Spin polarization-scaling quantum maps and channels

Kamil Yu. Magadov; Sergey N. Filippov

arxiv: 1907.11878 · v1 · pith:RSMEQPPAnew · submitted 2019-07-27 · 🪐 quant-ph

Spin polarization-scaling quantum maps and channels

Sergey N. Filippov , Kamil Yu. Magadov This is my paper

Pith reviewed 2026-05-24 15:04 UTC · model grok-4.3

classification 🪐 quant-ph

keywords spin polarization-scaling mapquantum mapscomplete positivityentanglement breakingspin-1 particlespositive mapsanisotropic decoherencequantum channels

0 comments

The pith

A linear map that scales the three Cartesian polarization components of a spin-1 particle remains positive and completely positive only inside explicit bounds on its three scaling parameters.

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

The paper defines a quantum map for spin-j particles that leaves the identity operator untouched while multiplying each of the three orthogonal spin-polarization components by its own scaling factor. It then derives the precise inequalities these three factors must obey for the map to be positive, completely positive, entanglement-breaking, 2-tensor-stable positive, and 2-locally entanglement-annihilating. The inequalities are stated explicitly for spin-1 particles and shown to differ from the spin-1/2 case. A reader cares because such maps describe anisotropic loss of spin coherence, and the derived bounds tell exactly when the map can still serve as a valid quantum operation or channel.

Core claim

The authors introduce the spin polarization-scaling map on the space of Hermitian operators for a spin-j system; the map fixes the identity component and multiplies the three Cartesian polarization operators by independent scaling parameters. They determine the conditions on these parameters under which the map satisfies positivity, complete positivity, entanglement breaking, 2-tensor-stable positivity, and 2-local entanglement annihilation, supplying the explicit parameter regions for the spin-1 case and noting the contrast with spin-1/2.

What carries the argument

The spin polarization-scaling map, a linear map on Hermitian operators that scales the three Cartesian polarization components while leaving the identity unchanged.

If this is right

Inside the positivity region the map sends every valid density operator to another valid density operator.
Inside the complete-positivity region the map admits a Kraus representation and can be realized physically as a quantum channel.
Inside the entanglement-breaking region the map destroys all quantum correlations when applied to one subsystem of any bipartite state.
The 2-tensor-stable positivity region guarantees that the map remains positive even when applied simultaneously to two copies of the system.
The explicit bounds for spin-1 particles are stricter than the corresponding spin-1/2 bounds in at least one direction.

Where Pith is reading between the lines

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

These bounds could be used to calibrate anisotropic noise models in spin-1 based quantum sensors or memories.
The structural difference from spin-1/2 suggests that higher-spin particles need separate channel-design rules when polarization decay is the dominant error.
One could numerically sample random scaling triples and verify that the map's complete-positivity boundary matches the analytic inequalities given in the paper.

Load-bearing premise

The map is defined from the outset as a uniform scaling of the three polarization components with no cross terms and with the identity component fixed.

What would settle it

For a chosen triple of scaling parameters lying just outside one of the derived boundaries, compute the smallest eigenvalue of the associated Choi operator and check whether it is negative.

Figures

Figures reproduced from arXiv: 1907.11878 by Kamil Yu. Magadov, Sergey N. Filippov.

read the original abstract

We introduce a spin polarization-scaling map for spin-$j$ particles, whose physical meaning is the decrease of spin polarization along three mutually orthogonal axes. We find conditions on three scaling parameters under which the map is positive, completely positive, entanglement breaking, 2-tensor-stable positive, and 2-locally entanglement annihilating. The results are specified for maps on spin-$1$ particles. The difference from the case of spin-$\frac{1}{2}$ particles is emphasized.

