# Casimir forces in the flatland: interplay between photo-induced phase
transitions and quantum Hall physics

Y. Muniz<EMAIL_ADDRESS>Instituto de Física, Universidade Federal
do Rio de Janeiro, Caixa Postal 68528, Rio de Janeiro 21941-972, RJ, Brazil
C. Farina<EMAIL_ADDRESS>Instituto de Física, Universidade Federal do Rio
de Janeiro, Caixa Postal 68528, Rio de Janeiro 21941-972, RJ, Brazil  W. J. M.
Kort-Kamp<EMAIL_ADDRESS>Theoretical Division, Los Alamos National
Laboratory, MS B262, Los Alamos, New Mexico 87545, United States

###### Abstract

We investigate how photo-induced topological phase transitions and the
magnetic-field-induced quantum Hall effect simultaneously influence the
Casimir force between two parallel sheets of staggered two-dimensional (2D)
materials of the graphene family. We show that the interplay between these two
effects enables on-demand switching of the force between attractive and
repulsive regimes while keeping its quantized characteristics. We also show
that doping these 2D materials below their first Landau level allows one to
probe the photoinduced topology in the Casimir force without the difficulties
imposed by a circularly polarized laser. We demonstrate that the magnetic
field has a huge impact on the thermal Casimir effect for dissipationless
materials, where the quantized aspect of the energy levels leads to a strong
repulsion that could be measured even at room temperature.

The Casimir effect is an interaction induced by quantum vacuum fluctuations of
the electromagnetic field and exists between material bodies [1, 2, 3, 4, 5,
6]. It was originally proposed by H.B.G. Casimir in 1948 as an attractive
force between two neutral perfectly conducting plates [7]. Fundamentally, it
is a dispersive force between two bodies at length scales where
electromagnetic retardation becomes relevant. The ever-present dispersive
interactions are one of the most intriguing kind of intermolecular forces and
play an important role not only in different areas of physics, but also in
chemistry, biology, and engineering [8]. For instance, dispersive forces are
crucial to understand the stability of colloids [9], the drug binding in
proteins and the doublehelix structure of DNA [10], and even the adhesion of
geckos to walls [11, 12]. In the manufacturing and operation of devices at the
micro- and nanoscales these forces also have severe consequences since they
dominate at very small distances and may cause stiction due to their tipically
attractive nature [13, 14, 15, 16, 17]. Therefore, it is very important to
study dispersive interactions and master their manipulation not only in
magnitude but also in sign, i.e., to tune the attractive or repulsive
character of the force. In this direction, real materials have shown to
support a wealth of different dispersive interactions [18, 5], and the strong
development of materials science over the last few decades has given a new
impetus to the study of the Casimir force and its possible control [19, 20].
In particular, switching between repulsive and attractive Casimir forces using
external agents can be realized with topological insulator plates [21, 22, 23,
24], and superparamagnetic metametamaterials [25].

The reduced dimensionality of graphene and its related nanostructures has also
led to the discover of novel behaviours in the Casimir interaction, which have
a strong dependence on temperature and doping [26, 27, 28, 29, 30, 31, 32, 33,
34]. Meanwhile, quantum Hall physics arising from externally applied magnetic
fields result in a complex and rich magneto-optical response of graphene [35,
36, 37]. This was already exploited for tuning and screening dispersive
interactions, which were shown to be quantized and significantly weaker when
compared to the situation without the field [39, 40]. On the other hand, two-
dimensional (2D) Dirac materials such as silicene, germanene, stanene, and
plumbene are representatives of the graphene family and present several
possible topological phase transitions under external fields due to their
significant spin-orbit coupling and finite staggering [41, 42, 43, 44, 45].
These phase transitions strongly impact their Casimir forces, in which
different scaling laws, repulsion, and force quantization are possible [46].
Nevertheless, the effects that can be brought to the Casimir force as a
consequence of coexistent topological and quantum Hall features are yet to be
unveiled.

Here we demonstrate that the introduction of a magnetic field is very
convenient for switching the sign of the Casimir force between graphene family
materials while keeping its quantized behaviour. We show that the zero-
temperature long distance Casimir force between two identical parallel sheets
is proportional to the product of their corresponding Chern numbers for every
non-trivial topological state, and can be made attractive or repulsive by
properly doping each plate. Furthermore, all photo-induced topologies are
mimicked by tuning the chemical potential and do not require energy absorption
from a circulary polarized laser, in contrast to what is feasible without the
magnetic field. We also investigate the Casimir effect for finite temperatures
and show that no matter how we model the materials with or without
dissipation, a huge change in their Casimir energies may be achieved. For a
dissipative system, the Casimir force at high temperatures is attractive and
does not depend on any material properties, while for non-dissipative
monolayers we observe repulsion in a wide range of temperatures. Due to their
direct parallel to the Drude and plasma models for metals, this contrasting
behaviour of the Casimir interaction in both situations touches fundamental
questions underlying Casimir physics, and could be useful in future
investigations.

This paper is organized as follows. In Section I we describe the framework in
which we carried our calculations of the Casimir energy. We also give a brief
overview on the graphene family materials main properties and magneto-optical
response. In Section II we present our results on the zero-temperature Casimir
interaction. We then discuss the long-distance behaviour of the Casimir energy
in the phase space, its distance dependence, and how changing physical
quantities such as magnetic field, chemical potential or the dissipation
affects the force. In section III we discuss the thermal effects on the
Casimir energy for dissipative and non-dissipative monolayers. Section IV is
left for final remarks and conclusions.

##  I Model and methods

Figure 1: (a) Top and side views of the graphene family honeycomb structure.
The red and blue colored atoms belong to inequivalent sublattices with a
finite staggering $2\ell$ between them. (b) Schematics of the system under
study. Two graphene family monolayers separated by a distance $d$ and
subjected to externally applied electric ($E_{z}$) and magnetic ($B_{z}$)
fields, and a circulary polarized laser ($\Lambda$); c) Low-energy band
structure around a given $K$ or $K^{\prime}$ point. For a non-zero magnetic
field, the quantized Landau levels (red circles) are built on top of the Dirac
cone.

In this section, we focus our attention on the Casimir energy per unit area
(denoted by $E$) between two layers of the graphene family materials. Like
graphene, these materials have a honeycomb structure, however, the two atoms
in the unit cell are arranged in staggered layers separated by a distance
$2\ell$, as depicted in Fig. 1a [44]. The schematics of the system under study
is shown in Fig. 1b, where the layers are separated by a distance $d$ and
subjected to a circulary polarized laser field of frequency $\omega_{0}$ and
intensity $I_{0}$, and an out-of-plane static and uniform electric ($E_{z}$)
and magnetic ($B_{z}$) fields. The Casimir energy can be obtained by the
Lifshitz formula, which is valid for $d$ much larger than the interatomic
distances of the materials, and is given by [5, 47]

${E}=k_{B}T\sideset{}{{}^{\prime}}{\sum}_{n}\int\frac{d^{2}{\bf
k}_{\parallel}}{(2\pi)^{2}}\mbox{log
det}\left(1-\mathbb{R}_{1}\cdot\mathbb{R}_{2}e^{-2k_{z,n}d}\right),$ (1)

where $k_{z,n}=\sqrt{{\bf k}_{\parallel}^{2}+\xi^{2}_{n}/c^{2}}$,
$\xi_{n}=2\pi nk_{B}T/\hbar$ with $n=0,1,...$ are the Matsubara frequencies,
and the prime on the summation symbol indicates that the term $n=0$ must be
multiplied by $1/2$. Also, $\mathbb{R}_{i}$ are the reflection matrices for
each sheet, with rows given by the Fresnel coefficients for incident s- and p-
polarized waves, which are given by [46]

$\displaystyle\mathbb{R}_{ss}$
$\displaystyle=-\frac{2\pi}{\delta_{n}}\left[\frac{\sigma_{xx}\xi_{n}}{k_{z,n}c^{2}}+\frac{2\pi}{c^{2}}\left(\sigma_{xx}^{2}+\sigma_{xy}^{2}\right)\right],$
(2) $\displaystyle\mathbb{R}_{sp}$
$\displaystyle=\mathbb{R}_{ps}=\frac{2\pi\sigma_{xy}}{\delta_{n}c},$ (3)
$\displaystyle\mathbb{R}_{pp}$
$\displaystyle=\frac{2\pi}{\delta_{n}}\left[\frac{\sigma_{xx}k_{z,n}}{\xi_{n}}+\frac{2\pi}{c^{2}}\left(\sigma_{xx}^{2}+\sigma_{xy}^{2}\right)\right],$
(4)

where
$\delta_{n}=1+\frac{2\pi\sigma_{xx}}{c}\left(\frac{\xi_{n}}{k_{z,n}}+\frac{k_{z,n}}{\xi_{n}}\right)+\frac{4\pi^{2}}{c^{2}}\left(\sigma_{xx}^{2}+\sigma_{xy}^{2}\right)$
and the conductivity components are evaluated at the imaginary Matsubara
frequencies $i\xi_{n}$. It is important to notice that since the reflections
occur inside the cavity, we must perform the transformation
$\sigma_{xy}\rightarrow-\sigma_{xy}$ for one of the monolayers.

