Fundamental concepts

To state our result below, we need to explain three different concepts. First, we need some properties of generalized physical theories (for example, refs 13,14,15,16,17). Second, we recall the concept of uncertainty relations, and finally the second law of thermodynamics.

General theories

Although it is not hard to prove our result for quantum theory, we extend our result to some more general physical theories. These are described by a probabilistic framework that makes the minimal assumptions that there are states and measurements, which can be made on a physical system (for example, refs 18,19). Even for general theories, we denote a state as , where is a convex state space. In quantum mechanics, is simply a density matrix. The assumption that the state space is convex is thereby generally made17 and says that if we can prepare states and , then the probabilistic mixture prepared by tossing a coin and preparing or with probability each is also an element of . A state is called pure if it cannot be written as a convex combination of other states. Measurements consist of linear functionals called effects. We call an effect pure if it cannot be written as a positive linear combination of any other allowed effects. Intuitively, each effect corresponds to a possible measurement outcome, where is the probability of obtaining ‘outcome’ given the state . More precisely, a measurement is thus given by . For quantum mechanics, we will simply label effects by measurement operators. For example, a projective measurement in the eigenbasis of the Pauli operator is denoted by . The assumption that effects are linear, that is, is linear in , is essentially made for all probabilistic theories17 and says that when we prepared a probabilistic mixture of states, the distribution of measurement outcomes scales accordingly.

Uncertainty relations

A modern way of quantifying uncertainty20,21 is by means of entropic uncertainty relations (see ref. 22 for a survey), or the closely related fine-grained uncertainty relations12. Here, we will use the latter. As for our cycle, we will only need two measurements with two outcomes, and each measurement is chosen with probability , we state their definition only for this simple case. Let and denote the two measurements with effects and , respectively. A fine-grained uncertainty relation for these measurements is a set of inequalities

To see why this quantifies uncertainty, note that if for some , then we have that if the outcome is certain for one of the measurements (for example, ), it is uncertain ( ) for the other. As an example from quantum mechanics, consider measurements in the and eigenbases (we use the common convention of labelling the and eigenbases states as and , respectively). We then have for all pure quantum states

The same relation holds for all other pairs of outcomes , and . Depending on , the eigenstates of either or saturate these inequalities. A state that saturates a particular inequality is also called a maximally certain state12.

For any theory such as the quantum mechanics, in which there is a direct correspondence between states and measurements, uncertainty relations can also be stated in terms of states instead of measurements. More precisely, uncertainty relations can be written in terms of states if pure effects and pure states are dual to each other in the sense that for any pure effect there exists a corresponding pure state , and conversely for every pure state an effect such that . Here, we restrict ourselves to theories that exhibit such a duality. This is often (but not always) assumed17,19. As a quantum mechanical example, consider the effect and the state . We then have with and .

For measurements and consisting of pure effects, let and denote the corresponding dual states. The equation 1 then takes the dual form

For our quantum example of measuring in the and eigenbasis, we have , , and . We then have that for all pure quantum effects

The same relation holds for all other pairs , and . Again, measurement effects from the eigenstates of either or saturate these inequalities. In analogy, with maximally certain states, we refer to effects that saturate the inequalities (3) as maximally certain effects. From now on, we will always consider uncertainty relations in terms of states.

Second law of thermodynamics

Finally, the second law of thermodynamics is usually stated in terms of entropies. One way to state it is to say that the entropy of an isolated system cannot decrease. These entropies can be defined for general physical theories, even for systems that are not described by the quantum formalism19,23,24 (Supplementary Methods). However, for our case, it will be sufficient to consider one operational consequence of the second law of thermodynamics25,26: there cannot exist a cyclic physical process with a net work gain over the cycle.

Main findings

Our main result is that if it was possible to violate the fine-grained uncertainty relations as predicted by quantum physics, then we could create a cycle with net work gain. This holds for any two projective measurements with two outcomes on a qubit. By the results of ref. 12, which showed that the amount of non-locality is solely determined by the uncertainty relations of quantum mechanics and our ability to steer, our result extends to a link between the amount of non-locality and the second law of thermodynamics (note that there is no violation of uncertainty classically, it is rather that classical measurements have at most probabilistic notions of uncertainty to begin with).

In the following, we focus on the quantum case, that is, in the situation where all the properties except the uncertainty relations hold as for quantum theory. In the Supplementary Methods, we extend our result to more general physical theories that satisfy certain assumptions. In essence, different forms of entropies coincide in quantum mechanics, but can differ in more general theories19,23,24. This has consequences on whether a net work gain in our cycle is due to a violation of uncertainty alone, or can also be understood as the closely related question of whether certain entropies can differ.

Let us now first state our result for quantum mechanics more precisely. We consider the following process as depicted in Fig. 1. We start with a box, which contains two types of particles described by states and in two separated volumes. The state is the equal mixture of the eigenstates and of two measurements (observables) and . The state is the equal mixture of and . We choose the measurements such that the equal mixture is the completely mixed state in dimension . We then replace the wall separating from by two semi-transparent membranes, that is, membranes which measure any arriving particle in a certain basis and only let it pass for a certain outcome. In the first part of the cycle, we separate the two membranes until they are in equilibrium, which happens when the state everywhere in the box can be described as . Then, in the second part of the cycle, we separate again into its different components.

Figure 1: The impossible process. At the start (a), two states ρ 0 and ρ 1 are in separated volumes, containing a total of N such states, N/2 in ρ 0 and N/2 in ρ 1 . Second (b), the wall separating them is replaced by two semi-transparent membranes, of which M 0 is transparent for and M1 for . These membranes then move apart until they reach their position at equilibrium (c). This happens when the state in the entire volume can be described by the mixture ρ. New membranes are inserted on the side, and the state ρ=Σ j q j |σ j 〉〈σ j | is then (d,e) separated into its pure components σ j by membranes, which are exactly opaque to σ j . In panel f, the volumes are then further subdivided to obtain sizes proportional to and , that is, the weights occuring in the eigen decomposition of and . We then unitarily transform the pure component σ j in each volume into the corresponding pure component or of ρ 0 and ρ 1 . Finally, the components of ρ 0 (ρ 1 ) are mixed to reach the initial configuration (a). Full size image

We find that the total work, which can be extracted by performing this cycle is given by

Here, is the von Neumann entropy of the state. The entropy appearing in the above expression is simply the Shannon entropy of the distribution over measurement outcomes when measuring in the basis and , respectively (the Shannon entropy of a probability distribution is given by . All logarithms in this paper are to base ).

Example

To illustrate our result, consider the concrete quantum example, where the states are given by

The work that can be extracted from the cycle then becomes

The fine-grained uncertainty relations predict in the quantum case that and are at most . We see that a theory, which can violate this uncertainty relation, that is, reach a larger value of , would lead to —a violation of the second law of thermodynamics.