Fluctuating States: What is the Probability of a Thermodynamical Transition?

If the second law of thermodynamics forbids a transition from one state to another, then it is still possible to make the transition happen by using a sufficient amount of work. But if we do not have access to this amount of work, can the transition happen probabilistically? In the thermodynamic limit, this probability tends to zero, but here we find that for finite-sized and quantum systems it can be finite. We compute the maximum probability of a transition or a thermodynamical fluctuation from any initial state to any final state and show that this maximum can be achieved for any final state that is block diagonal in the energy eigenbasis. We also find upper and lower bounds on this transition probability, in terms of the work of transition. As a by-product, we introduce a finite set of thermodynamical monotones related to the thermomajorization criteria which governs state transitions and compute the work of transition in terms of them. The trade-off between the probability of a transition and any partial work added to aid in that transition is also considered. Our results have applications in entanglement theory, and we find the amount of entanglement required (or gained) when transforming one pure entangled state into any other.

Popular Summary

Of all the laws of physics, the second law of thermodynamics likely has the largest impact on our everyday lives. We witness this law in action all of the time: A cup of hot coffee in a cold room will cool down rather than heat up; if we have a collection of coins, all tails facing up, and give it a shake, some will flip to heads. It is thanks to the second law of thermodynamics that we instantly recognize when we are watching a movie backwards. But what is the probability that we will witness a particular process that appears to violate the second law of thermodynamics? Is it possible to observe a collection of random coins that spontaneously flip to all tails, or any process that appears to happen backwards in time? The short answer is no. The probability of this situation occurring is so small that we could wait far longer than the age of the Universe and still not observe these rare events. However, in this study, we theoretically demonstrate that we may actually observe a particular process that violates the second law in microscopic or quantum systems.

These events only occur with a certain probability, which decreases as the size of the system being studied increases. We compute the maximum probability for witnessing any particular violation, which in many cases turns out to be significant for small systems. We then show that it is possible to achieve this violation and provide a method for doing so. We also provide bounds on this probability in terms of the work one needs to make the transition happen with certainty. We shed light on which energetic processes can be seen at quantum scales; these processes appear to be fundamentally different from those that occur on macroscopic scales. In particular, we show how spontaneous processes can go in the opposite direction, with respect to time, than what we would normally expect.

We anticipate that our findings will pave the way for further exploration of probabilistic state transformations in the context of quantum thermodynamics.

Figure 1

Lorenz curves. (a) The Lorenz curve for $\rho$ is defined by plotting the points: $\{(k/n,\sum _{i=1}^k{\eta }_i){\}}_{k=1}^n$. (b) The transition from $\sigma$ to $\rho$ is possible under NO, as the curve for $\sigma$ is never below that of $\rho$. (c) The Lorenz curve for a maximally mixed state is given by the dashed line from (0,0) to (1,1). All other states majorize it. (d) $s_{\log (5/4)}$ is an example of a sharp state. (e) $s_{I_{\infty }(\sigma )}$ is the least sharp state that majorizes $\sigma$.

Figure 2

We plot the curves of $\rho$, $\sigma$, and $\rho$ compressed by $p^{*}$ (with respect to the $x$ axis). The points A and B at which the vertical ratio between the curves of $\rho$ and $\sigma$ is maximum (which sets $l_1$ and $p^{*}$), and the sharp states that pass through those points, are also shown as dashed lines. After compressing the Lorenz curve of $\rho$ by a ratio of $p^{*}$, the point B will be taken to C, which will always either be below the curve of $\sigma$ or just touching it. This proves the lower bound in Eq. (33).

Figure 3

We plot the curves of $\rho$, $\sigma$, and $\rho$ compressed by $p^{*}$ (with respect to the $x$ axis). The points A and B at which the vertical ratio between the curves of $\rho$ and $\sigma$ is maximum (which sets $l_1$ and $p^{*}$) and the sharp states $I_{\infty }(\rho )$ and $I_{\infty }(\sigma )$ are also shown as dashed lines. Given that for sharp states all bounds are saturated, the appropriate maximum vertical and horizontal ratios coincide, and are ${\eta }_1/{\zeta }_1$, the ratio of the heights of ${\mathrm{B}}^{\prime }$ and ${\mathrm{A}}^{\prime }$. But this ratio is, by definition, bigger than or equal to $p^{*}$, the ratio between A and B. This means that if the curve of $\rho$ is compressed by $p^{*}$, the point ${\mathrm{B}}^{\prime }$ is mapped to C just above or touching the curve of $\sigma$, proving the upper bound of Eq. (33).

Figure 4

We show the $\beta$-ordered thermomajorization diagrams for various states of the system. Note that different states may have different $\beta$-orderings, and the markings on the $x$ axis correspond to one particular $\beta$-ordering. The curves always end at $(Z,1)$. The thermomajorization criteria states that we can take a state to another under thermal operations if and only if the curve of the initial state is above that of the final state. Hence, in this case (provided $\rho$ is block diagonal in the energy eigenbasis) there is a set of operations such that $\sigma \stackrel{\mathrm{TO}}{\to }\rho$, but not for the reverse process.

Figure 5

Here, we illustrate the construction of the state ${\rho }_{\sigma }$ used in the proof of Theorem 4. The points of the curve $\rho$ that are at the same horizontal position as the elbows of $\sigma$ are joined, and by concavity the resultant curve is always below $\rho$.

Figure 6

Here, we show graphically the steps of the proof of the first part of Theorem 5. In the decomposition of Eq. (44) the curve $p\sigma$ must always be below that of ${\rho }^{\prime }$ and, hence, also $\rho$. This sets the maximum probability $p^{*}$ as defined in Eq. (45). Both $p\sigma$ and the disordered ${\rho }^{\prime }$ have the same $\beta$-ordering.

Figure 7

We show the thermomajorization curves of a state to which a work qubit in one of two pure states has been tensored. Adding this work system takes $Z\to Z(1+e^{-\beta W})$, extending the $x$ axis. When we tensor with the ground state to form $\rho \otimes |0⟩⟨0|$, the curve is the same as for $\rho$ alone, but when the excited state is tensored, there is a change in the energy levels of the $\beta$-ordering, and as a result the curve of $\rho$ is compressed by a ratio of $e^{-\beta W}$.

Figure 8

Here, we show how $p^{*}(W)$ varies as a function of $W$ for qubits under noisy operations when $W_{\rho \to \sigma }<0$. Note the behavior at $W=0$, indicating the function is not convex in $W\ge W_{\rho \to \sigma }$.