Selected Topics in Mathematical Physics:
Quantum Information Theory

Master-Pflichtseminar im Wintersemester 2013/14. (Di., 16.15 - 18.00, Philosophenweg 19, Seminarraum).
Sprache des Seminars ist englisch. Start: 15. Oktober.

Vortrag Nr. 14 wird nicht am 4.2., sondern am 11.2.2014 stattfinden (Ort und Zeit wie immer).

Contact: Prof. Manfred Salmhofer, Markus Müller
Sprechstunde Markus Müller: Mo, 14-16 Uhr, Philosophenweg 19, Büro von Prof. Komnik. Achtung: Am 20.1. bin ich verreist (Thermodynamik-Konferenz Berlin).

The goal of this seminar is to get an overview on current research which tries to understand the foundations of statistical mechanics from a quantum information point of view. To this end, we will start with some basics of quantum information theory (entanglement, its quantification and operational meaning), then turn to some basic group representation theory and Lévy's Lemma, which allows to compute "Hilbert space averages" and prove concentration of measure for random quantum states. Finally, these technical tools will be applied to obtain strong statements about the thermalization of closed quantum systems.
If time permits, we will also discuss some closely related topics, such as Hayden and Preskill's work on the black hole information paradox, quantum pseudorandomness, or the evolving field of single-shot non-equilibrium thermodynamics.

Here is a nice review article: V. I. Yukalov, Equilibration and thermalization in finite quantum systems (arXiv).


List of talks

1. Overview and introduction. (15.10., Markus Müller; download PDF)
This will be an overview on the topics that we are going to hear in the seminar, but also a short introduction to some basic concepts that we need:
Classical versus quantum state space, composite systems, partial trace and entangled pure states.

[1] M. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, 2000.
[2] M. Piani, Lecture Notes on Entanglement Theory, IQC Waterloo, 2012. Part 1, part 2.
[3] J. Preskill, "Physics 219" Lecture Notes (scroll down to "Lecture Notes").
[4] D. Bruß and G. Leuchs (eds.), Lectures on Quantum Information, Wiley-VCH, 2007.

2. Definition of entanglement for pure and mixed states. (22.10., Marius Krumm, download PDF)
Schmidt decomposition, entanglement for pure and mixed states, von Neumann entropy, entanglement entropy for pure states.

3. Quantum operations and dense coding. (29.10., Markus Müller)
Quantum operations: axiomatic approach, completely positive trace-preserving maps, operator sum form, Stinespring dilation theorem. Dense Coding.

4. Operational aspects of entanglement. (5.11., Jakob Scharlau)
LOCC transformations, Nielsen's majorization criterion, entanglement cost and distillable entanglement.

5. Basics of group representation theory. (12.11., Markus Döring, download PDF)
Lie groups, the Haar measure, group representations, unitarity, statement and proof of a version of Schur's Lemma.

[5] B. Simon, Representations of Finite and Compact Groups, Graduate Studies in Mathematics vol. 10, American Mathematical Society, 1996.
[6] W. Fulton, J. Harris, Representation Theory, Graduate Texts in Mathematics, Springer, 2004.
[7] M. P. Müller, Random quantum states, measure concentration, and the additivity conjecture, IQC Lecture Notes, 2012.

6. Averages over the unitary group. (19.11., Yuyan Li, download PDF)
Examples how to compute averages of quantum information quantities over the unitary group via Schur's Lemma. Average reduced purity of random pure quantum states on a biartite Hilbert space.

[8] P. Hayden, D. Leung, and A. Winter, Aspects of generic entanglement, Commun. Math. Phys. 265, 95-117 (2006).

7. Measure concentration: Levy's Lemma. (26.11., Manuel Gerken, download PDF)
Statement of the lemma and proof sketch (assuming without proof the isoperimetric inequality on the sphere). Lipschitz continuity; concentration function.

[9] M. Ledoux, The concentration of measure phenomenon, Mathematical Surveys and Monographs 89, AMS, USA.
[10] V. D. Milman and G. Schechtman, Asymptotic theory of finite-dimensional normed spaces, Lecture Notes in Mathematics 1200, Springer, 2001.

8. Almost all pure states are almost maximally entangled. (3.12., Thomas Gläßle, download PDF)
Prove that random pure states are locally close to a maximally mixed state with high probablity. Fannes inequality and typical entanglement entropy of pure quantum states. Definition and quantum information significance of the trace distance resp. 1-norm, and how to bound it via the 2-norm.

