A note on the quantum Wielandt inequality

Owen Ekblad

arxiv: 2504.21638 · v3 · submitted 2025-04-30 · 🪐 quant-ph · cs.IT· math.IT· math.OA

A note on the quantum Wielandt inequality

Owen Ekblad This is my paper

Pith reviewed 2026-05-22 17:39 UTC · model grok-4.3

classification 🪐 quant-ph cs.ITmath.ITmath.OA

keywords quantum Wielandt inequalityindex of primitivityunital Schwarz mapscompletely positive mapsquantum channelsmatrix algebrasprimitivity bound

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The pith

The index of primitivity for any primitive unital Schwarz map on D by D matrices is at most 2(D-1) squared.

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

This note establishes a bound on how quickly a certain class of quantum maps becomes strictly positive under iteration. The author adapts an existing proof technique to cover unital Schwarz maps that are not trace preserving, showing the same numerical bound still holds. A sympathetic reader would care because the result immediately extends to all primitive completely positive maps, which model many physical quantum operations. The bound therefore supplies a uniform control on convergence or mixing behavior across a wider family of maps than previously known.

Core claim

In this note, we prove that the index of primitivity of any primitive unital Schwarz map is at most 2(D-1)^2, where D is the dimension of the underlying matrix algebra. This inequality was first proved by Rahaman for Schwarz maps which were both unital and trace preserving. As we show, the assumption of unitality is basically innocuous, but in general not all primitive unital Schwarz maps are trace preserving. Therefore, the precise purpose of this note is to showcase how to apply the method of Rahaman to unital primitive Schwarz maps that don't preserve trace. As a corollary, the same bound holds for any primitive 2-positive map and hence for arbitrary primitive completely positive maps.

What carries the argument

The index of primitivity, the smallest positive integer k such that the k-fold composition of the map has a strictly positive kernel on the matrix algebra.

If this is right

The bound applies directly to every primitive 2-positive map.
The bound applies to every primitive completely positive map.
The result supplies a uniform estimate for the number of iterations needed for positivity in a broader class of quantum operations.
The bound connects to questions about the minimal number of steps guaranteeing that a quantum channel reaches full support.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same adaptation technique might apply to other relaxations of the trace-preserving condition, such as maps that preserve a different positive functional.
For small matrix sizes the bound could be checked by exhaustive search over the coefficients of candidate maps to see whether it is ever tight.
If the conjecture of Perez-Garcia, Verstraete, Wolf and Cirac concerns the same quantity, this note gives an explicit finite upper bound that any counter-example would have to exceed.

Load-bearing premise

The method developed by Rahaman for unital trace-preserving Schwarz maps can be adapted to the case of unital Schwarz maps that do not preserve trace.

What would settle it

Exhibit a concrete primitive unital Schwarz map on 2 by 2 matrices whose index of primitivity exceeds 2.

read the original abstract

In this note, we prove that the index of primitivity of any primitive unital Schwarz map is at most $2(D-1)^2$, where $D$ is the dimension of the underlying matrix algebra. This inequality was first proved by Rahaman for Schwarz maps which were both unital and trace preserving. As we show, the assumption of unitality is basically innocuous, but in general not all primitive unital Schwarz maps are trace preserving. Therefore, the precise purpose of this note is to showcase how to apply the method of Rahaman to unital primitive Schwarz maps that don't preserve trace. As a corollary of this theorem, we show that the index of primitivity of any primitive 2-positive map is at most $2(D-1)^2$, so in particular this bound holds for arbitrary primitive completely positive maps. We briefly discuss of how this relates to a conjecture of Perez-Garcia, Verstraete, Wolf and Cirac.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This note shows Rahaman's primitivity bound carries over to unital Schwarz maps without trace preservation, with the same 2(D-1)^2 index.

