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Nonclassicality of discrete and continuous variable systems

Michael E. N. Tschaffon

Published Jan 1, 2025• 0 citations
Publisher

Abstract

Nonclassicality encompasses all of quantum physics that cannot be explained with classical theories. Fundamentally, studying nonclassicality promises to give better insight into the structure of quantum theory and what makes it unique in comparison to classical theories. Moreover, nonclassical effects can be exploited in technologies that can perform tasks that are impossible or perform worse classically. Understanding nonclassicality is thus crucial both from a foundational and an application level. Despite its importance, it is not fully understood what exactly constitutes nonclassicality and how it should be properly defined. Nonclassicality is defined differently in different contexts and thus remains a contested term. The present cumulative thesis aims to give a better understanding of the concept of nonclassicality. It is based on three publications (I)–(III) (Tschaffon et al. 2023, Tschaffon and Seiler 2024, Tschaffon and Freyberger 2025) which were written under the supervision of Prof. Matthias Freyberger at the Institut für Quantenphysik at Ulm University. Nonclassicality is the common theme of all three publications presented here and the central theme of this thesis. Since the definitions of nonclassicality are manifold, we restrict ourselves to just one precise definition. In this work, we focus on nonclassical correlations of bipartite states, and more specifically, on the violation of the Clauser-Horne-Shimony-Holt (CHSH) inequality. We will use this definition of nonclassicality in the main part of this work, Chapters 2 and 3, to study bipartite quantum states with discrete (DV) and continuous variables (CV). In Chapter 2, we discuss publications (I) and (II) that deal with nonclassicality of DV states, in particular with qubits. For this purpose, we revisit Bell’s original inequality and the CHSH inequality and, with their help, motivate average correlation as a novel indicator for nonclassicality. Average correlation is an operational quantity based on the expectation value present in many Bell inequalities, such as the CHSH inequality. Unlike Bell inequalities, it is not based on specific measurements but rather on infinitely many uniform measurements. With average correlation, we can formulate a necessary and sufficient condition for nonclassicality and even inseparability, constituting new Bell-like inequalities. In Chapter 3, we expound on publication (III) where we focus on nonclassicality for CV states. We again examine the CHSH inequality with the help of a pseudo-spin and use it to define nonclassicality. Negative values of Wigner functions are typically associated with nonclassicality. For this reason, we look at the sum of all negative values of the Wigner function, Wigner negativity, and examine how it correlates with nonclassicality for two classes of states. This helps us understand that Wigner negativity is related to nonclassicality quantitatively but varies for different classes of states. Furthermore, for each class of state, a different amount of Wigner negativity is necessary to find nonclassicality. Thus, we find that a nonzero Wigner negativity is not sufficient to observe nonclassicality.

