Quantifying imaginarity in terms of pure-state imaginarity

Shuanping Du; Zhaofang Bai

REVIEW 2 major objections 5 minor 1 cited by

Quantifying imaginarity in terms of pure-state imaginarity

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Pith's one-line read The paper proves that every imaginarity measure comes from a decreasing concave function of the pure-state overlap, and that no finite set of such measures decides mixed-state conversion under real operations.

desk verdict The convex roof and least-input pure-state imaginarity constructions are solid and useful, but the no-go theorem (Theorem 2) is not proven as written because the f_k family is not concave and the final inequality step is invalid. read the letter →

arxiv 2411.12215 v3 pith:YFMJAQFB submitted 2024-11-19 quant-ph

Shuanping Du , Zhaofang Bai This is my paper · ORCID

classification quant-ph PACS 03.65.Ud03.67.-a03.65.Ta

keywords resourcetheoryofimaginarityconvexroofconstructionmeasurerealoperationsstateconversiondecreasingconcavefunctionqubitformulamonotone

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

The paper sets up a general framework for measuring imaginarity, the amount of a quantum state that genuinely requires complex numbers rather than real ones. It shows that any reasonable imaginarity measure on pure states is encoded by a single decreasing concave function $f$ of the overlap $|\langle\psi^*|\psi\rangle|$, and that extending $f$ to mixed states by the convex roof (the least average pure-state imaginarity over all decompositions) always yields a valid measure. It then introduces a second, operationally motivated quantifier: the least imaginarity of a pure state that a real operation can convert into the given mixed state. For qubits this quantifier has a closed form $f\bigl(\sqrt{1-4(\operatorname{Im} b)^2}\bigr)$ in terms of the off-diagonal element. Finally, the paper proves that no finite collection of imaginarity measures can completely determine which mixed states can be converted into which by real operations in dimension four or higher.

What carries the argument

The central object is the conjugate overlap $x=|\langle\psi^*|\psi\rangle|$ of a pure state, together with a decreasing concave shape function $f(x)$ satisfying $f(1)=0$. Two constructions carry the argument: the convex roof $I_f(\rho)=\min_{\sum_i p_i|\psi_i\rangle\langle\psi_i|=\rho}\sum_i p_i f(|\langle\psi_i^*|\psi_i\rangle|)$, and the conversion cost $\tilde I_f(\rho)=\min_{|\phi\rangle\in R(\rho)} f(|\langle\phi^*|\phi\rangle|)$. The bridge between them is the fact that any pure-state ensemble can be produced from one canonical qubit pure state by a real operation, which turns the minimization over input states into a maximization over ensembles and yields the closed qubit formula.

What would settle it

A concrete check of the no-go proof: the step from $p_2 a^k \le b^k$ to $a\le b$ is not valid for $0<p_2<1$; for example $a=0.9$, $b=0.5$, $k=0.5$, $p_2=0.5$ satisfies the premise while violating the conclusion, so the claimed contradiction is not forced.

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Extended reading notes

Core claim

The central claim is that the whole family of convex-roof imaginarity measures is parameterized by decreasing concave functions $f:[0,1]\to[0,1]$ with $f(1)=0$, with pure-state value $f(|\langle\psi^*|\psi\rangle|)$. The converse also holds: any imaginarity measure satisfying axioms (I1), (I3), and (I4) must restrict to pure states as such an $f$. The paper further constructs the monotone $\tilde I_f(\rho)=\min_{|\phi\rangle\in R(\rho)} f(|\langle\phi^*|\phi\rangle|)$, where $R(\rho)$ is the set of pure states convertible to $\rho$ by real operations, and proves it is a valid imaginarity monotone equal to $f\bigl(\max_{\{p_i,|\phi_i\rangle\}}\sum_i p_i |\langle\phi_i^*|\phi_i\rangle|\bigr)$. For qubits the optimal decomposition is explicit, giving $\tilde I_f(\rho)=f\bigl(\sqrt{1-4(\operatorname{Im} b)^2}\bigr)$ with $b$ the off-diagonal element. Theorem 2 asserts that for dimension $d\ge 4$ no finite set of imaginarity measures can classify mixed-state convertibility under real operations.

