Pith. sign in

REVIEW 4 major objections 4 minor 49 references

A Geometric Quantum Speed Limit: Theoretical Insights and Photonic Implementation

T0 review · 4 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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

Pith's one-line read A new quantum speed limit based on the Bloch angle is a true lower bound on evolution time and is measurable without tomography.

desk verdict A correct but largely tautological Bloch-angle speed limit, useful mainly for the instantaneous velocity formula and the swap-test measurement; the saturation claim is unproved beyond qubits. read the letter →

arxiv 2506.00354 v1 pith:3E2UDKKB submitted 2025-05-31 quant-ph

Qianyi Wang , Ben Wang , Jun Wang , Lijian Zhang This is my paper · ORCID
classification quant-ph
keywords quantumspeedlimitBlochanglegeodesicevolutionswaptestLandau-Zenermodelmixedstatesunitaldynamicsphotonicexperiment
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

Using the angle between Bloch vectors, the paper defines an instantaneous velocity $v(t)$ and a time $\tau_\Theta$ at which the accumulated path length first reaches the target Bloch angle $\Theta(\rho_0,\rho_T)$. It argues that $\tau_\Theta$ is always a lower bound on the actual evolution time $T$, with equality exactly when the evolution path is a geodesic, i.e., when the Bloch vector executes uniform rotation with its tip tracing a great circle. The practical payoff is that the bound needs measurements only up to $\tau_\Theta$, not over the whole interval, and those measurements are swap-test overlaps rather than full state tomography. The authors demonstrate the method with a photonic qubit experiment for a time-independent Hamiltonian and for the Landau-Zener model, where accelerated dynamics make the new bound tighter than the earlier Bloch-angle QSL.

What carries the argument

The central object is the instantaneous Bloch-angle velocity $v(t)$, which converts infinitesimal Bloch-angle increments into a path-length integral. The argument then runs through the triangle inequality for the Bloch angle and the geodesic equation $\ddot r=-\alpha^2 r$ for the Bloch vector; the latter encodes the condition under which the path length equals the distance and the bound saturates. The experimental enabler is the swap test: for two copies of the evolving state, the overlap $\operatorname{Tr}(\rho_{t_1}\rho_{t_2})$ equals $1-2p_A$, where $p_A$ is the probability of measuring the antisymmetric state, so the path length can be accumulated from such overlap measurements without tomography.

What would settle it

For a single qutrit with a constant Hamiltonian, take $r(t)=\cos(\alpha t)r(0)+\sin(\alpha t)r'$ with $|r'|=|r(0)|$ and $r'\cdot r(0)=0$, and compute the eigenvalues of the corresponding density matrix at intermediate times; finding any negative eigenvalue would show this geodesic leaves the physical state space and invalidate the saturation claim beyond qubits.

Watch

Extended reading notes

Core claim

For a unitary evolution $\rho_t$ with Bloch vector $r(t)$, the paper defines $v(t)=\lim_{dt\to 0}\Theta(\rho_t,\rho_{t+dt})/dt$ and then defines $\tau_\Theta$ as the solution of $\Theta(\rho_0,\rho_T)=\int_0^{\tau_\Theta} v(t)\,dt$. Because the triangle inequality gives $\Theta(\rho_0,\rho_T)\le \int_0^T v(t)\,dt$, the paper concludes $\tau_\Theta\le T$. It identifies the saturating condition with the geodesic equation $\ddot r=-\alpha^2 r$, whose qubit solutions are uniform rotations with the Bloch vector perpendicular to the rotation axis, and shows that under this condition $\tau_\Theta=T$. For time-independent Hamiltonians the velocity is constant, so both the new and the prior Bloch-angle QSL reduce to $\Theta/v(0)$; for the Landau-Zener model the velocity typically grows, making the new bound tighter. The paper further claims that the overlaps $\operatorname{Tr}(\rho_t\rho_{t+dt})$ can be obtained photonically by a swap test from the probability of the antisymmetric outcome, bypassing full quantum state tomography.

Load-bearing premise

The saturation result holds if the assumed geodesic equation for the Bloch vector stays within the set of physical states in all dimensions; the paper demonstrates that only for qubits.

