Information AboutDensity Matrix |
| CATEGORIES ABOUT DENSITY MATRIX | |
| quantum mechanics | |
| functional analysis | |
| quantum information science | |
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A density Matrix is a Self-adjoint (or Hermitian ) Positive-semidefinite Matrix , (possibly infinite dimensional), of Trace one, that describes the statistical state of a Quantum System . The formalism was introduced by John Von Neumann (according to other sources independently by Lev Landau and Felix Bloch ) in 1927. It is the quantum-mechanical analogue to a Phase-space Probability Measure (probability distribution of position and momentum) in classical statistical mechanics. The need for a statistical description via density matrices arises when one considers either an ''ensemble'' of systems, or one system when its preparation history is uncertain. Situations in which a density matrix is used include the following: a quantum system in thermal equilibrium (at finite temperatures); nonequilibrium time-evolution that starts out of a mixed equilibrium state; and Entanglement between two subsystems, where each individual system must be described, via the Partial Trace operation, by a density matrix even though the complete system may be in a pure state; and in analysis of Quantum Decoherence . See also Quantum Statistical Mechanics . A density operator is an Operator corresponding to a density matrix under some Orthonormal Basis . Thus it is a Non-negative , Self-adjoint , Trace Class operator of trace one. THE NEED FOR A STATISTICAL DESCRIPTION In Quantum Mechanics , the State Vector ψWe assume ψ is a pure state in this example. of a system completely determines the statistical behavior of an Observable O. This means that if O is represented by an operator ''A'' on the Hilbert Space ''H'' of the system, then for any real-valued function ''F''Technically, ''F'' must be a Borel function defined on the real numbers, the expectation value of ''F''(O) ''F''(O) is defined to be the result of measuring O and then applying ''F'' to the outcome. is the quantity : or written as : in Dirac Notation . Now consider the example of a "mixed quantum system" prepared by statistically combining two different pure states φ, ψ each with probability 1/2. The preparation process for such a system consists in tossing an unbiased coin and using the preparation process for φ or for ψ depending on whether the toss outcome is heads or tails. It is not hard to show that the statistical properties of the observable O for the system prepared in such a mixed state are completely determined. However, there is no vector ξ which determines this statistical behavior in the sense that the expectation value of ''F''(O) is : Nevertheless: there is a unique operator ρ such that the expectation value can be written as : The operator ρ is the density operator of the mixed system. A simple calculation shows that for the example mentioned above: : FORMULATION For a finite dimensional Hilbert space, the most general density matrix is of the form | ||
| Where The Coefficients ''p''<sub>''j''</sub> Are Non-negative And Add Up To One This Represents A Statistical Mixture Of Pure States If The Given System Is Closed, Then One Can Think Of A Mixed State As Representing A Single System With An Uncertain Preparation History, As Explicitly Detailed Above ''or'' We Can Regard The Mixed State As Representing An | "http://wwwinformationdelightinfo/information/entry/Statistical_ensemble_(mathematical_physics)" class="copylinks">Ensemble of systems, ie large number of copies of the system in question, where ''p''<sub>''j''</sub> is the proportion of the ensemble being in the state <math>\psi_j
ang </math> An ensemble is described by a pure state if every copy of the system in that ensemble is in the same state, ie it is a ''pure ensemble'' |
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| '''Example''' Consider A Quantum Ensemble Of Size ''N'' With Occupancy Numbers ''n''<sub>1</sub>, ''n''<sub>2</sub>,,''n<sub>k</sub>'' Corresponding To The Orthonormal States 1>,,''k''>, Respectively, Where ''n''<sub>1</sub>++''n<sub>k</sub>'' | ''N'', and, thus, the coefficients ''p<sub>j</sub>'' = ''n<sub>j</sub>'' /''N'' For a pure ensemble, where all ''N'' particles are in state ''i''>, we have ''n<sub>j</sub>'' = 0, for all ''j'' &ne ''i'', from which we recover the corresponding density matrix ''&rho'' = ''i'' >< ''i'' |
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| :<math> Ho | \sum_j p_j \psi_j
ang \lang \psi_j </math> |
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| :<math> \lang A Ang | \sum_j p_j \lang \psi_jA\psi_j
ang = \operatorname{tr}[
ho A]</math> |
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| :<math>A | \sum_i a_i a_i
ang \lang a_i = \sum _i a_i P_i,</math> |
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| Where <math>P I | a_i
ang \lang a_i</math>, the corresponding density operator after the measurement is given by: |
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