A measure of majorization emerging from single-shot statistical mechanics

Abstract

The use of the von Neumann entropy in formulating the laws of thermodynamics has recently been challenged. It is associated with the average work whereas the work guaranteed to be extracted in any single run of an experiment is the more interesting quantity in general. We show that an expression that quantifies majorization determines the optimal guaranteed work. We argue it should therefore be the central quantity of statistical mechanics, rather than the von Neumann entropy. In the limit of many identical and independent subsystems (asymptotic i.i.d) the von Neumann entropy expressions are recovered but in the non-equilbrium regime the optimal guaranteed work can be radically different to the optimal average. Moreover our measure of majorization governs which evolutions can be realized via thermal interactions, whereas the non-decrease of the von Neumann entropy is not sufficiently restrictive. Our results are inspired by single-shot information theory.Statistical mechanics is a corner-stone of modern physics. Many of its basic paradigms and mathematical methods were set in an era where the experimental abilities were much more limited and modern information theory not developed. Accordingly there is currently significant momentum in investigating the theory's foundations in the quantum and nano regimes, see e.g. al (2012) to mention but a few recent contributions. We here derive an alternative type of statistical mechanics from scratch. Our approach is inspired by recent results in information theory (Renner and Wolf 2004, Renner 2005) and builds on (Dahlsten et al 2011, Rio et al 2011, Aberg 2012, Horodecki and Oppenheim 2013).We argue this approach is both significantly more general than the standard theory and addresses questions more relevant to modern experiments.It is more general in that we will not assume that the states of systems of interest are thermal, but rather just that there is a heat bath which when interacting with a system gradually takes that system towards a thermal state. Thus the system of interest is not necessarily in equilibrium. In fact we will allow for any probability distribution over energy levels. We do in particular not assume that the system under consideration is large or that internal correlations are negligible. This makes the approach significantly more relevant to modern experiments where small subsystems can be addressed individually and in time-scales faster than the thermalization time.A key difference regarding which questions are addressed is that we focus not on averages of distributions as in standard statistical mechanics. Instead we ask, for any given single run of an experiment, which threshold values are guaranteed to be exceeded, or more generally guaranteed to be exceeded up to some probability ε, not necessarily small. This is referred to as the single-shot paradigm, as opposed to the average paradigm. This distinction is important when distributions of quantities have a significant spread around the average, as is often the case for small systems.

Resource Theory of Quantum States Out of Thermal Equilibrium

“…[8]. Results similar to the former were also reported independently of [7] and the present work by Å berg [9] and later by Egloff et al [10], who investigated the work extractable from finite resources. For a different approach in the same quantum setting as ours, but more along the lines of statistical mechanics, see Ref.…”

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Unitary work extraction from a generalized Gibbs ensemble using Bragg scattering

“…Over the last decade, an increasing amount of attention has been devoted to the interplay of thermodynamics, quantum mechanics and information theory [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]. While most of the work is driven by fundamental questions regarding the validity of statistical mechanics in the quantum realm, recent technological advances have made it possible to actually fabricate and study thermal machines on the level of single-atoms.…”