The conductivity components for graphene family materials are well known and
can be cast as [38, 48],

$\displaystyle\sigma_{xx}$
$\displaystyle=\sigma_{yy}=\frac{iE_{B}^{2}\sigma_{0}}{\pi}\sum_{\eta,s}\sum_{n,m}\frac{f_{m}-f_{n}}{\varepsilon_{n}-\varepsilon_{m}}$
(5)
$\displaystyle\times\frac{(A_{m}^{+}A_{n}^{-})^{2}\delta_{|n|,|m|-\tilde{\eta}}+(A_{m}^{-}A_{n}^{+})^{2}\delta_{|n|,|m|+\tilde{\eta}}}{\varepsilon_{m}-\varepsilon_{n}+i\hbar(\xi+\Gamma)},$
(6) $\displaystyle\sigma_{xy}$
$\displaystyle=-\sigma_{yx}=-\frac{E_{B}^{2}\sigma_{0}}{\pi}\sum_{\eta,s}\sum_{n,m}\eta\frac{f_{m}-f_{n}}{\varepsilon_{n}-\varepsilon_{m}}$
(7)
$\displaystyle\times\frac{(A_{m}^{+}A_{n}^{-})^{2}\delta_{|n|,|m|-\tilde{\eta}}-(A_{m}^{-}A_{n}^{+})^{2}\delta_{|n|,|m|+\tilde{\eta}}}{\varepsilon_{m}-\varepsilon_{n}+i\hbar(\xi+\Gamma)}.$
(8)

Here,
$\varepsilon_{n}=\mbox{sgn}(n)\sqrt{|n|E_{B}^{2}+(\Delta_{s}^{\eta})^{2}}$ for
$n\neq 0$ and $\varepsilon_{0}=-\tilde{\eta}\Delta_{s}^{\eta}$ are the
quantized eigenenergies of the system, with $\tilde{\eta}=\mbox{sgn}(eB)\eta$,
$E_{B}=\sqrt{2v_{F}^{2}\hbar|eB_{z}|}$ being the relativistic analog of the
cyclotron energy, $v_{F}$ is the Fermi velocity, and $\Delta_{s}^{\eta}$ is
half the mass gap. The Dirac mass naturally arises from the spin orbit
coupling (SOC) $\lambda_{SO}$, but is modified by the applied electric field
and circularly polarized laser, leading to $\Delta_{s}^{\eta}=-\eta
s\lambda_{SO}+e\ell E_{z}+\eta\Lambda$, where $\Lambda=\pm 8\pi\alpha
v_{F}^{2}I_{0}/\omega_{0}^{3}$ and $\alpha\approx 1/137$ is the fine structure
constant. The SOC for silicene, germanene, stanene, and plumbene are
$\lambda_{SO}\approx 3.9,43,100,200$ meV [44, 49], respectively. Also,
$A_{n}^{\pm}=\sqrt{|\varepsilon_{n}|\pm\mbox{sgn}(n)\Delta_{s}^{\eta}/2|\varepsilon_{n}|}$
for $n\neq 0$, $A_{0}^{\pm}=(1\mp\tilde{\eta})/2$, $\sigma_{0}=\alpha c/4$ is
graphene’s universal conductivity.
$f_{n}=1/[e^{(\varepsilon_{n}-\mu)/k_{B}T}+1]$ denotes the Fermi Dirac
distribution, $\mu$ is the chemical potential, $T$ is the absolute
temperature, and $\Gamma$ is the dissipation rate. The summation indices
$\eta$ and $s$ are the valley and spin degrees of freedom, respectively, and
only admit values of $\pm 1$. The Landau energy levels are illustrated in Fig.
1c for a single Dirac cone. Terms originating from Rashba physics are not
considered due to their small contribution [43]. We finally notice that the
conductivity tensor is a real function of $\xi$.

##  II Zero-temperature Casimir energy

In all subsequent discussions, we shall always normalize the Casimir energy
between graphene family materials by the zero-temperature neutral
graphene/graphene Casimir interaction in the long distance regime, namely,
[27]

$E_{g}=-\frac{\hbar c\alpha}{32\pi d^{3}}.$ (9)

Since the Casimir force per unit area is given by $F=-\partial E/\partial d$,
we have $F_{g}<0$, which results in an attraction between the plates. We start
our analysis by investigating the Casimir energy in the long distance regime.
For $T=0$, the Casimir energy is given by an integral over imaginary
frequencies $i\xi$, which can be obtained by the transformation
$k_{B}T\sideset{}{{}^{\prime}}{\sum}_{n}\rightarrow\frac{\hbar
c}{2\pi}\int_{0}^{\infty}d\xi$. Since the integrand inside the Lifshitz
formula decays with $e^{-2\xi d/c}$, for large separations ($d>>\hbar
c/\lambda_{SO}$) we expect the graphene family low-frequency optical response
to be dominant. Therefore, a good approximation is to expand the integrand as
a power series of $i\xi$ and retain only the leading contribution (while
keeping the exponential factor). Since the reflection matrices depend on
frequency through the conductivity tensor, we shall first expand $\sigma_{xx}$
and $\sigma_{xy}$. In the absence of dissipation, eqs. (6) and (8) show that
the longitudinal conductivity is an odd function of frequency, while the Hall
conductivity is an even function of $\xi$. Hence,
$\sigma_{xx}(i\xi)=\sigma_{xx}^{\prime}(0)\xi+O(\xi^{3})$ and
$\sigma_{xy}(i\xi)=\sigma_{xy}(0)+O(\xi^{2})$. For $k_{B}T=\hbar\Gamma=0$, one
can show that the DC Hall conductivity presents clear signatures of topology
and quantum Hall effect. It is given by
$\sigma_{xy}=2\sigma_{0}(C_{\text{ph}}+C_{\text{QH}})/\pi$, where
$C_{\text{ph}}$ is the Chern number associated with photoinduced topology, and
$C_{\text{QH}}$ is the Chern number associated with the quantum Hall effect,
namely [48]

$\displaystyle C_{\text{ph}}$
$\displaystyle=-\frac{1}{2}\sum_{\eta,s}\theta\left(\varepsilon_{1}-|\mu|\right)\eta\,\mbox{sgn}\left(\Delta_{s}^{\eta}+\tilde{\eta}\mu\right),$
(10) $\displaystyle C_{\text{QH}}$
$\displaystyle=-\mbox{sgn}(eB\mu)\sum_{\eta,s}\theta\left(|\mu|-\varepsilon_{1}\right)\left(N_{s}^{\eta}+1/2\right),$
(11)

with $N_{s}^{\eta}$ being the number of filled Landau levels per Dirac cone.
By keeping only the DC Hall conductivity we have
$\mathbb{R}_{sp}=\mathbb{R}_{ps}\approx\alpha(C_{\text{ph}}+C_{\text{QH}})$
and $\mathbb{R}_{ss}=\mathbb{R}_{pp}\approx 0$, where we discarded
$\sigma_{xy}^{2}$ since it gives a higher order contribution in $\alpha$. By
applying the relation $\mbox{log det}\left(1+A\right)=\mbox{Tr
log}\left(1+A\right)\approx\mbox{Tr}(A)$ to the integrand of the Lifshitz
formula, we obtain

$\frac{E^{(0)}}{E_{g}}=-\frac{4\alpha}{\pi}(C_{1,\text{ph}}+C_{1,\text{QH}})(C_{2,\text{ph}}+C_{2,\text{QH}}).$
(12)

Since $E^{(0)}$ has the same distance dependence as $E_{g}$, the Casimir force
normalized by the graphene-graphene force ($-\partial E_{g}/\partial d$) is
also given by the previous equation. As a consequence, a negative (positive)
normalized energy implies a repulsive (attractive) force. Hence, for two
identical layers in the same non-trivial topological phase the Casimir force
in the long distance regime is repulsive and proportional to the square of the
full Chern number ($C=C_{\text{ph}}+C_{\text{QH}}$) characterizing the
corresponding phase of the materials. However, the Casimir force is attractive
if the materials are in different topological phases, which in this
configuration could only be achieved by doping the materials at different
levels since the external fields applied to both monolayers are the same. We
mention that the interplay between photo-induced topology and quantum Hall
physics requires $E_{B}\sim\lambda_{SO}$, which can be accessed in the
graphene family for magnetic fields less than or of the order of a few teslas.

