Density matrix and its application

Density matrix An alternative of state-vector (ket) representation for a certain set of state-vectors appearing with certain probabilities

3 Ensembles – pure and mixed states Pure sate Mixed state: set of pure quantum states with given probabilities Mixing: weighting with classical probabilities Superposition: weighting with quantum probability amplitudes E.g. a pure sate can be a superposition

Are density matrices unique? Density matrices are not unique. This is the price for being able to decompose entangled systems

The Trace where are the elements of the main diagonal Eigenvalues and eigenvectors

Properties of the Trace

Trace- properties of density matrices The trace of any density matrix is equal to one for a pure state for a mixed state for a pure entangled system for any mixed subsystem of an EPR pair

8 2 nd Postulate (evolution) The evolution of any closed physical system in time can be characterized by means of unitary transforms

9 3rd Postulate (measurement) Any quantum measurement can be described bymeans of a set of measurement operators {M m }, where m stands for the possible results of the measurement. The probability of measuring m if the system is in state v can be calculated as and the system after measuring m gets in state

Illustration Measurement basis:

Decomposing a system - Partial trace In general

Decomposing a system - Partial trace For product state systems

Decomposing a system - Partial trace For entangled systems pure!

Max mixed! Contains no information

Geometrical interpretation of density matrices Bloch sphere

Pure and mixed states The density matrix is not unique!