Measure of not-completely-positive qubit maps: the general case

Francesco Petruccione; R. Srikanth; Vinayak Jagadish

arxiv: 1907.10111 · v1 · pith:C7MUWQD3new · submitted 2019-07-23 · 🪐 quant-ph

Measure of not-completely-positive qubit maps: the general case

Vinayak Jagadish , R. Srikanth , Francesco Petruccione This is my paper

Pith reviewed 2026-05-24 17:11 UTC · model grok-4.3

classification 🪐 quant-ph

keywords not-completely-positive mapsqubit mapsdynamical matrixvolume measurequantum operationspositivity domain

0 comments

The pith

The volume measure of qubit maps including not-completely-positive ones is not well defined because their dynamical matrices have unbounded eigenvalue spectra.

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

The paper proposes a definition for valid not-completely-positive maps on qubits because these maps may lack a full positivity domain. Under this definition it proves that the eigenvalue spectrum of the dynamical matrix has no bound. This unbounded spectrum means the set of such maps cannot be given a finite volume. A reader would care because volume measures are used to quantify the relative size of different classes of quantum operations.

Core claim

The set of not-completely-positive (NCP) maps is unbounded unless further assumptions are made. This follows from proposing a reasonable definition of a valid NCP map motivated by specific examples where NCP maps lack a full positivity domain. For valid NCP maps the eigenvalue spectrum of the corresponding dynamical matrix is not bounded. Therefore in general the volume measure of qubit maps including NCP maps is not well defined.

What carries the argument

The dynamical matrix of a map, whose eigenvalue spectrum is shown to be unbounded for valid NCP maps.

If this is right

Any volume assigned to the full set of qubit maps diverges to infinity once valid NCP maps are admitted.
Finite-volume calculations of qubit maps must either exclude NCP maps or impose extra bounds on their parameters.
The space of all trace-preserving qubit maps cannot be treated as a finite-measure set when NCP maps are retained without restriction.

Where Pith is reading between the lines

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

Similar divergence may occur when attempting volume measures on maps of higher-dimensional systems.
Practical computations of map volumes will require additional regularization or cutoffs on the NCP sector.
Arguments that rely on integrating over all possible qubit maps will need to specify which subset of NCP maps is allowed.

Load-bearing premise

The definition of a valid NCP map proposed here is the appropriate one for deciding whether a volume measure exists.

What would settle it

A family of valid NCP maps in which the eigenvalues of every dynamical matrix remain bounded above and below by constants independent of any free parameter would falsify the unboundedness result.

Figures

Figures reproduced from arXiv: 1907.10111 by Francesco Petruccione, R. Srikanth, Vinayak Jagadish.

read the original abstract

We show that the set of not-completely-positive (NCP) maps is unbounded, unless further assumptions are made. This is done by first proposing a reasonable definition of a valid NCP map, which is nontrivial because NCP maps may lack a full positivity domain. The definition is motivated by specific examples. We prove that for valid NCP maps, the eigenvalue spectrum of the corresponding dynamical matrix is not bounded. Based on this, we argue that in general the volume measure of qubit maps, including NCP maps, is not well defined.

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

The paper defines a class of valid NCP maps and proves their dynamical matrices have unbounded eigenvalues, but the conclusion on volume measures rests on whether that definition is the right one.

read the letter

The main point is that the authors propose a definition of valid not-completely-positive qubit maps, motivated by cases where these maps lack a full positivity domain, and then prove that the eigenvalues of the associated dynamical matrix are unbounded under this definition. From there they conclude that volume measures over the space of qubit maps including NCP ones are not well defined in general. That is the concrete technical result they deliver. They do a clear job laying out why a special definition is needed and grounding it in examples, which makes the setup readable. The proof of unbounded spectrum follows from their chosen conditions, and there is no sign of circularity or parameter fitting since the work is a direct argument rather than data-driven. The math holds up on its own terms once the definition is accepted. The softer part is exactly the step the stress-test note flags: the paper shows the outcome for this particular definition but does not show that every reasonable alternative definition would produce the same unbounded spectrum. Without that, the claim that volume measures are ill-defined in general stays tied to one choice of what counts as valid. The abstract does not supply the derivation steps, so a referee would need to verify the details of the proof. This is a narrow technical note aimed at people already working on measures or geometry of quantum maps. A specialist reader might pick up the point about NCP maps and use it to think more carefully about how to restrict the space, but it is unlikely to change broader practice. It deserves peer review because the claim is precise and the underlying issue with measures is worth checking, even if the definition needs more justification in revision. I would send it out.

Referee Report

2 major / 0 minor

Summary. The manuscript proposes a definition of 'valid' not-completely-positive (NCP) qubit maps (motivated by examples, since NCP maps need not have a full positivity domain), proves that the eigenvalue spectrum of the associated dynamical matrix is unbounded under this definition, and concludes that the volume measure of qubit maps including NCP maps is therefore not well-defined in general.

