pith. sign in

arxiv: 2606.01405 · v1 · pith:SPD7EMZFnew · submitted 2026-05-31 · 🪐 quant-ph · hep-ph· hep-th

Trajectories of Critical Unstable Qubits in and on the Bloch Sphere

Snehit Panghal , Apostolos Pilaftsis This is my paper

Pith reviewed 2026-06-28 16:34 UTC · model grok-4.3

classification 🪐 quant-ph hep-phhep-th
keywords critical unstable qubitsBloch spheretrajectoriesco-decaying framenon-Hermitian Hamiltoniansdensity matrixanharmonic oscillationsstationary points
0
0 comments X

The pith

Critical unstable qubits follow explicit geometric trajectories on the Bloch sphere revealing stationary points.

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

The paper extends prior work on critical unstable qubits as a class of unstable two-level systems. In a co-decaying frame these systems display indefinite anharmonic oscillations between states and coherence-decoherence oscillations for mixed states, in contrast to the harmonic Rabi oscillations of Hermitian two-level systems. The density-matrix approach tracks the Bloch vector, and the authors supply the first explicit geometric constructions for the trajectories of both pure and mixed states inside and on the Bloch sphere. These constructions locate the stationary points at which the state remains fixed in time within the chosen frame.

Core claim

In an appropriately defined co-decaying frame, critical unstable qubits exhibit indefinite anharmonic oscillations between two states and coherence-decoherence oscillations of mixed states. Explicit geometric constructions obtain the trajectories of pure and mixed CUQs in and on the Bloch sphere, identifying stationary points at which the states do not evolve in time.

What carries the argument

The Bloch vector and its time evolution under the density-matrix formalism for CUQs, together with geometric constructions that trace the vector's paths.

If this is right

  • Trajectories of pure and mixed CUQs can be constructed geometrically inside and on the Bloch sphere.
  • Stationary points exist where the quantum state does not evolve with time in the co-decaying frame.
  • Mixed states undergo coherence-decoherence oscillations distinct from Hermitian Rabi behavior.
  • The findings bear on particle cosmology and quantum simulations of non-Hermitian Hamiltonians.

Where Pith is reading between the lines

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

  • The geometric constructions may simplify numerical modeling of non-Hermitian two-level dynamics in open systems.
  • Analog quantum simulators could test whether the predicted stationary points remain fixed under controlled decay.
  • Connections to cosmological models of unstable particles could be explored by mapping the Bloch trajectories to field evolution equations.

Load-bearing premise

The co-decaying frame is defined such that the claimed anharmonic oscillations and stationary points appear as described.

What would settle it

Direct computation or measurement of the Bloch vector at the constructed stationary points to check whether it remains constant over time in the co-decaying frame.

Figures

Figures reproduced from arXiv: 2606.01405 by Apostolos Pilaftsis, Snehit Panghal.

