Existence of Kraus decomposition in infinite dimension via strongly-convergent direct process tomography

Paul E. Lammert

REVIEW 2 major objections 4 minor 16 references

Existence of Kraus decomposition in infinite dimension via strongly-convergent direct process tomography

T0 review · 2 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read For every completely positive map between separable Hilbert spaces, an iterative reduction algorithm produces a Kraus decomposition whose partial sums converge in strong-operator topology.

desk verdict Genuinely constructive Kraus algorithm with a repairable but real proof gap: Eq. (7c) is false, though a containment version suffices. read the letter →

arxiv 2608.07207 v1 pith:ICEAJQ6B submitted 2026-08-07 quant-ph math-phmath.MP

Paul E. Lammert This is my paper · ORCID

classification quant-phmath-phmath.MP

keywords KrausdecompositioncompletelypositivemapsseparableHilbertspacesstrongoperatortopologyprocesstomographyoperator-sumrepresentationkernelrelationsconstructiveproof

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 shows that a Kraus decomposition, the operator-sum representation of a completely positive (CP) map, exists constructively for every CP map from trace-class operators on one separable Hilbert space to those on another. Instead of relying on nonconstructive representation theory, it gives an iterative algorithm that subtracts one rank-one pure operation at a time, each step forcing a chosen matrix element of the remainder to zero. The accompanying proof establishes that the partial sums converge to the original map in strong-operator topology, so the infinite sum is a genuine Kraus decomposition. This fuses the existence question with practical process tomography, since the only inputs are the ground matrix elements of the map. If correct, it turns a deep existence theorem into a directly usable computational scheme.

What carries the argument

The argument is carried by two devices. The kernel relation $K_\Lambda = \{(k,h) : \langle \Pi(k) | \Lambda \cdot \Pi(h)\rangle = 0\}$ records which ground matrix elements of a CP map vanish, and the $J$-transform, a basis-free variant of channel-state duality, turns $\Lambda$ into a positive operator $J\Lambda$ on the Hilbert–Schmidt space $\mathcal{B}_2(\mathcal{H},\mathcal{K})$. The reduction step invokes Lemma 1: for a positive operator $T$ with $T\phi\neq 0$, subtracting the rank-one projector $\Pi(\eta)$ with $\eta = T\phi/\sqrt{\langle \phi|T\phi\rangle}$ forces $T'\phi = 0$ and gives $\ker T' = \ker T + \mathbb{C}\phi$. Translating this through the $J$-transform produces the kernel-relation identity $K_{\Lambda'} = K_\Lambda \vee_S (k,h)$, which guarantees that an entry zeroed in one step is never undone. That permanence is what makes the remainders converge strongly to zero.

What would settle it

Run the algorithm on a concrete CP map and a fixed enumeration of basis pairs, checking after each step whether any matrix element previously forced to zero has become nonzero; if such a re-population occurs, the identity $K_{\Lambda'} = K_\Lambda \vee_S (k,h)$ fails and the convergence proof collapses.

Watch

Extended reading notes

Core claim

The paper claims that for every completely positive map $\Lambda \in \mathcal{B}(\mathcal{B}_1(\mathcal{H}), \mathcal{B}_1(\mathcal{K}))$ with $\mathcal{H}$ and $\mathcal{K}$ separable Hilbert spaces, the iterative reduction algorithm produces bounded operators $A_1, A_2, \ldots$ in $\mathcal{B}(\mathcal{H},\mathcal{K})$ such that $\Lambda = \sum_{i=1}^\infty \Theta(A_i)$ with $\Theta(A)\rho = A\rho A^\dagger$, and the partial sums converge to $\Lambda$ in the strong operator topology. Equivalently, the remainder $\Lambda_n := \Lambda - \sum_{i=1}^n \Theta(A_i)$ tends to zero in trace norm on every input state $\rho$. Each reduction step forces one ground matrix element $\langle \Pi(k) | \Lambda_n \cdot \Pi(h) \rangle$ to zero for a chosen basis pair $(k,h)$, and the proof shows that this zero is permanent. The algorithm therefore outputs a coherent family of exact Kraus decompositions of the restrictions of $\Lambda$ to ever-larger finite-dimensional subspaces.

Load-bearing premise

The convergence proof hinges on the permanence of forced zeros: once a ground matrix element of the remainder is set to zero, later reduction steps must leave it zero.

