Aspects of entropy in classical and in quantum physics

The entropy of entanglement provides an operational entanglement measure, i.e., has an operational meaning for pure states. It gives the asymptotic conversion rates to maximally entangled qubit pairs $\vert {\phi }^{+}\rangle =(\vert 0\rangle \vert 0\rangle +\vert 1\rangle \vert 1\rangle )/\sqrt{2}$, and back. Consider n copies of a bipartite pure state |ψ⟩ shared between A and B with entropy of entanglements S_ψ . Then, one can find a sequence of local operations and classical communication—i.e., operations that act only on the qubits in A and the qubits in B, and A and B can communicate classically to share e.g. the results of measurements—that provide the transformations $\vert \psi {\rangle }^{\otimes n}{\leftrightarrow}\vert {\phi }^{+}{\rangle }^{\otimes n{S}_{\psi }}$ in the limit of large n. Notice that the transformations are approximate, but the error vanishes in the limit of n → ∞. So the entropy of entanglement tells us how many maximally entangled pairs one can generate from non-maximally entangled pairs, i.e. how useful the non-maximally entangled pairs are for applications such as teleportation (which requires one maximally entangled pair per teleportation). An entropy of S_ψ = 1/4 just means that n non-maximally entangled states |ψ⟩ contain the same amount of entanglement as n/4 maximally entangled pairs—and it is possible to extract this number of maximally entangled pairs. In turn, the reverse transformation is also possible. That is, the entropy of entanglement also provides the entanglement cost, i.e. the amount of entanglement that is required to prepare multiple copies of the non-maximally entangled states. Considering again S_ψ = 1/4, this means that n/4 maximally entangled pairs are necessary to prepare n copies of |ψ⟩ by local operations and classical communication only. The protocols are introduced and described in [11], further studied in [13].

We also remark that for exact transformations between bipartite pure states, a single measure such as the entropy of entanglement does not suffice. Majorization criteria based on the Schmidt vectors of source and target state determine the possibility of a transformation [14], and its maximal success probability.

In case of mixed states, the situation is not so straightforward anymore. The mixedness of the reduced density operator of one system can result either from the fact that the two systems are entangled, or because the original overall state of the two systems has been mixed itself. Thus, the entropy of the reduced density operator is not directly related to the entanglement of the state [10].

Consider the simplest case of a classical source that produces a bit either with value 0 at probability p₀, or 1 with probability p₁. Since all classical information can be encoded in bits, the results derived here can easily be generalized. A random sequence of n such bits will be binomially distributed, i.e. the probability that a given sequence contains k₁ ones and k₀ = n − k₁ zeros is given by ${p}_{1}^{{k}_{1}}{p}_{0}^{n-{k}_{1}}$ with p₀ = 1 − p₁, and there are binomial (n, k₁) such bit strings with exactly k₁ ones. Using Stirlings formula, we approximate the phase space partitions with k₁ ones as

$$\begin{equation}{\Omega}{({k}_{1})}_{n}=(n,{k}_{1})\simeq \frac{n!}{{k}_{1}!{k}_{2}!}\simeq {\left(\frac{{k}_{1}}{{n}_{1}}\right)}^{-{k}_{1}}{\left(\frac{{k}_{2}}{{n}_{1}}\right)}^{-{k}_{2}}.\end{equation} \tag{ 13 }$$

The binomial distribution is centered at ${\bar{k}}_{\!1}={p}_{1}n$, see figure 3. This center of the Gaussian corresponds to the relevant part of the phase space, given by

$$\begin{equation}{\Omega}{({\bar{k}}_{\!1})}_{n}\simeq {({{p}_{1}}^{{p}_{1}}{{p}_{2}}^{{p}_{2}})}^{-n}={2}^{nS({p}_{1})}.\end{equation} \tag{ 14 }$$

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From the fact that a binomial distribution approaches a Gaussian distribution in the large n limit, one sees that the bulk of the distribution lies in the interval ${k}_{1}\in ({\bar{k}}_{\!1}-O(\sqrt{n}),{\bar{k}}_{\!1}+O(\sqrt{n}))$. Taking the neighboring k₁s into account gives rise to a (vanishing) overhead, i.e. n(S(p₁) + δ), where δ → 0. In this (thermodynamic) limit, all fluctuations become irrelevant, and only the peak value remains, drastically reducing the relevant phase space volume. The number of bit strings in this interval is of the order of ${2}^{nS({p}_{1})}$, and labeling all these bit strings using nS(p₁) bits provides a way to successfully transmit any of the possible sequences with vanishing error probability.

Similarly, as in classical information theory, entropy also plays a key role in quantum information theory. One of the earliest and nicest examples is the quantum version of the noiseless coding theorem put forward by B Schumacher [15]. For any given source of quantum systems described by a set of pure states with associated probabilities, {p_j , |ψ_j ⟩}, it provides the number of qubits that have to be transmitted to successfully relay a message of length n. The entropy of the source, i.e. the von Neumann entropy of the mixed state ρ = ∑_j p_j |ψ_j ⟩⟨ψ_j |, S_ρ = −tr(ρ log ρ), gives the channel capacity. The transmission of n states from the source requires the transmission of nS_ρ qubits. The basic idea behind the reduction is similar as for classical sources. However, in this case, not simply bit strings need to be transmitted as possible microstates in phase space, but whole subspaces in Hilbert space. The idea that the number of phase space cells in classical physics is generalized to the dimension of the Hilbert space will appear again in section 4.3. Consider first the case of the source producing orthogonal states, i.e. ⟨ψ_i |ψ_j ⟩ = 0 whenever i ≠ j—and for simplicity also just two states |0⟩ and |1⟩ with probabilities p₀ and p₁, respectively. Following the same line of argumentation as in the classical case, there are likely sequences containing states of the form $\vert 0{\rangle }^{\otimes n-{k}_{1}}\otimes \vert 1{\rangle }^{\otimes {k}_{1}}$ with ${k}_{1}\approx {\bar{k}}_{\!1}={p}_{1}n$ (and permutations thereof). The basic idea is now to transmit the whole subspace spanned by all these basis vectors. The dimension of this subspace is ${2}^{nS({p}_{1})}$ as in the classical case, where we have used that the entropy of ρ coincides with the entropy of the classical probability distribution for orthogonal vectors. One can then show that submitting this subspace—which requires the transmission of nS(p₁) qubits—suffices to successfully transmit the quantum information of all possible strings the source emits—with a vanishing error in the limit n → ∞.

