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Local preprocessing on the system qubit can make correlated two-qubit dynamics match the product-state prescription, with a single local unitary conjectured always to suffice.

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2026-08-04 14:47 UTC pith:D2QH72XK

load-bearing objection Solid side results, but the headline conjecture is unproven and Theorem 3 has a direct counterexample. the 4 major comments →

arxiv 2509.23083 v2 pith:D2QH72XK submitted 2025-09-27 quant-ph

An Interacting System and Environment with Prior Correlations Plus Local Operations Can Mimic Uncorrelated Evolution

Daniel Dilley , Alvin Gonzales , Jeffrey Larson , Mark Byrd This is my paper
classification quant-ph MSC 81P4581P40 PACS 03.65.Yz03.67.-a
keywords open quantum systemscompletely positive mapsnon-completely positive dynamicssystem-environment correlationslocal unitary operationstwo-qubit dynamicsKraus channelproduct state dynamics matching
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper asks when the reduced dynamics of a system that starts correlated with its environment can be reproduced by the same global unitary acting on an uncorrelated product state. It shows that applying a local operation to the system before the joint evolution — a measurement, a unitary, or a two-term Kraus channel — can enforce this dynamics-matching condition. Measurements always work but necessarily disturb the system, lowering fidelity. Local unitaries succeed in all 402 numerically tested instances, with average fidelity 99.8% and minimum 94.2%, and the paper conjectures this always holds. A two-term Kraus channel implemented with one ancilla qubit achieves unit fidelity in all tested cases; the authors also show a single fixed rotation can prevent non-CP dynamics over an entire time interval.

Core claim

The central claim is that initial system-environment correlations need not prevent a completely positive, product-state description of the observed subsystem dynamics: one can insert a local operation on the system before the global unitary and choose a valid environment state ζ_E such that tr_E(U ρ_SE U†) = tr_E(U ρ_S ⊗ ζ_E U†). For local measurements the paper gives an analytic prescription that always satisfies this condition, though at the cost of state fidelity; for local unitaries it proves the condition is always satisfiable for one- and two-parameter families of two-qubit unitaries via Givens rotations on the correlation matrix, and numerical work supports the same for fully general

What carries the argument

The dynamics-matching identity tr_E(U ρ_SE U†) = tr_E(U ρ_S ⊗ ζ_E U†) is the load-bearing test: it turns the question into the existence of a valid environment Bloch vector ζ_E. The proof machinery is the Cartan KAK decomposition of two-qubit unitaries into local rotations and three nonlocal parameters α_i, together with Givens rotations (counterclockwise rotations in coordinate planes of the correlation matrix, implemented by local unitaries on the system) that zero selected correlation-matrix entries, making the matching equations (44)–(46) solvable. For the time-dependent example, the realigned B-matrix eigenvalue of the dynamical matrix is the NCP witness. The two-term Kraus channel, der

Load-bearing premise

The paper's 'always' claims rest on the untested assumption that the 402 numerically sampled states and unitaries are representative of all two-qubit states and unitaries, and that the heuristic optimizer found globally feasible solutions.

What would settle it

Run a certified global optimization over local unitaries V and environment states ζ_E for a dense grid of two-qubit states ρ_SE and global unitaries U; if any instance fails to satisfy tr_E(U V ρ_SE V† U†) = tr_E(U V ρ_S V† ⊗ ζ_E U†) with a valid ζ_E, the conjecture is false. A cheaper check is to re-optimize the 402 reported cases with a branch-and-bound solver — a single failure at fidelity below 1 would falsify the two-term Kraus claim.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • Correlated two-qubit experiments can be reinterpreted as CP dynamics after a simple basis change on the system, without any environmental control.
  • In the time-dependent family, preparing the system in the R_Y(π/2)-rotated basis prevents all non-CP dynamics, so the avoided NCP behavior is a basis-preparation effect, not an environmental intervention.
  • A single ancilla qubit plus a two-term Kraus channel is a universal fidelity-restoring preprocessor for the 402 tested cases, making product-state modeling exact in those instances.
  • If Conjecture 1 holds, any two-qubit correlated state and any global unitary admit a local unitary preprocessing that makes the product-state prescription valid, collapsing the correlated-versus-product distinction for single-qubit reduced dynamics.
  • The constructive prescriptions assume knowledge of the joint correlation matrix and global unitary; the paper identifies removing this knowledge requirement as the next step.

