Fluctuating Work: From Quantum Thermodynamical Identities to a Second Law Equality

We investigate the connection between recent results in quantum thermodynamics and fluctuation relations by adopting a fully quantum mechanical description of thermodynamics. By including a work system whose energy is allowed to fluctuate, we derive a set of equalities that all thermodynamical transitions have to satisfy. This extends the condition for maps to be Gibbs preserving to the case of fluctuating work, providing a more general characterization of maps commonly used in the information theoretic approach to thermodynamics. For final states, block diagonal in the energy basis, this set of equalities is a necessary and sufficient condition for a thermodynamical state transition to be possible. The conditions serve as a parent equation that can be used to derive a number of results. These include writing the second law of thermodynamics as an equality featuring a fine-grained notion of the free energy. It also yields a generalization of the Jarzynski fluctuation theorem which holds for arbitrary initial states, and under the most general manipulations allowed by the laws of quantum mechanics. Furthermore, we show that each of these relations can be seen as the quasiclassical limit of three fully quantum identities. This allows us to consider the free energy as an operator, and allows one to obtain more general and fully quantum fluctuation relations from the information theoretic approach to quantum thermodynamics.

Popular Summary

In nature, there is always a preferred direction in which physical processes occur: A cup of hot coffee left alone will cool down, and a piece of glass that shattered against the floor will not reassemble. This situation is a consequence of the second law of thermodynamics, the universal principle that fixes this preferred direction. The second law of thermodynamics implies that the useful energy that physical systems contain tends to decrease with time. This principle, however, only holds on average; although processes that obey it occur with very high probability, others that break it are not ruled out, in principle. In our everyday, macroscopic lives, these rule-breaking processes are never seen because the probability of such anomalous fluctuations is almost completely suppressed. This same situation is not the case, however, for systems on microscopic and quantum scales.

Here, we study the general features of such statistical fluctuations on microscopic and quantum scales.

The second law of thermodynamics in the macroscopic regime is expressed as an inequality (e.g., “The amount of energy flowing from the cup to the air has to be larger than zero.”) However, when fluctuations become important on small scales, we show that one can state the second law as an equality instead and prove that it holds in the most general process allowed by the laws of quantum mechanics. This formulation of the second law contains a very large amount of information, dramatically constraining the probability and size of fluctuations, and, for instance, it tells us that the particular fluctuations that break the second law only occur with exponentially low probability. We then find an analogous set of fully quantum identities that govern thermodynamics at the quantum level.

We expect that our findings will pave the way for additional studies in the context of thermodynamics.

Figure 1

Given a system in state $\rho ={\sum }_{s=1}^n{p}_s|s⟩⟨s|$ with Hamiltonian $H_S={\sum }_{s=1}^n{E}_s|s⟩⟨s|$, its thermomajorization diagram (see Ref. [22] for more details) is formed by first relabeling the pairs of occupation probabilities and energy levels so that $p_1{e}^{\beta E_1}\ge p_2{e}^{\beta E_2}\ge \cdots \ge p_n{e}^{\beta E_n}$ and then plotting the points $\{{\sum }_{s=1}^k{e}^{-\beta E_s},{\sum }_{s=1}^k{p}_s{\}}_{k=1}^n$, joining them together to form a concave curve. Here, we show examples for a qubit. In the absence of a work storage system, $(\rho ,H_S)$ can be transformed into $(\sigma ,H_S)$ using a thermal operation if and only if the curve associated with $\rho$ is never below that of $\sigma$. In this example, the curve of $\rho$ crosses that of $\sigma$, so the transformation is not possible. When all values in a work distribution have the form $w_{s{s}^{\prime }}={\alpha }_{s^{\prime }}-{\gamma }_s$, the existence of a thermal operation mapping a quasiclassical state $\rho$ to quasiclassical state $\sigma$ while producing such a work distribution can be determined by considering the curves associated with $(\rho ,{\sum }_{s=1}^n(E_s+{\gamma }_s)|s⟩⟨s|)$ and $(\sigma ,\sum _{s=1}^n(E_s+{\alpha }_s)|s⟩⟨s|)$. In this example, the curve associated with $\rho$ and $\{{\gamma }_s\}$ lies above that of $\sigma$ and $\{{\alpha }_s\}$, so the transformation from $\rho$ to $\sigma$ is possible with respect to this work distribution. By adjusting ${\gamma }_s$ and ${\alpha }_s$ so that both curves are straight lines that overlap, one can make the average work of the transformation equal to the change in free energy, and the transformation becomes reversible.

Figure 2

As a simple example of the second law equality, one can think of single qubit erasure. In the limit of perfect erasure, the second law equality reads in this case $e^{\beta w_0}+e^{\beta w_1}=1$, and the average work spent is $⟨w⟩=\frac{1}{2}(w_0+w_1)$. Here, we show the trade-off between $w_0$ and $⟨w⟩$ for such perfect erasure. The optimal work value for erasure is the usual Landauer cost at $w_0=w_1=-T\log 2$. As seen in Eq. (53), perfect or near-perfect erasure requires the work cost to fluctuate arbitrarily.