Pith. sign in

REVIEW 1 major objections 3 minor 51 references

A thermofield-double model of Uhlmann anholonomy

T0 review · 1 major / 3 minor · reviewed 2026-08-06 · deepseek-v4-flash

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

Pith's one-line read Uhlmann parallel transport of an entangled state is optimal measurement on one side and holonomic computation on the other.

desk verdict Solid generalization of Uhlmann anholonomy to N qubits with a fixable parameter error in the iSWAP example; the core results hold up. read the letter →

arxiv 2507.12071 v1 pith:2ZT3BNYW submitted 2025-07-16 quant-ph hep-th

Péter Lévay , Csaba Velich This is my paper · ORCID
classification quant-phhep-th
keywords Uhlmannanholonomythermofielddoublemixed-stategeometricphaseholonomicquantumcomputationBuresmetricThomasrotationoptimalstatediscriminationinstantongaugefields
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

The paper studies a 2N-qubit entangled state in the thermofield double form and assumes its evolution is governed purely by Uhlmann parallel transport, the mixed-state generalization of Berry phase. It claims that this single geometric evolution has two faces: an observer coupled to the left subsystem sees a sequence of non-unitary filtering measurements, which turn out to be the optimal measurements for distinguishing the mixed states involved, while an observer on the right sees the same steps as a sequence of unitary gates of a holonomic quantum computation. The paper works out the Uhlmann connection for an N-qubit Bloch-ball-like family of thermal states, identifies it with a restriction of higher-dimensional instanton gauge fields, and shows the anholonomy around geodesic triangles is a Thomas rotation in Spin(2N+1). It then derives concrete interferometric signatures, phase shifts and visibility, in which these geometric effects should be observable, and exhibits a loop of geodesic segments that produces the iSWAP gate, one of the ingredients for universal quantum computation.

What carries the argument

The paper's central object is the purification $W(\tau,\mathbf r,n) = (1/\sqrt{2^N(1+\tau^2+r^2)})[(1+i\tau)I - r\,\not n]$ of an $N$-qubit thermal density matrix $\varrho = (I-\tanh\beta\,\not n)/2^N$, transported via the Uhlmann parallelism condition $W_1^* W_2 > 0$. The mechanism that carries the argument is the geometric-mean building block $L_{ij} = \varrho_i \sharp \varrho_j^{-1}$, which acts as a positive filtering operator on the left while its unitary counterpart $U_{ij}$ acts on the right; for the model these unitaries are computed explicitly as Thomas rotations $U_{uv}= (I + \not a\not b)/\sqrt{1+2ab+a^2b^2}$ in $\mathrm{Spin}(2N+1)$, and the Uhlmann connection becomes $A = \tfrac14(1-\mathrm{sech}\,\beta)[\hat u, d\hat u]$, which coincides on the $\tau=0$ slice with the higher-dimensional instanton/monopole gauge fields. The Bures metric on the state space is the one that makes the geodesic triangles well-defined; it is conformal to the hyperbolic metric.

What would settle it

Conduct the proposed Mach-Zehnder interferometry on the right subsystem with a geodesic-triangle loop, and compare the measured phase shift and visibility with equations (12.6) and (12.15): the claim predicts visibility multiplied by $\cos(\delta/2)$ from the Thomas-rotation angle and no anholonomy contribution to the phase shift for $N\ge 2$; a measured visibility or phase that disagrees with this formula would falsify the model's central duality.

Watch

Extended reading notes

Core claim

The central claim is that Uhlmann parallel transport of a thermofield-double state is a duality: with the right local unitaries gauged away, the left subsystem's reduced evolution is literally a chain of Hermitian positive filtering operators of the form $\varrho_i \sharp \varrho_j^{-1}$ (geometric means), and these are exactly the operators that maximize the statistical distance for distinguishing the pair of mixed states; the right subsystem's reduced evolution is the same chain read in reverse as a product of unitary Thomas rotations $U_{ij}\in\mathrm{Spin}(2N+1)$. For the paper's model the Uhlmann connection pulls back to the $\tau=0$ slice of the higher-dimensional monopole (instanton) gauge fields, and the anholonomy acquired around a Bures-geodesic triangle is a single Thomas rotation whose angle is expressed through the fidelities of the three vertices. These claims are made concrete by computing the phase shift and visibility that an interferometer attached to the right subsystem would measure, and by constructing a four-triangle sequence that produces the iSWAP gate.