Where Pith is reading between the lines

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

  • If the 'always' conjecture extends beyond two qubits, a similar Givens-zeroing argument on higher-rank correlation tensors would be needed; the paper leaves multipartite extension open.
  • The two-term Kraus result suggests that a single ancillary qubit may be a universal resource for matching correlated single-qubit dynamics with unit fidelity — a statement stronger than Conjecture 1 that could be tested by a brute-force search over all two-qubit unitaries.
  • A certified global-optimization pass over the 402 instances (rather than heuristic optimization) would either confirm the 99.8% average fidelity or reveal counterexamples, settling the numerical conjecture with computational rigor.
  • The NCP-prevention example implies that choosing the system's preparation basis can act as a control knob for non-Markovianity, which could be probed experimentally with current two-qubit platforms without needing full tomography.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. The paper asks when the reduced dynamics of a two-qubit system, initially correlated with its environment and then evolved under a global unitary U, can be reproduced as if the initial joint state had been a product state. It studies three local preprocessing strategies on the system: weak/projective measurements, local unitaries, and two-term Kraus channels. For measurements it gives constructive conditions and examples (Bell states, Werner states, SWAP∘CNOT); for unitaries it proves a theorem for one- and two-parameter families of nonlocal two-qubit unitaries, proves a theorem for diagonally correlated states, and presents numerical evidence for the full three-parameter case, culminating in Conjecture 1 that a local unitary always suffices. It also gives a time-dependent example in which a fixed local rotation prevents non-CP reduced dynamics, and it shows that a two-term Kraus channel (one ancilla) achieves unit fidelity in all 402 numerical cases. The manuscript includes code and data in a GitHub repository.

Significance. If the analytic claims are fully established, the paper gives a useful operational toolkit for deciding when correlated initial states can be replaced by a product-state model after local preprocessing. The constructive measurement strategy, the one/two-parameter unitary theorem, and the time-dependent NCP-prevention example are concrete contributions. The numerical study is an existence check rather than a parameter fit, and the authors are careful to label the full three-parameter statement as a conjecture. The availability of code and data is a strength. However, the result's headline strength depends on Conjecture 1, which is not proven; the paper's conclusion explicitly defers a formal classification of three-parameter unitaries to future work. The analytic theorem also has an unstated derivation at its core.

major comments (4)
  1. [III.H, Theorem 1, Eqs. (44)-(46)] The proof of Theorem 1 rests on the matching equations (44)-(46), but the text says only 'We derived these equations by solving Eq. (43) exactly.' No derivation, intermediate steps, or verification of the sign conventions are given. Since these equations are load-bearing for the one- and two-parameter unitary results, the derivation must be supplied (at least in an appendix), or the theorem is incomplete.
  2. [III.H, Theorem 1 proof, Givens rotations] The proof asserts that Givens rotations can always be chosen to make t32=0 and t31=0, and then a further rotation to zero expression (52). The simultaneous solvability of the two angle equations for the first two rotations is not demonstrated, and the formulas after the first rotation are not given. The final G12 rotation is stated to leave the first two equalities unchanged, but the required angle z is not constructed. Please provide explicit angle/existence proofs, including the parameter constraints, so the theorem is checkable.
  3. [III.K, Numerical evidence for Conjecture 1] The 402-case numerical study does not, as stated, establish Conjecture 1. The sampling imposes t_ii=0, eliminating diagonal correlation degrees of freedom before the search, and the retained cases are those for which ζ initially has magnitude >1; the tested set may not cover the hardest states. The optimization over local unitaries is not described (parametrization, objective, solver, tolerances, or any global-optimality certificate). The abstract's 'all tested instances' claim is accurate, but the conjecture's 'always' requires either a proof or a much more rigorous sample/optimization report.
  4. [III.J, Theorem 3] The proof of Theorem 3 is a single paragraph: the entangling-power condition is written, and it is asserted that if entanglement does not change then the dynamics can always be U-generated by a product state because non-entangling gates are locally equivalent to SWAP. This argument needs to be made explicit for mixed states and for the local-unitary (identity) class. As written, the theorem is not fully proved.
minor comments (4)
  1. [III.I, Eq. (55)-(56)] The matrix S is printed as diag(s2s3, s1s3, s2s3); the third diagonal entry should be s1s2. Also Eq. (58) writes 'b2' where the context indicates ||b||^2.
  2. [III.K] There is a typo: 't_ii = for all i' should be 't_ii = 0 for all i'.
  3. [III.H, Theorem 1 one-parameter case] In the one-parameter case the proof says 'set σ_1=0' but σ_1 is not defined; presumably ζ_1=0. Please state this explicitly and confirm that the resulting ζ remains a valid environmental state.
  4. [III.G, Eq. (36)-(38)] The ordering of the vectorized basis for the dynamical matrix A_S is not specified. Please define the basis (e.g., |00>,|01>,|10>,|11> ordering) so that the realigned-matrix criterion is unambiguous.

