pith. sign in

arxiv: 2605.23852 · v1 · pith:6DHHHOKRnew · submitted 2026-05-22 · 🪐 quant-ph · math-ph· math.MP

Convexity and non-Markovianity of Weyl Maps

Wen Xu , Vinayak Jagadish This is my paper

Pith reviewed 2026-05-25 04:11 UTC · model grok-4.3

classification 🪐 quant-ph math-phmath.MP
keywords Weyl dynamical mapsnon-Markovian dynamicsconvex combinationseternal non-Markovianitydiscrete phase spaceHermite normal formquantum dephasingfinite open systems
0
0 comments X

The pith

Convex combinations of eternally non-Markovian Weyl dephasing maps can generate Markovian semigroups, and irreducible eternally non-Markovian Weyl maps exist individually.

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

The paper classifies subgroups of the discrete phase space Z_d × Z_d via Hermite normal form to underpin Weyl dynamical maps in finite-dimensional systems. It shows that isotropic maps can form Markovian semigroups while anisotropic ones with nonuniform weights cannot, and that mixing eternally non-Markovian dephasing maps can cancel memory effects to produce Markovian evolution. It also proves a general condition for mixtures of distinct semigroups to become eternally non-Markovian and identifies single maps that display eternal non-Markovianity without any mixing, unlike the qubit Pauli case. Explicit qutrit examples demonstrate transitions among the regimes. A sympathetic reader would care because this reveals how algebraic structure and convexity together control memory effects in open quantum dynamics beyond low-dimensional Pauli channels.

Core claim

Weyl dynamical maps are built from subgroups of Z_d × Z_d classified completely by Hermite normal form. Isotropic Weyl maps generate Markovian semigroups under suitable conditions, but anisotropic maps with nonuniform weights never do. Convex combinations of eternally non-Markovian Weyl dephasing maps can produce Markovian semigroups, showing non-Markovianity is not additive. Mixtures of N distinct Weyl semigroups satisfy a general condition for eternal non-Markovianity. Irreducible eternally non-Markovian Weyl dephasing maps exist as individual maps without mixing.

What carries the argument

The Hermite normal form classification of subgroups of Z_d × Z_d that define Weyl maps, together with the isotropic/anisotropic distinction that governs the semigroup property.

If this is right

  • Convex combinations can suppress eternal non-Markovianity and yield Markovian dynamics.
  • Some single Weyl dephasing maps exhibit eternal non-Markovianity without requiring any mixing.
  • Anisotropic maps with nonuniform weights are excluded from semigroup generation.
  • A general algebraic condition determines when mixtures of N Weyl semigroups become eternally non-Markovian.
  • Qutrit examples show explicit transitions among Markovian, non-Markovian, and eternally non-Markovian regimes.

Where Pith is reading between the lines

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

  • The phase-space subgroup structure may offer a systematic way to engineer desired memory effects by choosing weights and mixing.
  • Similar convexity-driven suppression of non-Markovianity could appear in other discrete algebraic representations of quantum channels.
  • Higher-dimensional Weyl systems allow memory control that is unavailable in the Pauli qubit setting.

Load-bearing premise

The Hermite normal form gives a complete list of all subgroups of Z_d × Z_d that can underlie Weyl maps, and the isotropic versus anisotropic distinction alone decides whether the associated maps can form semigroups.

What would settle it

An explicit anisotropic Weyl map with nonuniform weights that nevertheless satisfies the semigroup property, or a convex combination of eternally non-Markovian Weyl dephasing maps whose mixture remains eternally non-Markovian.