An extension to non-orthogonal states, and a larger alphabet (or more possibilities) is straightforward. One has to look at the mixed state that describes the source, ρ = ∑_j p_j |ψ_j ⟩⟨ψ_j |. One can use the spectral decomposition to write this mixed state in the form ρ = ∑_k λ_k |k⟩⟨k|—and it turns out that the eigenvalues and eigenstates of the spectral decomposition characterize the source completely. Thus, it does not matter how a given density matrix is obtained or realized—as a mixture of non-orthogonal states, or by mixing eigenstates with probabilities specified by the eigenvalues. All sources that are described by the same density matrix are indistinguishable, and the same number of qubits is required to successfully transmit the information over a noiseless quantum channel. A successful transmission of quantum information produced by the source requires that an entangled state specified by a purification of ρ, $\vert {\psi }_{\rho }\rangle ={\sum }_{k}\sqrt{{\lambda }_{k}}\vert k\rangle \vert k\rangle$ is preserved when one part of the state is transmitted through the quantum channel, and similarly for multiple copies. The probability distribution is a multinomial distribution rather than a binomial one as we treated in our example. However, there is still an expectation value for each of the basis states, given by ${\bar{k}}_{\!j}={p}_{j}n$. The multinomial distribution approaches a multi-dimensional Gaussian, and it is easy to see from the central limit theorem that the bulk of the distribution is contained around the expected values k_j ≈ p_j n. So states of the form ${\bigotimes}_{j}\vert {\psi }_{j}{\rangle }^{\otimes {k}_{j}}$ with k_j ≈ p_j n span the likely subspace, where multinomial coefficients describe how many states of this form exist (simply counting the number of permutations). From there, one can again determine the size of the likely subspace, which is given by 2^nS(ρ) leading to the necessity to transmit n[S(ρ) + δ] qubits, where δ is some positive constant that determines the error probability. In the limit of large n → ∞, this is necessary and sufficient to guarantee a vanishing error probability for δ → 0.

Not surprisingly, entropies also appear in other problems of quantum information theory, in particular in quantum communication. We will only mention the important role of the Holevo quantity, quantum conditional entropy and quantum mutual information, but not go into further detail here (see reference [10] for an overview on the role of entropy in entanglement theory, [16] for an overview on quantum channel capacities, and [17] for an overview regarding entropic uncertainty relations, which express the inequivalence of several measurements performed on a quantum state and play an important role in the context of proving security of cryptographic protocols).

Another interesting instance where entropies appear in quantum protocols are entanglement purification protocols. Entanglement is an important resource for various tasks in quantum information processing, e.g. to perform teleportation. However, pure state entanglement in the form of maximally entangled pairs is required, while due to the influence of noise, decoherence and imperfections, quantum states will typically be mixed and noisy. However, one can use several copies of a mixed state—which are e.g. generated by distributing them over some noisy channel—to distill fewer copies with higher fidelity. To this end, consider an ensemble of noisy entangled states of two qubits, where each pair is described by a density operator that is diagonal in the Bell basis |ϕ_j ⟩ = I ⊗ σ_j |ϕ₀⟩ with $\vert {\phi }_{0}\rangle =(\vert 0\rangle \vert 0\rangle +\vert 1\rangle \vert 1\rangle )/\sqrt{2}$. We thus have ρ = ∑_j λ_j |ϕ_j ⟩⟨ϕ_j | and Γ = ρ^⊗n , where λ₀ = F is the fidelity of the state. The key idea is to view the ensemble as a mixture of all possible combinations of tensor products of Bell states with probabilities given by the coefficients λ_j , where one lacks the information which of the configuration is present. In order to purify the ensemble and obtain multiple copies of maximally entangled pure states, one needs to figure out which of the configurations one is dealing with—i.e. one needs to learn the missing information. This is achieved by so-called random hashing, see reference [11].

In order to illustrate the basic idea, consider mixed states of rank 2, i.e. states with λ₀ = F, λ₁ = (1 − F) and λ₂ = λ₃ = 0. We refer to |ϕ₀⟩ as the desired state, and to |ϕ₁⟩ as the error state. We consider a bilateral CNOT operation, i.e. a controlled-X gate applied at party A and party B on two qubits. This is a local operation in the sense of the parties A and B, but joint operations on two copies. The action of this gate on two Bell pairs of the considered kind is as follows: |ϕ_i ⟩|ϕ_j ⟩ → |ϕ_i ⟩|ϕ_i⊕j ⟩. That is, information about the error in the first entangled pair is transferred to the second entangled pair. By using such a bilateral CNOT gate with several different pairs as source, and always the same pair as target, the target pair actually shows the parity information about the total number of error states in this sub-ensemble. This parity information, i.e. whether one has an even or odd number of errors in the whole considered sub-ensemble, can be read out by performing local Z measurements on the qubits of the target pair, and classically communicating the results. If the measurement results coincide, there was an even number of errors, if not, the number of errors is odd. Notice that the entanglement in the target pair is destroyed by the measurement, i.e. the pair is consumed, but one bit of information on the sub-ensemble is revealed. We will use this as a tool to learn the desired information about a bigger ensemble, and determine the actual configuration of error states, i.e. the number of error states and their position. Once this is known, the ensemble is actually purified, as one then knows which of the possible configurations is present, and one can correct error states to the desired ones by applying a σ_x operation on the qubit in B. It is actually sufficient to measure the parity information on the number of error states in randomly selected sub-ensembles, as from sufficiently many such measurements the distribution of the error states can be determined. Each of the measurements reveal one bit of information, and for large ensembles and randomly selected sub-ensembles, these bits are independent [11].

It remains to show how much information is required, i.e. how many pairs of the ensemble need to be measured. For this, we can again use the insight on likely sequences and likely subspaces discussed in the context of the noisy (quantum) coding theorem. The mixed state ρ^⊗n can be seen as a mixture of states of the form $\vert {\phi }_{0}{\rangle }^{\otimes {k}_{0}}\vert {\phi }_{1}{\rangle }^{\otimes {k}_{1}}$ with k₀ + k₁ = n, with corresponding probability ${\lambda }_{0}^{{k}_{0}}{\lambda }_{1}^{{k}_{1}}$, and there are binomial (n, k₁) permutations of this kind. The probability distribution is centered around ${\bar{k}}_{0}={\lambda }_{0}n=Fn$, ${\bar{k}}_{\!1}={\lambda }_{1}n=(1-F)n$, where the neighborhood correspond to the likely configurations. Assuming for simplicity that the number of error states is exactly ${\bar{k}}_{\!1}=(1-F)n$, there are ${2}^{nS({\lambda }_{0},{\lambda }_{1})}$ configurations of this kind, and one needs nS(λ₀, λ₁) bits of information to reveal which of the configurations is present. Further taking the neighboring configurations ${k}_{1}\approx {\bar{k}}_{\!1}$ into account leads to a small correction nδ. Hence by measuring n[S(λ₀, λ₁) + δ] pairs, one can determine which of the configuration is present, thereby purifying the remaining ensemble of n[1 − S(λ₀, λ₁) − δ] pairs. A similar analysis holds true for general Bell-diagonal states.