[11] K. M. R. Audenaert, A Sharp Fannes-type Inequality for the von Neumann entropy, J. Phys. A: Math. Theor. 40, 8127-8136 (2007).

9. Entanglement and the foundations of statistical mechanics. (10.12., Malte Probst)
Present the results in the paper by Popescu, Short, and Winter with that title, and part of the proof.

[12] S. Popescu, A. J. Short, and A. Winter, Entanglement and the foundations of statistical mechanics, Nature Physics 2, 754 (2006). See also on arXiv.

10. Dynamical equilibration of closed quantum systems. (17.12., Gaofeng Huang)
Discuss the Poincare recurrence theorem as a "no-go theorem" for equlibration. Then, present the results of the paper by Short and Farrelly (arXiv:1110.5759) with parts of the proof.

[13] A. J. Short and T. C. Farrelly, Quantum equilibration in finite time, New J. Phys. 14, 013063 (2012).

11. Canonical typicality and equivalence of ensembles. (14.1., Markus Müller)
Present the "canonical typicality" paper by Goldstein, Lebowitz, Tumulka, and Zanghi, which gives the main (physical) ideas. Then discuss what can be proven rigorously in the special case of translation-invariant quantum spin systems with finite-range interaction.

[14] S. Goldstein, J. L. Lebowitz, R. Tumulka, and N Zanghi, Canonical Typicality, Phys. Rev. Lett. 96, 050403 (2006).
[15] M. P. Müller, E. Adlam, Ll. Masanes, and N. Wiebe, Thermalization and canonical typicality in translation-invariant quantum lattice systems, arXiv:1312.7420.

12. Black-hole information and the decoupling theorem. (21.1., Jan Blume)
This will be mainly based on the paper by Hayden and Preskill with the title "Black Holes as Mirrors". In particular, it involves the statement and sketch of proof of the decoupling theorem which has general significance in quantum information theory (not just for the black hole information problem), and the discussion of Uhlmann's theorem which proves that quantum information "cannot hide" in the correlations.

[16] D. N. Page, Information in Black Hole Radiation, Phys. Rev. Lett. 71, 3743 (1993).
[17] P. Hayden and J. Preskill, Black holes as mirrors: quantum information in random subsystems, JHEP 0709:120 (2007).
[18] P. Hayden, Decoupling: A building block for quantum information theory, QIP 2011 tutorial.

13. "Single-shot" thermodynamics: extracting work from "few" Szilard engines. (28.1., Jakob Scharlau)
We define and give some properties of the "smooth min- and max-entropies" from quantum information theory. Then, these entropies are used  to compute the work that can be extracted from "small" quantum systems. This will be mainly about the paper by Dahlsten et al. below.

[18] O. C. O. Dahlsten, R. Renner, E. Rieper, and V. Vedral, Inadequacy of von Neumann entropy for characterizing extractable work, New J. Phys. 13, 053015 (2011).
[19] R. Renner, Security of quantum key distribution, PhD thesis, ETH Zürich (2005).
[20] M. Tomamichel, A Framework for Non-Asymptotic Quantum Information Theory, PhD thesis, ETH Zürich (2012).

14. The resource theory of informational nonequilibrium in thermodynamics. (11.2., Marius Krumm, download PDF)
The goal is to approach the modern field of "single-shot thermodynamics", where people study the thermodynamics of (small) quantum systems. Here, we consider for simplicity only systems with trivial Hamiltonian, i.e. H=0, and we ask what state transformations are possible according to certain rules of the game: unitaries are free, and so are maximally mixed states (that is, thermal states), but every kind of "purity" is a resource. This will give us a refined version of the second law, recover Shannon, min-, max-, and Renyi entropies, and will give us a hint an a different, agent-based approach
to thermodynamics and its possible results.

[21] M. Horodecki, P. Horodecki, and J. Oppenheim, Reversible transformations from pure to mixed states, and the unique measure of information, Phys. Rev. A 67, 062104 (2003).
[22] G. Gour, M. P. Müller, V. Narasimhachar, R. W. Spekkens, and N. Yunger Halpern, The resource theory of informational nonequilibrium in thermodynamics, arXiv:1309.6586.
[23] F. G. S. L. Brandao, M. Horodecki, N. H. Y. Ng, J. Oppenheim, and S. Wehner, The second laws of quantum thermodynamics, arXiv:1305.5278.