Integral fluctuation theorems and trace-preserving map

Since we are mainly concerned with CP mappings in this paper, we use the superoperator form for simplicity. Furthermore, the FT necessitates the consideration of inverse mappings and measurements. These components have simpler expressions within the operator-state formalism, which we also employ. It’s noteworthy that these formalisms maintain mathematical equivalence with the conventional framework of quantum mechanics.

Let us first introduce the operator state $|O)$ for an operator $O$ on the Hilbert space $\mathcal{H}_{S}$ of quantum states of a system $S$, which belongs to a new Hilbert space $\mathsf{H}$ with an inner product defined by

An orthonormal basis for $\mathsf{H}$ is $|\Pi_{ij})$ where $\Pi_{ij}=\ket{i}\bra{j}$, as one can check that

The completeness of this basis $\{|\Pi_{ij})\}$ is

If only the diagonal elements are summed, the corresponding superoperator is actually a dephasing map

The conjugate relation reads

The unitary superoperator acting on the operator state gives

The conjugate relation shows that

The trace of an operator $\rho$ can be given by $(I|\rho)$. The superoperator $\mathcal{N}$ is trace-preserving if and only if it satisfies $(I|\mathcal{N}=(I|$.

Now we briefly introduce the FT. The FT can be framed in a unifying language in terms of entropy production [9]. The entropy production $\Delta\sigma$ is generally related to thermodynamic quantities such as Gibbs-von Neumann entropy, work, heat, and free energy. Its formulation depends on the underlying physical system. In detailed FT (Crooks FT), the forward entropy production is exponentially more likely than the reverse

where $P_{F}$ is the forward probability distribution of entropy production and $P_{B}$ is the backward one. They are given by the probability density function of the difference between two measurements

Its generating function

is often used. The parameter $\lambda$ is a complex number, and people are generally concerned about the properties of the generating function when $\lambda=i$. The Crooks FT implies the fundamental symmetry on the generating function

An integral FT immediately follows from the normalization of $P_{B}$

Combining it with Jensen’s inequality $\braket{e^{-X}}\geq e^{-\braket{X}}$, we can get a generalization of the second law of thermodynamics

In this section, we explore how to construct a CP map from the generating function and connect the FT with the TP property of this constructed CP map.

Consider an isolated, possible driven, quantum system evolves according to the unitary evolution. The probability density function of the difference between two measurements can be expressed as

where $\rho_{0}$ is the initial density matrices for the forward process. The observable $\hat{\sigma}_{t}$ is a Hermitian operator. The eigenvalues (eigenbasis) of $\hat{\sigma}_{t}$ are denoted by $\sigma_{t}$ ($\Pi_{\sigma_{t}}=\ket{\sigma_{t}}\bra{\sigma_{t}}$): $\hat{\sigma}_{t}=\sum_{\sigma_{t}}\sigma_{t}\Pi_{\sigma_{t}}$. The generating function of probability (14) is

The factor $e^{-i\lambda\hat{\sigma}}$ rescaling the measurement results indeed. It is related to the rescaling map $\mathcal{J}^{\alpha}_{O}(\cdot):=O^{\alpha}(\cdot)O^{\alpha\dagger}$, where $O$ is arbitrary operator and $\alpha$ is the power index. Here we only consider the cases where the real part of $\lambda$ is zero, and then we have

Using it, we can rewrite some items in the generating function as follows

where $\mathcal{M}_{\hat{\sigma}_{0}}:=\sum_{\sigma_{0}}|\Pi_{\sigma_{0}})(\Pi_{\sigma_{0}}|$ is a dephasing map. The rescaling map is CP, but generally not TP. It is worth pointing out that the rescaling map here is very similar to the re-weighting approach used in [10], where this approach is used to prepare the thermal state. With Eq. 17, we can rewrite Eq. 15 as

