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Density operator : ウィキペディア英語版
Density matrix

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 | \psi_1 \rangle with probability ''p''1, or it may be found in state | \psi_2 \rangle with probability ''p''2, or it may be found in state | \psi_3 \rangle with probability ''p''3, and so on. The density operator for this system is
: \hat\rho = \sum_i p_i |\psi_i \rangle \langle \psi_i|,
where \ need not be orthogonal and \sum_i p_i=1. By choosing an orthonormal basis \, one may resolve the density operator into the density matrix, whose elements are〔
: \rho_ = \sum_i p_i \langle u_m | \psi_i \rangle \langle \psi_i | u_n \rangle
= \langle u_m |\hat \rho | u_n \rangle.
The density operator can also be defined in terms of the density matrix,
: \hat\rho = \sum_ |u_m\rangle \rho_ \langle u_n| .
For an operator \hat A (which describes an observable A of the system), the expectation value \langle A \rangle is given by〔
: \langle A \rangle = \sum_i p_i \langle \psi_i | \hat | \psi_i \rangle
= \sum_ \langle u_m | \hat\rho | u_n \rangle \langle u_n | \hat | u_m \rangle
= \sum_ \rho_ A_
= \operatorname(\rho A).
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 |\psi_i\rangle weighted by the probabilities ''pi'' and can be computed as the trace of the product of the density matrix with the matrix representation of A 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) | \psi \rangle . A quantum system with a state vector | \psi \rangle 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 | \psi_1 \rangle and a 50% chance that the state vector is | \psi_2 \rangle . 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 | \psi \rangle = (| \psi_1 \rangle + | \psi_2 \rangle)/\sqrt .
A state is pure if and only if its density matrix \rho satisfies tr(\rho^2)=1 .

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
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