Editorial extensions

If this is right

  • The new bound can be computed without knowing the actual evolution time $T$, while the previously proposed Bloch-angle bound requires the average velocity over $[0,T]$.
  • For accelerated dynamics, such as the Landau-Zener model, the new bound is typically tighter than the existing Bloch-angle QSL, as confirmed by numerical sampling over initial states and purity levels.
  • The experimental protocol uses swap-test overlaps only, so it avoids full quantum state tomography and works for any dynamics for which the overlap can be measured.
  • When the evolution path is a geodesic, the bound is tight: $\tau_\Theta=T$, giving a geometric criterion for when a given unitary dynamics is as fast as possible.
  • Because the Bloch angle is robust under unital evolutions, the bound remains informative for mixed states where Bures-angle-based QSLs become loose.

Reading between the lines

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

  • The bound is path-dependent, so it is best interpreted as a certificate for a known trajectory rather than a universal state-pair limit; two different evolutions between the same states can have different lower bounds.
  • Extending the geodesic-saturation result beyond qubits requires verifying that the solution of $\ddot r=-\alpha^2 r$ stays within the set of physical Bloch vectors in dimensions $N>2$, which the paper does not demonstrate.
  • The same integral construction could be transferred to open-system dynamics using the Lindblad velocity derived in Appendix A, yielding an experimentally measurable speed limit for nonunitary evolution.
  • An online implementation could stop the evolution as soon as the accumulated swap-test path length reaches the target Bloch angle, turning the theoretical bound into a stopping rule.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. This paper introduces a quantum speed limit based on the Bloch angle. For unitary dynamics it defines an instantaneous velocity v(t) in Eq. (5), the accumulated path length s(T)=∫_0^T v(t)dt, and a new bound τΘ as the time at which the accumulated path length reaches a target angle Θ, Eq. (8). Because Θ(ρ0,ρT)≤s(T) by Eq. (6) and v(t)≥0, the paper concludes τΘ≤T, with equality when the actual path is a geodesic. The authors also compare this bound with an existing Bloch-angle bound, analyze the Landau-Zener model, and report a photonic experiment in which swap-test measurements of Tr(ρ_t ρ_{t+dt}) are used to determine path lengths without full quantum state tomography.

Significance. The experimental demonstration is a genuine asset: direct swap-test measurement of the overlap is a practical way to determine path lengths for the implemented unitary qubit dynamics, and Eq. (10) correctly gives the overlap for qubits. The lower-bound inequality itself is correct, and the qubit saturation example is clean. However, the paper's central theoretical novelty is limited: τΘ≤T follows almost immediately from the definition of τΘ and the triangle inequality, and the claimed characterization of geodesics in Appendix B is only verified for a one-parameter family of curves and is not established for the physical state space when N>2. The general claims of applicability to arbitrary dynamics and higher-dimensional systems are not supported by the analytical or experimental arguments. With appropriate restrictions and revised claims the manuscript could be acceptable, but in its present form the main theoretical claims require substantial work.

