw the following conclusions. Proposition 2 For any two single-qubit statesρ1 and ρ2 in the form given by formula (5), the imaginarity...

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[6] [6]

W. K. Wootters, Found. Phys. 42, 19 (2012)

work page 2012

[7] [7]

Hardy, W

L. Hardy, W. K. Wootters, Found. Phys. 42, 454 (2012)

work page 2012

[8] [8]

Aleksandrova, V

A. Aleksandrova, V . Borish, W. K. Wootters, Phys. Rev. A 87, 052106 (2013)

work page 2013