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 parametrizes spin-1 maps by scaling the three polarization components and supplies concrete conditions on those scales for positivity and related properties, but the map is only partially defined on the 9D space.

read the letter

The main takeaway is that this work defines a linear map on spin-1 operators that scales the three Cartesian polarization components by independent factors while fixing the identity, then works out the ranges of those three parameters that make the map positive, completely positive, entanglement-breaking, 2-tensor-stable positive, and 2-locally entanglement-annihilating. The explicit conditions for j=1 and the contrast with the qubit case are the new pieces relative to earlier literature on polarization maps. The paper does a service by turning the abstract properties into concrete intervals that a reader could plug into calculations. That part is straightforward and potentially handy for people already modeling higher-spin channels. The soft spot sits in the construction itself. The operator space for spin-1 is nine-dimensional, yet the definition only fixes the action on the identity and the three polarization operators. The five remaining directions (quadrupolar terms and the like) are left unspecified. No master equation, symmetry argument, or explicit matrix is supplied to pin them down, so the reported thresholds are conditional on whatever implicit choice was made for those coefficients. Altering that choice would move the boundaries, which makes the results less robust than they first appear. The work is aimed at quantum-information specialists who already care about maps on spin-j systems. Someone in that niche might extract usable parameter ranges, but the incomplete specification limits how far the claims can be trusted without the full matrices. I would bring the paper to a reading group to see the derivations, but I would not cite it in the next year because the motivation for the map form is thin. It deserves peer review because the attempt to characterize the properties is serious and the j=1 case is worth checking, even though the definition needs to be completed and justified.

Referee Report

1 major / 0 minor

Summary. The paper introduces a spin polarization-scaling map for spin-j particles that acts as a linear map on Hermitian operators by fixing the identity component and independently scaling the three Cartesian polarization operators Jx, Jy, Jz by parameters λx, λy, λz. For spin-1 particles it derives explicit conditions on these three parameters under which the map is positive, completely positive, entanglement-breaking, 2-tensor-stable positive, and 2-locally entanglement-annihilating, and contrasts the results with the spin-1/2 case.

Significance. The explicit threshold conditions on the scaling parameters, once the map is fully defined, would supply concrete, tunable examples of higher-spin quantum maps with controlled entanglement properties and could be useful for benchmarking numerical or analytic techniques in quantum information. The emphasis on the structural difference from qubits is a clear strength.

major comments (1)

[Map definition (§2) and positivity criteria (§3)] The definition of the map (stated in the abstract and §2) specifies its action only on the identity and the three polarization operators. For spin-1 the Hermitian operator space is 9-dimensional, so the map’s action on the five orthogonal operators (e.g., the quadrupolar terms proportional to {Jx,Jy}, Jx²−Jy², etc.) must be prescribed to obtain a complete linear map. The positivity, CP, and entanglement-breaking criteria derived for λx,λy,λz are therefore conditional on an unstated choice for those matrix elements; altering the choice moves the reported boundaries.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the need for a fully specified linear map. We address the single major comment below.

read point-by-point responses

Referee: [Map definition (§2) and positivity criteria (§3)] The definition of the map (stated in the abstract and §2) specifies its action only on the identity and the three polarization operators. For spin-1 the Hermitian operator space is 9-dimensional, so the map’s action on the five orthogonal operators (e.g., the quadrupolar terms proportional to {Jx,Jy}, Jx²−Jy², etc.) must be prescribed to obtain a complete linear map. The positivity, CP, and entanglement-breaking criteria derived for λx,λy,λz are therefore conditional on an unstated choice for those matrix elements; altering the choice moves the reported boundaries.

Authors: We agree that a complete linear map on the 9-dimensional space of Hermitian operators for spin-1 requires an explicit prescription for the action on the five quadrupolar operators orthogonal to span{I, Jx, Jy, Jz}. The original manuscript stated the action only on the identity and polarization operators and left the complement implicit, which is an oversight. In the revised version we will explicitly define the map to act as the zero map on that orthogonal complement. With this concrete choice the reported threshold conditions on λx, λy, λz become well-defined for the resulting map; we will also insert a brief remark acknowledging that other choices for the complement would generally shift the boundaries, exactly as the referee notes. The revision will appear in §§2–3 and does not alter the physical motivation or the contrast with the qubit case. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained from explicit map definition

full rationale

The paper defines the polarization-scaling map by its action on the three Cartesian components and identity, then applies standard definitions of positivity, complete positivity, and entanglement-breaking to derive parameter conditions. No quoted steps reduce by construction to fitted inputs renamed as predictions, self-citation chains, or ansatzes smuggled via prior work. The central claims remain independent of the inputs once the map is stipulated, consistent with the reader's assessment of score 2 or lower.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The three scaling parameters are the central free parameters whose admissible ranges constitute the main result. The work relies on the standard axiomatic definitions of positivity and complete positivity for quantum maps.