When one layer is in a trivial topological phase ($C=0$), which can be
achieved for external fields such that $C_{\text{ph}}=-C_{\text{QH}}$ or
$C_{\text{ph}}=C_{\text{QH}}=0$ (the latter only being possible if
$|\mu|<\varepsilon_{1}$ for every Dirac cone), the first contribution to the
Casimir energy arises from the longitudinal conductivity. In the situation
where $C_{1}=C_{2}=0$, the reflection matrices are given by
$\mathbb{R}_{sp}=\mathbb{R}_{ps}\approx 0$,
$\mathbb{R}_{ss}\approx-\frac{2\pi\sigma_{xx}^{\prime}\xi^{2}}{c^{2}k_{z}}$,
and $\mathbb{R}_{pp}\approx 2\pi\sigma_{xx}^{\prime}k_{z}$, where
$\sigma_{ij}^{\prime}=\partial\sigma_{ij}/\partial\xi|_{\xi\rightarrow 0}$,
hence

$\frac{E^{(0)}}{E_{g}}\approx\frac{144\pi}{5\alpha
d^{2}}\sigma_{1,xx}^{\prime}\sigma_{2,xx}^{\prime}.$ (13)

In the above formula, it is important to notice that, since the conductivity
is proportional to the fine structure constant, $E/E_{g}$ is linear in
$\alpha$, such as the case where $C_{1},C_{2}\neq 0$. However, the normalized
energy goes with $1/d^{2}$, decaying much faster with the distance. If only
$C_{1}=0$ but $C_{2}\neq 0$ (for instance, by considering monolayers of two
different graphene family materials), we must also consider the terms
proportional to $\sigma_{2,xy}^{2}$ inside the reflection matrices in order to
obtain the first non zero contribution. By doing that, we obtain

$\frac{E^{(0)}}{E_{g}}\approx\frac{8\alpha}{d}\sigma_{1,xx}^{\prime}C_{2}^{2}.$
(14)

This time, the normalized energy is proportional to $\alpha^{2}/d$, which is
also much smaller than the situation where the Chern numbers are finite. In
summary, the message from the previous equations is that if $C_{1}$ or $C_{2}$
vanish, the long distance Casimir force is attractive, smaller in magnitude,
and decreases faster with the distance than the case of non-trivial
topological states. Therefore, eq (12) is the dominant approximation to the
Casimir energy in the long distance regime, in contrast to what we get in the
abscence of a magnetic field, where the Casimir force near the boundaries of
the phase diagram is attractive and much higher than far from them[46].

(

Figure 2: Casimir energy in the $(E_{z},\Lambda)$ plane for (a)
$\mu=\lambda_{SO}$ and $E_{B}=1.2\lambda_{SO}$ and (b)
$E_{B}=0.8\lambda_{SO}$. (c) Normalized Casimir energy in the $(E_{z},\mu)$
plane for $\Lambda=0$ and $E_{B}=\lambda_{SO}$. (d) Normalized Casimir energy
as a function of the chemical potential for $\Lambda=0$. In all plots the
distance between the plates is given by $d\lambda_{SO}/\hbar c=10$ and the
dissipation is set to zero.

In Fig. 2a we plot the Casimir energy for two identical layers separated by
$d\lambda_{SO}/\hbar c=10$ in the $(E_{z},\Lambda)$ plane for
$\mu_{1}=\mu_{2}=\lambda_{SO}$ (denoted by $\mu$) and $E_{B}=1.2\lambda_{SO}$.
In all regions where $C\neq 0$ the normalized Casimir energy is negative and
approximately given by eq. (12), presenting clear signatures of topology. When
$C=0$, the right-hand side of eq. (12) vanishes, but we get a small attractive
force between the plates, which arises from further expansions of the
reflection matrices in the Lifshitz equation. Since $|\mu|<\varepsilon_{1}$
for every cone (which is achieved in our plot with the stronger condition
$|\mu|<E_{B}$), we have $C_{\text{QH}}=0$, implying that we are only able to
probe the photo-induced topology. This feature has its roots in the quantum
anomaly of the zeroth Landau level [50], which has a twice smaller degeneracy
and its energy does not depend on the magnetic field. As a consequence, the
Chern number depends only on
$\Delta^{\eta}_{s}\left[\Lambda\right]+\tilde{\eta}\mu=\Delta^{\eta}_{s}\left[\Lambda+\mbox{sgn}(eB)\mu\right]$,
and thus changing the chemical potential induces the same effect in the phase
diagram as modifying the circularly polarized laser intensity. Furthermore,
the diagram is also symmetric with respect to $E_{z}=0$ and antisymmetric with
respect to $\Lambda=-\mbox{sgn}(eB)\mu$. In the situation where $\mu=0$, the
Chern number is given by
$C=-\sum_{\eta,s}\eta\,\mbox{sgn}\left(\Delta_{s}^{\eta}\right)/2$, being the
same as in the case without the magnetic field [43]. For $|\mu|>E_{B}$, the
first Landau level may be occupied in a given cone depending on the values of
$E_{z}$ and $\Lambda$, resulting in $C_{\text{QH}}\neq 0$ and a wealth of
quantum Hall topological phases (Fig. 2b). Fig 2c is a plot of the normalized
Casimir energy in the $(E_{z},\mu)$ space for $\Lambda=0$. The shape of the
diagram in the region $|\mu|<E_{B}$ resembles Fig. 2a due to the similar role
played by $\mu$ and $\Lambda$ under this condition.

This shows that the chemical potential can be used as a substitute of the
circularly polarized laser field to probe the photo-induced topological
features of the Casimir effect. Furthermore, the chemical potential presents
an advantage over the laser field since it does not cause an increase of the
system’s temperature. For highly doped materials ($|\mu|>E_{B}$), the quantum
Hall effect dominates the Casimir interaction and hyperbola-like curves define
the boundaries between different topological phases. In Fig 2d we display the
normalized Casimir energy as a function of the chemical potential. When $\mu$
crosses one of the Landau levels in a single Dirac cone from below the Casimir
interaction increases, becoming more repulsive. This same behaviour is found
in the graphene-graphene interaction in the presence of an external magnetic
field [39]. However, here the position and height of the quantum Hall jumps
can be tuned with an externally applied electric field, which can move the
energy levels upwards or downwards.

(

Figure 3: Casimir energy as a function of the chemical potential of a single
monolayer for $\\{e\ell E_{z},\mu_{2}\\}/\lambda_{SO}$ = {(1,0.5), (1,-0.5),
(1.5,0.5)} (solid blue, dashed red, and dash-dotted green, respectively) and
$E_{B}=\lambda_{SO}$. For all curves, the distance between the plates is long
enough so that Eq. (12) is valid and $\Lambda,\Gamma=0$.

In Fig. 3 we plot the long distance Casimir energy for two identical layers as
a function of the chemical potential $\mu_{1}$ of one of the plates. In this
situation, the Chern numbers of the monolayers do not always coincide, which
opens up the possibility of changing the attractive or repulsive character of
the quantized Casimir force. The sign of the force can be tuned by properly
choosing $\mu_{1}$, while its magnitude could also be tailored by modifying
$E_{z}$ and $\mu_{2}$. We emphasize that this fine control of the Casimir
interaction is only possible due to the energy level quantization arising from
the introduction of a magnetic field. As pointed out in ref. [46], without
this agent switching the sign of the dispersive Casimir force from repulsion
to attraction breaks its quantized behaviour. We also comment that for
$|\mu_{1}|>E_{B}$ and $|\mu_{2}|<E_{B}$ the Casimir force is a full mix of
photo-induced topology and quantum Hall physics since $C_{1}=C_{1,\text{QH}}$
and $C_{2}=C_{2,\text{ph}}$.

The accuracy of the long distance approximation is not the same in all regions
of the phase space. In fact, if we look at the upper and bottom portions of
the phase diagram in Fig 2c and compare it to the numerical values obtained
from Eq. (12), it becomes evident that this approximation is not as good as it
is for smaller values of $\mu$. In Fig. 4a we plot the difference between the
exact Casimir energy, obtained by full numerical integration of the Lifshitz
formula, and the approximated solution in the long distance limit for
different values of $\mu$. All curves tend to zero for large distances, but it
is evident that the Casimir energy converges faster to its asymptotic
expression for smaller values of the chemical potential. The reason for this
discrepance is due to the fact that the asymptotic solution is obtained by
neglecting the contribution arising from the longitudinal conductivity.
However, as we can see in the inset of Fig. 4a, for larger values of $\mu$ the
longitudinal conductivity increases near $i\xi=0$ and, in this case, needs to
be accounted in the low frequency expansion for the distances considered. Fig.
4b shows how our result is affected by introducing a non-negligible
dissipation $\Gamma\neq 0$ in the monolayers. In this situation, the Casimir
energy becomes less repulsive, and can be attractive for sufficiently high
values of dissipation. To understand this from an analytical point of view, we
once again expand the integrand of Eq. (1) for small values of $\xi$. Since
both $\sigma_{xx}$ and $\sigma_{xy}$ are non zero at $\xi=0$, the dominant
contribution of the Casimir energy comes from the DC conductivity tensor. For
sufficiently small values of the dissipation ($\hbar\Gamma<<\lambda_{SO}$), we
have $\sigma_{xx}(0)\approx\sigma_{xx}^{\prime}(0)\Gamma$ and
$\sigma_{xy}(0)\approx\frac{\alpha c}{2\pi}C$, where we took advantage of the
similar role played by $\Gamma$ and $\xi$ in Eqs. (6) and (8). By retaining
only the longitudinal conductivity and expanding the reflection matrices for
small $\xi$ we obtain