major comments (4)
  1. [§II, Eq. (8)] The statement that τΘ≤T is a consequence of the definition of τΘ together with the triangle inequality and v(t)≥0: once Θ≤s(T) and s(t)=∫_0^t v(t)dt is nondecreasing, the solution of Θ=s(τ) trivially satisfies τ≤T. This is not an independent quantum speed limit in the sense of earlier bounds such as the Mandelstam-Tamm bound, which are constructed from initial and final state data and the Hamiltonian. The paper should either reposition the result as a trajectory-dependent bound that is useful for the experimental protocol, or prove a genuinely non-tautological property such as a bound on τΘ in terms of initial data alone.
  2. [Appendix B, Eqs. (B1)-(B5)] The geodesic equation r¨=−α²r is not derived; the authors only verify that the assumed solution (B2) saturates the inequality (6). This establishes sufficiency of that curve family, not that these curves are geodesics of the physical state space. For N>2 the generalized Bloch vector of a physical state satisfies positivity constraints beyond fixed norm (for example, a pure qutrit has Tr(ρ³)=1), and great circles in R^{N²−1} can leave the set of physical states at intermediate times. Therefore the saturation condition 'when the path connecting ρ0 and ρT follows a geodesic line, τΘ=T' is demonstrated only for qubits. Please restrict the saturation claim to N=2 or provide a derivation on the physical submanifold.
  3. [§II and Fig. 1(b)] The protocol claims to determine dΘ(ρ_t,ρ_{t+dt}) from the swap-test overlap Tr(ρ_t ρ_{t+dt}) alone, but Eq. (3) also requires Tr(ρ_t²) and Tr(ρ_{t+dt}²). For unitary dynamics these purities are constant and can be measured once, but the statement that the measurement framework 'is highly versatile and applicable to any type of dynamics' is not justified: for general non-unital dynamics the purities must be measured at every time step, and the overlap alone is insufficient. The experimental demonstration in this paper is for qubit unitary evolution, where this simplification is valid; the general claim should be qualified.
  4. [§III and Appendix D, Eq. (10)] Equation (10) relates Tr(ρ_{t1}ρ_{t2}) to the probability of the single anti-symmetric state |c⟩. This is exact only for qubits, where the anti-symmetric subspace is one-dimensional. For N>2 the anti-symmetric subspace has dimension N(N−1)/2>1, and measuring one anti-symmetric component does not give the full swap expectation value. The claims of applicability to higher-dimensional systems should be revised accordingly.
minor comments (4)
  1. [§II, Eq. (6)] The notation ∫_0^T Θ(ρ_t,ρ_{t+dt}) is informal; the integrand is not a function of dt as written. It would be clearer to write s(T)=∫_0^T v(t)dt with v(t)=Θ(ρ_t,ρ_{t+dt})/dt.
  2. [Figs. 4 and 5] The figures report experimental points without visible error bars or quantitative uncertainty estimates, and the time axis has no units. Please add error bars and specify the units and experimental parameters.
  3. [Fig. 3 caption] There is a typo: 'optial axis' should be 'optical axis'. Also, 'Schr¨ odinger' should be 'Schrödinger' in the text.
  4. [§III] The statement 'The results are in accord with the theoretical predictions within experimental errors' would be more informative with a quantitative measure of agreement, such as chi-squared values or maximum deviations.

Circularity Check

2 steps flagged · score 6.0 of 10

The new QSL's lower-bound property is installed by its defining integral equation: τΘ is defined as the hitting time of the cumulative speed integral, so τΘ≤T follows from monotonicity rather than from independent physics.

  1. self definitional [Section II, Eq. (8) and the following paragraph]
    "our bound is defined as the solution of the integral equation: Θ = ∫_0^{τΘ} v(t)dt = ∫_0^{τΘ} Θ(ρt, ρt+dt). (8) ... Since Θ≤∫_0^T v(t)dt=s(T) and v(t) is non-negative, we conclude that τΘ in our definition is less than or equal to T."

    Equation (8) defines τΘ as the time at which the cumulative integral of v(t) first reaches Θ. Because Eq. (6) gives s(T)≥Θ and v(t)≥0 makes the cumulative integral nondecreasing, the hitting time automatically satisfies τΘ≤T. The lower-bound property is therefore installed by the definition itself, not derived from initial- and final-state data or from any independent constraint on the dynamics. The only independent input is the pointwise velocity v(t); the advertised bound is just the hitting time of the accumulated path length.

  2. self definitional [Section II, immediately after Eq. (8)]
    "Moreover, when the path connecting the initial state ρ0 and the final state ρT follows a geodesic line, τΘ=T."

    This equality is also a direct consequence of the defining equation. On a geodesic, Eq. (6) is saturated, so the path length s(T) equals Θ. Since τΘ is defined as the time at which the cumulative integral reaches Θ, it must coincide with T. The saturation condition is thus a restatement of the definition under the geodesic assumption, not an independent derivation.

full rationale

The central definitional reduction is in Eq. (8): the new QSL is defined as the solution of Θ=∫_0^{τΘ} v(t)dt, and the claimed τΘ≤T follows immediately from monotonicity of the integral and the triangle inequality in Eq. (6). This is a self-definitional lower bound rather than an independently predicted speed limit. The saturation claim τΘ=T is likewise a restatement of the defining equation when the path length equals Θ. The paper's other components — the explicit formula for v(t) in Eq. (5), the Lindblad generalization in Appendix A, and the swap-test measurement — are not circular and are externally checkable. However, the geodesic claim in Appendix B is not circular but is a separate correctness gap: the authors verify only that the great-circle family r(t)=cos(αt)r(0)+sin(αt)r′ saturates Eq. (6), which proves sufficiency, not that all physical geodesics for N>2 have this form; the saturation condition is therefore unproved beyond qubits. No load-bearing self-citation or fitted-input-as-prediction pattern was found. Because the paper's headline lower-bound property itself reduces by construction, the circularity score is 6.