Significance. If the central claim holds, the result would establish that standard volume measures on the space of qubit maps cannot be extended to include NCP maps without additional restrictions, which bears on attempts to quantify the 'size' of sets of quantum maps or channels in quantum information. The manuscript supplies a proof of unboundedness once the definition is fixed, which is a concrete technical contribution, but the significance is limited by the lack of argument that the chosen definition is canonical.

major comments (2)

[Abstract] Abstract: the step from 'eigenvalue spectrum unbounded under the proposed definition of valid NCP map' to 'volume measure of qubit maps including NCP maps is not well defined' is load-bearing and rests entirely on the appropriateness of that definition; the text supplies no independent argument that every reasonable alternative definition would also produce an unbounded set.
[Abstract] Abstract: the manuscript states that a proof exists that the spectrum is unbounded for valid NCP maps, but supplies neither the derivation steps, the explicit form of the dynamical matrix, nor any error analysis or counter-example verification, preventing direct assessment of the proof's correctness.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. The central claim is that a reasonable definition of valid NCP qubit maps, motivated by examples, leads to an unbounded eigenvalue spectrum for the dynamical matrix, implying that general volume measures on the space of qubit maps are ill-defined without further restrictions. We address each major comment point by point below.

read point-by-point responses

Referee: [Abstract] Abstract: the step from 'eigenvalue spectrum unbounded under the proposed definition of valid NCP map' to 'volume measure of qubit maps including NCP maps is not well defined' is load-bearing and rests entirely on the appropriateness of that definition; the text supplies no independent argument that every reasonable alternative definition would also produce an unbounded set.

Authors: We agree that the conclusion depends on the chosen definition of a valid NCP map. The definition is introduced as reasonable because it is directly motivated by concrete physical examples of NCP maps that lack a full positivity domain, as explained in the manuscript. We do not claim or prove that every conceivable alternative definition would necessarily produce an unbounded set; our result shows that, under this natural definition, the set is unbounded and therefore general volume measures cannot be extended without additional constraints. If the referee has a specific alternative definition in mind, we would be happy to examine whether the unboundedness persists. revision: partial
Referee: [Abstract] Abstract: the manuscript states that a proof exists that the spectrum is unbounded for valid NCP maps, but supplies neither the derivation steps, the explicit form of the dynamical matrix, nor any error analysis or counter-example verification, preventing direct assessment of the proof's correctness.

Authors: The full derivation, including the explicit construction of the dynamical matrix for the proposed valid NCP maps and the steps establishing that its eigenvalue spectrum is unbounded, appears in the body of the manuscript. The abstract is intended only as a concise summary of the result. To improve accessibility, we will revise the abstract to include a brief reference to the relevant section containing the proof details and the explicit matrix form. The proof itself incorporates the necessary verification steps; we can add a short remark on this in the revised text if the referee finds it helpful. revision: yes

standing simulated objections not resolved

Demonstrating that every possible reasonable alternative definition of valid NCP maps must also yield an unbounded spectrum, since this would require an exhaustive enumeration or classification of all conceivable definitions, which lies outside the scope of the work.

Circularity Check

0 steps flagged

No significant circularity; derivation follows from an explicitly proposed definition without reduction to inputs by construction.

full rationale

The paper states it proposes a definition of valid NCP map (motivated by examples), proves the dynamical matrix eigenvalues are unbounded under that definition, and concludes the volume measure is ill-defined in general. No equations, fitted parameters, or self-citations are visible that would reduce the central claim to its own inputs. This matches none of the enumerated circularity patterns and is self-contained as a direct consequence of the stated definitional assumption.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available, so the ledger is populated from the abstract statements alone. The definition of valid NCP map is introduced without independent justification beyond motivation by examples; the claim that the spectrum is unbounded rests on that definition.

axioms (1)

domain assumption A reasonable definition of valid NCP map exists even though NCP maps may lack a full positivity domain.
Stated in the abstract as the starting point for the proof.

pith-pipeline@v0.9.0 · 5617 in / 1178 out tokens · 18256 ms · 2026-05-24T17:11:22.819948+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/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

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

We show that the set of not-completely-positive (NCP) maps is unbounded, unless further assumptions are made. This is done by first proposing a reasonable definition of a valid NCP map...
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

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

Theorem 1. The set of maps (including NCP maps) acting on a qubit is neither closed nor bounded.

