Entanglement increase from local interactions which lead to non-positive local reduced dynamics

Iman Sargolzahi

arxiv: 2605.08923 · v1 · submitted 2026-05-09 · 🪐 quant-ph

Entanglement increase from local interactions which lead to non-positive local reduced dynamics

Iman Sargolzahi This is my paper

Pith reviewed 2026-05-12 02:13 UTC · model grok-4.3

classification 🪐 quant-ph

keywords entanglementlocal interactionsnon-positive mapsreduced dynamicsbipartite systemquantum channelsentanglement increaselocal environments

0 comments

The pith

Local interactions can increase entanglement in a bipartite quantum system when reduced dynamics are non-positive maps.

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

A bipartite quantum system has each part interacting only with its own local environment. Normally this keeps entanglement from growing if the reduced dynamics are completely positive trace-preserving maps, that is, quantum channels. The paper shows that the reduced dynamics can instead take the form of a product of two local non-positive maps. In those cases entanglement between the parts can exceed its initial value. The work maps out the general conditions for such behavior and supplies a new general construction that produces the required non-positive maps.

Core claim

For a bipartite system AB with purely local interactions, the reduced dynamics on the system can be given by a map of the form Ψ_A ⊗ Ψ_B where each Ψ is a non-positive linear map rather than a quantum channel. Under this condition the entanglement between A and B can increase beyond its starting value, as first illustrated in the Jordan et al. example; the paper identifies the general circumstances under which this occurs and introduces a further systematic procedure that generates additional families of such local non-positive maps.

What carries the argument

The tensor-product map Ψ_A ⊗ Ψ_B formed from two local non-positive maps that nevertheless arise from purely local system-environment interactions.

If this is right

Entanglement can exceed its initial value in any system whose reduced dynamics match the Jordan et al. example.
General families of local non-positive maps exist that produce the same entanglement increase.
A new constructive procedure yields further local non-positive maps with the same property.
The usual expectation that local interactions cannot increase entanglement fails precisely when the reduced map is non-positive.

Where Pith is reading between the lines

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

Protocols that assume entanglement is non-increasing under local noise may need to verify that the actual reduced dynamics remain completely positive.
Distinguishing physical local evolutions from abstract non-positive maps becomes experimentally relevant for entanglement control.
The same mechanism could be checked in open multipartite systems to see whether non-positive local reductions produce similar growth.

Load-bearing premise

The overall evolution can still be produced by local interactions with separate environments even when the reduced dynamics on the system are given by a product of non-positive maps.

What would settle it

An explicit model of local interactions on a bipartite system whose reduced dynamics turn out to be completely positive trace-preserving maps, together with a numerical check that entanglement never exceeds its initial value.

Figures

Figures reproduced from arXiv: 2605.08923 by Iman Sargolzahi.

read the original abstract

Consider a bipartite quantum system S=AB such that each part interacts only with its local environment. Under such circumstances, one expects that the entanglement between parts A and B does not exceed its initial value during the time evolution. In fact, this is the case if the reduced dynamics of the system is given by $\mathcal{E}_{A}\otimes \mathcal{E}_{B}$, where $\mathcal{E}_{A}$ and $\mathcal{E}_{B}$ are quantum channels, i.e., completely positive trace-preserving maps. But, the reduced dynamics of the system may be given by a map as $\Psi_{A}\otimes \Psi_{B}$, where $\Psi_{A}$ and $\Psi_{B}$ are local non-positive maps. Then, the entanglement between A and B can exceed its initial value, as was shown in the case studied by Jordan et al. [Phys. Rev. A 76, 022102 (2007)]. In this paper, we first explore the general circumstances under which one can find such cases as they found. Next, we introduce another general procedure which leads to local non-positive maps that cause entanglement exceeding.

