Although the principle of superposition lies at the heart of quantum mechanics and is the root of almost all quantum phenomena such as coherence and entanglement, its quantification, except for that related to the resource theory of coherence and interference, remains relatively less studied. In this work, we address quantification of superposition from an information-theoretic perspective. We introduce a family of quantifiers of superposition, the phase-sensitive superposition, by taking into account the phases of amplitudes in the superposition of a fixed basis states (e.g., computational basis states). We establish a conservation relation for the phase-sensitive superposition, which is a kind of complementary relation and is reminiscent of wave-particle duality. We evaluate explicitly the second moment of phase-sensitive superposition and show that it is intrinsically related to the $l^2$-norm coherence. We characterize the dephasing channel induced by the maximally superposed states. We investigate the minimum and maximum superpositions, reveal their basic properties, and illustrate them through various examples. We further explore the dynamics of superposition in the Grover search algorithm, and demonstrate a complementary relation between superposition and success probability of the search algorithm. These results and quantifiers offer tools for analyzing structural features and implications of quantum superposition.
Although the principle of superposition lies at the heart of quantum mechanics and is the root of almost all quantum phenomena such as coherence and entanglement, its quantification, except for that related to the resource theory of coherence and interference, remains relatively less studied. In this work, we address quantification of superposition from an information-theoretic perspective. We introduce a family of quantifiers of superposition, the phase-sensitive superposition, by taking into account the phases of amplitudes in the superposition of a fixed basis states (e.g., computational basis states). We establish a conservation relation for the phase-sensitive superposition, which is a kind of complementary relation and is reminiscent of wave-particle duality. We evaluate explicitly the second moment of phase-sensitive superposition and show that it is intrinsically related to the l2-norm coherence. We characterize the dephasing channel induced by the maximally superposed states. We investigate the minimum and maximum superpositions, reveal their basic properties, and illustrate them through various examples. We further explore the dynamics of superposition in the Grover search algorithm, and demonstrate a complementary relation between superposition and success probability of the search algorithm. These results and quantifiers offer tools for analyzing structural features and implications of quantum superposition.
Abstract The principle of superposition is universal and lies at the heart of quantum theory. Although ever since the inception of quantum mechanics a century ago, superposition has occupied a central and pivotal place, rigorous and systematic studies of the quantification issue have attracted significant interests only in recent years, and many related problems remain to be investigated. In this work we introduce a figure of merit which quantifies superposition from an intuitive and direct perspective, investigate its fundamental properties, connect it to some coherence measures, illustrate it through several examples, and apply it to analyze wave-particle duality.
Quantum coherence quantifies the amount of superposition in a quantum system, and is the reason and resource behind several phenomena and technologies. It depends on the natural basis in which the quantum state of the system is expressed, which in turn hinges on the physical set-up being analyzed and utilized. While quantum coherence has hitherto been conceptualized by employing different categories of complete bases, there do exist interesting physical situations, where the natural basis is an incomplete one, an example being an interferometric set-up with the observer controlling only a certain fraction of all the slits. We introduce a quantification of quantum coherence with respect to an arbitrary incomplete basis for general quantum states, and develop the corresponding resource theory, identifying the free states and operations. Moreover, we obtain a complementarity relation between the so-defined quantum coherence and the which-path information in an interferometric set-up with several slits, of which only a section is in control of the observer or is accessible to her. This therefore provides us with another face of the wave-particle duality in quantum systems, demonstrating that the complementarity is functional in more general set-ups than thus far considered.
