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arXiv:2105.08273 · Physical Review Research

Hidden nonmacrorealism: Reviving the Leggett-Garg inequality with stochastic operations

Huan-Yu Ku, Hao‐Cheng Weng, Yen-An Shih, Po-Chen Kuo, Neill Lambert, Franco Nori, Chih‐Sung Chuu, Yueh-Nan Chen

Published Oct 29, 2021• 7 citations
arXivHTMLPDF

Abstract

The Leggett-Garg inequality (LGI) distinguishes nonmacrorealistic channels from macrorealistic ones by constraining the experimental outcomes of the underlying system. In this work, we propose a class of channels which, initially, cannot violate the LGI (in the form of the temporal Bell inequality) but can violate it after the application of stochastic pre- and post-operations (SPPOs). As a proof-of-principle experiment, we demonstrate the stochastic pre- and post-operations in an amplitude-damping channel with photonic qubits. We denote the above phenomenon as hidden nonmacrorealistic channels. We also discuss the relationship between these hidden nonmacrorealistic channels [in terms of the temporal Clauser-Horne-Shimony-Holt (CHSH) inequality] and the strongly nonlocality-breaking channel, which breaks the hidden spatial CHSH nonlocality for arbitrary states. In general, if the channel satisfies hidden nonmacrorealism, it is not a strongly CHSH nonlocality-breaking channel.

