Work and entropy of mixing in isolated quantum systems

Mixing two types of particles is one of the most common occurrences in nature. Whether it is the down-to-earth act of putting milk into our coffee or the release of $\text{CO}_{2}$ into the atmosphere, the underlying physical laws remain the same. Such processes also have practical implications—most notably in determining how much energy is needed to separate two compounds once mixed. For gases, this minimal energy has a fundamental bound, revealing that capturing $\text{CO}_{2}$ before it is released is far more efficient than attempting to remove it afterward [1]. A similar principle applies to other modern environmental challenges, such as desalination [2] or filtering microplastics [3] and hormones [4] from our water supply, where proactive measures can significantly reduce both costs and ecological impacts.

In classical physics, where gases are assumed to be in thermal equilibrium at a well-defined temperature, the energy cost of mixing is well understood. However, in systems that do not equilibrate or are not connected to a thermal bath the usual assumptions break down. Consequently, it becomes unclear how much energy is required to separate the gases. This challenge is especially pronounced in isolated quantum systems, which are now routinely realized and studied in experimental settings [5, 6, 7].

These questions are also fundamentally observer-dependent: to study how much energy it costs to separate the two gases, one must first define what it means “to separate.” Typically, this notion carries a connotation of spatial separation, such as one type of gas being confined within a box while the other remains outside. Even in this case, however, the energy cost depends on the size of the box, which is determined by the observer or the agent attempting to separate the gases [8]. This definition is inherently tied to what the agent can achieve, shaped by their specific capabilities and limitations in carrying out the separation [9].

This interplay between the observer’s experimental capabilities and knowledge lies at the heart of the Gibbs mixing paradox (GMP). Consider two observers, one who can distinguish the two gases and the other who cannot. Recognizing the difference, the first observer assigns a higher entropy to the observed state, based on the larger number of microstates in compliance with any macroscopic observation. The second observer perceives no distinction and assigns a lower entropy, asserting that all particles are of the same type. Crucially, the first observer’s knowledge provides an advantage—if the gases are spatially separated, they can extract energy by allowing the gases to mix (classically, through semi-permeable membranes). The second observer, unaware of the distinction, cannot exploit this opportunity. As E.T. Jaynes noted [9]: “We would observe, however, that the number of fish that you can catch is an ‘objective experimental fact’; yet it depends on how much ‘subjective’ information you have about the behavior of fish.”

While the role of different observers is widely acknowledged in these discussions, their treatment remains somewhat ambiguous in the absence of a formal framework defining what constitutes an “observer.” Instead, the situation is typically described qualitatively, with explanations stating that one observer can distinguish the two gases while the other cannot. From this premise, the appropriate definition of equilibrium entropy is either postulated or argued for, but it is not systematically deduced from the original statement. This lack of rigorous derivation leaves the connection between the observer’s capabilities and the entropy definition somewhat imprecise. Does “being able to distinguish” imply the ability to probe each particle individually and determine its type, or does it refer to probing a specific region of space to ascertain the proportion of particles of each type within it? These two interpretations are fundamentally different and lead to distinct outcomes, both in terms of the observer’s knowledge and the resulting thermodynamic description of the system.

This also highlights another aspect of the paradox: the abrupt, discontinuous change in entropy that occurs when particles are distinguishable, even if differences between them are minimal. The mere potential for distinction leads to a significantly larger associated entropy. This apparent paradox arises from treating entropy as a fundamental property of the physical system. However, if entropy is instead understood as an informational measure—reflecting the observer’s perceived ignorance about the system and defining the limits of their ability to extract energy—the paradox dissolves. In this interpretation, entropy is relational, shaped by the observer’s knowledge and capabilities rather than an intrinsic characteristic of the system.

In this paper, we formalize the concept of an observer and their capabilities in precise mathematical terms. This systematic approach removes the need for overly intuitive reasoning. Instead, it allows us to directly approach the question with higher clarity and precision. Thereby it allows us to quantify the additional energy that an observer with a higher resolution can extract from a system compared to an observer with a lower resolution. As an application, we analyze the difference in extracted energy between two observers in the Gibbs mixing scenario, in the generic 1D lattice model with fermions of two colors. Additionally, we focus on isolated quantum systems and compute the difference in energy that an observer can extract before and after the two gases mix, thereby establishing a lower bound on the energy required to separate them again.

Interestingly, this result bears a strong mathematical resemblance to Landauer’s bound, with a key difference: instead of the bath temperature, an “observational temperature” emerges, defined by the observer’s experimental capabilities and limitations. Furthermore, since equivalent reasoning applies to classical systems, the insights gained extend beyond the quantum case. The framework allows us to discuss the energy required to separate isolated classical gases out of equilibrium, broadening the applicability of the results.