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

The formalism rests on standard quantum mechanics and vector-space geometry. The only nonstandard input is the assumption that great-circle Bloch-vector motion is the relevant geodesic in all dimensions, which is unproved for N > 2. No parameters are fitted to data; the Landau-Zener parameters V and Delta are experimental settings, not extracted constants.

assumptions (4)
  • standard math The angular distance between Bloch vectors satisfies the triangle inequality, so the geodesic distance is no larger than any piecewise path length.
    Used in Eq. (6) to relate Theta(rho_0, rho_T) to the integrated infinitesimal angles s(T).
  • domain assumption The time evolution of density matrices follows the von Neumann equation for closed systems and the Lindblad equation for open systems.
    Used throughout Section II and Appendix A to expand rho_{t+dt} and compute the instantaneous velocity v(t).
  • domain assumption The generalized Bloch representation rho = I/N + (1/2) r . lambda with Tr(lambda_i lambda_j) = 2 delta_ij gives Eq. (3) for the Bloch angle.
    Used to define Theta and to derive the velocity expression in Eq. (5).
  • ad hoc to paper For N > 2, the solution of r-double-dot = -alpha^2 r remains an admissible unitary orbit of a quantum state.
    Appendix B treats this as the geodesic equation; for N > 2 the Bloch vector of a state is constrained beyond fixed length, so this is an unproved assumption when the paper generalizes beyond qubits.

how reviews work

0 comments
Cite this review

Pith. "Pith review of A Geometric Quantum Speed Limit: Theoretical Insights and Photonic Implementation." pith.science (2026). https://pith.science/paper/3E2UDKKB

@misc{pith2026250600354,
  author       = {Pith},
  title        = {Pith review of: A Geometric Quantum Speed Limit: Theoretical Insights and Photonic Implementation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3E2UDKKB}},
  note         = {Machine review of arXiv:2506.00354}
}
read the original abstract

Quantum mechanics imposes a lower bound on the time required for a quantum system to reach certain given targets. In this paper, from a geometric perspective, we introduce a new quantum speed limit (QSL) based on the Bloch angle and derive the condition for it to saturate. Experimentally, we demonstrate the feasibility of measuring this QSL using a photonic system through direct Bloch angle measurements via a swap test, bypassing the need for comprehensive quantum state tomography. Compared to the existing Bloch-angle-based QSL mentioned in prior work, our QSL requires fewer computational and experimental resources and provides tighter constraints for specific dynamics. Our work underscores the Bloch angle's effectiveness in providing tighter and experimentally accessible QSLs and advances the understanding of quantum dynamics.

Figures

Figures reproduced from arXiv: 2506.00354 by the authors.

Figure 1
Figure 1. FIG. 1. Illustration of a quantum system’s evolution path and its measurement. (a) The blue dashed line represents the geodesic [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Evolution of a Bloch vector [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Experimental setup for measurement of quantum speed limit. Photon pairs are generated through the parametric [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (2 more)
Figure 5
Figure 5. Figure 5: FIG. 5. The experimental path length [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Comparison of the two QSL bounds (the exist [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

49 extracted references · 32 canonical work pages

  1. [38]

    Giovannetti, S

    V. Giovannetti, S. Lloyd, and L. Maccone, Europhysics Letters62, 615 (2003)

    work page 2003

  2. [1]

    (5) 4 State Preparation and Time Evolution Swap TestSingle Photon Source PBS BD Mirror HWP APDQWP DT BBO FIG

    and Tr(Htρt)2 = Tr(Hρ0)2, making the velocityv(t) constant according to Eq. (5) 4 State Preparation and Time Evolution Swap TestSingle Photon Source PBS BD Mirror HWP APDQWP DT BBO FIG. 3. Experimental setup for measurement of quantum speed limit. Photon pairs are generated through the parametric downconversion process by pumping beta-barium-borate (BBO) ...

    work page 2000

  3. [2]