DOI
10.1103/physrevresearch.3.043083

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The Leggett-Garg inequality (LGI) distinguishes nonmacrorealistic channels from macrorealistic ones by constraining the experimental outcomes of the underlying system. In this work, we propose a class of channels which, initially, cannot violate the LGI (in the form of the temporal Bell inequality) but can violate it after the application of stochastic pre- and post-operations (SPPOs). As a proof-of-principle experiment, we demonstrate the stochastic pre- and post-operations in an amplitude-damping channel with photonic qubits. We denote the above phenomenon as hidden nonmacrorealistic channels. We also discuss the relationship between these hidden nonmacrorealistic channels [in terms of the temporal Clauser-Horne-Shimony-Holt (CHSH) inequality] and the strongly nonlocality-breaking channel, which breaks the hidden spatial CHSH nonlocality for arbitrary states. In general, if the channel satisfies hidden nonmacrorealism, it is not a strongly CHSH nonlocality-breaking channel.
Nonclassicality of temporal correlations pertaining to noncommutative sequential measurements is defined through the violation of macrorealistic inequalities, known as Leggett-Garg inequalities (LGI). We investigate the energy cost of the process associated with the Leggett-Garg test in the context of noiseless and Markovian noise for arbitrary initial states. We prove that in noiseless and in certain noisy scenarios, the maximal violations of LGI under the energy constraint occurs when the average energy of the process is equal to the negative of the energy of the initial state. Such a dependence of LGI on the choice of the initial state is not seen in the unconstrained case. Moreover, we find that in the presence of a moderate amount of Markovian noise, the amount of violation of LGI remains almost unaltered with a suitable choice of the evolution and dephasing operators in the neighborhood of the maximal violation line, thereby showing the robustness of temporal correlations under environmental effects.
Abstract In two-party, two-input and two-output measurement scenario only relevant Bell’s inequality is the Clauser–Horne–Shimony–Holt (CHSH) form. They also provide the necessary and sufficient conditions (NSCs) for local realism. Any other form, such as, Clauser–Horne and Wigner forms reduce to the CHSH one. The standard Leggett–Garg inequalities, proposed for testing incompatibility between macrorealism and quantum theory, are often considered to be the temporal analog of CHSH inequalities. However, they do not provide the NSCs for macrorealism. There is thus scope of formulating new macrorealist inequalities inequivalent and stronger than the standard Leggett–Garg inequalities. In this paper, we propose two different classes of macrorealistic inequalities. A class of inequalities which are equivalent to the standard ones in macrorealist model but inequivalent and stronger in quantum theory, and the other class of inequalities are inequivalent to the all the other formulations of Leggett–Garg inequalities both in macrorealist model and in quantum theory. The latter class of macrorealist inequalities reveals the incompatibility between macrorealism and quantum theory for specific cases even when any other formulation of Leggett–Garg inequalities fails to do so. We extend the formulations of inequivalent Leggett–Garg inequalities to the four-time and two-time measurement scenarios. Further, we provide a brief discussion about the alternative formulation of macrorealism known as the no-signaling in time (NSIT) conditions.
Quantum theory contravenes classical macrorealism by allowing a system to be in a superposition of two or more physically distinct states, producing physical consequences radically different from that of classical physics. We show that a system, upon subjecting to transform under superposition of unitary operators, exhibits enhanced non-macrorealistic feature - as quantified by violation of the Leggett-Garg inequality (LGI) beyond the temporal Tsirelson bound. Moreover, this superposition of unitaries also provides robustness against decoherence by allowing the system to violate LGI and thereby retain its non-macrorealistic behavior for a strikingly longer duration. Using an NMR register, we experimentally demonstrate the superposition of unitaries with the help of an ancillary qubit and verify these theoretical predictions.
A generalisation of the Leggett-Garg inequality (LGI) and the Bell-CHSH inequality is proposed in this work, by replacing a classical probability notion with an extended probability notion. The extended probability serves as an underlying layer of reality to the classical probability. Within the consistent history framework, the underlying layer is so-called a non-settleable history. Hence, the bounds of the LGIs and Bell-CHSH inequalities are extended. The resulting generalised inequalities are satisfied without violation for arbitrary measurement settings. Furthermore, generalised Macro- realism and generalised Local realism are introduced and analysed within the extended probability framework without relying on an explicit measurement. This opens up further possibilities on one of the great pursuits in physics, understanding on how a quantum system can possess classical properties.
In two-party, two-input and two-output measurement scenario only relevant Bell's inequality is the Clauser-Horne-Shimony-Holt (CHSH) form. They also provide the necessary and sufficient conditions for local realism. Any other form, such as, Clauser-Horne and Wigner forms reduce to the CHSH one. Standard Leggett-Garg inequalities are often considered to be the temporal analog of CHSH inequalities. But, they do not provide the necessary and sufficient conditions for macrorealism. There is thus scope of formulating new macrorealist inequalities different and stronger than the standard Leggett-Garg inequalities for testing compatibility between the macrorealism and quantum theory. In this paper, we propose three different classes of macrorealistic inequalities in three-time scenario; (i) The ones equivalent to standard Leggett-Garg inequalities in both macrorealist model and in quantum theory. (ii) A class of inequalities which are equivalent to the standard ones in macrorealist model but inequivalent and stronger in quantum theory (iii) Another class of inequalities which are inquivalent to the all formulations of Leggett-Garg inequalities both in macrorealist model and in quantum theory. This class of macrorealist inequalities reveals the incompatibility between macrorealism and quantum theory for specific cases even when any other formulation of Leggett-Garg inequalities fail to do so. We extend the formulations of the class (ii) inequalities to the four-time and two-time measurement scenario. Further, we provide a brief discussion about the alternate formulation of macrorealism which was derived based on the no-signaling in time conditions.
The temporal Clauser-Horne-Shimony-Holt (CHSH) inequality witnesses the nonclassicality of temporal correlations, but its violation is generally degraded by environmental noise. Here, we show that violation of the temporal CHSH inequality can be revived through coherent control of noisy quantum evolutions. We compare two physically distinct implementations: coherent control of noisy evolutions induced by interaction of the system with independent environments, and coherent control of two physically distinct, unitarily equivalent Stinespring dilations of the same noisy channel. Although these constructions generate identical deterministic system dynamics, they induce distinctly different post-selected evolutions. With a focus on the amplitude damping channel (ADC), we show that coherent control of equivalent dilations extend the range of temporal CHSH inequality violation well beyond both the incoherently controlled, or deterministic, scenario and what is achievable with independent environments. Under setting-independent post-selection of the coherent control implementation, the resulting violation further certifies that the channel is not strongly CHSH nonlocality-breaking. Our results identify the choice of Stinespring dilation as an operationally relevant resource in coherently controlled tests of temporal quantum correlations.