Open quantum dynamics without Complete Positivity: a criticism

Dariusz Chru\'sci\'nski; Fabio Benatti; Saverio Pascazio

arxiv: 2605.18639 · v1 · pith:KJD6OQNCnew · submitted 2026-05-18 · 🪐 quant-ph · math-ph· math.MP

Open quantum dynamics without Complete Positivity: a criticism

Fabio Benatti , Dariusz Chru\'sci\'nski , Saverio Pascazio This is my paper

Pith reviewed 2026-05-20 11:31 UTC · model grok-4.3

classification 🪐 quant-ph math-phmath.MP

keywords open quantum systemscomplete positivitynon-completely positive mapsdomain restrictionisotropic statescompatibility

0 comments

The pith

Restricting non-completely positive maps to compatible states grows too restrictive as system dimension increases.

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

The paper examines why complete positivity is often required for maps describing open quantum dynamics and tests an alternative that drops this requirement but limits the maps to only certain compatible initial states. Using isotropic states as the test case, it demonstrates that the allowed set of initial states shrinks rapidly as the dimension of the system grows. A reader would care because this shows the compatibility restriction cannot serve as a general workaround, forcing a choice between complete positivity or accepting very narrow applicability in larger systems.

Core claim

Proposals to describe open quantum dynamics with non-completely positive maps by restricting their domain to subsets of compatible initial states suffer an intrinsic weakness: for isotropic states the allowed domain becomes increasingly small with growing system dimension, limiting the practical usefulness of the approach.

What carries the argument

Domain restriction of non-completely positive maps to subsets of compatible initial states, which is shown to contract sharply for isotropic states as dimension rises.

If this is right

In high-dimensional systems fewer initial states can be paired with a given non-completely positive map.
The compatibility-based approach cannot scale to arbitrary open-system descriptions without additional constraints.
Models of open dynamics that drop complete positivity must either accept severe domain limits or find other consistency conditions.

Where Pith is reading between the lines

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

Similar domain-shrinkage effects may appear for other common families of states such as Werner states or random mixed states.
The result suggests that complete positivity might be harder to avoid in realistic many-body or high-dimensional settings than previously hoped.
One could test the claim numerically by sampling random initial states and counting the compatible fraction for fixed non-positive maps at increasing dimensions.

Load-bearing premise

The behavior seen with isotropic states under these restricted maps is representative of the general case for arbitrary states and maps.

What would settle it

Exhibit a family of initial states other than isotropic ones for which the fraction of compatible states remains large or constant as dimension grows under the same non-completely positive maps.

read the original abstract

The requirement of complete positivity is very often regarded as a fundamental consistency condition for the description of open quantum dynamics. We critically examine this requirement and discuss both its physical motivations and its limitations. We analyze proposals based on restricting the domain of non-completely positive maps to subsets of compatible initial states. Using isotropic states as a concrete example, we show that such domain restrictions become increasingly severe with growing system dimension, revealing an intrinsic weakness of the compatibility-based approach.

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 gives a concrete scaling example with isotropic states showing domain restrictions on non-CP maps tighten fast with dimension, but the claim of an intrinsic weakness rests on how representative that family is.

read the letter

The punchline is that restricting non-completely positive maps to compatible initial states runs into increasingly tight constraints as dimension grows, at least when you test it on isotropic states. The authors walk through the usual physical reasons people want complete positivity, then turn to the compatibility workaround and show with this family how the allowed set of starting states shrinks in a way that looks problematic for practical use in higher dimensions.

Referee Report

2 major / 1 minor

Summary. The paper critically examines the requirement of complete positivity (CP) for maps describing open quantum dynamics. It discusses physical motivations and limitations of CP, then analyzes proposals that restrict the domain of non-CP maps to subsets of 'compatible' initial states. Using isotropic states as a concrete example, the manuscript argues that such domain restrictions become increasingly severe with growing system dimension, thereby revealing an intrinsic weakness of the compatibility-based approach.