read the letter

The main thing to know is that this short note extends Rahaman's bound on the index of primitivity to primitive unital Schwarz maps that are not trace-preserving. The author adapts the combinatorial argument based on support graphs or iterated images and recovers the same bound of 2(D-1)^2. Unitality turns out to be enough, and the note walks through why the counting steps do not require trace preservation. As a corollary it immediately gives the bound for primitive 2-positive maps, which includes all primitive completely positive maps. That part is clean and directly useful for work on quantum channel mixing times. The discussion of the Perez-Garcia-Verstraete-Wolf-Cirac conjecture is mentioned but stays brief. The paper does a solid job of making the adaptation explicit rather than hand-waving it. The central claim holds up on the terms given, and the extension is not trivial even if the core technique is borrowed. The soft spots are modest. This is mostly a technical clarification rather than a new method or tighter bound. It leans heavily on the earlier Rahaman result with no independent verification or numerical checks provided. The note is short, so it does not test the bound on concrete examples or explore how much room there is between 2(D-1)^2 and actual indices in practice. If the adaptation has any edge case in the non-trace-preserving setting, it would need a referee to check the details carefully, but nothing in the outline suggests a load-bearing gap. This paper is for people already working on quantum information theory, specifically those who use primitivity indices or Wielandt-type bounds when analyzing convergence of quantum channels. A reader who needs the precise statement for unital but non-trace-preserving maps will find it convenient as a reference. It deserves a serious referee if submitted anywhere, because the extension is well-motivated, the argument is internally consistent, and the corollary for CP maps adds a small but clear piece to the literature.

Referee Report

0 major / 3 minor

Summary. The manuscript proves that the primitivity index of any primitive unital Schwarz map on the D-dimensional matrix algebra is at most 2(D-1)^2. It adapts the combinatorial support-graph argument of Rahaman (originally for unital trace-preserving Schwarz maps) to the general unital case, shows that trace preservation is not required, and obtains the same bound as a corollary for primitive 2-positive maps (hence for primitive CP maps). The note briefly relates the result to the Perez-Garcia–Verstraete–Wolf–Cirac conjecture.

Significance. If the adaptation holds, the result usefully widens the quantum Wielandt inequality to a larger class of maps that arise in quantum information. The explicit verification that the counting argument on iterated images carries over without trace preservation is a concrete strength, as is the immediate corollary for completely positive maps. The manuscript thereby supplies a modest but clean extension of an existing bound.

minor comments (3)

The phrase 'basically innocuous' in the introduction is informal; replace it with a one-sentence statement of which properties of the support graph survive without trace preservation.
Add a short remark after the main theorem clarifying whether the bound is known to be sharp for non-trace-preserving examples.
The reference list should include the precise citation for Rahaman’s original argument so readers can compare the two proofs directly.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, including the recognition that the adaptation of Rahaman's combinatorial support-graph argument to unital Schwarz maps (without trace preservation) is a concrete strength, and for noting the immediate corollary for primitive completely positive maps. The referee's summary accurately captures the purpose of the note as a modest but clean extension of the existing bound. Since the recommendation is minor revision but no specific major comments are listed, we have conducted an additional review of the text for any minor clarifications or typographical issues.

Circularity Check

0 steps flagged

No significant circularity; derivation adapts external prior result

full rationale

The paper's central result adapts Rahaman's combinatorial counting argument on iterated images and support graphs for the primitivity index, explicitly showing that unitality alone suffices without trace preservation. The bound 2(D-1)^2 follows from the same step-counting to full support as in the cited prior work, with the adaptation exhibited directly in the manuscript rather than assumed or fitted. Rahaman's result is an independent external reference (distinct authorship), providing non-circular support. No self-definitional loops, fitted inputs renamed as predictions, or load-bearing self-citations appear in the derivation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The proof rests on standard properties of unital Schwarz maps and primitivity in finite-dimensional operator algebras; no new free parameters, ad-hoc axioms, or invented entities are introduced.

axioms (1)

domain assumption Standard properties of unital Schwarz maps on matrix algebras hold as in prior literature
Invoked to extend Rahaman's method to the non-trace-preserving case

pith-pipeline@v0.9.0 · 5688 in / 1129 out tokens · 28903 ms · 2026-05-22T17:39:56.774248+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem A. q(ϕ) ≤ 2(D−1)^2 for primitive unital Schwarz maps ϕ : M_D → M_D.
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We prove that the index of primitivity of any primitive unital Schwarz map is at most 2(D−1)^2

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

[1]

Generalized Choi maps in three-dimensional matrix alge- bra

S. J. Cho, S.-H. Kye, and S. G. Lee. “Generalized Choi maps in three-dimensional matrix alge- bra”. In:Linear Algebra and its Applications171 (July 1992), pp. 213–224.issn: 00243795. DOI: https://doi.org/10.1016/0024-3795(92)90260-H

work page doi:10.1016/0024-3795(92)90260-h 1992
[2]

