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arXiv:2606.01405

Trajectories of Critical Unstable Qubits in and on the Bloch Sphere

Snehit Panghal, Apostolos Pilaftsis

quant-phhep-phhep-thSubmitted May 31, 2026• Announced Jun 2, 2026• 0 citations
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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.

Comments
35 pages, 11 figures

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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.
We study in detail the dynamics of unstable two-level quantum systems by adopting the Bloch-vector representation. We identify a novel class of critical scenarios in which the so-called energy-level and decay-width vectors, E{\bf E}E and Γ{\bf\Gamma}Γ, are orthogonal to one another, and the parameter r=∣Γ∣/(2∣E∣)r = |{\bf \Gamma}|/(2|{\bf E}|)r=∣Γ∣/(2∣E∣) is less than 1. Most remarkably, we find that critical unstable qubit systems exhibit atypical behaviours like coherence--decoherence oscillations when analysed in an appropriately defined co-decaying frame of the system. By making use of a Fourier series decomposition, we define anharmonicity observables that quantify the degree of non-sinusoidal oscillation of a CUQ. We apply the results of our formalism to the neutral-meson systems and derive generic upper limits on these new observables. In particular, we provide a compilation table of all well-explored meson--antimeson two-level systems in terms of Bloch-sphere parameters.
We study in detail the dynamics of unstable two-level quantum systems by adopting the Bloch-vector representation. We identify a novel class of critical scenarios in which the so-called energy-level and decay-width vectors, E{\bf E}E and Γ{\bfΓ}Γ, are orthogonal to one another, and the parameter r=∣Γ∣/(2∣E∣)r = |{\bf Γ}|/(2|{\bf E}|)r=∣Γ∣/(2∣E∣) is less than~1. Most remarkably, we find that critical unstable qubit systems exhibit atypical behaviours like coherence--decoherence oscillations when analysed in an appropriately defined co-decaying frame of the system. By making use of a Fourier series decomposition, we define anharmonicity observables that quantify the degree of non-sinusoidal oscillation of a CUQ. We apply the results of our formalism to the neutral-meson systems and derive generic upper limits on these new observables. In particular, we provide a compilation table of all well-explored meson--antimeson two-level systems in terms of Bloch-sphere parameters.
We extend our previous work on a novel class of unstable qubits which we have identified recently and called them critical unstable qubits (CUQs). The characteristic property of CUQs is that the energy-level and decay-width vectors, E and Γ , are orthogonal to one another, and the key parameter r = | Γ | / | 2 E | is less than 1. Most remarkably, CUQs exhibit two atypical behaviors: (i) they display coherence-decoherence oscillations in a co-decaying frame of the system described by a unit Bloch vector b , and (ii) the unit Bloch vector b describing a pure CUQ sweeps out unequal areas during equal intervals of time, while rotating about the vector E . The latter phenomenon emerges beyond the usual oscillatory pattern due to the energy-level difference of the two-level quantum system, which governs an ordinary qubit. By making use of a Fourier series decomposition, we define anharmonicity observables that quantify the degree of oscillation of a CUQ. We apply the results of our formalism to the B 0 B ¯ 0 -meson system and derive, for the first time, generic upper limits on these new observables.
We study in detail the dynamics of unstable two-level quantum systems by adopting the Bloch-sphere formalism of qubits. By employing the Bloch-vector representation for such unstable qubit systems, we identify a novel class of critical scenarios in which the so-called energy-level and decay-width vectors, E\mathbf{E}E and $\mathbf{\ensuremath{\Gamma}}$, are orthogonal to one another, and the parameter $r=|\mathbf{\ensuremath{\Gamma}}|/(2|\mathbf{E}|)$ is less than 1. Most remarkably, we find that critical unstable qubit systems exhibit atypical behaviors like coherence-decoherence oscillations when analyzed in an appropriately defined codecaying frame of the system. In the same frame, a unit Bloch vector b\mathbf{b}b describing a pure critical qubit will sweep out unequal areas during equal intervals of time, while rotating about the vector E\mathbf{E}E. These phenomena emerge beyond the usual oscillatory pattern due to the energy-level difference of the two-level quantum system. Interestingly enough, we observe that these new features will persist even for quasicritical scenarios, in which the vectors E\mathbf{E}E and $\mathbf{\ensuremath{\Gamma}}$ are not perfectly orthogonal to each other. Applications of our results to quantum information and to unstable meson-antimeson and other systems are discussed.
We establish a geometric generalization of the Bloch sphere for two-level quantum systems with non-Hermitian Hamiltonians using the Spacetime Algebra (STA) formulation. By lifting the state density operator from the even subalgebra to the full STA, we show that the state space expands from the unit 2-sphere to a future light cone. The non-unitary time evolution generated by a general non-Hermitian Hamiltonian corresponds to proper orthochronous Lorentz transformations on the null vectors. We classify the Hamiltonian dynamics into four distinct geometric classes: spatial rotations (corresponding to PT\mathcal{PT}PT-symmetric systems), pure boosts (anti-PT\mathcal{PT}PT-symmetric systems), null rotations (exceptional points), and general mixtures. We also use this formulation to study several results from PT-symmetric quantum mechanics, including the topological features of the exceptional points.
Abstract In the geometry of quantum evolutions, a geodesic path is viewed as a path of minimal statistical length connecting two pure quantum states along which the maximal number of statistically distinguishable states is minimum. In this paper, we present an explicit geodesic analysis of the dynamical trajectories that emerge from the quantum evolution of a single-qubit quantum state. The evolution is governed by an Hermitian Hamiltonian operator that achieves the fastest possible unitary evolution between given initial and final pure states. Furthermore, in addition to viewing geodesics in ray space as paths of minimal length, we also verify the geodesicity of paths in terms of unit geometric efficiency and vanishing geometric phase. Finally, based on our analysis, we briefly address the main hurdles in moving to the geometry of quantum evolutions for open quantum systems in mixed quantum states.