This entanglement distillation protocol is optimal for the discussed rank-2 states, but not for general rank-4 states. Further details can be found in reference [18]. Nevertheless, it is interesting to see that the same concept of likely sequences and subspaces, which rely on the simple combination of multinomials and the central limit theorem as illustrated in figure 3, shows up again in a broader context.

Next, we want to show that binomial distributions are also a key ingredient for the historically important derivation of Planck's radiation formula and its generalization in the framework of entanglement entropy. In 1900, Max Planck derived his famous formula describing black-body radiation using a simple oscillator model using the concept of classical phase space. In this section, we review and extend this model and compare phase space and Hilbert space approach.

Consider P different oscillators, among which N energy portions are randomly distributed. The energy portion is proportional to the associated oscillator frequency, i.e. ∝ ν. In order to fit with the experiment, Planck empirically found

$$\begin{equation}E\equiv \langle U\rangle =\frac{N}{P}{\epsilon}=\frac{{\epsilon}}{{\mathrm{e}}^{{\epsilon}/({k}_{\text{B}}T)}-1}.\end{equation} \tag{ 15 }$$

This equation can be understood as an expression for the temperature T = T(E) as a function of energy. Using some kind of mathematical reengineering, he derived from ∂S/∂E = 1/T(E) an expression for the entropy of the black-body radiation at temperature T = T(E) by integration of the empirical result (15), leading to

$$\begin{align}\hfill {S}_{\text{black}-\text{body}}& ={\int }^{E}\mathrm{d}{E}^{\prime }\frac{1}{T({E}^{\prime })}={k}_{\text{B}}\,\mathrm{ln}\,\frac{{\left(1+\frac{E}{{\epsilon}}\right)}^{1+\frac{E}{{\epsilon}}}}{{\left(\frac{E}{{\epsilon}}\right)}^{\frac{E}{{\epsilon}}}}\hfill \\ \hfill & =\frac{{k}_{\text{B}}}{P}\,\mathrm{ln}\,\frac{{(P+N)}^{P+N}}{{P}^{P}{N}^{N}}\simeq \frac{{k}_{\text{B}}}{P}\,\mathrm{ln}\left(\genfrac{}{}{0pt}{}{P+N-1}{N}\right)\hfill \end{align} \tag{ 16 }$$

meaning that the total number of microstates of the black-body radiation at temperature T within the oscillator model is larger than the usual value Ω_Boltzmann = P^N /N! from the Boltzmann statistics. When Planck derived the entropy (15) from the experimental data of the black-body radiation, the interpretation was not clear yet. In particular, the fundamental meaning of the proportionality constant in = hν had not been settled yet.

Quantum physics provides an interesting interpretation of Planck's constant h as the smallest countable unit in phase space. The classical phase space Ω(E, N, V) is thus divided into a discrete set of P 'Planck cells' with P = Ω(E, N, V)/(2πℏ)^3N in phase space ⁵ .

Now, suppose that N energy quanta will be filled into these P Planck cells. Then, we can distinguish three physically relevant expressions for the number of possible microstates:

$$\begin{equation}\begin{aligned}\hfill {{\Omega}}_{\text{Fermi\mbox{--}Dirac}}(E)& =\left(\genfrac{}{}{0pt}{}{P}{N}\right),\hfill \\ \hfill {{\Omega}}_{\text{Boltzmann}}(E)& =\frac{{P}^{N}}{N!},\hfill \\ \hfill {{\Omega}}_{\text{Bose\mbox{--}Einstein}}(E)& =\left(\genfrac{}{}{0pt}{}{P+N-1}{N}\right).\hfill \end{aligned}\end{equation} \tag{ 17 }$$

Indeed, repeating the steps shown after equation (15) for the black-body radiation, using Stirlings formula, we recover Fermi–Dirac statistics, Boltzmann statistics, and Bose–Einstein-statistics from these binomial distributions.

The subtle statistical differences between fermions, bosons, and the classical Boltzmann statistics can be read off from (17) as follows: in case of Boltzmann statistics, the energy quanta have no influence on each other. For fermions, each Planck cell may only be occupied at most once. Thus, the first energy quantum has P choices, the second (P − 1), and so on. As the energy quanta are indistinguishable for a given energy, we must divide by 1/N!, leading to P(P − 1)(P − 2)...(P − N + 1)/N! microstates. This is the well-known interpretation of Pauli's principle in Hilbert space, leading to antibunching.

For bosons, the reverse is true: the first energy quantum again has P choices, but in contrast to fermions, subsequent bosons have not less, but more choices: the second qubit has (P + 1), the third (P + 2), and so on. In the ansatz of Einstein [19], this statistical behavior is called stimulated emission: bosons of the same energy and in the same phase space cell stimulate each other, which is expressed by a larger phase space volume. The number of bosons that may occupy the same phase space cell is unlimited, and a given energy quantum in a cell increases the probability for the next quantum to occupy the same cell. This behavior is called bunching.

We may ask which of these two key ideas—Pauli's principle or stimulated emission—can be derived from more general principles. In both cases, experimental data has been the starting point for the statistical expression given by (17), and in both cases, up to know, the quest for 'deeper' or more fundamental reasons for this behavior are a topic of active research.

Note that the definition of a temperature as a statistical property is only possible in the thermodynamic limit, where Stirlings approximation in (17) can be applied. However, the statistics for bosons and fermions as described by the binomials in (17) remains valid also beyond this thermodynamic limit. We will discuss this point further in the following section, when proceeding to entanglement entropy in Hilbert space.

Classical entropy is defined in phase space, while quantum entropy is a function of amplitudes in an extended Hilbert space. At first, these two approaches seem to be very different. Here, we want to show how the concept of entanglement entropy can provide a unified viewpoint for both approaches.