Load-bearing premise

The load-bearing premise is that some system, environment interaction actually implements Uhlmann parallel transport for the entangled 2N-qubit state; the paper explicitly says it does not attempt to reveal the physical mechanism (Section 1), and if no physical dynamics enforces the Uhlmann condition, the predicted duality and interferometric signatures would not describe any real evolution.

Editorial extensions

If this is right

  • The same cyclic geometric evolution of an entangled state can be viewed as an optimal discrimination protocol on one side and a holonomic quantum computation on the other.
  • The Uhlmann connection for the model is not arbitrary: it is the restriction of higher-dimensional instanton gauge fields, so the geometry of mixed-state transport and topological gauge fields are tied together.
  • The anholonomy for Bures-geodesic triangles in this model always lies in $\mathrm{Spin}(2N+1)$ and is a Thomas rotation, meaning special-relativistic boost composition underlies mixed-state holonomies.
  • Interference experiments on one subsystem can reveal the computation happening on the other: the phase shift and visibility formulas (12.6) and (12.15) are directly measurable predictions.
  • A loop made of geodesic segments can realize the iSWAP gate, so Uhlmann anholonomy is in principle a resource for universal quantum computation.

Reading between the lines

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

  • The paper leaves open what physical dynamics enforces the Uhlmann condition; a natural extension would be to engineer it via continuous measurement or reservoir engineering, and then test whether the dual description survives when the transport is not perfectly parallel.
  • The left-measurement/right-computation duality suggests a general compilation principle: any cyclic mixed-state holonomy can be compiled into a quantum circuit on a purification partner, with the filtering steps as syndrome-like measurements; testing this on small systems, N=2 or 3, would separate the geometric claim from the model's special symmetries.
  • The connection between the Uhlmann connection and instantons hints that the topological charge of the parent gauge field could classify the computational power of the resulting holonomic gates, a classification not worked out in the paper.
  • For N=3, the Spin(6) ≅ SU(4) embedding might allow the construction of two-qubit gates inside three-qubit systems using only the special states of this model, which could be a more direct route to experimental tests than the iSWAP construction.
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.

Referee Report

1 major / 3 minor

Summary. The paper introduces a 2N-qubit thermofield-double family W(τ,r,n) whose left and right marginals are N-qubit density matrices of the form (I+u·Γ)/2^N. It assumes that the system evolves by Uhlmann parallel transport and derives the Uhlmann connection for this family (Eq. (6.7)), identifies its τ=0 restriction with higher-dimensional instanton/monopole gauge fields (Eq. (7.13)), and computes the holonomy around Bures geodesic triangles as an element of Spin(2N+1) (Eq. (9.16)). It then shows that, with the right unitaries gauged, the left-subsystem description is a sequence of optimal filtering measurements while the right-subsystem description is a reversed sequence of unitary gates; it proposes an interferometric setup for detecting the resulting phase and visibility (Eq. (12.6)), and it ends with a construction of the iSWAP gate from four geodesic triangles (Sec. 12.3).

Significance. The paper's central formalism is a useful explicit bridge between Uhlmann anholonomy, optimal mixed-state discrimination, and holonomic quantum computation. The derivations are largely self-contained and analytic: the connection, Bures metric, fidelity formulas, and interferometric expressions are parameter-free, and the N=1 limit correctly reduces to the known Thomas-rotation result of Ref. [23]. The left/right duality and the explicit phase-shift/visibility formulas give concrete, checkable predictions for a Mach-Zehnder experiment, provided the Uhlmann parallel-transport condition is physically implemented. The unresolved iSWAP parameter inconsistency in Sec. 12.3 is local, but it currently blocks the paper's universality claim.

