Mutual compatibility/incompatibility of quasi-Hermitian quantum observables

Miloslav Znojil

REVIEW 2 major objections 5 minor 33 references

Mutual compatibility/incompatibility of quasi-Hermitian quantum observables

T0 review · 2 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read Two non-Hermitian observables are compatible exactly when a diagonal rescaling makes their frame link unitary.

desk verdict The exact compatibility criterion (Lemma 2) is sound and is the real takeaway, but the paper's practical criteria in Section 4.2 mis-count constraints and the N=2 'parametric freedom survives' claim is false. read the letter →

arxiv 2505.00791 v1 pith:AJ3W4QYU submitted 2025-05-01 math-ph math.FAmath.MPquant-ph

Miloslav Znojil This is my paper · ORCID

classification math-phmath.FAmath.MPquant-ph MSC 81Q1247B50

keywords quasi-Hermitianobservablessharedinner-productmetricnon-HermitianquantummechanicsDysonmapmatrixcompatibilitycriterionrealspectrafinite-dimensionalHilbertspacepositivediagonalscaling

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 asks when two non-Hmitian operators with real spectra can both be observables of one quantum system. In quasi-Hermitian quantum mechanics, an observable need only be Hermitian after a change of physical inner product, but a single metric must serve every observable. The paper proves that for a pair of $N\times N$ candidates with real non-degenerate spectra, a shared metric exists if and only if positive diagonal matrices $c_1,c_2$ can rescale the matrix $M=\Omega_1\Omega_2^{-1}$ into a unitary matrix $U=c_1Mc_2^{-1}$. It exhibits a three-by-three pair, one real asymmetric and one complex symmetric, whose spectra are real yet which fails the test, so no metric makes them simultaneously quasi-Hermitian. The payoff is that the result reduces a question about operators and metrics to a finite algebraic test, and it warns that real spectra alone do not make a set of observables mutually compatible.

What carries the argument

The load-bearing object is the frame-relation matrix $M=\Omega_1\Omega_2^{-1}$ between the two diagonalizing Dyson maps, together with the positive diagonal matrices $c_1,c_2$ that encode the arbitrary normalization of each eigenvector. Since each physical metric for $A_j$ is $\Theta_j=\Omega_j^\dagger c_j^2\Omega_j$, requiring one shared metric is the same as asking for $c_1,c_2$ with $\Omega_1^\dagger c_1^2\Omega_1=\Omega_2^\dagger c_2^2\Omega_2$; this equality is equivalent to $M^\dagger c_1^2M$ being diagonal and, in turn, to $c_1Mc_2^{-1}$ being unitary. The unitary form is the paper's compact criterion and the practical route to constructing compatible pairs from a chosen unitary matrix.

What would settle it

Solve the linear system (14) for the paper's $3\times3$ example at $s=1/2$, $a=1/2$: the unique solution forces $x_1=-\rho/12+i\sqrt3\,\rho/12$, $x_2=\rho/9$, $x_3=\rho$, with $\rho$ a free complex parameter, so the $c_j$ cannot be real positive; that is direct evidence against compatibility. To refute the general criterion, one would need either a pair whose only scaling solution is non-positive yet which admits a shared metric outside the $\Theta_j=\Omega_j^\dagger c_j^2\Omega_j$ family, or a pair that passes the unitary-scaling test but still has no shared metric.

Watch

Extended reading notes

Core claim

Let $A_1$ and $A_2$ be two $N\times N$ matrices with real, non-degenerate spectra, and choose Dyson maps $\Omega_1,\Omega_2$ such that $a_j=\Omega_j A_j\Omega_j^{-1}$ is real diagonal. Set $M=\Omega_1\Omega_2^{-1}$. The paper's central claim is that $A_1$ and $A_2$ are compatible—there exists a positive-definite metric $\Theta$ with $A_j^\dagger\Theta=\Theta A_j$ for $j=1,2$—if and only if there are positive diagonal matrices $c_1,c_2$ for which $U=c_1Mc_2^{-1}$ is unitary. Equivalently, the off-diagonal entries of $M^\dagger c_1^2 M$ must vanish, with its diagonal entries defining $c_2^2$. The proof runs through the residual normalization freedom of the eigenvector matrices: every eligible metric for $A_j$ has the form $\Theta_j=\Omega_j^\dagger c_j^2\Omega_j$, so a shared metric is exactly a choice of $c_1,c_2$ making these two expressions equal. The paper's worked $3\times3$ example has real spectra but forces the scaling parameters to be non-real, demonstrating an incompatible pair.

Load-bearing premise

The if-and-only-if claim rests on treating positive rescaling of each eigenvector as the only metric freedom, which requires real, non-degenerate spectra; with degenerate eigenvalues the admissible metrics form a strictly larger family and the diagonal-scaling test stops being exhaustive.