Significance. If the scaling result for isotropic states generalizes, the work would usefully highlight a practical limitation of domain-restriction strategies for non-CP dynamics in high-dimensional quantum systems. The concrete example and focus on dimension dependence are strengths, but the significance is tempered by the lack of evidence that the observed behavior is representative rather than symmetry-specific.

major comments (2)

Abstract and isotropic-states section: the claim that domain restrictions 'become increasingly severe with growing system dimension, revealing an intrinsic weakness' rests on the unstated assumption that the shrinkage observed for isotropic states (which are invariant under U⊗U*) is representative of arbitrary states and maps. No comparison with less symmetric families or general argument is provided to support the 'intrinsic' label; if the domain size scales more favorably for generic states, the central criticism does not follow from the example alone.
The manuscript does not include explicit derivations or numerical checks of the positivity constraint on Φ(ρ) for the isotropic family, making it impossible to verify the claimed severity scaling or to assess whether the restriction is load-bearing for the compatibility approach in general.

minor comments (1)

Notation for the restricted domain and the precise definition of 'compatible initial states' could be clarified with a short formal definition early in the text.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We are grateful to the referee for the thorough review and insightful comments, which have helped us improve the manuscript. Below we address the major comments point by point.

read point-by-point responses

Referee: Abstract and isotropic-states section: the claim that domain restrictions 'become increasingly severe with growing system dimension, revealing an intrinsic weakness' rests on the unstated assumption that the shrinkage observed for isotropic states (which are invariant under U⊗U*) is representative of arbitrary states and maps. No comparison with less symmetric families or general argument is provided to support the 'intrinsic' label; if the domain size scales more favorably for generic states, the central criticism does not follow from the example alone.

Authors: We thank the referee for pointing this out. The isotropic states were selected as a concrete, analytically solvable example due to their symmetry and relevance in quantum information theory. We agree that without a general argument, the claim of an 'intrinsic weakness' may be overstated. In the revised version, we have modified the abstract and the discussion in the isotropic states section to clarify that the observed severe domain restrictions for this family indicate a potential limitation of the compatibility approach, and we have added a remark noting that further investigation with other state families would be valuable to assess generality. revision: partial
Referee: The manuscript does not include explicit derivations or numerical checks of the positivity constraint on Φ(ρ) for the isotropic family, making it impossible to verify the claimed severity scaling or to assess whether the restriction is load-bearing for the compatibility approach in general.

Authors: We acknowledge this shortcoming. To address it, we have included in the revised manuscript an appendix providing the explicit derivation of the positivity condition for Φ(ρ) when ρ is an isotropic state. Furthermore, we have added numerical results for dimensions up to d=5 to illustrate the scaling of the allowed domain size with dimension. These additions should make the severity of the restrictions verifiable and help evaluate the implications for the compatibility approach. revision: yes

Circularity Check

0 steps flagged

No circularity: criticism relies on explicit isotropic-state calculation in standard QIT

full rationale

The paper's central move is to take the compatibility-based approach to non-CP maps, restrict its domain to states compatible with a given map, and then compute the size of that domain explicitly for the family of isotropic states. This is a direct, parameter-free calculation using the known form of isotropic states and the positivity condition on the output; it does not fit any parameter to data and then relabel the fit as a prediction, nor does it define any quantity in terms of the result it claims to derive. The argument is self-contained against external benchmarks (standard definitions of isotropic states and complete positivity) and does not rest on a load-bearing self-citation chain. The observed shrinkage with dimension is therefore an independent observation rather than a tautology.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard axioms of quantum mechanics and quantum information theory such as the definition of completely positive maps and the properties of isotropic states; no free parameters or invented entities are introduced in the abstract.

axioms (2)

standard math Completely positive maps preserve positivity when extended to larger systems including an ancillary space.
Invoked as the standard consistency condition being criticized.
domain assumption Isotropic states form a one-parameter family that can be used to test map compatibility.
Used as the concrete example to demonstrate dimension dependence.

pith-pipeline@v0.9.0 · 5601 in / 1237 out tokens · 35247 ms · 2026-05-20T11:31:56.885072+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.