major comments (1)
  1. [§12.3, Eq. (12.18)] The advertised parameters do not produce the iSWAP gate. From Eq. (12.18) with a=0, /p=cΓ_j and /q=bΓ_i, one obtains R(0,b,c) = (I + (1/2)bc[Γ_i,Γ_j])/√(1+b²c²), so the rotation angle satisfies tan(δ/2)=bc. For δ=π/4 (which Eq. (12.21) requires, since φ=δ/2=π/8), one needs bc=tan(π/8)=√2−1≈0.4142. However, the text sets b=c=tanh(β/2)=√2−1 with β=log(√2+1), giving bc=(√2−1)²≈0.1716. Then sin(δ/2)=bc/√(1+b²c²)≈0.1686, so δ≈0.339 rad, not π/4. Consequently the four-triangle sequence in Eq. (12.21) does not implement the iSWAP gate, and the associated computational-universality claim is not supported as written. The fix is local — for example, choosing b=c=√(√2−1), or an appropriate β — but the explicit construction and the surrounding text must be corrected.
minor comments (3)
  1. [Eq. (9.8)] The typesetting of Eq. (9.8) as "p p2 = ..." is very hard to parse; please rewrite it as p/|p|² = ... and define all quantities explicitly.
  2. [Sec. 6] The same symbol A is used for the connection form on the bundle and for its pullback to the base; the footnote explains this, but a distinct notation in the main text would improve clarity.
  3. [Sec. 3] The term "quasi-classicality" could mislead readers, since the bound-entangled states can still violate Bell-type inequalities, as the authors themselves note; a brief terminological clarification would be helpful.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular reduction: connection, holonomy, filtering and interference results are closed-form derivations; self-citations ([23], [37]) serve as anchors with in-paper re-derivations, not load-bearing steps; the §12.3 iSWAP parameter values are internally inconsistent (a correctness defect, not circularity).

full rationale

The derivation chain is self-contained and does not reduce outputs to inputs. The purification family W(τ,r,n) (Eq. 4.7) is an explicit ansatz; from it the thermal density matrix (Eq. 4.10), the Uhlmann connection A=¼(1−sechβ)[n̂,dn̂] (Eq. 6.7) and the parallel-translation operator G (Eq. 6.8) are computed in Appendix 15.3 from the standard pull-back formula (6.5), cited to the external group Wang et al. [36]. The instanton identification A|_{τ=0}=A±|_{τ=0} (Eq. 7.13) is re-derived in Appendix 15.3 (Eqs. 15.27–15.32), not merely quoted. The Thomas-rotation form of the anholonomy Uuv (Eqs. 8.8–8.10) is proved in Appendix 15.5 (Eqs. 15.43–15.54); the same-author N=1 paper [23] is used only as a special-case anchor and later as a cross-check (Eq. 12.16 reduces to Eq. (34) of [23] for χ=0). The geodesic-triangle holonomy R(a,b,c) (Eq. 9.7) is stated as a generalization of Eq. (29) of [23], but its ingredients — the fidelity identity (9.15) and the Uuv representation (8.21) — are derived in this paper, so the self-citation is an anchor, not load-bearing. The filtering/optimal-measurement claim is supported by the explicit rank-n projectors (Eqs. 10.13–10.15) and the eigenvalue computation (11.2), with the optimality theorem attributed to external work [30,31]. The interferometric phase shifts and visibilities (Eqs. 12.8, 12.15, 12.16) are closed-form traces containing no fitted constants; the paper fits no parameter to data and then renames a fit a prediction. The left/right 'filtering vs. holonomic computation' duality is an interpretation of the mathematically derived identity (5.13), not a result put in by assumption. The paper also explicitly declares its main premise — 'we are not attempting to reveal the physical mechanism which implements such a condition' — so the Uhlmann evolution is a stated assumption, not a hidden input. One flagged defect (per the reviewing rule): in §12.3 the text says 'with the choice δ = π/4 we get β = log(√2 + 1). This means that |b| = |c| = √2 − 1', but Eq. (12.18) implies sin(δ/2) = bc/√(1+b²c²), i.e. bc = tan(δ/2) = √2−1; with b=c=√2−1 one has bc=(√2−1)²≈0.1716, giving δ≈0.34 rad instead of π/4. The advertised symmetric parameters therefore do not yield the iSWAP gate as written (the correct symmetric choice is b=c=√(√2−1)≈0.6436). This is an internal-consistency defect in the universality demonstration — a missing-support/correctness concern — not a circular reduction, and it does not raise the circularity score.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The central derivations rely on standard Uhlmann formalism, gamma matrix algebra, and known optimal-discrimination results. The only ad hoc element is the assumption that Uhlmann parallel transport is physically realized. The model's external parameters (β, χ, n) are inputs, and the iSWAP example chooses specific triangle parameters by hand. No new physical entities are introduced.