Consider N spins of a paramagnet with Hamiltonian $H=\mu {\sum }_{k=1}^{N}\,\vec{\mathbf{B}}{\vec{\mathbf{S}}}_{k}$ in Hilbert space. Suppose that on average, n₁ spins are parallel, and n₂ antiparallel to the magnetic field at given temperature T. The total energy is then given by $U=\mu \vert \vec{B}\vert (N-2{n}_{1})=\mu \vert \vec{B}\vert ({n}_{2}-{n}_{1})\equiv {\epsilon}n$, and the mean energy per spin is given by $\langle E\rangle =\frac{n}{N}{\epsilon}$. Then, the phase space volume reads

$$\begin{equation}{{\Omega}}_{\text{Paramagnet}}(E)=\left(\genfrac{}{}{0pt}{}{N}{{n}_{1}}\right)=\frac{N!}{\left(\frac{N-n}{2}\right)!\left(\frac{N+n}{2}\right)!}.\end{equation} \tag{ 18 }$$

From this expression, we can derive the probability for spin up/down with respect to the direction of the magnetic field as ${p}_{\mathrm{u}\mathrm{p}/\mathrm{d}\mathrm{o}\mathrm{w}\mathrm{n}}=\frac{1}{2}(1\pm \frac{n}{N})=\frac{1}{2}(1\pm \mathrm{tanh}(\beta {\epsilon}))$ with β = 1/(k_B T). As the degree of freedom of spin is defined in Hilbert space, at first, this problem looks quite different as compared to degrees of freedom in phase space. Since no phase space description is available, the spin is sometimes considered to be prototypical for quantum physics. However, we will show in what follows, that this is not the case, as the description in phase space can also be generalized using entanglement entropy in an extended Hilbert space for bosonic systems. In this sense, the spin system is nothing special.

While the Hamiltonian $H=\mu {\sum }_{k=1}^{N}\,\vec{\mathbf{B}}{\vec{\mathbf{S}}}_{k}$ does not include interactions between the spins, the mere fact that the spins share the same temperature necessarily includes thermalization due to interaction with a heat bath. Whatever small this interaction might be, any such incoherent interaction leads to entanglement with the environment. Thus, we may view the mixed state ρ = p_up|0⟩⟨0| + p_down|1⟩⟨1| with entropy (8) in Hilbert space ${\mathcal{H}}_{A}$ as a result of entanglement with the environment in the extended Hilbert space ${\mathcal{H}}_{A}\times {\mathcal{H}}_{B}$. More in general, as shown in figure 4, we may divide the spin system with N = N_A + N_B spins into two subsystems A, B. For the number of possible microstates in each subsystem A, B, we obtain

$$\begin{align}\hfill {d}_{E}^{\text{Fermion}}& =\left(\genfrac{}{}{0pt}{}{N}{{n}_{1}}\right)=\sum\limits _{{n}_{1}^{A}=0}^{{n}_{1}}\left(\genfrac{}{}{0pt}{}{{N}_{A}}{{n}_{1}^{A}}\right)\left(\genfrac{}{}{0pt}{}{{N}_{B}}{{n}_{1}^{B}}\right)\hfill \\ \hfill & \equiv \sum\limits _{{n}_{1}^{A}=0}^{{n}_{1}}{\,d}_{A}^{\text{Fermion}}({n}_{A}){d}_{B}^{\text{Fermion}}({n}_{B})=\sum \underset{j{}}{\overset{\text{Fermion}}{d}}.\hfill \end{align} \tag{ 19 }$$

Thus, the Hilbert space is divided into a direct sum of the type

$$\begin{equation}H(E)=\underset{j}{\bigotimes}{H}_{j}({{\epsilon}}_{j})=\underset{j}{\bigotimes}[{H}_{A}({{\epsilon}}_{j})\times {H}_{B}(E-{{\epsilon}}_{j})],\end{equation} \tag{ 20 }$$

where d_E is the dimension is the full Hilbert space, and d_j = H_j (_j ) the dimension of a particular partition. How can we calculate the entanglement entropy for this system? The number of microstates d_A/B (n_A/B ) defines the dimension of the Hilbert spaces ${\mathcal{H}}_{A/B}$. Indeed, there are d_A/B (n_A/B ) states composed from n_A/B qubits |0⟩ and N − n_A/B qubits |1⟩. We label these states as |a_i ⟩ and |b_k ⟩, with i = 1, ...d_A (n_A ) and k = 1, ...d_B (n_B ). Similar as in (9), we express a pure state of the composite system as

$$\begin{equation}\begin{aligned}\hfill \vert {\psi }_{AB}\rangle & =\prod\limits _{i=1}^{{d}_{A}({n}_{A})}\prod\limits _{k=1}^{{d}_{B}({n}_{B})}{z}_{ik}\vert {a}_{i}{b}_{k}\rangle ,\hfill \\ \hfill {\rho }_{A\times B}& =\vert {\psi }_{AB}\rangle \langle {\psi }_{AB}\vert ,\hfill \\ \hfill \mathrm{t}\mathrm{r}\,{\rho }_{A\times B}& =\mathrm{t}\mathrm{r}\,{\rho }_{A\times B}^{2}=1.\hfill \end{aligned}\end{equation} \tag{ 21 }$$

The entanglement entropy for a particular configuration described by the amplitudes z_ik is then obtained by tracing out the environment ρ_A ≡ tr_B (ρ_A×B ) and calculating S_A (z_ik ) = −tr_A (ρ_A  log ρ_A ). Unitary time development in the full Hilbert space leads to time-dependent amplitudes z_ik (t), and in turn to a time-dependent entanglement entropy, which can be calculated from the specific Hamiltonian of the system. We assume that time development leads to quantum ergodicity in Hilbert space if all degrees of freedom do interact uniformly with each other. Numerically, it can be shown in simple model systems with all-all interaction that a very small number of qubits is sufficient to reach this limit. Then, the entropy of entanglement converges to the mean value obtained by integrating over the full Hilbert space with equal measure

$$\begin{equation}\langle {S}_{A}^{B}\rangle =-Z\int \prod\limits _{ik}\mathrm{d}{z}_{ik}\,\mathrm{d}{\bar{z}}_{ik}\,{\mathrm{t}\mathrm{r}}_{A}({\rho }_{A}\,\mathrm{log}\,{\rho }_{A})\delta \left(1-\sum\limits _{ik}\vert {z}_{ik}{\vert }^{2}\right).\end{equation} \tag{ 22 }$$

Note the analogy to the approach in phase space: also in phase space, the time development of the coordinates (x(t), p(t)) derived from the classical Hamiltonian in case of a fully chaotic system can be replaced by a phase space integral dp dx under the assumption of ergodicity.