free parameters (1)

three scaling parameters
The parameters that scale the x, y, z polarization components; conditions on their values are the output of the paper.

axioms (2)

standard math A linear map on Hermitian operators is positive if it sends positive semidefinite operators to positive semidefinite operators.
Invoked when stating the positivity condition for the scaling map.
standard math Complete positivity is defined via the Choi matrix or via the tensor product with the identity map.
Used to obtain the completely-positive conditions.

pith-pipeline@v0.9.0 · 5598 in / 1319 out tokens · 24337 ms · 2026-05-24T15:04:05.475079+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/RealityFromDistinction.lean reality_from_one_distinction unclear

?

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

We introduce a spin polarization-scaling map for spin-j particles... Φ[X] = 1/(2j+1) tr[X] I + 3/(j(j+1)(2j+1)) Σ λi tr[X Ji] Ji
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

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

The map is trace-preserving and unital... physical meaning is the transformation of the spin polarization pi ↦ λi pi

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

39 extracted references · 39 canonical work pages · 1 internal anchor

[1]

map Φ is known to be positive if and only if|λi| ⩽ 1, completely positive if and only if 1± λ3 ⩾|λ1± λ2|, entanglement breaking if and only if |λ1| +|λ2| +|λ3| ⩽ 1, 2-local-entanglement-annihilating if and only ifλ2 1+λ2 2+λ2 3 ⩽ 1, 2-tensor-stable positive if and only if1± λ2 3 ⩾|λ2 1± λ2 2| [4, 5, 7]. Similar characterization for higher spins (j ⩾ 1) is...

work page
[2]

Spin polarization-scaling quantum maps and channels

Therefore, the minimal eigenvalue of Φ[X] reads 1 2j + 1  tr[X]− 3 j + 1 √ 3∑ i=1 (λitr[XJ i])2   . (3) Suppose X ⩾ 0. As|tr[XJ i]| ⩽ jtr[X], the minimal value of (3) is non-negative if1− 3j j+1 √∑3 i=1 λ2 i ⩾ 0. Thus, we have found suﬃcient condition for positivity of the map Φ. Proposition 1. Spin-polarization-scaling map Φ is posi- tive if ∑3 i=...

work page internal anchor Pith review Pith/arXiv arXiv 1907
[3]

For higher spins (j ⩾ 1) the result of Example 5 can be extended as follows

is 2-tensor-stable positive if and only if Φ2 is completely positive, i.e.1± λ2 3 ⩾|λ2 1± λ2 2|. For higher spins (j ⩾ 1) the result of Example 5 can be extended as follows. Proposition 5. If the spin polarization-scaling map Φ is 2-tensor-stable positive, then Φ2 is completely positive. Proof. Consider a positive-semideﬁnite operator |ψ+⟩⟨ψ+|, where|ψ+⟩ ...

work page
[4]

+ 27λ2 1λ2 2λ2 3 ⩾ 0, which is depicted in Fig. 1(b). A linear map Φ :B(H)↦→B (H) is called 2-locally en- tanglement annihilating [7–9, 23] if(Φ⊗ Φ)[|ψ⟩⟨ψ|] is sep- arable for all|ψ⟩∈H⊗H . The same line of reasoning as for 2-tensor-stable positive maps leads to the following result. Proposition 6. If the spin polarization-scaling map Φ is 2-locally entang...

work page
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S. N. Filippov, A. A. Melnikov, and M. Ziman,Dissocia- tion and annihilation of multipartite entanglement struc- ture in dissipative quantum dynamics, Phys. Rev. A 88, 062328 (2013)

