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Journal of Physics Conference Series

Violation of Leggett—Garg inequalities for quantum-classical hybrids

Marcin Łobejko, Jerzy Dajka

Published Jul 3, 2015• 1 citation
Publisher

Abstract

Violation of Leggett-Garg inequalities can serve as a signature of a failure of (macroscopic) realism. We investigate violation of the simplest Leggett-Garg inequality for a qubit coupled to an integer j spin (angular momentum). Such a system effectively reveals quantum-classical hybrid behavior in the limit of large j values. We show that a maximal violation of the Leggett- Garg inequality is larger for quantum-classical hybrids than for fully quantum systems.

DOI
10.1088/1742-6596/626/1/012038

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Violation of Leggett-Garg inequalities can serve as a signature of a failure of (macroscopic) realism. We investigate violation of the simplest Leggett-Garg inequality for a qubit coupled to an integer j spin (angular momentum). Such a system effectively reveals quantum-classical hybrid behavior in the limit of large j values. We show that a maximal violation of the Leggett- Garg inequality is larger for quantum-classical hybrids than for fully quantum systems.
The violation of the Leggett–Garg inequality is studied for a quantum top (with angular momentum JzJ_zJz​ J z of integer or half-integer size), being driven by classical Gaussian white noise. The form of a longitudinal (Jz)(J_z)(Jz​) ( J z ) or a transverse (Jx)(J_x)(Jx​) ( J x ) coupling of noise to the angular momentum affects both (i) to what extent the Leggett–Garg inequality is violated and (ii) how this violation is influenced by the size j of the spinning top and direction of a coupling (transverse or longitudinal). We introduce j-independent method, using two- dimensional invariant subspace of the system Hilbert space, which allows us to find out strict analytical solution for a noise-free system and with longitudinal coupling and to extract from the whole dynamics effects purely induced by a noise. It is demonstrated that in the semi-classical limit of a large angular momentum j and for the transverse coupling, the Leggett–Garg inequalities become more strongly violated as compared to the deep quantum regime of small j.
Abstract This study investigates the emergence of macroscopic classical behavior from quantum foundations via the entropic Leggett--Garg inequality. We introduce a geometric framework for deriving entropic Leggett--Garg inequalities with higher-order temporal correlations and demonstrate their advantages over conventional formulations. Numerical analyses show that entropic Leggett--Garg inequalities offer a robust and complementary criterion to standard approaches, providing a transparent information theoretic interpretation that facilitates the characterization of coherent quantum processes. By applying the WKB approximation, we prove that violations for maximally mixed states remain bounded by a constant in the macroscopic limit, indicating that macrorealism dominates in generic parameter regimes. We further explain previously reported maximal violations at specific parameter regimes as a consequence of the breakdown of the WKB approximation. Our findings indicate that quantum and classical descriptions remain macroscopically incompatible, while violations persist only in fine-tuned regimes, clarifying the conditions for detecting macroscopic quantum phenomena.
Investigating the boundary between a quantum mechanical description and a classical realistic view is of fundamental importance. The Leggett-Garg (LG) inequality provides a criterion to distinguish between quantum systems and classical systems, and can be used to prove the macroscopic superposition state. A larger upper bound of the LG function can be obtained in a multilevel system. Here, we present an experimental violation of the Leggett-Garg inequality in a three-level system using a nitrogen-vacancy center in diamond by ideal negative result measurements. The experimental maximum value of the Leggett-Garg function is ${K}_{3}^{\text{expt}}=1.625\ifmmode\pm\else\textpm\fi{}0.022$, which exceeds the L\"uders bound with a $5\ensuremath{\sigma}$ level of confidence.
Macrorealistic description governing the classical realm is known to get uprooted in the quantum domain. The Leggett-Garg inequality [A. J. Leggett and A. Garg, Phys. Rev. Lett. 54, 857 (1985)] proposes to probe the macrorealistic limit emerging from the quantum scenario. It places a bound on the linear combinations of temporal correlations of observables measured sequentially at different time intervals. Violation of the Leggett-Garg inequality, up to the so-called temporal Tsirelson bound (TTB), has been realized in the quantum domain. It is important to understand whether quantum theory permits larger violation beyond the TTB and up to the algebraic maximum value of the Leggett-Garg inequality. To this end, it has been shown [C. Budroni and C. Emary, Phys. Rev. Lett. 113, 050401 (2014)] that a three-term Leggett-Garg inequality can be violated beyond the TTB 3/23/23/2 in the time evolution of N>2N>2N>2 level quantum systems by choosing suitable measurement schemes and state-update rules. Furthermore, it is revealed that the algebraic maximum value 3 can be realized in the limit $N\ensuremath{\rightarrow}\ensuremath{\infty}$. Their measurement scheme rules out violation of the Leggett-Garg inequality beyond the TTB by a qubit. Being genuinely quantum in nature, we ask whether such enhanced violation of the Leggett-Garg inequality could be witnessed in a qubit itself. Here, we show that a qubit undergoing time evolution generated by a non-Hermitian PT\mathcal{P}\mathcal{T}PT-symmetric Hamiltonian indeed violates the three-term Leggett-Garg inequality beyond the TTB and up to its algebraic maximum value 3.
As is well known, the macroscopic realism and the noninvasive measurability together lead to Leggett—Garg inequalities violated by quantum mechanics. We consider tests of the Leggett—Garg type with use of the q-entropies. For all q ≥ 1, quantum mechanics predicts violations of an entire family of q-entropic inequalities of the Leggett—Garg type. Violations are exemplified with a quantum spin-s system. In general, entropic Leggett—Garg inequalities give only necessary conditions that some probabilistic model is compatible with the macrorealism in the broader sense. The presented q-entropic inequalities allow to widen a class of situations, in which an incompatibility with the macrorealism can be tested. In the considered example, both the strength and range of violations are somehow improved by varying q. We also examine q-entropic inequalities of the Leggett—Garg type in the case of detection inefficiencies, when the no-click event may occur in each observation. With the use of the q-entropic inequalities, the required amount of efficiency may be reduced.
Leggett and Garg derived inequalities that probe the boundaries of classical and quantum physics by putting limits on the properties that classical objects can have. Historically, it has been suggested that Leggett-Garg inequalities are easily violated by quantum systems undergoing sequences of strong measurements, casting doubt on whether quantum mechanics correctly describes macroscopic objects. Here I show that Leggett-Garg inequalities cannot be violated by any projective measurement. The perceived violation of the inequalities found previously can be traced back to an inappropriate assumption of non-invasive measurability. Surprisingly, weak projective measurements cannot violate the Leggett-Garg inequalities either because even though the quantum system itself is not fully projected via weak measurements, the measurement devices are.