    The probability of obtaining the out- come|1⟩is (1− |⟨ψ 1|ψ2⟩|2)/2, which directly yields the overlap of the two systems

    Subsequently, a Hadamard gate is applied to the ancillary qubit, followed by a measurement. The probability of obtaining the out- come|1⟩is (1− |⟨ψ 1|ψ2⟩|2)/2, which directly yields the overlap of the two systems. 8 (a) (c) (b) (d) 1 3 4 5 6 0 2 0.0 0.5 1.0 1.5 2.0 2.5 3.0 1 3 4 5 6 0 2 1 3 4 5 6 0 2 1 3 4 5 6 0 2 0.0 0.5 1.0 1.5 2.0 2.5 3.00.0 0.5 1.0 1....

    work page 2000

  4. [3]

    Lloyd, Nature406, 1047 (2000)

    S. Lloyd, Nature406, 1047 (2000)

    work page 2000

  5. [4]

    J. D. Bekenstein, Phys. Rev. Lett.46, 623 (1981)

    work page 1981

  6. [5]

    Deffner and E

    S. Deffner and E. Lutz, Phys. Rev. Lett.105, 170402 (2010)

    2010

  7. [6]

    Abah and E

    O. Abah and E. Lutz, Europhysics Letters118, 40005 (2017)

    work page 2017

  8. [7]

    Mukherjee, W

    V. Mukherjee, W. Niedenzu, A. G. Kofman, and G. Kur- izki, Phys. Rev. E94, 062109 (2016)

    work page 2016

Show all 49 references
  1. [8]

    F. C. Binder, S. Vinjanampathy, K. Modi, and J. Goold, New Journal of Physics17, 075015 (2015)

    2015

  2. [9]

    Campaioli, F

    F. Campaioli, F. A. Pollock, F. C. Binder, L. C´ eleri, J. Goold, S. Vinjanampathy, and K. Modi, Phys. Rev. Lett.118, 150601 (2017)

    2017

  3. [10]

    Juli` a-Farr´ e, T

    S. Juli` a-Farr´ e, T. Salamon, A. Riera, M. N. Bera, and M. Lewenstein, Phys. Rev. Res.2, 023113 (2020)

    2020

  4. [11]

    Giovannetti, S

    V. Giovannetti, S. Lloyd, and L. Maccone, Nature Pho- tonics5, 222 (2011)

    2011

  5. [12]

    Caneva, M

    T. Caneva, M. Murphy, T. Calarco, R. Fazio, S. Mon- tangero, V. Giovannetti, and G. E. Santoro, Phys. Rev. Lett.103, 240501 (2009)

    2009

  6. [13]

    Lloyd and S

    S. Lloyd and S. Montangero, Phys. Rev. Lett.113, 010502 (2014)

    2014

  7. [14]

    G. C. Hegerfeldt, Phys. Rev. Lett.111, 260501 (2013)

    2013

  8. [15]

    Carlini, A

    A. Carlini, A. Hosoya, T. Koike, and Y. Okudaira, Phys. Rev. Lett.96, 060503 (2006)

    2006

  9. [16]

    D. C. Brody and D. M. Meier, Phys. Rev. Lett.114, 100502 (2015)

    2015

  10. [17]

    X. Wang, M. Allegra, K. Jacobs, S. Lloyd, C. Lupo, and M. Mohseni, Phys. Rev. Lett.114, 170501 (2015)

    2015

  11. [18]

    Mandelstam, J

    L. Mandelstam, J. Phys.(USSR)9, 249 (1945)

    1945

  12. [19]

    Margolus and L

    N. Margolus and L. B. Levitin, Physica D: Nonlinear Phenomena120, 188 (1998), proceedings of the Fourth Workshop on Physics and Consumption

    1998

  13. [20]

    L. B. Levitin and T. Toffoli, Phys. Rev. Lett.103, 160502 (2009)

    2009

  14. [21]

    Giovannetti, S

    V. Giovannetti, S. Lloyd, and L. Maccone, Phys. Rev. A67, 052109 (2003)

    2003

  15. [22]

    Giovannetti, S

    V. Giovannetti, S. Lloyd, and L. Maccone, Journal of Optics B: Quantum and Semiclassical Optics6, S807 (2004)

    2004

  16. [23]