Editorial extensions

If this is right

Every CP map on separable spaces acquires an explicit, ordered Kraus list, so existence is established constructively without representation theory.
Each new Kraus operator $A_n$ carries one more guaranteed zero matrix entry, so truncating after $N$ steps yields an exact decomposition of the map restricted to an $N$-dimensional subspace, with a uniform bound on the remainder.
The remainders are always CP, uniformly bounded by $\|\Lambda\|$, and converge strongly to zero, so the infinite sum converges without auxiliary Hilbert-space extensions.
Because the inputs are only ground matrix elements, the algorithm is directly executable as a process tomography scheme for channels on infinite-dimensional systems.

Reading between the lines

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

If the permanence identity holds, a natural numerical stopping rule becomes certified: stop when the trace norm of the remainder drops below tolerance, and the truncated sum is a guaranteed approximation.
The same reduction strategy might extend to maps on non-separable spaces by transfinite iteration, or to generating Kraus decompositions adapted to an arbitrary net of subspaces, though the paper does not address these.
The kernel-relation view suggests a structural characterization of CP maps through their zero patterns, possibly linking to matrix-completion problems in quantum tomography.

Share X Bluesky LinkedIn Reddit HN

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

Dear colleague,

The paper's core idea is genuine: an iterative algorithm that produces Kraus operators one at a time for a CP map on separable Hilbert spaces, with the remainder tending to zero in strong-operator topology. That is a real improvement over the nonconstructive Kraus existence proof, and it connects directly to practical process tomography. The exposition of the matrix picture and the uniform bounds is clear, and the infinite-dimensional lifting arguments look careful. No free parameters, no invented entities, no circularity; the one self-citation to earlier notes supplies standard finite-dimensional J-transform machinery.

The proof has a real gap. Proposition 1's identity (7c) is false as stated, even in finite dimension. The stress-test counterexample works: with H=K=C^2 and Λ=Θ(A1)+Θ(A2)+Θ(A3) for the three operators given, reduction at (e1,e1) yields Λ'=Θ(A2)+Θ(A3), which has (e2,e2) in its kernel relation, while the S-closure of the original kernel relation with (e1,e1) does not contain (e2,e2). The underlying kernel statement in Lemma 1 appears true, though its proof contains a false implication about ψ⊥kerT+Cϕ; but even with a corrected Lemma 1, translating kernel equality to equality of S-closed kernel relations is not valid. What the convergence proof actually requires is only the containment K_{Λ'} ⊇ K_Λ ∨_S (k,h), and that does follow from the true kernel statement plus complete positivity. So the theorem is very likely correct, but the written proof is not.

The rest of the infinite-dimensional argument—the quasi-Cauchy-Schwarz bound, the projection-lifting lemma, the SOT convergence proof—reads as sound. Once the authors replace (7c) with the containment version and fix the Lemma 1 proof, this is a solid contribution. As it stands, it deserves a serious referee but not acceptance without revision. I would send it out, and I would cite the algorithm once the proof is repaired; the gap is instructive enough that I would also bring it to a reading group. The intended reader is someone doing process tomography in infinite dimensions or large finite systems, or anyone teaching Kraus decomposition from a constructive angle.

Recommendation: revise.

Best

Referee Report

2 major / 4 minor

Summary. This paper proposes a constructive algorithm for Kraus decompositions of completely positive maps Λ: B_1(H) → B_1(K) on separable Hilbert spaces. Starting from ground-matrix elements of Λ, the algorithm repeatedly extracts one Kraus operator A_m via Eq. (6), zeroing one ground-matrix element of the remainder at each step. The paper claims that the partial sums converge strongly (in trace norm on every input state) to Λ, and that finite-dimensional projections of Λ are exactly represented by finite initial segments of the generated operators, Eq. (23). The proofs are organized around a J-transform reduction to Lemma 1, a statement about positive operators, from which Proposition 1 derives kernel-relation identities.

Significance. An elementary, algorithmic proof of the infinite-dimensional Kraus decomposition theorem would be a genuinely useful contribution, both as a foundationally transparent alternative to the standard C*-algebraic existence proof and as a practical process-tomography scheme. The paper is commendably free of fitted parameters, does not assume the target decomposition, and the finite-dimensional compatibility statement in Eq. (23) is valuable. However, the two technical issues detailed below concern exactly the mechanism that guarantees persistence of zeroed matrix elements and hence drives the convergence argument; until they are repaired, the central claim is not rigorously established as written. The errors appear repairable: the stated kernel-relation equality can be weakened to a containment that still suffices for the convergence proof.

major comments (2)

