Abstract
Thermodynamics can be formulated in either of two approaches, the phenomenological approach, which refers to the macroscopic properties of systems, and the statistical approach, which describes systems in terms of their microscopic constituents. We establish a connection between these two approaches by means of a new axiomatic framework that can take errors and imprecisions into account. This link extends to systems of arbitrary sizes including very small systems, for which the treatment of imprecisions is pertinent to any realistic situation. Based on this, we identify the quantities that characterise whether certain thermodynamic processes are possible with entropy measures from information theory. In the error-tolerant case, these entropies are so-called smooth min and max entropies. Our considerations further show that in an appropriate macroscopic limit there is a single entropy measure that characterises which state transformations are possible. In the case of many independent copies of a system (the so-called i.i.d. regime), the relevant quantity is the von Neumann entropy. Transformations among microcanonical states are characterised by the Boltzmann entropy.
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Notes
- 1.
Throughout this chapter, we follow Lieb and Yngvason’s convention regarding the notation of the order relation, where \(X \prec Y\) means that X can be transformed into Y with an adiabatic process. Note that this is the reverse convention of what a reader acquainted with the literature on quantum resource theories might expect.
- 2.
The composition operation is assumed to be associative and commutative.
- 3.
The scaling is taken to obey \(1 X = X\) as well as \(\alpha _1(\alpha _2 X) = (\alpha _1 \alpha _2) X\), thus \(1 \Gamma _{\mathrm {EQ}}= \Gamma _{\mathrm {EQ}}\) and \(\alpha _1(\alpha _2 \Gamma _{\mathrm {EQ}}) = (\alpha _1 \alpha _2) \Gamma _{\mathrm {EQ}}\).
- 4.
In general, \(\prec \) may not be a total preorder and the ordering of values assigned to states that cannot be compared by means of \(\prec \) may be ambiguous.
- 5.
This definition extends to states obeying \(X \prec X_0\) or \(X_1 \prec X\) [5], where it has to be interpreted slightly differently. To see this, note that the expression
$$\begin{aligned} ((1-\alpha ) X_0, \alpha X_1) \prec X \end{aligned}$$(32.5)is equivalent to \((((1-\alpha ) X_0, \alpha X_1), \alpha ' X_1) \prec (X, \alpha ' X_1)\) for any \(\alpha '\ge 0\) and hence also to \(((1-\alpha ) X_0, (\alpha '+\alpha ) X_1) \prec (X, \alpha ' X_1)\). This allows us to consider negative \(\alpha \) (while still using only positive coefficients as scaling factors). If \(\alpha <0\), by choosing \(\alpha '=-\alpha \), condition (32.2) is to be understood as a transformation \(((1-\alpha ) X_0) \prec (X, |\alpha | X_1)\). A similar argument shows that for \(\alpha >1\) the condition \(((1-\alpha ) X_0, \alpha X_1) \prec X\) can be understood as \(\alpha X_1 \prec ((\alpha -1) X_0, X)\).
- 6.
The bounds given in [7] are
$$\begin{aligned} S_{\mathrm {-}}(X)&= \sup \left\{ S(X') \,\right. \mid \left. X' \in \Gamma _{\mathrm {EQ}}, \ X' \prec X\;\;\right\} \end{aligned}$$(32.6)$$\begin{aligned} S_{\mathrm {+}}(X)&= \inf \, \left\{ S(X'') \right. \mid \left. X'' \in \Gamma _{\mathrm {EQ}}, \ X \prec X''\,\right\} \ . \end{aligned}$$(32.2)In phenomenological thermodynamics, where equilibrium states are traditionally described in terms of continuous parameters (such as the internal energy and the volume for instance), there exists an equilibrium state \(X_\alpha \) for each \(((1-\alpha ) X_0, \alpha X_1)\) that obeys \(X_\alpha \sim ((1-\alpha ) X_0, \alpha X_1)\). Under these circumstances the bounds (32.5) and (32.6) coincide with (32.3) and (32.4) respectively. This is, however, not implied by the axioms and may not hold if systems are described in a different way (e.g. in the statistical approach taken in Sect. 32.3). Note also that for (32.3) and (32.4) the inequality in (32.7) need not be strict.
- 7.
We alert the reader to the non-standard notation for the order relation (see also Footnote 1).
- 8.
Notice that approximations are also relevant for macroscopic systems. However, we do not usually explicitly mention them there, as they are extremely accurate.
- 9.
For \( \epsilon =0\) the usual majorisation relation is recovered.
- 10.
We aim to achieve the transformation \(\rho \otimes \frac{\mathbbm {1}_2}{2} \prec _{\mathrm {M}}^ \epsilon \rho \otimes |0\rangle \! \langle 0|\), where the ordered spectra of the two states are \((\frac{p}{2},\frac{p}{2},\frac{1-p}{2},\frac{1-p}{2})\) and \((p,1-p,0,0)\).
- 11.
The largest eigenvalues of \(\rho \otimes \chi _{\lambda _1}\) and \(\rho \otimes \chi _{\lambda _2}\) are \(p 2^{-\lambda _1}\) and \(p 2^{-\lambda _ 2}\) respectively, where p is the maximal eigenvalue of \(\rho \). Up to \(2^{\lambda _2}\), the sums of the eigenvalues of the two states are \(p 2^{\lambda _2-\lambda _1}\) and p respectively, thus the transition is only possible if \(p 2^{\lambda _2-\lambda _1} \ge p- \epsilon \).
- 12.
- 13.
Axiom (M1), for instance, follows since \((X,\tau _\lambda ) \prec ^ \epsilon (Y,\tau _\lambda )\) implies \((X,\tau _\lambda ,\chi _{\lambda _1},\chi _{\lambda _{2'}}) \prec ^ \epsilon (Y,\tau _\lambda ,\chi _{\lambda _1},\chi _{\lambda _{2'}})\) by (A4) and since by applying (32.36) and the property (A5) for \(X_\mathrm {ref}\) this can be shown to also imply \(X \prec ^ \epsilon Y\). (M2), (M3) and (M4) also follow from (32.36) and the axioms.
- 14.
Axiom (E5) is, however, not implied by this.
- 15.
- 16.
This follows as the state \(\tau _{\lambda _n^{-}}\) has to be chosen such that it majorises \((1- \epsilon ) \rho ^{\otimes n}\). Furthermore, \(\rho ^{\otimes n}\) has to majorise \((1- \epsilon ) \tau _{\lambda _n^{+}}\), hence its rank has to be smaller than that of \(\tau _{\lambda _n^{+}}\).
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Acknowledgements
Philippe Faist is not listed as an author for editorial reasons. MW is supported by the EPSRC (grant number EP/P016588/1). RR acknowledges contributions from the Swiss National Science Foundation via the NCCR QSIT as well as project No. 200020 165843.
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Weilenmann, M., Krämer, L., Renner, R. (2018). Smooth Entropy in Axiomatic Thermodynamics. In: Binder, F., Correa, L., Gogolin, C., Anders, J., Adesso, G. (eds) Thermodynamics in the Quantum Regime. Fundamental Theories of Physics, vol 195. Springer, Cham. https://doi.org/10.1007/978-3-319-99046-0_32
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