Figure 1
Figure 1. Figure 1: Comparative Bloch-sphere trajectories for pure state (a) Hermitian (Rabi) and (b) [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Comparative Bloch-sphere trajectories for mixed state Rabi and CUQ oscillations. [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Bloch-sphere trajectories b(t) of the maximally-mixed CUQ for different values r. Moreover, we have β0 = α0 [cf. (2.16)], and since cos δ0 = 1, the general solution (2.27) reduces to (2.18), as should be. We note that β0 and δ0 can be related to each other as they are transformations of the initial parameters b0 and φ0 through (2.23) and (2.24). Unlike in the previous section, the way β0 and δ0 are defined… view at source ↗
Figure 4
Figure 4. Figure 4: Eigenvectors of the CUQ Hamiltonian on the Bloch sphere [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Plots of b(t) with an initial mixed state b0, for different values of r. 4.1 Bloch Sphere Orbits for b(0) ⊥ e Using the analytical result of the CUQ Bloch vector b(t) in (2.22), we can compute the equation of the curve traced by b(t) in the {γ, e × γ} plane. To this end, we define X ≡ √ 1 − r 2 sin (2θ + β0) cos δ0 1 − r cos (2θ + β0) cos δ0 , Y ≡ cos (2θ + β0) cos δ0 − r 1 − r cos (2θ + β0) cos δ0 . (4.1)… view at source ↗
Figure 6
Figure 6. Figure 6: Plot of b(t) with initial mixed state of the form b0 · γ = 0 and b0 · (e × γ) < 0, for different values of r With some algebra, we obtain the following relation between X and Y : X2 a 2 + (Y + e 2 ) 2 a 2 (1 − e 2 ) = 1 , (4.2) which is the equation for an ellipse. In (4.2), a is the semi-major axis given by a =  1 − r 2 1 − r 2 cos2 δ0 1/2 cos δ0 , (4.3) and a √ 1 − e 2 is the semi-minor axis, conventio… view at source ↗
Figure 7
Figure 7. Figure 7: CUQ anharmonicity and coherence-decoherence oscillations for different values of [PITH_FULL_IMAGE:figures/full_fig_p016_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Comparative Bloch-sphere trajectory for different pure state oscillations in (a) Her [PITH_FULL_IMAGE:figures/full_fig_p017_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Bloch-sphere trajectories of pure states with [PITH_FULL_IMAGE:figures/full_fig_p018_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Geometric construction of CUQ orbit b(t) for an initial state b0 = e. The plane cuts the Bloch sphere along a circular patch. Since b0 is a pure state, the boundary of the circular patch is also the Bloch-sphere trajectory of this pure CUQ state. of the Bloch-sphere trajectories can be computed analytically by performing a Frenet-Serret construction, as done in Appendix A of [7]. Nonetheless, there is a g… view at source ↗
Figure 11
Figure 11. Figure 11: Bloch-sphere trajectory of an initial mixed CUQ state [PITH_FULL_IMAGE:figures/full_fig_p021_11.png] view at source ↗
read the original abstract

We extend previous studies on a novel class of unstable two-level systems which were called Critical Unstable Qubits (CUQs). In an appropriately defined co-decaying frame, the CUQs exhibit striking phenomena of indefinite anharmonic oscillations between two states and coherence-decoherence oscillations of mixed states. These features are distinct from the usual Rabi oscillations observed in the Hermitian counterpart of two-level systems, which are harmonic and preserve the coherence of the quantum state. We employ the density matrix formalism to study these phenomena for mixed states and delve into the nature of the trajectory traversed by these states in the Bloch sphere by studying the time evolution of the Bloch vector that describes the quantum state of the unstable qubit. In particular, we provide for the first time explicit geometric constructions to obtain trajectories of both pure and mixed CUQs in and on the Bloch sphere. This enables us to identify the stationary points of CUQs, at which the states do not evolve in time in the co-decaying frame. The potential implications of our findings for particle cosmology and quantum simulations of non-Hermitian Hamiltonians are discussed.

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

0 major / 2 minor

Summary. The manuscript extends prior work on Critical Unstable Qubits (CUQs) by employing the density matrix formalism to analyze their time evolution in a co-decaying frame. It claims to supply the first explicit geometric constructions for the trajectories of both pure and mixed CUQs on and inside the Bloch sphere, identifies stationary points at which the states cease to evolve, and contrasts the resulting indefinite anharmonic and coherence-decoherence oscillations with the harmonic, coherence-preserving Rabi oscillations of the Hermitian case. Potential implications for particle cosmology and quantum simulations of non-Hermitian Hamiltonians are noted.

