††thanks<EMAIL_ADDRESS><EMAIL_ADDRESS>
# Time resolved quantum tomography in molecular spectroscopy by the Maximal
Entropy Approach

Varun Makhija ‡ Department of Chemistry and Physics, University of Mary
Washington, Fredericksburg, Virginia 22401, USA  Rishabh Gupta‡ Department of
Chemistry, Purdue University, West Lafayette, Indiana, 47907, USA  Simon
Neville  National Research Council Canada, 100 Sussex Drive, Ottawa, Ontario,
K1A 0R6, Canada
Micheal Schuurman  National Research Council Canada, 100 Sussex Drive, Ottawa,
Ontario, K1A 0R6, Canada
Department of Chemistry, University of Ottawa, Ottawa, Ontario, K1N 6N5,
Canada
Joseph Francisco  Department of Earth and Environmental Science and Department
of Chemistry, University of Pennsylvania, Philadelphia, PA, USA  Sabre Kais
Department of Chemistry, Department of Electrical and Computer Engineering,
and Purdue Quantum Science and Engineering Institute, Purdue University, West
Lafayette, Indiana, 47907, USA

(August 28, 2024)

###### Abstract

Attosecond science offers unprecedented precision in probing the initial
moments of chemical reactions, revealing the dynamics of molecular electrons
that shape reaction pathways. A fundamental question emerges: what role, if
any, do quantum coherences between molecular electron states play in
photochemical reactions? Answering this question necessitates quantum
tomography—the determination of the electronic density matrix from
experimental data, where the off-diagonal elements represent these coherences.
The Maximal Entropy (MaxEnt) based Quantum State Tomography (QST) approach
offers unique advantages in studying molecular dynamics, particularly with
partial tomographic data. Here, we explore the application of MaxEnt-based QST
on photoexcited ammonia, necessitating the operator form of observables
specific to the performed measurements. We present two methodologies for
constructing these operators: one leveraging Molecular Angular Distribution
Moments (MADMs) which accurately capture the orientation-dependent vibronic
dynamics of molecules; and another utilizing Angular Momentum Coherence
Operators to construct measurement operators for the full rovibronic density
matrix in the symmetric top basis. A key revelation of our study is the direct
link between Lagrange multipliers in the MaxEnt formalism and the unique set
of MADMs. Additionally, we visualize the electron density within the molecular
frame, demonstrating charge migration across the molecule. Furthermore, we
achieve a groundbreaking milestone by constructing, for the first time, the
entanglement entropy of the electronic subsystem—a metric that was previously
inaccessible. The entropy vividly reveals and quantifies the effects of
coupling between the excited electron and nuclear degrees of freedom.
Consequently, our findings open new avenues for research in ultrafast
molecular spectroscopy within the broader domain of quantum information
science, offering profound implications for the study of molecular systems
under excitation using quantum tomographic schemes.

‡‡footnotetext: These authors contributed equally to this work.

Keywords: Ultrafast molecular dynamics, Quantum state tomography, Maximal
Entropy formalism

