Quantum Information and the Second Law

Title: Irreversible work and Maxwell demon in terms of quantum thermodynamic force

Authors: B. Ahmadi, S. Salimi, A. S. Khorashad

Institutions: Department of Physics, University of Kurdistan, P.O.Box 66177-15175, Sanandaj, Iran and International Centre for Theory of Quantum Technologies, University of Gdansk, Wita Stwosza 63, 80-308, Gdansk, Poland

Manuscript: Published in Nature Scientific Reports (open access)

Decoherence is the phenomenon that successfully explains the so-called quantum-classical transition: quantum coherence, which allows systems to maintain uniquely quantum superposition between states, is lost to an external environment. Once coherence is lost, the system effectively acts as though it is classical. This effect explains the rarity of macroscopic quantum phenomena, and it can be understood as a quantum information flow process: quantum information which is originally localised in the system of interest, describing the superpositions initially present between the eigenstates of the system, is dissipated via the system’s interaction with a large reservoir.

However, some classes of ‘system + reservoir’ dynamics are characterised by a backflow of information into the system: information flows out from the system into the reservoir, and after a finite time, some amount of this information returns. This is often described as memory, because the information describes past states of the system. Dynamics of this kind are called non-Markovian, and can often be seen when the reservoir has a small size, or is structured in some way. The presence or absence of information in a system is closely linked to the system’s entropy. In fact, entropy can be thought of as the amount of information about a system which is inaccessible. So decoherence – the irreversible loss of information to a reservoir – is associated with a large entropy increase, whereas non-Markovian information backflow is associated with entropy decrease.

The Second Law of thermodynamics states that entropy production is always positive for irreversible processes, and zero for reversible processes. It does not allow a system’s entropy to globally decrease – although small local decreases may be observed in thermal fluctuations, they provide a negligible contribution to the whole system. In quantum information theory terms, we’d say that the Second Law describes the tendency of information to diffuse out of systems. So it is not completely clear how non-Markovian dynamics, in which information becomes more localised, can be consistent with the Second Law.

Thermodynamics as we usually understand it cannot be straightforwardly applied to quantum systems. It relies on the ability to average over thermal fluctuations, which are negligible with respect to the large systems considered. In quantum systems, no such guarantee can be made: both thermal fluctuations and quantum fluctuations can play a very significant role in the dynamics of the whole system, and systems can be made of very few component parts. Therefore, in order to understand how local entropy reductions can be consistent with the Second Law, we need to rethink our understanding of thermodynamics to explicitly include quantum information. In a paper published in early 2021, Ahmadi et al [1] explicitly incorporate quantum information into an expression of the Second Law, and give information flow and backflow a thermodynamic interpretation.

Thermodynamic Efficiency

The connection between thermodynamics and information theory can be encapsulated by the Maxwell’s demon thought experiment. In the thought experiment, there is a box filled with gas at temperature $T$, and a partition in the centre of the box. A small demon stands by a massless door in the partition, and strategically opens the door so that all the particles pass into one half of the box, and the other half of the box is a vacuum. Maxwell intended his demon to challenge the interpretation of the Second Law: without doing any work, the demon has lowered the entropy of the box. However, it can be understood in a different way: the demon is only able to change the system’s entropy via possession of information about the particles – i.e. their position and momentum. Therefore, information can be used to perform more work than expected by the Second Law. This understanding led to the famous slogan“information is physical” [2] – meaning it has to be accounted for in thermodynamics.

The authors address the question of incorporating information into the Second Law by considering the work that can be done by a variety of thermodynamic systems. The fundamental relation of thermodynamics can be written as

$\textrm{d} F = \textrm{d} U - T \textrm{d} S = \textrm{d} W - T \textrm{d} S,$

where $F$ is the Helmholtz energy, describing the maximum extractable work, $U$ the internal energy of the system, $W$ the actual extracted work, and $S$ the entropy production during the process in which work is extracted. We can see from this relation that when entropy increases, $\textrm{d} S > 0$, the extracted work is smaller than the maximum extractable work. But when entropy decreases, $\textrm{d} S < 0$ – i.e. because of a quantum non-Markovian process – the extracted work is higher than the maximum.