Circularity Check

0 steps flagged

No significant circularity: the derivations solve the stated equations directly, and the numerical results are feasibility/existence searches rather than fitted parameters relabeled as predictions.

full rationale

The paper's central condition, Eq. (3), is existential (existence of a valid ζE), and the authors' constructions solve for the local operation and the environment state from the matching equations rather than defining the target into existence. The local-measurement 'always works' result is a constructive theorem with a one-line proof (a projective measurement leaves a product state, so Eq. (3) holds with ζE the post-measurement environment state); this is a theorem, not a circular definition. The unitary and Kraus-channel results are searched over V and ζE (3+3 and 12 degrees of freedom) against the three components of the reduced-state equality; the reported fidelities are outputs of these searches, not fitted constants later called predictions. The 402-case study selects hard cases (ζE norm > 1 for V=I) and then finds V, so the reported success is an existence check; the lack of a global-optimality certificate and non-random sampling are evidence-strength concerns, not circularity. The self-citations are to independently published tools: Ref. [17] for measurement update formulas, Ref. [18] for the general single-qubit TP-map parametrization (Eq. 59) used to compute the NCP example's dynamical matrix, and Ref. [26] for explicit SO(3)-to-SU(2) maps used to realize Givens rotations; these are real, paper-checkable lemmas and do not smuggle in the conclusion. The paper explicitly flags its own limitations: Theorem 1's Eqs. (44)-(46) are asserted to be "derived by solving Eq. 43 exactly" with the algebra omitted, the asserted existence of Givens rotations zeroing t32 and t31 is not exhibited, and Conjecture 1 (local unitaries always suffice) is left open with "a formal classification of two-qubit three-parameter nonlocal unitaries" listed as future work in the Conclusion. These are proof gaps and scope limits in the claim of universality, not circular reductions; the headline 'always' claim is weaker than the proven content, but the derivation chain itself is self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 6 axioms · 0 invented entities

The central claims rest on standard CP/Kraus theory, the Cartan decomposition of two-qubit unitaries, the single-qubit environment assumption, and the dynamics-matching definition; no free parameters are fitted to data and no new entities are postulated.

axioms (6)
  • standard math CP maps admit Kraus representations; partial trace over a product state yields CP dynamics (Eqs. 1-5).
    Used as the background formalism throughout Sections II and III.
  • standard math Every two-qubit unitary decomposes as U = (L1⊗L2)Ω(R1⊗R2) via Cartan decomposition (Section III H).
    The core reduction step for Theorems 1 and 2 and for the numerical treatment of three-parameter unitaries.
  • domain assumption System and environment are each one qubit; all analytic results and numerics are for two-qubit joint states.
    The entire framework, including the NCP-prevention example, is built on a single-qubit environment; extensions are left open.
  • domain assumption The dynamics-matching condition (Eq. 3) is the operative definition of mimicking uncorrelated evolution; matching one input-output pair does not imply the dynamical map is CP for all inputs (Section II).
    Central to the paper's 'U-generated by a product state' concept; the paper explicitly notes this limitation.
  • domain assumption The realigned B_S-matrix criterion (Ref. [19]) is a valid test for complete positivity of the reduced dynamics (Section III G).
    Used to certify NCP dynamics in the time-dependent example; relies on the maximally mixed state being in the compatibility domain.
  • domain assumption The measurement update formulas (Eq. 9) from the authors' prior work (Ref. [17]) are correct.
    Used without re-derivation for the measurement-based preprocessing results.