Significance. If the claimed geometric constructions are rigorously derived and reproducible, the work supplies a concrete visualization tool for non-Hermitian qubit dynamics that is absent from the Hermitian literature. The identification of stationary points in the co-decaying frame constitutes a falsifiable geometric prediction that could be tested in quantum simulations.

minor comments (2)
  1. The abstract states that the co-decaying frame is 'appropriately defined' but does not restate its explicit transformation rule or the non-Hermitian Hamiltonian used; a one-sentence reminder in the introduction would aid readers who have not consulted the cited prior studies.
  2. Figure captions (if present) should explicitly label which curves correspond to pure versus mixed states and which points are the newly identified stationary points.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision for our manuscript on Critical Unstable Qubits. No specific major comments were listed in the report.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper's central contribution consists of new explicit geometric constructions for CUQ trajectories on the Bloch sphere, derived from the density matrix formalism and the time evolution of the Bloch vector. These constructions are presented as novel extensions that allow identification of stationary points in the co-decaying frame. Although the work references prior studies on CUQs and the co-decaying frame, the abstract and claims do not reduce the new geometric results to definitions or fits from those priors by construction. No load-bearing step equates a prediction or first-principles result to its own inputs via self-citation chains, ansatzes, or renaming. The derivation chain remains self-contained with independent content.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based solely on the abstract; the central claims rest on the prior definition of CUQs and the co-decaying frame from referenced previous studies, with no free parameters, axioms, or invented entities specified here.

pith-pipeline@v0.9.1-grok · 5726 in / 1175 out tokens · 33454 ms · 2026-06-28T16:34:47.033815+00:00 · methodology

discussion (0)

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

Reference graph

Works this paper leans on

27 extracted references · 9 linked inside Pith

  1. [1]

    I. I. Rabi,Space Quantization in a Gyrating Magnetic Field,Phys. Rev.51(1937), no. 8 652

    1937

  2. [2]

    Weisskopf and E

    V. Weisskopf and E. Wigner,Berechnung der nat¨ urlichen linienbreite auf grund der diracschen lichttheorie,Zeitschrift f¨ ur Physik63(Jan, 1930) 54–73

    1930

  3. [3]

    T. D. Lee, R. Oehme, and C.-N. Yang,Remarks on Possible Noninvariance Under Time Reversal and Charge Conjugation,Phys. Rev.106(1957) 340–345

    1957

  4. [4]

    Pilaftsis,Resonant CP violation induced by particle mixing in transition amplitudes, Nucl

    A. Pilaftsis,Resonant CP violation induced by particle mixing in transition amplitudes, Nucl. Phys. B504(1997) 61–107, [hep-ph/9702393]

    Pith/arXiv arXiv 1997

  5. [5]

    Karamitros, T

    D. Karamitros, T. McKelvey, and A. Pilaftsis,Quantum coherence of critical unstable two-level systems,Phys. Rev. D108(2023), no. 1 016006, [arXiv:2212.06031]

    arXiv 2023

  6. [6]

    Karamitros, T

    D. Karamitros, T. McKelvey, and A. Pilaftsis,Critical unstable qubits: An application to the B0B¯0 meson system,Phys. Rev. D111(2025), no. 9 096020, [arXiv:2502.15625]

    arXiv 2025

  7. [7]

    W. D. Heiss,The physics of exceptional points,J. Phys. A45(2012) 444016, [arXiv:1210.7536]

    Pith/arXiv arXiv 2012

  8. [8]

    D. C. Brody and E.-M. Graefe,Mixed-State Evolution in the Presence of Gain and Loss, Phys. Rev. Lett.109(2012), no. 23 230405, [arXiv:1208.5297]

    Pith/arXiv arXiv 2012

  9. [9]

    Sergi and K

    A. Sergi and K. G. Zloshchastiev,Non-Hermitian quantum dynamics of a two-level system and models of dissipative environments,Int. J. Mod. Phys. B27(2013) 1350163, [arXiv:1207.4877]

    Pith/arXiv arXiv 2013

  10. [10]

    Kawabata, Y

    K. Kawabata, Y. Ashida, and M. Ueda,Information Retrieval and Criticality in Parity-Time-Symmetric Systems,Phys. Rev. Lett.119(2017), no. 19 190401, [arXiv:1705.04628]

    Pith/arXiv arXiv 2017

  11. [11]

    Kowalski and J

    K. Kowalski and J. Rembieli´ nski,Integrable nonlinear evolution of the qubit,Annals of Physics411(2019) 167955

    2019

  12. [12]