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The integral in Hilbert space can be calculated using random matrix theory as shown in reference [20]. The result is the celebrated Page formula for the mean entanglement entropy in Hilbert space ${\mathcal{H}}_{A}\times {\mathcal{H}}_{B}$, which is an analytic and exact result, depending only on the dimensions d_A and d_B of ${\mathcal{H}}_{A}\times {\mathcal{H}}_{B}$ [21]

$$\begin{equation}\langle {S}_{A}^{B}\rangle ={\Psi}({d}_{A}{d}_{B}+1)-{\Psi}({d}_{B}+1)-\frac{{d}_{A}-1}{2{d}_{B}}\equiv \langle {S}_{A}^{B}{\rangle }_{\text{Page}}\end{equation} \tag{ 23 }$$

where Ψ(x) = Γ'(x)/Γ(x) is the logarithmic derivative of the gamma function Γ(x). In the limit of large heat bath (d_B ≫ d_A ), we obtain the approximation

$$\begin{equation}\langle {S}_{A}^{B}\rangle \simeq \mathrm{log}\,{d}_{A}-\frac{{d}_{A}^{2}-1}{2{d}_{A}{d}_{B}}.\end{equation} \tag{ 24 }$$

It follows that the typical entropy of entanglement is close to the maximum value. For an environment B with k qubits, the corrections fade out as $\propto {\!d}_{B}^{-1}={2}^{-k}$.

When we divide our system of N spins with n₁ spins 'up' into parts A and B, in order to obtain the mean entanglement entropy, there is one more technical step to be considered: the Page equation holds true for one partition with ${n}_{1}={n}_{1}^{B}+{n}_{1}^{A}$ as shown in equation (20). Summing up all partitions, the exact result for the full mean entropy has been calculated analytically in references [20, 21] as

$$\begin{equation}\langle {S}_{A}^{B}\rangle =\sum\limits _{j}\frac{{d}_{j}}{{d}_{E}}[\langle {S}_{{A}_{j}}^{B}\rangle +{\Psi}({d}_{E}+1)-{\Psi}({d}_{j}+1)].\end{equation} \tag{ 25 }$$

The entanglement entropy only depends on the dimensions of the Hilbert spaces in each partition. For the spin system, these partitions are given by equation (19). In case of a Bose gas, we find partitions of type

$$\begin{align}\hfill {d}_{E}^{\text{Boson}}& =\left(\genfrac{}{}{0pt}{}{P+N-1}{N}\right)=\sum\limits _{{N}_{A}=0}^{N}\left(\genfrac{}{}{0pt}{}{{P}_{A}+{N}_{A}-1}{{N}_{A}}\right)\left(\genfrac{}{}{0pt}{}{{P}_{B}+{N}_{B}-1}{{N}_{B}}\right)\hfill \\ \hfill & \equiv \sum\limits _{{N}_{A}=0}^{N}{d}_{A}^{\text{Boson}}({n}_{A}){d}_{B}^{\text{Boson}}({n}_{B})=\sum \underset{j{}}{\overset{\text{Boson}}{d}}\hfill \end{align} \tag{ 26 }$$

with N = N_A + N_B , P = P_A + P_B and where P is the number Planck cells in phase space, and N number of bosons. For fermions, the partitions are given by (19) while for the Boltzmann statistics, the partitions read

$$\begin{equation}\frac{{P}^{N}}{N!}=\sum\limits _{{N}_{A}=0}^{N}\frac{{P}_{A}^{{N}_{A}}}{{N}_{A}!}\frac{{P}_{B}^{{N}_{B}}}{{N}_{B}!}.\end{equation} \tag{ 27 }$$

In all these cases, the mean entanglement entropy (25) reproduces the results for the entropy obtained in phase space in the limit of a large environment P_B . For example, in the spin system, we obtain for a large total number of spins N [20]

$$\begin{equation}{\langle {S}_{A}^{B}\rangle }_{\text{spins}}\simeq -\mathrm{min}({N}_{A},{N}_{B})[{p}_{\text{up}}\,\mathrm{ln}\,{p}_{\text{up}}+{p}_{\text{down}}\,\mathrm{ln}\,{p}_{\text{down}}]\end{equation} \tag{ 28 }$$

with ${p}_{\mathrm{u}\mathrm{p}/\mathrm{d}\mathrm{o}\mathrm{w}\mathrm{n}}=\frac{1}{2}(1\pm \frac{n}{N})$ with n ≡ n₁ − n₂. Again, only a very small subspace of the Hilbert space (20) finally turns out to be relevant, and again the argument boils down just to restrict to expectation values of binomial distributions as in equation (14).

Note that the derivation and definition of temperature from entanglement entropy is only possible for N_A ≪ N, as only in this (thermodynamic) limit, the usual arguments of statistical mechanics allowing to ignore fluctuations of the entropy can be applied. Exact (and rather involved) expressions for the fluctuations of the entropy of entanglement have been derived in [20]. The fluctuations for a given partition can be approximated by

$$\begin{equation*}{\Delta}{S}_{A}^{B}\simeq \sqrt{({d}_{A}^{2}-1)/(2{d}_{A}^{2}{d}_{B}^{2})}\end{equation*}$$

for d_B ≫ 1 and indeed fade out in the thermodynamic limit [20] as 1/d_B . Only in this limit, a temperature can be defined in the sense of thermodynamics. However, the partitions (19) and (26) are valid beyond this thermodynamic limit. Under the assumption of quantum ergodicity, that is, integrating the whole Hilbert space with uniform measure as defined in (22), using the exact expressions for the entanglement entropy (23) and (25) and the pertinent expressions for fluctuations from the mean entropy [20], quantum corrections to the thermodynamics of fermions and boson can be discussed for N_A ≃ N_B and for very small system sizes.

To summarize, we have shown that the entropy of entanglement provides a unified viewpoint for physical systems in Hilbert space even beyond the thermodynamic limit. The number of relevant Planck cells in phase space defines the dimension of pertinent Hilbert space. Note that this is a simple counting argument, which, much the same as Weyl's law [6], does not make any assumptions concerning the localization of eigenfunctions of the pertinent Hamiltonian within the Hilbert space.

