Quantum Theory

Sommersemester 2016

Prof. Dr. Michael M. Wolf, Prof. Dr. Robert König

Time & Location: Thursdays, 14.15-15.45, Seminarraum M5 (03.10.011). First meeting on April 14. No meetings on May 26 (holiday) and June 9 (absence).

List of topics and suggested references

1. Hilbert space, bounded, self-adjoint, unitary and positive operators, spectral theorem, trace, trace class operators. (14.4.)
2. Dirac notation, pure and mixed states, purity, density operators, Bloch sphere. (21.4.)
3. Measurements: observables, POVMs, Born's rule/postulate, post-measurement states (collapse) and quantum Zeno effect, informationally complete measurements and state tomography. (28.4.)
4. Uncertainty relations and joint measurability. (12.5.)
5. Tensor products, composite systems, partial trace, Schmidt decomposition, purification. (19.5.)
6. Product, separable and entangled states. Entanglement witnesses. LOCC operations, definition of entanglement measures. (2.6.)
7. Polytope of classical (local hidden variable) correlations, quantum correlations, classical and quantum values of games, Bell inequalities, Tsirelson bound. (16.6.)
8. Unitary and non-unitary dynamics, quantum channels (complete positivity), one-parameter groups and generators, Schrödinger equation. [16] (23.6.)
9. Choi-Jamiolkowski isomorphism, Stinespring dilation, Naimark's theorem. (30.6.)
10. Teleportation, dense coding, no-cloning, no-signaling, impossible machines. [11,14] (7.7.)
11. Fidelity, entropies, Fannes' inequality, strong subadditivity. [2,4,7,12]
12. Schumacher compression, Holevo-Schumacher-Westermoreland theorem. [18]
13. Quantum capacity, zero-capacity channels, superactivation. [18,13]

Suggested references are as follows: References [5,10,15] give general background material on quantum information/computation. References [1,9,17] cover most of the mathematical concepts necessary for topics 1-9. Additional topic-specific references are given in the list above.

Bibliography

1

R. Alicki and M. Fannes, Quantum dynamical systems, Oxford University Press, 2001.

2

K. Audenaert, A Sharp Fannes-type Inequality for the von Neumann Entropy, 2006.

3

M. Fannes, A continuity property of the entropy density for spin lattice systems, Comm. Math. Phys., Vol. 31, No. 4, 291-294 (1973).

4

E. Lieb and M. Ruskai, Proof of the strong subadditivity of quantum-mechanical entropy, J. Math. Phys. 14, 1938 (1973)

5

M. A. Nielsen, I. L. Chuang, Quantum computation and quantum information, Cambridge University Press, 2000.

6

D. Gottesman, An Introduction to Quantum Error Correction and Fault-Tolerant Quantum Computation, 2009

7

M. Junge, R. Renner, D. Sutter, M. Wilde and A. Winter, Universal recovery from a decrease of quantum relative entropy, 2015

8

R. Horodecki, P. Horodecki, M. Horodecki and K. Horodecki, Quantum entanglement, Rev. Mod. Phys. 81, 865-942, 2009.

9

M. Keyl, Fundamentals of Quantum Information Theory, Phys. Rep. 369, 431-548, 1999-2015

10

J. Preskill, Quantum Computation, Lecture Notes, 2015.

11

R. F. Werner, All teleportation and dense coding schemes, J. Phys. A: Math. Gen. 34 7081, 2001.

12

M. Ruskai, Another short and elementary proof of strong subadditivity of quantum entropy, Rep. Math. Phys., vol. 60, no. 1, August 2007, Pages 1-12

13

G. Smith, Quantum Channel Capacities, 2010

14

R. F. Werner, Optimal Cloning of Pure States, 1998.

15

R. F. Werner, Quantum Information Theory - an Invitation, 2001.

16

M. Wolf, Quantum Channels and Operations, Lecture Notes, 2012.

17

M. Wolf, Quantum effects, Lecture Notes, 2014.

18

M. Wilde, Quantum Information Theory, Cambridge University Press, 2013.

---