A Schwarz Inequality for Positive Linear Maps on C*-Algebras

M.-D. Choi. “A Schwarz Inequality for Positive Linear Maps on C*-Algebras”. In:Illinois J. Math.18.4 (1974), pp. 565–574. DOI:https://doi.org/10.1215/ijm/1256051007

work page doi:10.1215/ijm/1256051007 1974
[3]

Some Assorted Inequalities for Positive Linear Maps on C*-algebras

M.-D. Choi. “Some Assorted Inequalities for Positive Linear Maps on C*-algebras”. In:Journal of Operator Theory4.2 (1980), pp. 271–285.issn: 03794024, 18417744

work page 1980
[4]

Spectral Properties of Positive Maps on C*-Algebras

D. E. Evans and R. Høegh-Krohn. “Spectral Properties of Positive Maps on C*-Algebras”. In: Journal of the London Mathematical Societys2-17.2 (1978), pp. 345–355. DOI:https://doi. org/10.1112/jlms/s2-17.2.345

work page doi:10.1112/jlms/s2-17.2.345 1978
[5]

Spectral properties of tensor products of channels

S. Jaques and M. Rahaman. “Spectral properties of tensor products of channels”. In:Journal of Mathematical Analysis and Applications465.2 (Sept. 2018), pp. 1134–1158.issn: 0022247X. DOI:https://doi.org/10.1016/j.jmaa.2018.05.052

work page doi:10.1016/j.jmaa.2018.05.052 2018
[6]

A generic quantum Wielandt’s inequality

Y. Jia and A. Capel. “A generic quantum Wielandt’s inequality”. In:Quantum8 (May 2024), p. 1331.issn: 2521-327X. DOI:https://doi.org/10.22331/q-2024-05-02-1331

work page doi:10.22331/q-2024-05-02-1331 2024
[7]

Quantum Version of Wielandt’s Inequality Revisited

M. Micha lek and Y. Shitov. “Quantum Version of Wielandt’s Inequality Revisited”. In:IEEE Transactions on Information Theory65.8 (2019), pp. 5239–5242. DOI:https://doi.org/10. 1109/TIT.2019.2897772

work page arXiv 2019
[8]

Matrix product state representations

D. Perez-Garcia et al. “Matrix product state representations”. In:Quantum Info. Comput.7.5 (July 2007), pp. 401–430.issn: 1533-7146

work page 2007
[9]

A New Bound on Quantum Wielandt Inequality

M. Rahaman. “A New Bound on Quantum Wielandt Inequality”. In:IEEE Transactions on Information Theory66.1 (Jan. 2020), pp. 147–154.issn: 0018-9448. DOI:https://doi.org/ 10.1109/TIT.2019.2945776

work page doi:10.1109/tit.2019.2945776 2020
[10]

A Quantum Version of Wielandt’s Inequality

M. Sanz et al. “A Quantum Version of Wielandt’s Inequality”. In:IEEE Transactions on Infor- mation Theory56.9 (Sept. 2010), pp. 4668–4673.issn: 0018-9448. DOI:https://doi.org/10. 1109/TIT.2010.2054552

work page arXiv 2010
[11]

Shitov.Growth in matrix algebras and a conjecture of Perez-Garcia, Verstraete, Wolf and Cirac.https://vixra.org/pdf/2308.0028v1.pdf

Y. Shitov.Growth in matrix algebras and a conjecture of Perez-Garcia, Verstraete, Wolf and Cirac.https://vixra.org/pdf/2308.0028v1.pdf. 2024

work page arXiv 2024
[12]

Størmer.Positive Linear Maps of Operator Algebras

E. Størmer.Positive Linear Maps of Operator Algebras. Berlin, Heidelberg: Springer Berlin Hei- delberg, 2013.isbn: 978-3-642-34368-1. DOI:https://doi.org/10.1007/978-3-642-34369-8

work page doi:10.1007/978-3-642-34369-8 2013
[13]

Unzerlegbare, nicht negative Matrizen

H. Wielandt. “Unzerlegbare, nicht negative Matrizen”. In:Mathematische Zeitschrift52.1 (Dec. 1950), pp. 642–648.issn: 0025-5874. DOI:https://doi.org/10.1007/BF02230720

work page doi:10.1007/bf02230720 1950

[1] [1]

Generalized Choi maps in three-dimensional matrix alge- bra

S. J. Cho, S.-H. Kye, and S. G. Lee. “Generalized Choi maps in three-dimensional matrix alge- bra”. In:Linear Algebra and its Applications171 (July 1992), pp. 213–224.issn: 00243795. DOI: https://doi.org/10.1016/0024-3795(92)90260-H

work page doi:10.1016/0024-3795(92)90260-h 1992

[2] [2]