##  1 Introduction

The field of Molecular Reaction Dynamics seeks to understand the detailed
evolution of chemical reactions at the molecular level [1, 2]. The development
of attosecond physics allows for experimental access to such molecular details
during the initial moments of photochemical processes with unprecedented time
resolution, with the promise of exerting control on such reactions at these
timescales [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20,
21, 10]. As suggested by experimental and theoretical studies, these initial
dynamics are driven by quantum mechanical coherences between molecular
electronic states, and coupling of the electronic motion to the nuclear
degrees of freedom [6, 10, 22, 23, 24, 25, 26, 20]. The direct measurement of
such coherences necessitates time resolved quantum tomography on these
timescales [27, 28], which then also facilitates imaging the resulting
dynamics [29, 30]. Thus far, this has only been successfully achieved in the
electronic ground state [31, 32, 33, 34, 27] and the limited case of an
excited two-electronic-state system [30]. Furthermore, the latter
demonstration relied on prior knowledge of symmetry selection rules into the
excited states, and applies only to a specific measurement technique. In
quantum information science (QIS), quantum tomography has been a subject of
intensive study, as quantum computation necessitates the robust determination
of quantum states prepared in the lab [35, 36, 37, 38, 39]. Within QIS,
Quantum State Tomography (QST) endeavors to derive statistical inferences
regarding an unknown quantum state within a system. This process hinges on
extracting insights from the expectation values of a comprehensive set of
observables [40, 41, 42, 43, 44]. Traditionally, these observables correspond
to the mean measurement values of an informationally complete set of Hermitian
operators in order to uniquely reconstruct an unknown quantum state. However,
the conventional approach to QST faces significant challenges when applied to
modern large-scale quantum platforms [45]. This challenge stems from the
exponential increase in the number of measurements required for complete state
reconstruction relative to the size of the system. Consequently, performing
QST for even moderately sized systems represents a substantial difficulty. In
addition to the formidable challenge of exponential scaling in analyzing
complex systems, the quest for accurate estimation of quantum states
encounters yet another hurdle due to the pervasive noise inherent in today’s
Noisy Intermediate-Scale Quantum (NISQ) devices [46]. Consequently, the
fidelity of available measurements is compromised, with some measurements even
proving inaccessible. To circumvent these limitations, researchers have
proposed diverse approaches diverging from traditional QST. These innovative
methods include Neural Network Tomography [47, 48, 49, 50], Matrix Product
State Tomography [44], Quantum Overlapping Tomography [51], and Shadow
Tomography [52, 53]. Recognizing the noisy reality of quantum systems and the
challenge of obtaining high-fidelity measurements across all parameters,
efforts have emerged to conduct QST based on incomplete data. Methodologies
such as Maximum Likelihood Estimation (MLE) [54, 55, 56, 57, 58, 59], Bayesian
Mean Estimation (BME) [60, 61, 62, 63], and Maximal Entropy Formalism-based
QST [36, 37, 38, 64] have been explored in pursuit of more robust estimation
methodologies.
To the best of our knowledge, tomographic methods developed in QIS have not
yet been applied to dynamics in a photoexcited molecule. Potentially, these
methods could generalize to several techniques of ultrafast measurement. Here,
we demonstrate the application of the Maximal Entropy Approach (MaxEnt) to
quantum state tomography for the photoexcited ammonia molecule, the same two-
electronic-state system studied in [30]. This represents four important
advances in ultrafast molecular spectroscopy: (i) MaxEnt does not require
prior knowledge of symmetry selection rules in the excited state; (ii) To
apply MaxEnt, we determine an operator form of the observable used in the
experiment in [30], which can potentially be applied to other ultrafast
measurement techniques; (iii) MaxEnt can be used in cases of ‘incomplete’
information [65], when a unique tomography is not possible, to determine a
compatible quantum state, allowing its extension to cases with more degrees of
freedom; (iv) MaxEnt allows construction of the time evolving entropy of the
electronic subsystem directly revealing and quantifying the correlation of
electronic motion with nuclear degrees of freedom, and tracking the dynamics
of quantum information in the molecule [66, 67]. These advances set the
foundation for the generalization of quantum tomography in ultrafast molecular
reaction dynamics. Moreover, the results presented here provide the first
physical interpretation for the parameters in the MaxEnt approach to quantum
tomography. This demonstrates the possibility of feedback between time-
resolved quantum dynamics and QIS that can be generated by future applications
of our methods in ultrafast science.

##  2 Maximal Entropy Approach to Quantum Tomography

The MaxEnt formalism is a powerful and elegant framework in statistical
physics and information theory that allows us to derive the most unbiased
probability distribution, given a set of constraints [68, 69]. At its core,
MaxEnt seeks to maximize the entropy of a system while honoring the
constraints that capture our prior knowledge about the system. This formalism
provides a robust and principled approach to infer the probability
distribution of a classical system when we have incomplete information or
limited data.
One of the key insights of MaxEnt lies in its seamless integration with the
method of Lagrange multipliers ($\lambda_{k}\in\mathbf{C}^{2}$) [70, 71, 72,
73], which introduces a set of parameters that enable us to craft probability
distributions that satisfy specific constraints. When applied to quantum
systems, MaxEnt leads us to a remarkable expression for the density operator
($\hat{\rho}_{ME}$) of an unknown quantum state as it seeks to maximize the
von Neumann entropy of the system. This density operator encapsulates the
essence of MaxEnt’s power—it yields a density matrix that is both least biased
as well as consistent with our preconceived knowledge, as represented by the
known expectation values of certain operators $\hat{f}_{k}$.

$\hat{\rho}_{ME}(\vec{\lambda},\hat{f})=\frac{1}{Z(\lambda_{1},\ldots,\lambda_{k})}\exp\\{-\sum_{k}\lambda_{k}\hat{f}_{k}\\}$
(1)