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

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S. N. Filippov and M. Ziman, Bipartite entanglement- annihilating maps: Necessary and suﬃcient conditions, Phys. Rev. A88, 032316 (2013)

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[1] [1]

map Φ is known to be positive if and only if|λi| ⩽ 1, completely positive if and only if 1± λ3 ⩾|λ1± λ2|, entanglement breaking if and only if |λ1| +|λ2| +|λ3| ⩽ 1, 2-local-entanglement-annihilating if and only ifλ2 1+λ2 2+λ2 3 ⩽ 1, 2-tensor-stable positive if and only if1± λ2 3 ⩾|λ2 1± λ2 2| [4, 5, 7]. Similar characterization for higher spins (j ⩾ 1) is...

work page

[2] [2]

Spin polarization-scaling quantum maps and channels

Therefore, the minimal eigenvalue of Φ[X] reads 1 2j + 1  tr[X]− 3 j + 1 √ 3∑ i=1 (λitr[XJ i])2   . (3) Suppose X ⩾ 0. As|tr[XJ i]| ⩽ jtr[X], the minimal value of (3) is non-negative if1− 3j j+1 √∑3 i=1 λ2 i ⩾ 0. Thus, we have found suﬃcient condition for positivity of the map Φ. Proposition 1. Spin-polarization-scaling map Φ is posi- tive if ∑3 i=...

work page internal anchor Pith review Pith/arXiv arXiv 1907

[3] [3]

For higher spins (j ⩾ 1) the result of Example 5 can be extended as follows

is 2-tensor-stable positive if and only if Φ2 is completely positive, i.e.1± λ2 3 ⩾|λ2 1± λ2 2|. For higher spins (j ⩾ 1) the result of Example 5 can be extended as follows. Proposition 5. If the spin polarization-scaling map Φ is 2-tensor-stable positive, then Φ2 is completely positive. Proof. Consider a positive-semideﬁnite operator |ψ+⟩⟨ψ+|, where|ψ+⟩ ...

work page

[4] [4]

+ 27λ2 1λ2 2λ2 3 ⩾ 0, which is depicted in Fig. 1(b). A linear map Φ :B(H)↦→B (H) is called 2-locally en- tanglement annihilating [7–9, 23] if(Φ⊗ Φ)[|ψ⟩⟨ψ|] is sep- arable for all|ψ⟩∈H⊗H . The same line of reasoning as for 2-tensor-stable positive maps leads to the following result. Proposition 6. If the spin polarization-scaling map Φ is 2-locally entang...

work page

[5] [5]

M. S. Byrd, C. A. Bishop, and Y.-C. Ou,General open- system quantum evolution in terms of aﬃne maps of the polarization vector, Phys. Rev. A83, 012301 (2011)

work page 2011

[6] [6]

A.Ch¸ ecińskaandK.Wydkiewicz,Complete positivity con- ditions for quantum qutrit channels, Phys. Rev. A 80, 032322 (2009)

work page 2009

[7] [7]

Choi,Completely positive linear maps on complex matrices, Linear Algebra Appl.10, 285 (1975)

M.-D. Choi,Completely positive linear maps on complex matrices, Linear Algebra Appl.10, 285 (1975)

work page 1975

[8] [8]

S. N. Filippov, PPT-Inducing, distillation-prohibiting, and entanglement-binding quantum channels , J. Russ. Laser Res. 35, 484 (2014)

work page 2014

[9] [9]

S. N. Filippov and K. Yu. Magadov, J. Phys. A: Math. Theor. 50, 055301 (2017)

work page 2017

[10] [10]

S. N. Filippov, A. A. Melnikov, and M. Ziman,Dissocia- tion and annihilation of multipartite entanglement struc- ture in dissipative quantum dynamics, Phys. Rev. A 88, 062328 (2013)

work page 2013

[11] [11]

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

work page 2012

[12] [12]

S. N. Filippov and M. Ziman, Bipartite entanglement- annihilating maps: Necessary and suﬃcient conditions, Phys. Rev. A88, 032316 (2013)

work page 2013

[13] [13]