[Section 3.2, Lemma 1] The proof of the reverse inclusion in Eq. (18) uses the implication 'ψ⊥kerT+Cφ ⇒ <η|ψ>∝<φ|Tψ>=0', which is false. For T=diag(4,1,0), φ=(1,2,1)/√6 and ψ=(2,-1,0), one has ψ⊥φ and ψ⊥kerT, yet <φ|Tψ>=√6≠0. The lemma itself is nevertheless true: from T'ψ=0 one obtains Tψ=(<η|ψ>/<η|φ>)Tφ and hence T(ψ-cφ)=0 with c=<η|ψ>/<η|φ>, so ψ∈kerT+Cφ. The proof should be rewritten accordingly, since Eq. (18) is the basis for the kernel-relation translation in Proposition 1.
[Section 3.3.1, Eq. (7c)] The identity K_{Λ'}=K_Λ∨_S(k,h) is false as stated, already in finite dimension. Let H=K=C^2 with ONB {e1,e2}, and take Λ=Θ(A1)+Θ(A2)+Θ(A3) with A1=(|e1><e1|-|e2><e2|)/√2, A2=|e1><e2|, A3=|e2><e1|. Here K_Λ contains no nonzero pair; reducing at (e1,e1) returns A=A1, so Λ'=Θ(A2)+Θ(A3) and (e2,e2)∈K_{Λ'}. However the smallest S-relation containing K_Λ and (e1,e1) is, up to zero pairs, {(αe1,βe1)}, which does not contain (e2,e2). Thus Eq. (7c) fails. The convergence argument in Section 3.4 needs only the containment K_{Λ'}⊇K_Λ∨_S(k,h), which follows from the corrected Lemma 1 and complete positivity; Proposition 1 and the zero-pattern statements in Sections 2.3–2.4 and 3.4 should be revised to use this weaker statement.

minor comments (4)

[Section 2.4] The sentence 'The (k(m),h(m)) entries of A_n ... are guaranteed zero as soon as m≤n' is incorrect at m=n: A_n is chosen so that its matrix element at (k(n),h(n)) is nonzero unless that element was already zero in Λ_{n-1}. The statement should say m<n for A_n and m≤n for Λ_n.
[Section 3.3.2] In the bound '∥A∥≤√(∥Λ_n∥_{1,1})', the index n is undefined at that point of the proof; it should refer to the input map of the reduction step, i.e., ∥Λ∥_{1,1} or ∥Λ_{m-1}∥_{1,1} in the iteration.
[Section 3.4, Lemma 3] The proof says 'dPa,sΛ=0 for large enough m'; this should read '\widehat{P}_{a,s}Λ_m=0' (the remainder, not the original map), since the original map is never zero.
[Abstract] The phrase 'the generated sum convergences' should be 'the generated sum converges'.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the Kraus-generating algorithm is a genuine constructive derivation; the only self-citation is non-load-bearing, and the flagged Lemma 1 proof gap is a correctness issue, not circularity.

full rationale

The paper's central claim is not assumed as input. In Prop. 1 (Sec. 2.2), the candidate operator A is defined by the explicit ground-matrix formula (6) in terms of the given CP map Λ, and Λ'=Λ−Θ(A) is then shown CP with the claimed kernel relation (7). The iteration (9) uses only these constructed A_m's; the convergence proof in Sec. 3.4 uses positivity, the quasi-CS inequality (5), and finite-dimensional lifting, none of which postulates a Kraus decomposition. There are no fitted parameters, no data subset, and no post-hoc exclusion. The only self-citation is Ref. [13] in Sec. 3.1.1 for the J-transform/Choi-Jamiolkowski statement (12), but the same statement is attributed to Refs. [14–16] and is standard channel-state duality; it is therefore independent support and does not make the argument circular. For completeness, and as explicitly requested by the reviewing rule, I flag (not as circularity) a genuine proof gap: Sec. 3.2's proof of Lemma 1 contains the implication 'ψ⊥kerT+Cϕ⇒⟨η|ψ⟩∝⟨ϕ|Tψ⟩=0', which is false as written; and the Skeptic's finite-dimensional counterexample indicates that Prop. 1's Eq. (7c) fails in general, though the containment K_{Λ'}⊇K_Λ∨S(k,h) and the kernel statement of Lemma 1 are enough for the convergence argument after repair. These issues concern correctness/repairability, not circular equivalence: no claim of the paper reduces by construction to its own inputs.