The expression for $\hat{\rho}_{ME}$, as shown in Eq. (1), elegantly combines
the Lagrange multipliers $\lambda_{k}$ and the known operators $\hat{f}_{k}$.
The exponential term captures the interplay between these parameters, allowing
us to precisely balance the desire for maximizing entropy with the constraints
imposed by our prior information. The normalization factor
$Z(\lambda_{1},\ldots,\lambda_{k})=\Tr\\{\exp\\{-\sum_{k}\lambda_{k}\hat{f}_{k}\\}\\}$
guarantees that the resulting density operator represents a valid density
matrix. In essence, MaxEnt empowers us to strike a harmonious balance between
complete neutrality, represented by maximum entropy, and the real-world
constraints that govern our understanding of a system.
Within the context of quantum tomography, reconstruction of an unknown quantum
state relies on experimental measurements of Hermitian operators, typically
defined as Positive-Operator-Valued Measures (POVMs). These measurements
provide essential information about the quantum system, enabling the inference
of its characteristics through reconstruction techniques. The MaxEnt formalism
has proven effective in reconstructing quantum states prepared on quantum
devices across various N-qubit systems. In these cases, constraints are
imposed by the expectation values of operators representing probabilities and
coherences. This approach has also been implemented on prototype quantum
devices provided by IBM that also corroborates the MaxEnt method for quantum
tomography [36, 37, 38].
It is important to note that, unlike typical QST procedures for quantum
devices, the operators used in MaxEnt-based QST for molecular dynamics do not
need to be POVMs. For molecular dynamics a time and orientation angle-
dependent Lab Frame Density Matrix (LFDM), $\rho_{ij}(\mathbf{\Omega},t)$,
with $\mathbf{\Omega}$ representing the orientation angles, can be defined in
a basis of electronic-vibrational (vibronic) states $\ket{i}$, $\ket{j}$ [29].
Application of MaxEnt to determine this LFDM requires the definition of
operators $\hat{O}_{i}(\mathbf{\Omega})$, representing observables, such that

$\langle\hat{O}_{i}\rangle(t)=\Tr\left\\{\hat{\rho}(\mathbf{\Omega},t)\hat{O}_{i}(\mathbf{\Omega})\right\\},$
(2)

where the trace is over all molecular coordinates including orientation
angles, which we refer to as the ‘full trace’ here in. If a number of time
resolved measurements $\langle O_{i}\rangle(t)$ are made, the maximal entropy
density matrix satisfying Eq. (2) may be represented as [36, 65]

$\rho(\mathbf{\Omega},t)=\frac{e^{-\sum_{i}\lambda_{i}(t)\hat{O}_{i}(\mathbf{\Omega})}}{Z}$
(3)

where $Z=\Tr\\{e^{-\sum_{i}\lambda_{i}(t)\hat{O}_{i}(\mathbf{\Omega})}\\}$,
with $\Tr$ being the full trace, and the $\lambda_{i}(t)$ are Lagrange
multipliers determined by fitting to the experimental data. This approach has
properties that make it particularly suitable to the molecular case.
Specifically, it is applicable to mixed states and it can be used to find a
state compatible with Eq. (2) and (3) in cases where complete tomographic
information is not available [65]. The application of MaxEnt to molecular
dynamics requires knowing the form of $\hat{O}_{i}(\mathbf{\Omega})$ for the
particular measurement performed. This is constructed in Section 3 below.

##  3 The Operator form of the observable

We discuss two possible ways to construct operators for time resolved
photoelectron angular distributions measured from a resonantly excited
molecule [23]. One uses the previously studied Molecular Angular Distribution
Moments (MADMs) [74, 29, 30] to explicitly select the $\ket{\mathbf{\Omega}}$
\- diagonal elements of the vibronic density matrix, the LFDM, probed by the
experiment [29, 23]. This representation has previously been used to determine
molecular frame quantities from lab frame measurements [75, 74, 76, 15, 77,
78, 79, 80]. For a recent review see [81]. It may be considered an irreducible
representation of the measurement operator, which provides the necessary
information to fully determine only the molecular frame quantities.
Alternatively, a measurement operator which comprises the full rovibronic
density matrix in the symmetric top basis can be constructed using the Angular
Momentum Coherence Operators introduced in [79]. This representation contains
redundant information in molecular frame, but new lab frame information. Only
the LFDM has been successfully constructed from experimental data [30], but
both options can potentially be addressed by MaxEnt [36, 37, 38]. While we
develop both options here, we focus on reconstructing the LFDM leaving the
rovibronic problem for future study.

