# Generalized quantum fluctuation-dissipation identities

Boris Maulén  Departamento de Ciencias Químicas, Facultad de Ciencias Exactas,
Universidad Andres Bello.  Departamento de Física, Facultad de Ciencias
Exactas, Universidad Andres Bello. Sazié 2212, piso 7, 8370136, Santiago,
Chile<EMAIL_ADDRESS>Sergio Davis  Research Center in the
intersection of Plasma Physics, Matter and Complexity (P2mc), Comisión Chilena
de Energía Nuclear, Casilla 188-D, Santiago, Chile.  Departamento de Física,
Facultad de Ciencias Exactas, Universidad Andres Bello. Sazié 2212, piso 7,
8370136, Santiago, Chile.  Daniel Pons  Departamento de Matemáticas, Facultad
de Ciencias Exactas, Universidad Andres Bello, Santiago, Chile.

###### Abstract

In this article, we derive the fluctuation-dissipation theorem from Bayesian
statistics in the language of quantum statistical mechanics, using density
matrices representing the respective state of knowledge about the system. This
theorem, which we have called the quantum-fluctuation-dissipation theorem
(Q-FDT), allows us to obtain expectation identities between quantum
observables depending on a continuous parameter $\gamma\in\mathbb{R}$. In this
sense, we consider parameters contained in the observables (e.g. a
perturbative parameter), or parameters such as Lagrange multipliers only,
excluding parameters that modify the underlying Hilbert space. Using both
canonical and grand canonical density matrices in the Q-FDT, we explore the
resulting fluctuation-dissipation identities. In this sense, we found that
some of the expectation identities in common use in traditional quantum
statistics and quantum chemistry, such as the thermodynamical fluctuation-
dissipation theorem, the Ehrenfest and the Hellmann-Feynman theorems, among
others, are particular instances of the Q-FDT. Lastly, using a generalized
density matrix arising from a Maximum-Entropy procedure, we derive generalized
fluctuation-dissipation identities: these generalized identities allow us to
group all the previous cases in a unitary scheme.

##  1 Introduction

The concept of expectation value plays a central role both in statistical
sciences and physics. It corresponds to the prediction that a theory makes on
a variable or an observable of a system under study. Examples of expectation
values in physics are the ensemble average in classical statistical mechanics,
the pure-state mean value of basic quantum mechanics, and the mixed-state mean
value of quantum statistics [1]. In order to give a number to an expectation
value, it is necessary to count with a statistical distribution of the
accessible states of the respective physical system. For example, to calculate
the ensemble average of a phase-space function we only need to integrate that
function weighted by a phase-space distribution over all coordinates and all
the classical momenta. However, in several cases, the resulting integral can
not be evaluated straightforwardly.

On the other hand, following the Bayesian formulation of statistics [2], it is
possible to derive an algebraic identity that relates the derivative of an
expectation value taken in a certain state of knowledge with another
expectation taken in the same state of knowledge. Specifically, for an
arbitrary function $\omega(\mathbf{X};\gamma)$ of a random variable
$\mathbf{X}$, which depends parametrically on a continuous parameter $\gamma$
and follows a nowhere vanishing statistical distribution
$P(\mathbf{X}|\gamma,I)$, we have

$\dfrac{\partial}{\partial\gamma}\left\langle\omega(\mathbf{X};\gamma)\right\rangle_{\gamma,I}=\int_{U}d\mathbf{x}\left[\dfrac{\partial\omega(\mathbf{X};\gamma)}{\partial\gamma}+\omega(\mathbf{X};\gamma)\dfrac{\partial\ln
P(\mathbf{X}|\gamma,I)}{\partial\gamma}\right]P(\mathbf{X}|\gamma,I),$ (1)

where $\gamma$ is also a parameter contained in the statistical distribution,
$U$ is some region such as $\mathbf{x}\in U$, and $I$ points out a certain
state of knowledge. It is worth noting that in Eq. (1), the expectation values
are taken not only in the state of knowledge given by $I$, but also by the
continuous parameter $\gamma$. The identity shown in Eq. (1) is called, for
historical reasons, the fluctuation-dissipation theorem (FDT) [3]. With the
aid of the FDT, it is possible, for example, to evaluate indirectly the mean
value and the variance associated to the discrete variable $k$ in the Poisson
distribution without the need to evaluate any integral [3]. Applied to
statistical physics, the FDT allows us to recover certain identities such as
the relationship between the mean energy with the logarithmic derivative of
the canonical partition function and the relationship between the specific
heat of a thermodynamical system with its energy fluctuation: this relation
corresponds to the thermodynamical version of the fluctuation-dissipation
theorem [4].

In the case of quantum mechanics, we have two types of expectation values
depending on whether we are dealing with pure quantum states or mixed quantum
states. In the first case, we use the wave function $\psi_{n}$ associated with
a certain quantum number $n$ as a probability amplitude, to calculate the
expectation value of a Hermitian operator, while in the case of mixed states,
it is necessary to know the density matrix $\hat{\rho}$ of the system. Thus,
for mixed-state quantum systems, the density matrix becomes a key quantity
since it, being the equivalent of the phase-space distribution in quantum
mechanics, acts as a probability distribution of the accessible quantum
states.

Depending on the type of the system, we have different methodologies to obtain
the density matrix $\hat{\rho}$ associated with the quantum state of such a
system. For example, in atomic and molecular systems, we can reconstruct the
density matrix using some fundamental parameters of the respective model, such
as the total charge and the number of electrons, and also imposing well-known
constraints to the statistical weights (probabilities) [5]. On the other hand,
within the inferential statistical approach of E. T. Jaynes [6, 7], we can
build the least biased quantum distribution only maximizing the von-Neumann
entropy

$S=-k\Tr\left(\hat{\rho}\log\hat{\rho}\right),$ (2)

under specific constraints related to whatever we know about the system. This
procedure is, indeed, the application of the Maximum Entropy Principle
(MaxEnt) to quantum statistics [8]. In practice, the constraints that are used
in the MaxEnt procedure are given by different Lagrange multipliers, such as
the inverse temperature $\beta$ or the product $\alpha=-\beta\mu$, etc., where
each of them is associated with respective expectation values of key
observables of the system (energy and mean particle number, respectively).
Additionally, the knowledge of the expectation values of quantum observables
allows us to reconstruct the density matrix of a microscopic system. Such a
process is known as Quantum State Tomography [9, 10, 11], which, nowadays,
comprises an important ground to quantum computing [12].

Once we have the density matrix of a system, we only need a states base in
order to expand the trace operator in the expression

$\left\langle\hat{A}\right\rangle_{\hat{\rho}}=\Tr\left\\{\hat{A}\hat{\rho}\right\\},$
(3)

where $\hat{A}=\hat{A}^{{\dagger}}$ is a Hermitian linear operator acting on
the Hilbert space $\mathcal{H}$ of the accessible states of the system. It is
important to stress that we have used the density matrix $\hat{\rho}$ as a
subscript in the expectation value (3) because, in quantum statistics and
following the Bayesian nomenclature, it represents the state of knowledge in
which we are calculating the expectation of $\hat{A}$. If the density matrix
$\hat{\rho}$ used in Eq. (3) comes from the MaxEnt principle, it depends on a
collection of Lagrange multipliers, so that, the expectation value (3) depends
on the same parameters too. We will call that collection of Lagrange
multipliers statistical-like parameters, since these determine the form of the
statistical model given by the density matrix. Moreover, there are cases in
which the observables of the system may depend on extra parameters, different
from those already mentioned, such as an oscillation frequency, a perturbative
parameter, time, etc. We will call such a collection of parameters quantum-
like parameters, because these are only contained in the quantum observables
(e.g. the Hamiltonian) and never appear outside of them.

The main goal of this article is to adapt to the quantum world the classical
FDT Eq. (1), taking special care in the way how the concepts are translated
from classical to quantum statistics. Additionally, we will show that several
traditional identities of quantum statistics and quantum mechanics, such as
the Hellmann-Feynman and the Ehrenfest theorems, the thermodynamical FDT, and
the generalization of the thermodynamic integration identity (useful in
determining the Helmholtz free energy difference in numerical simulations)
among others, become different instances of our quantum fluctuation-
dissipation theorem. Finally, by means of a generalized density matrix coming
from the MaxEnt principle we obtain generalized fluctuation-dissipation
identities: we will conclude that all of the previous FD identities, such as
the canonical and the grand canonical FD identities, are particular instances
of such generalized relationships.