free parameters (1)
  • b, c (geodesic triangle vertices in Section 12.3) = b=c=√2−1 (claimed to yield δ=π/4)
    Chosen by hand to realize the iSWAP gate. The stated values are inconsistent with Eq. (12.18), which gives sin(δ/2)=bc/√(1+b²c²)≈0.169 instead of sin(π/8)≈0.383.
assumptions (5)
  • ad hoc to paper Uhlmann parallel transport is physically implemented by the system-environment interaction
    Assumed throughout; the authors state in Section 1 they do not attempt to reveal the physical mechanism. If no dynamics enforces the Uhlmann condition, the model's physical predictions do not apply.
  • domain assumption Choice of gauging the right subsystem's local unitaries
    The Uhlmann condition admits two gauge choices; the paper fixes right-side unitaries as gauge degrees of freedom, breaking left-right symmetry. The other choice would exchange the roles of the two subsystems.
  • domain assumption Restriction to full-rank thermal states in the generalized Bloch ball B
    Density matrices are restricted to ρ=(I+u·Γ)/2^N with |u|<1, limiting the holonomies to Spin(2N+1) ⊂ SU(2N). This is acknowledged as a special class.
  • standard math Geometric mean properties of positive matrices and the Uhlmann connection formula of Ref. [36]
    Background results used in Sections 5 and 6 without proof.
  • standard math Fuchs-Caves and Ericsson theorems on optimal quantum state discrimination
    Used to identify the geodesic operator as the optimal measurement for distinguishing two mixed states (Refs. [30,31]).

how reviews work

0 comments
Cite this review

Pith. "Pith review of A thermofield-double model of Uhlmann anholonomy." pith.science (2026). https://pith.science/paper/2ZT3BNYW

@misc{pith2026250712071,
  author       = {Pith},
  title        = {Pith review of: A thermofield-double model of Uhlmann anholonomy},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2ZT3BNYW}},
  note         = {Machine review of arXiv:2507.12071}
}
read the original abstract

A simple parametrized family of quantum systems consisting of two entangled subsystems, dubbed left and right ones, both of them featuring N qubits is considered in the thermofield double formalism. We assume that the system evolves in a purely geometric manner based on the parallel transport condition due to Uhlmann. We explore the different interpretations of this evolution relative to observers either coupled to the left or to the right subsystems. The Uhlmann condition breaks the symmetry between left and right by regarding one of the two possible sets of local unitary operations as gauge degrees of freedom. Then gauging the right side we show that the geometric evolution on the left manifests itself via certain local operations reminiscent of non-unitary filtering measurements. On the other hand on the right the basic evolutionary steps are organized into a sequence of unitary operations of a holonomic quantum computation. We calculate the Uhlmann connection governing the transport for our model which turns out to be related to higher dimensional instantons. Then we evaluate the anholonomy of the connection for geodesic triangles with geodesic segments defined with respect to the Bures metric. By analysing the explicit form of the local filtering measurements showing up on the left side we realize that they are also optimal measurements for distinguishing two given mixed states in the statistical sense. We also point out that by conducting an interference experiment on the right side one can observe the physical effects of the anholonomic quantum computation. We demonstrate this by calculating explicit examples for phase shifts and visibility patterns arising in such interference experiments. Finally a sequence of geodesic triangles producing the iSWAP gate via anholonomy needed for computational universality is presented.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

51 extracted references · 46 canonical work pages

  1. [23]

    L´ evay,Thomas rotation and the mixed state geometric phase , Journal of Physics A: Mathematical and General 37 (2004) 4593

    P. L´ evay,Thomas rotation and the mixed state geometric phase , Journal of Physics A: Mathematical and General 37 (2004) 4593

    work page 2004

  2. [1]