###  3.1 Option One: The LFDM

The LFDM, determining the subspace diagonal in the Euler angle basis of
molecular orientations $\ket{\mathbf{\Omega}}$, is [29]

$\hat{\rho}(\mathbf{\Omega},t)=\sum_{ij}\int{d\mathbf{\Omega}}\sum_{KQS}A^{K}_{QS}(i,j;t)D^{K*}_{QS}(\mathbf{\Omega})\ket{i}\ket{\mathbf{\Omega}}\bra{\mathbf{\Omega}}\bra{j}.$
(4)

Here, the $A^{K}_{QS}(i,j;t)$ are the previously studied MADMS [79] associated
with vibronic states $\ket{i}$ and $\ket{j}$ and $D^{K}_{QS}(\mathbf{\Omega})$
are elements of the Wigner D-matrix representation of the rotation operator.
In this basis, we also define a measurement operator

$\hat{O}_{LM}(\mathbf{\Omega};\epsilon)=\sum_{ij}\int{d\mathbf{\Omega}}\sum_{KQS}\frac{2K+1}{8\pi^{2}}C^{LM}_{KQS}(j,i;\epsilon)D^{K}_{QS}(\mathbf{\Omega})\ket{i}\ket{\mathbf{\Omega}}\bra{\mathbf{\Omega}}\bra{j}.$
(5)

The coefficients $C^{LM}_{KQS}(j,i;\epsilon)$ are determined by the nature of
the probe process, $\epsilon$ being parameters associated with this process.
For example, in the case of ejected photoelectrons from an excited molecule,
$\epsilon$ is the kinetic energy of the ejected electron. In general, it is
assumed that the scattered particle used as a probe has some anisotropy in the
lab frame, allowing a multipole expansion of the detected signal in spherical
harmonics $Y_{LM}(\theta_{e},\phi_{e})$, $\theta_{e}$ and $\phi_{e}$ being lab
frame ejection angles. From this, we can determine the time-dependent measured
quantity
$\Tr\\{\hat{\rho}(\mathbf{\Omega},t)\hat{O}_{LM}(\mathbf{\Omega};\epsilon)\\}$.
This can be achieved using the following well-known orthonormality relations:

$\int{d\mathbf{\Omega}D^{K*}_{QS}(\mathbf{\Omega})D^{K^{\prime}}_{Q^{\prime}S^{\prime}}(\mathbf{\Omega})}=\frac{8\pi^{2}}{2K+1}\delta_{KK^{\prime}}\delta_{QQ^{\prime}}\delta_{SS^{\prime}};$
(6)

which, along with Eq. 4 and Eq. 5, lead to

$\left(\hat{\rho}(\mathbf{\Omega},t)\hat{O}_{LM}(\mathbf{\Omega};\epsilon)\right)_{ii}=\sum_{j}\sum_{KQS}C^{LM}_{KQS}(i,j;\epsilon)A^{K}_{Q,S}(i,j;t)$
(7)

for the diagonal elements of
$\hat{\rho}(\mathbf{\Omega},t)\hat{O}_{LM}(\mathbf{\Omega};\epsilon)$. The
resulting trace yields the desired measured quantity:

$\beta_{LM}(\epsilon;t)=\sum_{ij}\sum_{KQS}C^{LM}_{KQS}(i,j;\epsilon)A^{K}_{QS}(i,j;t).$
(8)

For the case of measurement by one-photon ionization the coefficients
$C^{LM}_{KQS}(i,j;\epsilon)$ contain matrix elements of the dipole operator
between the states $\ket{i}$, $\ket{j}$ and the continuum states used as a
template for measurement [23]. These are determined by ‘complete’
photoionization experiments, typically using high-resolution resonant
spectroscopies [82, 83]. We note that Eq. 5 does not represent the measurement
of an equivalent quantity for a molecule fixed at orientation
$\mathbf{\Omega}$ relative to the lab frame, but instead rigorously defines
the operator associated with the lab frame measurement of the quantity in Eq.
8. In some cases, such as if only the $L=0$, $M=0$ term is considered (the
ejection-angle integrated yield), these are equivalent. For one-photon
ionization, a transformation matrix exists for symmetric molecules that
provides the molecule-fixed $\beta_{LM}(\epsilon;\mathbf{\Omega})$ if the
$C^{LM}_{KQS}(i,j;\epsilon)$ are known. Recent studies of one and few-photon
ionization have shown that these time resolved measurements are also
parameterized by Eq. 8, and that the $C^{LM}_{KQS}(i,j;\epsilon)$ can be
determined for ionization of the electronic ground state through ionization of
an excited rotational wavepacket [78, 79, 80]. Similarly, ultrafast X-ray and
electron diffraction measurements can also be parameterized by Eq. 8 as shown
in [84], in which case the $C^{LM}_{KQS}(i,j;\epsilon)$ were also determined
in the ground state by diffraction from a rotational wavepacket. Eq. 8 has
also been extensively used to parameterize measurements of strong field
ionization and dissociation [75, 74, 76, 15, 77, 78, 79, 80]. Thus, Eq. 8 can
serve as a general way to parameterize ultrafast measurements in terms of
matrix elements of the operator Eq. 4, the two being directly related by the
LFDM:
$\beta_{LM}(\epsilon;t)=\Tr\\{\hat{\rho}(\mathbf{\Omega},t)\hat{O}_{LM}(\mathbf{\Omega};\epsilon)\\}$
. This provides a general route to quantum tomography for orientation
dependent vibronic dynamics using MaxEnt.