First, we review the background including mixing and observational entropy (Sec. II). Then, we derive the difference between extracted energy by two observers (Sec. III), with a specific result resembling Landauer’s bound for observers that have similar capabilities (Sec. III.1), followed by the difference in extracted energy by the same observer at different times (Sec. III.2). In Sec. IV, we analyze entropy in the Gibbs mixing scenario in quantum systems for an observer that can distinguish the particles (Sec. IV.1) and three versions of an observer who cannot (Sec. IV.2, IV.3, IV.4), and compare them (Sec. IV.5). We also analyze the change in entropy for these observers when an isolated quantum system evolves IV.6, showing that for all observers with various degrees of ignorance, the observed change in entropy is the same. Then we combine the previous results to show the difference between extracted energy in the Gibbs mixing scenario (Sec. V). In particular, we outline the assumptions under which we prove that an observer unaware of the two particle types extracts the same amount of energy as one who is aware, thereby resolving the Gibbs mixing paradox. We also include two simulations: first, the difference between two observers, and second, the difference for the same observer but at different times. Finally, we conclude (Sec. VI).

Consider two ideal gases—initially separated by a partition, both with identical volume, pressure, and temperature—that are allowed to mix. The Gibbs Mixing Paradox (GMP) arises because of the differing observer abilities. An Observer Rick, who can distinguish the gases notices an entropy increase from mixing. Morty, on the other hand, cannot distinguish them and thus attributes no entropy change. This discontinuity depends on the gases being distinguishable, challenging classical thermodynamic notions of entropy (see Fig. 1). Note here, that we consider distinguishability between the gases rather than their constituent individual particles, which may be fundamentally indistinguishable in quantum physics. In second quantization, this corresponds to being able to distinguish different spatial modes or different values of spin.

Gibbs Mixing Paradox. Jaynes [9] framed entropy as a subjective measure tied to an observer’s coarse-graining of microstates. For Rick, reverting the system to its original state requires work proportional to the entropy change; for Morty, simply reinserting the partition suffices. This perspective anchors entropy in information. Building on Jaynes’ perspective, many authors have explored the GMP from both philosophical and physical standpoints [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26]. These approaches can be further categorized. Philosophical approaches focus on particle identity and the foundation of entropy. For example, Maslov [14] emphasizes contextual particle distinguishability, while Ihnatovych [24] critiques classical entropy additivity. From an information-theoretic perspective, Lairez [26] aligns with Jaynes. Operational approaches resolve the discontinuity pragmatically. Allahverdyan and Nieuwenhuizen [11] replace coarse-graining with ergotropy (extractable work), showing it decreases continuously as gases become indistinguishable. Peters [19] attributes entropy changes to uncertainty in particle locations. Also, in agreement with Jaynes, Yadin et al. [27] reconcile observer-dependent work extraction with respect to a single heat bath in a first-quantization description. With a setup similar to the Hong-Ou-Mandel experiment, they argue that Morty can also extract work, contrary to the predictions of the classical GMP.

Mixing Beyond GMP. Mixing has also been studied independently. Yoshida and Nakagawa [28] quantify free energy changes in small quantum systems, while Takakura [29] explore entropy in generalized probabilistic theories. Sasa et al. [30] analyze quasistatic work in quantum thermal processes. These works highlight broader thermodynamic implications of mixing beyond classical paradoxes.

Observational entropy. Von Neumann argued that von Neumann entropy does not accurately represent thermodynamic entropy, as it assigns zero entropy to all pure states and fails to capture the information loss that arises when only macroscopic degrees of freedom are observed. As an alternative, he introduced macroscopic entropy [31, 32], which uses state-counting similar to Boltzmann entropy. This concept has recently gained renewed interest and been generalized to observational entropy [33, 34, 35, 36, 37, 38, 39], leading to a range of applications [40, 41, 42, 43, 44, 45, 46, 47, 48].

For a projective measurement defining a complete set of orthogonal projectors, known as coarse-graining ${\mathcal{C}}=\{{P}_{i}\}$, observational entropy is defined as

Here, $p_{i}=\tr[{P}_{i}{\rho}]$ represents the probability of the system being in macrostate $i$, while $V_{i}=\tr[{P}_{i}]$ denotes the volume of the macrostate, i.e., the number of microstates (pure states) contained within it. Each projector also corresponds to a subspace, defined by its support. Thus, we can equivalently write that a coarse-graining corresponds to a partition of the Hilbert space into subspaces, ${\mathcal{H}}=\bigoplus_{i}{\mathcal{H}}_{i}$, where $V_{i}=\dim{\mathcal{H}}_{i}$. The first term in observational entropy corresponds to the Shannon entropy, capturing the uncertainty in identifying the macrostate to which the system belongs. The second term represents the mean Boltzmann entropy, quantifying the residual uncertainty in determining the specific microstate due to the coarse-grained nature of the measurement.

