
Mathematical structures of quantum theory
 Faculty of Mathematics, Physics and Informatics, Comenius University, Bratislava (5 credits)
 code 2FTF227, winter semester (2+1h/week)
 lecturer: Mário Ziman
 undergraduate and graduate students
 prerequisities: quantum mechanics, linear algebra,
 room F2125, Fridays 12:2015:00 (subject to change)
Content:
§ 1. Quantum Theory Refresher
§ 2. States and effects
 basic statistical model of an experiment, density operators, convexity,
pure and mixed states, purity
§ 3. Effects and superposition
 duality, convexity, Gleason's theorem, doubleslit interference
§ 4. Purification and automorphisms
 tensor product, purification, group of automorphisms, Wigner's theorem
§ 5. Observables
 POVM, convexity, sharp observables,
§ 6. Estimation and discrimination
 informational completeness, minimumerror discrimination, unambiguous
discrimination
§ 7. Quantum channels
 complete positivity, Stinespring dilation theorem,
§ 8. Representations of quantum channels
 Kraus decomposition, ChoiJamiolkowski isomorphism, convexity
§ 9. Classes of channels
 contractive, qubit channels,
§ 10. Time evolution
 time, master equations, Schrodinger equation
§ 11. Instruments
 quantum operation, projection "postulate", repeatability
§ 12. Measurement models
 SternGerlach apparatus, photodetector
Exam:
 homeworks (20 p + at least 5 bonus points), written test (10b) plus (optional) oral exam
 A (>25 p), B(>20 p), C(>15 p), D(>10 p), E(>5 p)
Literature:
T.Heinosaari, M.Ziman:
The Mathematical Language of Quantum theory (Cambridge, 2012)
T.Heinosaari, M.Ziman:
Guide to mathematical concepts of quantum theory,
Acta Physica Slovaca 58, 487674 (2008)
Homeworks:
Set 5 (ideally due 07/11/2014)
5.1 Consider a threevalued polarization observable
F_{0}=k V⟩⟨V,
F_{+}=k +⟩⟨+,
F_{}=k ⟩⟨,
where V⟩, ±⟩ = ± cosηV⟩+ sinηH⟩
are unit vectors (V/H⟩ stands for vertical/horizontal polarization)
and k is some positive constant. Find values of k and η
for which these operators determines a valid POVM. (1p)
5.2 Illustrate the statistical map for the observable from the
Task 5.1 above. (1p)
Set 4 (ideally due 24/10/2014)
4.1 Consider a measurement apparatus being internally a uniform mixture
of ideal SternGerlach apparatuses oriented along the axis Z, or X.
Find POVM description for such measurement. (1p)
4.2 Find conditions (in terms of its eigenvalues) under which an effect
E can describe two different outcomes of the same measurement. (1p)
Set 3 (ideally due 17/10/2014)
3.1 Is it possible to increase the purity of states by mixing them together? (1p)
3.2 Suppose you are give two sources. First producing systems in a state ψ⟩ and the second producing systems in a state φ⟩. How to check whether one of them is a superposition of the other. (1p)
Set 2 (ideally due 10/10/2014)
2.1 Consider a physical system associated with twodimensional Hilbert space. Find explicit range of parameters (in the basis of Pauli operators) determining the set of states. (1p)
2.2 Consider a physical system associated with twodimensional Hilbert space. Find explicit range of parameters (in the basis of Pauli operators) determining the set of states. (1p)
Set 1 (ideally due 03/10/2014)
1.1 Suppose we obtain a box containing 10 photons. We know that either there
are 5 horizontally and 5 vertically polarized photons, or 5 leftcircular and 5 rightcircular polarized photons. Is it possible to say which box we received? If you have an answer, please describe why? (1p)
1.2 Let S be the swap operator acting on tensor product of isomorphic Hilbert spaces as follows S(φ⊗ψ⟩)=(ψ⊗φ⟩) on any pair of vectors φ⊗ψ⟩. Show that for all operators ρ and ξ the following identity holds: trS(ρ⊗ξ)=tr[ρξ]. (1p)
Bonus 1
What is measured by a realistic photonresolving photodetector? In other words: determine effects describing the registration of N photons providing
that the efficiency of detecting individual photons is η. (2p)
Bonus 2
Use an observable from Taks 5.1. to discriminate between three pure states
V⟩, +⟩, ⟩. In particular, when an outcome
F_{j} is observed then we conclude the state j⟩ is prepared.
Evaluate the probability of error? (2p)
Bonus 3
Prove that measurements of position and momentum (using in a random manner)
are not informationally complete. (2p)