Consider a classical engine which uses two reservoirs, at temperatures $T_h$ and $T_c$, as in the above image. It absorbs an amount of heat $\Delta Q_h$ from the hot reservoir, performs an amount of work $\Delta W$, and rejects an amount of heat $\Delta Q_c$ to the cold reservoir. During the most efficient possible process – the Carnot cycle, which is reversible and generates no entropy – the efficiency of the engine is

$\eta_C = \frac{W}{\Delta Q_H} = 1 - \frac{T_c}{T_h}.$

Other, less efficient, processes cannot reach the Carnot efficiency, due to irreversibility and entropy increase. If the entropy production during the cycle is $\Delta S = \Delta S_1 + \Delta S_2$, the efficiency is

$\eta = \eta_C - \frac{T_c \Delta S}{\Delta Q_h},$

with $\eta = 1 - \frac{T_3}{T_1}$. The goal now is to determine a similar expression for a corresponding quantum engine, which may well have $\Delta S < 0$.

Reversible and Irreversible Work

In order to analyse the work done by a quantum thermodynamic system, the authors partition the work into two parts: the reversible work $\Delta W_{\rm rev}$ and the irreversible work $\Delta W_{\rm irr}$. The reversible work is

$\Delta W_{\rm rev} = k T \Delta I + \Delta F_{\infty},$

where $F_{\infty}$ is the Helmholtz energy of the equilibrium state – essentially, the work that can be extracted from the system after it has reached thermal equilibrium with its environment –
and $I(t) = S(\rho(t) || \rho_{\infty})$ is the relative von Neumann entropy between the state of the system at time $t$ and the equilibrium state $\rho_{\infty}$. Relative entropy is a measure of how much information is shared between two quantum states – how easy it is to distinguish the two states, so $I(t)$ tells us how far away $\rho(t)$ is from the equilibrium state.
In general, the reversible work should be negative because it is being done by, not on, the system.

The irreversible work is $\Delta W_{\rm irr} = k T \Delta S,$ positive when $\Delta S > 0$, which means it reduces the magnitude of the reversible work.

These definitions can be understood as follows. The reversible work $\Delta W_{\rm rev}$ is the maximum amount of internal energy which would be “spent” by the system if the system was undergoing a reversible process – e.g. a Carnot cycle. This is directly dependent on the information content of the system, via the relative entropy. The irreversible work is the amount of internal energy which cannot be spent, due to loss of information from the system. Therefore, we can refer to $\Delta W_{\rm irr}$ as encoded information, because it is inaccessible.

Quantum Decoding

Let’s think about an example quantum system. Because it interacts with an environment, it must be described by a density matrix $\rho(t)$ rather than a wavefunction $\psi(t)$. The density matrix is a much more general description of a quantum state, and describes systems which can lose information. When we are using the density matrix, expectation values of operators are found by taking the trace – for example, the energy expectation value is $\langle E \rangle = {\rm Tr} (\hat{H}\rho).$
Our example system has initial state $\rho_0$ at time $t=0$, and it interacts with a bath at temperature $T$, according to Hamiltonian $H$. After the system has evolved over a time $t$, the state of the system is $\rho_t$. The irreversible work after this time is

$\Delta W_{\rm irr} = k T [S(\rho_0 || \rho_{\infty}) - S(\rho_t || \rho_{\infty})].$

The first term is the information shared by the initial state and the equilibrium state, which represents an information minimum. The second term is the information shared between the current state and the equilibrium state – so the quantity $\Delta W_{\rm irr}$ quantifies the information which has been lost between time $0$ and $t$, relative to the total amount of information the system can contain. In other words, it is the entropy which has been gained over the evolution from $\rho_0$ to $\rho_t$. It is worth noting that the information lost during an open quantum system evolution is usually information about the coherence – i.e. which superpositions the initial state contained.