In this section, we derive the difference in work extracted by two observers with different observational capabilities.

Consider two observers, Rick and Morty, with two measuring devices of different precision. These measuring devices correspond to coarse-grainings ${\mathcal{C}_{\mathrm{R}}}$ and ${\mathcal{C}_{\mathrm{M}}}$, We define the difference in observational entropies as

We assume that the measurement of Rick has a higher precision as measured by lower observational entropy, meaning that $\Delta S\geq 0$. For what follows, we require this assumption only for the particular quantum state from which the observers will extract energy, while for other states this inequality may be reversed. This will make all other quantities of type $\Delta$ introduced below positive.

The work extracted by performing a joint extraction on a large number of copies of the system, which is characterized by the measurement probabilities in a single coarse-grained basis ${\mathcal{C}}$, is given by observational ergotropy [49],

There, ${\rho}_{\beta}=\exp(-\beta{H})/Z$ is the thermal state with inverse temperature $\beta=1/kT$ implicitly defined by the von Neumann entropy, as a solution to $S_{{\mathrm{vN}}}({\rho}_{\beta})=S_{\mathcal{C}}$, and $k$ is the Boltzmann constant. We call temperature $T$ obtained by solving this equation an observational temperature. Expression $\tr[{H}{\rho}]$ represents the initial energy of the state from which the energy is extracted, while $\tr[{H}{\rho}_{\beta}]$ is the average energy of the final state after the extraction protocol is employed.

The work extraction protocol is partially random, meaning that one first employs a randomizing unitary which washes away the information about the degrees of freedom that cannot be obtained from the coarse measurement ${\mathcal{C}}$, and then the global work extraction unitary that is optimized for the average state. In the language of quantum reference frames, the first step is known as “twirling” and encodes the lack of a quantum reference frame [50]. Observational ergotropy, Eq. (3), then represents the average extracted work upon repetition of this protocol many times. See Fig. 2 for the sketch of this extraction protocol. For more details see Refs. [49, 51, 44].

The difference in extracted energies by the two observers is equal to,

where $S_{{\mathrm{vN}}}({\rho}_{{\beta_{\mathrm{R}}}})=S_{\mathcal{C}_{\mathrm{R}}}$ and $S_{{\mathrm{vN}}}({\rho}_{{\beta_{\mathrm{M}}}})=S_{{\mathcal{C}_{\mathrm{M}}}}$.

This formula holds completely generally, for any two coarse-grainings. The positive difference in entropies $\Delta S$ implies that work extracted $\Delta W$ is also positive.

The formula for the observational ergotropy fundamentally depends on the initial energy, which is unknowable to the observers given that they obtain only limited information by measuring in their basis, ${\mathcal{C}_{\mathrm{R}}}$ and ${\mathcal{C}_{\mathrm{M}}}$, respectively. It can, however, be estimated from these measurements [52]. On the other hand, the formula for the difference no longer depends on this initial energy, only on the respective observational entropies, which allows us to study it in more detail.

Note that in some scenarios, especially when the initial state energy is small, execution of the protocol may cost more energy than is being extracted for both observers. The positive difference in extracted work $\Delta W$ then means that Rick has lost less energy than Morty by executing the protocol.

We derive the entropy difference $\Delta S$, which we can then input into the formula for the ergotropy difference, for a quantum version of the Gibbs mixing paradox.

We consider observers illustrated in Fig. 3: Rick, who is able to determine the total number of particles of each color in both the left and the right side of the box; and three clones of Morty, who can identify the total number of particles in each part of the box but not their color, but have varying degrees of knowledge about colors of particles in the full system. Morty 1: knows the total number of blue particles in the full system; Morty 2: does not know the total number of particles in the full system, but is aware that these two types exist; and Morty 3: is not even aware of the existence of the two types of colors and as a result, will associate incorrect volumes to the observed macrostates.

The ability to determine or identify is described by a specific coarse-graining representing a macroscopic measurement that goes into the definition of observational entropy that either of them associates to the observed system.