read the original abstract

We investigate the emergence of non-Markovian dynamics in finite-dimensional open quantum systems governed by Weyl dynamical maps and their convex combinations. Using the Hermite normal form, we provide a complete classification of the subgroups of the discrete phase space $\mathbb{Z}_d \times \mathbb{Z}_d$, establishing the algebraic framework underlying the Weyl maps. We characterize isotropic Weyl dynamical maps that generate Markovian semigroups and show that anisotropic Weyl maps with nonuniform weight distributions cannot possess the semigroup property. Furthermore, we analyze the role of convexity in the generation and suppression of memory effects. Remarkably, we prove that convex combinations of eternally non-Markovian Weyl dephasing maps can generate Markovian semigroups, demonstrating that non-Markovianity is not additive under mixing. Conversely, we establish a general condition under which convex mixtures of $N$ distinct Weyl semigroups exhibit eternal non-Markovianity. In contrast to the qubit Pauli setting, we further identify the existence of irreducible eternally non-Markovian Weyl dephasing maps, namely, individual dynamical maps that display eternal memory effects without requiring any mixing mechanism. Finally, explicit qutrit examples illustrate the transition among Markovian, non-Markovian and eternally non-Markovian regimes. Our results uncover a fundamental connection among finite phase-space algebra, convex structures, and quantum memory effects, thereby extending the theory of non-Markovian dynamics beyond the Pauli framework.

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.

Referee Report

2 major / 1 minor

Summary. The paper classifies subgroups of the discrete phase space Z_d × Z_d via the Hermite normal form to underpin Weyl dynamical maps. It characterizes isotropic maps generating Markovian semigroups, proves anisotropic maps with nonuniform weights lack the semigroup property, shows convex combinations of eternally non-Markovian Weyl dephasing maps can yield Markovian semigroups (non-additivity of non-Markovianity), gives a general condition for mixtures of N Weyl semigroups to be eternally non-Markovian, identifies irreducible eternally non-Markovian Weyl dephasing maps (unlike the Pauli case), and illustrates transitions with explicit qutrit examples.

Significance. If the algebraic classification and convexity results hold, the work meaningfully extends non-Markovianity theory beyond the qubit Pauli setting by linking finite phase-space subgroups, weight distributions, and convex mixing to memory effects. The demonstration of non-additivity under mixing and the existence of irreducible eternal non-Markovian maps are notable contributions with potential implications for higher-dimensional open-system dynamics.

major comments (2)
  1. [Abstract] Abstract and the classification section: the central claims that anisotropic Weyl maps with nonuniform weights cannot possess the semigroup property, that irreducible eternally non-Markovian maps exist, and that convex combinations of eternal NM maps can produce Markovian semigroups all rest on the Hermite normal form supplying an exhaustive list of subgroups of Z_d × Z_d together with the isotropic/anisotropic distinction plus weight uniformity fully determining the semigroup property. The manuscript must explicitly verify that the HNF classification captures every relevant subgroup (including lifts from Z^2 containing dZ^2) and that no counterexample anisotropic nonuniform case satisfies the semigroup condition under the paper's own definition of the dynamical map.
  2. [Abstract] The proof that convex combinations of eternally non-Markovian maps can generate Markovian semigroups is load-bearing for the non-additivity claim; it should be checked against the same algebraic partition to confirm that the mixing does not inadvertently select only isotropic components or uniform weights.
minor comments (1)
  1. Notation for the weight distributions and the precise definition of 'eternally non-Markovian' should be introduced with an equation number early in the text for clarity when reading the qutrit examples.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting points that merit additional clarification. We address each major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract and the classification section: the central claims that anisotropic Weyl maps with nonuniform weights cannot possess the semigroup property, that irreducible eternally non-Markovian maps exist, and that convex combinations of eternal NM maps can produce Markovian semigroups all rest on the Hermite normal form supplying an exhaustive list of subgroups of Z_d × Z_d together with the isotropic/anisotropic distinction plus weight uniformity fully determining the semigroup property. The manuscript must explicitly verify that the HNF classification captures every relevant subgroup (including lifts from Z^2 containing dZ^2) and that no counterexample anisotropic nonuniform case satisfies the semigroup condition under the paper's own definition of the dynamical map.

    Authors: The Hermite normal form furnishes a complete enumeration of all subgroups of Z_d × Z_d, including those obtained as quotients of lattices in Z^2 that contain dZ^2; this is a standard result in the theory of finite abelian groups and lattice subgroups. Our classification section already invokes this exhaustiveness to partition maps into isotropic and anisotropic cases with uniform or nonuniform weights. To make the verification explicit, we will insert a short paragraph (with a reference to the relevant theorem on HNF) confirming that every subgroup is accounted for and that direct inspection of the resulting list yields no counterexample anisotropic nonuniform map satisfying the semigroup property under the definitions given in the paper. revision: yes

  2. Referee: [Abstract] The proof that convex combinations of eternally non-Markovian maps can generate Markovian semigroups is load-bearing for the non-additivity claim; it should be checked against the same algebraic partition to confirm that the mixing does not inadvertently select only isotropic components or uniform weights.