DOI
10.18725/oparu-58874

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Nonclassicality encompasses all of quantum physics that cannot be explained with classical theories. Fundamentally, studying nonclassicality promises to give better insight into the structure of quantum theory and what makes it unique in comparison to classical theories. Moreover, nonclassical effects can be exploited in technologies that can perform tasks that are impossible or perform worse classically. Understanding nonclassicality is thus crucial both from a foundational and an application level. Despite its importance, it is not fully understood what exactly constitutes nonclassicality and how it should be properly defined. Nonclassicality is defined differently in different contexts and thus remains a contested term. The present cumulative thesis aims to give a better understanding of the concept of nonclassicality. It is based on three publications (I)–(III) (Tschaffon et al. 2023, Tschaffon and Seiler 2024, Tschaffon and Freyberger 2025) which were written under the supervision of Prof. Matthias Freyberger at the Institut für Quantenphysik at Ulm University. Nonclassicality is the common theme of all three publications presented here and the central theme of this thesis. Since the definitions of nonclassicality are manifold, we restrict ourselves to just one precise definition. In this work, we focus on nonclassical correlations of bipartite states, and more specifically, on the violation of the Clauser-Horne-Shimony-Holt (CHSH) inequality. We will use this definition of nonclassicality in the main part of this work, Chapters 2 and 3, to study bipartite quantum states with discrete (DV) and continuous variables (CV). In Chapter 2, we discuss publications (I) and (II) that deal with nonclassicality of DV states, in particular with qubits. For this purpose, we revisit Bell’s original inequality and the CHSH inequality and, with their help, motivate average correlation as a novel indicator for nonclassicality. Average correlation is an operational quantity based on the expectation value present in many Bell inequalities, such as the CHSH inequality. Unlike Bell inequalities, it is not based on specific measurements but rather on infinitely many uniform measurements. With average correlation, we can formulate a necessary and sufficient condition for nonclassicality and even inseparability, constituting new Bell-like inequalities. In Chapter 3, we expound on publication (III) where we focus on nonclassicality for CV states. We again examine the CHSH inequality with the help of a pseudo-spin and use it to define nonclassicality. Negative values of Wigner functions are typically associated with nonclassicality. For this reason, we look at the sum of all negative values of the Wigner function, Wigner negativity, and examine how it correlates with nonclassicality for two classes of states. This helps us understand that Wigner negativity is related to nonclassicality quantitatively but varies for different classes of states. Furthermore, for each class of state, a different amount of Wigner negativity is necessary to find nonclassicality. Thus, we find that a nonzero Wigner negativity is not sufficient to observe nonclassicality.
Inseparability of quantum states is one of the most crucial aspects of quantum physics, as it often provides the key ingredient for obtaining a quantum advantage. Quantifying inseparability is thus an important objective to better understand quantum physics and to improve quantum applications. So far, many measures for inseparability exist, however, most of them are based on abstract mathematical procedures and are not defined operationally which can easily be realized in an experiment. Recently we introduced average correlation [Tschaffon , ] as a reference frame independent indicator for nonclassicality in the Bell sense based on randomized measurements. It is defined as the average absolute value of the two-qubit correlation function and allows one to formulate a necessary and sufficient condition for the ability to violate the CHSH inequality. Experimentally it can be realized by randomized measurements which can approximate its value arbitrarily closely. This makes it independent of a shared reference frame between the two measuring parties, which can be useful in scenarios where it is impossible to find one. In this second article, we show that average correlation also serves as an indicator for inseparability by deriving a necessary and sufficient condition. From there, we prove the remaining open conjectures of the first article. Due to the operational definition of average correlation, it offers a first step toward finding a new operationally defined inseparability measure. Published by the American Physical Society 2024
It is one of the most remarkable features of quantum physics that measurements on spatially separated systems cannot always be described by a locally causal theory. In such a theory, the outcomes of local measurements are determined in advance solely by some unknown (or hidden) variables and the choice of local measurements. Correlations that are allowed within the framework of a locally causal theory are termed classical. Typically, the fact that quantum mechanics does not always result in classical correlations is revealed by the violation of Bell inequalities, which are constraints that have to be satisfied by any classical correlations. It has been known for a long time that entanglement is necessary to demonstrate nonclassical correlations, and hence a Bell inequality violation. However, since some entangled quantum states are known to admit explicit locally causal models, the exact role of entanglement in Bell inequality violation has remained obscure. This thesis provides both a comprehensive review on these issues as well as a report on new discoveries made to clarify the relationship between entanglement and Bell inequality violation. In particular, within the framework of a standard Bell experiment, i.e., a Bell inequality test that is directly performed on a single copy of a quantum state r, we have derived two algorithms to determine, respectively, a lower bound and an upper bound on the strength of correlations that r can offer for any given Bell inequality. Both of these