The above suggests the following reflection: Are the FD-like identities
(Ehrenfest theorem, thermodynamical FDT, etc.) physical relationships or, on
the contrary, are only straightforward applications of the Bayesian reasoning
to quantum statistical mechanics? [13, 14, 15, 16]. Without giving a
definitive answer, we, at least, will seed the question.

This article is organized as follows: Section 2 is devoted to the derivation
of the quantum-FDT (Q-FDT). In Section 3 we review the Hellmann-Feynman
theorem as a particular case of the Q-FDT, alluding to its importance in
quantum chemistry. Section 4 is dedicated to review applications of Q-FDT to
the canonical and grand canonical models from statistical mechanics. Section 5
provides a view of the Ehrenfest theorem from our Q-FDT. At last Section 6 is
devoted to the generalized FD identities coming from a MaxEnt density matrix.

A remark about the notation: in this work we use the symbol $\ln$ for the
natural logarithm of a scalar function, and the symbol $\log$ for the natural
logarithm of an operator. The circumflex accent or hat over a symbol $A$,
namely $\hat{A}$, denotes an operator on the underlying Hilbert space, as is
customary in the quantum chemistry and chemical physics literature [17].

##  2 Derivation

Consider an observable $\hat{A}(\gamma)$ and a mixed quantum state represented
by a density matrix $\hat{\rho}(\gamma)$, acting both on a Hilbert space
$\mathcal{H}$ with a fixed domain $\Omega$. Both observables are dependent on
a continuous parameter $\gamma\in\mathbb{R}$ which can be of the following
types:

  * $\mathbf{1}$)

Quantum-like parameters: Are parameters that can be contained in the
Hamiltonian $\hat{H}(\gamma)$ or other observables of the system, such as a
perturbative parameter, time or some coordinate.

  * $\mathbf{2}$)

Statistical-like parameters: Are parameters that are not contained in the
observables of the system, and appear as Lagrange multipliers associated to
some constraint. Examples of this type of parameters are the inverse
temperature $\beta$, chemical potential $\mu$, etc.

In this article, we exclude parameters that change the underlying Hilbert
space. For example, those that change the size or shape of the system (such as
the length of the well in the model of the potential-well) are not considered
in this work.

We are interested in calculating the derivative of the expectation value of
$\hat{A}(\gamma)$ in the state $\hat{\rho}(\gamma)$ with respect to the
parameter $\gamma$, i.e.

$\dfrac{\partial}{\partial\gamma}\left\langle\hat{A}(\gamma)\right\rangle_{\hat{\rho}(\gamma)}=\dfrac{\partial}{\partial\gamma}\Tr\left\\{\hat{A}(\gamma)\hat{\rho}(\gamma)\right\\}.$
(4)

Due to the property of the invariance of the expectation with respect to the
basis, we are free to choose, in Eq. (4), a continuous or a discrete basis
kets. In this way, in order to expand the trace operator in (4), we use the
eigenstates of $\hat{\rho}(\gamma)$, and we assume that they form a countable
basis $\left\\{\ket{n;\gamma}\right\\}_{n\in\mathbb{N}}$ labeled both by a
quantum number $n\in\mathbb{N}$ and by the parameter $\gamma$. We can write

$\begin{split}\dfrac{\partial}{\partial\gamma}\Tr\left\\{\hat{A}(\gamma)\hat{\rho}(\gamma)\right\\}&=\dfrac{\partial}{\partial\gamma}\sum_{n\in\mathbb{N}}\bra{n;\gamma}\hat{A}(\gamma)\hat{\rho}(\gamma)\ket{n;\gamma},\\\
&=\sum_{n\in\mathbb{N}}\left[\dfrac{\partial}{\partial\gamma}\bra{n;\gamma}\right]\hat{A}(\gamma)\hat{\rho}(\gamma)\ket{n;\gamma}\\\
&+\sum_{n\in\mathbb{N}}\bra{n;\gamma}\hat{A}(\gamma)\hat{\rho}(\gamma)\left[\dfrac{\partial}{\partial\gamma}\ket{n;\gamma}\right]\\\
&+\sum_{n\in\mathbb{N}}\bra{n;\gamma}\dfrac{\partial}{\partial\gamma}\left[\hat{A}(\gamma)\hat{\rho}(\gamma)\right]\ket{n;\gamma}.\end{split}$
(5)

Now, if we restrict to certain observables $\hat{A}$ compatible with the
density matrix of the system, i.e., $\left[\hat{A},\hat{\rho}\right]=0$, we
have that $\hat{A}$ and $\hat{\rho}$ have the common eigenbasis
$\left\\{\ket{n;\gamma}\right\\}$. In this way, the spectral decompositions of
$\hat{A}$ and $\hat{\rho}$ are given by

$\begin{split}&\hat{A}(\gamma)=\sum_{n\in\mathbb{N}}a_{n}(\gamma)\hat{P}_{n}(\gamma),\\\
&\hat{\rho}(\gamma)=\sum_{n\in\mathbb{N}}\rho_{n}(\gamma)\hat{P}_{n}(\gamma),\end{split}$
(6)

where $a_{n}(\gamma)$ and $\rho_{n}(\gamma)$ are the eigenvalues of $\hat{A}$
and $\hat{\rho}$, respectively. Moreover, $\rho_{n}(\gamma)$ represents the
associated probabilities to each pure state
$\hat{P}_{n}(\gamma)=\ket{n;\gamma}\bra{n;\gamma}$ to which the system has
access, and whose form depends on the particular statistical model (e.g.
canonical, grand canonical, etc.). Knowing the eigenvalues of
$\hat{A}(\gamma)$ and $\hat{\rho}(\gamma)$, we can use them in the first two
sums of the left-hand side of $\left(\ref{FD1}\right)$, thus obtaining

$\begin{split}&\left[\dfrac{\partial}{\partial\gamma}\bra{n;\gamma}\right]\hat{A}(\gamma)\hat{\rho}(\gamma)\ket{n;\gamma}+\bra{n;\gamma}\hat{A}(\gamma)\hat{\rho}(\gamma)\left[\dfrac{\partial}{\partial\gamma}\ket{n;\gamma}\right],\\\
&=\rho_{n}(\gamma)\left\\{\left[\dfrac{\partial}{\partial\gamma}\bra{n;\gamma}\right]\hat{A}(\gamma)\ket{n;\gamma}+\bra{n;\gamma}\hat{A}(\gamma)\left[\dfrac{\partial}{\partial\gamma}\ket{n;\gamma}\right]\right\\},\\\
&=\rho_{n}(\gamma)a_{n}(\gamma)\left\\{\left[\dfrac{\partial}{\partial\gamma}\bra{n;\gamma}\right]\ket{n;\gamma}+\bra{n;\gamma}\left[\dfrac{\partial}{\partial\gamma}\ket{n;\gamma}\right]\right\\},\end{split}$
(7)

and assuming normalized states $\ket{n;\gamma}$, i.e.
$\innerproduct{n;\gamma}{n;\gamma}=1$, we recall that

$\dfrac{\partial}{\partial\gamma}\left[\innerproduct{n;\gamma}{n;\gamma}\right]=\left\\{\left[\dfrac{\partial}{\partial\gamma}\bra{n;\gamma}\right]\ket{n;\gamma}+\bra{n;\gamma}\left[\dfrac{\partial}{\partial\gamma}\ket{n;\gamma}\right]\right\\}=0.$
(8)

Thus, Eq. $\left(\ref{FD1}\right)$ is reduced to

$\dfrac{\partial}{\partial\gamma}\left\langle\hat{A}(\gamma)\right\rangle_{\hat{\rho}(\gamma)}=\sum_{n\in\mathbb{N}}\bra{n;\gamma}\dfrac{\partial\hat{A}(\gamma)}{\partial\gamma}\hat{\rho}(\gamma)\ket{n;\gamma}+\sum_{n\in\mathbb{N}}\bra{n;\gamma}\hat{A}(\gamma)\dfrac{\partial\hat{\rho}(\gamma)}{\partial\gamma}\ket{n;\gamma},$
(9)

where the first sum in Eq. (9) is easily related to the expectation of
$\partial\hat{A}/\partial\gamma$ in the state $\hat{\rho}(\gamma)$,