or its conjugated form

Where $\mathcal{M}_{\hat{\sigma}_{t}}$ is omitted because it shares the same basis as $e^{i\lambda\hat{\sigma}_{t}}$. Additionally, we omitted the ${\dagger}$ on $\mathcal{M}$ and $\mathcal{J}_{e^{-\hat{\sigma}}}$ because they are self-conjugate. If $\{\Pi_{\sigma_{0}}\}$ is the diagonal basis of $\rho_{0}$, then $\mathcal{M}_{\hat{\sigma}_{0}}$ can also be omitted. In Eqs. 18 and 19, the generating function is completely rewritten with the operator state and completely positive maps. As will be shown later, the integral FT will be determined by the TP property of these maps.

Take $\hat{\sigma}_{t}=-\ln\hat{\rho}_{t}$, then we have

In such cases, the $\mathcal{M}_{\hat{\sigma}_{0}}$ can be omitted

Combining Eqs. 18 and 21, we get

Therefore, for a closed system, the entropy production $\ln(\hat{\rho}_{0}/\hat{\rho}_{t})$ satisfies integral FT. Since $\braket{-\ln\hat{\rho}_{t}}=S(\rho_{t})$, the average of this entropy production is the same as the change of the Gibbs-von Neumann entropy of the system.

For all cases in this paper, we have $\text{Tr}e^{-\hat{\sigma}_{t}}=1$, which can also be achieved in other cases by adding a normalization factor. Under this condition, according to Eq. 19, the integral FT is satisfied if and only if the following constructed CP map is also a TP map

This map provides a new perspective for considering the integral FT. The mapping (23) is not necessarily a TP mapping, it depends on the initial state $\rho_{0}$ as well as the initial measurements $\hat{\sigma}_{0}$. If observable are taken as Eq. 20, it is easy to prove that the constructed map (23) is TP

which is consistent with the previous conclusion that the integral FT holds.

Now consider an open quantum system $S$ interacting with an environment $E$; $S$ and $E$ form a closed system with unitary evolution. If there is no correlation between the initial system and the environment, we can set $\hat{\sigma}_{t}=-\ln\hat{\rho}_{S}(t)-\ln\hat{\rho}_{E}$, and then we have

Similar to the case of closed systems, it is easy to prove that open systems also satisfy integral FT

This integral FT is completely general, but only contains quantities like entropy, which require knowledge of all information about the state of the environment. In open quantum systems, the environment is usually considered to be relatively large and obtaining all information about it is difficult. Therefore, it is necessary to introduce some thermodynamic quantities such as work, free energy, and heat to simplify the measurement. For example, it is usually assumed that the initial state of the heat bath is the canonical ensemble $\hat{\rho}_{E}=e^{-\beta\hat{H}_{E}}/Z_{E}$, so we can set

If we make no assumptions about the initial state of the environment, but still set $\hat{\sigma}_{t}$ as Eq. 27, then we have

Since the environment $\rho_{E}(0)$ can deviate from the canonical ensemble $\rho^{\text{can}}_{E}$, the integral FT will have errors and $G_{F}(i)$ can deviate from 1.

The deviation of the environment state is not a sufficient condition for the violation of the integral FT, which is also related to the system-environment interaction. For non-driven systems, the Hamiltonian of the whole system can generally be written as $H=H_{S}+H_{E}+H_{I}$. If there is no system-environment interaction $H_{I}=0$, then the overall generating function can be separated into a system part and an environment part $G_{F}(\lambda)=G^{S}_{F}(\lambda)G^{E}_{F}(\lambda)$. Since the system state does not deviate, we have $G^{S}_{F}(i)=1$. And the environment part gives

Since $[U_{E}(t),\rho^{\text{can}}_{E}]=0$, there is always $G^{E}_{F}(i)=1$, no matter whether $\rho_{E}(0)$ deviates from $\rho^{\text{can}}_{E}$ or not. This inspires us to decompose the rescaling map in section II.3 as follows

$\mathcal{J}^{-1}_{\rho^{\text{can}}_{S}}\circ\mathcal{J}_{\rho_{S}(t)}$ is a local superoperator of the system. Using decomposition (30), we can rewrite the generating function (II.3) as

where

and $\rho^{\prime}_{E}(0)=\mathcal{M}_{\hat{\sigma}_{0}}^{E}[\rho_{E}(0)]$ is the environment density matrix resulting from the decoherence of the environment due to the measurement.