$\frac{\Delta
E^{(0)}}{E_{g}}=\frac{4\alpha\Gamma}{c}\frac{\mbox{log}(\sigma_{1,xx}^{\prime}/\sigma_{2,xx}^{\prime})}{(\sigma_{2,xx}^{\prime})^{-1}-(\sigma_{1,xx}^{\prime})^{-1}}.$
(15)

Figure 4: (a) Long distance approximated expression given by Eq. (12)
subtracted from the Casimir energy as a function of distance. The chosen
parameters are $\Lambda=0$, $\Gamma=0$, $e\ell E_{z}=\lambda_{SO}$, and
$\mu=\\{0.5,1.25,1.5\\}\lambda_{SO}$ (solid blue, dashed red, and dash-dotted
green lines, respectively). The inset shows the longitudinal and Hall
conductivity at frequency $\hbar\xi=0.01\lambda_{SO}$ as a function of the
chemical potential. (b). Casimir energy for $\Lambda=0$, $e\ell
E_{z}=\lambda_{SO}$, $\mu=1.25\lambda_{SO}$, and
$\hbar\Gamma=\\{0,0.01,0.02\\}\lambda_{SO}$ (solid, dashed, and dash-dotted
lines, respectively) as a function of distance. The gray dotted lines are the
approximated solutions in the long distance regime, given by the sum of Eqs.
(12) and (15).

For two identical layers, this expression simplifies to $\Delta
E^{(0)}/E_{g}=4\alpha\Gamma\sigma_{xx}^{\prime}/c$. The dotted gray lines in
Fig. 4b are given by the sum of eq. (12) with eq. (15), and have a good
agreement with the non approximated results (red curves).

##  III Finite-temperature Casimir energy

Figure 5: (a) Dissipationless Casimir energy as a function of temperature for
a separation distance of $d\lambda_{SO}/\hbar c=10$. The dotted lines
correspond to the approximated value of the $n=0$ Matsubara frequency
contribution presented in Eq. (16). The left inset is a zoom of the plot,
showing that each curve goes to its correspondent zero-temperature limit
(dashed gray lines). The right inset shows the Casimir force in the region
where it becomes attractive.(b) Casimir energy as a function of distance for
$k_{B}T=10^{-3}\lambda_{SO}$ and $\hbar\Gamma=0.01\lambda_{SO}$. The gray line
corresponds to the $n=0$ Matsubara contribution (Eq. (17)). In both plots,
$\Lambda=0$, $e\ell E_{z}=\lambda_{SO}$, and
$\mu=\\{0.5,1.25,1.5\\}\lambda_{SO}$ (solid blue, dashed red, and dash-dotted
green lines, respectively).

Now we consider thermal effects in the Casimir energy between two parallel
graphene family layers. For finite temperatures the Casimir interaction energy
follows from Eq. (1) by considering the temperature dependent conductivity of
the monolayers. The finite-temperature Casimir energy exhibits a rather
distinct behaviour depending on whether or not we neglect dissipation in the
materials. We first consider two identical dissipationless layers
($\Gamma=0$). In Fig. 5a we show the normalized Casimir energy in the long
distance regime as a function of temperature for different values of $\mu$.
For sufficiently small temperatures ($k_{B}T/\lambda_{SO}\lesssim 10^{-2}$),
the Casimir energy is well described by the zero-temperature limit (left
inset). As we increase the temperature, however, the Casimir force becomes
more repulsive (right inset), reaching values two orders of magnitude greater
than those at zero temperature in the region $k_{B}T/\lambda_{SO}\in[0.1,1]$
(below, near, and above room temperature for silicene, germanene, and stanene,
respectively). This contrasts with the results found in the absence of a
magnetic field, where even for smaller values of temperature
($k_{B}T/\lambda_{SO}\simeq 10^{-3}$, which corresponds to $T\sim 2,3K$ for
plumbene and smaller for other materials) the force is attractive, reaching
the zero-temperature limit only for $k_{B}T/\lambda_{SO}\simeq
10^{-4}$.[46]For $k_{B}T\simeq\lambda_{SO}$, the strength of the interaction
reaches a maximum value and then rapidly becomes attractive due to thermal
transitions between different Landau levels.

This behaviour of the finite-temperature Casimir energy can be described
analytically by considering only the contribution of the $n=0$ Matsubara
frequency in Eq. (1). Since for $\Gamma=0$ and small values of $\xi$ we have
$\sigma_{xx}=\sigma_{xx}^{\prime}(0)\xi$, we get a contribution from both the
longitudinal and Hall conductivities. By expanding the reflection matrices in
$\xi$, taking the limit $\xi\rightarrow 0$, and keeping the lowest order
contribution in $\alpha$, we get $\mathbb{R}_{sp}=\mathbb{R}_{ps}\approx
2\pi\sigma_{xy}/c$, $\mathbb{R}_{ss}\approx 0$, and $\mathbb{R}_{pp}\approx
2\pi\sigma_{xx}^{\prime}k_{\parallel}$ (notice that the conductivities depend
on temperature). Hence, we obtain

$\frac{E}{E_{g}}=-\frac{16\pi^{2}\sigma_{1,xy}\sigma_{2,xy}}{\hbar
c\alpha}k_{B}Td+\frac{12\pi^{2}\sigma_{1,xx}^{\prime}\sigma_{2,xx}^{\prime}}{\hbar
c\alpha}\frac{k_{B}T}{d}.$ (16)

The dotted lines in Fig. 5a were obtained from the previous equation, and we
note that there is a good agreement between this approximation and the
numerical calculations. Eq. (16) highlights that the repulsive Casimir
interaction arises from the Hall conductivity, which holds information about
topology and quantum Hall physics, while the attractive counterpart is due to
the longitudinal conductivity. In contrast to the zero-temperature limit, the
positive and negative terms of eq. (16) have different distance scaling laws.
As a consequence, the Casimir force becomes attractive at a temperature
slightly lower than that associated to the situation $E=0$, as shown in the
right inset of Fig. 5a. In the regime where $k_{B}T\ll\lambda_{SO}$ but $T$ is
high enough so that keeping only the $n=0$ term is still a good approximation,
we can discard the second term of eq. (16) and assume that
$\sigma_{i,xy}\approx 2\sigma_{0}C_{i}/\pi$, which leads to $E/E_{g}=-(4\alpha
ck_{B}Td/\hbar)C_{1}C_{2}$. This shows that applying a magnetic field on the
materials increases the robustness of the topological features arising in the
Casimir force, which are now present even at high temperatures ($k_{B}T\sim
0.1\lambda_{SO}$). On the left inset of Fig. 5a, the slope of each curve is
proportional to the square of its corresponding Chern number, being higher for
larger values of $\mu$.

The picture is not the same if we add dissipation to the system. In this case,
$\sigma_{xx}(0)\neq 0$ and the limit $\xi\rightarrow 0$ of the reflection
matrices yields $\mathbb{R}_{ss}=\mathbb{R}_{sp}=\mathbb{R}_{ps}=0$ and
$\mathbb{R}_{pp}=1$. Hence, regardless of the topological phase of the
materials, the $n=0$ contribution to the Casimir energy is given by

$\frac{E}{E_{g}}=\frac{2\zeta(3)}{\hbar c\alpha}k_{B}Td,$ (17)

which is exactly the same result as the one found in the absence of a magnetic
field. Note that there is no dependence on the conductivities and, more
importantly, the result is the same for any non-zero value of $\Gamma$. Fig.
5b shows that, in this situation, even for temperatures of
$k_{B}T/\lambda_{SO}=10^{-3}$ the Casimir force is always attractive, as the
case when $B=0$. Hence, in order to see repulsion in a dissipative system one
has to decrease the temperature to the sub-kelvin regime, which poses a
serious difficult in measuring the Casimir force.

