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New Journal of Physics

Reply to Comment on ‘A scattering quantum circuit for measuring Bell's time inequality: a nuclear magnetic resonance demonstration using maximally mixed states’

Alexandre M. Souza, Ivan S. Oliveira, Roberto S. Sarthour

Published May 9, 2012• 1 citation
Publisher

Abstract

We have proposed an experiment for testing the violation of the Leggett–Garg (LG) inequality using an auxiliary probe [1]. In our scheme, a single microscopic qubit (ancillary) is correlated to a quantum ensemble. The operation of the method was demonstrated using liquid-state nuclear magnetic resonance (NMR) [1]. Recently, a new experiment was performed using a very similar approach [2]. A special case in [2], the three-correlation case, is identical to our proposal [1].

DOI
10.1088/1367-2630/14/5/058002

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A recent paper by Souza, Oliveira and Sarthour (SOS) reports the experimental violation of a Leggett–Garg (LG) inequality (sometimes referred to as a temporal Bell inequality). The inequality tests for quantum mechanical superposition: if the inequality is violated, the dynamics cannot be explained by a large class of classical theories under the heading of macrorealism. Experimental tests of the LG inequality are beset by the difficulty of carrying out the necessary so-called 'non-invasive' measurements (which for the macrorealist will extract information from a system of interest without disturbing it). SOS argue that they nevertheless achieve this difficult goal by putting the system in a maximally mixed state. The system then allegedly undergoes no perturbation during their experiment. Unfortunately, the method is ultimately unconvincing to a skeptical macrorealist and so the conclusions drawn by SOS are unjustified.

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We have proposed an experiment for testing the violation of the Leggett–Garg (LG) inequality using an auxiliary probe [1]. In our scheme, a single microscopic qubit (ancillary) is correlated to a quantum ensemble. The operation of the method was demonstrated using liquid-state nuclear magnetic resonance (NMR) [1]. Recently, a new experiment was performed using a very similar approach [2]. A special case in [2], the three-correlation case, is identical to our proposal [1].
Working on quantum systems entail different interests, for example, working on fundamental \nunderstanding of quantum systems also lay foundation for better quantum computation \ntechniques. A test for whether a system is behaving quantum mechanically or \nclassically is devised by Leggett and Garg in form of inequalities, called Leggett-Garg Inequalities \n(LGI). Such Inequalities are violated by a system whose evolution is governed \nby quantum mechanics. A precise experiment to violate LGIs require a guarantee that the \nmeasurement does not affect the system or its future dynamics. These Inequalities were \nproposed for dichotomic systems,systems which can have two outcomes. Here we present \nan LG experiment on a three-level quantum system, which theoretically have larger quantum \nupper bound than that of a two-level quantum system. This larger violation also \nprovides a bigger buffer to taking in account of the various experimental imperfections. \nPerforming a quantum computing task requires precise level of control to initialize, \nperform and measure the quantum system. With increasing size of the quantum processor \nthe challenge is to maintain optimal control. Nuclear Magnetic Resonance (NMR) has \nalways been a very faithful test-bed for quantum processing ideas. In NMR, we perform \nRadio Frequency (RF) pulses to control and steer the system to the desired state. Most \nused method to derive the exact frequency and amplitude of these pulses for a given \ntask is based on gradient. Although systematic, one have to simulate these pulses on \na classical computer first, which makes the task very inefficient. We report a a way of \nperforming optimization with a hybrid quantum-classical scheme. This scheme helps us \nperform classically harder computational tasks on the quantum processor. We optimize \npulses which drive our system from 7-coherence state to 12-coherence state on a 12-qubit \nNMR processor. \nElectron Spin Resonance (ESR) employs the same techniques as of NMR but having \nadvantage in larger polarization compared to later. Although this does not imply better \ncontrol, cause the frequency at which pulses are required to control an ESR system fall into \nmicrowave region. Microwave frequency are harder to control electronically, thus making \nit harder for performing ESR quantum computing. The hybrid scheme used in NMR \nexperiment relies on some ideal pulses which are needed to be optimized classically. We \nalleviate this requirement by using finite difference method of calculating gradient. We \ncompare these methods with the earlier methods to show the superiority of such a scheme. \nState-to-state transfer pulses are sufficient for most of the quantum computing task, \nbut, an universal quantum information implementation requires state independent pulses. \nThe techniques used in optimizing state-to-state pulses can be modified to optimize for a \nstate independent pulse. We show that this methods scale polynomially with the number \nof qubits and is general in terms of its implementation. We further reduce the resource \nrequirement by using a NMR related implementation.
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.
A recent paper by Souza, Oliveira and Sarthour (SOS) reports the experimental violation of a Leggett–Garg (LG) inequality (sometimes referred to as a temporal Bell inequality). The inequality tests for quantum mechanical superposition: if the inequality is violated, the dynamics cannot be explained by a large class of classical theories under the heading of macrorealism. Experimental tests of the LG inequality are beset by the difficulty of carrying out the necessary so-called 'non-invasive' measurements (which for the macrorealist will extract information from a system of interest without disturbing it). SOS argue that they nevertheless achieve this difficult goal by putting the system in a maximally mixed state. The system then allegedly undergoes no perturbation during their experiment. Unfortunately, the method is ultimately unconvincing to a skeptical macrorealist and so the conclusions drawn by SOS are unjustified.
We report experimental implementation of a protocol for testing the Leggett-Garg inequality (LGI) for nuclear spins precessing in an external magnetic field. The implementation involves certain controlled operations, performed in parallel on pairs of spin-1/2 nuclei (target and probe) from molecules of a nuclear magnetic resonance ensemble, which enable evaluation of temporal correlations from an LG string. Our experiment demonstrates violation of the LGI for time intervals between successive measurements, over which the effects of relaxation on the quantum state of target spin are negligible. Further, it is observed that the temporal correlations decay, and the same target spin appears to display macrorealistic behavior consistent with LGI.