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 paper generalizes the Jordan et al. example by outlining conditions and a new construction for local non-positive maps that can increase entanglement under local interactions.

read the letter

The core takeaway is that the authors move from the single 2007 counterexample to broader conditions under which local interactions produce non-positive reduced maps on each subsystem, plus a separate constructive procedure for generating such maps. That extension is the main addition over the cited prior work. They correctly note that when the reduced dynamics take the form of a tensor product of non-positive maps, entanglement between A and B can grow, which runs counter to the usual expectation for completely positive maps. The paper stays within standard open-systems quantum mechanics and avoids circular definitions, which is a plus. The constructive angle could make it easier for others to generate further examples. The main limitation is the one flagged in the stress-test note. Non-positivity of the effective maps generally requires initial system-environment correlations. In that setting the reduced state of AB does not evolve under a fixed linear map that can be applied to arbitrary initial states; the output depends on the specific correlations. The authors need to demonstrate, with explicit derivations or numerical checks, that their entanglement calculations use the actual partial trace over the environments rather than simply feeding states into the formal non-positive tensor-product map. If that step is handled carefully, the claims hold; if not, the generalization is weaker than presented. This is aimed at researchers already working on entanglement dynamics in open quantum systems who are familiar with the Jordan result. A reader looking for new counterexamples or construction methods will find value, while someone outside that niche will not. The work is coherent enough on its own terms to merit peer review, even if revisions are likely needed on the map-validity point. I would send it to referees.

Referee Report

2 major / 2 minor

Summary. The paper considers a bipartite system AB where each subsystem interacts locally with its own environment. It argues that while completely positive trace-preserving reduced dynamics (quantum channels) preserve the no-increase property for entanglement, the reduced dynamics can instead take the form of a tensor product of non-positive maps Ψ_A ⊗ Ψ_B. Under this condition entanglement between A and B can exceed its initial value. The authors first characterize the general circumstances in which such non-positive local maps arise from local interactions (generalizing the Jordan et al. 2007 example), and then present an explicit general construction that produces families of such maps leading to entanglement growth.

Significance. If the central construction is valid and the resulting maps are shown to arise from unitary evolution on the full system-environment space with only local interactions, the result supplies a systematic method for generating counter-examples to the expectation that local open-system dynamics cannot increase bipartite entanglement. This would be relevant to the study of non-Markovianity, initial system-environment correlations, and the boundary between positive and completely-positive maps in open quantum systems.

major comments (2)

[§3] §3 (general procedure): the claim that the reduced dynamics is given by a fixed linear map Ψ_A ⊗ Ψ_B that can be applied to arbitrary initial ρ_AB is not demonstrated. When the non-positivity originates from initial A-E_A (B-E_B) correlations, the partial trace after the local unitaries U_A(t) ⊗ U_B(t) yields an output that depends on the specific form of those correlations; hence the map is not a dynamical map on the system alone. The entanglement-increase calculation must be performed on the actual physical reduced state rather than by formally applying the non-positive map to an arbitrary input.
[§2.2] §2.2 (characterization of Jordan-type cases): the conditions under which the reduced map becomes non-positive are stated in terms of the initial joint state, but no explicit verification is given that the resulting time-evolved ρ_AB(t) indeed violates the entanglement bound while remaining consistent with a global unitary evolution on AB E_A E_B. A concrete numerical or analytic example with the full Hilbert-space evolution would be required to confirm the claim.

minor comments (2)

[Introduction] Notation for the non-positive maps is introduced as Ψ_A, Ψ_B without an explicit statement of the domain (Hermitian operators, trace-preserving or not) and without a comparison to the standard definition of positive maps used in the literature.
[Abstract] The abstract states that the reduced dynamics “may be given by” Ψ_A ⊗ Ψ_B; this phrasing should be clarified to indicate under which initial conditions the equality holds.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the constructive major comments. We agree that additional clarifications and explicit verifications will strengthen the presentation. We address each point below and will incorporate the suggested revisions in the next version of the manuscript.

read point-by-point responses

Referee: [§3] §3 (general procedure): the claim that the reduced dynamics is given by a fixed linear map Ψ_A ⊗ Ψ_B that can be applied to arbitrary initial ρ_AB is not demonstrated. When the non-positivity originates from initial A-E_A (B-E_B) correlations, the partial trace after the local unitaries U_A(t) ⊗ U_B(t) yields an output that depends on the specific form of those correlations; hence the map is not a dynamical map on the system alone. The entanglement-increase calculation must be performed on the actual physical reduced state rather than by formally applying the non-positive map to an arbitrary input.