    Rembieli´ nski and P

    J. Rembieli´ nski and P. Caban,Nonlinear extension of the quantum dynamical semigroup, Quantum5(2021) 420, [arXiv:2003.09170]

    arXiv 2021

  13. [13]

    C. M. Bender, S. Boettcher, and P. Meisinger,PT symmetric quantum mechanics,J. Math. Phys.40(1999) 2201–2229, [quant-ph/9809072]. 34

    Pith/arXiv arXiv 1999

  14. [14]

    C. M. Bender and S. Boettcher,Real spectra in nonHermitian Hamiltonians having PT symmetry,Phys. Rev. Lett.80(1998) 5243–5246, [physics/9712001]

    Pith/arXiv arXiv 1998

  15. [15]

    C. M. Bender,Making sense of non-Hermitian Hamiltonians,Rept. Prog. Phys.70 (2007) 947, [hep-th/0703096]

    Pith/arXiv arXiv 2007

  16. [16]

    A. K. Alok, S. Banerjee, N. R. S. Chundawat, and S. U. Sankar,Probing quantum decoherence at Belle II and LHCb,JHEP05(2024) 124, [arXiv:2402.02470]

    arXiv 2024

  17. [17]

    Cheng, T

    K. Cheng, T. Han, M. Low, and T. A. Wu,Quantum Tomography in Neutral Meson and Antimeson Systems,arXiv:2507.12513

    arXiv

  18. [18]

    Rembieli´ nski and J

    J. Rembieli´ nski and J. Ciborowski,On nonlinear description of neutrino flavour evolution in solar matter,Annals Phys.458(2023) 169481

    2023

  19. [19]

    Kowalski,Linear and integrable nonlinear evolution of the qutrit,Quant

    K. Kowalski,Linear and integrable nonlinear evolution of the qutrit,Quant. Inf. Proc.19 (2020), no. 5 145, [arXiv:2006.10322]

    arXiv 2020

  20. [20]

    Moiseyev,Non-Hermitian Quantum Mechanics

    N. Moiseyev,Non-Hermitian Quantum Mechanics. Cambridge University Press, 2011

    2011

  21. [21]

    Karamitros, T

    D. Karamitros, T. McKelvey, and A. Pilaftsis,Varying entropy degrees of freedom effects in low-scale leptogenesis,Phys. Rev. D109(2024), no. 5 055007, [arXiv:2310.03703]

    arXiv 2024

  22. [22]

    Zhang, D

    G.-L. Zhang, D. Liu, X.-M. Wang, and M.-H. Yung,Observation of exceptional point in a PT broken non-Hermitian system simulated using a quantum circuit,Sci. Rep.11(2021) 13795, [arXiv:2005.13828]

    arXiv 2021

  23. [23]

    Dogra, A

    S. Dogra, A. A. Melnikov, and G. S. Paraoanu,Quantum simulation of parity-time symmetry breaking with a superconducting quantum processor,Commun. Phys.4(2021) 26, [arXiv:2111.12036]

    arXiv 2021

  24. [24]

    Koukoutsis, P

    E. Koukoutsis, P. Papagiannis, K. Hizanidis, A. K. Ram, G. Vahala, O. Amaro, L. I. Inigo Gamiz, and D. Vallis,Quantum Implementation of Non-unitary Operations with Biorthogonal Representations,Quant. Inf. Comput.25(2025), no. 2 141–155, [arXiv:2410.22505]

    arXiv 2025

  25. [25]

    Jebraeilli and M

    A. Jebraeilli and M. R. Geller,Quantum simulation of a qubit with a non-Hermitian Hamiltonian,Phys. Rev. A111(2025), no. 3 032211, [arXiv:2502.13910]

    arXiv 2025

  26. [26]

    D. C. Brody,Biorthogonal quantum mechanics,J. Phys. A47(2013), no. 3 035305, [arXiv:1308.2609]

    Pith/arXiv arXiv 2013

  27. [27]

    D. H. Carlson,On real eigenvalues of complex matrices,Pacific Journal of Mathematics 15(1965) 1119–1129. 35

    1965