CP-preserving channels

Indu Bala; Sourav Das; Swapan Rana

REVIEW 2 major objections 7 minor 14 references

CP-preserving channels

T0 review · 2 major / 7 minor · reviewed 2026-07-31 · grok-4.5

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Pith's one-line read CP-preserving channels are fully characterized in dimensions up to 4, and the trace distance of non-negativity fails strong monotonicity.

desk verdict Solid niche follow-up that settles the strong-monotonicity question with an explicit counter-example and extends the d≤4 characterization; the claimed alternative CPDNN⇒CPCP proof has a one-sentence gap that is repairable from the paper’s own facts but is not written down. read the letter →

arxiv 2607.24682 v1 pith:OMH6QUTH submitted 2026-07-27 quant-ph math-phmath.MP

Indu Bala , Sourav Das , Swapan Rana This is my paper · ORCID

classification quant-phmath-phmath.MP MSC 81P4015B4890C22

keywords CP-preservingchannelsCPCPmapsCPDNNtracedistanceofnon-negativitycopositivematricesquantumresourcetheoryChoimatrix

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper works inside the resource theory of non-negativity of quantum amplitudes, where completely positive (CP) states are free and the free operations are channels that map CP states to CP states. It supplies necessary and sufficient conditions for a channel to be CP-preserving when the local dimensions are at most 4, recovers the known qubit criteria as a special case, and notes that the same conditions remain necessary in higher dimensions. It also proves that every CPDNN channel from n-by-n matrices into 2-by-2 matrices is automatically CPCP, and that every unital CPDNN map in the opposite direction is likewise CPCP. Finally it exhibits an explicit counter-example showing that the natural trace-distance measure of non-negativity is not strongly monotonic under free operations. A sympathetic reader cares because these results close several concrete open questions left by the original resource-theory paper and clarify which free operations and which monotones can safely be used.

What carries the argument

The dual-map copositivity criterion (Theorem 2.1 and Observation 2.4) together with the comparison-matrix argument that turns a DNN dual Choi matrix into a CP matrix (Theorem 3.6).

What would settle it

Construct a concrete CPDNN channel from n-by-n matrices to 2-by-2 matrices whose dual Choi matrix has a comparison matrix that is not positive semidefinite; if such a channel exists the equivalence CPDNN = CPCP collapses.

Watch

Extended reading notes

Core claim

For dimensions 2 through 4 a quantum channel is CP-preserving if and only if the symmetric real parts of the dual images of the matrix units are copositive; the same statements become only necessary once the dimension exceeds 4. Independently, every CPDNN channel into qubits is CPCP, every unital CPDNN map out of qubits is CPCP, and the trace distance of non-negativity fails strong monotonicity under CPCP channels.

Load-bearing premise

The proof that a CPDNN channel into qubits is CPCP rests on an unproved claim that the comparison matrix of the dual Choi operator is positive semidefinite.

Editorial extensions

If this is right

CP-preserving channels on systems of size at most 4 can be certified by a finite list of copositivity checks or by a single SDP.
The free operations of the resource theory may safely be taken to be the CPCP maps whenever the output is a qubit.
Unital CPDNN maps from qubits to higher dimensions are automatically free operations in the strongest (CPCP) sense.
The trace-distance monotone cannot be used for asymptotic or multi-copy resource conversion arguments.
Symmetry reduction under permutations yields closed-form values of the trace-distance measure for the uniform pure states up to dimension 4.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The same block-projection counter-example that kills strong monotonicity of the trace distance will likely work for any other distance-based monotone that does not automatically average under direct sums.
If the comparison-matrix step can be repaired or replaced, the equivalence CPDNN = CPCP may extend to a larger class of output dimensions.
The SDP feasibility program already written for state conversion under CP-preserving channels can be reused to search systematically for a golden resource state in dimension 3 or higher.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper cleanly answers two concrete questions left open by Johnston–Sikora: it gives necessary-and-sufficient copositivity criteria for CP-preserving channels when both dimensions are at most 4, and it supplies an explicit counter-example showing that the trace-distance measure of non-negativity fails strong monotonicity under CPCP (hence also under CP-preserving) operations. Both of those pieces are usable as written.