Authors: We agree with the referee that the effective maps Ψ_A ⊗ Ψ_B arising from initial system-environment correlations are not universal dynamical maps that can be applied to arbitrary initial states ρ_AB. In the manuscript, these maps are derived specifically for the class of initial joint states that encode the relevant A-E_A and B-E_B correlations. The entanglement increase is shown for states belonging to this class. To address the concern, we will revise §3 to explicitly clarify that the construction yields effective (non-positive) maps valid only for the considered family of initial conditions, and we will recompute the entanglement measures directly from the reduced states obtained via the global unitary evolution on the full AB E_A E_B space rather than by formal application of the maps to arbitrary inputs. This will ensure physical consistency is transparent. revision: yes
Referee: [§2.2] §2.2 (characterization of Jordan-type cases): the conditions under which the reduced map becomes non-positive are stated in terms of the initial joint state, but no explicit verification is given that the resulting time-evolved ρ_AB(t) indeed violates the entanglement bound while remaining consistent with a global unitary evolution on AB E_A E_B. A concrete numerical or analytic example with the full Hilbert-space evolution would be required to confirm the claim.

Authors: We thank the referee for highlighting this point. While §2.2 provides a general characterization of the conditions leading to non-positive reduced maps in the Jordan-type setting, we acknowledge that an explicit verification with the full Hilbert-space evolution would make the result more convincing. In the revised manuscript we will add a concrete numerical example (with small-dimensional Hilbert spaces) that specifies the initial joint state of A B E_A E_B, the local unitaries, the global unitary evolution, the resulting reduced ρ_AB(t), and direct computation of an entanglement monotone (e.g., negativity) showing an increase, while confirming that the global evolution remains unitary. This will explicitly demonstrate consistency with the physical setup. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation relies on external reference and explicit construction

full rationale

The paper's core argument starts from the standard fact that local CPTP maps preserve entanglement bounds and then contrasts this with the possibility of non-positive local maps arising from initial system-environment correlations. It cites Jordan et al. (external, 2007) for the concrete example and proceeds by exploring general conditions and constructing another procedure. No quantity is defined in terms of the target entanglement increase, no fitted parameter is relabeled as a prediction, and no load-bearing step reduces to a self-citation or ansatz smuggled from the authors' prior work. The derivation chain remains self-contained against the external benchmark of open-system quantum mechanics.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard quantum mechanics of open systems and the definition of reduced dynamics; no free parameters, ad-hoc axioms, or new invented entities are introduced in the abstract.

axioms (2)

domain assumption Reduced dynamics of a bipartite system interacting locally with separate environments can be expressed as a tensor product of local maps.
Invoked in the abstract when stating that the reduced dynamics may be given by Ψ_A ⊗ Ψ_B.
standard math Completely positive trace-preserving maps cannot increase entanglement.
Standard result in quantum information used as the baseline expectation.

pith-pipeline@v0.9.0 · 5495 in / 1085 out tokens · 32412 ms · 2026-05-12T02:13:03.698209+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.

the reduced dynamics of the system may be given by a map as Ψ_A ⊗ Ψ_B, where Ψ_A and Ψ_B are local non-positive maps
IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

?

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

entanglement between A and B can exceed its initial value

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

32 extracted references · 32 canonical work pages

[1]

They considered a systemSincluding two qubitsAandB

to illustrate the results of the previous section. They considered a systemSincluding two qubitsAandB. Each qubit is a two-level atom interacting with its local environment (reservoir) constructed from the quantized modes of the electromagnetic field in a cavity. So, the dy- namics of the whole system-environment is as Eq. (4). In addition, since the init...

work page
[2]

(12), the mapE A(t) is reversible, and so isE B(t)

Therefore, for allt̸=t n in Eq. (12), the mapE A(t) is reversible, and so isE B(t). So, fort i ̸=t n in Eq. (12), we can write Eqs. (7) and (8). Now, we consider Fig. 4 of Ref. [5] and chooset iγ0 = 20, which is an appropriate initial instant, i.e.,t i ̸=t n in Eq. (12). We see that for thist i the stateρ AB(ti) is separable. Then, if we choose, e.g.,t f ...

work page
[3]

Yu and J

T. Yu and J. H. Eberly, Sudden death of entanglement, Science323, 598 (2009)

work page 2009
[4]

Yu and J

T. Yu and J. H. Eberly, Finite-time disentanglement via spontaneous emission, Phys. Rev. Lett.93, 140404 7 (2004)

work page 2004
[5]