    Berry, Quantal phase factors accompanying adiabatic changes , Proceedings of the Royal Society of London

    M.V. Berry, Quantal phase factors accompanying adiabatic changes , Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences 392 (1984) 45. – 45 –

    work page 1984

  3. [2]

    Haag, N.M

    R. Haag, N.M. Hugenholtz and M. Winnink, On the equilibrium states in quantum statistical mechanics, Communications in Mathematical Physics 5 (1967) 215

    work page 1967

  4. [3]

    E. Witten, Why does quantum field theory in curved spacetime make sense? and what happens to the algebra of observables in the thermodynamic limit? , arXiv preprint (2022) [2112.11614]

    arXiv 2022

  5. [4]

    Geodesics for mixed quantum states via their geometric mean operator

    P.M. Alsing, C. Cafaro and S. Ray, Geodesics for mixed quantum states via their geometric mean operator, arXiv preprint (2024) [ 2404.04136]

    work page Pith review arXiv 2024

  6. [5]

    Uhlmann, Parallel transport and quantum holonomy along density operators , Reports on Mathematical Physics 24 (1986) 229

    A. Uhlmann, Parallel transport and quantum holonomy along density operators , Reports on Mathematical Physics 24 (1986) 229

    work page 1986

  7. [6]

    Uhlmann, The metric of bures and the geometric phase , in Quantum Groups and Related Topics, R

    A. Uhlmann, The metric of bures and the geometric phase , in Quantum Groups and Related Topics, R. Gielerak, J. Lukierski, Z. Popowicz et al., eds., vol. 13 of Mathematical Physics Studies, (Dordrecht), pp. 267–274, Kluwer Academic Publishers (1992), DOI

    work page 1992

  8. [7]

    Uhlmann, Gauge field governing parallel transport along mixed states , Lett

    A. Uhlmann, Gauge field governing parallel transport along mixed states , Lett. Math. Phys. 21 (1991) 229

    work page 1991

Show all 51 references
  1. [8]

    D. Bures, An extension of Kakutani’s theorem on infinite product measures to the tensor product of semifinite w∗-algebras, Transactions of the American Mathematical Society 135 (1969) 199

    1969

  2. [9]

    Perlmutter, A universal feature of cft r´ enyi entropy, Journal of High Energy Physics 2014 (2014) 117 [ 1308.1083]

    E. Perlmutter, A universal feature of cft r´ enyi entropy, Journal of High Energy Physics 2014 (2014) 117 [ 1308.1083]

    2014 arXiv

  3. [10]

    Casini, M

    H. Casini, M. Huerta and R.C. Myers, Towards a derivation of holographic entanglement entropy, Journal of High Energy Physics 2011 (2011) 036 [ 1102.0440]

    2011 arXiv

  4. [11]

    Belavin, A.M

    A.A. Belavin, A.M. Polyakov, A.S. Schwartz and Y.S. Tyupkin, Pseudoparticle solutions of the yang-mills equations , Physics Letters B 59 (1975) 85

    1975

  5. [12]

    monopoles

    Z. Horv´ ath and L. Palla,Spontaneous compactification and “monopoles” in higher dimension, Nuclear Physics B 142 (1978) 327

    1978

  6. [13]

    Horodecki, P

    M. Horodecki, P. Horodecki and R. Horodecki, Mixed-state entanglement and distillation: Is there a “bound” entanglement in nature? , Physical Review Letters 80 (1998) 5239–5242

    1998

  7. [14]

    V´ ertesi and N

    T. V´ ertesi and N. Brunner,Quantum nonlocality does not imply entanglement distillability , Physical Review Letters 108 (2012)

    2012

  8. [15]

    Maldacena and L

    J. Maldacena and L. Susskind, Cool horizons for entangled black holes , Fortschritte der Physik 61 (2013) 781

    2013

  9. [16]

    Witten, Aps medal for exceptional achievement in research: Invited article on entanglement properties of quantum field theory , Rev

    E. Witten, Aps medal for exceptional achievement in research: Invited article on entanglement properties of quantum field theory , Rev. Mod. Phys. 90 (2018) 045003