Editorial extensions

If this is right

Compatibility of two quasi-Hermitian observables with real non-degenerate spectra can be decided by the linear constraints (16) on $c_1^2$; no search over all metrics is needed.
At $N\geq4$, the $N(N-1)/2$ off-diagonal constraints outnumber the $N$ free parameters, so a generic pair of non-Hermitian candidates is incompatible.
At $N=2$ there is only one constraint on two parameters, so compatible pairs remain typical within the class considered.
For $K$ observables the same test requires $K-1$ equations of type (16) on $N$ parameters, so adding observables makes shared metrics rapidly more restrictive.
The explicit $3\times3$ example shows that each operator having a real spectrum does not imply the pair is jointly observable; hidden Hermiticity of the individual operators is insufficient.

Reading between the lines

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

A numerical route the paper does not develop: for a given $M$, search over positive diagonal $c_1,c_2$ by reducing the deviation of $c_1Mc_2^{-1}$ from unitarity; the same algorithm would serve as a practical compatibility test for larger $N$.
The parameter count suggests a rigidity principle the paper does not state: when compatibility holds for $N\geq3$, the shared metric is generically overdetermined rather than freely chosen, so adding observables fixes the physical inner product almost uniquely.
A testable extension: for $N=\infty$ approximations by finite truncations, the scaling parameters' behavior under refinement could indicate which infinite-dimensional pairs are compatible, although the paper warns the infinite limit is nontrivial.

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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 one thing worth knowing about this paper: the exact, non-perturbative compatibility criterion in Lemma 2 is correct, and it fills a real gap left by the earlier perturbative no-go. The paper shows that two finite-dimensional quasi-Hermitian observables with real, non-degenerate spectra are compatible iff the frame-relation matrix M = Ω1 Ω2^{-1} can be diagonal-scaled to unitarity. That is a clean and useful reformulation, and the accompanying reduction to linear equations (15)–(16) is practical. The explicit 3x3 pair that fails the test, even though both operators have real spectra, is a valuable counterexample; I spot-checked the eigenvector relations and they do satisfy Eq. (19).

The problems are concentrated in Section 4.2. The count of constraints in (16) treats each complex off-diagonal equality as a single real condition. Each of those is two real conditions. That is not a pedantic distinction: it breaks the N=2 claim. The paper says a single constraint on two real parameters leaves parametric freedom. But for the counterexample with A1 = diag(1,-1), A2 = [[1,2],[0,-1]], the off-diagonal equation forces x1 = 0, so no positive diagonal c1 exists and the pair is incompatible. The same miscount shifts the generic-incompatibility threshold from N ≥ 4 to N ≥ 3. So Lemma 2 survives, but the paper's advertised guidance about when compatibility is easy or generic is unreliable.

Minor points: the paper relies on non-degenerate spectra and finite dimension, and it says so, so those are limitations rather than hidden flaws. The example's decisive algebra is asserted rather than shown, but it checks out. The text is discursive and has typos, but it is readable.

Who is this for? People building quasi-Hermitian models with a second observable—cosmology, PT-symmetric contexts, model builders. They should read Lemma 2 and the example, and ignore the threshold statements until corrected. The core result is serious enough to deserve referee time, but not without requiring a fix to the constraint counting and the N=2 discussion. My recommendation: send it to peer review, with the practical criteria section flagged for major revision.

Referee Report

2 major / 5 minor

Summary. The manuscript addresses the problem of when two bounded non-Hermitian operators A1 and A2 on a finite-dimensional Hilbert space, with real spectra, can be observables of a single quasi-Hermitian quantum theory, i.e. admit a common positive metric Θ satisfying A_j^† Θ = Θ A_j. The author's strategy is to diagonalize each A_j via a Dyson map Ω_j and to parametrize all metrics for A_j by Θ_j(c_j) = Ω_j† c_j^2 Ω_j with positive diagonal c_j. Lemma 1 states that compatibility is equivalent to the existence of c1,c2 with Θ1(c1)=Θ2(c2); Lemma 2 restates this as unitarity of U(c1,c2)=c1 Ω1 Ω2^{-1} c2^{-1}. The paper then discusses counts of constraints, claims generic incompatibility for N≥4 and surviving parametric freedom at N=2, and illustrates the results with a 3x3 pair with real spectra that is shown to be incompatible.

