Third Law of Thermodynamics as a Single Inequality

Abstract

The third law of thermodynamics in the form of the unattainability principle states that exact ground-state cooling requires infinite resources. Here we investigate the amount of non-equilibrium resources needed for approximate cooling. We consider as resource any system out of equilibrium, allowing for resources beyond the i.i.d. assumption and including the input of work as a particular case. We establish in full generality a sufficient and a necessary condition for cooling and show that for a vast class of non-equilibrium resources these two conditions coincide, providing a single necessary and sufficient criterion. Such conditions are expressed in terms of a single function playing a similar role for the third law to the one of the free energy for the second law. From a technical point of view we provide new results about concavity/convexity of certain Renyi-divergences, which might be of independent interest.Comment: Extended discussion on approximated and exact catalysts and models for the source of work. 22 pages. 2 Figure

Cooling to Absolute Zero: The Unattainability Principle

“…In this section we will summarize the results of Ref. [7]. There, cooling processes are considered which involve arbitrary heat bath and catalyst.…”

“…where K(ρ R , H R , ρ S , H S , β) → 0 as T → 0. (See [7] for a definition of K.) Hence, in the limit of very cold final states where the UP applies, both inequalities converge to a single one ruled by the vacancy. These conditions can be also re-expressed for the multi-copy case ρ R = ρ ⊗n , where each copy has a local Hamiltonian h, to obtain lower bound for the final temperature T ≥ k(nV β (ρ, h)) −1 where k is a constant.…”

“…[U, H] = 0 finite bath resource catalyst Allahverdyan (2011) [15] finite d B and J B no yes work no Reeb (2014) [16] finite J B no yes work no Scharlau (2016) [17] finite d B and J B yes yes work no Masanes (2017) [8] finite C B (E) and W wc no yes work no Wilming (2017) [7] finite resources yes no non-eq. yes Müller (2017) [18] finite catalyst yes no both yes…”

“…Although this seems to be in contradiction with the third law of thermodynamics, we note that this procedure only works if the initial resource ρ R is exactly pure. If instead we consider slightly noisy work ρ R = |0 0 | + (1 − ) |1 1 |, then perfect cooling is impossible for any > 0, regardless of the value of W (even if it diverges) [7]. In this sense, perfect cooling is only possible if we have already as a resource a state which is not full-rank.…”

Work estimation and work fluctuations in the presence of non-ideal measurements

“…In particular, it was shown in [14] that ideal projective measurements are not exactly realisable in practise because they require the preparation of initially pure pointer states (at least in some nontrivial subspace) to satisfy condition (ii). However, the third law of thermodynamics prevents one from reaching the ground-state of any system with finite resources [15,33,34,[39][40][41][42]. Since any other (pure) state necessarily has higher energy than the ground state, the third law thus excludes ideal measurements.…”