Load-bearing premise

The no-go theorem's construction assumes that the mixing probabilities in the prepared state can be chosen so that every finite set of measures sees the mixture as at least as imaginary as the target while the specially constructed measure sees it as strictly less imaginary; the proof never demonstrates those two requirements are compatible.

Editorial extensions

If this is right

Any decreasing concave $f$ with $f(1)=0$ yields a valid imaginarity measure by convex roof, and every measure satisfying (I1), (I3), and (I4) is of this form on pure states.
The pure-state restriction of a convex-roof imaginarity measure depends only on $|\langle\psi^*|\psi\rangle|$, so pure states with equal conjugate overlap are equally imaginary under all such measures.
The quantifier $\tilde I_f$ is a valid imaginarity monotone and is the largest monotone that agrees with $I_f$ on pure states; it equals $f\bigl(\max_{\{p_i,|\phi_i\rangle\}}\sum_i p_i |\langle\phi_i^*|\phi_i\rangle|\bigr)$.
For qubits, $\tilde I_f(\rho)=f\bigl(\sqrt{1-4(\operatorname{Im} b)^2}\bigr)$, where $b$ is the off-diagonal matrix element, making all these monotones explicitly computable on one qubit.
For dimensions $d\ge 4$, no finite collection of imaginarity measures is complete for deciding mixed-state convertibility under real operations (Theorem 2).

Reading between the lines

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

Taking Theorem 1's converse at face value, imaginarity is a one-dimensional resource on pure states, in contrast to coherence and entanglement where pure-state order is governed by majorization; this makes candidate measures comparable by a single scalar function.
The conversion-cost construction $\tilde I_f$ looks like a one-shot formation cost for imaginarity; for well-chosen $f$ it may coincide with an asymptotic conversion rate, giving a second operational meaning beyond the geometric measure.
The qubit formula implies that for one qubit every monotone in this family is a monotone function of $|\operatorname{Im} b|$, so they all impose the same state order; genuinely different orderings can only appear in dimension three or higher.
If the no-go theorem survives a repaired proof, it sharpens the general impossibility result for continuous faithful monotones by showing that even dropping those regularity requirements leaves an infinite hierarchy of conditions.

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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

The core quantification results are the real value here. Theorem 1 gives a clean characterization of convex roof imaginarity measures in terms of decreasing concave functions on [0,1], with a converse that is genuinely useful. Theorems 3–6 develop the least-imaginarity-of-input-pure-states monotone ~I_f, give it an operational reading as a formation cost under real operations, and provide a neat qubit formula. I checked the proofs of these theorems and they look sound, given the cited lemmas from the imaginarity and coherence literature. Table I is a handy summary. The paper is worth reading for these tools alone.

The soft spot is Theorem 2, the no-go result on mixed-state conversion. The proof constructs measures from f_k(x) = 1 - (x^k ∧ 1) for k ∈ (0,1]. For 0

The paper would benefit from a careful revision of Theorem 2. The central characterization (Theorem 1) and the ~I_f monotone are separable and should survive intact. I would send this to a serious referee, because the main tools are valuable and mostly correct, and the flaws in Theorem 2 are isolated and fixable in principle, either by finding a genuinely concave family of functions for the no-go construction or by citing a different route. The authors should be asked to address the concavity issue and the p2 step directly.

For a reader working in resource theories of imaginarity or coherence, this is a useful addition. The flaws are real but contained. I'd bring it to a reading group to discuss the convex roof characterization, and I might cite it in my own work on imaginarity measures — though I would avoid relying on Theorem 2 until it is repaired.

Referee Report

2 major / 5 minor

Summary. This paper develops two approaches to quantifying imaginarity. First, for any decreasing concave f with f(1)=0, it defines I_f(rho) by the convex roof over pure-state values f(|<psi*|psi>|) and proves (Theorem 1) that I_f satisfies (I1)-(I5), with a converse stating that every imaginarity measure satisfying (I1),(I3),(I4) restricts to such an f on pure states. Second, for a state rho and the set R(rho) of pure states convertible to rho under real operations, it defines ~I_f(rho)=min_{|phi> in R(rho)} f(|<phi*|phi>|) and proves it is an imaginarity monotone (Theorem 3), gives a decomposition formula (Theorem 4), an analytic qubit expression (Theorem 5), and a characterization of convexity (Theorem 6). The paper also states a no-go theorem (Theorem 2) claiming that no finite set of imaginarity measures can decide mixed-state convertibility in dimension at least 4.

