Transcription of Chapter 9 Density Matrices - univie.ac.at
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Chapter 9 Density Matrices - univie.ac.at
Chapter 9 Density MatricesIn this Chapter we want to introduce Density Matrices , also called Density operators,which conceptually take the role of the state vectors discussed so far, as they encode allthe (accessible) information about a quantum mechanical system. It turns out that the pure states, described by state vectors1| on Hilbert space, are idealized descriptionsthat cannot characterize statistical (incoherent) mixtures, which often occur in the ex-periment, in Nature. These objects are very important for the theory of quantuminformation and quantum communication. More detailed information about the densitymatrix formalism can be found in [17]. General Properties of Density MatricesConsider an observableAin the pure state| with the expectation value given by A = |A| ,( )then the following definition is obvious:Definition matrix for the pure state| is given by :=| |This Density matrix has the following properties:I) 2= projector( )II) = hermiticity( )III) Tr = 1 normalization( )IV)
164 CHAPTER 9. DENSITY MATRICES Example: Density matrix for spin 1 2 Generally, this will be a 2 2 matrix that can be written as linear combination of the identity 1 and the Pauli matrices ˙ x;˙ y and ˙ z, as ˆ= 1 2 (1 + ~a~˙) : (9.25) The coe cient ~ais named the Bloch vector and can be calculated as the expectation value of the Pauli ...
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