Yonac, T

M. Yonac, T. Yu and J. H. Eberly, Pairwise concurrence dynamics: a four-qubit model, J. Phys. B: At. Mol. Opt. Phys.40, S45 (2007)

work page 2007
[6]

Aolita, F

L. Aolita, F. de Melo and L. Davidovich, Open-system dynamics of entanglement: a key issues review, Rep. Prog. Phys.78, 042001 (2015)

work page 2015
[7]

Bellomo, R

B. Bellomo, R. Lo Franco and G. Compagno, Non- Markovian effects on the dynamics of entanglement, Phys. Rev. Lett.99, 160502 (2007)

work page 2007
[8]

D’Arrigo, R

A. D’Arrigo, R. Lo Franco, G. Benenti, E. Paladino and G. Falci, Recovering entanglement by local operations, Ann. Phys.350, 211 (2014)

work page 2014
[9]

Chen, G.-Y

S.-L. Chen, G.-Y. Chen and Y.-N. Chen, Increase of en- tanglement by local PT -symmetric operations, Phys. Rev. A90, 054301 (2014)

work page 2014
[10]

Anderloni, F

S. Anderloni, F. Benatti, R. Floreanini, Redfield reduced dynamics and entanglement, J. Phys. A: Math. Theor. 40, 1625 (2007)

work page 2007
[11]

Breuer and F

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

work page 2002
[12]

Rivas and S

A. Rivas and S. F. Huelga,Open Quantum Systems: An Introduction(Springer, Heidelberg, Germany, 2011)

work page 2011
[13]

Sargolzahi and S

I. Sargolzahi and S. Y. Mirafzali, Entanglement increase from local interaction in the absence of initial quantum correlation in the environment and between the system and the environment, Phys. Rev. A97, 022331 (2018)

work page 2018
[14]

Roszak and L

K. Roszak and L. Cywinski, Equivalence of qubit- environment entanglement and discord generation via pure dephasing interactions and the resulting conse- quences, Phys. Rev. A97, 012306 (2018)

work page 2018
[15]

T. F. Jordan, A. Shaji and E. C. G. Sudarshan, Entangle- ment increase from local interactions and not completely positive maps, Phys. Rev. A76, 022102 (2007)

work page 2007
[16]

M. A. Nielsen and I. L. Chuang,Quantum Computation and Quantum Information(Cambridge University Press, Cambridge, England, 2000)

work page 2000
[17]

J. M. Dominy, A. Shabani and D. A. Lidar, A general framework for complete positivity, Quant. Inf. Process. 15, 465 (2016)

work page 2016
[18]

Stelmachovic and V

P. Stelmachovic and V. Buzek , Dynamics of open quan- tum systems initially entangled with environment: Be- yond the Kraus representation, Phys. Rev. A64, 062106 (2001);ibid.67, 029902(E)(2003)

work page 2001
[19]

Sargolzahi, Necessary and sufficient condition for the reduced dynamics of an open quantum system interacting with an environment to be linear, Phys

I. Sargolzahi, Necessary and sufficient condition for the reduced dynamics of an open quantum system interacting with an environment to be linear, Phys. Rev. A102, 022208 (2020)

work page 2020
[20]

J. M. Dominy and D. A. Lidar, Beyond complete posi- tivity, Quant. Inf. Process.15, 1349 (2016)

work page 2016
[21]

Buscemi, Complete positivity, Markovianity, and the quantum data-processing inequality, in the presence of initial system-environment correlations, Phys

F. Buscemi, Complete positivity, Markovianity, and the quantum data-processing inequality, in the presence of initial system-environment correlations, Phys. Rev. Lett. 113, 140502 (2014)

work page 2014
[22]

Lu, Structure of correlated initial states that guar- antee completely positive reduced dynamics, Phys

X.-M. Lu, Structure of correlated initial states that guar- antee completely positive reduced dynamics, Phys. Rev. A93, 042332 (2016)

work page 2016
[23]

Sargolzahi and S

I. Sargolzahi and S. Y. Mirafzali, When the assignment map is completely positive, Open Sys. Info. Dyn.25, 1850012 (2018)

work page 2018
[24]

Sargolzahi, Positivity of the assignment map implies complete positivity of the reduced dynamics, Quant

I. Sargolzahi, Positivity of the assignment map implies complete positivity of the reduced dynamics, Quant. Inf. Process.19, 310 (2020)

work page 2020
[25]