Similar to Eq. 23, the conjugated form of section II.3 gives ¹¹1In Eq. 33, we omit the $I_{E}$ in the right parenthesis, which will change the meaning of the first $\mathcal{J}_{\rho_{E}}^{1/2}$ in $\mathcal{N}_{\text{FT}}$: it becomes an assignment map $\rho_{S}\to\rho_{S}\otimes\rho_{E}$, which is linear and CPTP. The similar approaches are also used in section IV.

and the integral FT holds if and only if the following constructed map is a TP map

The key parts of the formulae (34) are the mapping $\mathcal{N}_{\beta}(t)$ related to the evolution and the mapping $\mathcal{J}_{\rho^{\prime}_{E}(0)}^{1/2}$ related to the initial state and the initial measurement. If the initial state of the environment is the canonical ensemble, then we have $\mathcal{N}_{\text{FT}}(t)=\mathcal{U}^{\dagger}_{SE}(t)\circ\mathcal{J}_{\rho^{\text{can}}_{E}}^{1/2}$. It acts on the operator state and gives $\mathcal{N}_{\text{FT}}(t)|\rho_{S}(t))=\mathcal{U}^{\dagger}_{SE}(t)|\rho_{S}(t)\otimes\rho^{\text{can}}_{E})$. Therefore, it is indeed a TP map in this case. When the environment $\rho_{E}(0)$ deviates from the canonical ensemble $\rho^{\text{can}}_{E}$, the constructed map $\mathcal{N}_{\text{FT}}(t)$ is generally not a TP map. The integral FT will have errors and $G_{F}(i)$ can deviate from 1. The inverse temperature in Eq. 34 remains arbitrary, allowing for optimization of the deviation estimate by selecting it based on the state of the environment. For instance, when considering an environment $X$ initialized in a single-energy eigenstate $\ket{E^{X}_{a}}$, the choice of inverse temperature is typically determined by aligning the total energy, often set as $\beta_{a}$ such that

When the evolution time tends to zero, it is obvious that $\mathcal{N}_{\text{FT}}(t)=\mathcal{J}_{\rho_{E}(0)}^{1/2}$. It acts on the operator state and gives $\mathcal{N}_{\text{FT}}(t)|\rho_{S}(t))=|\rho_{S}(t)\otimes\rho_{E}(0))$. Therefore, it is a TP map and the FT holds. If the evolution time is short, it is foreseeable that the error can be bounded, as we will discuss in detail below.

When the evolution time is short, the two measurements about the system can only be influenced by a small piece of environment due to the limitation of information flow speed. Considering the ETH, it is difficult to distinguish the overall pure state of the environment from the canonical bath in this small area. This makes the integral FT approximately valid. From the perspective of constructed map $\mathcal{N}_{\text{FT}}$, we can rewrite it as

where $\mathcal{N}_{\beta}^{T}(t)$ is defined in section B.2. It is a truncated version of $\mathcal{N}_{\beta}(t)$ based on the speed limit of information flow. The last term of formula (III.1) is indeed a TP mapping. Therefore, the deviation of the integral FT can be estimated using the bounds of other terms. The terms in the first and third lines of section III.1 represent the error due to truncation, which can be bounded by the Lieb-Robinson bound. See section C.1 for specific calculations. The term in the second line of section III.1 represents the error caused by the deviation of the initial state. Since the mapping $\mathcal{J}_{\rho_{S}^{\text{can}}}^{1/2}\circ\mathcal{N}_{\beta}^{T\dagger}(t)\circ\mathcal{J}^{-1/2}_{\rho^{\text{can}}_{S}}$ acts on state $|\rho_{S}(t))$ and gives a local operator, this part of the error can be bounded by ETH. See section C.2 for details. Finally, in the large-environment limit, by combining the error terms in section III.1, we obtain