This discrepance between the results obtained for a dissipationless and a
dissipative system is no coincidence. Having a finite dissipation, even if
arbitrarily small, is fundamentally different from having no dissipation. For
metallic materials, this is equivalent to describe the system using a plasma
($\Gamma=0$) or a Drude ($\Gamma\neq 0$) model, which leads to significantly
different final results [51]. In principle, the Drude model is the most
straightforward approach for taking into account the relaxation properties of
conduction electrons, and should be applicable in the quasistatic limit (which
corresponds to the high temperature regime of the Casimir force). In fact, a
few experiments [52, 53] show that the Casimir force is consistent with the
calculations using this model. However, the plasma model approach, which
usually appears in the literature in infrared optical phenomena, was shown to
best describe the Casimir force between metals in most of the experiments [54,
55, 56, 57]. Which model is the correct one to use when calculating the
Casimir force is an open problem that still awaits its ending chapters. Here
we provide a full description of the Casimir interaction between dissipative
and dissipationless graphene family topological insulators and comment that
the Drude versus plasma discussion can be extended to non-metallic systems. We
emphasize that this extremely different behaviour of the dissipationless
Casimir force in the graphene family does not show up when $B=0$, where it was
shown that both models differ only by a factor $2$ (as the case of metallic
plates) [46]. Hence, the addition of a magnetic field to the problem allows
one not only to avoid difficulties concerning temperature but also to clearly
discriminate both descriptions and investigate fundamental questions in
physics.

##  IV Conclusions and final remarks

In this work we presented a complete study of the Casimir effect between
graphene family materials under the influence of an external magnetic field.
We have shown that, for sufficiently large distances, the zero-temperature
Casimir force between two monolayers is proportional to the product of their
corresponding Chern numbers and can be made attractive or repulsive by
properly doping the materials. The remarkable possibility of switching the
sign of the Casimir force in the flatland is exclusively due to the interplay
between photo-induced phase transitions and quantum Hall physics brought by
the presence of a magnetic field. Due to the intrinsically two-dimensional
nature of the graphene family materials, in order to test this prediction in
an experiment one should be able to measure the Casimir force in a parallel
surface configuration. In the last few years we have witnessed important
advances regarding the alignment and stability in Casimir force measurement
setups with parallel plates [58, 59, 60], which may be extended to two-
dimensional topological insulators in the near future. On the other hand, a
metallic sphere close to the graphene family sheet could be used to explore
the topological features of the system, and is a more adopted scheme for
measuring the Casimir force in laboratory [61]. Furthermore, we can use the
chemical potential as a substitute of the circulary polarized laser field,
which can be very useful to avoid increasing the materials temperature due to
photo-absorption. We believe this may help experimental investigations of the
Casimir force between graphene family materials near zero temperature.

Finally, we analyzed the thermal effects on the Casimir energy, which proved
to be significantly different from previous studies using the graphene family
materials. We have shown that the dissipative model imply a rather trivial
high temperature Casimir force, where no topological physics is present.
However, the picture is very different if we assume a dissipationless system.
In this case, we found repulsive forces of the order of the zero-temperature
graphene-graphene interaction, even at room temperature, and the topological
features are not completely lost. For instance, let us consider two stanene
monolayers at $T=300K$ and separated by a distance $d=20\mu$m. By evaluating
the Casimir force per unit area for $\Lambda=0$ and $e\ell
E_{z}=\lambda_{SO}$, we obtain $F\approx 0.1$nPa at $\mu=1.25\lambda_{SO}$ and
$F\approx 0.2$nPa at $\mu=1.5\lambda_{SO}$, while the zero-temperature
graphene-graphene Casimir pressure is given by $F_{g}\approx 0.05$nPa.
Finally, since some experiments show that the Casimir force is in agreement
with the plasma model and others with the Drude model, we emphasize the
importance of our findings in the dissipationless limit. In fact, a huge
difference between both situations allows drawing more confident conclusions
when testing which model correctly describes the Casimir interaction in the
extended graphene family.

##  V Acknowledgements

The authors thank V. Henning and R. Guimarães for enlightening discussions.
Y.M. and. C.F. acknowledge funding by the Coordenação de Aperfeiçoamento de
Pessoal de Nível Superior (CAPES). C.F. acknowledges Conselho Nacional de
Desenvolvimento Científico e Tecnológico (CNPq) - research grant
310365/2018-0. W.K.-K. acknowledges the Laboratory Directed Research and
Development program of Los Alamos National Laboratory for funding under
project number 20190574ECR.

