Physical setup

Our goal is to provide ultimate quantitative bounds applicable to any cooling procedure—namely, we wish to find a lower bound for the temperature that a system can reach after any process which uses some given resources or lasting some given time t. Therefore, we must allow for the most general quantum transformation, that is, those that respect total energy conservation and are microscopically reversible (unitary). This general setup includes thermodynamically irreversible protocols and also unrealistic protocols where total control of the microscopic degrees of freedom of the bath is required. Surprisingly, we will find here, as was found for the case of the second law25,27,29,30, that having such unrealistic degree of control does not appear to give one an advantage over having very crude control.

We will show that the density of states of the reservoir assisting the cooling process has an important impact on how fast a system can be cooled. (The density of states Ω(E) is the number of states with energy E.) We see that the faster Ω(E) grows, the lower the temperature that can be achieved with fixed resources or in a fixed amount of time. Even more: if Ω(E) grows exponentially or faster, then cooling to absolute zero in finite time is in principle possible, allowing for a violation of the third law. However, we will see that exponential or super-exponential Ω(E) should be regarded as unphysical. This becomes more intuitive when expressed in terms of the (micro-canonical) heat capacity C(E), related to S(E)=ln Ω(E) via

where primes represent differentials. If Ω(E) grows exponentially or faster, then C(E) is infinite or negative, which is regarded as unphysical. If Ω(E) is sub-exponential, then C(E) is positive. And, the faster Ω(E) grows, the larger C(E) is. Only a reservoir with infinite-dimensional Hilbert space can keep S(E) growing for all E. And indeed, infinite-dimensional reservoirs are the ones that allow for faster cooling. However, our results are general and also apply to the finite-dimensional case.

Suppose that we want to cool a quantum system with Hilbert space dimension d, and Hamiltonian H S having ground-state degeneracy g, gap above the ground state Δ and largest energy J. What are the resources required to do so?

Fundamental assumptions

Let us specify the setup more concretely and collect the assumptions we will adopt (those which come from first principles):

(i) We consider the start of the process to be when the system has not yet been put in contact with the work storage system (the weight) nor the reservoir, so that initially, the global state is ρ S ⊗ρ B ⊗ρ W . While other initial starting scenario may be of interest, its consideration is beyond the scope of the current paper.

(ii) We allow for the most general quantum transformation on system, bath and weight, which is reversible (unitary) and preserves total energy. This might appear restrictive compared with the paradigms that allow arbitrary interaction terms, however this is not the case, since arbitrary interactions can be incorporated into the model as shown in Appendix H of ref. 27 and in ref. 25, simply by allowing the energy of the work system to fluctuate. In many paradigms, this is implicitly enforced by assuming that all missing energy is counted as work. Paradigms which relax this condition are essentially ignoring the energy transferred to other systems, or treat these other systems as classical. Essentially, we impose energy conservation to ensure we properly account for all energy costs associated with the interaction while the various unitaries or interaction terms simply transfer or take energy from the weight to compensate. The cooling process is thus any transformation of the form

where U is a global unitary satisfying

(iii) The work that is consumed within the transformation is taken from the weight. Since we are interested in ultimate limitations, we consider an idealized weight with Hamiltonian having continuous and unbounded spectrum . Any other work system can be simulated with this one30. We denote by w max the worst-case value of the work consumed, that is,

w max will generally be much larger than the average work 〈W〉. In any physically reasonable process carried out in finite time, one expects it to be finite.

(iv) We also require, as in ref. 29, that the cooling transformation commutes with the translations on the weight. In other words, the functioning of the thermal machine is independent of the origin of energies of the weight, so that it just depends on how much work is delivered from the weight. This can be understood as defining what work is—it is merely the change in energy we can induce on some external system. This also ensures that the weight is only a mechanism for delivering or storing work, and is not, for instance, an entropy dump (see Result 1 in the Supplementary Discussion). It also ensures that the cooling process always leaves the weight in a state that can be used in the next run or the process. Thus

where the Hermitian operator Π acts as for all . Beyond this, we allow the initial state of the weight ρ W to be arbitrary. In particular, it can be coherent, which provides an advantage27.

(v) We assume that the bath has volume V and is in the thermal state at given inverse temperature , with Z B the partition function of the bath. We denote the free energy density of the bath (in the canonical state ρ B ) by .

(vi) The micro-canonical heat capacity (2) is not negative C(E) for all energies E. This implies that S(E) is sublinear in E. We also prove in the Supplementary Methods that if S(E) grows linearly or faster, then perfect cooling in finite time is possible.

With these assumptions, we show that to perfectly cool the system to absolute zero, at least one of these two resources, the volume of the bath V, or the worst-case value of the work consumed w max has to be infinite. Also, we bound the lowest achievable temperature of the system in terms of V and w max .

Quantifying unattainability from first principles

With assumptions (i)–(vi), we consider two cases, one where the initial and final state are thermal, and one where we allow arbitrary initial and final states. Our first result concerns the former, and states that in any process where the worst-case work injected is w max , the final temperature of the system cannot be lower than

in the large w max ,V limit. The micro-canonical free-energy density at inverse temperature β 0 is defined by

where E 0 is the solution of equation S′(E 0 )=β 0 . Recall that, when the volume of the bath V is large, it is usually the case that f mic (β 0 )=f can (β 0 ) and these are independent of V.

Let us analyse the behaviour of equation (7) in terms of the resources invested. As w max grows, β 0 decreases and f mic increases, yielding a lower final temperature . Since all the volume dependence in equation (7) is explicit, hence, a larger V also translates into a lower final temperature.