Next, we consider quantum thermodynamic heat engines (QHEs) (in the limit N_A ≪ N) that—in analogy to classical heat engines—produce power by transferring heat from a hot into a cold heat reservoir. Such (quantum) thermodynamic cycles reveal a cornerstone in modern physics and have substantially changed human's life since the first industrial revolution in the 19th century.

The central model used in this historical context is that of an ideal gas of N particles confined in a box of volume V and with kinetic energy $E={\sum }_{k}{p}_{\mathrm{K}}^{2}/(2m)$. Then, the phase space volume

$$\begin{equation}{\Omega}(E,N,V)\propto \frac{1}{N!}\prod\limits _{k}\int {d}^{3N}{q}_{k}{d}^{3N}{p}_{k}\delta (H(p,q)-E)\end{equation} \tag{ 29 }$$

leads to the entropy of the ideal gas

$$\begin{equation}{S}_{\text{Ideal}\;\text{gas}}(U,V,N)={k}_{\text{B}}\,\mathrm{ln}\left(\frac{{V}^{N}}{N!}{U}^{3N/2}\right)\end{equation} \tag{ 30 }$$

with U = ⟨E⟩. For a fixed number of particles, the change of entropy reads

$$\begin{equation}\mathrm{d}{S}_{\text{Ideal}\;\text{gas}}=N{k}_{\text{B}}\frac{\mathrm{d}V}{V}+\frac{3}{2}N{k}_{\text{B}}\frac{\mathrm{d}U}{U}.\end{equation} \tag{ 31 }$$

With $U=\frac{3}{2}N{k}_{\text{B}}T$, and pV = Nk_B T this expression is equivalent to dU = T dS − p dV. Comparing this with the first law of thermodynamics dU = δQ + δW, we arrive at the expression of Clausius for the change of entropy $\mathrm{d}S=\frac{\delta Q}{T}$ which is the starting point to describe heat engines, e.g. the Otto cycle. Within classical physics, these results do not change if we divide the volume V into small but constant subvolumes V₀. We may read B = V/V₀ as number of possible regions where N particles can be arranged, which would again lead to Ω_Boltzmann = B^N /N! as discussed in the previous section. In 1905, Einstein used this analogy to the ideal gas in configuration space in his first attempts to describe the statistics of photons starting from Wien's law.

In what follows, we will discuss the few-particle regime (N_A ≃ 1), where quantum effects take place and whose theoretical [22, 23] investigation has opened an active, current research direction. Recent experiments on quantum heat engines based on a single qubit could be successfully implemented using various setups [24–29]. In this field, the thermodynamic definition of entropy, dS = dQ/T of equation (30), plays a crucial role to describe the properties of the thermodynamic system of interest. Note that for the definition of the temperature and ignoring fluctuations, the heat bath must be much larger than the system size (N_A ≪ N). An illustration of such a quantum Otto cycle [30, 31] can be found in figure 5.

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In analogy to classical heat engines, its quantum version also consists of four strokes: in the two isentropic strokes (1 and 3) the quantum WM, e.g. a quantum harmonic oscillator, or an electron's spin, exchanges work with an environment by being 'compressed' and 'expanded' by changing the working parameter λ, which is in our example the magnetic field strength. During the isentropic strokes, the dynamics are considered to describe an isolated system where no heat is exchanged with the environment, thus keeping its entropy constant.

In the two heat-exchange strokes, the WM is alternating coupled to two heat reservoirs at cold T_c and hot temperature T_h, respectively. Thus, the WM exchanges heat with the environment until it reaches thermal equilibrium, where the entropy flows in the same direction as the heat.

For our illustration, we choose a quantum WM that consists of a single spin and which can be parametrized by the state Hamiltonian

$$\begin{equation}H(t)=\mu B(t){\sigma }_{z},\end{equation} \tag{ 32 }$$

where the magnetic field acts as a 'working parameter', i.e. B ∝ λ (figure 5). We assume that the magnetic field varies linearly between a small value B_c and a large value B_h. The energy splitting thus varies between _c ≡ μB_c for the points A, D and _h ≡ μB_h > _c for B, C of the quantum Otto cycle of figure 5. The entanglement entropy of the single spin is given by equation (28) with N_A = 1, thus

$$\begin{equation}S({p}_{\pm })=-{p}_{+}\;\mathrm{l}\mathrm{n}\;{p}_{+}-{p}_{-}\;\mathrm{l}\mathrm{n}\;{p}_{-},\end{equation} \tag{ 33 }$$

where the probabilities p_± to find the spin values ± are then given by ${p}_{\pm }=\frac{1}{2}\left\{1\pm \mathrm{tanh}[\beta (t){\epsilon}(t)]\right\}=\frac{1}{2}\left(1\pm E/{\epsilon}(t)\right]$. Here, E = tr(ρH) is the expectation value for the energy.

The strokes of the quantum Otto cycle read as follows:

• (a)  
  Isentropic 'compression' (A → B): in A, the spin is in its thermal equilibrium state at temperature T_A = T_c. The 'compression' for the quantum Otto cycle is achieved by increasing the magnetic field and thus the energy splitting from _c to _h. The isentropic stroke can be described as a sequence of quasi-static unitary transformations and thus no heat is exchanged with the environment. In analogy to the compression stroke of a classical Otto cycle, the entropy remains constant and in turn the probabilities for spin up/down as defined by the binomial distribution in (14) and (33). The mixedness of the qubit remains fixed, while the energy splitting increases with constant β.
• (b)  
  Isochoric heating: (B → C): in B, the spin is not in thermal equilibrium anymore. The temperature T_B is given by T_B = (_h/_c)T_A . The WM is brought in contact with the hot reservoir at temperature T_h. While the energy splitting remains constant, the probabilities for spin up/down as defined in (14) and (33) change and in turn the mixedness of the qubit and the entropy, until thermalization is achieved. During this heat-exchange stroke, the heat ${Q}_{\text{h}}^{\text{ad}}$ as the difference in energy between the points B and C is imparted by the hot reservoir into the cold. The duration of this isochoric stroke is in general much shorter than the isentropic stroke due to the requirement on quantum adiabaticity (see below).
• (c)  
  Isentropic 'expansion' (C → D): in C, the spin is in its thermal equilibrium state at temperature T_C = T_h. During this 'power stroke', the WM exchanges the work ${W}_{3}^{\text{ad}}$ while the magnetic field quantum adiabatically decreases. While the energy splitting decreases, much the same as during the isentropic 'compression', the mixedness of the qubit remains fixed, and in turn β.
• (d)  
  Isochoric cooling: (D → A): in D, similar as to (b), the spin is not in thermal equilibrium anymore. The temperature T_D is given by T_D = (_c/_h)T_C . The WM is brought in contact with the cold reservoir at temperature T_c until it thermalizes back into its original initial thermal state ρ_A . During this 'cooling' stroke, the heat ${Q}_{\text{c}}^{\text{ad}}$ as the energy difference between the points A and D is transferred into the cold reservoir, consequently decreasing the entropy.