We want to consider a possible cycle that this quantum system can undergo. Let’s construct a cycle between two reservoirs at temperature $T_h$ and $T_c$. There are four steps:

1. The system begins in state $\rho_0$, and interacts with the hot reservoir over time $t_1$, until it is in state $\rho_1$. The interaction is described by the Hamiltonian $H_0$. The change in the system’s energy expectation value over the interaction is equivalent to the amount of heat it absorbs: $\Delta Q_h ={\rm Tr}[H_0(\rho_0) - {\rm Tr}[H_0(\rho_1)]= {\rm Tr}[H_0(\rho_0 - \rho_1)]$. The entropy of the system changes by $\Delta S_h$, which has a contribution from the heat absorption, $\frac{\Delta Q_h}{k T_h}$, and a contribution from the evolution of the state of the system: $S(\rho_0||\rho_{\infty}) - S(\rho_1 || \rho_{\infty}) - \int_0^{t_1} {\rm Tr}\left (\rho(t) d_t \ln \rho(t) \right) \textrm{d} t.$
2. The system decouples from reservoir. While staying in the same state $\rho_1$, the interaction Hamiltonian is slowly (adiabatically) changed from $H_0$ to $H_1$. No entropy is produced.
3. Much like in Step 1, the system interacts with the cold reservoir according to the Hamiltonian $H_1$ and evolves from state $\rho_1$ to state $\rho_0$. The heat rejected to the cold reservoir during this step is $\Delta Q_c = {\rm Tr}[H_1(\rho_0 - \rho_1)]$. The entropy of the system changes by $\Delta S_c$, which – just like in Step 1 – has a contribution from the heat rejection, and a contribution from the evolution of the state of the system.
4. The system decouples from the cold reservoir and, while the system remains in state $\rho_0$, the interaction Hamiltonian is adiabatically changed from $H_1$ to $H_0$. No entropy is produced.

The work done during this cycle is $\Delta W = \Delta W_{\rm rev} + \Delta W_{\rm irr}$. The irreversible work is

$\Delta W_{\rm irr} = k T_h \Delta S_h + k T_c \Delta S_c,$

which contains a contribution from Step 1, the hot reservoir interaction, and a contribution from Step 3, the cold reservoir interaction.
Therefore, the efficiency is

$\eta_Q = \eta_C - \frac{k T_h \Delta S_h + k T_c \Delta S_c}{\Delta Q_h}$

When the system is non-Markovian, the evolution of the state of the system can cause a negative entropy contribution. For a sufficiently non-Markovian system, $k T_h \Delta S_h + k T_c \Delta S_c < 0$, and then $\eta_Q > \eta_C$. Therefore, non-Markovian dynamics can be used to construct engines which are more efficient than a Carnot engine.

This can never happen in classical equilibrium thermodynamics – unless there is an external feedback mechanism, i.e. a Maxwell demon. Maxwell’s demon can be thought of as an information decoder – it takes information which was inaccessible to the system, and makes it accessible. This information decoding describes a negative entropy production, and therefore an efficiency greater than the Carnot efficiency. However, this cannot be achieved without a demon in classical thermodynamics, whereas in quantum thermodynamics non-Markovianity can play the role of the demon.

The Second Law of Thermodynamics

The Second Law can now be extended to quantum systems:

In a quantum thermodynamic process, information can be encoded and also decoded for the system to do work, and this encoded (decoded) work equals temperature T times entropy production of the system, i.e.

This explicitly incorporates information into the Second Law. For classical macroscopic thermodynamics, it reduces to just the encoded part. This definition emphasises the connection between thermodynamics and information, instead of focusing on defining an arrow of time dependent on positive entropy production, and ensures that there are no violations in the presence of quantum non-Markovianity or Maxwell demons.

Summary

The authors of this paper aimed to explicitly include information into a more general definition of the Second Law of thermodynamics. They divided the work done by a thermodynamic system into two contributions: the reversible work, which is the maximum available work in the absence of information flow at all, and irreversible work, which quantifies the amount of work that is gained or lost due to information flow into or out of the system. This partitioning allowed the authors to derive the generic efficiency of an engine, which in the quantum non-Markovian case can be higher than the Carnot efficiency. This was given a thermodynamic interpretation: negative entropy production corresponds to information being decoded, so that it becomes accessible to the system, and more work is performed than expected by the usual formulation of the Second Law. Based on this analysis, the authors have introduced a novel formulation of the Second Law which incorporates information and is not violated by quantum non-Markovian systems.

References

[1] B. Ahmadi, S. Salimi, A. Khorashad,Scientific reports2021,11, 1–9.

[2] R. Landauer et al.,Physics Today1991,44, 23–29.

Sapphire Lally works on modelling non-Markovian effects in open quantum systems.

Thanks go to Akash Dixit for his many helpful edits and suggestions.