In the following calculations, we consider both the blue and the red particles to be either fermions or hard-core bosons, which will affect the counting, however, a very similar derivation can also be done with bosons. The results for these will coincide when the gas is sufficiently dilute.

We consider the total Hilbert space to be that of any possible configuration of any number of blue and red particles on the lattice of $L$ sites – i.e., spatial modes in second quantization. Here, color could be a spin degree of freedom (immutable within the observers capability; evolved with a spin-preserving Hamiltonian). The actual size of the total Hilbert space will not actually matter, as long as every state that we consider can be contained within it. For instance, when considering Hamiltonians that preserve the total number of particles of each type, we can restrict to a Hilbert space that respects this symmetry. However, both of these approaches will give the same results since these come from counting the dimensions of the macrostates and not the dimension of the full Hilbert space.

For simplicity, we assume that both sides are of the same lattice size, $w$. The non-symmetric case, of $L_{A}$ and $L_{B}$ being the size of the left and the right side, respectively, can be obtained by substituting $\ln w\rightarrow a\ln L_{A}+(1-a)\ln L_{B}$ into the results below ($a$ is defined below).

Inserting the entropy difference, Eq. (20), into Eq. (7), we obtain the difference between the amount of work Rick and Morty can extract in both quantum and classical Gibbs mixing scenario,

where $\tilde{c}_{E}={C_{\!E}}/(Nk)$ is the dimensionless specific heat. We reiterate that temperature $T$ is not a temperature of a thermal bath. Instead, it is the observational temperature given by $S({\rho}_{T})=S_{{\mathcal{C}}_{M}}$. For an ideal gas described in Table 1, we obtain $\Delta W=NkT\ln 2(1-\frac{1}{3}\ln 2+\frac{2}{27}(\ln 2)^{2}+\cdots)\approx 0.805\times NkT\ln 2$.

We observe that the formula for Rick’s entropy difference at different times, Eq. (24), is the same as the aforementioned entropy difference. Following the arguments of Section III.2, this shows that Rick will extract $\Delta W$ less energy if he chooses to perform the extraction at a later time, as opposed to immediately.

While different types of Morty associate different entropy, this difference remains constant throughout the evolution, Eq. (28). Consequently, the increase in entropy will be the same for all of them, Eq. (26). The extracted work difference depends only on this entropy increase. There is no advantage for either Morty 1 or 2; they will extract the same amount of energy.

Morty 3 requires much more care. This is because he lacks a correct understanding of the underlying Hilbert space, resulting from his unawareness of the existence of the two particles, due to his inability to distinguish between different colors (which is his defining feature). This we explained in Section IV.4. As a result, he further misjudges the actual unitary he applies to the system when attempting to extract work, misjudges the measurement he performs, and would infer a density matrix from quantum tomography that deviates from the true one. However, in Appendix C we prove that under some reasonable assumptions, he will extract the same amount of energy as Morty 2. This includes three consistency assumptions: one on the density matrix, stating that the actual density is macroscopically consistent with the one he would infer by performing any measurement that is unable to distinguish the colors of the particles, which we call an admissible measurement. Second, on the measurements, saying that the probability of distribution outcomes he predicts is the same as the actual one when he performs any admissible measurement. Third, on the unitary operations, saying that the actual unitary measurement is consistent with what is expected and detectable by admissible measurements. Furthermore, we have the super-selection rule assumptions, which state that it is not possible to have superpositions between different types of particles. This means that states such as $\ket{0+}+\ket{-0}$ are not allowed, nor can they be created by applying either measurement or unitary operations, or by letting the system evolve naturally under the Hamiltonian. So far, these assumptions have been relatively weak. Finally, we have one non-trivial assumption that will not be true for many systems, namely that the Hamiltonian does not depend on the color of the particles but only on the configuration—describing whether the particle is or is not at a given lattice site, irrespective of its color. This means that every two states with the same configuration, such as $\ket{0+}$ and $\ket{0-}$ corresponding to $\ket{01}$, must have the same energy. This ensures that Morty 3 correctly understands the system’s energy, given the constraints of what he is able to measure.

With these assumptions, Morty 3 extracts exactly the same amount of energy as Morty 2. Thus, the difference in extracted work between the two times will be the same.

To conclude, there is no advantage for either clone of Morty. In other words, it does not matter whether Morty knows or does not know whether there are two types of particles in the system rather than just one, or how many blue particles there are in total. In either case, he will extract the same amount of energy. This resolves the Gibbs mixing paradox: while absolute values of associated entropy might be different, there is no difference in terms of what different clones of Morty can practically do.