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

28 extracted references · 28 canonical work pages

[1]

Furthermore, for the noise to be described by a quantum dynamical semigroup, λ must be a positive constant

For Markovianity to hold in the sense of CP divisibility, λ must always be positive 7 as a function of q . Furthermore, for the noise to be described by a quantum dynamical semigroup, λ must be a positive constant. In a non-Markovian dephasing channel such as quantum random telegraph noise (RTN) [21], the negativity of λ arises from the presence of terms ...

work page
[2]

If we restrict to the case of Pauli channels, subject to the condition of positivity (i.e., having full positivity domain), then the volume measure of NCP maps is twice that of CP maps [15]. 10

work page
[3]

This corresponds to the simplex whose four vertices are U P U†, where P is a Pauli matrix

A similar result also holds if the Pauli channels are rotated by a ﬁxed unitary U. This corresponds to the simplex whose four vertices are U P U†, where P is a Pauli matrix. This may be considered as a rotated version of the Pauli tetrahedron T , discussed in [15]. Finally, note that the double-CNOT, or equivalently a double-C-phase gate, provides perhaps...

work page arXiv
[4]

Breuer and F

H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems (Oxford University Press, 2002)

work page 2002
[5]

R. F. Simmons and J. L. Park, Found. Phys. 11, 47 (1981)

work page 1981
[6]

G. A. Raggio and H. Primas, Found. Phys. 12, 433 (1982)

work page 1982
[7]

R. F. Simmons and J. L. Park, Found. Phys. 12, 437 (1982). 11

work page 1982
[8]

Pechukas, Phys

P. Pechukas, Phys. Rev. Lett. 73, 1060 (1994)

work page 1994
[9]

Alicki, Phys

R. Alicki, Phys. Rev. Lett. 75, 3020 (1995)

work page 1995
[10]

Pechukas, Phys

P. Pechukas, Phys. Rev. Lett. 75, 3021 (1995)

work page 1995
[11]

T. F. Jordan, A. Shaji, and E. C. G. Sudarshan, Phys. Rev. A 70, 052110 (2004)

work page 2004
[12]

H. A. Carteret, D. R. Terno, and K. ˙Zyczkowski, Phys. Rev. A 77, 042113 (2008)

work page 2008
[13]

Breuer, E.-M

H.-P. Breuer, E.-M. Laine, J. Piilo, and B. Vacchini, Rev. Mod. Phys. 88, 021002 (2016)

work page 2016
[14]

L. Li, M. J. W. Hall, and H. M. Wiseman, Phys. Rep. 759, 1 (2018)

work page 2018
[15]

S. J. Szarek, E. Werner, and K. ˙Zyczkowski, J. Math. Phys. 49, 032113 (2008)

work page 2008
[16]

M. E. Cuﬀaro and W. C. Myrvold, Philos. Sci. 80, 1125 (2013)

work page 2013
[17]

Rivas, S

A. Rivas, S. F. Huelga, and M. B. Plenio, Phys. Rev. Lett. 105, 050403 (2010)

work page 2010
[18]

Jagadish, R

V. Jagadish, R. Srikanth, and F. Petruccione, Phys. Rev. A 99, 022321 (2019)

work page 2019
[19]

Shrikant, R

U. Shrikant, R. Srikanth, and S. Banerjee, Phys. Rev. A 98, 032328 (2018)

work page 2018
[20]

E. C. G. Sudarshan, P. M. Mathews, and J. Rau, Phys. Rev. 121, 920 (1961)

work page 1961
[21]

Choi, Linear Algebra Appl

M.-D. Choi, Linear Algebra Appl. 10, 285 (1975)

work page 1975
[22]

Schmid, K

D. Schmid, K. Ried, and R. W. Spekkens, arXiv:1806.02381 (2018)

work page arXiv 2018
[23]

Jagadish and F

V. Jagadish and F. Petruccione, Quanta 7, 54 (2018)

work page 2018
[24]

Daﬀer, K

S. Daﬀer, K. W´ odkiewicz, J. D. Cresser, and J. K. McIver, Phys. Rev. A 70, 010304(R) (2004)

work page 2004
[25]

Andersson, J

E. Andersson, J. D. Cresser, and M. J. W. Hall, J. Mod. Opt. 54, 1695 (2007)

work page 2007
[26]

Rivas, S

A. Rivas, S. F. Huelga, and M. B. Plenio, Rep. Prog. Phys. 77, 094001 (2014)

work page 2014
[27]

Jamio lkowski, Rep

A. Jamio lkowski, Rep. Math. Phys. 3, 275 (1972)

work page 1972
[28]

Chru´ sci´ nski and A

D. Chru´ sci´ nski and A. Kossakowski, Phys. Rev. Lett.104, 070406 (2010)

work page 2010

[1] [1]

Furthermore, for the noise to be described by a quantum dynamical semigroup, λ must be a positive constant