pith-pipeline@v1.3.0-alltime-deepseek · 19354 in / 25269 out tokens · 178622 ms · 2026-08-04T14:47:48.075250+00:00 · methodology

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read the original abstract

Initial system-environment correlations can induce reduced dynamics that depart from the standard completely positive (CP) description. We study when such dynamics can be reproduced by the same global unitary acting on an initial system-environment product state. We show that local preprocessing on the system, applied prior to the joint evolution, can lead to dynamics which are completely positive. We focus on two-qubit system-environment states, with a three-qubit extension when a single ancilla is used to implement Kraus channel preprocessing. We analyze three classes of local operations$-$measurements, unitaries, and Kraus channels$-$and characterize their ability to satisfy dynamics matching while minimally perturbing the initial system state. Local measurements can always enforce dynamics matching but necessarily reduce state fidelity. We numerically investigated local unitary preprocessing. Local unitaries enforce dynamics matching in all tested instances and typically preserve the system state: in 402 numerical cases the average fidelity is 99.8%, more than 92% of cases exceed 99% fidelity, and the minimum fidelity is 94.2%. We show that a two-term Kraus channel, realizable with a single ancillary qubit coupled to the system, achieves dynamics matching with unit fidelity in all tested cases. As a second contribution, we demonstrate that local preprocessing can prevent the emergence of non--completely positive (NCP) reduced dynamics in a time-dependent setting. For a correlated two-qubit experiment with a time-dependent global unitary U(t), the induced compatibility domain is fixed by the initial correlations. For every t in a nontrivial continuous interval, the reduced dynamics are NCP when defined over this domain. We then show that a single, time-independent local unitary applied prior to the joint evolution renders the reduced dynamics CP over the entire interval.

Figures

Figures reproduced from arXiv: 2509.23083 by Alvin Gonzales, Daniel Dilley, Jeffrey Larson, Mark Byrd.

Figure 1
Figure 1. Figure 1: FIG. 1. Plot of optimal fidelity of [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: shows the minimum ϵ-values that are required to U-generate the dynamics of the system using a prod￾uct state for θ ∈ [0, π/2]. These data points illustrate how optimal fidelity always decreases as the strength of the measurement increases. As would be expected, lo￾cal measurements can never completely preserve fidelity. On the other hand, when θ = π/2, no measurement is necessary and we can U-generate the … view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Plot of the optimal fidelity for each of the 402 experi [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗

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Reference graph

Works this paper leans on

36 extracted references · 1 linked inside Pith

  1. [1]

    U-generation by a product state,

    Access to Both Qubits Now imagine if we can apply local unitaries to both qubits inρ S1S2 instead of treating it as a system and environment. One may be interested in preventing cer- tain dynamics between two accessible data qubits that are negatively affected by their local environment, caus- ing the two data qubits to interact in a non-standard way. The...

  2. [2]

    Using the reduced form in Eq. 43, we must show that the three equalities (t32 −a 3ζ2) cos(2α3) sin(2α2) + [(−t23 +a 2ζ3) cos(2α2) + (b1 −ζ 1) sin(2α2)] sin(2α3) = 0,(44) (−t31 +a 3ζ1) cos(2α3) sin(2α1) + [(t13 −a 1ζ3) cos(2α1) + (b2 −ζ 2) sin(2α1)] sin(2α3) = 0,(45) and (t21 −a 2ζ1) cos(2α2) sin(2α1) + [(−t12 +a 1ζ2) cos(2α1) + (b3 −ζ 3) sin(2α1)] sin(2α2...

