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A density matrix is a matrix that describes a quantum system in a ''mixed state'', a statistical ensemble of several quantum states. This should be contrasted with a single state vector that describes a quantum system in a ''pure state''. The density matrix is the quantum-mechanical analogue to a phase-space probability measure (probability distribution of position and momentum) in classical statistical mechanics. Explicitly, suppose a quantum system may be found in state with probability ''p''1, or it may be found in state with probability ''p''2, or it may be found in state with probability ''p''3, and so on. The density operator for this system is : where need not be orthogonal and . By choosing an orthonormal basis , one may resolve the density operator into the density matrix, whose elements are〔 : The density operator can also be defined in terms of the density matrix, : For an operator (which describes an observable of the system), the expectation value is given by〔 : In words, the expectation value of ''A'' for the mixed state is the sum of the expectation values of ''A'' for each of the pure states weighted by the probabilities ''pi'' and can be computed as the trace of the product of the density matrix with the matrix representation of in the same basis. Mixed states arise in situations where the experimenter does not know which particular states are being manipulated. Examples include a system in thermal equilibrium (or additionally chemical equilibrium) or a system with an uncertain or randomly varying preparation history (so one does not know which pure state the system is in). Also, if a quantum system has two or more subsystems that are entangled, then each subsystem must be treated as a mixed state even if the complete system is in a pure state. The density matrix is also a crucial tool in quantum decoherence theory. The density matrix is a representation of a linear operator called the ''density operator''. The close relationship between matrices and operators is a basic concept in linear algebra. In practice, the terms ''density matrix'' and ''density operator'' are often used interchangeably. Both matrix and operator are self-adjoint (or Hermitian), positive semi-definite, of trace one, and may be infinite-dimensional. The formalism was introduced by John von Neumann in 1927 and independently, but less systematically by Lev Landau and Felix Bloch in 1927 and 1946 respectively. ==Pure and mixed states== In quantum mechanics, a quantum system is represented by a state vector (or ket) . A quantum system with a state vector is called a ''pure state''. However, it is also possible for a system to be in a statistical ensemble of different state vectors: For example, there may be a 50% probability that the state vector is and a 50% chance that the state vector is . This system would be in a ''mixed state''. The density matrix is especially useful for mixed states, because any state, pure or mixed, can be characterized by a single density matrix. A mixed state is different from a quantum superposition. In fact, a quantum superposition of pure states is another pure state, for example . A state is pure if and only if its density matrix satisfies . 抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)』 ■ウィキペディアで「density matrix」の詳細全文を読む スポンサード リンク
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