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arXiv:2603.12061

Critical Unstable Qubits in Particle Physics

Dimitrios Karamitros, Thomas McKelvey, Snehit Panghal, Apostolos Pilaftsis

quant-phhep-phSubmitted Mar 12, 2026• Announced Mar 13, 2026• 0 citations
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Abstract

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, ${\bf E}$ and ${\bfΓ}$, are orthogonal to one another, and the parameter $r = |{\bf Γ}|/(2|{\bf E}|)$ 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.

Comments
21 pages, 5 figures, 5 tables. Proceedings of Science conference

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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 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 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 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-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.
The Bloch equation and its variants constitute the fundamental dynamical model for arbitrary two-level systems. Many important processes, including those in more complicated systems, can be modeled and understood through the two-level approximation. It is therefore of widespread relevance, especially as it relates to understanding dissipative processes in current cutting-edge applications of quantum mechanics. Although the Bloch equation has been the subject of considerable analysis in the 70 years since its inception, there is still, perhaps surprisingly, significant work that can be done. This paper extends the scope of previous analyses. It provides a framework for more fully understanding the dynamics of dissipative two-level systems. A solution is derived that is compact, tractable, and completely general, in contrast to previous results. Any solution of the Bloch equation depends on three roots of a cubic polynomial that are crucial to the time dependence of the system. The roots are typically only sketched out qualitatively, with no indication of their dependence on the physical parameters of the problem. Degenerate roots, which modify the solutions, have been ignored altogether. Here the roots are obtained explicitly in terms of a single real-valued root that is expressed as a simple function of the system parameters. For the conventional Bloch equation, a simple graphical representation of this root is presented that makes evident the explicit time dependence of the system for each point in the parameter space. Several intuitive, visual models of system dynamics are developed. A Euclidean coordinate system is identified in which any generalized Bloch equation is separable, i.e., the sum of commuting rotation and relaxation operators. The time evolution in this frame is simply a rotation followed by relaxation at modified rates that play a role similar to the standard longitudinal and transverse rates. These rates are functions of the applied field, which provides information towards control of the dissipative process. The Bloch equation also describes a system of three coupled harmonic oscillators, providing additional perspective on dissipative systems.
Abstract We study an analogous Bloch sphere representation of higher-level quantum systems using the Heisenberg-Weyl operator basis. We introduce a parametrization method that will allow us to identify a real-valued Bloch vector for an arbitrary density operator. Before going into arbitrary d-level (d ≥ 3) quantum systems (qudits), we start our analysis with three-level ones (qutrits). It is well known that we need at least eight real parameters in the Bloch vector to describe arbitrary three-level quantum systems (qutrits). However, using our method we can divide these parameters into four weight, and four angular parameters, and find that the weight parameters are inducing a unit sphere in four-dimension. And, the four angular parameters determine whether a Bloch vector is physical. Therefore, unlike its qubit counterpart, the qutrit Bloch sphere does not exhibit a solid structure. Importantly, this construction allows us to define different properties of qutrits in terms of Bloch vector components. We also examine the two and three-dimensional sections of the sphere, which reveal a non-convex yet closed structure for physical qutrit states. Further, we apply our representation to derive mutually unbiased bases (MUBs), characterize unital maps for qutrits, and assess ensembles using the Hilbert-Schmidt and Bures metrics. Moreover, we extend this construction to qudits, showcasing its potential applicability beyond the qutrit scenario.