$\sum_{n\in\mathbb{N}}\bra{n;\gamma}\dfrac{\partial\hat{A}(\gamma)}{\partial\gamma}\hat{\rho}(\gamma)\ket{n;\gamma}=\left\langle\dfrac{\partial\hat{A}(\gamma)}{\partial\gamma}\right\rangle_{\hat{\rho}(\gamma)}.$
(10)

Then, in order to rewrite as an expectation value the second sum in Eq. (9),
we need to assume that $\hat{\rho}$ is a non-singular operator. This can
always be done by discarding all possible zero eigenvalues of $\hat{\rho}$ and
considering only those that are non-zero (i.e. non-zero probabilities). In
this way, if we multiply $\partial\hat{\rho}/\partial\gamma$ by
$\mathds{1}=\hat{\rho}^{-1}\hat{\rho}$ from the right in the second sum of Eq.
(9), and using the fact that

$\dfrac{\partial\hat{\rho}}{\partial\gamma}\hat{\rho}^{-1}=\dfrac{\partial\log\hat{\rho}}{\partial\gamma}$
(11)

which follows from the Taylor series of $\hat{\rho}^{-1}$ and
$\log\hat{\rho}$, we finally get

$\begin{split}\sum_{n\in\mathbb{N}}\bra{n;\gamma}\hat{A}(\gamma)\dfrac{\partial\hat{\rho}(\gamma)}{\partial\gamma}\ket{n;\gamma}&=\sum_{n\in\mathbb{N}}\bra{n;\gamma}\hat{A}(\gamma)\dfrac{\partial\hat{\rho}(\gamma)}{\partial\gamma}\hat{\rho}^{-1}\hat{\rho}\ket{n;\gamma},\\\
&=\sum_{n\in\mathbb{N}}\bra{n;\gamma}\hat{A}(\gamma)\dfrac{\partial\log\hat{\rho}(\gamma)}{\partial\gamma}\hat{\rho}\ket{n;\gamma},\\\
&=\left\langle\hat{A}(\gamma)\dfrac{\partial\log\hat{\rho}(\gamma)}{\partial\gamma}\right\rangle_{\hat{\rho}(\gamma)}.\end{split}$
(12)

We conclude that the derivative with respect to a continuous parameter
$\gamma$ of the expectation value of an observable $\hat{A}(\gamma)$ is given
by

$\dfrac{\partial}{\partial\gamma}\left\langle\hat{A}(\gamma)\right\rangle_{\hat{\rho}(\gamma)}=\left\langle\dfrac{\partial\hat{A}(\gamma)}{\partial\gamma}\right\rangle_{\hat{\rho}(\gamma)}+\left\langle\hat{A}(\gamma)\dfrac{\partial\log\hat{\rho}(\gamma)}{\partial\gamma}\right\rangle_{\hat{\rho}(\gamma)}.$
(13)

Eq. (13) corresponds to a quantum version of the fluctuation-dissipation
theorem (Q-FDT) for observables compatible with the density matrix. By means
of Eq. (13) we will derive not only standard results of quantum mechanics and
thermodynamics (the relation between the partition function and the mean
energy of a quantum system, the Ehrenfest theorem, and the thermodynamical
version of the fluctuation-dissipation theorem, among others), but also new
identities in the field (see Section 4.2 below). This supports the idea of
using the name of the fluctuation-dissipation theorem for Eq. (13). At last,
it is important to remark that a simple way to ensure that
$\left[\hat{A},\hat{\rho}\right]=0$, is when both $\hat{A}$ and $\hat{\rho}$
are functions of the Hamiltonian $\hat{H}$, i.e. $\hat{A}=A(\hat{H})$ and
$\hat{\rho}=\rho(\hat{H})$.

##  3 Hellmann-Feynman from quantum fluctuation-dissipation theorem

In the context of classical statistics, the derivative with respect to a
continuous parameter $\gamma$ on the expectation value of a function
$\omega(\mathbf{X};\gamma)$ in the state of knowledge given only by $I$, is

$\dfrac{\partial}{\partial\gamma}\left\langle\omega(\mathbf{X};\gamma)\right\rangle_{I}=\left\langle\dfrac{\partial\omega(\mathbf{X};\gamma)}{\partial\gamma}\right\rangle_{I}.$
(14)

We observe that Eq. (14) is a particular case of the classical FDT (Eq. (1)):
the last reduces to the former in the case when the statistical distribution
$P(\mathbf{X}|I)$ does not depend on the parameter $\gamma$ used in the
derivative.

In quantum mechanics, we have an analogous situation. Consider the Q-FDT Eq.
(13) for an operator $\hat{A}(\gamma)$ and for the instance where the density
matrix $\hat{\rho}$ represents a mixed state but does not contain the
parameter $\gamma$ used in the derivative. In such a case, the Q-FDT reduces
to

$\dfrac{\partial}{\partial\gamma}\left\langle\hat{A}(\gamma)\right\rangle_{\hat{\rho}}=\left\langle\dfrac{\partial\hat{A}(\gamma)}{\partial\gamma}\right\rangle_{\hat{\rho}}.$
(15)

Observe that the above identity is the quantum equivalent of the Eq. (14).
Then, in the special case in which the operator is the Hamiltonian of the
system, i.e. $\hat{A}(\gamma)=\hat{H}(\gamma)$, where $\gamma$ may be some
parameter contained in it (mass, force constant, etc.), we have

$\dfrac{\partial}{\partial\gamma}\left\langle\hat{H}\right\rangle_{\hat{\rho}}=\left\langle\dfrac{\partial\hat{H}(\gamma)}{\partial\gamma}\right\rangle_{\hat{\rho}}.$
(16)

Eq. (16), which is called the Hellmann-Feynman theorem, is widely used in the
context of quantum chemistry and molecular physics to explore the chemical
bonding in molecular systems [17, 18].

Not only the Hellmann-Feynman theorem is recovered when the density matrix
$\hat{\rho}$ does not depend on $\gamma$, but also when it represents a pure
state projector, namely $\hat{\rho}=\ket{n;\gamma}\bra{n;\gamma}$ (which may
depend on $\gamma$). In this case, we have

$\begin{split}\dfrac{\partial}{\partial\gamma}\left\langle\hat{H}(\gamma)\right\rangle_{n}&=\dfrac{\partial}{\partial\gamma}\Tr{\hat{H}\ket{n;\gamma}\bra{n;\gamma}},\\\
&=\dfrac{\partial}{\partial\gamma}\sum_{m\in\mathbb{N}}\bra{m;\gamma}\hat{H}(\gamma)\ket{n;\gamma}\delta_{n,m},\\\
&=\dfrac{\partial}{\partial\gamma}\left[\bra{n;\gamma}\hat{H}(\gamma)\ket{n;\gamma}\right],\\\
&=\bra{n;\gamma}\dfrac{\partial\hat{H}(\gamma)}{\partial\gamma}\ket{n;\gamma},\end{split}$
(17)

where we have assumed that $\left\\{\ket{n;\gamma}\right\\}_{n\in\mathbb{N}}$
is an orthonormal eigenbasis of $\hat{H}(\gamma)$ (cf. Eq. (8)). Hence

$\dfrac{\partial
E_{n}(\gamma)}{\partial\gamma}=\left\langle\dfrac{\partial\hat{H}(\gamma)}{\partial\gamma}\right\rangle_{n},$
(18)

where $E_{n}(\gamma)$ is the energy eigenvalue associated to the state
$\ket{n;\gamma}$.

