## Abstract

We study Leggett-Garg inequalities (LGIs) for a two level system (TLS) undergoing Markovian dynamics described by unital maps. We first find the analytic expression of LG parameter K _{3} (simplest variant of LGIs) in terms of the parameters of two distinct unital maps representing time evolution for intervals: t _{1} to t _{2} and t _{2} to t _{3}. We then show that the maximum violation of LGI for all possible unital maps can never exceed well known Lüders bound of K 3 L u ¨ d e r s = 3 / 2 over the full parameter space. We further show that if the map for the time interval t _{1} to t _{2} is non-unitary unital then irrespective of the choice of the map for interval t _{2} to t _{3} we can never reach Lüders bound. On the other hand, if the measurement operator eigenstates remain pure upon evolution from t _{1} to t _{2}, then depending on the degree of decoherence induced by the unital map for the interval t _{2} to t _{3} we may or may not obtain Lüders bound. Specifically, we find that if the unital map for interval t _{2} to t _{3} leads to the shrinking of the Bloch vector beyond half of its unit length, then achieving the bound K 3 L u ¨ d e r s is not possible. Hence our findings not only establish a threshold for decoherence which will allow for K 3 = K 3 L u ¨ d e r s , but also demonstrate the importance of temporal sequencing of the exposure of a TLS to Markovian baths in obtaining Lüders bound.

Original language | English |
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Article number | 205302 |

Journal | Journal of Physics A: Mathematical and Theoretical |

Volume | 56 |

Issue number | 20 |

DOIs | |

State | Published - 19 May 2023 |

Externally published | Yes |

## Keywords

- CPTP maps
- Legget-Garg inequality
- Lüders bound
- Markovian dynamics
- unital maps

## ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Statistics and Probability
- Modeling and Simulation
- Mathematical Physics
- General Physics and Astronomy