## References

  * [1] Bordag, M. et al. Advances in the Casimir effect. OUP Oxford Vol. 145 (2009).
  * [2] Dalvit, D. et al. eds. Casimir physics. Springer Vol. 834 (2011).
  * [3] Buhmann, S. Y. Dispersion Forces I: Macroscopic quantum electrodynamics and ground-state Casimir, Casimir-Polder and Van der Waals forces. Springer Vol. 247 (2013).
  * [4] Bordag, M., Mohideen, U., Mostepanenko, V. M. New developments in the Casimir effect. Physics reports 353 1-3 (2001).
  * [5] Woods, L. M. et al. A materials perspective on Casimir and van der Waals interactions. Rev. Mod. Phys. 88, 045003 (2016).
  * [6] Farina, C. The casimir effect: some aspects. Brazilian journal of physics 36, 1137-1149 (2006).
  * [7] Casimir, H. B. G. On the attraction between two perfectly conducting plates. Proc. Kon. Nederland Akad. Wetensch 51,  793-795 (1948).
  * [8] Parsegian, V. A. Van der Waals Forces: A Handbook for Biologists, Chemists, Engineers, and Physicists Cambridge U.P., New York (2006)
  * [9] Israelashivili, J. Intermolecular and Surface Forces. 3rd ed. Academic Press, New York (2011).
  * [10] DiStasio, R. A., Jr., von Lilienfeld, O. A., Tkatchenko, A. Collective many-body van der Waals interactions in molecular systems. Proc. Natl. Acad. Sci. U. S. A. 109, 14791 (2012).
  * [11] Autumn, K. et al. Adhesive force of a single gecko foot-hair. Nature 405, 681 (2000).
  * [12] Autumn, K. et al. Evidence for van der Waals Adhesion in Gecko Setae. Proc. Natl. Acad. Sci. U. S. A. 99, 12252 (2002).
  * [13] Serry, F. M., Walliser, D., Maclay, G. J. The role of the Casimir effect in the static deflection and stiction of membrane strips in microelectromechanical systems (MEMS). J. Appl. Phys. 84, 2501 (1998).
  * [14] Buks, E., Roukes, M. L. Stiction, adhesion energy, and the Casimir effect in micromechanical systems, Phys. Rev. B 63, 033402 (2001).
  * [15] Broer, W. et al. Significance of the Casimir force and surface roughness for actuation dynamics of MEMS. Phys. Rev. B 87, 125413 (2013).
  * [16] Sedighi, M., Palasantzas, G. Influence of low optical frequencies on actuation dynamics of microelectromechanical systems via Casimir forces. J. Appl. Phys. 117, 144901 (2015).
  * [17] Rodrigues, J. R., Gusso, A., Rosa, F. S. S., Almeida, V. R. Rigorous analysis of Casimir and van der Waals forces on a silicon nano-optomechanical device actuated by optical forces. Nanoscale 10, 3945-3952 (2018).
  * [18] Klimchitskaya, G. L., Mohideen, U., Mostepanenko, V. M. The Casimir force between real materials: experiment and theory. Rev. Mod. Phys. 81,  1827-1885 (2009).
  * [19] Rodriguez, A. W., Capasso, F., Johnson, S. G. The Casimir effect in microstructured geometries. Nat. Photon. 5, 211-221 (2011).
  * [20] Tang, L. et al. Measurement of non-monotonic Casimir forces between silicon nanostructures. Nat. Photon. 11, 97-101 (2017).
  * [21] Grushin, A. G., Cortijo, A. Tunable Casimir repulsion with three-dimensional topological insulators. Phys. Rev. Lett 106, 020403 (2011).
  * [22] Grushin, A. G., Rodriguez-Lopez, P., Cortijo, A. Effect of finite temperature and uniaxial anisotropy on the Casimir effect with three-dimensional topological insulators, Phys. Rev. B 84, 045119 (2011).
  * [23] Rodriguez-Lopez, P., Grushin, A. G. Repulsive Casimir effect with Chern insulators. Phys. Rev. Lett. 112, 056804 (2014).
  * [24] Wilson, J. H., Allocca, A. A., Victor Galitski, Repulsive Casimir force between Weyl semimetals, Phys. Rev. B 91, 235115 (2015).
  * [25] Ma, J., Zhao, Q., Meng, Y., Magnetically controllable Casimir force based on a superparamagnetic metametamaterial. Phys. Rev. B 89, 075421 (2014)
  * [26] Gómes-Santos, G. Thermal van der Waals interaction between graphene layers. Phys. Rev. B 80,  245424 (2009).
  * [27] Drosdoff, D., Woods, L. M. Casimir forces and graphene sheets. Phys. Rev. B 82, 155459 (2010).
  * [28] Sarabadani, J. et al. Many-body effects in the van der Waals-Casimir interaction between graphene layers. Phys. Rev. B 84, 155407 (2011)
  * [29] Bordag, M., Klimchitskaya, G. L., Mostepanenko, V. M. Thermal Casimir effect in the interaction of graphene with dielectrics and metals. Phys. Rev. B 86 165429 (2012).
  * [30] Klimchitskaya, G. L., Mostepanenko, V. M. Van der Waals and Casimir interactions between two graphene sheets. Phys. Rev. B 87 075439 (2013).
  * [31] Tsoi, S. et al. van der Waals screening by single-layer graphene and molybdenum disulfide. ACS Nano 8, 12410-12417 (2014).
  * [32] Gobre, V. V., Tkatchenko, A. Scaling laws for van der Waals interactions in nanostructured materials. Nat. Commun. 4, 2341 (2014).
  * [33] Bimonte, G., Klimchitskaya, G. L., Mostepanenko, V. M. Thermal effect in the Casimir force for graphene and graphene-coated substrates: Impact of nonzero mass gap and chemical potential. Phys. Rev. B 96 115430 (2017).
  * [34] Klimchitskaya, G. L., Mostepanenko, V. M. Quantum field theoretical description of the Casimir effect between two real graphene sheets and thermodynamics. Phys. Rev. D 102 016006 (2020).
  * [35] Gusynin, V. P., Sharapov, S. G. Unconventional Integer Quantum Hall Effect in Graphene. Phys. Rev. Lett. 95, 146801 (2005).
  * [36] Gusynin, V. P., Sharapov, S. G., Carbotte, J. P. Magneto-optical conductivity in graphene. J. Phys.: Condens. Matter 19, 026222 (2007).
  * [37] Goerbig, M. O. Electronic Properties of Graphene in a Strong Magnetic Field. Rev. Mod. Phys. 83, 1193 (2011).
  * [38] Tabert, C. J., Nicol, E. J. Magneto-optical conductivity of silicene and other buckled honeycomb lattices Phys. Rev. B 88, 085434 (2013).
  * [39] Tse, W. K., MacDonald, A. H. Quantized Casimir force. Phys. Rev. Lett. 109, 236806 (2012).
  * [40] Cysne, T. et al. Tuning the Casimir-Polder interaction via magneto-optical effects in graphene. Phys. Rev. A 90, 052511 (2014).
  * [41] Ni, Z. et al. Tunable bandgap in Silicene and Germanene. Nano Lett. 12, 113-118 (2012).
  * [42] Stille, L., Tabert, C. J., Nicol, E. J. Optical signatures of the tunable band gap and valley-spin coupling in silicene. Phys. Rev. B 86, 195405 (2012).
  * [43] Ezawa, M. Photoinduced Topological Phase Transition and a Single Dirac-Cone State in Silicene. Phys. Rev. Lett. 110, 026603 (2013).
  * [44] Ezawa, M. Monolayer Topological Insulators: Silicene, Germanene and Stanene. J. Phys. Soc. Jpn. 84, 121003 (2015).
  * [45] Kort-Kamp, W.J. M. Topological phase transitions in the photonic spin Hall effect. Phys. Rev. Lett. 119, 147401 (2017)
  * [46] Rodriguez-López, P., Kort-Kamp, W. J. M., Dalvit, D. A. R., Woods, L. M. Casimir force phase transitions in the graphene family. Nat. Commun. 8, 14699 (2017).
  * [47] Lambrecht, A., Neto, P. A. M., Reynaud, S. The Casimir effect within scattering theory. New Journal of Phys. 8,243 (2006).
  * [48] Ledwith, P., Kort-Kamp, W. J. M., Dalvit, D. A. R. Topological phase transitions and quantum Hall effect in the graphene family. Phys. Rev. B 97, 165426 (2018).
  * [49] Yu. X.-L., Huang, L., Wu, J. From a normal insulator to a topological insulator in plumbene. Phys. Rev. B 95, 125113 (2017).
  * [50] Karch, J. et al. Dynamic Hall Effect Driven by Circularly Polarized Light in a Graphene Layer. Phys. Rev. Lett. 105,  227402 (2010)
  * [51] Klimchitskaya, G. L., Mostepanenko, V. M. Casimir free energy of metallic films: discriminating between Drude and plasma model approaches. Phys. Rev. A 92,  042109 (2015).
  * [52] Garcia-Sanchez, D. et al. Casimir force and in situ potential measurements on nanomembranes. Phys. Rev. Lett. 109,  027202 (2012).
  * [53] Sushkov, A. O. et al. Observation of the thermal Casimir force. Nat. Phys. 7,  230-233 (2011)
  * [54] Decca, R. S. et al. Improved tests of extra-dimensional physics and thermal quantum field theory from new Casimir force measurements. Phys. Rev. D 68,  116003 (2003)
  * [55] Decca, R. S. et al. Tests of new physics from precise measurements of the Casimir pressure between two gold-coated plates. Phys. Rev. D 75,  077101 (2007).
  * [56] Banishev, A. A. et al. Casimir interaction between two magnetic metals in comparison with nonmagnetic test bodies. Phys. Rev. B 88,  155410 (2013)
  * [57] Bimonte, G., López, D., Decca, R. S. Isoelectronic determination of the thermal Casimir force. Phys. Rev. B 93,  184434 (2016)
  * [58] Norte, R. A. et al. Platform for Measurements of the Casimir Force between Two Superconductors. Phys. Rev. Lett 121,  030405 (2018)
  * [59] Sedmik, R., Brax, P. Status Report and first Light from Cannex: Casimir Force Measurements between flat parallel Plates. J. Phys.: Conf. Ser. 1138 012014 (2018)
  * [60] Mostepanenko, V. M., Klimchitskaya, G. L. Recent measurements of the Casimir force: Comparison between experiment and theory Mod. Phys. Lett. A 35,  2040007 (2020)
  * [61] Hartmann, M., Ingold, G.-L., Neto, P. A. M. Plasma versus Drude modeling of the Casimir force: beyond the proximity force approximation. Phys. Rev. Lett. 119,  043901 (2017).


# Maximum entropy methods for quantum state compatibility problems

Shi-Yao Hou  College of Physics and Electronic Engineering, Center for
Computational Sciences, Sichuan Normal University, Chengdu 610068, China
Shenzhen SpinQ Technology Co., Ltd., Shenzhen, China  Zipeng Wu  Department of
Physics, The Hong Kong University of Science and Technology, Clear Water Bay,
Kowloon, Hong Kong, China  Jinfeng Zeng  Department of Physics, The Hong Kong
University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong,
China  Beijing Academy of Quantum Information Sciences, Beijing 100193, China
Ningping Cao  Institute for Quantum Computing, University of Waterloo,
Waterloo, Ontario, Canada  Chenfeng Cao  Department of Physics, The Hong Kong
University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong,
China  Youning Li<EMAIL_ADDRESS>College of Science, China Agricultural
University, Beijing, 100080, People’s Republic of China  Bei Zeng
<EMAIL_ADDRESS>Department of Physics, The Hong Kong University of Science and
Technology, Clear Water Bay, Kowloon, Hong Kong, China

###### Abstract

Inferring a quantum system from incomplete information is a common problem in
many aspects of quantum information science and applications, where the
principle of maximum entropy (MaxEnt) plays an important role. The quantum
state compatibility problem asks whether there exists a density matrix $\rho$
compatible with some given measurement results. Such a compatibility problem
can be naturally formulated as a semidefinite programming (SDP), which
searches directly for the existence of a $\rho$. However, for large system
dimensions, it is hard to represent $\rho$ directly, since it needs too many
parameters. In this work, we apply MaxEnt to solve various quantum state
compatibility problems, including the quantum marginal problem. An immediate
advantage of the MaxEnt method is that it only needs to represent $\rho$ via a
relatively small number of parameters, which is exactly the number of the
operators measured. Furthermore, in case of incompatible measurement results,
our method will further return a witness that is a supporting hyperplane of
the compatible set. Our method has a clear geometric meaning and can be
computed effectively with hybrid quantum-classical algorithms.

##  I Introduction

The connection of statistical mechanics and information theory encodes a
fundamental logic of science. In standard statistical mechanicsPathria (2016),
it has been generally accepted that canonical distribution of equilibrium
corresponds to the state with maximum entropy, after early work of
BoltzmanBoltzmann (1877). With the foundation and boom of information theory
in 1940sShannon (1948), Jaynes developed the connotation of entropy in
statistical mechanics and endow new information-theoretical meaning of this
conceptJaynes (1957a, b). Lying at the heart is the principle of maximum
entropy (MaxEnt), saying that the probability distribution that best
represents the current state of knowledge is the one with maximum entropy. In
information theory, MaxEnt is excellent in dealing with the inference problem
of incomplete information, which is ubiquitous in science. With linear
constraints, the general solution for MaxEnt adopts an exponential form, which
is widely studied in terms of the information geometry methods Csiszár  _et
al._ (2004). MaxEnt finds a wide range of applications in areas such as data
classification Palmieri and Ciuonzo (2013), density estimation Botev and
Kroese (2011), natural language processing Ratnaparkhi (1998), and parameter
estimation in artificial intelligence Malouf (2002).