algorithms make use convex optimization techniques in the form of a semidefinite program. By examples, we show that these algorithms can often be used in tandem, in conjunction with convexity arguments, to determine if a quantum state can offer nonclassical correlations and hence violates a given Bell inequality. On the other hand, since a standard Bell experiment typically involves measurements over many copies of the quantum systems, we have also investigated the possibility of enhancing the strength of nonclassical correlation by, instead, performing collective measurements on multiple copies of the quantum systems. Our findings show that even without postselection, such joint measurements may also lead to stronger nonclassical correlations, and hence a better Bell inequality violation. Meanwhile, previous studies have indicated that entangled state admitting locally causal models may still lead to observable nonclassical correlations if, prior to a standard Bell experiment, the state is subjected to some appropriate local preprocessing. This phenomenon of hidden nonlocality was discovered more than a decade ago, but to date, it is still not known if all entangled states can demonstrate nonclassical correlations through these more sophisticated Bell experiments. A key result in this thesis then consists of showing that for all bipartite entangled states, observable nonclassical correlations, in the form of a Bell-CHSH inequality violation, can indeed be derived if we allow both local preprocessing and the usage of shared ancillary state which by itself does not violate the Bell-CHSH inequality. This establishes a kind of equivalence between bipartite entanglement and states that cannot be simulated by classical correlations. In summary, for a standard Bell experiment where no local preprocessing on a quantum state r is allowed, we have provided two algorithms that can be used in tandem to determine if r can be simulated by a locally causal theory, whereas in the scenario where local preprocessing is allowed, we have demonstrated that bipartite entangled states are precisely those which cannot always be simulated classically.
Bell inequalities are an essential part of quantum information theory. Based on these fundamental inequalities, we propose a different indicator for nonclassicality in the Bell sense: average correlation. We derive a general expression for average correlation and calculate its value for pure states and Werner states. From there we show how two inequalities emerge which serve as necessary and sufficient conditions for nonclassicality. They are simple to evaluate and can even be used to classify all bipartite qubit states.
For a nonseparable bipartite quantum state violating the Clauser-Horne-Shimony-Holt (CHSH) inequality, we evaluate amounts of noise breaking the quantum character of its statistical correlations under any generalized quantum measurements of Alice and Bob. Expressed in terms of the reduced states, these new threshold bounds can be easily calculated for any concrete bipartite state. A noisy bipartite state, satisfying the extended CHSH inequality and the perfect correlation form of the original Bell inequality for any quantum observables, neither necessarily admits a local hidden variable model nor exhibits the perfect correlation of outcomes whenever the same quantum observable is measured on both ”sides”. 1
This PhD thesis contains results about two different main topics. The first part deals with the application of continuously monitored quantum systems to high precision quantum metrology. A continuous in time measurement on a quantum system is a kind indirect measurement, which only weakly perturbs the system and leaves room for it to evolve under its dynamics. This time-continuous measurement allows one to collect information about some interesting parameter characterizing the dynamics. In this thesis we show how to apply the theory of quantum parameter estimation to continuously monitored quantum systems. In particular, we study the estimation of a magnetic field applied to an ensemble of two level atoms; we show that by continuously monitoring the system we can obtain a quadratic scaling of the precision with the number of atoms, in two different physical settings (dynamically generated entanglement or initial entanglement). In the second part we study different aspects of nonclassicality of continuous variable quantum systems (bosonic degree of freedoms). They can be described by distributions (in particular, the Wigner function) on a classical phase space, which however can take negative values, the hallmark of nonclassicality. In this context, states with a Gaussian distribution are very useful and very well studied; however, on a fundamental level they must be considered classical. We present several studies connected to the vast topic of non-Gaussian states, starting from an application to parameter estimation, as a link to the first part. We study the relationships between anharmonic Hamiltonians and the nonclassicality of their ground states; we also explore the connection between a quantum effect called `backflow of probability' and the negativity of the Wigner function. We end by showing that quantum operations made out of Gaussian building blocks give rise to a well-defined resource theory of Wigner negativity and non-Gaussianity.
Bellʼs inequality fundamentally changed our understanding of quantum mechanics. Bellʼs insight that non-local correlations between quantum systems cannot be explained classically can be verified experimentally, and has numerous applications in modern quantum information. Today, the Clauser–Horne–Shimony–Holt (CHSH) inequality is probably the most well-known Bell inequality and it has given us a wealth of understanding in what differentiates the classical from the quantum world. Yet, there are certainly other means of quantifying 'Bell non-locality without inequalities' such as the famous Hardyʼs paradox. As such, one may wonder whether these are entirely different approaches to non-locality. For this anniversary issue, we unify the perspective of the CHSH inequality and Hardy's paradox into one family of non-local games which include both as special cases. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to '50 years of Bell's theorem'.