Using isotropic states as a concrete example, we show that such domain restrictions become increasingly severe with growing system dimension, revealing an intrinsic weakness of the compatibility-based approach.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

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

the volume V_comp(μ,d) of entangled Λ_μ-compatible isotropic states ... vanishes as d^{-1} for large d

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

22 extracted references · 22 canonical work pages · 1 internal anchor

[1]

J. N. Norton,Causation as Folk Science, Philosophers’ Imprint3, 1 (2003)

work page 2003
[2]

Gorini, A

V. Gorini, A. Kossakowski, and E. C. G. Sudarshan,Completely positive dynamical semigroups ofN-level systems, J. Math. Phys.17, 821 (1976)

work page 1976
[3]

Lindblad,On the generators of quantum dynamical semigroups, Commun

G. Lindblad,On the generators of quantum dynamical semigroups, Commun. Math. Phys.48, 119 (1976)

work page 1976
[4]

Chru´ sci´ nski and S, Pascazio,A Brief History of the GKLS Equation, Open Sys

D. Chru´ sci´ nski and S, Pascazio,A Brief History of the GKLS Equation, Open Sys. Inf. Dyn. 24, 1740001 (2017). 9

work page 2017
[5]

ˇStelmachoviˇ c, V

P. ˇStelmachoviˇ c, V. Buˇ zek,Structure and parametrization of stochastic maps of density ma- trices, Phys. Rev. A64, 062106 (2001)

work page 2001
[6]

Sudarshan, A

E.C.G. Sudarshan, A. Shaji,Structure and parametrization of stochastic maps of density ma- trices, J. Phys. A: Math. Gen.36, 5073 (2003)

work page 2003
[7]

Shaji, E.C.G

A. Shaji, E.C.G. Sudarshan,Who’s afraid of not completely positive maps?, Phys. Lett. A341 48-54 (2005)

work page 2005
[8]

Carteret, D.R

H.A. Carteret, D.R. Terno, and K. ˙Zyczkowski,Dynamics beyond completely positive maps: Some properties and applications, Phys. Rev. A77, 042113 (2008)

work page 2008
[9]

J. M. Dominy and D. A. Lidar,Beyond Complete Positivity, arXiv:1503.05342 (2015)

work page internal anchor Pith review Pith/arXiv arXiv 2015
[10]

Hartmann, W.T

R. Hartmann, W.T. Strunz,Accuracy assessment of perturbative master equations: Embracing nonpositivityPhys. Rev. A101, 012103 (2020)

work page 2020
[11]

Su´ arez, R

A. Su´ arez, R. Silbey, I. Oppenheim,Memory effects in the relaxation of quantum open systems J. Chem. Phys.97, 5101 (1992)

work page 1992
[12]

Gaspard, M

P. Gaspard, M. Nagaoka,Slippage of initial conditions for the Redfield master equation, J. Chem. Phys.1115668 (1999)

work page 1999
[13]

W. F. Stinespring,Positive functions onC ∗-algebras, Proc. Amer. Math. Soc.6, 211 (1955)

work page 1955
[14]

KrausStates, Effects and Operations: Fundamental Notions of Quantum Theory, Lect

K. KrausStates, Effects and Operations: Fundamental Notions of Quantum Theory, Lect. Notes Phys., Springer-Verlag, Berlin, 1983

work page 1983
[15]

E. B. Davies,Quantum Theory of Open Systems, New York, Academic Press, 1976

work page 1976
[16]

D¨ umcke, H

R. D¨ umcke, H. Spohn,The proper form of the generator in the weak coupling limitZ. Phys. B 34, 419 (1979)

work page 1979
[17]

Alicki and K

R. Alicki and K. Lendi,Quantum Dynamical Semigroups and Applications, Springer, 1987

work page 1987
[18]

A. G. Redfield,The theory of relaxation processes, Adv. Magn. Res.1, 1 (1965)

work page 1965
[19]

M. D. Choi,Completely positive linear maps on complex matrices, Lin. Alg. Appl.10, 285 (1975)

work page 1975
[20]

M. D. Choi,Positive linear maps onC ∗-algebras, J. Canad. Math.24, 520 (1972)

work page 1972
[21]

Horodecki, P

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

work page 2009
[22]

Horodecki, P

M. Horodecki, P. Horodecki,Reduction criterion of separability and limits for a class of dis- tillation protocols, Phys. Rev. A59, 4206-4216 (1999) 10

work page 1999

[1] [1]