where the difference in the generating function corresponds one-to-one with the error terms in section III.1. The superscript ”can” represents replacing the initial environment state in $\mathcal{N}_{\text{FT}}(t)$ with the canonical ensemble, while ”$T$” represents replacing $\mathcal{N}_{\beta}$ in $\mathcal{N}_{\text{FT}}(t)$ with $\mathcal{N}^{T}_{\beta}$. We illustrate these replacements in fig. 1. From the figure, we can observe that when using the truncated mapping $\mathcal{N}_{\beta}^{T}(t)$, only a portion of the environment will participate, which is why ETH can be applied.

When the evolution time is long, the system can gather sufficient information about the environment, allowing it to distinguish the pure state environment from the thermal state environment. Consequently, under the long-term evolution, the integral FT may exhibit significant deviations. However, while large deviations are possible, they are typically transient. This transient nature stems from the system eventually reaching equilibrium as the evolution time lengthens. For most times, the total system tends to approach a fixed steady state [13]. Considering the ETH, distinguishing between these steady states with local observations becomes challenging. As a result, the integral FT remains approximately valid in the long-term average.

For any quantity $O(t)$, its long-time average is defined as follows

In the generating function, not only does $\mathcal{N}_{\text{FT}}(t)$ change with time, but the final state of the system also evolves over time

Combining it with Eqs. 33 and III.1, we have

where $\mathcal{N}_{\beta}^{T^{\prime}}(t)$ is defined in Eq. 87, representing another truncated version of $\mathcal{N}_{\beta}(t)$ based on the limit imposed by information flow speed under the imaginary time evolution. By substituting $\mathcal{N}_{\beta}$ with $\mathcal{N}_{\beta}^{T^{\prime}}$ in $\mathcal{N}_{\text{FT}}$, we obtain $\mathcal{N}^{T^{\prime}}_{\text{FT}}$. The last term of section III.2 is equivalent to $\overline{\mathcal{U}^{\dagger}_{SE}(t)\circ\mathcal{J}_{\rho^{\text{can}}_{E}}^{1/2}\circ\mathcal{E}_{t}}$, which is also a TP mapping. The terms in the first and third lines of section III.2 represent the error due to truncation, which can be constrained by the Lieb-Robinson bound. Detailed calculations are provided in section C.1. Under long-time averaging, a single unitary evolution yields the dephasing map [14]

where $\Pi_{a}=\ket{E_{a}}\bra{E_{a}}$ is the energy eigenstate of the total Hamiltonian $H$. A unitary evolution $\mathcal{U}^{\dagger}_{SE}(t)$ together with a dynamical evolution $\mathcal{E}_{t}$ give the following map

where $\Pi_{ab}=\ket{E_{b}}\bra{E_{a}}$. Correspondingly, the error caused by the second line in section III.2 can be divided into two parts. The first part is

We discuss its bound in detail in section C.3. The second part is

We discuss its bound in detail in section C.4. Finally, in the large-environment limit, by combining sections III.2, 108 and 113, we have

where ”can” represents replacing the initial environment state in $\mathcal{N}_{\text{FT}}(t)$ with the canonical ensemble and ”$T^{\prime}$” denotes the substitution of $\mathcal{N}_{\beta}$ in $\mathcal{N}_{\text{FT}}(t)$ with $\mathcal{N}^{T^{\prime}}_{\beta}$. In conclusion, the first and third terms of section III.2 depends on the locality of the imaginary time evolution component of the map $\mathcal{N}_{\beta}(t)$ and can be constrained by the Lieb-Robinson bound. The second term of section III.2 depends on the indistinguishability of the states within a typical energy shell and can be restricted by ETH.