For Markovianity to hold in the sense of CP divisibility, λ must always be positive 7 as a function of q . Furthermore, for the noise to be described by a quantum dynamical semigroup, λ must be a positive constant. In a non-Markovian dephasing channel such as quantum random telegraph noise (RTN) [21], the negativity of λ arises from the presence of terms ...

work page

[2] [2]

If we restrict to the case of Pauli channels, subject to the condition of positivity (i.e., having full positivity domain), then the volume measure of NCP maps is twice that of CP maps [15]. 10

work page

[3] [3]

This corresponds to the simplex whose four vertices are U P U†, where P is a Pauli matrix

A similar result also holds if the Pauli channels are rotated by a ﬁxed unitary U. This corresponds to the simplex whose four vertices are U P U†, where P is a Pauli matrix. This may be considered as a rotated version of the Pauli tetrahedron T , discussed in [15]. Finally, note that the double-CNOT, or equivalently a double-C-phase gate, provides perhaps...

work page arXiv

[4] [4]

Breuer and F

H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems (Oxford University Press, 2002)

work page 2002

[5] [5]

R. F. Simmons and J. L. Park, Found. Phys. 11, 47 (1981)

work page 1981

[6] [6]

G. A. Raggio and H. Primas, Found. Phys. 12, 433 (1982)

work page 1982

[7] [7]

R. F. Simmons and J. L. Park, Found. Phys. 12, 437 (1982). 11

work page 1982

[8] [8]

Pechukas, Phys

P. Pechukas, Phys. Rev. Lett. 73, 1060 (1994)

work page 1994

[9] [9]

Alicki, Phys

R. Alicki, Phys. Rev. Lett. 75, 3020 (1995)

work page 1995

[10] [10]

Pechukas, Phys

P. Pechukas, Phys. Rev. Lett. 75, 3021 (1995)

work page 1995

[11] [11]

T. F. Jordan, A. Shaji, and E. C. G. Sudarshan, Phys. Rev. A 70, 052110 (2004)

work page 2004

[12] [12]

H. A. Carteret, D. R. Terno, and K. ˙Zyczkowski, Phys. Rev. A 77, 042113 (2008)

work page 2008

[13] [13]

Breuer, E.-M

H.-P. Breuer, E.-M. Laine, J. Piilo, and B. Vacchini, Rev. Mod. Phys. 88, 021002 (2016)

work page 2016

[14] [14]

L. Li, M. J. W. Hall, and H. M. Wiseman, Phys. Rep. 759, 1 (2018)

work page 2018

[15] [15]

S. J. Szarek, E. Werner, and K. ˙Zyczkowski, J. Math. Phys. 49, 032113 (2008)

work page 2008

[16] [16]

M. E. Cuﬀaro and W. C. Myrvold, Philos. Sci. 80, 1125 (2013)

work page 2013

[17] [17]

Rivas, S

A. Rivas, S. F. Huelga, and M. B. Plenio, Phys. Rev. Lett. 105, 050403 (2010)

work page 2010

[18] [18]

Jagadish, R

V. Jagadish, R. Srikanth, and F. Petruccione, Phys. Rev. A 99, 022321 (2019)

work page 2019

[19] [19]

Shrikant, R

U. Shrikant, R. Srikanth, and S. Banerjee, Phys. Rev. A 98, 032328 (2018)

work page 2018

[20] [20]

E. C. G. Sudarshan, P. M. Mathews, and J. Rau, Phys. Rev. 121, 920 (1961)

work page 1961

[21] [21]

Choi, Linear Algebra Appl

M.-D. Choi, Linear Algebra Appl. 10, 285 (1975)

work page 1975

[22] [22]

Schmid, K

D. Schmid, K. Ried, and R. W. Spekkens, arXiv:1806.02381 (2018)

work page arXiv 2018

[23] [23]

Jagadish and F

V. Jagadish and F. Petruccione, Quanta 7, 54 (2018)

work page 2018

[24] [24]

Daﬀer, K

S. Daﬀer, K. W´ odkiewicz, J. D. Cresser, and J. K. McIver, Phys. Rev. A 70, 010304(R) (2004)

work page 2004

[25] [25]

Andersson, J

E. Andersson, J. D. Cresser, and M. J. W. Hall, J. Mod. Opt. 54, 1695 (2007)

work page 2007

[26] [26]

Rivas, S

A. Rivas, S. F. Huelga, and M. B. Plenio, Rep. Prog. Phys. 77, 094001 (2014)

work page 2014

[27] [27]

Jamio lkowski, Rep

A. Jamio lkowski, Rep. Math. Phys. 3, 275 (1972)

work page 1972

[28] [28]

Chru´ sci´ nski and A

D. Chru´ sci´ nski and A. Kossakowski, Phys. Rev. Lett.104, 070406 (2010)

work page 2010