We provide two simulations of the ergotropy difference: Static, applied only to the initial state which is localized in one of Rick’s macrostates but for varying system sizes, see Fig. 4; and dynamic, where the system starts in an initial macrostate of Rick but is then evolved with Hamiltonian that preserves the total number of particles of each kind and distributes them uniformly in the long-time limit, see Fig. 5.

The Hamiltonian is given by

with periodic boundary conditions. $c_{i}^{\dagger}$ and $c_{i}$ represent creation and annihiliation operators at site $i$, respectively, and $n_{i}$ represents the number operator. The numerical computations are done in the chaotic phase, with $t_{1}=v_{1}=1$ and $t_{2}=v_{2}=0.96$.

To minimize the error, we compute the entropy exactly from the binomial coefficients, Eqs. (9) and (12). Further, we compute the ergotropy difference by inserting this entropy difference into the exact formula, Eq. (4), and then from varying levels of approximation using the perturbative expansion, Eq. (7).

In this work, we investigated how two observers—each characterized by a distinct ability to distinguish macrostates—assign entropy to an isolated quantum or classical system, and how this observer-dependent entropy influences the energy they can extract. Focusing on the difference in extractable energy between the two observers, we found that its perturbative expansion closely resembles Landauer’s bound, with higher-order corrections determined by the ratio of entropy to heat capacity. Notably, rather than the usual temperature of a thermal bath, an “observational temperature” emerges, governed by the observers’ knowledge of the system.

In the case of critical systems, we identified a critical amount of knowledge—reached when the observational temperature equals the critical temperature—beyond which the amount of extractable work jumps discontinuously. We further demonstrated that the same formula for the extracted energy difference applies when considering a single observer extracting energy at different times.

We then applied these ideas to the Gibbs mixing paradox, in which one observer can distinguish two types of particles while another cannot. A closer examination revealed three distinct subtypes of the latter observer, each associating a different amount of entropy to the system.

However, all of them observe the same entropy increase. For the two observers who are aware of the existence of two types of particles, this means that they extract the same amount of work. For the third, who is not aware of their existence, we assume consistency of their description, superselection rules, and independence of the Hamiltonian on different particle types. With these assumptions, we show that they extract exactly the same amount of energy as the second observer, who is aware of the existence of two particle types, but unable to distinghuish them. This resolves the Gibbs mixing paradox as follows. Even though an observer who discovers the presence of “two colors” of particles experiences a discontinuous jump in the entropy they assign to the system (counterintuitively, knowing more but associating higher entropy!), this new knowledge does not provide a practical advantage for work extraction if the particles remain indistinguishable in their measurements.

We further argued that the loss of extractable energy caused by mixing must also place a lower bound on the energy required to separate the particles, though there are subtleties in reversing the protocol that merit future investigation. Our analysis was restricted to projective measurements; generalizing to POVMs [38, 61], including non-commuting measurements [62, 63] would be an interesting avenue for follow-up work. While free energy is the relevant quantity in standard mixing scenarios, we showed that “observational ergotropy” plays an analogous role in the isolated systems considered here. It would be worthwhile to explore whether these concepts overlap, potentially through studying local equilibrium states where a combination of the local particle number (as considered here) and the local energy coarse-grainings define a “dynamical thermodynamic entropy [64].” Further, as dynamical thermodynamic entropy corresponds with thermodynamic entropy for systems in equilibrium, it would be interesting to compare the work difference derived from this work and that from the adiabatic expansion of ideal gas. This would serve as a check for the consistency of the derived theory. Another promising direction is to examine Fisher information associated with temperature measurements, given its relation to the energy variance that appears in the perturbative expansion of the extracted work.

Finally, we propose two experimental platforms which allow the manipulation of two conserved, distinguishable particle states to test these ideas in a quantum setting. First, we can use Bose-Einstein condensates (BECs), e.g. of ${}^{87}\text{Rb}$, where hyperfine states (as “colors”) are spatially separated by applying a magnetic field to induce Zeeman splitting [65, 66, 67]. The evolution conserves the density of each type, and mixing can be created by modulating magnetic field [68]. The second promising platform is a 2D optical lattice with Lithium atoms in hyperfine states to denote $\ket{\uparrow}$ and $\ket{\downarrow}$. Each lattice site holds either no atom, one atom of either spin, or two atoms of opposite spin. Using a spin-removal beam and site-resolved imaging [69, 70], we can determine the spin number, type, and mixing at each site, enabling the computation of observational entropy and extractable work. To realize the work-extraction protocol in practice would require further modifications—such as developing controls for the random and extraction unitaries—or adapting the protocol to fit experimental constraints.