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Journal of Hengyang Normal University

Pure State and Mixed State in Quantum Information

Deng-Yu Zhang

Published Jan 1, 2003• 0 citations
OpenAlex

Abstract

The quantum pure slates and mixed states are the most important quantum states in quantum information. The two states define and differences of the two states density operator are given in quantum theory. Quantum entanglement state and decoherence are discussed. Differences and connection in the two stales are analyzed.

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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.
Abstract In this chapter we discuss statistical mixtures of pure quantum states, resulting in so-called mixed states, and their representation using density operators. The properties of density operators for pure and mixed states are analysed and we examine their time evolution governed by the Liouville-von Neumann equation. As particular examples of interest we consider the density operators for thermal states, focusing on harmonic oscillators and the application of this model to calculate the average photon number for the modes of the electromagnetic field at thermal equilibrium. We then turn to two-level systems, or qubits, which are of particular relevance in quantum information theory and we inspect the geometry of the state space of density operators as a convex subspace of the Hilbert-Schmidt space. Finally, we take a closer look at density operators for bipartite systems which forms the mathematical basis for considering entanglement in later chapters.
In this chapter, the concepts of states and ensembles are critically reviewed. Among other things, it is shown that if the state of every composite quantum system contains all information that can be known about a system, states cannot be pure states. In Section 15.1, we criticize the standard practice of treating the state vector (that is, a pure state) as the complete description of a (single, or frequently prepared) quantum system. Section 15.2 shows that the density operator contains objective information about single quantum systems. This conclusion is reinforced in Section 15.3, which discusses Gibbs’ in his time revolutionary notion of fictitious ensembles for representing single macroscopic systems in equilibrium. The final Section 15.4 shows that pure states for objects in a bounded region of space are also questionable from the perspective of relativistic quantum field theory.