Significance. The core algebraic reformulation is attractive and, under the stated non-degeneracy assumption, correct: Lemma 1 follows from the standard characterization of metrics for diagonalizable operators, and Lemma 2's equivalence to Eq. (15) is verified by direct multiplication. The 3x3 example is a concrete demonstration of incompatibility, and the paper's Lemma 2, if properly scoped, would provide a useful criterion for simultaneous quasi-Hermiticity. However, the paper's practical counting criterion contains a load-bearing error that affects the claimed threshold and the N=2 discussion, and Lemma 2 as stated is missing the non-degeneracy hypothesis. With those fixed, the criterion would be a useful contribution.

major comments (2)

[Section 4.2, Eq. (16) (and its use in Section 5)] The constraint count is incorrect. Each off-diagonal equation (M† c1² M)_{mn}=0 for m<n is one complex equation, i.e. two real equations, and the diagonal entries of M† c1² M must be positive; the paper counts N(N-1)/2 complex constraints and ignores positivity. The correct real count is N(N-1) equations for N positive variables, so generic incompatibility already starts at N=3, not N≥4. The claim that at N=2 'the parametric freedom survives' is false: with A1=diag(1,-1), A2=[[1,2],[0,-1]], Ω1=I and Ω2=[[1,1],[0,1]], one obtains M=[[1,-1],[0,1]] and M† c1² M=[[x1,-x1],[-x1,x1+x2]], so Eq. (16) forces x1=0 and no positive c1 exists. The pair is indeed incompatible, since for non-degenerate A1 any shared Θ must be diagonal and then fails for A2. Thus the practical criterion as stated misfires already in the minimal case.
[Lemma 2 (Section 4.2)] The statement is missing the non-degeneracy hypothesis that is used to make the c_j parametrization exhaustive in Lemma 1. For degenerate spectra, positive operators commuting with a_j are not only diagonal, so the equivalence can fail. Example: A1=I and A2=[[1,2],[0,-1]] are compatible—A1 is Hermitian and A2 is quasi-Hermitian with shared metric Θ=[[1,1],[1,2]]. Fix Ω1=I and Ω2=[[1,1],[0,1]] as above; then M=[[1,-1],[0,1]] and c2²=M†c1²M forces the off-diagonal entry -x1 to vanish, so no positive diagonal c1 exists. The shared metric is not of the form Ω1†c1²Ω1 with diagonal c1, showing that the 'only if' direction of Lemma 2 requires the non-degeneracy assumption (or a block-diagonal generalization).

minor comments (5)

[Section 4.1, above Lemma 1] 'c1 and c1' should read 'c1 and c2'.
[Section 5] 'aadditional' is a typo for 'additional'.
[Section 6] 'Ai and A2' should read 'A1 and A2'.
[Section 4.2] The phrase 'Both sides of this relation are Hermitian matrices which have two parts' is vague; it would be clearer to state that Eq. (15) is equivalent to the vanishing of all off-diagonal entries of M† c1² M together with positivity of its diagonal entries.
[Abstract and Section 3.1] The abstract's unqualified 'criteria of existence' should mention the finite-dimensional, non-degenerate setting in which the criteria are proved, since Section 3.1 explicitly restricts to N<∞ and Section 6 warns that N=∞ is nontrivial.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Lemmas 1 and 2 follow by direct algebraic rewrite of the quasi-Hermiticity and shared-metric equations; self-citations are motivational or restate an elementary residual-freedom fact derived in the text.

full rationale

The paper's central claims are Lemma 1 (shared metric iff positive diagonal scalings make Theta_1(c1)=Theta_2(c2)) and Lemma 2 (equivalently, U=c1 M c2^{-1} is unitary). Neither claim is an input: Eq. (14) is just the quasi-Hermiticity condition A_j^dagger Theta = Theta A_j written in the eigenbases fixed by Eqs. (8)-(10); Eqs. (15) and (17) are algebraic rearrangements using M=Omega_1 Omega_2^{-1}. The residual-freedom statement that all metrics are Omega_j^dagger c_j^2 Omega_j is derived in Section 4.1 from the fact that each column of Omega_j^dagger is an eigenvector and can be scaled independently; the citation to the author's earlier work [22] is not the load-bearing argument. The example in Section 5 explicitly solves the linear equations and finds no positive real solution, so the incompatibility verdict is computed, not assumed. Section 3.1's finite-dimension restriction and Section 4.1's non-degeneracy assumption are stated hypotheses, not circular definitions. The skeptical objection about counting complex off-diagonal constraints as real equations is a mathematical-correctness concern about Section 4.2's parametric count, not a circularity of the derivation: Lemma 2's equivalence itself is unaffected. There is therefore no step where a prediction reduces to a fitted input, a defined quantity is defined in terms of the target, or a self-citation carries the argument without independent support.