  3. [3]

    The Theory of Open Quantum Systems

    Heinz-Peter Breuer and Francesco Petruccione. The Theory of Open Quantum Systems. Oxford University Press, Oxford, UK, 2002

    2002

  4. [4]

    Angel Rivas and Susana F. Huelga. Open Quantum Systems: An Introduction. Springer, Berlin, Heidelberg, 2012

    2012

  5. [5]

    Rodr ´ ıguez-Rosario, and Al´ an Aspuru-Guzik

    Kavan Modi, C´ esar A. Rodr ´ ıguez-Rosario, and Al´ an Aspuru-Guzik. Positivity in the presence of ini- tial system-environment correlation. Phys. Rev. A, 86(6):064102, 2012

    2012

  6. [6]

    Rodriguez-Rosario

    Aharon Brodutch, Animesh Datta, Kavan Modi, An- gel Rivas, and Cesar A. Rodriguez-Rosario. Vanishing quantum discord is not necessary for completely positive maps. Phys. Rev. A, 87:042301, Apr 2013

    2013

  7. [7]

    Longjiang Liu and D. M. Tong. Completely positive maps within the framework of direct-sum decomposition of state space. Phys. Rev. A, 90:012305, Jul 2014

    2014

  8. [8]

    Completely pos- itive maps and classical correlations

    Cesar A Rodriguez-Rosario, Kavan Modi, Aik-meng Kuah, Anil Shaji, and E C G Sudarshan. Completely pos- itive maps and classical correlations. Journal of Physics A: Mathematical and Theoretical, 41(20):205301, apr 2008

    2008

  9. [9]

    Restrictions on initial system-environment correlations based on the dynamics of an open quantum system

    Eric Chitambar, Ali Abu-Nada, Russell Ceballos, and Mark Byrd. Restrictions on initial system-environment correlations based on the dynamics of an open quantum system. Phys. Rev. A, 92:052110, Nov 2015

    2015

  10. [10]

    (Github Repository) Dynamics Matching.https: //github.com/qubit-qutrit/Dynamics-Matching.git, Sep 2025

    Daniel Dilley, Alvin Gonzales, Jeffrey Larson, and Mark Byrd. (Github Repository) Dynamics Matching.https: //github.com/qubit-qutrit/Dynamics-Matching.git, Sep 2025

    2025

  11. [11]

    Theory of quan- tum error-correcting codes

    Emanuel Knill and Raymond Laflamme. Theory of quan- tum error-correcting codes. Phys. Rev. A, 55:900–911, Feb 1997

    1997

  12. [12]

    Quantum Computation and Quantum Information

    Michael A Nielsen and Isaac L Chuang. Quantum Computation and Quantum Information. Cambridge University Press, 2011

    2011

  13. [13]

    Reduced dynamics need not be com- pletely positive

    Philip Pechukas. Reduced dynamics need not be com- pletely positive. Phys. Rev. Lett., 73:1060–1062, Aug 1994

    1994

  14. [14]

    reduced dynamics need not be completely positive

    Robert Alicki. Comment on “reduced dynamics need not be completely positive”. Phys. Rev. Lett., 75:3020–3020, Oct 1995

    1995

  15. [15]

    Jordan, Anil Shaji, and E

    Thomas F. Jordan, Anil Shaji, and E. C. G. Sudarshan. Dynamics of initially entangled open quantum systems. Phys. Rev. A, 70:052110, Nov 2004

    2004

  16. [16]

    Bennett, David P

    Charles H. Bennett, David P. DiVincenzo, John A. Smolin, and William K. Wootters. Mixed-state entan- glement and quantum error correction. Phys. Rev. A, 54:3824–3851, Nov 1996

    1996

  17. [17]

    Pechukas replies:

    Philip Pechukas. Pechukas replies:. Phys. Rev. Lett., 75:3021–3021, Oct 1995

    1995

  18. [18]

    Todd A. Brun. A simple model of quantum trajectories. American Journal of Physics, 70(7):719–737, 2002

    2002

  19. [19]

    Some entanglement survives most measurements

    Alvin Gonzales, Daniel Dilley, and Mark Byrd. Some entanglement survives most measurements. Phys. Rev. A, 109:022233, Feb 2024

    2024

  20. [20]

    Guaran- teeing completely positive quantum evolution

    Daniel Dilley, Alvin Gonzales, and Mark Byrd. Guaran- teeing completely positive quantum evolution. Journal of Physics A: Mathematical and Theoretical, 54(50):505302, nov 2021