Significance. If the central theorems are correct, the paper provides a clean, unifying characterization of convex-roof imaginarity measures and a new operational monotone with a direct state-conversion interpretation, including an analytic qubit formula. The equivalence in Theorem 1 is elegant, and the proof strategy via Proposition 2 is largely self-contained. The numerical examples and the connection to geometric imaginarity add value. However, the no-go application in Theorem 2 is the advertised headline result and it is not currently established as written; the remaining theorems may still be sound, but the paper's main claim of a no-go theorem needs repair.

major comments (2)

[Section I, Theorem 2] The functions f_k(x)=1-(x^k and 1) do not satisfy condition (iii) for 0<k<1. Since x^k is concave on [0,1] for k in (0,1), f_k is convex; the displayed inequality (lambda a+(1-lambda)b) and 1 >= lambda(a and 1)+(1-lambda)(b and 1) proves concavity of x and 1, which is the opposite of what is needed for f_k=1-(x and 1) to be concave. A concrete violation for k=1/2, x=0.04, y=0.64, lambda=1/2 is f_k(0.34) ≈ 0.417 < 0.5 = (f_k(x)+f_k(y))/2. Therefore Theorem 1 does not license the claim that each I_{f_k} is an imaginarity measure, and the monotonicity inequality used in the proof of Theorem 2 is unsupported.
[Section I, Theorem 2 proof] The step 'Choosing k = 1 - |<phi*_{eta0}|phi_{eta0}>|, one can get |<psi*_2|psi_2>| <= |<phi*_{eta0}|phi_{eta0}>|' does not follow. From the displayed inequality, using p1=1-p2, one obtains p2 |<psi*_2|psi_2>|^k <= |<phi*_{eta0}|phi_{eta0}>|^k; the inference to the unpowered inequality requires p2 >= 1, while p1>0 and p1+p2=1 imply p2<1. The proof also fails to show that the lower bound on p1 (needed for the existing measures I_j) can be satisfied simultaneously with the requirement that p2 be large enough for the contradiction; these two requirements may conflict. Theorem 2 is therefore not established as written.

minor comments (5)

[Page 2] The word 'imagenarity' should be 'imaginarity'.
[Page 5] The phrase 'for a tupe of quantum states' should be 'for a tuple of quantum states'.
[Theorem 6] In statement (3), there is a mismatched parenthesis in 'f (sum_i ~p_i|<~psi*_i|~psi_i>|)) <= ...'; one parenthesis should be removed.
[Proof of Theorem 3] The expression 'Phi(rho)=Phi(Phi_0(|psi>))' omits the state |psi><psi|; it should read Phi(rho)=Phi(Phi_0(|psi><psi|)).
[Proposition 2] The definition of K_j contains a denominator sqrt(1-x), which is undefined when x=1; the case x=1 (all pure states real) should be handled separately or excluded explicitly, even though the statement can be recovered by a limiting argument.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the convex-roof and least-imaginarity constructions are proved from stated axioms, and the authors' self-citations are not load-bearing.

full rationale

The paper's central results are self-contained derivations rather than restatements of inputs. Theorem 1 defines I_f from an arbitrary decreasing concave f and proves the measure axioms from that definition; the converse constructs f from a given measure, which is a characterization, not a circular assumption. The monotone ~I_f is defined via the conversion set R(ρ) and its monotonicity is proven using Proposition 2 and the properties of f, with no hidden reliance on the conclusion. Theorem 6's equivalence is the content of the theorem, not a premise used to prove earlier results. The authors' prior works [38], [41], [45] are cited for motivation or as pointers to related frameworks, and the proofs do not reduce to those citations. The mathematical concerns raised independently about the concavity of f_k(x)=1-(x^k∧1) and about the inference from p2 a^k > b^k are correctness or rigor issues, not circularity; per the review rules they do not affect the circularity score.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The constructions are mathematical and rely on standard resource-theory definitions plus a few cited lemmas. No new physical entities are introduced. The only adjustable inputs are the functions f and parameters k used to define particular measures.

free parameters (2)

f(x) = not fitted; arbitrary decreasing concave function with f(1)=0
The convex roof measure I_f and the monotone ~I_f are defined for any f satisfying (i)-(iii); the paper gives examples but no data selects f.
k in f_k(x)=1 - x^k and 1 = chosen in proof of Theorem 2
The family f_k is used to construct the counterexample in Theorem 2; k is a free tuning parameter, not fixed by any principle.