The counter-example in §5 is the cleanest contribution. The block-diagonal state ρ = p|Υ₂⟩⟨Υ₂| ⊕ (1-p)|Υ₃⟩⟨Υ₃| and the two-block projection channel are elementary; the upper bound on N_T(ρ) via ||ρ-I/5||_1 is correct, and the elementary inequality (4-p)/3 > (|1-5p|+|4-5p|+3)/5 holds on (4/25,2/5). That settles the question negatively and can be trusted immediately. The d≤4 characterization (Thm 2.1 and the dual/Choi/copositivity observations) is the natural extension once CP=DNN is granted; the qubit recovery matches the known conditions, and the SDP formulation is a practical bonus. The symmetry-reduction argument for N_T(|Υ_d⟩) is also neat and immediately gives the values in Table 1.

The soft spot is Theorem 3.6 (and the one-line dual statement 3.7). The entire DNN o CP step rests on the bare assertion that the comparison matrix of J_{Φ*} is PSD. That does not follow from DNN-ness alone, so a reader cannot verify the alternative proof without supplying the missing three-line lemma: trace-preservation forces the diagonal blocks of J_{Φ*} to be diagonal, after which M(J_{Φ*}) is a signature congruence of J_{Φ*} and therefore PSD. The claim is true and uses only ingredients already in the text, but as written the proof has a hole. Cha’s independent argument remains the only complete published proof for the qubit-output direction. Writing quality is rough (typos, incomplete sentences), yet the logical skeleton is recoverable.

This is for people already working inside the Johnston–Sikora resource theory or anyone who needs low-dimensional tests or a concrete non-monotone example. It deserves a serious referee; the monotonicity failure and the d≤4 criteria should survive, and the comparison-matrix gap is easily fixed. I would accept it for peer review.

Referee Report

2 major / 7 minor

Summary. The paper works within the Johnston–Sikora resource theory of non-negativity, in which CP matrices are free states and CP-preserving channels are free operations. It gives four main results: (i) necessary and sufficient conditions for a channel to be CP-preserving in dimensions d≤4 (Theorem 2.1 and Observations 2.3–2.8), phrased as copositivity of symmetrized dual-map images, recovering the known qubit conditions and yielding an SDP formulation; (ii) an alternative proof that every CPDNN channel Φ : M_n→M_2 is CPCP (Theorem 3.6, a result recently obtained by Cha) and a new dual statement that every unital CPDNN map M_2→M_n is CPCP (Theorem 3.7); (iii) symmetry reductions showing the nearest CP/DDN/DNN state to a permutation-invariant state can be assumed to share the symmetry, with evaluation of the trace-distance measure for the uniform states |Υ_d⟩; (iv) an explicit counterexample showing the trace-distance of non-negativity violates strong monotonicity under CPCP operations, using ρ = p|Υ₂⟩⟨Υ₂| ⊕ (1−p)|Υ₃⟩⟨Υ₃| and the block-projection channel, for p ∈ (4/25, 2/5). Items (i), (iii), and (iv) are sound on inspection; item (ii) contains an unproved load-bearing assertion.

Significance. If the results stand, the paper closes two concrete open problems from Johnston–Sikora (2022): the failure of strong monotonicity for the trace-distance of non-negativity is demonstrated by a fully explicit, falsifiable counterexample (state, channel, and elementary inequality all written out), and the CPCP/CPDNN equivalence is given an alternative, arguably more transparent route via the comparison-matrix criterion, plus a new dual direction (unital maps M_2→M_n). The d≤4 characterization is checkable via known copositivity criteria, and the symmetry reduction of §4 makes measure evaluations tractable. These are modest but solid and verifiable contributions to a small resource-theoretic literature; the explicit counterexample is the most durable item.

major comments (2)

[§3, proof of Theorem 3.6] The proof asserts, without argument, that 'the comparison matrix of J_{Φ*} is positive semidefinite, so J_{Φ*} is CP' (via Theorem 3.5). This cannot follow from DNN-ness alone: for a general DNN matrix A, M(A) need not be PSD, otherwise DNN=CP in all dimensions, contradicting the d≥5 counterexamples the paper itself relies on in §4/Table 1. The bridge must therefore use the channel structure, and it can: Φ trace-preserving implies Φ* unital, so Σ_m Φ*(E_{mm}) = I; each summand Φ*(E_{mm}) is PSD and entrywise nonnegative, and since the sum has zero off-diagonal entries, each Φ*(E_{mm}) is diagonal. Hence in J_{Φ*} = [[D, F],[F^T, I−D]] both diagonal blocks are diagonal, so M(J_{Φ*}) = S J_{Φ*} S with S = diag(I_n, −I_n), a signature congruence, and PSD follows immediately. As written, however, the paper proves none of this, and the assertion is the sole load-bearing step. Please add this
[§3, Theorem 3.7] Theorem 3.7 is a one-line reduction to Theorem 3.6 (duality plus Lemmas 3.3–3.4) and therefore inherits the unproved comparison-matrix step in full. It cannot be considered established until the gap in Theorem 3.6 is closed. The repair indicated above suffices for both results, so this requires no new ideas — but the revision must be made before either theorem's proof is complete.