An interesting use of the Hellmann-Feynman theorem, in the context of quantum
chemistry, is in the case in which the parameter $\gamma$ corresponds to a
certain nuclear coordinate $R_{i,\alpha}$ ($ith$ coordinate of the nucleus
$\alpha$) of a $N$-electron molecular system. Considering the molecular
Schrödinger equation within the Born-Oppenheimer approximation [17]

$\left[\hat{H}_{el}+\hat{V}_{NN}\right]\ket{\psi_{el}}=U(\mathbf{R})\ket{\psi_{el}},$
(19)

where $\hat{H}_{el}=\hat{T}_{el}+\hat{V}_{el}$ is the electronic Hamiltonian
(which includes the electronic kinetic energy operator $\hat{T}_{el}$ and the
electronic potential operator $\hat{V}_{el}$), $\hat{V}_{NN}$ is the potential
due to nuclear repulsion and $\ket{\psi_{el}}$ are the electronic states of
the molecule. The energy $U(\mathbf{R})$ corresponds to the electronic energy
plus the internuclear repulsion and is called the Born-Oppenheimer energy (BO)
of the molecule [19]. We notice that the BO energy depends parametrically on
the coordinates of the nuclei in the molecule throughout the nuclear
coordinate vector
$\mathbf{R}=\left(R_{x,1},R_{y,1},R_{z,1},...,R_{x,M},R_{y,M},R_{z,M}\right)$,
where $M$ is the total number of nuclei in the molecule. In such a case, the
Hellmann-Feynman theorem can be written as (for $\gamma=R_{i,\alpha}$)

$\dfrac{\partial U(\mathbf{R})}{\partial R_{i,\alpha}}=\int
d\mathbf{r}_{1}...d\mathbf{r}_{N}\hskip
2.84544pt\psi_{el}^{*}(\mathbf{r}_{1},...,\mathbf{r}_{N})\left[\dfrac{\partial
V_{el}}{\partial R_{i,\alpha}}+\dfrac{\partial V_{NN}}{\partial
R_{i,\alpha}}\right]\psi_{el}(\mathbf{r}_{1},...,\mathbf{r}_{N}),$ (20)

where $\mathbf{r}_{i}$ denotes the coordinates of the $ith$ electron. The
derivative of the BO energy with respect to $R_{i,\alpha}$ can be seen as the
$ith$ component of a force on the $\alpha th$ nucleus due to the other nuclei
and the electron cloud. See [20] for more details.

##  4 Canonical and grand canonical identities

Before giving the general results obtained from the Q-FDT for a generalized
density matrix, we illustrate the use of (13) in the canonical and grand
canonical statistical models. According to Section 2, we are free to choose
$\hat{A}$ as any observable compatible with $\hat{\rho}$ such as the identity
operator $\mathds{1}$, the Hamiltonian $\hat{H}$ and the density matrix
$\hat{\rho}$ itself, or even an arbitrary function $F(\hat{\rho})$ of
$\hat{\rho}$. For the sake of simplicity, we only consider the the cases
$\hat{A}=\mathds{1}$ and $\hat{A}=\hat{H}$.

###  4.1 Canonical model

We consider a canonical density matrix of the form

$\hat{\rho}_{c}(\beta)=\dfrac{\exp{-\beta\hat{H}}}{\mathcal{Z}(\beta)},$ (21)

where $\beta$ is the inverse temperature and

$\mathcal{Z}(\beta)=\Tr\left(e^{-\beta\hat{H}}\right)$ (22)

is the canonical partition function, which depends on $\beta$ and, in certain
cases, on a quantum-like parameter $\lambda$. If we use the canonical model
Eq. (21) in the Q-FDT, we obtain

$\dfrac{\partial}{\partial\gamma}\left<\hat{A}\right>_{\hat{\rho}_{c}}=\left<\dfrac{\partial\hat{A}}{\partial\gamma}\right>_{\hat{\rho}_{c}}-\left<\hat{A}\dfrac{\partial}{\partial\gamma}\left(\beta\hat{H}\right)\right>_{\hat{\rho}_{c}}-\dfrac{\partial\ln\mathcal{Z}(\beta)}{\partial\gamma}\left<\hat{A}\right>_{\hat{\rho}_{c}},$
(23)

where $\gamma$ may be a quantum-like ($\lambda$) or a statistical-like
parameter ($\beta$). In the following, we will use in the above relation Eq.
(23) (which we have called the canonical Q-FDT) $\hat{A}=\mathds{1}$ and
$\hat{A}=\hat{H}$ for both types of parameters, obtaining four canonical
fluctuation-dissipation identities (which are summarized in Table 1):

  * $i)$

$\hat{A}=\mathds{1}$ and $\gamma$ is a quantum-like parameter ($\lambda$):

In this case, we have that

$\dfrac{\partial}{\partial\lambda}\left\langle\mathds{1}\right\rangle=0\hskip
14.22636pt\text{and}\hskip
14.22636pt\left\langle\dfrac{\partial\mathds{1}}{\partial\lambda}\right\rangle=0,$

so the second term of the right-hand side of Eq. (13) becomes

$\begin{split}0=\left\langle\dfrac{\partial\log\hat{\rho}_{c}}{\partial\lambda}\right\rangle_{\hat{\rho}_{c}}&=\left\langle\dfrac{\partial}{\partial\lambda}\left[-\beta\hat{H}(\lambda)-\ln\mathcal{Z}(\beta,\lambda)\mathds{1}\right]\right\rangle_{\hat{\rho}_{c}},\\\
&=-\beta\left\langle\dfrac{\partial\hat{H}(\lambda)}{\partial\lambda}\right\rangle_{\hat{\rho}_{c}}-\dfrac{\partial\ln\mathcal{Z}(\beta,\lambda)}{\partial\lambda},\end{split}$
(24)

and we obtain

$\left\langle\dfrac{\partial\hat{H}(\lambda)}{\partial\lambda}\right\rangle_{\hat{\rho}_{c}}=-\dfrac{1}{\beta}\dfrac{\partial\ln\mathcal{Z}(\beta,\lambda)}{\partial\lambda}\bigg{|}_{\beta}.$
(25)

Identifying in the above identity the Helmholtz Free energy,

$\mathcal{F}=-\dfrac{1}{\beta}\ln\mathcal{Z}(\beta,\lambda),$ (26)

we can rewrite this (Eq. (25)) in the form

$\left\langle\dfrac{\partial\hat{H}(\lambda)}{\partial\lambda}\right\rangle_{\hat{\rho}_{c}}=\dfrac{\partial\mathcal{F}(\beta,\lambda)}{\partial\lambda}.$
(27)

Finally, we can obtain the free energy difference $\Delta\mathcal{F}$
integrating with respect to
$\lambda\in\left[\lambda_{min},\lambda_{max}\right]$ on both sides:

$\Delta\mathcal{F}=\int_{\lambda_{min}}^{\lambda_{max}}d\lambda\left\langle\dfrac{\partial\hat{H}(\lambda)}{\partial\lambda}\right\rangle_{\hat{\rho}_{c}}.$
(28)

This procedure is used to obtain free energy differences via molecular
dynamics, such as the ionic hydration free energy difference, and is called
the thermodynamic integration technique [21, 22].

  * $ii)$

$\hat{A}=\mathds{1}$ and $\gamma$ is a statistical-like parameter ($\beta$):

As in the previous case, we have

$\dfrac{\partial}{\partial\beta}\left\langle\mathds{1}\right\rangle=0\hskip
14.22636pt\text{and}\hskip
14.22636pt\left\langle\dfrac{\partial\mathds{1}}{\partial\beta}\right\rangle=0,$

and the Q-FDT gives

$\begin{split}0=\left\langle\dfrac{\partial\log\hat{\rho}_{c}}{\partial\beta}\right\rangle_{\hat{\rho}_{c}}&=\left\langle\dfrac{\partial}{\partial\beta}\left[-\beta\hat{H}-\ln\mathcal{Z}(\beta)\mathds{1}\right]\right\rangle_{\hat{\rho}_{c}},\\\
&=-\left\langle\hat{H}\right\rangle_{\hat{\rho}_{c}}-\dfrac{\partial\ln\mathcal{Z}(\beta)}{\partial\beta},\end{split}$
(29)

where we have used the fact that $\hat{H}$ does not depend on the inverse
temperature $\beta$. In this way, we get

$\left\langle\hat{H}\right\rangle_{\hat{\rho}_{c}}=-\dfrac{\partial\ln\mathcal{Z}(\beta)}{\partial\beta}\bigg{|}_{\lambda}.$
(30)

It is important to observe that the identity shown in Eq. (30) is the well-
known relationship between the partition function and the average energy which
arises in the context of canonical ensemble in traditional statistical
mechanics [1].