    2021

  21. [21]

    A. R. Usha Devi, A. K. Rajagopal, and Sudha. Open- system quantum dynamics with correlated initial states, not completely positive maps, and non-markovianity. Phys. Rev. A, 83:022109, Feb 2011

    2011

  22. [22]

    Differential Geometry, Lie Groups, and Symmetric Spaces, volume 34 of Pure and Applied Mathematics

    Sigurdur Helgason. Differential Geometry, Lie Groups, and Symmetric Spaces, volume 34 of Pure and Applied Mathematics. Academic Press, New York / London / San Francisco, 1978

    1978

  23. [23]

    Anthony W. Knapp. Lie Groups Beyond an Introduction, volume 140 of Progress in Mathematics. Birkh¨ auser, Boston / Basel / Berlin, 2 edition, 2002

    2002

  24. [24]

    Robert R. Tucci. An introduction to Car- tan’s KAK decomposition for QC programmers. arXiv:quant-ph/0507171, 2005

    Pith/arXiv arXiv 2005

  25. [25]

    Navin Khaneja and Steffen J. Glaser. Cartan decompo- sition of su(2n) and control of spin systems. Chemical Physics, 267(1):11–23, 2001

    2001

  26. [26]

    Kraus and J

    B. Kraus and J. I. Cirac. Optimal creation of entangle- ment using a two-qubit gate. Phys. Rev. A, 63:062309, May 2001

    2001

  27. [27]

    Unitary transformations can be distinguished lo- cally

    Xiang-Fa Zhou, Yong-Sheng Zhang, and Guang-Can Guo. Unitary transformations can be distinguished lo- cally. Phys. Rev. Lett., 99:170401, Oct 2007

    2007

  28. [28]

    Identi- fying quantum correlations using explicit so(3) to su(2) maps

    Daniel Dilley, Alvin Gonzales, and Mark Byrd. Identi- fying quantum correlations using explicit so(3) to su(2) maps. Quantum Information Processing, 21(10):343, Oc- tober 2022

    2022

  29. [29]

    Wootters

    Scott Hill and William K. Wootters. Entanglement of a pair of quantum bits. Phys. Rev. Lett., 78:5022–5025, Jun 1997

    1997

  30. [30]

    En- tangling power of quantum evolutions

    Paolo Zanardi, Christof Zalka, and Lara Faoro. En- tangling power of quantum evolutions. Phys. Rev. A, 62:030301, Aug 2000

    2000

  31. [31]

    Entanglement of quantum evolutions

    Paolo Zanardi. Entanglement of quantum evolutions. Phys. Rev. A, 63:040304, Mar 2001

    2001

  32. [32]

    Nonlocal properties of two-qubit gates and mixed states, and the optimization of quantum com- putations

    Yuriy Makhlin. Nonlocal properties of two-qubit gates and mixed states, and the optimization of quantum com- putations. Quantum Information Processing, 1(4):243– 252, 2002

    2002

  33. [33]

    A. J. Scott. Multipartite entanglement, quantum-error- correcting codes, and entangling power of quantum evo- lutions. Phys. Rev. A, 69:052330, May 2004

    2004

  34. [34]

    Entangling power of multipartite uni- tary gates

    Tomasz Linowski, Grzegorz Rajchel-Mieldzio´ c, and Karol ˙Zyczkowski. Entangling power of multipartite uni- tary gates. Journal of Physics A: Mathematical and Theoretical, 53(12):125303, mar 2020

    2020

  35. [35]

    Forrest Stinespring

    W. Forrest Stinespring. Positive functions onc ∗-algebras. Proceedings of the American Mathematical Society, 6(2):211–216, April 1955

    1955

  36. [36]

    Orthogo- nalization and the Gram–Schmidt Process

    Gene H. Golub and Charles F. Van Loan. Matrix Computations. Johns Hopkins University Press, Balti- more, MD, 3rd edition, 1996. See Chapter 5, “Orthogo- nalization and the Gram–Schmidt Process”. 15 The submitted manuscript has been created by UChicago Ar- gonne, LLC, Operator of Argonne National Laboratory (“Ar- gonne”). Argonne, a U.S. Department of Ener...

    1996