In analogy to their classical counterpart, we can find an appropriate expression for the efficiency of such quantum heat engines, defined as the useful work divided by the heat ${Q}_{\text{h}}^{\text{ad}}$ imparted by the hot bath, i.e.

$$\begin{equation}\eta =\frac{\mathrm{u}\mathrm{s}\mathrm{e}\mathrm{f}\mathrm{u}\mathrm{l}\ \mathrm{w}\mathrm{o}\mathrm{r}\mathrm{k}}{\mathrm{i}\mathrm{m}\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{e}\mathrm{d}\ \mathrm{h}\mathrm{e}\mathrm{a}\mathrm{t}}=-\frac{{W}_{1+3}^{\text{ad}}}{{Q}_{\text{h}}^{\text{ad}}}.\end{equation} \tag{ 34 }$$

For the quantum Otto cycle of figure 5, we find

$$\begin{equation}\eta =1-\frac{{{\epsilon}}_{\mathrm{c}}}{{{\epsilon}}_{\mathrm{h}}}\leqslant 1-\frac{{T}_{\mathrm{c}}}{{T}_{\mathrm{h}}}={\eta }_{\text{Carnot}}.\end{equation} \tag{ 35 }$$

In analogy to classical Otto heat engines, the ratio of energy splittings _c/_h due to the oscillating magnetic field can be regarded as the 'compression coefficient'. A detailed derivation of the efficiency η (35) can be found in appendix A.1.

Such quantum thermodynamic cycles depict an irreversible process in which, according to the second law of (quantum) thermodynamics, the thermodynamic entropy S_irr. ⩾ 0 of the whole composite system AB, consisting of the quantum Otto cycle (subsystem A) and the environment (subsystem B), increases. In such thermodynamic processes, the notion of entropy can be viewed as a measure of irreversibility, that is, 'closing' the cycle by getting back to the initial point A of figure 5 usually requires an irreversible change in the environment which manifests itself as flown heat Q.

Although the strokes and working mechanisms of the quantum Otto cycle look very similar to its classical analog, there exists a fundamental difference between the former and the latter. Whereas in the classical case, the adiabatic work-exchange strokes require no heat transfer due to a fast stroke—such that the latter remains quasi-static and the system does not have time to get out-of-equilibrium—the same, however, is not true in the quantum case. Here, the work-exchange strokes are adiabatic in a quantum sense, that is, they are so slow such that the system remains in its instantaneous eigenstate during the whole isentropic stroke, thus keeping their populations constant during the latter. This feature constitutes a severe hindrance in the development of efficient quantum heat engines that operate at finite time while still producing a high power output and efficiency and has been object of current research [32–36].

In such finite-time quantum heat engines and refrigerators, the excitation of coherences in the work-exchange causes the system to be driven out-of-equilibrium. The information on these excitations, that are stored in the off-diagonal elements of the corresponding density matrix, are dissipated away in the heat-exchange strokes. The latter feature is analogous to classical mechanics and has been dubbed quantum friction [37–39] and which is equivalent to an increase in the thermodynamic entropy, equation (30), of the composite system. In a nutshell, this increase in the thermodynamic entropy can be calculated from the change of probabilities for spin up/down using the time dependent binomial distribution in (14) and (33) for the four steps (a)–(d) of the cycle.

As a future work, using the ansatz described in section 4.3, it would be interesting to shrink the size of the heat bath beyond the thermodynamic limit with only a few qubits N_B , such that fluctuations of the entropy cannot be neglected anymore.

In this appendix, we provide additional information on QHEs as highlighted in section 5. In more detail, we derive the efficiency of a QHE whose WM is a single spin that is subject to a time-dependent magnetic field B(t) in z-direction, respectively. The time-dependent Hamiltonian that describes the working parameter λ of the WM reads

$$\begin{equation}H[t]=-\mu B(t){\sigma }^{z},\qquad B(t)={B}_{\mathrm{c}}+f(t){\Delta}B={B}_{\mathrm{c}}+f(t)({B}_{\text{h}}-{B}_{\mathrm{c}}),\end{equation} \tag{ 38 }$$

where f(t) is the control function that fulfills f(t = 0) = 0, i.e. providing the cold working parameter λ_c for the points A and D of figure 5, and f(t = τ_l ) = 1 for B and C with hot working parameter λ_h, respectively, with τ_l the isentropic stroke duration and l ∈ {2; 4}. As in the main text, we introduce the two relevant energy splittings _c = μB_c, _h = μB_h.

Due to the requirement of quantum adiabaticity in the latter, that is, we perform reversible quasi-static processes during these isentropic strokes, the thermodynamic system at each point i ∈ {A, B, C, D} of the cycle can be written as a thermal state

$$\begin{equation}{\rho }_{i}=\frac{\mathrm{exp}[-{\beta }_{i}{H}_{0}({\lambda }_{i})]}{{Z}_{i}},\quad {Z}_{i}=\mathrm{Tr}\left\{\mathrm{exp}[-{\beta }_{i}{H}_{0}({\lambda }_{i})]\right\}\end{equation} \tag{ 39 }$$

with Z_i the quantum partition function and β_i = 1/T_i the inverse temperature where we have set the Boltzmann constant to one. Consequently, this Gibbs state gives each eigenenergy E_j of the corresponding eigenstate of the Hamiltonian H₀(λ_i ) a probability 0 ⩽ p_j ⩽ 1 of occurrence, thus describing a canonical statistical ensemble of eigenstates. The average energy of this ensemble reads

$$\begin{equation}{\langle E\rangle }_{i}=\sum\limits _{j}{E}_{j}{p}_{j}=\mathrm{T}\mathrm{r}[{H}_{0}({\lambda }_{i}){\rho }_{i}]=-{E}_{i}\;\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}({\beta }_{i}{E}_{i}),\end{equation} \tag{ 40 }$$