Horodecki, P

M. Horodecki, P. W. Shor and M. B. Ruskai, Entan- glement breaking channels, Rev. Math. Phys.15, 629 (2003)

work page 2003
[26]

Khatri, L

S. Khatri, L. Lami and M. M. Wilde, Principles of quan- tum communication theory: a modern approach (2025)

work page 2025
[27]

S. N. Filippov, T. Rybar, and M. Ziman, Local two- qubit entanglement-annihilating channels, Phys. Rev. A 85, 012303 (2012)

work page 2012
[28]

Moravcikova and M

L. Moravcikova and M. Ziman, Entanglement- annihilating and entanglement-breaking channels, J. Phys. A: Math. Theor.43, 275306 (2010)

work page 2010
[29]

Horodecki, P

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

work page 2009
[30]

Vidal, Entanglement monotones, J

G. Vidal, Entanglement monotones, J. Mod. Opt.47, 355 (2000)

work page 2000
[31]

Khatri, K

S. Khatri, K. Sharma and M. M. Wilde, Information- theoretic aspects of the generalized amplitude-damping channel, Phys. Rev. A102, 012401 (2020)

work page 2020
[32]

Masajada, M

P. Masajada, M. Fellous-Asiani and A. Streltsov, Op- timizing entanglement distribution via noisy quantum channels, Phys. Rev. A113, 052414 (2026)

work page 2026

[1] [1]

They considered a systemSincluding two qubitsAandB

to illustrate the results of the previous section. They considered a systemSincluding two qubitsAandB. Each qubit is a two-level atom interacting with its local environment (reservoir) constructed from the quantized modes of the electromagnetic field in a cavity. So, the dy- namics of the whole system-environment is as Eq. (4). In addition, since the init...

work page

[2] [2]

(12), the mapE A(t) is reversible, and so isE B(t)

Therefore, for allt̸=t n in Eq. (12), the mapE A(t) is reversible, and so isE B(t). So, fort i ̸=t n in Eq. (12), we can write Eqs. (7) and (8). Now, we consider Fig. 4 of Ref. [5] and chooset iγ0 = 20, which is an appropriate initial instant, i.e.,t i ̸=t n in Eq. (12). We see that for thist i the stateρ AB(ti) is separable. Then, if we choose, e.g.,t f ...

work page

[3] [3]

Yu and J

T. Yu and J. H. Eberly, Sudden death of entanglement, Science323, 598 (2009)

work page 2009

[4] [4]

Yu and J

T. Yu and J. H. Eberly, Finite-time disentanglement via spontaneous emission, Phys. Rev. Lett.93, 140404 7 (2004)

work page 2004

[5] [5]

Yonac, T

M. Yonac, T. Yu and J. H. Eberly, Pairwise concurrence dynamics: a four-qubit model, J. Phys. B: At. Mol. Opt. Phys.40, S45 (2007)

work page 2007

[6] [6]

Aolita, F

L. Aolita, F. de Melo and L. Davidovich, Open-system dynamics of entanglement: a key issues review, Rep. Prog. Phys.78, 042001 (2015)

work page 2015

[7] [7]

Bellomo, R

B. Bellomo, R. Lo Franco and G. Compagno, Non- Markovian effects on the dynamics of entanglement, Phys. Rev. Lett.99, 160502 (2007)

work page 2007

[8] [8]

D’Arrigo, R

A. D’Arrigo, R. Lo Franco, G. Benenti, E. Paladino and G. Falci, Recovering entanglement by local operations, Ann. Phys.350, 211 (2014)

work page 2014

[9] [9]

Chen, G.-Y

S.-L. Chen, G.-Y. Chen and Y.-N. Chen, Increase of en- tanglement by local PT -symmetric operations, Phys. Rev. A90, 054301 (2014)

work page 2014

[10] [10]

Anderloni, F

S. Anderloni, F. Benatti, R. Floreanini, Redfield reduced dynamics and entanglement, J. Phys. A: Math. Theor. 40, 1625 (2007)

work page 2007

[11] [11]

Breuer and F

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

work page 2002

[12] [12]

Rivas and S

A. Rivas and S. F. Huelga,Open Quantum Systems: An Introduction(Springer, Heidelberg, Germany, 2011)

work page 2011

[13] [13]

