For noncommutative macroscopic observables, the measurement outcomes are not defined naively. To overcome this difficulty, one considers their approximation by commuting observables. For more details, see Sec. 12b.

Some readers would be concerned about the relation to results by Iyoda et al. [28]. They showed $\mathrm{\Delta }S-\beta Q\ge 0$, a form of the second law for isothermal operations, in a system attached to a heat reservoir. By contrast, we are interested in the second law for adiabatic operations.

In this paper, we make use of the standard Bachmann-Landau notations $O(•)$, $o(•)$, $\mathrm{\Omega }(•)$, $\omega (•)$, and $\mathrm{\Theta }(•)$ to describe the asymptotic behavior of functions. Let $f$ and $g$ be real functions, and let $g(x)$ be strictly positive for all large enough values of $x$. We say $f(x)=O(g(x))$ if $\lim {\sup }_{x\to \mathrm{\infty }}|f(x)|/g(x)<\mathrm{\infty }$ and $f(x)=o(g(x))$ if ${\lim }_{x\to \mathrm{\infty }}f(x)/g(x)=0$. We also say $f(x)=\mathrm{\Omega }(g(x))$ if $\lim {inf}_{x\to \mathrm{\infty }}f(x)/g(x)>0$ and $f(x)=\omega (g(x))$ if ${\lim }_{x\to \mathrm{\infty }}f(x)/g(x)=\mathrm{\infty }$. If both $f(x)=O(g(x))$ and $f(x)=\mathrm{\Omega }(g(x))$, we write $f(x)=\mathrm{\Theta }(g(x))$.

Generalization to other lattices is straightforward.

If one would like the theory to be independent of the choice of $K$, one can take the limit $K\to \mathrm{\infty }$ after the thermodynamic limit $L\to \mathrm{\infty }$. See Sec. 14b for details.

Because a translation of $\alpha$ causes no change in $A_{{\mathcal{S}}_L}(\alpha )$, we can restrict the support of $\alpha$ to $C^{\ell }$ without loss of generality.

Here, “additive quantities in thermodynamics” mean additive mechanical quantities in thermodynamics such as the total magnetization. That is, we exclude genuine thermodynamic quantities such as the free energy.

A similar concept has been investigated in Ref. [135] and references therein.

This was proven in Supplemental Material of [125] under the same assumption as the quantum central limit theorem of the characteristic function.

Any thermodynamic quantities can be measured by measuring additive quantities, since they are functions of additive observables.

In this paper, we only consider the case where the back-reaction effects from the system to the fields, such as electric or magnetic field, are negligible.

Our results also hold even when $f^{\mu }(t)$’s depend on $L$ but their magnitudes are bounded by a constant independent of $L$.

When an external field $\mathcal{F}(t)$ couples to an additive observable $B$ as $-g(\mathcal{F}(t))B$ in $H_L(t)$, where $g(\mathcal{F})$ is a nonlinear function of $\mathcal{F}$, take $g(\mathcal{F}(t))$ as $f(t)$.

According to Refs. [125, 126], if the measurement is performed appropriately, the backactions on the densities of additive observables are macroscopically negligible. If so, it is sufficient to prepare only a single quantum state representing iMATE.

Note that the inverse temperatures ${\beta }_0$ and ${\beta }_1$ in Eq. (80) are not necessarily positive. In other words, this result applies to transitions between positive and negative inverse temperature states [136, 137, 138, 139], which are beyond standard thermodynamics.

We do not know exactly how fast the divergence of ${\ell }_L$ can be. However, at least when ${\ell }_L=\mathrm{\Theta }(L)$, the local equivalence between the microcanonical and the canonical Gibbs states breaks down [110], which is outside the scope of our interest. Therefore, ${\ell }_L=o(L)$ should be imposed.

Note that, as discussed in Refs. [36, 140], $T$ should be taken exponentially large with respect to $N$ to make their upper bound meaningful.

We can also take $K$ smaller. For example, when $M^{\max }=6$ it is sufficient to take $K=60$ instead of $K=6!$.

This property is the physical motivation of the adjustment (93). Since the same property is not necessarily satisfied for all $M$ such that $M^{\max }<M\le M^{\max }!$, we focus on the case of $M\le M^{\max }$.

This is because Eq. (19) for $K=(M^{\max })!$ implies Eq. (19) for $K=M^{\max }$.

Precisely, when $v^{\mu }$ and $(v^{\mu }{)}^{†}$ are linearly dependent, $(v^{\mu }{)}^{†}=e^{i\theta }{v}^{\mu }$, so we use $e^{i\theta /2}{v}^{\mu }$ instead.

This result implies that the completion of $\mathcal{A}$, $\overline{\mathcal{A}}$, is invariant under macroscopic operations.

The above proofs also imply that we can replace $\mathcal{A}$ in Definitions t9–t11 by its completion $\overline{\mathcal{A}}$ without causing any change to these definitions.

Note that, since post-measurement state is not used, the disturbance of measurement is not problematic, and ${\delta }_{\mathit{j},{\mathit{j}}^{\prime }}^{(k),\mathrm{error}}$ can be taken arbitrarily small.