Stability of Quantum Statistical Ensembles with Respect to Local Measurements Boris Fine

1. Stability of Quantum Statistical Ensembles with Respect to Local Measurements Boris Fine Skolkovo Institute of Science and Technology University of Heidelberg W. Hahn and B. V. Fine, arXiv:1601.06402.

3. Skoltech – June 2016 Postdoctoral and Ph.D. positions are available in the group of B.F.

4. Microcanonical Ensemble Derivation of the Boltzmann-Gibbs distribution Psubsystem (E) νe(-E) ~ e- β E Boltzmann-Gibbs 0 νe(E) Esubsystem Around E = 0 νe(E) ≅νe(0) eβ E D. C. Brody and L. P. Hughston, (1998). G. Aarts, G. F. Bonini, and C. Wetterich, (2000). J. Naudts and E. Van der Straeten, (2006) G. Jona-Lasinio and C. Presilla, (2006) B.V. Fine, (2009). B. Fresch and G. J. Moro, (2009). M. Müller, D. Gross, and J. Eisert, (2011). 0 Eenvironment νtot(E) snow Ψ (t) = Σi Cie-i Ei t φi | Cie-i Ei t|do not depend on time. boiling water 0 Etotal Why?

5. Quantum micro-canonical (QMC) ensemble: W. K. Wootters, Found. Phys. 20, 1365 (1990). D. C. Brody and L. P. Hughston, J. Math. Phys. 39, 6502 (1998). C. M. Bender, D. C. Brody, and D. W. Hook, J. Phys. A 38, L607 (2005). B.V. Fine, Phys. Rev. E 80, 051130 (2009). B. Fresch and G. J. Moro, J. Phys. Chem. A 113, 14 502 (2009). M. Müller, D. Gross, and J. Eisert, Commun. Math. Phys. 303, 785 (2011). unrestricted participation of eigenstates

6. Results:QMC-based statistics for an isolated system with N>>1 B.F., PRE 80, 051130 (2009) small-pk approximation confirmed by the direct Monte-Carlo sampling in B.F. and F. Hantschel, Phys. Scr. T151, 014078 (2012) Condensation for macroscopic systems Density matrix elements for a small subsystem Not Boltzmann-Gibbs!

7. Ensembles emerging in thermally isolated clusters of spins ½ under multiple non-adiabatic perturbations [K. Ji & B.F., PRL 107, 050401 (2011)] 16 spins ½ ~ 65 000 quantum states Emergence of the QMC-like statistics

8. Implications of the QMC result: For macroscopic systems: Existence of a new fundamental limit for the applicability of the conventional thermodynamics associated with the energy window for the eigenstates participating in statistical ensembles. - What enforces it under everyday conditions? For non-macroscopic systems with large number of quantum levels: The QMC ensemble might be realizable under generic non-adiabatic perturbations. - How can it be protected?

9. Probability distribution for the total energy:

11. Stability of quantum statistical ensembles with respect to local measurements Similar approach in another context: A. Shimizu and T. Miyadera, Phys. Rev. Lett. 89, 270403 (2002).

13. Lattices of spins 1/2

14. Formalism:

16. Narrowing effect:

18. Calculations for the initial two-peak energy distribution:

20. Narrowing vs. broadening for a Gaussian g(E) narrowing: broadening:

21. System of interacting spins: Two single-spin measurements sufficiently close in space and time are necessary to get outcome correlated with the total energy. Characteristic narrowing time for single-spin measurements in non-magnetic phase:

22. Direct numerical simulations: Spin-½ chain with anisotropic nearest neighbor interaction, Initial state is a superposition of two thermal states: T1 = 0.1 and T2 = - 0.1

23. Direct numerical simulations - heating:

24. Conclusions: 1. Rare random measurements impose strict constraints on quantum statistical ensembles. 2. Ensembles with broad energy distributions are unstable with respect to local measurements. This result justifies the use of ensembles with narrow energy distributions, such as canonical or microcanonical. 3. In experiments on preparing unconventional ensembles in finite systems, one should try to avoid: (i) external static fields (ii) long-range order (iii) local constants of motion