With the emergence of quantum information theory in 1970sHolevo (1973), MaxEnt
developed its own new meaning and applications in this new field, by extending
the information entropy to von Neumann entropy of quantum states. A natural
problem in quantum information theory is the quantum state inference from
local information. Inference with complete information is the so-called
quantum tomography Bužek  _et al._ (1998), while the incomplete information
version is quantum maximum entropy inference (qMaxEnt). qMaxEnt is applied to
a broad range of problem in quantum information, such as entanglement
detectionŘeháček  _et al._ (2001). Later studies on quantum information
geometry further connected the qMaxEnt to correlations in many-body quantum
systemsZhou (2008); Niekamp  _et al._ (2013).

In this paper we focus on the use of qMaxEnt to solve the quantum state
compatibility problem. For a quantum system of $n$ qubits, consider a set of
Hermitian operators $\\{A_{1},A_{2},\cdots,A_{m}\\}$, and corresponding a set
of real numbers $\\{a_{1},a_{2},\cdots,a_{m}\\}$. The quantum state
compatibility problem asks whether there exists a quantum state $\rho$ of the
system, such that $a_{i}=\operatorname{Tr}(\rho A_{i})$. This problem can be
naturally formulated as a semidefinite programming (SDP), to find the
solutions to the following linear matrix inequalities (LMIs)

$\displaystyle\quad\forall A_{i}\quad\operatorname{Tr}(\rho A_{i})=a_{i},$ (1)
$\displaystyle\quad\rho\succeq 0.$ (2)

If the solution to the LMIs exist, that is all $a_{i}$s are compatible, we
will obtain a density matrix $\rho$ satisfying $\operatorname{Tr}(\rho
A_{i})=a_{i}$. Otherwise the quantum state compatibility problem has no
solution. When $n$ is large, however, it is hard to represent $\rho\succeq 0$
which has exponential number of parameters in $n$.

In practice, normally $m$ is only polynomial in $n$ (e.g. $A_{i}$s are local
operators), we can instead to use the qMaxEnt method. That is, we can consider
an operator $H$ of the form

$\displaystyle H=\sum c_{i}A_{i},$ (3)

and state of the form

$\displaystyle\rho^{\prime}=\frac{e^{-H}}{\operatorname{Tr}(e^{-H})},$ (4)

then by MaxEnt, a compatible state $\rho$ exists if and only if
$\rho^{\prime}$ exists, which satisfies
$a_{i}=\operatorname{Tr}(\rho^{\prime}A_{i})$. An immediate advantage of this
qMaxEnt compared to the SDP algorithm is that $\rho^{\prime}$ only needs $m$
parameters to represent.

However, if the algorithm cannot return any $\rho^{\prime}$, then it could due
to the effectiveness and the convergence of numerical algorithms used. In
other words, if we can find $\rho^{\prime}$ satisfying
$a_{i}=\operatorname{Tr}(\rho^{\prime}A_{i})$ to certain precision, we can
then say that the answer to the compatibility problem is “yes” and
$\rho^{\prime}$ is a compatible state. However, if we fail to find
$\rho^{\prime}$ satisfying $a_{i}=\operatorname{Tr}(\rho^{\prime}A_{i})$ to
certain precision, we will need to further justify that the answer to the
compatibility problem is indeed “no”. To do so, we will further find a
witness. The witness distinguishes the given incompatible $a_{i}$s and all the
compatible $a_{i}$s. The idea of our method is illustrated in Fig. 1. Notice
that when the SDP failed, it only returns “no”, without any further
information.

Figure 1: Geometric interpretation of the maximum entropy method.
$\mathcal{A}$ denotes the set of all $a_{i}$s. $\mathcal{B}$ denotes the set
of all $a_{i}$s corresponding to some compatible state $\rho$, which is a
convex set. The boundary of $\mathcal{B}$ represents the ground states of some
Hamiltonian $H=\sum c_{i}A_{i}$. Flat part of the boundary of $\mathcal{B}$
(e.g., $T_{1}$-$T_{2}$) corresponding to degenerate ground states of some $H$.
For $a_{i}$s in $\mathcal{B}$ (e.g., point $S$), the qMaxEnt method finds
compatible states of the form
$\rho^{\prime}=\frac{e^{-H}}{\operatorname{Tr}(e^{-H})}$. If the corresponding
quantum state compatibility problem has no solution, then $a_{i}$s must be
outside of $\mathcal{B}$ (e.g., points $S^{\prime}$, $S^{\prime\prime}$). In
this case, the qMaxEnt algorithm returns some density matrices corresponding
to the boundary points of $\mathcal{B}$ (e.g., points $P^{\prime}$,
$P^{\prime\prime}$). $P^{\prime}$ corresponds to the unique ground state of
some Hamiltonian $H^{\prime}$, which is the witness detecting the
incompatibility of $S^{\prime}$ (the supporting hyperplane is tangent to
$\mathcal{B}$ at point $P^{\prime}$). $P^{\prime\prime}$ corresponds to a
mixture state in the degenerate ground state space of some $H^{\prime\prime}$
since it falls to the flat region on the boundary of $\mathcal{B}$. In this
case, a further analysis of the spectrum of the corresponding density matrix
of $P^{\prime\prime}$ will lead to the witness $H^{\prime\prime}$, which is
the supporting hyperplane of $\mathcal{B}$ that intersects $P^{\prime\prime}$.

The most difficult part of our method is to calculate
$\operatorname{tr}(A_{i}\frac{e^{-H}}{\operatorname{Tr}(e^{-H})})$ for some
$H$, where one can use some machine learning algorithms. In recent years,
machine learning was introduced in to help with quantum information
processingCao  _et al._ (2020); Zuo  _et al._ (2022). Quantum machine learning
concepts, which is directly related to qMaxEnt, has been developed, such as
quantum Boltzmann machine (QBM)Amin  _et al._ (2018), especially for the case
when there is no hidden layer. Furthermore, with the incoming of the so-called
noisy-intermedia-scale quantum (NISQ) era Preskill (2018), many hybrid
quantum-classical algorithms (such as the variational QBMsZoufal  _et al._
(2021); Shingu  _et al._ (2021), the variational quantum code searcherCao  _et
al._ (2022a)), which may be more efficient on near-term quantum devices and be
able to replace the subroutines of traditional algorithm, has been developed.
Of particular relevance are the hybrid quantum-classical algorithms that
prepare the thermal state of the form
$\frac{e^{-H}}{\operatorname{Tr}(e^{-H})}$ Zeng  _et al._ (2021); Verdon  _et
al._ (2019); Wu and Hsieh (2019); Wang  _et al._ (2021); Chowdhury  _et al._ ;
Zoufal  _et al._ (2021). All these algorithms can then help to scale our
method with the existence of near-term quantum computers.

We organize our paper as follows: in Sec. II, we introduce the details of our
new algorithms that solve the quantum state compatibility problems based on
qMaxEnt; in Sec. III, we apply our algorithms to solve some quantum state
compatibility problems for random measurement operators; in Sec. IV, we
further apply our method to solve a special kind of the quantum state
compatibility problem, i.e. the quantum marginal problem; in Sec. V, we
discuss the hybrid quantum-classical algorithms, which can help to scale our
method; finally we conclude in Sec. VI.

##  II The maximum entropy method

For a quantum system of $n$ qubits, consider a set of Hermitian operators
$\\{A_{1},A_{2},\cdots,A_{m}\\}$, and a corresponding set of real numbers
$\\{a_{1},a_{2},\cdots,a_{m}\\}$. The quantum state compatibility problem asks
whether there exists a quantum state $\rho$ of the system, such that
$a_{i}=\operatorname{Tr}(\rho A_{i}),\forall i$. The MaxEnt states that there
exists a state $\rho$, which satisfies

$a_{i}=\operatorname{Tr}(\rho A_{i}),$ (5)

if and only if there exists a maximum entropy state $\rho^{\prime}$ such that

$\operatorname{Tr}(\rho^{\prime}A_{i})=a_{i}.$ (6)

The maximum entropy state has the form

$\rho^{\prime}=\frac{e^{-H}}{\operatorname{Tr}(e^{-H)}},$ (7)

where

$H=\sum_{i}c_{i}A_{i}.$ (8)

In this setting, $\rho^{\prime}$ is determined by vecter
$\vec{c}=\\{c_{1},c_{2},\cdots,c_{m}\\}$. Therefore, if we can find the
maximum entropy state, we can conclude that there is a solution to the
compatibility problem. Otherwise, the compatible $\rho$ does not exist. Now
the compatibility problem is transformed into finding the $c_{i}$s.