    2018

  10. [17]

    Maldacena, The large n limit of superconformal field theories and supergravity , International Journal of Theoretical Physics 38 (1999) 1113

    J.M. Maldacena, The large n limit of superconformal field theories and supergravity , International Journal of Theoretical Physics 38 (1999) 1113

    1999

  11. [18]

    Verlinde, Wormholes in Quantum Mechanics , 2105.02129

    H. Verlinde, Wormholes in Quantum Mechanics , 2105.02129

    arXiv

  12. [19]

    Nogueira, S

    F.S. Nogueira, S. Banerjee, M. Dorband, R. Meyer, J. van den Brink and J. Erdmenger, Geometric phases distinguish entangled states in wormhole quantum mechanics , Phys. Rev. D 105 (2022) L081903

    2022

  13. [20]

    Banerjee, M

    S. Banerjee, M. Dorband, J. Erdmenger, R. Meyer and A. Weigel, Berry phases, wormholes and factorization in ads/cft , Journal of High Energy Physics 2022 (2022) 162 [ 2202.11717]. – 46 –

    2022 arXiv

  14. [21]

    Zanardi and M

    P. Zanardi and M. Rasetti, Holonomic quantum computation , Physics Letters A 264 (1999) 94 [quant-ph/9904011]

    1999 arXiv

  15. [22]

    Wilczek and A

    F. Wilczek and A. Zee, Appearance of gauge structure in simple dynamical systems , Phys. Rev. Lett. 52 (1984) 2111

    1984

  16. [24]

    Yang, Generalization of dirac’s monopole to su(2) gauge fields , Journal of Mathematical Physics 19 (1978) 320

    C.N. Yang, Generalization of dirac’s monopole to su(2) gauge fields , Journal of Mathematical Physics 19 (1978) 320

    1978

  17. [25]

    Dittmann and G

    J. Dittmann and G. Rudolph, On a connection governing parallel transport along 2 ×2 density matrices, Journal of Geometry and Physics 10 (1992) 93

    1992

  18. [26]

    Sj¨ oqvist, A.K

    E. Sj¨ oqvist, A.K. Pati, A. Ekert, J.S. Anandan, M. Ericsson, D.K.L. Oi et al., Geometric phases for mixed states in interferometry , Physical Review Letters 85 (2000) 2845

    2000

  19. [27]

    Everett, ”relative state” formulation of quantum mechanics , Rev

    H. Everett, ”relative state” formulation of quantum mechanics , Rev. Mod. Phys. 29 (1957) 454

    1957

  20. [28]

    Kirklin, Islands and uhlmann phase: explicit recovery of classical information from evaporating black holes , Journal of High Energy Physics 2022 (2022) 119

    J. Kirklin, Islands and uhlmann phase: explicit recovery of classical information from evaporating black holes , Journal of High Energy Physics 2022 (2022) 119

    2022

  21. [29]

    Kirklin, The holographic dual of the entanglement wedge symplectic form , Journal of High Energy Physics 2020 (2020) 71

    J. Kirklin, The holographic dual of the entanglement wedge symplectic form , Journal of High Energy Physics 2020 (2020) 71

    2020

  22. [30]

    ˚Asa Ericsson, Geodesics and the best measurement for distinguishing quantum states , Journal of Physics A: Mathematical and General 38 (2005) L725 [ quant-ph/0508133]

    2005 arXiv

  23. [31]

    Fuchs and C.M

    C.A. Fuchs and C.M. Caves, Mathematical techniques for quantum communication theory , Open Systems and Information Dynamics 3 (1995) 345

    1995

  24. [32]

    Bhatia, Positive Definite Matrices , Princeton University Press (2006)

    R. Bhatia, Positive Definite Matrices , Princeton University Press (2006)

    2006

  25. [33]

    Bhatia, T

    R. Bhatia, T. Jain and Y. Lim, On the bures–wasserstein distance between positive definite matrices, Expositiones Mathematicae 37 (2019) 165

    2019

  26. [34]

    Peres, Separability criterion for density matrices , Physical Review Letters 77 (1996) 1413

    A. Peres, Separability criterion for density matrices , Physical Review Letters 77 (1996) 1413

    1996

  27. [35]

