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arXiv:2401.10602 · Journal of Physics A Mathematical and Theoretical

Fractional conformal map, qubit dynamics and the Leggett–Garg inequality

Sourav Paul, Anant Vijay Varma, Sourin Das

Published Aug 27, 2024• 1 citation
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Abstract

Abstract A pure state of a qubit can be geometrically represented as a point on the extended complex plane through stereographic projection. By employing successive conformal maps on the extended complex plane, we can generate an effective discrete-time evolution of the pure states of the qubit. This work focuses on a subset of analytic maps known as fractional linear conformal maps. We show that these maps serve as a unifying framework for a diverse range of quantum-inspired conceivable dynamics, including (i) unitary dynamics,(ii) non-unitary but linear dynamics and (iii) non-unitary and non-linear dynamics where linearity (non-linearity) refers to the action of the discrete time evolution operator on the Hilbert space. We provide a characterization of these maps in terms of Leggett–Garg inequality complemented with no-signaling in time and arrow of time conditions.

DOI
10.1088/1751-8121/ad742a

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The time-evolution operator obtained from the fractional-time Schrödinger equation (FTSE) is said to be nonunitary since it does not preserve the norm of the vector state in time. As done in the time-dependent non-Hermitian quantum formalism, for a traceless non-Hermitian two-level quantum system, we demonstrate that it is possible to map the nonunitary time-evolution operator in a unitary one. It is done by considering a dynamical Hilbert space with a time-dependent metric operator, constructed from a Hermitian time-dependent Dyson map, in respect to which the system evolves in a unitary way, and the standard quantum mechanics interpretation can be made properly. To elucidate our approach, we consider three examples of Hamiltonian operators and their corresponding unitary dynamics obtained from the solutions of FTSE, and the respective Dyson maps.
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Lüders Bounds of Leggett–Garg Inequalities, PTPT\mathcal {PT}PT‐ Symmetric Evolution and Arrow‐of‐Time
Abstract Leggett–Garg inequalities (LGIs) test the incompatibility between the notion of macrorealism and quantum theory. For unitary dynamics, the optimal quantum violation of a Leggett–Garg inequality is constrained by the Lüders bound. However, the LGIs do not provide the necessary and sufficient condition of macrorealism. A suitably formulated set of no‐signaling in time conditions along with the arrow‐of‐time condition provides the same. In this paper, two formulations in the three‐time Leggett–Garg scenario, are studied namely, the standard LGIs and the recently formulated variant of LGIs when the system evolves under PT‐symmetric Hamiltonian. It is first demonstrated that the quantum violations of both forms of LGIs exceed their respective Lüders bounds and can even reach their algebraic maximum. It is further shown that for the case of standard Leggett–Garg inequality, the violation of Lüders bound can be obtained when both no‐signaling in time and arrow‐of‐time conditions are violated. Interestingly, for the case of a variant of LGIs, for suitable choices of relevant parameters, the quantum violation can even be obtained when only the arrow‐of‐time is violated. However, all no‐signaling in time conditions are satisfied. Such a feature is hitherto unexplored.