minor comments (7)

[§2.2, Proposition 2.9] This no-go claim ('there exists no unital CP-preserving channel...') rests entirely on a numerical SDP being 'found to be strictly infeasible' in MATLAB. No code, solver, tolerance, or dual (Farkas) certificate is given. A non-existence claim supported only by an unreported numerical run is not verifiable; please deposit the CVX script, report the infeasibility certificate, or give an analytic argument.
[§4, Eq. (5)] The set DDN appears in Eq. (5), Observation 4.2, and Table 1 without definition. Please define it (presumably diagonally dominant nonnegative, from [3]) at first use.
[§4, Table 1] The d=5 row reports N_T^DDN = N_T^CP = (17−√5)/10 and N_T^DNN = 1 + 1/√5 with no derivation, in a section that advertises that the nearest CP state 'can be found without any computation.' Please provide at least a sketch (which a_k in Eq. (9) achieve the optimum, and the DNN lower bound) or a reference.
[§5] The logic is correct but should be made explicit: I/5 is a CP state, so N_T(ρ) ≤ ‖ρ − I/5‖₁, and since the post-measurement average (4−p)/3 strictly exceeds this upper bound for p ∈ (4/25, 2/5), it exceeds N_T(ρ) itself. One line stating that an upper bound suffices would prevent misreading. Also worth noting explicitly that K_1, K_2 have nonnegative entries, so Φ is CPCP by Theorem 3.1(iii), not merely CP-preserving.
[§2, Note 2.7] The claim that 'every symmetric matrix [is a] difference of two CP matrices' is stated without justification. It is true (A + λI is strictly diagonally dominant with nonnegative entries for large λ, and diagonally dominant DNN matrices are CP) but needs a line of argument or a citation to [2]/[13].
[§2, Observation 2.8] The Y_{r,s} defined here as (1/2)Σ Φ(|i⟩⟨j|+|j⟩⟨i|)_{r,s}|i⟩⟨j| differs from Observation 2.4's Y_{r,s} (which symmetrizes the matrix units); please check the factors and make the notation consistent. In the d=3 example after Observation 2.4, the (2,1) entry reads '⟨2⟩⟨11|' — a typo for |2⟩⟨1|.
[Throughout] Numerous typos: 'CP-preservinng', 'positve', 'quantumk', 'atlest', 'arbitray', 'chaannel', 'qubut', 'simialr', 'satifies', 'Johnstonet al.' (missing space), J_ϕ for J_Φ in Lemmas 3.3–3.4. Theorem 3.5 should carry a theorem/page number in [13]. Corollary 2.5's phrase 'trace preserving condition of Φ*' should read 'Φ* is trace-preserving since Φ is unital.'

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: mathematical resource-theory paper with independent proofs and an explicit counter-example; the only load-bearing gap is an unproved lemma, not a definitional loop.

full rationale

The paper works entirely inside the Johnston–Sikora free-state/free-operation framework and classical cone theory (Diananda, Berman–Shaked-Monderer, Choi). Section 2 characterizations rest on the classical equality CP=DNN for d≤4 together with the dual/Choi identities (1)–(2); nothing is defined in terms of the claimed output. Theorems 3.6–3.7 reduce CPDNN⇒CPCP by dual Choi structure plus the comparison-matrix theorem; the comparison-matrix PSD claim is asserted without proof (a soundness hole, not a circular step—the claim is not smuggled in by self-citation or by redefining the target). Section 4 symmetry reductions are ordinary convexity/triangle-inequality arguments. Section 5 supplies an explicit block-diagonal state and a CPCP block-projection channel whose numerical inequality violates strong monotonicity by direct calculation. There are no fitted parameters, no self-citation uniqueness theorems, and no renaming of known empirical patterns. The derivation chain is therefore free of the six circularity patterns.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The work sits entirely inside the Johnston–Sikora resource theory and classical cone theory. No free parameters are fitted. The load-bearing external facts are the equality CP=DNN for order ≤4, the comparison-matrix sufficient condition for complete positivity, and the definitions of CPCP/CPDNN via Choi matrices. No new physical entities are postulated.