  * $iii)$

$\hat{A}=\hat{H}$ and $\gamma$ is a quantum-like parameter ($\lambda$):

In this case we have

$\begin{split}\dfrac{\partial}{\partial\lambda}\left\langle\hat{H}(\lambda)\right\rangle_{\hat{\rho}_{c}}&=\left\langle\dfrac{\partial\hat{H}(\lambda)}{\partial\lambda}\right\rangle_{\hat{\rho}_{c}}+\left\langle\hat{H}(\lambda)\dfrac{\partial}{\partial\lambda}\left[-\beta\hat{H}(\lambda)-\ln\mathcal{Z}(\beta,\lambda)\mathds{1}\right]\right\rangle_{\hat{\rho}_{c}},\\\
&=\left\langle\dfrac{\partial\hat{H}(\lambda)}{\partial\lambda}\right\rangle_{\hat{\rho}_{c}}-\beta\left\langle\hat{H}\dfrac{\partial\hat{H}}{\partial\lambda}\right\rangle_{\hat{\rho}_{c}}-\dfrac{\partial\ln\mathcal{Z}(\beta,\lambda)}{\partial\lambda}\left\langle\hat{H}\right\rangle_{\hat{\rho}_{c}},\end{split}$
(31)

and using Eq. (25), we finally get

$\dfrac{\partial}{\partial\lambda}\left\langle\hat{H}(\lambda)\right\rangle_{\hat{\rho}_{c}}=\left\langle\dfrac{\partial\hat{H}(\lambda)}{\partial\lambda}\right\rangle_{\hat{\rho}_{c}}-\beta\hskip
1.42271pt\mathrm{Cov}\left(\hat{H},\dfrac{\partial\hat{H}}{\partial\lambda}\right),$
(32)

where
$\mathrm{Cov}\left(\hat{H},\dfrac{\partial\hat{H}}{\partial\lambda}\right)$
stands for

$\begin{split}\mathrm{Cov}\left(\hat{H},\dfrac{\partial\hat{H}}{\partial\lambda}\right)&\equiv\left<\Delta\hat{H}\hskip
1.42271pt\Delta\left(\dfrac{\partial\hat{H}}{\partial\lambda}\right)\right>_{\hat{\rho}_{c}},\\\
&=\left\langle\hat{H}\dfrac{\partial\hat{H}}{\partial\lambda}\right\rangle_{\hat{\rho}_{c}}-\left\langle\hat{H}\right\rangle_{\hat{\rho}_{c}}\left\langle\dfrac{\partial\hat{H}}{\partial\lambda}\right\rangle_{\hat{\rho}_{c}},\end{split}$
(33)

i.e., corresponds to the covariance between the Hamiltonian and its derivative
in the state of knowledge given by $\hat{\rho}$, and measures the correlation
between both operators in a statistical sense. In Eq. (33) $\Delta\hat{H}$
means the deviation of $\hat{H}$ from its expectation (mean) value,

$\Delta\hat{H}=\hat{H}-\left<\hat{H}\right>\mathds{1},$

and the same for $\partial\hat{H}/\partial\lambda$.

It is worth mentioning two compatible observables (i.e. whose commutator is
null), might not be uncorrelated. This can be seen if we recast the covariance
between any two observables $\hat{A}$ and $\hat{B}$ as [23]

$\mathrm{Cov}\left(\hat{A},\hat{B}\right)=\dfrac{1}{2}\left<\left[\hat{A},\hat{B}\right]\right>_{\hat{\rho}}+\dfrac{1}{2}\left<\left\\{\Delta\hat{A},\Delta\hat{B}\right\\}\right>_{\hat{\rho}},$
(34)

where

$\left\\{\Delta\hat{A},\Delta\hat{B}\right\\}=\Delta\hat{A}\Delta\hat{B}+\Delta\hat{B}\Delta\hat{A}$

stands for the anticommutator between the deviations of $\hat{A}$ and
$\hat{B}$. In this way, we can easily see that if
$\left[\hat{A},\hat{B}\right]=0$, the anticommutator term survives and so does
the covariance.

  * $iv)$

$\hat{A}=\hat{H}$ and $\lambda$ is a statistical-like parameter ($\beta$):

In this case, the Q-FDT gives

$\begin{split}\dfrac{\partial}{\partial\beta}\left\langle\hat{H}\right\rangle_{\hat{\rho}_{c}}&=\left\langle\hat{H}\dfrac{\partial}{\partial\beta}\left[-\beta\hat{H}-\ln\mathcal{Z}(\beta)\mathds{1}\right]\right\rangle_{\hat{\rho}_{c}}\\\
&=-\left\langle\hat{H}^{2}\right\rangle_{\hat{\rho}_{c}}-\dfrac{\partial\ln\mathcal{Z}(\beta)}{\partial\beta}\left\langle\hat{H}\
\right\rangle_{\hat{\rho}_{c}}.\end{split}$ (35)

where we have used that $\partial\hat{H}/\partial\beta=0$.

Using Eq. (30) in the second term of Eq. (35), we finally obtain

$\dfrac{\partial}{\partial\beta}\left\langle\hat{H}\right\rangle_{\hat{\rho}_{c}}=-\mathrm{Var}\left(\hat{H}\right),$
(36)

where $\mathrm{Var}\left(\hat{H}\right)$ is the variance in the energy of the
system, namely

$\mathrm{Var}\left(\hat{H}\right)=\left\langle\hat{H}^{2}\right\rangle_{\hat{\rho}_{c}}-\left\langle\hat{H}\right\rangle^{2}_{\hat{\rho}_{c}}.$
(37)

Then, by means of the chain rule and using $\beta=1/kT$, we finally obtain

$C=\dfrac{\partial}{\partial
T}\left\langle\hat{H}\right\rangle_{\hat{\rho}_{c}}=\dfrac{1}{kT^{2}}\mathrm{Var}\left(\hat{H}\right),$
(38)

where $C$ is the heat capacity of the system, $T$ is the absolute temperature
and $k$ is the Boltzmann constant. Traditionally, Eq. (38) is understood as
one instance of the fluctuation-dissipation theorem [4]; it relates an
external change in energy, driven by a change in temperature (hence,
dissipation) with the internal fluctuations of the energy in equilibrium [24].
It supports the name given to Eq. (13), namely the fluctuation-dissipation
theorem.

Observable |  $\gamma=\lambda$ (quantum-like parameter) |  $\gamma=\beta$ (statistical-like parameter)
---|---|---
$\hat{A}=\mathds{1}$ | $\left\langle\dfrac{\partial\hat{H}(\lambda)}{\partial\lambda}\right\rangle=-\dfrac{1}{\beta}\dfrac{\partial\ln\mathcal{Z}(\beta,\lambda)}{\partial\lambda}\bigg{|}_{\beta}$ | $\left\langle\hat{H}\right\rangle=-\dfrac{\partial\ln\mathcal{Z}(\beta)}{\partial\beta}\bigg{|}_{\lambda}$
$\hat{A}=\hat{H}$ | $\dfrac{\partial}{\partial\lambda}\left\langle\hat{H}(\lambda)\right\rangle=\left\langle\dfrac{\partial\hat{H}(\lambda)}{\partial\lambda}\right\rangle-\beta\hskip 1.42271pt\mathrm{Cov}\left(\hat{H},\dfrac{\partial\hat{H}}{\partial\lambda}\right)$ | $\dfrac{\partial}{\partial\beta}\left\langle\hat{H}\right\rangle=-\mathrm{Var}\left(\hat{H}\right)$
Table 1: Canonical quantum fluctuation-dissipation identities

###  4.2 Grand canonical model

The grand canonical density matrix corresponds to the following statistical
model:

$\hat{\rho}_{g}(\beta,\mathit{z})=\dfrac{\exp\left\\{-\beta\left(\hat{H}-\mu\hat{N}\right)\right\\}}{\Xi(\beta,\mathit{z})},$
(39)

where $\mu$ is the chemical potential, $\hat{N}$ is the number operator and
$\Xi(\beta,\mathit{z})$ is the grand partition function which depends on the
inverse temperature $\beta$ and the fugacity $z=e^{\beta\mu}$. As in the
canonical model, the grand partition function may also depend on a quantum-
like parameter $\lambda$.