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

No free parameters are fitted and no new physical entities are postulated. The construction is an algorithm that consumes ground matrix elements of the given CP map. The only nonstandard device is the bookkeeping concept of kernel relations, a mathematical shorthand, not an invented physical entity.

assumptions (4)

standard math Finite-dimensional channel-state duality: Lambda is CP iff JLambda is a positive operator on B2(H,K) (Eqs. (11)-(12)).
Used in Section 3.1.1 and 3.3.1 to translate the reduction step into a positive-operator deflation; cited to Refs. [13,16].
standard math Trace-norm approximation of density operators by finite-rank projections (Section 3.3.2, Eq. (20)).
Used to lift positivity and complete positivity from finite-dimensional compressions to the full infinite-dimensional map (Lemma (21b)).
standard math Cauchy-Schwarz and equality conditions for positive operators (Section 3.1.3, Lemma 1).
Used in the quasi-Cauchy-Schwarz inequality (5) and the deflation proof; the paper's proof of Lemma 1 contains a flawed implication.
domain assumption H and K are separable Hilbert spaces and Lambda is a bounded completely positive map on trace-class operators (Section 1).
The theorem's domain; the algorithm requires enumerable orthonormal bases and access to ground matrix elements.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Existence of Kraus decomposition in infinite dimension via strongly-convergent direct process tomography." pith.science (2026). https://pith.science/paper/ICEAJQ6B

@misc{pith2026260807207,
  author       = {Pith},
  title        = {Pith review of: Existence of Kraus decomposition in infinite dimension via strongly-convergent direct process tomography},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ICEAJQ6B}},
  note         = {Machine review of arXiv:2608.07207}
}

read the original abstract

An algorithm is presented for Kraus decomposition of a completely positive operator over separable (countably-infinite-dimensional) Hilbert spaces, together with an elementary proof that the generated sum convergences in strong-operator topology. This improves on the standard, nonconstructive, proof by fusing the abstract problem with practical process tomography. Kraus operators are generated one-by-one, each having one more guaranteed zero matrix entry than the previous one. In this way, the stream of outputs of the algorithm provides a coherent family of Kraus decompositions of restrictions of the target CP map to ever-larger subspaces.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

16 extracted references · 5 canonical work pages

[1]

Vacchini, Open quantum systems---foundations and theory, Graduate Texts in Physics

B. Vacchini, Open quantum systems---foundations and theory, Graduate Texts in Physics. Springer, Cham, ISBN 978-3-031-58217-2; 978-3-031-58218-9, doi:10.1007/978-3-031-58218-9 (2024)

doi:10.1007/978-3-031-58218-9 2024
[2]

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, ISBN 978-0-521-63503-5, doi:10.1017/CBO9780511976667 (2000)

doi:10.1017/cbo9780511976667 2000
[3]

D. A. Lidar, Lecture notes on the theory of open quantum systems, doi:10.48550/arXiv.1902.00967 (2020), 1902.00967

arXiv doi:10.48550/arxiv.1902.00967 2020
[4]

Chruscinski, Dynamical maps beyond markovian regime?, Physics Reports-Review Section of Physics Letters 992, 1 (2022), doi:10.1016/j.physrep.2022.09.003

D. Chruscinski, Dynamical maps beyond markovian regime?, Physics Reports-Review Section of Physics Letters 992, 1 (2022), doi:10.1016/j.physrep.2022.09.003

doi:10.1016/j.physrep.2022.09.003 2022
[5]

Breuer and F

H.-P. Breuer and F. Petruccione, Theory of Open Quantum Systems, Oxford University Press, ISBN 978-0-198-52063-4 (2002)

work page 2002
[6]

Benatti, Dynamics, information and complexity in quantum systems, Theoretical and Mathematical Physics

F. Benatti, Dynamics, information and complexity in quantum systems, Theoretical and Mathematical Physics. Springer, Berlin, ISBN 978-1-4020-9305-0, doi:10.1007/978-1-4020-9306-7 (2009)

work page doi:10.1007/978-1-4020-9306-7 2009
[7]

Hayashi, Quantum information theory, Graduate Texts in Physics

M. Hayashi, Quantum information theory, Graduate Texts in Physics. Springer-Verlag, Berlin, second edn., ISBN 978-3-662-49723-4; 978-3-662-49725-8, doi:10.1007/978-3-662-49725-8 (2017)

doi:10.1007/978-3-662-49725-8 2017
[8]

I. L. Chuang and M. A. Nielsen, Prescription for experimental determination of the dynamics of a quantum black box, Journal of Modern Optics 44(11-12), 2455 (1997), doi:10.1080/09500349708231894, https://www.tandfonline.com/doi/pdf/10.1080/09500349708231894

doi:10.1080/09500349708231894 1997

Show all 16 references

[9]

Jaeger, Quantum information, Springer, New York, ISBN 978-0-387-35725-6; 0-387-35725-4, doi:10.1007/978-0-387-36944-0, An overview, With a foreword by Tommaso Toffoli (2007)