###  3.2 Option 2: The Full Rovibronic Density Matrix

The full rovibronic density operator in the symmetric top basis, $\ket{JMK}$,
is

$\hat{\rho}(t)=\sum_{ij}\sum_{J_{i}M_{i}K_{i}}\sum_{J^{\prime}_{j}M^{\prime}_{j}K^{\prime}_{j}}\rho_{\omega_{i}\omega^{\prime}_{j}}(i,j;t)\ket{i}\ket{J_{i}M_{i}K_{i}}\bra{J^{\prime}_{j}M^{\prime}_{j}K^{\prime}_{j}}\bra{j}.$
(9)

Here $\omega\equiv\\{J,M,K\\}$, $J$ being the total angular momentum, $K$ its
projection onto the molecule fixed axis and $M$ its projection onto the space
fixed axis. In this basis the MADMs, $A^{K}_{QS}(i,j;t)$ can be written as

$A^{K}_{QS}(i,j;t)=\sum_{\omega_{i},\omega^{\prime}_{j}}\rho_{\omega_{i}\omega^{\prime}_{j}}(i,j;t)\left\langle{j\omega^{{}^{\prime}}_{j}}\left|\hat{A}^{K}_{QS}\right|{i\omega_{i}}\right\rangle,$
(10)

where $\hat{A}^{K}_{QS}$ are previously defined Angular Momentum Coherence
Operators. Their matrix elements
$\left\langle{j\omega^{{}^{\prime}}_{j}}\left|\hat{A}^{K}_{QS}\right|{i\omega_{i}}\right\rangle$
in the symmetric top basis are analytically known [29]. With this, the
measurement operator becomes,

$\hat{O}_{LM}(\mathbf{\Omega};\epsilon)=\sum_{ij}\sum_{\omega_{i}\omega^{\prime}_{j}}\sum_{KQS}C^{LM}_{KQS}(j,i;\epsilon)\left\langle{j\omega^{{}^{\prime}}_{j}}\left|\hat{A}^{K}_{QS}\right|{i\omega_{i}}\right\rangle\ket{i}\ket{J_{i}M_{i}K_{i}}\bra{J^{\prime}_{j}M^{\prime}_{j}K^{\prime}_{j}}\bra{j},$
(11)

and
$\beta_{LM}(\epsilon;t)=\Tr\\{\hat{\rho}(t)\hat{O}_{LM}(\mathbf{\Omega};\epsilon)\\}$
gives the same result as Eq. 8 for the measured quantity. Determining the
operator $\hat{\rho}(t)$ is a larger problem than determining
$\hat{\rho}(\mathbf{\Omega},t)$, since $\hat{\rho}(t)$ includes the
$\ket{\Omega}$-off diagonal contributions explicitly excluded in Eq. 4. While
this provides no new information on molecular frame dynamics, new lab frame
quantities such as the Entropy of the molecule as a whole, or lab frame
angular momenta become additionally accessible. Note, however, that
$\hat{\rho}(\mathbf{\Omega},t)$ contains sufficient information to construct
any molecular frame observable, and therefore the entanglement entropy of the
electronic subsystem can still be constructed.

##  4 Procedure

Here we apply MaxEnt to the NH3 molecule resonantly excited to its two-
electronic-state $\tilde{B}^{1}E^{\prime\prime}$ manifold by a femtosecond
laser pulse; and probed by a subsequent, photoionizing laser pulse as it
evolves in time. The LFDM was determined in [30] with the help of
spectroscopic selections rules. Specifically, the quantities
$\beta_{00}(\epsilon,t)$, $\beta_{20}(\epsilon,t)$ and
$\beta_{40}(\epsilon,t)$ from Eq. 8 were directly measured for a fixed kinetic
energy of the ejected electron ($\epsilon=0.2$ eV), as function of time after
the excitation. The matrix elements of the measurement operator
$C(i,j;\epsilon)$ are known from spectroscopic studies of this manifold, and
symmetry selection rules were used to determine that only three, unique, MADMs
$A^{K}_{QS}(i,j;t)$ were excited by the first pulse. Since three measurements
were made at each time delay ($\beta_{00}$, $\beta_{20}$ and $\beta_{40}$) Eq.
8 could be uniquely inverted to determine the MADMs and construct the LFDM
(Eq. 4).
The unique reconstruction of a quantum system’s density matrix in QIS, as
detailed in 2, necessitates measurements in the form of expectation values
derived from an informationally complete set of Hermitian operators. These
operators are typically composed of the Pauli string operators ($\sigma_{x}$,
$\sigma_{y}$, $\sigma_{z}$, $\sigma_{i}$). In contrast, here we use the
operator in Eq. 5 with the MaxEnt density operator Eq. 3 to directly determine
the LFDM. At each time step, the process entails constructing the MaxEnt
density matrix by parameterizing the measurement operators
($\hat{\beta}_{00}$, $\hat{\beta}_{20}$, and $\hat{\beta}_{40}$) with the
Lagrange multipliers, as follows:

$\hat{\rho}_{ME}(\Omega,\vec{\lambda}(t))=\frac{\exp\\{-\lambda_{00}(t)\hat{\beta}_{00}(\Omega,t)-\lambda_{20}(t)\hat{\beta}_{20}(\Omega,t)-\lambda_{40}(t)\hat{\beta}_{40}(\Omega,t)\\}}{Z(\lambda_{00}(t),\lambda_{20}(t),\lambda_{40}(t))}$
(12)

The measurement operators vary with both the orientation angle ($\Omega$) and
time, whereas the Lagrange multipliers are solely dependent on time. The
partition function $Z(\lambda_{00}(t),\lambda_{20}(t),\lambda_{40}(t))$
ensures the normalization of the density operator and is obtained by taking
the trace of the numerator in $\hat{\rho}_{ME}$ over all orientation angles at
a specific time. The unknown Lagrange multipliers, at each time step, are
subsequently determined using gradient descent optimization based on the three
measurements acquired from experiments, represented as ($\beta_{00}$,
$\beta_{20}$, and $\beta_{40}$). The associated cost function is defined as:

$\textrm{Cost}=(\sum_{LM}Tr\\{\hat{\rho}_{ME}(\Omega,\vec{\lambda}(t))\hat{\beta}_{LM}\\}-\beta_{LM})^{2}$
(13)

Essentially, the cost function quantifies the mean-squared error (MSE) between
the actual and computed expectation values of the measurement operators,
derived from the reconstructed MaxEnt density operator. The optimization
iterates until the cost falls below a predetermined threshold, resulting in
obtaining the converged set of Lagrange multipliers. Upon determining the
Lagrange multipliers, Eq. 12 represents the resulting LFDM.

##  5 Results

###  5.1 Quantum Tomography by the Maximum Entropy Approach

Figure 1: A comparison of the optimized $\lambda_{k}(t)$ parameters from the
MaxEnt procedure and the unique MADMs $A^{K}_{QS}(i,j;t)$ determined in
Morrigan et al. [30]. This comparison reveals that the MaxEnt procedure
automatically determines the unique set of MADMs that determine the excited
state molecular dynamics. The inset at the bottom right shows the geometry of
NH3 in the $\tilde{B}^{1}E^{\prime\prime}$ manifold with the relevant
orientation angles $\mathbf{\Omega}=\\{\theta,\chi\\}$ with respect to
polarization of the excitation laser pulse.

In Fig. 1 the optimized Lagrange Multipliers resulting from the MaxEnt
procedure alongside the three unique MADMs determined in [30]. Comparison of
these parameters provides an immediate interpretation of the $\lambda_{k}(t)$
in Eq. 3: they are the unique set of MADMs $A^{K}_{QS}(i,j;t)$ that determine
the excited state dynamics of the molecule. We reiterate that to perform the
tomography in [30], the number of unique non-zero MADMs and their labels
needed to be determined before hand. The MaxEnt procedure does not label the
MADMs; that is, the values of $K,Q,S$ and the pair of electronic states
$\ket{i}$, $\ket{j}$ associated with each $\lambda_{i}(t)$ are not known.
However, it automatically determines the correct set of MADMs to within a
scale factor. In this particular case of NH3 the three $\lambda_{k}(t)$
correspond to: (i) the coherence - $A^{2}_{0\pm 2}(\pm,\mp;t)$ \- between the
pair of electronic states here labelled $\ket{+}$, $\ket{-}$; (ii) the average
alignment of the 3-fold symmetry axis of NH3 \- $A^{2}_{00}(\pm,\pm;t)$ \- in
each electronic state; and (iii) the orientation integrated population -
$A^{0}_{00}(\pm,\pm;t)$ \- of each electronic state. Additional studies are
required to determine whether this correspondence between the $\lambda_{k}(t)$
and coherence, molecular alignment and population is general; but the results
presented here provide a first interpretation of these Lagrange multipliers.

Figure 2: A comparison of the LFDM determined by MaxEnt to that determined in
[30] for different orientations of the molecular frame relative to the
polarization of the excitation laser pulse (see Fig. 1 inset).