    Authors: The explicit construction of the convex combination is performed within the same algebraic partition: the mixture produces a Markovian semigroup if and only if the resulting weight distribution and isotropy class satisfy the semigroup criterion derived from the HNF classification. The proof therefore does not inadvertently restrict to isotropic uniform components; rather, the non-additivity arises precisely because certain anisotropic nonuniform mixtures can cancel memory effects while remaining outside the isotropic-uniform class. We will augment the proof paragraph with a one-sentence cross-reference to the partition, making this dependence fully transparent. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation rests on external algebraic classification

full rationale

The paper derives its claims about Markovian semigroups, eternal non-Markovianity, and convexity effects from the standard Hermite normal form classification of subgroups of Z_d × Z_d together with explicit map constructions and convex-combination arguments. This classification is an independent mathematical fact imported from algebra, not defined or justified inside the paper via self-citation chains, fitted parameters, or ansatzes that presuppose the target results. No equation or theorem reduces by construction to its own inputs, and the central distinctions (isotropic vs. anisotropic, uniform vs. nonuniform weights) are applied to the classified subgroups without circular redefinition.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard facts from finite group theory and convex analysis of quantum maps; no free parameters or new postulated entities are mentioned in the abstract.

axioms (2)
  • standard math The set Z_d × Z_d equipped with componentwise addition forms an abelian group whose subgroups can be completely classified by the Hermite normal form.
    Invoked to provide the algebraic framework for Weyl maps.
  • domain assumption Weyl dynamical maps are defined via the action of these subgroups on the finite-dimensional Hilbert space.
    Standard construction in the discrete phase-space formulation of quantum maps.

pith-pipeline@v0.9.0 · 5787 in / 1501 out tokens · 37684 ms · 2026-05-25T04:11:14.674454+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

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

46 extracted references · 46 canonical work pages

  1. [1]

    For an invertible time-local master equation of Eq

    CP-divisibility (Markovianity).A dynamical map{E(t)}t≥0is said to beCP-divisibleif, for any t≥s≥0, there exists a CPTP mapE(t,s )such that E(t) =E(t,s)E(s). For an invertible time-local master equation of Eq. (3), CP-divisibility is equivalent to the condition γα(t)≥0∀α,∀t≥0. Dynamical evolution satisfying this condition is termedMarkovianaccording to the...

    work page

  2. [2]

    This represents the strongest notion of Markovianity and corresponds to strictly memoryless evolution

    Markovian semigroup.If the decay rates are time-independent nonnegative constants, i.e., γα(t)≡γα≥0, then the generatorL(t)becomes time-independent and the corresponding dynamical map forms a one- parameter quantum dynamical semigroup, E(t+s) =E(t)E(s), t≥s≥0. This represents the strongest notion of Markovianity and corresponds to strictly memoryless evolution

    work page

  3. [3]

    For time-local master equation, this occurs if there exists at least one decay channelαand a timet> 0 such that γα(t)<0

    Non-Markovianity.The dynamics is classified as non-Markovianwhenever CP-divisibility is violated. For time-local master equation, this occurs if there exists at least one decay channelαand a timet> 0 such that γα(t)<0. Negative decay rates indicate that the failure of the generator to be of GKLS form at that time and imply that the intermediate mapE(t,s )...

    work page

  4. [4]

    A dynamics is said to be eternally non-Markovian if, for at least one decay channelαsuch that γα(0) = 0, γ α(t)<0∀t>0 +

    Eternal non-Markovianity (ENM).Eternal non-Markovianity denotes a particularly strong vio- lation of CP-divisibility. A dynamics is said to be eternally non-Markovian if, for at least one decay channelαsuch that γα(0) = 0, γ α(t)<0∀t>0 +. In this regime, the CP-divisibility is violated imme- diately after the initial time, and the evolution is non-Markovi...

    work page

  5. [5]

    Total count.The total number of subgroups of orderK is given by the sum over the parameters set SK: N(K) = ∑ (m,n)∈SK gcd ( n, d m ) ,(19) where gcd ( n, d m ) is the greatest common divisor of nand d m

    work page

  6. [6]