In what follows we provide a bound for the physically relevant family of entropies

where α>0 and ν∈[1/2, 1) are two constants. Such an entropy is extensive, and if we set it describes electromagnetic radiation (or any massless bosonic field) in a D-dimensional box of volume V. It is generally believed that there is no other reservoir that has a density of states growing faster with E than this36, and certainly none which has ν≥1. The later, corresponds to the bath with negative heat capacity discussed earlier, which enables cooling with finite w max . In the Supplementary Discussion, we adapt bound (7) to the entropy (9), obtaining

up to leading terms. Now, all the dependence on V and w max is explicit. In particular, we observe that larger values of V and w max allow for lower temperatures. And also, larger values of ν, which amount to a faster entropy growth, allowing for lower temperatures.

As mentioned above, the cooling processes that we consider are very general. In particular, they can alter the Hamiltonian of the system during the process, as long as the final Hamiltonian is identical to the initial one H S . This excludes the uninteresting cooling method consisting of re-scaling the Hamiltonian H S →0. However, our bounds can easily be adapted to process where the final Hamiltonian differs from the initial one, as we will discuss in the conclusion.

Let us now consider the more general case, where neither the initial or final state need be thermal, but can instead be arbitrary. As it is already well known14,15,17,18,30, the unattainability of absolute zero is not a consequence of the fact that the target state has low energy, but rather that it has low entropy. Hence, this directly translates to the unattainability of any pure state, or more generally, any state with rank g lower than the initial state. These type of processes are generally known as information erasure, or purification. Now we analyse the limitations of any processes which takes an arbitrary initial state ρ S and transforms it into a final state with support onto the g-rank projector P. We quantify the inaccuracy of the transformation by the error . For the sake of clarity, we assume that the system has trivial Hamiltonian H S =0 (the general case is treated in the Supplementary Discussion), and we denote by λ min and λ max the smallest and largest eigenvalues of ρ S . In the Supplementary Methods, we show that any process ρ S → has error

The results presented above, as well as others of more generality presented in the Supplementary Discussion, quantify our ability to cool a system (or more generally, put it into a reduced rank state), in terms of two resources: the volume of the bath V, and the worst-case fluctuation of the work consumed w max . They thus constitute a form of third law, in the sense that they place a bound on cooling, given some finite resources. We now wish to translate this into the time it would take to cool the system, and we will do so, by consider the notion of a thermal machine and making two physically reasonable assumptions.

Thermal machines

Let us recall that the field of computational complexity is based on the Church-Turing thesis—the idea that we consider a computer to be a Turing machine, and then explore how the time of computation scales with the size of the problem. Different machines may perform differently—the computer head may move faster or slower across the memory tape; information may be stored in bits or in higher dimensional memory units, and the head may write to this memory at different speeds. Nature does not appear to impose a fundamental limit to the dimension of a computer memory unit or the speed at which it may be written. However, for any physically reasonable realization of a computer, and whatever the speed of these operations, it is fixed and finite, and only then do we examine the scaling of time with problem size. And what is important is the overall scaling of the time with input (polynomial or exponential), rather than any constants. Likewise here, we will consider a fixed thermal machine, and we will assume that it can only transfer a finite amount of energy into the heat bath in finite time. Likewise, in a finite time, it cannot explore an infinite size heat bath. A thermal machine which did otherwise would be physically unreasonable.

We can consider both V and w max as monotonic functions of time t. The longer our thermal machine runs, the more work it can pump into the heat bath, and the larger the volume of the bath it can explore. For any particular thermal machine, one can put a finite bound on by substituting these functions into equation (10). In particular, if we assume that the interaction is mediated by the dynamics of a local Hamiltonian, then the interaction of a system with a bath of volume V and spacial dimension d will take time

where v is proportional to the speed of sound in the bath (or Lieb–Robinson velocity37), and V1/D the linear dimension of the bath. The implementation of general unitaries takes much longer than equation (12), but this serves as a lower bound. Since we are interested here in the scaling of temperature with time, rather than with constant factors, we need not be concerned by the fact that practical thermal machines operate at much slower speeds. Of course, just as with actual computers, thermal machines generally have speeds well below the Lieb–Robinson bound. Note that, despite V being finite, the Hilbert space of the bath can be infinite-dimensional. If one wanted to have a bound which was independent of the thermal machine, and independent on the speed of sound which is a property of the bath, then one could always take v to be the speed of light. While such a bound would not be practically relevant, it would be fundamental. This is similar to bounds on computation, where to get a fundamental bound, one should take the gate speed to be infinite (since there is no fundamental bound on this) and convert the number of bits used in the process to time by multiplying by the speed of light.

A relationship between worst-case work w max and time t is obtained by noticing the following. In finite t it is not possible to inject into the bath an infinite amount of work. For simplicity, here we assume a linear relationship

where the constant u will depend on the interactions between system and weight. However, we stress that, if a particular physical setup is incorrectly modelled by the relations (12) and (13), then any other bound t≥h 1 (w max ) and t≥h 2 (V) is also good. As long as h 1 and h 2 are strictly monotonic functions the unattainability principle will hold.

Limitations using thermal machines

For any particular thermal machine, we can now derive limitations on the temperature that can be reached in a given time t. Since the physical system with the fastest entropy growth that we are aware of is radiation, it is worthwhile to dedicate the next paragraph to the case in equation (9), because this should provide a bound with wide validity. Using the particular relations (12) and (13), and substituting them into equation (10), for the case of radiation, we obtain

in the large t limit. Our bound (14) can be straightforwardly adapted to any other relation t≥h 1 (w max ) and t≥h 2 (V). It is interesting to observe in equation (14) the relationship between the characteristic time (how long does it takes to cool to a fixed ) and the size of the system V S . Exploiting the usual relation ln d∝V S we obtain the sublinear scaling