The Quantum State

Lalit A. Patel

In the existing formalism of quantum states, probability amplitudes of quantum states are complex numbers. A composition of entangled quantum states, such as a Bell state, cannot be decomposed into its constituent quantum states, implying that quantum states lose their identities when they get entangled. This is contrary to the observation that a composition of entangled quantum states decays back to its constituent quantum states. To eliminate this discrepancy, this paper introduces a new type of numbers, called virtual numbers, which produce zero upon multiplication with complex numbers. In the proposed formalism of quantum states, probability amplitudes of quantum states are general numbers made of complex and virtual numbers. A composition of entangled quantum states, such as a Bell state, can then be decomposed into its constituent quantum states, implying that quantum states retain their identities when they get entangled.

Galina Nemova

Lev Deych

Deng-Yu Zhang

The quantum pure state and mixed state are the most important quantum states in quantum information.They are the foundation of studying density operator,quantum entanglement and quantum decoherence.The two states' definition and innate differences are given in quantum theory.The observable effect for quantum pure states' mutually phase is the most important characteristics in quantum interfer- ence.The transformation from quantum pure state to mixed state is analyzed through measurement of mechanic quantity (?)_z,(?)_x and (?)_n by quantum mechanics methods.The result shows that the measurement of mechanical quantity is anisotropic in the quantum pure state but for quantum mixed state is isotropic.

Maximilian Schlosshauer

Namrata Shukla, Naeem Akhtar, Barry C. Sanders

The well-studied quantum optical Schr\"odinger cat state is a superposition of two distinguishable states, with quantum coherence between these macroscopically distinguishable states being of foundational and, in the context of quantum-information processing, practical use. We refer to these quantum-optical cat states as quantum dichotomous states, reflecting that the state is a superposition of two options, and we introduce the term quantum multichotomous state to refer to a superposition of multiple macroscopically distinguishable options. For a single degree of freedom, such as position, we construct the quantum multichotomous states as a superposition of Gaussian states on the position line in phase space. Using this nomenclature, a quantum tetrachotomous state (QTS) is a coherent superposition of four macroscopically distinguishable states. We define, analyze, and show how to create such states, and our focus on the QTSs is due to their exhibition of much richer phenomena than for the quantum dichotomous states. Our characterization of the QTS involves the Wigner function, its marginal distributions, and the photon-number distribution, and we discuss the QTS's approximate realization in a multiple-coupled-well system.