    Maximality.The function N (K)attains its maxi- mum when K=d. In this specific maximal case, the setSd and the total countN(d)reduces to Sd ={(m,n)∈N2|mn=d}, N(d) =σ1(d) = ∑ m|d d m = s∏ i=1 pei+1 i −1 pi−1, respectively, whereσ1(d)is the divisor sum function. The proof of Lemma 2 is given in Appendix B. III. SEMIGROUP PROPERTY OF ISOTROPIC WEYL DYNAMICAL ...

    work page

  7. [7]

    However, the convex combination (23) is defined for sub- groups of order|G|≥3, which is a contradiction

    We 7 substitute the specific amplitude for the semigroup into this inequality: r= |G|−1 |G|≤1 2 =⇒ |G|≤2. However, the convex combination (23) is defined for sub- groups of order|G|≥3, which is a contradiction. Con- sequently, the constituent Weyl dephasing maps in the mixture are not ENM. Thus, the Markovian semigroup is generated by mixing maps that are...

    work page

  8. [8]

    Since the number of mixturesN must be an integer, this inequality enforcesN = 2

    Substituting these specific values into our bound(28) yields 2≤N <min {22−1 2−1,N(2) } = 3. Since the number of mixturesN must be an integer, this inequality enforcesN = 2. This recovers the well-known Pauli result: mixing two different Pauli dephasing semi- groups guarantees ENM[31]. However, mixing all three Pauli dephasing semigroups covers the entire ...

    work page

  9. [9]

    [19], demonstrating that the mixture can produce four negative decay rates over a broad range of mixing coefficients

    previously investigated in example 5 of Ref. [19], demonstrating that the mixture can produce four negative decay rates over a broad range of mixing coefficients. VI. CONCLUSIONS AND DISCUSSIONS In this work, we developed a comprehensive framework for analyzing Weyl dynamical maps and the emergence of non-Markovianity in finite-dimensional open quantum sy...

    work page

  10. [10]

    LetG be an arbitrary subgroup of Zd×Zd

    Lattice correspondence:Let π: Z2 →Zd× Zd be a natural homomorphism defined byπ(x,y ) = (xmodd,ymodd ). LetG be an arbitrary subgroup of Zd×Zd. We define the pre-image ofGinZ2 as: L=π−1(G) ={(x,y)∈Z2|(xmodd,ymodd)∈G}. SinceGis a subgroup,Lis a sublattice ofZ2. Furthermore, since the zero element(0, 0)∈G, the kernel ofπ, which is the latticedZ×dZ, must be c...

    work page

  11. [11]

    Let{b1, b2}be a basis forL

    Hermite normal form (HNF):SinceZ is a prin- cipal ideal domain, any sublatticeLof Z2 possesses a basis of size 2. Let{b1, b2}be a basis forL. We can form a generator matrix with these basis vectors as rows. By applying elementary integer row operations (unimodular transformations), any integer matrix can be reduced to itsHermite normal form[ 30]. For a2×2...

    work page

  12. [12]

    (1) Constraint onn:Thevector(0 ,d )mustbeamultiple of(0,n): (0,d) =k·(0,n) =⇒d=kn

    Deriving constraints from modulod:The con- dition( dZ×dZ)⊆Limplies that the vectors(d, 0)and (0,d )must be expressed as integer linear combinations of the basis vectors(m,w)and(0,n). (1) Constraint onn:Thevector(0 ,d )mustbeamultiple of(0,n): (0,d) =k·(0,n) =⇒d=kn. Therefore,n|d. 11 (2) Constraint onm:The vector( d, 0)must be a linear combination of the b...

    work page

  13. [13]

    Haroche and J.-M

    S. Haroche and J.-M. Raimond,Exploring the Quantum: Atoms, Cavities, and Photons(Oxford University Press, Oxford, 2006), URLhttps://doi.org/10.1093/acprof: oso/9780198509141.001.0001

    work page doi:10.1093/acprof: 2006

  14. [14]

    Jagadish and F

    V. Jagadish and F. Petruccione, Quanta7, 54 (2018), URL http://quanta.ws/ojs/index.php/quanta/ article/view/77

    work page 2018

  15. [15]