Using the grand canonical model Eq. (39) in the Q-FDT (13), we obtain the
grand canonical Q-FDT:

$\dfrac{\partial}{\partial\gamma}\left<\hat{A}\right>_{\hat{\rho}_{g}}=\left<\dfrac{\partial\hat{A}}{\partial\gamma}\right>_{\hat{\rho}_{g}}-\left<\hat{A}\dfrac{\partial}{\partial\gamma}\left(\beta\hat{H}\right)\right>_{\hat{\rho}_{g}}+\left<\hat{A}\dfrac{\partial}{\partial\gamma}\left(\beta\mu\hat{N}\right)\right>_{\hat{\rho}_{g}}-\dfrac{\partial\ln\Xi(\beta,\mathit{z})}{\partial\gamma}\left<\hat{A}\right>_{\hat{\rho}_{g}},$
(40)

where, in the present case, $\gamma$ may be a quantum-like ($\lambda$) or a
statistical-like parameter ($\beta$ or $\mu$). As above, we can replace
$\hat{A}$ with certain observables compatibles with the grand canonical
density matrix $\hat{\rho}_{g}$. In this case we have the following three
options:

  * 1)

$\hat{A}=\mathds{1}$

  * 2)

$\hat{A}=\hat{H}$

  * 3)

$\hat{A}=\hat{N}$

We obtain for the grand canonical statistical model nine different
fluctuation-dissipation identities, which are summarized in Table 2. Some of
these identities are well-known relations coming from quantum statistics while
others are new ones. At this point, it is worth mentioning that the number
operator $\hat{N}$ is, in general, a constant operator, so that a quantum-like
parameter $\lambda$ can only appear within the Hamiltonian. Since the
calculations concerning the grand canonical model follow in a very similar way
to the calculations of the canonical model, here we will omit the details.

Considering the first row in Table 2 for $\hat{A}=\mathds{1}$. We have that
these three cases (using $\gamma=\lambda$, $\gamma=\beta$ and $\gamma=\mu$)
lead to known relationships of quantum statistical mechanics. In particular
for the pair ($\mathds{1},\lambda$) we obtain

$\left<\dfrac{\partial\hat{H}}{\partial\lambda}\right>_{\hat{\rho}_{g}(\lambda)}=-\dfrac{1}{\beta}\dfrac{\partial\ln\Xi(\beta,\mathit{z},\lambda)}{\partial\lambda}\bigg{|}_{\beta,\mathit{z}},$
(41)

where the derivative has been taken fixing the fugacity and the inverse
temperature. As in the canonical model, we can identify another thermodynamic
potential: the grand potential $\Phi(\beta,\mathit{z},\lambda)$. It is defined
by

$\Phi(\beta,\mathit{z},\lambda)=-\dfrac{1}{\beta}\ln\Xi(\beta,\mathit{z},\lambda).$
(42)

If we integrate Eq. (41) with respect to
$\lambda\in\left[\lambda_{min},\lambda_{max}\right]$, using the definition of
$\Phi$, we can obtain the grand potential difference:

$\Delta\Phi(\beta,\mathit{z},\lambda)=\int_{\lambda_{min}}^{\lambda_{max}}d\lambda\left<\dfrac{\partial\hat{H}}{\partial\lambda}\right>_{\hat{\rho}_{g}}.$
(43)

Eq. (43) is another instance of the thermodynamic integration technique
applied to open quantum systems. Then, for the pairs ($\mathds{1},\beta$) and
$(\mathds{1},\mu)$, we have

$\left<\hat{H}\right>_{\hat{\rho}_{g}}=-\dfrac{\partial\ln\Xi(\beta,\mathit{z})}{\partial\beta}\bigg{|}_{\lambda,\mathit{z}}$
(44)

and

$\left<\hat{N}\right>_{\hat{\rho}_{g}}=\dfrac{1}{\beta}\dfrac{\partial\ln\Xi(\beta,\mathit{z})}{\partial\mu}\bigg{|}_{\lambda,\beta},$
(45)

respectively. The identities in Eqs. (44) and (45), which relate, in the first
case, the grand partition function with the mean energy for an open quantum
system, and in the second case, the grand partition function with the mean
particle number, are well-known identities in quantum statistics [1]. It is
important to highlight that the derivative in Eq. (44) has been taken fixing
both $\lambda$ and $\mathit{z}$, while in Eq. (45) both $\lambda$ and $\beta$
are fixed.

Consider now the second row in Table 2 for $\hat{A}=\hat{H}$. For the pairs
$(\hat{H},\lambda)$ and $(\hat{H},\beta)$ we obtain the same identities (Eqs.
(32) and (36)) as for the canonical model, but using $\hat{\rho}_{g}$ in the
calculations of the mean values. Because we have the same identities for both
canonical and grand canonical models, one may be tempted to interpret this as
a consequence of the principle of equivalence between different statistical
ensembles. However, such a principle is meaningless in our case, because we
are interested in small quantum systems in which it is not possible to employ
the thermodynamic limit ($N\rightarrow\infty$): such equivalence between these
canonical and grand canonical identities is an algebraic consequence of the
similitude between both statistical models; the grand canonical model can be
considered as the canonical model with an extra condition.

Next, for the pair $(\hat{H},\mu)$ we have the identity

$\dfrac{\partial}{\partial\mu}\left<\hat{H}\right>_{\hat{\rho}_{g}}=\beta\hskip
1.42271pt\mathrm{Cov}\left(\hat{H},\hat{N}\right),$ (46)

which accounts for the dependence of the variation of the mean energy with
respect to the chemical potential expressed as the covariance between the
Hamiltonian and the number operator: the stronger the correlation between
$\hat{H}$ and $\hat{N}$, the greater dependence of the energy with respect to
the chemical potential.

Consider finally the third row of Table 2 for $\hat{A}=\hat{N}$. As above, we
obtain three relations, namely for $\gamma=\lambda$, $\gamma=\beta$ and
$\gamma=\mu$. The first two cases ($\gamma=\lambda$ and $\gamma=\beta$), along
with Eq. (46), to the best of our knowledge, are new identities. Thus, for the
pair ($\hat{N},\lambda$) we obtain

$\dfrac{\partial}{\partial\lambda}\left<\hat{N}\right>_{\hat{\rho}_{g}}=-\beta\hskip
1.42271pt\mathrm{Cov}\left(\hat{N},\dfrac{\partial\hat{H}}{\partial\lambda}\right);$
(47)

identity that has been derived using Eq. (41). Then, for the pair
($\hat{N},\beta$), we obtain

$\dfrac{\partial}{\partial\beta}\left<\hat{N}\right>_{\hat{\rho}_{g}}=-\mathrm{Cov}\left(\hat{N},\hat{H}\right),$
(48)

where we have used Eq. (44). As in Eq. (46), the correlation between the
Hamiltonian and the number operator allows to predict the change between a
property of the system (the mean particle number) driven by the change in a
external parameter (the inverse temperature). At last, for the pair
($\hat{N},\mu$), and by means of Eq. (45), the following identity is obtained:

$\dfrac{\partial}{\partial\mu}\left<\hat{N}\right>_{\hat{\rho}_{g}}=\beta\hskip
1.42271pt\mathrm{Var}\left(\hat{N}\right).$ (49)

Eq. (49), which can be derived also in a traditional way by applying
derivatives with respect to the chemical potential on the grand partition
function, allows to predict the change in the mean particle number with an
external parameter (chemical potential) only measuring the internal
fluctuations in the particle number. For this reason, we can interpret Eq.
(49) as another instance of the thermodynamical FDT, but for the conjugate-
variable couple $(\hat{N},\mu)$.