In Fig. 2 we compare the LFDM determined by MaxEnt to that determined in [30]
at three selected orientations. A number of qualitative features are in
agreement: (i) the time evolving populations are identical at each of the
three orientations; (ii) the electronic coherence varies rapidly in the first
1.5 ps after excitation, after which it remains relatively stable with small
fluctuations; (iii) the phase of the coherence changes with orientation in the
same manner; and (iv) the real or imaginary parts of the coherence vary with
significantly lager amplitude than do the populations at all orientations.
These are the primary determining features of the electronic and rotational
dynamics of the molecule as discussed in detail below. The noteworthy
quantitative differences between the two results are an overall offset in the
populations, as well as a scale factor and greater stability in the coherence.
These are primarily due to the fact that the $\tilde{B}^{1}E^{\prime\prime}$
manifold of NH3 is known to be dissociative, which is reflected in the slow
decay of the population and coherence from [30]. The MaxEnt procedure strictly
requires that $\Tr\\{\rho(\mathbf{\Omega},t)\\}=1$, thus compensating for the
slow decay of experimental signal. As an advantage, this allows the
construction of the entanglement entropy of the excited electron, which was
not possible with the result from [30], as we show below.

###  5.2 Molecular Dynamics

Figure 3: The time dependent (a) one electron attachment density at
orientation $\mathbf{\Omega}=\\{\theta=\pi/2,\chi=0$} and (b) the molecular
axis distribution constructed from the MaxEnt LFDM. The former reveals the
electronic motion in the molecular frame as charge migration and the latter
tracks the rotational motion of the molecule.  Figure 4: (a) The entanglement
entropy of the electron as a function of time constructed using the MaxEnt
LFDM, indicating that the electronic state is in a maximally mixed or
entangled state. The orientation dependent entropy of the electron at (a) 0.06
ps and (b) 0.5 ps after the pump pulse, along with one-electron attachment
densities at $\mathbf{\Omega}=\\{\theta=0,\chi=0$} and
$\mathbf{\Omega}=\\{\theta=\pi/2,\chi=0$} constructed using the MaxEnt LFDM.
These maps show the effect of coupling between the electron and rotating
nuclei, with high entropy regions indicating a highly mixed electronic-nuclear
state.

With the LFDM determined, we can construct the electron density in position
space, $P(\mathbf{r},\mathbf{\Omega},t)=\\\
\sum_{ij}\rho_{ij}(\mathbf{\Omega},t)\phi_{i}(\mathbf{r})\phi_{j}(\mathbf{r})$,
with $\mathbf{r}$ the position vector of the electrons in the molecular frame
and $\phi_{i}(\mathbf{r})$ the wavefunction associated with the electronic
state $\ket{i}$. In Fig 3(a) we show the one-electron attachment density [79,
30] at selected time delays, for the NH3 molecule with polarization of the
excitation pulse along an NH bond. The attachment density quantifies the
accumulation of density in the molecular frame [30]. As in [30] we observe
charge migration across the molecule with a period of about 1 ps at this
orientation. In Fig 3(b) we show the molecular axis distribution
$P(\theta,t)=\sum_{i}\rho_{ii}(\theta,t)$ as a function of time, $\theta$
being the relative angle of the molecular 3-fold symmetry axis and the
polarization of the excitation pulse (see Fig. 1 inset). In agreement with
[30] we observe rapid rotation of the 3-fold axis in the first 1.5 ps,
synchronized with the charge migration in the molecular frame, followed by
small fluctuations around $\theta=\pi/2$ for the remainder of the time window.
In [30] this is attributed to ro-electronic coupling due to a molecular
Coriolis effect.

