## Quantum Thermodynamics

Quantum thermodynamics investigates which kind of tasks can be performed when, given access to an infinite bath of particles in a thermal state, we are allowed to implement arbitrary energy-conserving quantum gates. As it turns out, this is a very good model of how things actually work in a lab. Each no-go theorem in quantum thermodynamics can thus be interpreted as a limitation on our ability to transform the universe.

Consider this: in order to probe a quantum system, we must make it interact with a measurement device, see the figure below. While in the classical case the role of the device or battery is merely to provide energy to the target system, in the quantum case, the measurement device also acts as a *catalyst*. By providing a resource of coherence between different energy eigenstates, the measurement device is able to effect certain (otherwise impossible) evolutions on the system under observation.

In [1], we study how constraints on the energy spectrum of the measurement device translate into limitations on the measurements which we can effect on a target system with a non-trivial energy operator. We find that certain measurements of two-level systems require an infinite amount of coherence (and thus, an infinite amount of energy) to be implemented. We also derive a universal relation between the average energy of a measurement device and the maximum precision of the measurements it allows one to carry on two-level systems. More specifically, given a measurement device with average energy E ̅, used to probe a two-level system with energy gap Δ, the inaccuracy ε of the device satisfies

The measurement devices which saturate this relation are described by quantum states which have not appeared earlier in the literature. We call them *battery states*, and they perform substantially better than coherent states, the usual probe resource in quantum optics.

All the above applies only when the target is a two-level system. For higher dimensional targets, we devised algorithms to characterize the set of achievable measurements, given a promise on the energy spectrum of the probe, or its full state specification. Our work thus identifies the boundaries between what is possible or impossible to measure, i.e., between what we can see or not, when energy conservation is at stake.

Rather than focusing on how much we can know about a target system, a more conventional approach to quantum thermodynamics is to assess how much work we can extract from it. In this regard, past literature on quantum thermodynamics has mainly focused on the potential of non-Gibbsian quantum *states* for work extraction, overlooking the fact that non-thermal quantum transformations or *channels* also constitute a thermodynamical resource. In [2], given access to a number of quantum channels, we consider the problem of how to incorporate them into a thermal engine so as to distill a maximum amount of work.

We took an interest in this problem because certain theoretical proposals to solve the measurement problem in quantum mechanics postulate that closed quantum systems are subject to a non-unitary decohering process in the position basis. In principle, this non-thermal resource could be harnessed to extract work from nothingness via a suitable “collapse engine”. The importance of solving humanity’s long-standing energy problem is nicely put by the extra-terrestrial character Chocky in John Wyndham’s 1968 novel:

“[Your fuels] are your capital. When they are spent you will be back where you were before you found them. This is not progress, it is profligacy […]. Most of your power is being used to build machines to consume power faster and faster, while your sources of power remain finite. There can only be one end to that. You should be employing your resources, while you still have them, to tap and develop the use of a source of power which is not finite. Once you have access to an infinite supply of power you will have broken out of the closed circle of your solar-economy”. -John Wydnham,Chocky

Inspired by Wyndham’s prose, we managed to reduce the problem of maximizing the extractable work to a tractable optimization over finitely many variables. We find that, under the Ghirardi-Rhimini-Weber theory, collapse engines are indeed possible, but highly impractical: we estimate that 1 kiloton of hydrogen is required to power a 40-Watt lightbulb indefinitely [2].

[1] M. Navascués and S. Popescu, *How energy conservation limits our measurements*, Phys. Rev. Lett. 112, 140502 (2014).

[2] M. Navascués and L. P. García-Pintos, *Non-thermal quantum channels as a thermodynamical resource*, Phys. Rev. Lett. 115, 010405 (2015).

**Additional readings**

J. Goold, M. Huber, A. Riera, L. del Rio and P. Skrzypczyk, *The role of quantum information in thermodynamics --- a topical review*, J. Phys. A: Math. Theor. 49, 143001 (2016).

A. Bassi, K. Lochan, S. Satin, T. P. Singh and H. Ulbricht, *Models of Wave-function Collapse, Underlying Theories, and Experimental Tests*, Rev. Mod. Phys. 85, 471-527 (2013).