Obs. | $\gamma=\lambda$ | $\mathbf{\gamma=\beta}$ | $\mathbf{\gamma=\mu}$
---|---|---|---
$\hat{A}=\mathds{1}$ | $\left<\dfrac{\partial\hat{H}}{\partial\lambda}\right>=-\dfrac{1}{\beta}\dfrac{\partial\ln\Xi(\beta,\mathit{z},\lambda)}{\partial\lambda}\bigg{|}_{\beta,\mathit{z}}$ | $\left<\hat{H}\right>=-\dfrac{\partial\ln\Xi(\beta,\mathit{z})}{\partial\beta}\bigg{|}_{\lambda,\mathit{z}}$ | $\left<\hat{N}\right>=\dfrac{1}{\beta}\dfrac{\partial\ln\Xi(\beta,\mathit{z})}{\partial\mu}\bigg{|}_{\lambda,\beta}$
$\hat{A}=\hat{H}$ | $\dfrac{\partial}{\partial\lambda}\left\langle\hat{H}(\lambda)\right\rangle=\left\langle\dfrac{\partial\hat{H}(\lambda)}{\partial\lambda}\right\rangle-\beta\hskip 1.42271pt\mathrm{Cov}\left(\hat{H},\dfrac{\partial\hat{H}}{\partial\lambda}\right)$ | $\dfrac{\partial}{\partial\beta}\left\langle\hat{H}\right\rangle=-\mathrm{Var}\left(\hat{H}\right)$ | $\dfrac{\partial}{\partial\mu}\left<\hat{H}\right>=\beta\hskip 1.42271pt\mathrm{Cov}\left(\hat{H},\hat{N}\right)$
$\hat{A}=\hat{N}$ | $\dfrac{\partial}{\partial\lambda}\left<\hat{N}\right>=-\beta\hskip 1.42271pt\mathrm{Cov}\left(\hat{N},\dfrac{\partial\hat{H}}{\partial\lambda}\right)$ | $\dfrac{\partial}{\partial\beta}\left<\hat{N}\right>=-\mathrm{Cov}\left(\hat{N},\hat{H}\right)$ | $\dfrac{\partial}{\partial\mu}\left<\hat{N}\right>=\beta\hskip 1.42271pt\mathrm{Var}\left(\hat{N}\right)$
Table 2: Grand canonical quantum fluctuation-dissipation identities.

##  5 Ehrenfest theorem from quantum fluctuation-dissipation theorem

An important result of quantum mechanics is the Eherenfest theorem, which
states that [23]

$\dfrac{d}{dt}\left\langle\hat{A}\right\rangle_{\ket{\psi}}=\left\langle\dfrac{\partial\hat{A}}{\partial
t}\right\rangle_{\ket{\psi}}+\dfrac{1}{i\hbar}\left\langle\left[\hat{A},\hat{H}\right]\right\rangle_{\ket{\psi}},$
(50)

and accounts for the time evolution of the expectation value of an observable
$\hat{A}$ in a closed quantum system with a Hamiltonian $\hat{H}$. Originally,
the expectations in Eq. (50) are taken in a pure state $\ket{\psi}\bra{\psi}$,
but this can easily be generalized to the case of mixed states.

In order to derive the Ehrenfest theorem (Eq. (50)) from the Q-FDT, we use in
Eq. (13) an arbitrary time-dependent density matrix $\hat{\rho}(t)$ with an
arbitrary time-dependent observable $\hat{A}(t)$,

$\dfrac{d}{dt}\left\langle\hat{A}(t)\right\rangle_{\hat{\rho}(t)}=\left\langle\dfrac{\partial\hat{A}(t)}{\partial
t}\right\rangle_{\hat{\rho}(t)}+\left\langle\hat{A}(t)\dfrac{\partial\log\hat{\rho}(t)}{\partial
t}\right\rangle_{\hat{\rho}(t)},$ (51)

where we have chosen the time $t$ as the parameter $\gamma$. Rewriting the
second term in the right side of Eq. (51) as

$\begin{split}\left\langle\hat{A}(t)\dfrac{\partial\log\hat{\rho}(t)}{\partial
t}\right\rangle_{\hat{\rho}(t)}&=\left\langle\hat{A}(t)\dfrac{\partial\hat{\rho}(t)}{\partial
t}\hat{\rho}^{-1}(t)\right\rangle_{\hat{\rho}(t)}\\\
&=\Tr\left\\{\hat{A}(t)\dfrac{\partial\hat{\rho}(t)}{\partial
t}\right\\},\end{split}$ (52)

where we have used $\hat{\rho}^{-1}\hat{\rho}=\mathds{1}$, and assuming that
$\hat{\rho}(t)$ evolves in an unitary way, i.e., according to the Liouville-
von Neumann equation of motion

$\dfrac{\partial\hat{\rho}(t)}{\partial
t}=\dfrac{1}{i\hbar}\left[\hat{H}(t),\hat{\rho}(t)\right],$ (53)

we obtain

$\left\langle\hat{A}(t)\dfrac{\partial\log\hat{\rho}(t)}{\partial
t}\right\rangle_{\hat{\rho}(t)}=\dfrac{1}{i\hbar}\Tr\left\\{\hat{A}(t)\left[\hat{H}(t),\hat{\rho}(t)\right]\right\\}.$
(54)

After some algebra, it is possible to recast the trace operator in (54) as an
expectation value in the state $\hat{\rho}(t)$,

$\Tr\left\\{\hat{A}\left[\hat{H},\hat{\rho}\right]\right\\}=\left\langle\left[\hat{A},\hat{H}\right]\right\rangle_{\hat{\rho}(t)},$
(55)

where we have used the cyclic property of the trace operator. Finally, the
last expression allows us to recover the Ehrenfest theorem from the the Q-FDT

$\dfrac{d}{dt}\left\langle\hat{A}(t)\right\rangle_{\hat{\rho}(t)}=\left\langle\dfrac{\partial\hat{A}(t)}{\partial
t}\right\rangle_{\hat{\rho}(t)}+\dfrac{1}{i\hbar}\left\langle\left[\hat{A},\hat{H}\right]\right\rangle_{\hat{\rho}(t)}.$
(56)

##  6 Generalized MaxEnt density matrix identities

Given a set $\left\\{\alpha_{i}\right\\}_{i=1,...,n}$ of statistical-like
parameters, and another set $\left\\{\lambda_{i}\right\\}_{i=1,...,m}$ of
quantum-like parameters, we can generalize the results of the previous
sections introducing a generalized density matrix of the form

$\hat{\rho}\left(\alpha_{1},...,\alpha_{n};\lambda_{1},...,\lambda_{m}\right)=\dfrac{1}{\mathcal{Z}\left(\alpha_{1},...,\alpha_{n};\lambda_{1},...,\lambda_{m}\right)}\exp\left[-\sum_{j=1}^{n}\alpha_{j}\hat{F}_{j}\left(\lambda_{1},...,\lambda_{m}\right)\right],$
(57)

where $\left\\{\hat{F}_{j}\right\\}_{j=1,...,n}$ is a set of observables
acting on the Hilbert space $\mathcal{H}$ of states of the system (e.g.
Hamiltonian, number operator, etc.). These observables are compatible with
each other and may depend on one or more quantum-like parameters. It is
important to notice that the statistical-like parameters depend on the
statistical model only and appear as Lagrange multipliers associated with
certain constraints in the MaxEnt procedure, while the quantum-like parameters
are related to the particular features of the system, and some of them can be
contained in the Hamiltonian or other observables of the system (e.g.,
perturbation parameters). We have used the appellative generalized for the
density matrix in the sense given in Ref. [6]: within the approach of MaxEnt
inference not only does it have the Hamiltonian the prime role, but all the
observables in the set $\left\\{\hat{F}_{j}\right\\}_{j=1,...,n}$ possess
equal weight.

The entropy associated to such a generalized density matrix is obtained
through the von-Neumann entropy (2), which in this case is given by

$S=\sum_{j=1}^{n}\alpha_{j}\left\langle\hat{F}_{j}\right\rangle_{\hat{\rho}}+\ln\mathcal{Z}\left(\alpha_{1},...,\alpha_{n};\lambda_{1},...,\lambda_{m}\right).$
(58)

We can easily identify the canonical and the grand-canonical cases encoded in
the entropy Eq. (58). For instance, considering only one statistical-like
parameter $\alpha$ and setting it equal to the inverse temperature $\beta$,
and choosing $\hat{F}=\hat{H}$, we obtain

$-kT\ln\mathcal{Z}(\beta;\lambda_{1},...,\lambda_{m})=E-TS=\mathcal{F}$ (59)

where $\mathcal{F}$ is the Helmholtz free energy (26). On the other hand, if
we consider two statistical-like parameters, namely, $\alpha_{1}=\beta$ and
$\alpha_{2}=-\beta\mu$, and setting $\hat{F}_{1}=\hat{H}$ and
$\hat{F}_{2}=\hat{N}$ for the corresponding observables, we can recover the
grand-canonical equation of state

$\ln\Xi(\beta,\mu;\lambda_{1},...,\lambda_{m})=\dfrac{pV}{kT}=-\dfrac{\Phi}{kT},$
(60)

where $\Xi(\beta,\mu;\lambda_{1},...,\lambda_{m})$ and $\Phi$ are the grand
partition function and the grand potential, respectively.