    Pusz and S.L

    W. Pusz and S.L. Woronowicz, Functional calculus for sesquilinear forms and the purification map, Reports on Mathematical Physics 8 (1975) 159

    1975

  28. [36]

    Wang, X.-Y

    X. Wang, X.-Y. Hou, Z. Zhou, H. Guo and C. Chien, Uhlmann phase of coherent states and the uhlmann-berry correspondence, SciPost Physics Core 6 (2023) 024

    2023

  29. [37]

    L´ evay,Non-abelian born–oppenheimer electric gauge force and the natural metric on hilbert subspaces, Physical Review A 45 (1992) 1339

    P. L´ evay,Non-abelian born–oppenheimer electric gauge force and the natural metric on hilbert subspaces, Physical Review A 45 (1992) 1339

    1992

  30. [38]

    Dirac, Quantised singularities in the electromagnetic field , Proceedings of the Royal Society of London

    P.A.M. Dirac, Quantised singularities in the electromagnetic field , Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character 133 (1931) 60

    1931

  31. [39]

    Mukunda, P.K

    N. Mukunda, P.K. Aravind and R. Simon, Wigner rotations, bargmann invariants and geometric phases, Journal of Physics A: Mathematical and General 36 (2003) 2347

    2003

  32. [40]

    A.K. Bhattacharyya, On a measure of divergence between two statistical populations defined by their probability distributions , Bulletin of the Calcutta Mathematical Society 35 (1943) 99

    1943

  33. [41]

    Wootters, Statistical distance and hilbert space , Phys

    W.K. Wootters, Statistical distance and hilbert space , Phys. Rev. D 23 (1981) 357. – 47 –

    1981

  34. [42]

    Echternach, C.P

    P. Echternach, C.P. Williams, S.C. Dultz, S.L. Braunstein and J.P. Dowling, Universal quantum gates for single cooper pair box based quantum computing , Quantum Information and Computation 1 (2001) 143

    2001

  35. [43]

    DiVincenzo, Two-bit gates are universal for quantum computation , Phys

    D.P. DiVincenzo, Two-bit gates are universal for quantum computation , Phys. Rev. A 51 (1995) 1015

    1995

  36. [44]

    Anandan, Non-adiabatic non-abelian geometric phase , Physics Letters A 133 (1988) 171

    J. Anandan, Non-adiabatic non-abelian geometric phase , Physics Letters A 133 (1988) 171

    1988

  37. [45]

    P. L´ evay,Quaternionic gauge fields and the geometric phase , Journal of Mathematical Physics 32 (1991) 2347 [https://pubs.aip.org/aip/jmp/article-pdf/32/9/2347/19271963/2347 1 online.pdf]

    1991

  38. [46]

    Grundberg, T.H

    J. Grundberg, T.H. Hansson and A. Karlhede, Geometrical phases from spinning particles , Phys. Rev. D 41 (1990) 2642

    1990

  39. [47]

    Polyakov, Fermi-Bose Transmutations Induced by Gauge Fields , Mod

    A.M. Polyakov, Fermi-Bose Transmutations Induced by Gauge Fields , Mod. Phys. Lett. A 3 (1988) 325

    1988

  40. [48]

    Korchemskaya and G.P

    I.A. Korchemskaya and G.P. Korchemsky, Polyakov spin factor, berry phase and random walks of spinning particles , Journal of Physics A: Mathematical and General 24 (1991) 4511

    1991

  41. [49]

    Strominger, Loop space solution of two-dimensional qcd , Physics Letters B 101 (1981) 271

    A. Strominger, Loop space solution of two-dimensional qcd , Physics Letters B 101 (1981) 271

    1981

  42. [50]

    Nowak, M

    M.A. Nowak, M. Rho and I. Zahed, Spin factors and geometric phases in arbitrary dimensions, Physics Letters B 254 (1991) 94

    1991

  43. [51]

    J. Liu, H. Yuan, X. Lu and X. Wang, Quantum fisher information matrix and multiparameter estimation, Journal of Physics A: Mathematical and Theoretical 53 (2020) 023001. – 48 –

    2020

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.