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
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
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.
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.
Bell nonlocality and quantum steering are recognized as two important physical resources and offer quantum advantages in modern quantum information processing. Inspired by the asymmetric relations of teleportation to Bell nonlocality and quantum steering, in this paper we unify the maximal violation of Bell inequality and the maximal violation of the three-setting linear steering inequality by introducing the concept of average correlation and manage to establish the relation of average correlation to quantum teleportation. Particularly, we propose a necessary and sufficient condition for any two-qubit state with nonclassicality in terms of average correlation for teleportation. Our results suggest that average correlation is not only used to detect the nonclassicality of quantum states, but also can be useful as a potential resource for teleportation.
Quantum nonlocality and nonclassicality are two remarkable characteristics of quantum theory, and offer quantum advantages in some quantum information processing. Motivated by recent work on the interplay between nonclassicality quantified by average correlation [Tschaffon et al., Phys. Rev. Res. 5,023063 (2023)] and Bell nonlocality, in this paper we aim to establish the relationship between the average correlation and the violation of the three-setting linear steering inequality for two-qubit systems. Exact lower and upper bounds of average correlation versus steering are obtained, and the respective states which suffice those bounds are also characterized. For clarity of our presentation, we illustrate these results with examples from well-known classes of two-qubit states. Moreover, the dynamical behavior of these two quantifiers is carefully analyzed under the influence of local unital and nonunital noisy channels. The results suggest that average correlation is closely related to the violation of three-setting linear steering, like its relationship with Bell violation. Particularly, for a given class of states, the hierarchy of nonclassicality-steering-Bell nonlocality is demonstrated.
Abstract Nonclassicality is seen as an important physical resource and offers quantum advantages in modern quantum information processing. Motivated by recent work on the interplay between Bell nonlocality and nonclassicality quantified by average correlation [ Phys. Rev. Res. 5 023063 (2023)], in this paper we aim to establish the intrinsic relationship between the average correlation and the violation of the three-setting linear steering inequality for arbitrary two-qubit states. The results suggest that average correlation is closely related to the violation of the three-setting linear steering inequality, like its relationship with Bell inequality violation. Moreover, the dynamical behaviors of average correlation and steering are carefully analyzed under the influence of local unital and nonunital noisy channels. Particularly, for any two-qubit states, the hierarchy of average correlation-steering-Bell nonlocality is demonstrated by using the singular value decomposition theorem.
This paper studies the quantification and structural properties of quantum average correlation based on average coherence. Motivated by two mathematically equivalent approaches to define average coherence: one by averaging over complete sets of mutually unbiased bases, and the other by integrating over all orthogonal bases under the Haar measure, we define an average correlation for bipartite systems as the difference between global and local skew information. This correlation measure is shown to satisfy essential properties including non negativity, contractivity under local quantum channels, and local unitary invariance. We further prove the equivalence between the average correlation defined via mutually unbiased bases and that defined via unitary groups. Finally, we derive a complementarity relation that connects wave-particle duality with the average correlation between a system and its environment.