Assumptions & free parameters 2 free parameters · 3 assumptions · 0 invented entities

The central criterion rests on three domain assumptions: non-degenerate real spectra (so metric freedom is exactly positive diagonal scaling), finite dimension, and positivity of the physical metric. The core mathematics is standard linear algebra. The hand-chosen example parameters s = a = 1/2 affect only the demonstration, not the general claim. No fitted parameters or invented entities appear.

free parameters (2)

s = 1/2
Example parameter in Eqs. (19) and (21); chosen by hand to make the 3 by 3 demonstration concrete. It is not fitted to data and does not enter the general criterion, but the example's incompatibility conclusion depends on this choice.
a = 1/2
Example parameter in Eqs. (20) and (22); chosen by hand inside the real-spectrum window |a| <= 1. Illustrative only, but the example's conclusion depends on it.

assumptions (3)

domain assumption Each observable candidate A_j is diagonalizable with real, non-degenerate spectrum, so the positive operators commuting with the diagonal eigenvalue matrix a_j are exactly the positive diagonal matrices (Section 4.1).
This exhaustiveness of the residual freedom underlies Lemma 1 and turns compatibility into the diagonal-scaling test (14)-(17). Degenerate spectra would enlarge the commutant and break the if-and-only-if.
domain assumption The Hilbert space is finite-dimensional, N < infinity (stated in Sections 3.1 and 6).
The linear-algebra criterion and the counting argument are finite-dimensional; the paper itself warns that the N to infinity limit is nontrivial.
domain assumption The physical metric Theta must be positive definite, bounded, with bounded inverse (from [3], used throughout).
Positivity is what rules out the example's complex solution and is the physical content of quasi-Hermiticity; the compatibility criterion is only valid for such admissible metrics.

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Cite this review

Pith. "Pith review of Mutual compatibility/incompatibility of quasi-Hermitian quantum observables." pith.science (2026). https://pith.science/paper/AJ3W4QYU

@misc{pith2026250500791,
  author       = {Pith},
  title        = {Pith review of: Mutual compatibility/incompatibility of quasi-Hermitian quantum observables},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AJ3W4QYU}},
  note         = {Machine review of arXiv:2505.00791}
}

abstract

In the framework of quasi-Hermitian quantum mechanics the eligible operators of observables may be non-Hermitian, $A_j\neq A_j^\dagger$, $j=1,2, \ldots,K$. In principle, the standard probabilistic interpretation of the theory can be re-established via a reconstruction of physical inner-product metric $\Theta\neq I$ guaranteeing the quasi-Hermiticity $A_j^\dagger \,\Theta=\Theta\,A_j$. The task is easy at $K=1$ because there are many eligible metrics $\Theta=\Theta(A_1)$. In our paper the next case with $K=2$ is analyzed. The criteria of the existence of a shared metric $\Theta=\Theta(A_1,A_2)$ are presented and discussed.

Discussion (0). Continue with ORCID to comment.

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Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

[1] [1]

Quantum Mechanics ; North Holland: Amsterdam, The Netherlands, 1961

Messiah, A. Quantum Mechanics ; North Holland: Amsterdam, The Netherlands, 1961

1961

[2] [2]

G.; Geyer, H

Scholtz, F. G.; Geyer, H. B.; Hahne, F. J. W. Quasi-Hermitian Oper ators in Quantum Mechanics and the Variational Principle. Ann. Phys. 1992, 213, 74 - 101

work page 1992

[3] [3]

On the metric operator for the imagina ry cubic oscillator

Siegl, P.; Krejˇ ciˇ r ´ ık, D. On the metric operator for the imagina ry cubic oscillator. Phys. Rev. 2012, D 86 , 121702(R)

work page 2012

[4] [4]

Quasi-Hermitian operators

Dieudonn´ e, J. Quasi-Hermitian operators. In Proceedings of t he International Sym- posium on Linear Spaces, Jerusalem, Israel, 5–12 July 1961; Perga mon: Oxford, UK, 1961; pp. 115–122

work page 1961

[5] [5]

Pseudo-Hermitian Quantum Mechanics

Mostafazadeh, A. Pseudo-Hermitian Quantum Mechanics. Int. J. Geom. Meth. Mod. Phys. 2010, 7, 1191-1306

2010

[6] [6]

Equivalence of unstable anharmonic oscilla tors and double wells

Buslaev, V.; Grecchi, V. Equivalence of unstable anharmonic oscilla tors and double wells. J. Phys. A Math. Gen. 1993, 26, 5541–5549

work page 1993

[7] [7]

Problem of the co- existence of several non-Hermitian observables in PT-symmetric q uantum mechanics

Znojil, M.; Semor´ adov´ a, I.; R˚ uˇ ziˇ cka, F.; Moulla, H.; Leghrib,I. Problem of the co- existence of several non-Hermitian observables in PT-symmetric q uantum mechanics. Phys. Rev. A 2017, 95, 042122

work page 2017

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M., Boettcher, S

Bender, C. M., Boettcher, S. Real spectra in non-Hermitian Ham iltonians having PT symmetry. Phys. Rev. Lett. 1998, 80, 5243

work page 1998