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On the generic increase of observational entropy in isolated systems
Published 18 Apr 2024 in quant-ph and cond-mat.stat-mech | (2404.11985v2)
Abstract: Observational entropy -- a quantity that unifies Boltzmann's entropy, Gibbs' entropy, von Neumann's macroscopic entropy, and the diagonal entropy -- has recently been argued to play a key role in a modern formulation of statistical mechanics. Here, relying on algebraic techniques taken from Petz's theory of statistical sufficiency and on a L\'evy-type concentration bound, we prove rigorous theorems showing how the observational entropy of a system undergoing a unitary evolution chosen at random tends to increase with overwhelming probability and to reach its maximum very quickly. More precisely, we show that for any observation that is sufficiently coarse with respect to the size of the system, regardless of the initial state of the system (be it pure or mixed), random evolution renders its state practically indistinguishable from the uniform (i.e., maximally mixed) distribution with a probability approaching one as the size of the system grows. The same conclusion holds not only for random evolutions sampled according to the unitarily invariant Haar distribution, but also for approximate 2-designs, which are thought to provide a more physically and computationally reasonable model of random evolutions.
- J. von Neumann, Mathematical foundations of quantum mechanics (Princeton university press, 1955).
- A. Wehrl, General properties of entropy, Rev. Mod. Phys. 50, 221 (1978).
- D. Šafránek, J. M. Deutsch, and A. Aguirre, Quantum coarse-grained entropy and thermodynamics, Phys. Rev. A 99, 010101 (2019a), arXiv:1707.09722 [quant-ph] .
- D. Šafránek, J. M. Deutsch, and A. Aguirre, Quantum coarse-grained entropy and thermalization in closed systems, Phys. Rev. A 99, 012103 (2019b), arXiv:1803.00665 [quant-ph] .
- P. Strasberg and A. Winter, First and second law of quantum thermodynamics: A consistent derivation based on a microscopic definition of entropy, PRX Quantum 2, 030202 (2021).
- A. Riera-Campeny, A. Sanpera, and P. Strasberg, Quantum systems correlated with a finite bath: Nonequilibrium dynamics and thermodynamics, PRX Quantum 2, 010340 (2021), arXiv:2008.02184 [quant-ph] .
- D. Šafránek, A. Aguirre, and J. M. Deutsch, Classical dynamical coarse-grained entropy and comparison with the quantum version, Phys. Rev. E 102, 032106 (2020), arXiv:1905.03841 [cond-mat.stat-mech] .
- J. M. Deutsch, D. Šafránek, and A. Aguirre, Probabilistic bound on extreme fluctuations in isolated quantum systems, Phys. Rev. E 101, 032112 (2020), arXiv:1806.08897 [gr-qc] .
- C. Nation and D. Porras, Taking snapshots of a quantum thermalization process: Emergent classicality in quantum jump trajectories, Phys. Rev. E 102, 042115 (2020), arXiv:2003.08425 [quant-ph] .
- P. Strasberg, M. G. Díaz, and A. Riera-Campeny, Clausius inequality for finite baths reveals universal efficiency improvements, Phys. Rev. E 104, L022103 (2021), arXiv:2012.03262 [quant-ph] .
- R. Hamazaki, Speed Limits for Macroscopic Transitions, PRX Quantum 3, 020319 (2022), arXiv:2110.09716 [cond-mat.stat-mech] .
- R. Modak and S. Aravinda, Observational-entropic study of anderson localization, Phys. Rev. A 106, 062217 (2022).
- S. PG, R. Modak, and S. Aravinda, Witnessing quantum chaos using observational entropy, Phys. Rev. E 107, 064204 (2023).
- J. Schindler and A. Winter, Continuity bounds on observational entropy and measured relative entropies, arXiv e-prints , arXiv:2302.00400 (2023), arXiv:2302.00400 [quant-ph] .
- D. Šafránek, D. Rosa, and F. C. Binder, Work extraction from unknown quantum sources, Phys. Rev. Lett. 130, 210401 (2023).
- D. Šafránek and D. Rosa, Measuring energy by measuring any other observable, Phys. Rev. A 108, 022208 (2023).
- D. Šafránek, Ergotropic interpretation of entanglement entropy, arXiv e-prints , arXiv:2306.08987 (2023), arXiv:2306.08987 [quant-ph] .
- J. von Neumann, Proof of the ergodic theorem and the H-theorem in quantum mechanics. Translation of: Beweis des Ergodensatzes und des H-Theorems in der neuen Mechanik, European Physical Journal H 35, 201 (2010).
- R. C. Tolman, The Principles of Statistical Mechanics (Dover Publications, 2010).
- D. Petz, Sufficient subalgebras and the relative entropy of states of a von neumann algebra, Communications in mathematical physics 105, 123 (1986).
- D. Petz, Sufficiency of channels over von neumann algebras, The Quarterly Journal of Mathematics 39, 97 (1988).
- D. Petz, Monotonicity of quantum relative entropy revisited, Reviews in Mathematical Physics 15, 79 (2003).
- A. Jenčová and D. Petz, Sufficiency in quantum statistical inference: a survey with examples, Infinite Dimensional Analysis, Quantum Probability and Related Topics 09, 331 (2006).
- M. Ledoux, The concentration of measure phenomenon (American Mathematical Society, 2001).
- P. Hayden, D. W. Leung, and A. Winter, Aspects of generic entanglement, Communications in Mathematical Physics 265, 95 (2006).
- R. A. Low, Large deviation bounds for k-designs, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 465, 3289 (2009).
- A. W. Harrow and S. Mehraban, Approximate unitary t-designs by short random quantum circuits using nearest-neighbor and long-range gates, Communications in Mathematical Physics 401, 1531–1626 (2023).
- M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information: 10th Anniversary Edition (Cambridge University Press, 2010).
- M. M. Wilde, Quantum Information Theory (Cambridge University Press, 2013).
- H. Umegaki, On information in operator algebras, Proc. Japan Acad. 37, 459 (1961).
- H. Umegaki, Conditional expectation in an operator algebra, iv (entropy and information), Kodai Mathematical Seminar Reports 14, 59 (1962).
- S. Kullback and R. A. Leibler, On information and sufficiency, Ann. Math. Stat. 22, 79 (1951).
- H. Martens and W. M. de Muynck, Nonideal quantum measurements, Foundations of Physics 20, 255 (1990).
- A. Polkovnikov, Microscopic diagonal entropy and its connection to basic thermodynamic relations, Annals of Physics 326, 486–499 (2011).
- F. Buscemi and V. Scarani, Fluctuation theorems from bayesian retrodiction, Phys. Rev. E 103, 052111 (2021).
- C. C. Aw, F. Buscemi, and V. Scarani, Fluctuation theorems with retrodiction rather than reverse processes, AVS Quantum Science 3, 045601 (2021).
- H. Tasaki, Macroscopic irreversibility in quantum systems: Eth and equilibration in a free fermion chain, arXiv preprint arXiv:2401.15263 (2024).
- F. Buscemi, K. Kobayashi, and S. Minagawa, A complete and operational resource theory of measurement sharpness, Quantum 8, 1235 (2024).
- I. Csiszár and Z. Talata, Context tree estimation for not necessarily finite memory processes, via bic and mdl, IEEE Transactions on Information theory 52, 1007 (2006).
- V. D. Milman and G. Schechtman, Asymptotic theory of finite dimensional normed spaces: Isoperimetric inequalities in riemannian manifolds, Vol. 1200 (Springer Science & Business Media, 1986).
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