G. Jaeger, Quantum information, Springer, New York, ISBN 978-0-387-35725-6; 0-387-35725-4, doi:10.1007/978-0-387-36944-0, An overview, With a foreword by Tommaso Toffoli (2007)

2007 doi
[10]

Kraus, General state changes in quantum theory, Annals of Physics 64(2), 311 (1971), doi:10.1016/0003-4916(71)90108-4

K. Kraus, General state changes in quantum theory, Annals of Physics 64(2), 311 (1971), doi:10.1016/0003-4916(71)90108-4

1971 doi
[11]

Kraus, States, effects, and operations, vol

K. Kraus, States, effects, and operations, vol. 190 of Lecture Notes in Physics, Springer-Verlag, Berlin, ISBN 3-540-12732-1, doi:10.1007/3-540-12732-1 (1983)

1983 doi
[12]

Mohseni, A

M. Mohseni, A. T. Rezakhani and D. A. Lidar, Quantum-process tomography: Resource analysis of different strategies , Physical Review A 77(3), 032322 (2008), doi:10.1103/PhysRevA.77.032322

2008 doi
[13]

P. E. Lammert, Notes on completely positive maps and continuous-time markovian cp evolution. a geometry-flavored perspective, doi:10.48550/arXiv.2507.11766 (2026), 2507.11766

2026 doi arXiv
[14]

Choi, Completely positive linear maps on complex matrices, Linear Algebra and its Applications 10(3), 285 (1975), doi:10.1016/0024-3795(75)90075-0

M. Choi, Completely positive linear maps on complex matrices, Linear Algebra and its Applications 10(3), 285 (1975), doi:10.1016/0024-3795(75)90075-0

1975 doi
[15]

Jiang, S

M. Jiang, S. Luo and S. Fu, Channel-state duality, Phys. Rev. A 87, 022310 (2013), doi:10.1103/PhysRevA.87.022310

2013 doi
[16]

Grabowski, M

J. Grabowski, M. Kus and G. Marmo, On the relation between states and maps in infinite dimensions, Open Systems & Information Dynamics 14(4), 355 (2007), doi:10.1007/s11080-007-9061-3

2007 doi

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [1]

Vacchini, Open quantum systems---foundations and theory, Graduate Texts in Physics

B. Vacchini, Open quantum systems---foundations and theory, Graduate Texts in Physics. Springer, Cham, ISBN 978-3-031-58217-2; 978-3-031-58218-9, doi:10.1007/978-3-031-58218-9 (2024)

doi:10.1007/978-3-031-58218-9 2024

[2] [2]

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, ISBN 978-0-521-63503-5, doi:10.1017/CBO9780511976667 (2000)

doi:10.1017/cbo9780511976667 2000

[3] [3]

D. A. Lidar, Lecture notes on the theory of open quantum systems, doi:10.48550/arXiv.1902.00967 (2020), 1902.00967

arXiv doi:10.48550/arxiv.1902.00967 2020

[4] [4]

Chruscinski, Dynamical maps beyond markovian regime?, Physics Reports-Review Section of Physics Letters 992, 1 (2022), doi:10.1016/j.physrep.2022.09.003

D. Chruscinski, Dynamical maps beyond markovian regime?, Physics Reports-Review Section of Physics Letters 992, 1 (2022), doi:10.1016/j.physrep.2022.09.003

doi:10.1016/j.physrep.2022.09.003 2022

[5] [5]

Breuer and F

H.-P. Breuer and F. Petruccione, Theory of Open Quantum Systems, Oxford University Press, ISBN 978-0-198-52063-4 (2002)

work page 2002

[6] [6]

Benatti, Dynamics, information and complexity in quantum systems, Theoretical and Mathematical Physics

F. Benatti, Dynamics, information and complexity in quantum systems, Theoretical and Mathematical Physics. Springer, Berlin, ISBN 978-1-4020-9305-0, doi:10.1007/978-1-4020-9306-7 (2009)

work page doi:10.1007/978-1-4020-9306-7 2009

[7] [7]

Hayashi, Quantum information theory, Graduate Texts in Physics

M. Hayashi, Quantum information theory, Graduate Texts in Physics. Springer-Verlag, Berlin, second edn., ISBN 978-3-662-49723-4; 978-3-662-49725-8, doi:10.1007/978-3-662-49725-8 (2017)

doi:10.1007/978-3-662-49725-8 2017

[8] [8]

I. L. Chuang and M. A. Nielsen, Prescription for experimental determination of the dynamics of a quantum black box, Journal of Modern Optics 44(11-12), 2455 (1997), doi:10.1080/09500349708231894, https://www.tandfonline.com/doi/pdf/10.1080/09500349708231894

doi:10.1080/09500349708231894 1997