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arXiv:1210.1970 · Physical Review A

Violation of entropic Leggett-Garg inequality in nuclear spins

Hemant Katiyar, Abhishek Shukla, ‎K‎. ‎R‎. ‎K‎. ‎Rao, T. S. Mahesh

Published May 2, 2013• 68 citations
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Abstract

We report an experimental study of recently formulated entropic Leggett-Garg inequality (ELGI) by Usha Devi et al. [Phys. Rev. A 87, 052103 (2013)]. This inequality places a bound on the statistical measurement outcomes of dynamical observables describing a macrorealistic system. Such a bound is not necessarily obeyed by quantum systems, and therefore provides an important way to distinguish quantumness from classical behavior. Here we study ELGI using a two-qubit nuclear magnetic resonance system. To perform the noninvasive measurements required for the ELGI study, we prepare the system qubit in a maximally mixed state as well as use the ``ideal negative result measurement'' procedure with the help of an ancilla qubit. The experimental results show a clear violation of ELGI by over four standard deviations. These results agree with the predictions of quantum theory. The violation of ELGI is attributed to the fact that certain joint probabilities are not legitimate in the quantum scenario, in the sense they do not reproduce all the marginal probabilities. Using a three-qubit system, we also demonstrate that three-time joint probabilities do not reproduce certain two-time marginal probabilities.

DOI
10.1103/physreva.87.052102

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We report an experimental study of recently formulated entropic Leggett-Garg inequality (ELGI) by Usha Devi et al. [Phys. Rev. A 87, 052103 (2013)]. This inequality places a bound on the statistical measurement outcomes of dynamical observables describing a macrorealistic system. Such a bound is not necessarily obeyed by quantum systems, and therefore provides an important way to distinguish quantumness from classical behavior. Here we study ELGI using a two-qubit nuclear magnetic resonance system. To perform the noninvasive measurements required for the ELGI study, we prepare the system qubit in a maximally mixed state as well as use the ``ideal negative result measurement'' procedure with the help of an ancilla qubit. The experimental results show a clear violation of ELGI by over four standard deviations. These results agree with the predictions of quantum theory. The violation of ELGI is attributed to the fact that certain joint probabilities are not legitimate in the quantum scenario, in the sense they do not reproduce all the marginal probabilities. Using a three-qubit system, we also demonstrate that three-time joint probabilities do not reproduce certain two-time marginal probabilities.
The Leggett–Garg (LG) test of macroscopic realism involves a series of dichotomic non-invasive measurements that are used to calculate a function which has a fixed upper bound for a macrorealistic system and a larger upper bound for a quantum system. The quantum upper bound depends on both the details of the measurement and the dimension of the system. Here we present an LG experiment on a three-level quantum system, which produces a larger theoretical quantum upper bound than that of a two-level quantum system. The experiment is carried out in nuclear magnetic resonance and consists of the LG test as well as a test of the ideal assumptions associated with the experiment, such as measurement non-invasiveness. The non-invasive measurements are performed via the modified ideal negative result measurement scheme on a three-level system. Once these assumptions are tested, the violation becomes small, despite the fact that the LG value itself is large. Our results showcase the advantages of using the modified measurement scheme that can reach higher LG values, since these leave a larger margin for violating the inequality in the face of experimental imperfections.
We formulate entropic Leggett-Garg inequalities, which place constraints on the statistical outcomes of temporal correlations of observables. The information theoretic inequalities are satisfied if macrorealism holds. We show that the quantum statistics underlying correlations between time-separated spin component of a quantum rotor mimics that of spin correlations in two spatially separated spin-sss particles sharing a state of zero total spin. This brings forth the violation of the entropic Leggett-Garg inequality by a rotating quantum spin-sss system in a similar manner as does the entropic Bell inequality [S. L. Braunstein and C. M. Caves, Phys. Rev. Lett. 61, 662 (1988)] by a pair of spin-sss particles forming a composite spin singlet state.
In 1985, Leggett and Garg (1985 Phys. Rev. Lett. 54 857) proposed a Bell-like inequality to test (in)compatibility between two fundamental concepts of quantum mechanics. The first concept is 'macroscopic realism', which is the quality of a physical property of a quantum system being independent of observation at the macroscopic level. The second concept is 'noninvasive measurability', which is the possibility of performing a measurement without disturbing the subsequent evolution of a system. One of the key requirement for testing the violation of the Leggett-Garg inequality, or Bell's time inequality, is the ability to perform noninvasive measurements over a qubit state. In this paper, we present a quantum scattering circuit that implements such a measurement for maximally mixed states. The operation of the circuit is demonstrated using liquid-state nuclear magnetic resonance (NMR) in chloroform, in which the time correlations of a qubit are measured on a probe (ancillary) qubit state. The results clearly show a violation region and are in excellent agreement with the predictions of quantum mechanics.
Leggett-Garg inequalities (LGIs) have been proposed in order to assess how far the predictions of quantum mechanics defy "macroscopic realism." With LGIs, correlations of measurements performed on a single system at different times are described. We report on an experiment that demonstrates the violation of an LGI with neutrons. The final measured value of the Leggett-Garg correlator K=1.120±0.007(stat)±0.019(sys), obtained in a neutron interferometric experiment, is clearly above the limit K=1 predicted by macrorealistic theories. The experimental results are analyzed within the framework of dynamical theory of neutron diffraction, evidently reproducing the obtained values.
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.
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.