S. N. Filippov and M. Ziman, Entanglement sensitivity to signal attenuation and ampliﬁcation, Phys. Rev. A90, 010301(R) (2014)

work page 2014

[14] [14]

S. K. Goyal, B. N. Simon, R. Singh, and S. Simon,Geom- etry of the generalized Bloch sphere for qutrits, J. Phys. A: Math. Theor.49, 165203 (2016)

work page 2016

[15] [15]

Heinosaari and M

T. Heinosaari and M. Ziman,The Mathematical Language of Quantum Theory, (Cambridge University Press, Cam- bridge, 2012)

work page 2012

[16] [16]

A. S. Holevo,Quantum coding theorems, Russ. Math. Sur- veys 53, 1295 (1998)

work page 1998

[17] [17]

Horodecki, P

M. Horodecki, P. Horodecki, and R. Horodecki,Separa- bility of mixed states: necessary and suﬃcient conditions, Phys. Lett. A223, 1 (1996)

work page 1996

[18] [18]

Horodecki, M

P. Horodecki, M. Horodecki, R. Horodecki,Binding en- tanglement channels, J. Mod. Opt.47, 347 (2000)

work page 2000

[19] [19]

Horodecki, P

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

work page 2009

[20] [20]

Horodecki, P

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

work page 2003

[21] [21]

Jamiołkowski,Linear transformations which preserve trace and positive semideﬁniteness of operators , Rep

A. Jamiołkowski,Linear transformations which preserve trace and positive semideﬁniteness of operators , Rep. Math. Phys. 3, 275 (1972)

work page 1972

[22] [22]

Jiang, S

M. Jiang, S. Luo, and S. Fu,Channel-state duality, Phys. Rev. A 87, 022310 (2013)

work page 2013

[23] [23]

Karimipour, A

V. Karimipour, A. Mani, and L. Memarzadeh,Character- ization of qutrit channels in terms of their covariance and symmetry properties, Phys. Rev. A84, 012321 (2011)

work page 2011

[24] [24]

King,Maximization of capacity andlp norms for some product channels, J

C. King,Maximization of capacity andlp norms for some product channels, J. Math. Phys.43, 1247 (2002)

work page 2002

[25] [25]

L. J. Landau and R. F. Streater,On Birkhoﬀ’s theorem for doubly stochastic completely positive maps of matrix algebras, Linear Algebra and its Applications 193, 107 (1993)

work page 1993

[26] [26]

W. A. Majewski and T. I. Tylec,Comment on Channel- state duality, Phys. Rev. A88, 026301 (2013)

work page 2013

[27] [27]

Moravčíková and M

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

work page 2010

[28] [28]

Müller-Hermes, D

A. Müller-Hermes, D. Reeb, and M. M. Wolf,Positivity of linear maps under tensor powers, J. Math. Phys. 57, 015202 (2016)

work page 2016

[29] [29]

Nathanson and M

M. Nathanson and M. B. Ruskai,Pauli diagonal chan- nels constant on axes, J. Phys. A: Math. Theor.40, 8171 (2007)

work page 2007

[30] [30]

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

work page 2000

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Peres,Separability criterion for density matrices, Phys

A. Peres,Separability criterion for density matrices, Phys. Rev. Lett. 77, 1413 (1996)

work page 1996

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Petz and H

D. Petz and H. Ohno,Generalizations of Pauli channels, Acta Math. Hungar.124, 165 (2009)

work page 2009

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de Pillis,Linear transformations which preserve Her- mitian and positive semideﬁnite operators, Paciﬁc J

J. de Pillis,Linear transformations which preserve Her- mitian and positive semideﬁnite operators, Paciﬁc J. of Math. 23, 129 (1967)

work page 1967

[34] [34]

M. B. Ruskai,Qubit entanglement breaking channels, Rev. Math. Phys. 15, 643 (2003)

work page 2003

[35] [35]

M. B. Ruskai, S. Szarek, and E. Werner,An analysis of completely-positive trace-preserving maps onM2, Linear Algebra Appl. 347, 159 (2002)

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