A Schwarz Inequality for Positive Linear Maps on C*-Algebras

M.-D. Choi. “A Schwarz Inequality for Positive Linear Maps on C*-Algebras”. In:Illinois J. Math.18.4 (1974), pp. 565–574. DOI:https://doi.org/10.1215/ijm/1256051007

work page doi:10.1215/ijm/1256051007 1974

[3] [3]

Some Assorted Inequalities for Positive Linear Maps on C*-algebras

M.-D. Choi. “Some Assorted Inequalities for Positive Linear Maps on C*-algebras”. In:Journal of Operator Theory4.2 (1980), pp. 271–285.issn: 03794024, 18417744

work page 1980

[4] [4]

Spectral Properties of Positive Maps on C*-Algebras

D. E. Evans and R. Høegh-Krohn. “Spectral Properties of Positive Maps on C*-Algebras”. In: Journal of the London Mathematical Societys2-17.2 (1978), pp. 345–355. DOI:https://doi. org/10.1112/jlms/s2-17.2.345

work page doi:10.1112/jlms/s2-17.2.345 1978

[5] [5]

Spectral properties of tensor products of channels

S. Jaques and M. Rahaman. “Spectral properties of tensor products of channels”. In:Journal of Mathematical Analysis and Applications465.2 (Sept. 2018), pp. 1134–1158.issn: 0022247X. DOI:https://doi.org/10.1016/j.jmaa.2018.05.052

work page doi:10.1016/j.jmaa.2018.05.052 2018

[6] [6]

A generic quantum Wielandt’s inequality

Y. Jia and A. Capel. “A generic quantum Wielandt’s inequality”. In:Quantum8 (May 2024), p. 1331.issn: 2521-327X. DOI:https://doi.org/10.22331/q-2024-05-02-1331

work page doi:10.22331/q-2024-05-02-1331 2024

[7] [7]

Quantum Version of Wielandt’s Inequality Revisited

M. Micha lek and Y. Shitov. “Quantum Version of Wielandt’s Inequality Revisited”. In:IEEE Transactions on Information Theory65.8 (2019), pp. 5239–5242. DOI:https://doi.org/10. 1109/TIT.2019.2897772

work page arXiv 2019

[8] [8]

Matrix product state representations

D. Perez-Garcia et al. “Matrix product state representations”. In:Quantum Info. Comput.7.5 (July 2007), pp. 401–430.issn: 1533-7146

work page 2007

[9] [9]

A New Bound on Quantum Wielandt Inequality

M. Rahaman. “A New Bound on Quantum Wielandt Inequality”. In:IEEE Transactions on Information Theory66.1 (Jan. 2020), pp. 147–154.issn: 0018-9448. DOI:https://doi.org/ 10.1109/TIT.2019.2945776

work page doi:10.1109/tit.2019.2945776 2020

[10] [10]

A Quantum Version of Wielandt’s Inequality

M. Sanz et al. “A Quantum Version of Wielandt’s Inequality”. In:IEEE Transactions on Infor- mation Theory56.9 (Sept. 2010), pp. 4668–4673.issn: 0018-9448. DOI:https://doi.org/10. 1109/TIT.2010.2054552

work page arXiv 2010

[11] [11]

Shitov.Growth in matrix algebras and a conjecture of Perez-Garcia, Verstraete, Wolf and Cirac.https://vixra.org/pdf/2308.0028v1.pdf

Y. Shitov.Growth in matrix algebras and a conjecture of Perez-Garcia, Verstraete, Wolf and Cirac.https://vixra.org/pdf/2308.0028v1.pdf. 2024

work page arXiv 2024

[12] [12]

Størmer.Positive Linear Maps of Operator Algebras

E. Størmer.Positive Linear Maps of Operator Algebras. Berlin, Heidelberg: Springer Berlin Hei- delberg, 2013.isbn: 978-3-642-34368-1. DOI:https://doi.org/10.1007/978-3-642-34369-8

work page doi:10.1007/978-3-642-34369-8 2013

[13] [13]

Unzerlegbare, nicht negative Matrizen

H. Wielandt. “Unzerlegbare, nicht negative Matrizen”. In:Mathematische Zeitschrift52.1 (Dec. 1950), pp. 642–648.issn: 0025-5874. DOI:https://doi.org/10.1007/BF02230720

work page doi:10.1007/bf02230720 1950