1. Density matrix and its application

2. Density matrix An alternative of state-vector (ket) representation for a certain set of state-vectors appearing with certain probabilities. 2016. 06. 08.2

3. 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

4. Are density matrices unique? Density matrices are not unique. This is the price for being able to decompose entangled systems 2016. 06. 08.4

5. The Trace where are the elements of the main diagonal Eigenvalues and eigenvectors 2016. 06. 08.5

6. Properties of the Trace 2016. 06. 08.6

7. 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 2016. 06. 08.7

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

9. 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

10. Illustration Measurement basis: 2016. 06. 08.10

11. Decomposing a system - Partial trace In general 2016. 06. 08.11

12. Decomposing a system - Partial trace For product state systems 2016. 06. 08.12

13. Decomposing a system - Partial trace For entangled systems 2016. 06. 08.13 pure!

14. 2016. 06. 08.14 Max mixed! Contains no information

15. Geometrical interpretation of density matrices Bloch sphere 2016. 06. 08.15

16. 2016. 06. 08.16

17. Pure and mixed states The density matrix is not unique! 2016. 06. 08.17