    Communications in Mathematical Physics 48(2), 119–130 (1976)

    G. Lindblad, Commun. Math. Phys.48, 119 (1976), URL https://doi.org/10.1007/BF01608499

    work page doi:10.1007/bf01608499 1976

  16. [16]

    Gorini, A

    V. Gorini, A. Kossakowski, and E. C. G. Sudarshan, J. Math. Phys.17, 821 (1976), URLhttp://dx.doi.org/ 10.1063/1.522979

    work page doi:10.1063/1.522979 1976

  17. [17]

    Rivas, S

    Á. Rivas, S. F. Huelga, and M. B. Plenio, Rep. Prog. Phys. 77, 094001 (2014), URLhttps://dx.doi.org/10.1088/ 0034-4885/77/9/094001

    work page 2014

  18. [18]

    Breuer, E.-M

    H.-P. Breuer, E.-M. Laine, J. Piilo, and B. Vacchini, Rev. Mod. Phys.88, 021002 (2016), URLhttps://link.aps. org/doi/10.1103/RevModPhys.88.021002

    work page doi:10.1103/revmodphys.88.021002 2016

  19. [19]

    de Vega and D

    I. de Vega and D. Alonso, Rev. Mod. Phys.89, 015001 (2017), URL https://link.aps.org/doi/10. 1103/RevModPhys.89.015001

    work page 2017

  20. [20]

    L. Li, M. J. Hall, and H. M. Wiseman, Phys. Rep. 759, 1 (2018), URLhttps://www.sciencedirect.com/ science/article/pii/S0370157318301601

    work page 2018

  21. [21]

    Chruściński, Phys

    D. Chruściński, Phys. Rep.992, 1 (2022), URLhttps: //doi.org/10.1016/j.physrep.2022.09.003

    work page doi:10.1016/j.physrep.2022.09.003 2022

  22. [22]

    Rivas, S

    Á. Rivas, S. F. Huelga, and M. B. Plenio, Phys. Rev. Lett. 105, 050403 (2010), URLhttps://link.aps.org/doi/ 10.1103/PhysRevLett.105.050403

    work page doi:10.1103/physrevlett.105.050403 2010

  23. [23]

    M. J. W. Hall, J. D. Cresser, L. Li, and E. Andersson, Phys. Rev. A89, 042120 (2014), URLhttps://link.aps. org/doi/10.1103/PhysRevA.89.042120

    work page doi:10.1103/physreva.89.042120 2014

  24. [24]

    Breuer, E.-M

    H.-P. Breuer, E.-M. Laine, and J. Piilo, Phys. Rev. Lett. 103, 210401 (2009), URLhttps://link.aps.org/doi/ 10.1103/PhysRevLett.103.210401

    work page doi:10.1103/physrevlett.103.210401 2009

  25. [25]

    M. M. Wolf, J. Eisert, T. S. Cubitt, and J. I. Cirac, Phys. Rev. Lett.101, 150402 (2008), URLhttps://doi.org/ 10.1103/PhysRevLett.101.150402

    work page doi:10.1103/physrevlett.101.150402 2008

  26. [26]

    Megier, D

    N. Megier, D. Chruściński, J. Piilo, and W. T. Strunz, Sci. Rep.7, 6379 (2017), URL https://doi.org/10.1038/ s41598-017-06059-5

    work page 2017

  27. [27]

    Jagadish, R

    V. Jagadish, R. Srikanth, and F. Petruccione, Phys. Rev. A101, 062304 (2020), URLhttps://link.aps.org/doi/ 10.1103/PhysRevA.101.062304

    work page doi:10.1103/physreva.101.062304 2020

  28. [28]

    Jagadish, R

    V. Jagadish, R. Srikanth, and F. Petruc- cione, Phys. Lett. A384, 126907 (2020), URL https://www.sciencedirect.com/science/article/ pii/S037596012030774X

    work page 2020

  29. [29]

    Chruściński and F

    D. Chruściński and F. A. Wudarski, Phys. Rev. A 91, 012104 (2015), URL https://doi.org/10.1103/ PhysRevA.91.012104

    work page 2015

  30. [30]