###  5.3 Entropy of the Electron: Electron-Nuclear Coupling

Here we can additionally provide direct evidence of ro-electroinc coupling by
constructing the entanglement entropy of the electronic subsystem,
$S_{el}(t)=-\Tr{\hat{\rho}(\Omega,t)\log{\hat{\rho}(\Omega,t)}}$, shown in
Fig. 4(a). Evidently, $S_{el}(t)\approx 1$ for the entire time window. To
interpret this we can consider the spectroscopic model of the
$B^{1}E^{\prime\prime}$ state developed by Western et al. [85, 86]. To
accurately model the absorption spectrum NH3 is treated as a rotating, planar,
symmetric top coupled to a single excited electron with states $\ket{+}$ and
$\ket{-}$. In this bi-partite model, the measured $S_{el}$ suggests that the
excited electron is in a maximally mixed, or maximally entangled state. Prior
to excitation, however, NH3 is in the ground electronic state, so
$S_{el}=0$.The excited state is prepared coherently by the excitation laser
pulse, implying that a measurement of $S_{el}\approx 1$ in fact suggests
maximal entanglement of the excited electron with the rotating molecular core.
To determine the characteristics of this ro-electronic coupling, we can
finally construct an orientation dependent entropy of the excited electron,
$S_{el}(\Omega,t)=-\Tr{\hat{\rho}(\Omega,t)\log{\hat{\rho}(\Omega,t)}}$,
without tracing over orientation angles, as shown in Fig. 4(b) and (c). This
provides a map of the degree of ro-electronic mixing as a function of
orientation and time, with the orientation averaged entropy remaining constant
(Fig. 4(a)). We observe that orientations of maximum entropy - where the
electronic state is highly mixed, or correlated with molecular rotation -
change considerably within the first 0.5 ps. Furthermore, at high entropy
orientations the electronic density is more spread out over the molecule as
shown in the insets of Fig. 4(b) and (c); at 0.06 ps for
$\\{\theta=0,\chi=0\\}$ and at 0.5 ps for $\\{\theta=\pi/2,\chi=0\\}$. On the
other hand , for orientations with minimum entropy, the electronic density is
more compact; for $\\{\theta=\pi/2,\chi=0\\}$ at 0.06 ps it is polarized along
the NH bond, parallel to the laser polarization and for
$\\{\theta=0,\chi=0\\}$ at 0.5 ps is localized at the N atom. By about 1.5 ps
$S_{el}(\Omega,t)$ reverts to the orientation dependence shown in Fig. 4(a),
and finally stabilizes to the dependence in Fig. 4(b) in concert with
stabilization of the rotational dynamics ($P(\theta,t)$ in Fig. 3). In sum,
this provides a complete picture of the electronic dynamics in the molecular
frame and its coupling to nuclear motion: the evolution of the electronic
density at any orientation is driven by coupling to the rotational subsystem
as quantified by $S_{el}(\Omega,t)$. When the electronic state is highly mixed
with that of the rotating molecule the charge density spreads over the entire
molecular frame and it localizes when the electronic state becomes
approximately separable.
Alternatively, $S_{el}(\Omega,t)$ provides a map of the flow of quantum
information in the space of molecular orientations. In this case, Fig. 4(a)
indicates that information does not dissipate from the electronic subsystem
within the time window of the experiment, even though its state is mixed with
that of the rotating nuclei. The ro-electronic coupling only causes the flow
of information between different molecular orientations. Quantum information
dynamics in molecules is a relatively new topic of research [87, 66, 88, 67],
and we show here the MaxEnt approach can provide access to metrics of quantum
information in isolated molecules.

##  6 Conclusion and Outlook

Our work provides a framework to apply quantum tomography methods developed in
QIS to the ultrafast dynamics of isolated molecules. In doing so it provides
the tools necessary to image electronic dynamics in the molecular frame, a
general goal in attosecond and molecular physics, providing insight into the
coupling of these dynamics to nuclei as well as the ability to monitor the
correlation of the electrons and the nuclei. Here, we have demonstrated these
capabilities by applying the MaxEnt approach to quantum tomography in the NH3
molecule resonantly excited to its $\tilde{B}^{1}E^{\prime\prime}$ manifold of
states. While being consistent with previous results, this approach also
provides the ability to construct the entropy of the electron in the molecular
frame revealing the consequences of coupling between the excited electron and
the rotating nuclei. These results potentially open new areas of inquiry
addressing the flow of quantum information in molecules. Here, we observe that
electronic-rotational coupling redistributes information in the space of
orientations as function of time, without causing dissipation. An immediate
application of this approach would be to study the dynamics of quantum
information in the nearby, Jahn-Teller active, ’$\tilde{C}$’ manifold [89] of
NH3 activating coupling between electronic motion and vibration, as well as
between vibrational modes. Recent theoretical studies suggest that such
interactions could lead to a scrambling of quantum information [66, 67].
Additionally we note that any approach to quantum tomography requires
determination of the observable operator
$\hat{O}_{LM}(\mathbf{\Omega};\epsilon)$ representing the probe of molecular
dynamics. This has been determined for one photon ionization of only a handful
of excited molecular states by high resolution spectroscopy [90, 91, 92, 93,
94, 82, 83, 95, 78]. However, recently developed methods in attosecond physics
that determine the state of an ejected photoelectron [96, 97, 98, 99, 100] can
potentially be applicable here. As such, our work presents numerous
opportunities for further investigation in the fields of ultrafast molecular
dynamics and QIS.

## Acknowledgements

We would like to thank Raphy D. Levine and Manas Sajjan for many useful
discussions. S.K. would also like to acknowledge funding from the U.S.
Department of Energy (DOE) (Office of Basic Energy Sciences), under Award No.
DE-SC0019215, and the National Science Foundation under Award No. 1955907.

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