assumptions (4)

standard math For real symmetric matrices of order at most 4, the completely-positive cone coincides with the doubly-nonnegative cone (CP=DNN).
Invoked throughout §2 to turn entrywise non-negativity of Φ(|x⟩⟨x|) into the CP-preserving property; classical (Maxfield–Minc / Gray–Wilson).
standard math If A is symmetric and entrywise non-negative and its comparison matrix M(A) is positive semidefinite, then A is completely positive (Berman–Shaked-Monderer).
Cited as Theorem 3.5 and applied as the sole bridge from DNN to CP in the proof of Theorem 3.6.
domain assumption A linear map is CPCP iff its Choi matrix is a CP matrix iff it admits a Kraus representation with entrywise non-negative operators (Johnston–Sikora, Thm 1).
Taken as the definition of the free operations with tensor-product structure; used in §3 and for the counter-example channel in §5.
domain assumption A map is CPDNN iff its Choi matrix is DNN (Johnston–Sikora, Thm 4).
Starting point of the CPDNN⇒CPCP arguments in Theorems 3.6–3.7.

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Cite this review

Pith. "Pith review of CP-preserving channels." pith.science (2026). https://pith.science/paper/OMH6QUTH

@misc{pith2026260724682,
  author       = {Pith},
  title        = {Pith review of: CP-preserving channels},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OMH6QUTH}},
  note         = {Machine review of arXiv:2607.24682}
}

abstract

Completely positive (CP) matrices are ubiquitous in modern science and technology with applications in optimization, graph theory, and quantum entanglement. Recently, Johnston \emph{et al.} [Linear Algebra and its Applications, 2022] have cast CP matrices into the framework of quantum resource theories, where CP states serve as free states and CP-preserving channels act as free operations. This work addresses several questions raised in their work. Specifically, we provide the necessary and sufficient conditions of CP-preserving channels in small dimensions, which are necessary in higher dimensions, and discuss the resource quantification via the trace distance of non-negativity. By constructing an explicit counterexample, we demonstrate that the trace-distance measure of non-negativity violates strong monotonicity. We also provide an alternative proof that every CPDNN channel $\Phi:\MM_n\to \MM_2$ is CPCP. Additionally, we show that any unital CPDNN map $\Phi:\MM_2\to \MM_n$ is also CPCP.

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Reference graph

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Pith tools

Reviewed July 31, 2026 · model on record in the stance chip above.

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On the computational complexity of membership problems for the completely positive cone and its dual

P. J. Dickinson and L. Gijben, “On the computational complexity of membership problems for the completely positive cone and its dual”, Computational Optimization and Applica- tions57, 403 (2014)

2014

[2] [2]

Shaked-Monderer and A

N. Shaked-Monderer and A. Berman,Copositive and completely positive matrices(WORLD SCIENTIFIC, 2021). 12

2021

[3] [3]

Completely positive completely positive maps (and a resource theory for non-negativity of quantum amplitudes)

N. Johnston and J. Sikora, “Completely positive completely positive maps (and a resource theory for non-negativity of quantum amplitudes)”, Linear Algebra and its Applications 653, 395 (2022)

2022

[4] [4]

Quantum resource theories

E. Chitambar and G. Gour, “Quantum resource theories”, Rev. Mod. Phys.91, 025001 (2019)

2019

[5] [5]

CPDNN quantum channels with qubit output are CPCP

H. Cha, “CPDNN quantum channels with qubit output are CPCP”, arXiv:2603.16962 (2026)

arXiv 2026

[6] [6]

Completely positive linear maps on complex matrices

M.-D. Choi, “Completely positive linear maps on complex matrices”, Linear Algebra and its Applications10, 285 (1975)

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J. Watrous,The theory of quantum information(Cambridge University Press, 2018)

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P. H. Diananda, “On non-negative forms in real variables some or all of which are non- negative”, Mathematical Proceedings of the Cambridge Philosophical Society58, 17 (1962)

1962