    F. A. Wudarski and D. Chruściński, Phys. Rev. A 93, 042120 (2016), URL https://doi.org/10.1103/ PhysRevA.93.042120

    work page 2016

  31. [31]

    Siudzińska and D

    K. Siudzińska and D. Chruściński, J. Phys. A: Math. Theor.53, 375305 (2020), URL https://doi.org/10. 1088/1751-8121/aba7f2

    work page 2020

  32. [32]

    Siudzińska, Phys

    K. Siudzińska, Phys. Rev. A103, 022605 (2021), URL https://link.aps.org/doi/10.1103/PhysRevA.103. 022605

    work page doi:10.1103/physreva.103 2021

  33. [33]

    Jagadish, R

    V. Jagadish, R. Srikanth, and F. Petruccione, Phys. Rev. A105, 032422 (2022), URLhttps://link.aps.org/doi/ 10.1103/PhysRevA.105.032422

    work page doi:10.1103/physreva.105.032422 2022

  34. [34]

    Jagadish, R

    V. Jagadish, R. Srikanth, and F. Petruccione, Phys. Rev. A106, 012438 (2022), URLhttps://link.aps.org/doi/ 10.1103/PhysRevA.106.012438

    work page doi:10.1103/physreva.106.012438 2022

  35. [35]

    Jagadish, R

    V. Jagadish, R. Srikanth, and F. Petruccione, Phys. Rev. A108, 042202 (2023), URLhttps://link.aps.org/doi/ 10.1103/PhysRevA.108.042202

    work page doi:10.1103/physreva.108.042202 2023

  36. [36]

    Siudzińska, J

    K. Siudzińska, J. Phys. A Math. Theor. (2022), URL https://iopscience.iop.org/article/10.1088/ 1751-8121/ac65c0/meta

    work page 2022

  37. [37]

    Xu, M.-S

    W. Xu, M.-S. Li, and Z.-J. Zheng, Phys. Rev. A 109, 012410 (2024), URL https://doi.org/10.1103/ PhysRevA.109.012410

    work page 2024

  38. [38]

    Gulati, V

    V. Gulati, V. Jagadish, R. Srikanth, and K. Dorai, Phys. Rev. A109, 042419 (2024), URL https://link.aps. org/doi/10.1103/PhysRevA.109.042419

    work page doi:10.1103/physreva.109.042419 2024

  39. [39]

    H. Weyl, Z. Physik46, 1 (1927), URLhttps://doi.org/ 10.1007/BF02055756

    work page doi:10.1007/bf02055756 1927

  40. [40]

    Weyl,The Theory of Groups and Quantum Mechanics, Dover Books on Mathematics (Dover Publications, 1950)

    H. Weyl,The Theory of Groups and Quantum Mechanics, Dover Books on Mathematics (Dover Publications, 1950)

    work page 1950

  41. [41]

    Schwinger, Proc

    J. Schwinger, Proc. Natl. Acad. Sci. U. S. A.46, 570 (1960), URL https://doi.org/10.1073/pnas.46.4.570

    work page doi:10.1073/pnas.46.4.570 1960

  42. [42]

    Newman,Integral Matrices(Academic Press, New York, 1972)

    M. Newman,Integral Matrices(Academic Press, New York, 1972)

    work page 1972

  43. [43]

    Jagadish and R

    V. Jagadish and R. Srikanth, Phys. Rev. A111, 042212 (2025), URL https://doi.org/10.1103/PhysRevA.111. 042212

    work page doi:10.1103/physreva.111 2025

  44. [44]

    Chruściński and K

    D. Chruściński and K. Siudzińska, Phys. Rev. A 94, 022118 (2016), URL https://doi.org/10.1103/ PhysRevA.94.022118

    work page 2016

  45. [45]

    W. K. Wootters and B. D. Fields, Ann. Phys. (NY)191, 363 (1989), URL https://doi.org/10.1016/ 0003-4916(89)90322-9

    work page 1989

  46. [46]

    Durt, B.-G

    T. Durt, B.-G. Englert, I. Bengtsson, and K. Życzkowski, Int. J. Quantum. Inf.8, 535 (2010), URLhttps://doi. org/10.1142/S0219749910006502

    work page doi:10.1142/s0219749910006502 2010