Sargolzahi and S

I. Sargolzahi and S. Y. Mirafzali, Entanglement increase from local interaction in the absence of initial quantum correlation in the environment and between the system and the environment, Phys. Rev. A97, 022331 (2018)

work page 2018

[14] [14]

Roszak and L

K. Roszak and L. Cywinski, Equivalence of qubit- environment entanglement and discord generation via pure dephasing interactions and the resulting conse- quences, Phys. Rev. A97, 012306 (2018)

work page 2018

[15] [15]

T. F. Jordan, A. Shaji and E. C. G. Sudarshan, Entangle- ment increase from local interactions and not completely positive maps, Phys. Rev. A76, 022102 (2007)

work page 2007

[16] [16]

M. A. Nielsen and I. L. Chuang,Quantum Computation and Quantum Information(Cambridge University Press, Cambridge, England, 2000)

work page 2000

[17] [17]

J. M. Dominy, A. Shabani and D. A. Lidar, A general framework for complete positivity, Quant. Inf. Process. 15, 465 (2016)

work page 2016

[18] [18]

Stelmachovic and V

P. Stelmachovic and V. Buzek , Dynamics of open quan- tum systems initially entangled with environment: Be- yond the Kraus representation, Phys. Rev. A64, 062106 (2001);ibid.67, 029902(E)(2003)

work page 2001

[19] [19]

Sargolzahi, Necessary and sufficient condition for the reduced dynamics of an open quantum system interacting with an environment to be linear, Phys

I. Sargolzahi, Necessary and sufficient condition for the reduced dynamics of an open quantum system interacting with an environment to be linear, Phys. Rev. A102, 022208 (2020)

work page 2020

[20] [20]

J. M. Dominy and D. A. Lidar, Beyond complete posi- tivity, Quant. Inf. Process.15, 1349 (2016)

work page 2016

[21] [21]

Buscemi, Complete positivity, Markovianity, and the quantum data-processing inequality, in the presence of initial system-environment correlations, Phys

F. Buscemi, Complete positivity, Markovianity, and the quantum data-processing inequality, in the presence of initial system-environment correlations, Phys. Rev. Lett. 113, 140502 (2014)

work page 2014

[22] [22]

Lu, Structure of correlated initial states that guar- antee completely positive reduced dynamics, Phys

X.-M. Lu, Structure of correlated initial states that guar- antee completely positive reduced dynamics, Phys. Rev. A93, 042332 (2016)

work page 2016

[23] [23]

Sargolzahi and S

I. Sargolzahi and S. Y. Mirafzali, When the assignment map is completely positive, Open Sys. Info. Dyn.25, 1850012 (2018)

work page 2018

[24] [24]

Sargolzahi, Positivity of the assignment map implies complete positivity of the reduced dynamics, Quant

I. Sargolzahi, Positivity of the assignment map implies complete positivity of the reduced dynamics, Quant. Inf. Process.19, 310 (2020)

work page 2020

[25] [25]

Horodecki, P

M. Horodecki, P. W. Shor and M. B. Ruskai, Entan- glement breaking channels, Rev. Math. Phys.15, 629 (2003)

work page 2003

[26] [26]

Khatri, L

S. Khatri, L. Lami and M. M. Wilde, Principles of quan- tum communication theory: a modern approach (2025)

work page 2025

[27] [27]

S. N. Filippov, T. Rybar, and M. Ziman, Local two- qubit entanglement-annihilating channels, Phys. Rev. A 85, 012303 (2012)

work page 2012

[28] [28]

Moravcikova and M

L. Moravcikova and M. Ziman, Entanglement- annihilating and entanglement-breaking channels, J. Phys. A: Math. Theor.43, 275306 (2010)

work page 2010

[29] [29]

Horodecki, P

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

work page 2009

[30] [30]

Vidal, Entanglement monotones, J

G. Vidal, Entanglement monotones, J. Mod. Opt.47, 355 (2000)

work page 2000

[31] [31]

Khatri, K

S. Khatri, K. Sharma and M. M. Wilde, Information- theoretic aspects of the generalized amplitude-damping channel, Phys. Rev. A102, 012401 (2020)

work page 2020

[32] [32]

Masajada, M

P. Masajada, M. Fellous-Asiani and A. Streltsov, Op- timizing entanglement distribution via noisy quantum channels, Phys. Rev. A113, 052414 (2026)

work page 2026