In a recent work [S. Asthana. New Journal of Physics 24.5 (2022): 053026], we have shown the interrelation of different nonclassical correlations in multiqubit systems with quantum coherence in a single logical qubit. In this work, we generalize it to higher-dimensional systems. For this, we take different choices of logical qudits and logical continuous-variable (cv) systems in terms of their constituent physical qudits and physical cv systems. Thereafter, we show reciprocity between conditions for coherence (in logical qudits and logical cv systems) and conditions for nonlocality and entanglement (in their underlying constituent qudits and cv systems). This shows that a single nonclassicality condition detects different types of nonclassicalities in different physical systems. Thereby, it reflects the interrelations of different nonclassical features of states belonging to Hilbert spaces of nonidentical dimensions.
In a recent work [S. Asthana. New Journal of Physics 24.5 (2022): 053026], we have shown the interrelation of different nonclassical correlations in multiqubit systems with quantum coherence in a single logical qubit. In this work, we generalize it to higher-dimensional systems. For this, we take different choices of logical qudits and logical continuous-variable (cv) systems in terms of their constituent physical qudits and physical cv systems. Thereafter, we show reciprocity between conditions for coherence (in logical qudits and logical cv systems) and conditions for nonlocality and entanglement (in their underlying constituent qudits and cv systems). This shows that a single nonclassicality condition detects different types of nonclassicalities in different physical systems. Thereby, it reflects the interrelations of different nonclassical features of states belonging to Hilbert spaces of nonidentical dimensions.
Scalable modern-time fault-tolerant quantum computation and quantum communication in a network employ a large number of physical qubits. For example, IBM is reported to have made a 127-qubit quantum computer. Unlike classical computation, quantum computation employs different types of logical qubits and qudits in terms of physical multiqubit and multiqudit systems respectively. Given this, of particular interest to us is to enquire on how quantum coherence in logical qubits is a manifestation of underlying quantum correlations in constituent physical multiqubit systems and vice-versa. In a recent work [Asthana, Sooryansh. New J Phys 24.5 (2022): 053026], we have shown that there is reciprocity in nonclassical correlations in physical multiqubit systems and coherence in a single logical qubit system. Subsequently, we have generalised the framework to higher dimensional quantum systems []. The crux of this study is that a single nonclassicality condition derived for quantum coherence in a logical system detects more than one type of nonclassicality in Hilbert spaces of nonidentical dimensions.
A unified characterization of continuous-variable quantum states is presented that gives general insight in their nonclassical features and, in particular, in their entanglement properties. Nonclassicality is characterized in terms of observable characteristic functions. Equivalently, the nonclassicality conditions are reformulated in terms of moments. The latter approach turns out to be a powerful tool for deriving a complete set of entanglement conditions for bipartite quantum systems, which are fully equivalent to the nonpositivity of the partially transposed density operator. The possibilities of observing the needed moments by balanced homodyne correlation measurements are analyzed.
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.
Universal quantum computers require fault-tolerant logical qudits, as qudits naturally align with the simulation of multi-level physical systems. Here, we present a general framework and working examples for encoding fault-tolerant logical qudits in finite-dimensional spin systems. We construct distance-3, distance-5 codewords, and general 2t+1-distance codes that can be implemented using a single physical qudit or a small number of coupled qudits for higher distances, while requiring a Hilbert space dimension significantly smaller than conventional constructions based on multiple logical qubits. Logical operations and error correction protocols can be implemented with polynomial scaling in the number of elementary operations. We further discuss schematic designs for physical implementation and required single-gate fidelities, which are compatible with current spin qudit platforms. This strategy provides a resource-efficient path toward realizing fault-tolerant logical qudits in finite multi-level physical systems.
Recent years have witnessed a rapidly growing interest in high-dimensional quantum entanglement for fundamental studies as well as towards novel applications. Therefore, the ability to verify entanglement between physical qudits, d-dimensional quantum systems, is of crucial importance. To show nonclassicality, Hardy's paradox represents ``the best version of Bell's theorem'' without using inequalities. However, so far it has only been tested experimentally for bidimensional vector spaces. Here, we formulate a theoretical framework to demonstrate the ladder proof of Hardy's paradox for arbitrary high-dimensional systems. Furthermore, we experimentally demonstrate the ladder proof by taking advantage of the orbital angular momentum of high-dimensionally entangled photon pairs. We perform the ladder proof of Hardy's paradox for dimensions 3 and 4, both with the ladder up to the third step. Our paper paves the way towards a deeper understanding of the nature of high-dimensionally entangled quantum states and may find applications in quantum information science.
We derive and implement a general method to characterize the nonclassicality in compound discrete- and continuous-variable systems. For this purpose, we introduce the operational notion of conditional hybrid nonclassicality which relates to the ability to produce a nonclassical continuous-variable state by projecting onto a general superposition of discrete-variable subsystem. We discuss the importance of this form of quantumness in connection with interfaces for quantum communication. To verify the conditional hybrid nonclassicality, a matrix version of a nonclassicality quasiprobability is derived and its sampling approach is formulated. We experimentally generate an entangled, hybrid Schrödinger cat state, using a coherent photon-addition process acting on two temporal modes, and we directly sample its nonclassicality quasiprobability matrix. The introduced conditional quantum effects are certified with high statistical significance.