assumptions (5)

domain assumption Definition of real states and real operations as free states and free operations
Taken from Refs. [1,28]; the entire resource-theoretic framework rests on this.
domain assumption Any quantum state can be produced from some pure state by a real operation (Ref. [32])
Needed for R(rho) to be nonempty in the definition of ~I_f.
domain assumption Proposition 1 (Ref. [28]): canonical form of pure states under real orthogonal operations
Used in Proposition 2 and in the proof of Theorem 1's converse.
domain assumption Equivalence of (I3)+(I4) with (I2)+(I5) (Ref. [29])
Used to justify additivity for direct sums in Theorem 2's proof.
standard math Standard convex analysis facts about concavity and Cauchy-Schwarz
Used throughout the proofs.

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Pith. "Pith review of Quantifying imaginarity in terms of pure-state imaginarity." pith.science (2026). https://pith.science/paper/YFMJAQFB

@misc{pith2026241112215,
  author       = {Pith},
  title        = {Pith review of: Quantifying imaginarity in terms of pure-state imaginarity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YFMJAQFB}},
  note         = {Machine review of arXiv:2411.12215}
}

read the original abstract

Complex numbers are widely used in quantum physics and are indispensable components for describing quantum systems and their dynamical behavior. The resource theory of imaginarity has been built recently, enabling a systematic research of complex numbers in quantum information theory. In this work, we develop two theoretical methods for quantifying imaginarity, motivated by recent progress within resource theories of entanglement and coherence. We provide quantifiers of imaginarity by the convex roof construction and quantifiers of the imaginarity by the least imaginarity of the input pure states under real operations. We also apply these tools to study the state conversion problem in resource theory of imaginarity.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

On the Bargmann invariants for quantum imaginarity
quant-ph 2024-12 conditional novelty 6.0 of 10

The set of Bargmann invariants from circulant Gram matrices is exactly the n-th power of a regular n-gon, and all such invariants can be realized by qubits.

Reference graph

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This shows that convex roof quantiﬁers of imaginarity are more handy for operation. In addition, if f is strictly decreasing, then If is faithful, that is, If (ρ) = 0 if and only if ρ is a real state. Using Theorem 1, one can construct many interesting imaginarity measures. We provide two examples here. Example 1: Let f (x) = 1−x 2 , then If (|ψ ⟩) = 1 − ...

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

In addition, if f is strictly decreasing, then If is faithful, that is, If (ρ) = 0 if and only if ρ is a real state

This shows that convex roof quantiﬁers of imaginarity are more handy for operation. In addition, if f is strictly decreasing, then If is faithful, that is, If (ρ) = 0 if and only if ρ is a real state. Using Theorem 1, one can construct many interesting imaginarity measures. We provide two examples here. Example 1: Let f (x) = 1−x 2 , then If (|ψ ⟩) = 1 − ...

work page

[2] [2]

Hickey and G

A. Hickey and G. Gour, Quantifying the imaginarity of quantum mechanics, J. Phys. A: Math. Theor. 51, 414009 (2018)

2018

[3] [3]

Chitambar and G

E. Chitambar and G. Gour, Quantum resource theories, Rev. Mod. Phys. 91, 025001 (2019)

2019

[4] [4]

E. C. Stueckelberg, Quantum theory in real Hilbert space, Helv. Phys. Acta 33, 458 (1960)

work page 1960

[5] [5]

E. C. G. Stueckelberg von Breidenbach and M. Guenin, Quantum theory in real Hilbert space II (Addenda and Errats), Helv. Phys. Acta 34, 621 (1961)

work page 1961

[6] [6]

W. K. Wootters, Entanglement sharing in real vector space quantum Theory, Foundations of Physics 42, 19 (2010)

work page 2010

[7] [7]

Hardy and W

L. Hardy and W. K. Wootters, Limited holism and real vector space quantum theory, Foundations of Physics 42, 454 (2011)

work page 2011

[8] [8]

Aleksandrova, V

A. Aleksandrova, V. Borish, and W. K. Wootters, Real vector space quantum theory with a universal quantum bit, Phys. Rev. A 87, 052106 (2013)

work page 2013