Superposition

Hai‐Bo Xing

The resource theory of superposition is a generalization of coherence theory. A rigorous framework for quantifying superposition of linear independent states has been given by previous studies. In this paper, the optimal superposition enhancement of a mixed state is studied. Based on the $l_{1}$ measure of superposition the optimal superposition enhancement value of a mixed state can be obtained under the stochastic diagonal superposition-free operations(or incomplete diagonal superposition-free operations). In addition, the optimal probability corresponding to the optimal superposition enhancement value of a mixed state is also discussed. The results show that although the superposition-free operations can decrease the superposition of a state, the superposition enhancement of a state is possible, a single mixed state can be probabilistically transformed into another state with higher superposition than before by superposition-free operations. These studies will give us a better understanding of the superposition manipulation and the resource theory of superposition.

Nigel Ward

Shuang-Ren Zhao

In electromagnetic theory and quantum theory, there are superposition principle. Traditionally there are only one kind superposition principle. However, in this author's theory the retarded wave and advanced wave are all involved, the superposition become multiple. For example this author need to deal the problem how to superpose the retarded wave and the advanced wave. This author found that there are 5 different kinds of superpositions. The superposition principles have some differences. The research about these differences is a key to open the door of many physical difficulties. For example the particle and wave duality problem, and to judge which interpretation of the quantum mechanics is a correct one. The first two superposition principles are the superpositions with and without the test charges. The slight different superposition principles are the superposition with a retarded wave alone and the superposition with the advanced wave alone. According to theory of this author, the emitter sends the retarded wave, the absorber sends the advance wave. Hence, normal electromagnetic field actually is consist of retarded wave and advanced wave. This two waves together become the traditional electromagnetic fields. This kind of electromagnetic fields can be seen approximately as retarded wave only, this kind wave also has its own superposition. This kind of superposition is also different with the superposition when we consider the retarded waves alone and also the advanced waves alone. In this article this author will discuss the differences of these different superpositions. This author will also discuss the different physical result with a few different superposition principles. In this article this author will prove the superposition with test charge and the superposition with retarded waves alone or advance wave alone can be see one kind of superposition. This superposition is correct and can be derived from the mutual energy principle. This will be referred as first kind of superposition. The superposition without test charge and the superposition with the retarded waves and advanced waves can be seen as one kind of superposition. For this kind of superpositions are not correct naturally. This kind of super position are applied to N-charge's Poynting theorem. In order to make this superposition work, the self-energy principle have to be applied. Without self-energy principle this kind of superpositions will violate the energy conservation law. Considering the self-energy conservation this kind of superposition become the superposition of first kind. This is referred as second kind of superposition. The third kind of superposition is for the traditional electromagnetic field. Here the traditional electromagnetic field means the electromagnetic field where the advanced waves are omit. When the advanced waves are omit doesn't mean the advanced waves do not exist, that is only means the retarded wave and the advanced wave are with nearly same intensity and can be treated together in some situation, for example, in wave guide, or in the free space where the absorber are uniformly distributed at infinite big sphere. Abstract Only when the self-energy principle is accept, all kinds of superposition can be accept. However different superposition has different physical meanings. Otherwise only the superposition with test charge or the superposition with only one kind wave either retarded waves or advanced waves can be accepted. Hence, the discussion about the superposition also support the concept of the self-energy principle which means there must exist the time reversal waves. That also means the waves do not collapse but collapse back. Wave collapse means collapse to its target, for example, the retarded wave will collapse to an absorber and the advanced wave will collapse to an emitter. Wave collapse back means the retarded wave sent from emitter will collapse back to an emitter; The advanced wave sent from the absorber will collapse back to an absorber. Hence, one purpose of this article is to clarify the superposition principles, and another purpose is to support this author's electromagnetic field theory which is started from two new axioms the self-energy principle and the mutual energy principle.

Xiaotong Wang, Shunlong Luo, Yue Zhang

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.

Ray LaPierre