In order to obtain generalized FD identities using the MaxEnt density matrix,
it is necessary to bear in mind the different choices for the continuous
parameter $\gamma_{k}$ in the Q-FDT, i.e.,
$\gamma_{k}\in\left\\{\alpha_{1},...,\alpha_{n}\right\\}\cup\left\\{\lambda_{1},...,\lambda_{m}\right\\}$,
with $k\in\left\\{1,2,...,n,n+1,...,n+m\right\\}$. In this way we have the
following ordering of parameters:

$\gamma_{1}=\alpha_{1},\gamma_{2}=\alpha_{2},...,\gamma_{n}=\alpha_{n},\gamma_{n+1}=\lambda_{1},...,\gamma_{n+m}=\lambda_{m}.$
(61)

Using the model given by (57) in (13) we obtain the Q-FDT in its generalized
form:

$\dfrac{\partial}{\partial\gamma_{k}}\left\langle\hat{A}\right\rangle_{\hat{\rho}}=\left\langle\dfrac{\partial\hat{A}}{\partial\gamma_{k}}\right\rangle_{\hat{\rho}}-\sum_{j=1}^{n}\left\langle\hat{A}\dfrac{\partial}{\partial\gamma_{k}}\left(\alpha_{j}\hat{F}_{j}\right)\right\rangle_{\hat{\rho}}-\dfrac{\partial\ln\mathcal{Z}}{\partial\gamma_{k}}\left\langle\hat{A}\right\rangle_{\hat{\rho}},$
(62)

where $\hat{A}$ is an observable depending on quantum-like parameters only and
must be compatible with the generalized density matrix $\hat{\rho}$. As in the
previous section, we can choose different operators for the observable
$\hat{A}$. The results for $\hat{A}=\mathds{1}$ and $\hat{A}=\hat{F}_{k}$, for
both types of continuous parameters, are summarized in Table 3. These four
cases encode all the possibilities for the Q-FDT. As a result of introducing a
generalized density matrix, we have gained breath and depth: breadth because
we can see all the above fluctuation-dissipation identities, using both the
canonical and grand canonical models, as particular instances of Eq. (62);
depth by being able to frame several quantum-statistical identities within an
inferential-statistic (and Bayesian) framework.

Consider the first row in Table 3 for $\hat{A}=\mathds{1}$. We have for the
pairs $\left(\mathds{1},\lambda_{k}\right)$ and
$\left(\mathds{1},\alpha_{k}\right)$ the identities

$\sum_{j=1}^{n}\alpha_{j}\left\langle\dfrac{\partial\hat{F}_{j}}{\partial\lambda_{k}}\right\rangle_{\hat{\rho}}=-\dfrac{\partial\ln\mathcal{Z}}{\partial\lambda_{k}}$
(63)

and

$\left\langle\hat{F}_{k}\right\rangle_{\hat{\rho}}=-\dfrac{\partial\ln\mathcal{Z}}{\partial\alpha_{k}},$
(64)

where Eq. (63) is a generalization of the thermodynamic integration formula
for $n$ Lagrange multipliers, while Eq. (64) establishes a relation between
the mean value of the $kth$ observable of the set with its respective
conjugate variable (parameter). It is worth mentioning that Eq. (64) agrees
with the identity previously obtained, by means of maximum-entropy inference
by E. T. Jaynes in the reference [8]. That is how our approach differs from
previous approaches: not only are we using a density matrix coming from a
MaxEnt procedure, but we are also systematizing all the FD identities within
the same general theorem (Eq. (13)).

Finally, for the pairs $(\hat{F}_{k},\lambda_{l})$ and
$(\hat{F}_{k},\alpha_{l})$ in the second row of Table 3 we have the following
identities:

$\dfrac{\partial}{\partial\lambda_{l}}\left\langle\hat{F}_{k}\right\rangle_{\hat{\rho}}=\left\langle\dfrac{\partial\hat{F}_{k}}{\partial\lambda_{l}}\right\rangle_{\hat{\rho}}-\sum_{j=1}^{n}\alpha_{j}\hskip
1.70709pt\mathrm{Cov}\left(\hat{F}_{k},\dfrac{\partial\hat{F}_{j}}{\partial\lambda_{l}}\right)$
(65)

and

$\dfrac{\partial}{\partial\alpha_{l}}\left\langle\hat{F}_{j}\right\rangle_{\hat{\rho}}=-\mathrm{Cov}\left(\hat{F}_{j},\hat{F}_{l}\right),$
(66)

where

$\begin{split}\mathrm{Cov}\left(\hat{F}_{k},\hat{F}_{l}\right)&\equiv\left\langle\Delta\hat{F}_{k}\Delta\hat{F}_{l}\right\rangle_{\hat{\rho}}\\\
&=\left\langle\hat{F}_{k}\hat{F}_{l}\right\rangle_{\hat{\rho}}-\left\langle\hat{F}_{k}\right\rangle_{\hat{\rho}}\left\langle\hat{F}_{l}\right\rangle_{\hat{\rho}}\end{split}$
(67)

are the covariances between the observables $\hat{F}_{k}$ and $\hat{F}_{l}$ in
the quantum state $\hat{\rho}$. We observe that Eq. (65) is the generalization
of the identity (32) for $n$ different Lagrange multipliers while, Eq. (66)
sets up a generalization of the thermodynamical-like FD identities (cf. Eqs
(36) for the canonical model, and Eqs. (46), (48) and (49) for the grand
canonical model).

Operator |  $\gamma_{k}=\lambda_{k}$ (quantum-like parameter) |  $\gamma_{k}=\alpha_{k}$ (statistical-like parameter)
---|---|---
$\hat{A}=\mathds{1}$ | $\sum_{j=1}^{n}\alpha_{j}\left\langle\dfrac{\partial\hat{F}_{j}}{\partial\lambda_{k}}\right\rangle_{\hat{\rho}}=-\dfrac{\partial\ln\mathcal{Z}}{\partial\lambda_{k}}$ | $\left\langle\hat{F}_{k}\right\rangle_{\hat{\rho}}=-\dfrac{\partial\ln\mathcal{Z}}{\partial\alpha_{k}}$
$\hat{A}=\hat{F}_{k}$ | $\dfrac{\partial}{\partial\lambda_{l}}\left\langle\hat{F}_{k}\right\rangle_{\hat{\rho}}=\left\langle\dfrac{\partial\hat{F}_{k}}{\partial\lambda_{l}}\right\rangle_{\hat{\rho}}-\sum_{j=1}^{n}\alpha_{j}\hskip 1.70709pt\mathrm{Cov}\left(\hat{F}_{k},\dfrac{\partial\hat{F}_{j}}{\partial\lambda_{l}}\right)$ | $\dfrac{\partial}{\partial\alpha_{l}}\left\langle\hat{F}_{j}\right\rangle_{\hat{\rho}}=-\mathrm{Cov}\left(\hat{F}_{j},\hat{F}_{l}\right)$
Table 3: Generalized fluctuation-dissipation identities. Depending on the type
of the continuous parameter $\gamma_{k}$ and the choice of the observable, we
obtain four different cases.

##  7 Concluding remarks

In this work we derived the fluctuation-dissipation theorem coming from
Bayesian statistics in the language and rules of quantum mechanics, and we
exposed the usefulness of this, the Q-FDT, by including all the
thermodynamical-like FD identities, among others, beneath its shade.

A prospective and interesting challenge would be exploring the scope of
another kind of expectation identities, such as the conjugate-variables
identities, in quantum statistics. In this sense, we would have a complete
program of Bayesian expectation identities in the context of quantum
statistical mechanics.

##  8 Acknowledgments

Ms. Vania Sáez is thanked for her advice and encouragement, and also for her
inspiring ideas concerning the nature of matter and time. Ms. Yasna Ortiz is
thanked for her grammatical advice, and for her assistance in the preparation
of the speech given in the IWOSP 2023 conference. Miss Guadalupe and Andina
are thanked for their silent and constant companion. By last, BM and SD
thankfully acknowledge funding from ANID FONDECYT 1220651 grant.

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