Quantum coherence, present whenever a quantum system exists in a superposition of multiple classically distinct states, marks one of the fundamental departures from classical physics.Quantum coherence has recently been investigated rigorously within a resource-theoretic formalism.However, the finer-grained notion of multilevel coherence, which explicitly takes into account the number of superposed classical states, has remained relatively unexplored.A comprehensive analysis of multilevel coherence, which acts as the singleparty analogue to multipartite entanglement, is essential for understanding natural quantum processes as well as for gauging the performance of quantum technologies.Here, we develop the theoretical and experimental groundwork for characterizing and quantifying multilevel coherence.We prove that nontrivial levels of purity are required for multilevel coherence, as there is a ball of states around the maximally mixed state that do not exhibit multilevel coherence in any basis.We provide a simple, necessary, and sufficient analytical criterion to verify the presence of multilevel coherence, which leads to a complete classification of multilevel coherence for three-level systems.We present the robustness of multilevel coherence, a bona fide quantifier, which we show to be numerically computable via semidefinite programming and experimentally accessible via multilevel coherence witnesses, which we introduce and characterize.We further verify and lower bound the robustness of multilevel coherence by performing a semi-device-independent phase discrimination task, which is implemented experimentally with four-level quantum probes in a photonic setup.Our results contribute to understanding the operational relevance of genuine multilevel coherence, also by demonstrating the key role it plays in enhanced phase discrimination-a primitive for quantum communication and metrology-and suggest new ways to reliably and effectively test the quantum behavior of physical systems.
In this section we provide a broad overview of the basic principles of quantum mechanics and how they enable modern quantum technologies. The starting point is the concept of superposition, which allows quantum systems to be in multiple states at the same time and forms the basis of quantum computation using qubits and parallel processing of information. The conversation then turns to quantum entanglement, a non-classical effect that creates strong correlations between particles, and has applications in secure communication and quantum teleportation. Also mentioned is the quantum coherence, which preserves phase relations between quantum states and makes it possible for quantum systems to function correctly. But it is also emphasized that quantum coherence is fragile to environmental perturbations. This chapter also covers quantum measurement, taking into account the probabilistic nature of quantum mechanics, and how the act of observation results in the collapse of a quantum state to a definite outcome. These principles together constitute the theoretical basis of quantum computing, where quantum gates and algorithms and qubits take advantage of superposition, entanglement and interference to offer computational benefits over classical systems. The most important algorithms like Shor’s and Grover’s algorithms that demonstrate a considerable speed-up in solving the problem are discussed along with different models of quantum computing, such as gate, adiabatic and measurement-based models. Furthermore, the section reviews some physical realizations of quantum systems, such as superconducting qubits, trapped ions, photonic systems and topological qubits, with their advantages and difficulties. Overall, this discussion shows how the basic principles of quantum mechanics are the foundation for the development of advanced quantum technologies, while also dealing with the practical difficulties that arise in their implementation.
While Bohr's complementarity principle constitutes a bedrock of quantum mechanics with profound implications, coherence, as a defining feature of the quantum realm originating from the superposition principle, pervades almost every quantum consideration. By exploiting the algebraic and geometric structure of state-channel interaction, we show that an information-theoretic measure of coherence and a quantitative symmetry-asymmetry complementarity emerge naturally from the formalism of quantum mechanics. This is achieved by decomposing the state-channel interaction into a symmetric part and an asymmetric part, which satisfy a conservation relation. The symmetric part is represented by the symmetric Jordan product, and the asymmetric part is synthesized by the skew-symmetric Lie product. The latter further leads to a significant extension of the celebrated Wigner--Yanase skew information, and has an operational interpretation as quantum coherence of a state with respect to a channel. This not only presents a basic and alternative framework for addressing complementarity, but also puts the study of coherence in a broad context involving channels. Fundamental properties of the symmetry-asymmetry complementarity are revealed, and applications and implications are illustrated via several prototypical channels as well as the Mach--Zehnder interferometry, in which the fringe visibility is linked to symmetry and the which-path is linked to asymmetry.
The superposition principle lies at the heart of many nonclassical properties of quantum mechanics. Motivated by this, we introduce a rigorous resource theory framework for the quantification of superposition of a finite number of linear independent states. This theory is a generalization of resource theories of coherence. We determine the general structure of operations which do not create superposition, find a fundamental connection to unambiguous state discrimination, and propose several quantitative superposition measures. Using this theory, we show that trace decreasing operations can be completed for free which, when specialized to the theory of coherence, resolves an outstanding open question and is used to address the free probabilistic transformation between pure states. Finally, we prove that linearly independent superposition is a necessary and sufficient condition for the faithful creation of entanglement in discrete settings, establishing a strong structural connection between our theory of superposition and entanglement theory.