where we used that the eigenenergies of H₀(λ_i ) are ±E_i as well as the relations

$$\begin{align}\hfill \mathrm{Tr}[H(t){\rho }_{i}]& =\frac{\mathrm{Tr}\left\{H(t)\mathrm{exp}[-{\beta }_{i}H(t)]\right\}}{{Z}_{i}}=\frac{{E}_{i}\,\mathrm{exp}[-{\beta }_{i}{E}_{i}]-{E}_{i}\,\mathrm{exp}[{\beta }_{i}{E}_{i}]}{{Z}_{i}}=\frac{-2{E}_{i}\,\mathrm{sinh}({\beta }_{i}{E}_{i})}{{Z}_{i}},\hfill \\ \hfill {Z}_{i}& =\mathrm{Tr}\left\{\mathrm{exp}[-{\beta }_{i}{H}_{0}({\lambda }_{i})]\right\}=\mathrm{exp}[-{\beta }_{i}{E}_{i}]+\mathrm{exp}[{\beta }_{i}{E}_{i}]=2\,\mathrm{cosh}[{\beta }_{i}{E}_{i}]\hfill \end{align} \tag{ 41 }$$

along with the fact that tanh(x) = sinh(x)/cosh(x). Calculating these average energies of equation (40) at each point i of the quantum Otto cycle, we can determine an expression for the efficiency η, equation (35), of the latter as the output work divided by the input heat, i.e.,

$$\begin{equation}\eta =-\frac{\langle E{\rangle }_{B}-\langle E{\rangle }_{A}+\langle E{\rangle }_{D}-\langle E{\rangle }_{C}}{\langle E{\rangle }_{C}-\langle E{\rangle }_{B}}=1-\frac{\langle E{\rangle }_{D}-\langle E{\rangle }_{A}}{\langle E{\rangle }_{C}-\langle E{\rangle }_{B}}.\end{equation} \tag{ 42 }$$

The average energies ⟨E⟩_i at each point i of the cycle with original Hamiltonian H₀(λ_i ), equation (38), read

$$\begin{align}\hfill A& :H[t=0]=-{{\epsilon}}_{\mathrm{c}}{\sigma }^{z}{\Rightarrow}\qquad \qquad\langle E{\rangle }_{A}=-{{\epsilon}}_{\mathrm{c}}\mathrm{tanh}\left(\frac{{{\epsilon}}_{\mathrm{c}}}{{T}_{\text{c}}}\right),\hfill \\ \hfill B& :H[t={\tau }_{1}]=-{{\epsilon}}_{\mathrm{h}}{\sigma }^{z}{\Rightarrow}\qquad \qquad\langle E{\rangle }_{B}=-{{\epsilon}}_{\mathrm{h}}\,\mathrm{tanh}\left(\frac{{{\epsilon}}_{\mathrm{c}}}{{T}_{\text{c}}}\right)=-{{\epsilon}}_{\mathrm{h}}\,\mathrm{tanh}\left(\frac{{{\epsilon}}_{\mathrm{h}}}{{T}_{B}}\right),\hfill \\ \hfill C& :H[t={\tau }_{1}+\left.{\tau }_{2}\right)]=-{{\epsilon}}_{\mathrm{h}}{\sigma }^{z}{\Rightarrow}\quad \langle E{\rangle }_{C}=-{{\epsilon}}_{\mathrm{h}}\,\mathrm{tanh}\left(\frac{{{\epsilon}}_{\mathrm{h}}}{{T}_{\text{h}}}\right),\hfill \\ \hfill D& :H[t={\tau }_{1}+{\tau }_{2}+{\tau }_{3}]=-{{\epsilon}}_{\mathrm{c}}{\sigma }^{z}{\Rightarrow}\langle E{\rangle }_{D}=-{{\epsilon}}_{\mathrm{c}}\,\mathrm{tanh}\left(\frac{{{\epsilon}}_{\mathrm{h}}}{{T}_{\text{h}}}\right)=-{{\epsilon}}_{\mathrm{c}}\,\mathrm{tanh}\left(\frac{{{\epsilon}}_{\mathrm{c}}}{{T}_{\text{C}}}\right).\hfill \end{align} \tag{ 43 }$$

With equation (42), we obtain the magnetic-field (energy splitting)-dependent efficiency

$$\begin{equation}{\eta }_{\text{LZ}}=1-\frac{{{\epsilon}}_{\mathrm{c}}}{{{\epsilon}}_{\text{h}}}\end{equation} \tag{ 44 }$$

which is upper bounded to one for the choice _c < _h and T_c < T_h.

For the quantum Otto cycle to work as a heat engine, the heat

$$\begin{align}\hfill {Q}_{\text{h}}^{\text{ad}}& =\langle E{\rangle }_{C}-\langle E{\rangle }_{B}=-{{\epsilon}}_{\text{h}}\,\mathrm{tanh}\left(\frac{{{\epsilon}}_{\text{h}}}{{T}_{\text{h}}}\right)+{{\epsilon}}_{\text{h}}\,\mathrm{tanh}\left(\frac{{{\epsilon}}_{\mathrm{c}}}{{T}_{\text{c}}}\right)\hfill \\ \hfill & ={{\epsilon}}_{\text{h}}\left[\mathrm{tanh}\left(\frac{{{\epsilon}}_{\mathrm{c}}}{{T}_{\text{c}}}\right)-\mathrm{tanh}\left(\frac{{{\epsilon}}_{\text{h}}}{{T}_{\text{h}}}\right)\right]\geqslant 0\hfill \end{align} \tag{ 45 }$$

is transferred from the hot into the cold reservoir. Due to the fact that tanh(x) is a monotonously increasing function for increasing x, equation (45) leads to the relation

$$\begin{equation}\frac{{{\epsilon}}_{\mathrm{c}}}{{{\epsilon}}_{\text{h}}}\geqslant \frac{{T}_{\text{c}}}{{T}_{\text{h}}}\end{equation} \tag{ 46 }$$

and further inserting into the expression (44) for the efficiency of this single-body quantum heat engine reveals the Carnot efficiency

$$\begin{equation}{\eta }_{\text{LZ}}=1-\frac{{{\epsilon}}_{\mathrm{c}}}{{{\epsilon}}_{\text{h}}}\leqslant 1-\frac{{T}_{\text{c}}}{{T}_{\text{h}}}={\eta }_{\text{Carnot}}\end{equation} \tag{ 47 }$$

as the ultimate upper bound, as expected from a thermodynamic cycle.