    P. J. Jones and P. Kok, Phys. Rev. A82, 022107 (2010)

    2010

  17. [24]

    Zwierz, Phys

    M. Zwierz, Phys. Rev. A86, 016101 (2012)

    2012

  18. [25]

    S. L. Braunstein and G. J. Milburn, Phys. Rev. A51, 1820 (1995)

    1995

  19. [26]

    S. L. Braunstein, C. M. Caves, and G. Milburn, Annals of Physics247, 135 (1996)

    1996

  20. [27]

    del Campo, I

    A. del Campo, I. L. Egusquiza, M. B. Plenio, and S. F. Huelga, Phys. Rev. Lett.110, 050403 (2013)

    2013

  21. [28]

    Deffner and E

    S. Deffner and E. Lutz, Phys. Rev. Lett.111, 010402 (2013)

    2013

  22. [29]

    M. M. Taddei, B. M. Escher, L. Davidovich, and R. L. de Matos Filho, Phys. Rev. Lett.110, 050402 (2013)

    2013

  23. [30]

    Mondal, C

    D. Mondal, C. Datta, and S. Sazim, Physics Letters A 380, 689 (2016)

    2016

  24. [31]

    C. M. Bender, D. C. Brody, H. F. Jones, and B. K. Meister, Phys. Rev. Lett.98, 040403 (2007)

    2007

  25. [32]

    Zheng, L

    C. Zheng, L. Hao, and G. L. Long, Phil. Trans. R. Soc. A.371, 20120053 (2013)

    2013

  26. [33]

    D. P. Pires, M. Cianciaruso, L. C. C´ eleri, G. Adesso, and D. O. Soares-Pinto, Phys. Rev. X6, 021031 (2016)

    2016

  27. [34]

    Kakutani, Annals of Mathematics49, 214 (1948)

    S. Kakutani, Annals of Mathematics49, 214 (1948)

    1948

  28. [35]

    Bures, Transactions of the American Mathematical Society135, 199 (1969)

    D. Bures, Transactions of the American Mathematical Society135, 199 (1969)

    1969

  29. [36]

    Uhlmann, Reports on Mathematical Physics9, 273 (1976)

    A. Uhlmann, Reports on Mathematical Physics9, 273 (1976)

    1976

  30. [37]

    Jozsa, Journal of Modern Optics41, 2315 (1994)

    R. Jozsa, Journal of Modern Optics41, 2315 (1994)

    1994

  31. [39]

    Deffner and S

    S. Deffner and S. Campbell, Journal of Physics A: Math- ematical and Theoretical50, 453001 (2017)

    2017

  32. [40]

    Campaioli, F

    F. Campaioli, F. A. Pollock, F. C. Binder, and K. Modi, Phys. Rev. Lett.120, 060409 (2018)

    2018

  33. [41]

    Altepeter, E

    J. Altepeter, E. Jeffrey, and P. Kwiat (Academic Press,

  34. [42]

    J. C. Garcia-Escartin and P. Chamorro-Posada, Phys. Rev. A87, 052330 (2013)

    2013

  35. [43]

    Cincio, Y

    L. Cincio, Y. Suba¸ sı, A. T. Sornborger, and P. J. Coles, New Journal of Physics20, 113022 (2018)

    2018

  36. [44]

    H. Zhan, B. Wang, M. Mi, J. Xie, L. Xu, A. Zhang, and L. Zhang, Light: Science & Applications14, 83 (2025)

    2025

  37. [45]

    M. A. Nielsen and I. L. Chuang,Quantum Computa- tion and Quantum Information: 10th Anniversary Edi- tion(Cambridge University Press, 2010)

    2010

  38. [46]

    Mirkin, F

    N. Mirkin, F. Toscano, and D. A. Wisniacki, Phys. Rev. A94, 052125 (2016)

    2016

  39. [47]

    Buhrman, R

    H. Buhrman, R. Cleve, J. Watrous, and R. de Wolf, Phys. Rev. Lett.87, 167902 (2001)

    2001

  40. [48]

    Landau, Phys

    L. Landau, Phys. Z. Sowjetunion2, 118 (1932)

    1932

  41. [49]

    Zener, Proc

    C. Zener, Proc. R. Soc. Lond. A137, 696 (1932)

    1932

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.