J. N. Norton,Causation as Folk Science, Philosophers’ Imprint3, 1 (2003)

work page 2003

[2] [2]

Gorini, A

V. Gorini, A. Kossakowski, and E. C. G. Sudarshan,Completely positive dynamical semigroups ofN-level systems, J. Math. Phys.17, 821 (1976)

work page 1976

[3] [3]

Lindblad,On the generators of quantum dynamical semigroups, Commun

G. Lindblad,On the generators of quantum dynamical semigroups, Commun. Math. Phys.48, 119 (1976)

work page 1976

[4] [4]

Chru´ sci´ nski and S, Pascazio,A Brief History of the GKLS Equation, Open Sys

D. Chru´ sci´ nski and S, Pascazio,A Brief History of the GKLS Equation, Open Sys. Inf. Dyn. 24, 1740001 (2017). 9

work page 2017

[5] [5]

ˇStelmachoviˇ c, V

P. ˇStelmachoviˇ c, V. Buˇ zek,Structure and parametrization of stochastic maps of density ma- trices, Phys. Rev. A64, 062106 (2001)

work page 2001

[6] [6]

Sudarshan, A

E.C.G. Sudarshan, A. Shaji,Structure and parametrization of stochastic maps of density ma- trices, J. Phys. A: Math. Gen.36, 5073 (2003)

work page 2003

[7] [7]

Shaji, E.C.G

A. Shaji, E.C.G. Sudarshan,Who’s afraid of not completely positive maps?, Phys. Lett. A341 48-54 (2005)

work page 2005

[8] [8]

Carteret, D.R

H.A. Carteret, D.R. Terno, and K. ˙Zyczkowski,Dynamics beyond completely positive maps: Some properties and applications, Phys. Rev. A77, 042113 (2008)

work page 2008

[9] [9]

J. M. Dominy and D. A. Lidar,Beyond Complete Positivity, arXiv:1503.05342 (2015)

work page internal anchor Pith review Pith/arXiv arXiv 2015

[10] [10]

Hartmann, W.T

R. Hartmann, W.T. Strunz,Accuracy assessment of perturbative master equations: Embracing nonpositivityPhys. Rev. A101, 012103 (2020)

work page 2020

[11] [11]

Su´ arez, R

A. Su´ arez, R. Silbey, I. Oppenheim,Memory effects in the relaxation of quantum open systems J. Chem. Phys.97, 5101 (1992)

work page 1992

[12] [12]

Gaspard, M

P. Gaspard, M. Nagaoka,Slippage of initial conditions for the Redfield master equation, J. Chem. Phys.1115668 (1999)

work page 1999

[13] [13]

W. F. Stinespring,Positive functions onC ∗-algebras, Proc. Amer. Math. Soc.6, 211 (1955)

work page 1955

[14] [14]

KrausStates, Effects and Operations: Fundamental Notions of Quantum Theory, Lect

K. KrausStates, Effects and Operations: Fundamental Notions of Quantum Theory, Lect. Notes Phys., Springer-Verlag, Berlin, 1983

work page 1983

[15] [15]

E. B. Davies,Quantum Theory of Open Systems, New York, Academic Press, 1976

work page 1976

[16] [16]

D¨ umcke, H

R. D¨ umcke, H. Spohn,The proper form of the generator in the weak coupling limitZ. Phys. B 34, 419 (1979)

work page 1979

[17] [17]

Alicki and K

R. Alicki and K. Lendi,Quantum Dynamical Semigroups and Applications, Springer, 1987

work page 1987

[18] [18]

A. G. Redfield,The theory of relaxation processes, Adv. Magn. Res.1, 1 (1965)

work page 1965

[19] [19]

M. D. Choi,Completely positive linear maps on complex matrices, Lin. Alg. Appl.10, 285 (1975)

work page 1975

[20] [20]

M. D. Choi,Positive linear maps onC ∗-algebras, J. Canad. Math.24, 520 (1972)

work page 1972

[21] [21]

Horodecki, P

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

work page 2009

[22] [22]

Horodecki, P

M. Horodecki, P. Horodecki,Reduction criterion of separability and limits for a class of dis- tillation protocols, Phys. Rev. A59, 4206-4216 (1999) 10

work page 1999