###  II.1 The loss function

We can then set up a loss function:

$f(\vec{c})=\sum_{i}\left[a_{i}-\operatorname{Tr}\rho^{\prime}(\vec{c})A_{i}\right]^{2}.$
(9)

Obviously, if there does exist a $\rho^{\prime}$, such that
$\operatorname{Tr}(\rho^{\prime}A_{i})=a_{i},\forall i$, the loss function $f$
will reach its minimum 0.Therefore, the problem becomes an optimization
problem. Nowadays, optimization problems are usually solved by using gradient-
based algorithms. The basic idea of these algorithms is to define an objective
function and find its minimum. Roughly, the gradient-based optimization
algorithms can be described as follows. For a function $f(\vec{c})$, where
$\vec{c}=(c_{1},c_{2},\cdots,c_{m})$ is a vector of parameters. In order to
find the minimum of such function, we can randomly choose an initial value
$\vec{c}^{0}$ and then iteratively update it to optimum. For the $k$-th
iteration, update $c^{k}$ to $c^{k+1}$ by

$\vec{c}^{k+1}=\vec{c}^{k}-\alpha\nabla f(\vec{c}),$ (10)

where $\alpha\in\mathbb{R}$ is a scalar. If $\alpha$ is small enough, it could
be guaranteed that $f(\vec{c}^{k+1})\leq f(\vec{c}^{k})$. Usually, $\alpha$ is
determined by line search algorithms. To approach better performance and
convergence, algorithms such as Broyden-Fletcher-Goldfarb-Shanno (BFGS)
algorithm BROYDEN (1970); Fletcher (1970); Goldfarb (1970); Shanno (1970) or
conjugate gradient Hestenes and Stiefel (1952) are used more frequently.

One alternative loss function is based on the quantum Boltzmann machine Amin
_et al._ (2018). Quantum Boltzmann machine is a new machine learning approach
proposed based on Boltzmann distribution of a quantum Hamiltonian. The basic
idea of quantum Boltzmann machine is that for a quantum system described by
Hamiltonian $H$ with adjustable parameters $H(\vec{\theta})$, of which the
general form is decided by the system, by measuring certain operator
$\Lambda_{\mathbf{v}}$, we can obtain the Boltzmann probability distribution
$P_{\mathbf{v}}$:

$P_{\mathbf{v}}=\frac{\operatorname{tr}(\Lambda_{\mathbf{v}}e^{-H})}{\operatorname{tr}e^{-H}}.$
(11)

The loss function is

$\mathcal{L}=-\sum_{\mathbf{v}}P_{\mathbf{v}}^{\text{data
}}\log\frac{\operatorname{tr}\left[\Lambda_{\mathbf{v}}e^{-H}\right]}{\operatorname{tr}\left[e^{-H}\right]}=-\sum_{\mathbf{v}}P_{\mathbf{v}}^{\text{data
}}\log P_{\mathbf{v}},$ (12)

where $P_{\mathbf{v}}^{\text{data}}$ is the probability distribution of
training data. This loss function is the cross entropy loss. If the two
probability distribution are exactly the same, the loss function reaches its
minimum.

For our problem, we can assume $P^{\text{data}}_{\mathbf{v}}$ be the measured
result $a_{i}=\operatorname{Tr}(\rho A_{i})$, and $P_{\mathbf{v}}$ be
$\operatorname{Tr}(\rho^{\prime}A_{i})$, then the loss function is

$\mathcal{L}=-a_{i}\log\operatorname{Tr}(\rho^{\prime}A_{i}),$ (13)

However, $P^{\text{data}}_{\mathbf{v}}$ and $P_{\mathbf{v}}$ in loss function
are probability distributions, that is, for each $\mathbf{v}$,

$\displaystyle 0\leq$ $\displaystyle P_{\mathbf{v}}\leq 1,$ (14)
$\displaystyle 0\leq$ $\displaystyle P^{\text{data}}_{\mathbf{v}}\leq 1,$ (15)

and

$\displaystyle\sum_{\mathbf{v}}P_{\mathbf{v}}$ $\displaystyle=1,$ (16)
$\displaystyle\sum_{\mathbf{v}}P^{\text{data}}_{\mathbf{v}}$
$\displaystyle=1.$ (17)

Obviously, if $A_{i}$s are chosen to be product of Pauli Matrices, the
expectation value for any density operator $\rho$ is

$-1\leq 1\operatorname{Tr}(\tilde{\rho}_{m}A_{i})\leq 1.$ (18)

Therefore, Eq. 13 can not be optimized (note that for logarithm function, the
variable $\operatorname{Tr}(\rho A_{i})$ must be positive). Therefore, we have
to make $a_{i}$ and $\operatorname{Tr}(\rho A_{i})$ probability distributions,
that is, apart from the condition that $A_{i}$s should be complete,

$\sum_{i}A_{i}=I,$ (19)

and for any $\rho$

$\operatorname{tr}(\rho A_{i})\geq 0.$ (20)

To satisfy Eq. 19 and Eq. 20, we just need to make $A_{i}$s being a collection
of positive semi-definite operators that sums to identity. A set of such
operators is also called positive operator valued measures (POVMs). The
condition of $A_{i}$s being complete means the POVMs should be chosen
information complete (IC). More specifically, we can choose $A_{i}$s to be
symmetric information complete POVMs (SIC-POVMs). One way of constructing SIC-
POVMs could be found in Ref. Salazar  _et al._ (2012). With $A_{i}$s being
chosen to be SIC-POVMs, both $a_{i}$ and $\operatorname{Tr}(\rho A_{i})$
become a probability distribution. The optimization methods could be used to
optimize the loss function Eq. 13.

Besides the the gradient descent, which changes the parameters $\vec{c}^{i}$
as a whole in every iteration, there are other ways of finding the minimum
value of a function. One algorithm for solving such a problem is proposed in
Niekamp  _et al._ (2013) which is based on alternating optimization. The basic
idea of alternating optimization is to optimize one parameter each time.

###  II.2 The witness

However, one problem with gradient-based algorithms is that they often fall
into local minima. For our problem, if the maximum entropy state is found, the
problem is solved. But, if the maximum entropy state is not found, we can not
arbitrarily say that the maximum entropy state does not exist. Although
according to our tests, it is highly probable that we can find the maximum
entropy state if it exists. Here, we give a geometric interpretation of this
problem.

Let $\mathcal{A}$ be the set of all possible $a_{i}$s, $\mathcal{B}$ be the
set for all $a_{i}$s with compatible $\rho$. The shape of $\mathcal{B}$ is
very complicated, but one thing known is that the set is complex. The boundary
of $\mathcal{B}$ are ground states of certain Hamiltonian with the form
$H=\sum_{i}x_{i}A_{i}$, as shown in Fig. 1. When the optimization is
performed, the maximum entropy state is obviously limited in space
$\mathcal{B}$. Therefore, if there exists a global state with compatible
$\rho$, e.g., point $S$, then we can find a solution. If the given $\rho$ is
not reduced density matrices of a certain global density matrix, a point $P$
which minimize the function $f(\vec{x})$ will be found. Since the definition
of function $f(\vec{x})$ is actually the Euclidean distance of optimum point
$P$ and the target point $S^{\prime}$, $P$ is actually the point nearest to
$S^{\prime}$, which means $P$ must be on the boundary of $\mathcal{B}$.
Therefore, $P$ represents a ground state (if $H$ is not degenerate) or a
mixture of degenerate ground states (if $H$ is degenerate). As mentioned
before, there is no guarantee that the optimization will approach the point
$S$ or $P$. But, since $P$ is a ground state or a mixture of the ground states
of the Hamiltonian $H$, we can form a witness. Through point $P^{\prime}$, we
can draw a tangent hyper plane (in a two dimension space shown in Fig. 1, the
hyper plane is a line) with function

$\frac{x-a_{1}}{c_{1}}+\frac{y-a_{2}}{c_{2}}+\cdots=0$ (21)

One thing has to be noticed here is the degeneracy. The boundary of
$\mathcal{B}$ is composed of all ground states. For certain cases, the
Hamiltonian $H$ could be degenerate. As shown in Fig. 1. If the point of the
maximum entropy state is point $P^{\prime}$, $H^{\prime}$ serves as the
witness. However, if the point of all reduced density matrices is the point
$S^{\prime\prime}$, then the nearest point is $P^{\prime\prime}$ will be
found. The Hamiltonian corresponding to $P^{\prime\prime}$ is degenerate, that
is, there are more than two states has the same lowest energy. By checking the
rank of the corresponding density matrix or just diagonalizing the density
matrix, we can decide whether the Hamiltonian is degenerate. As shown in Fig.
1, all the states on the boundary between the points $T_{1}$ and $T_{2}$ are
degenerated, and each is a mixture of different ground states. Those pure
ground states are the states corresponding to $T_{1}$ and $T_{2}$. Suppose the
pure states corresponding to $T_{1}$ and $T_{2}$ are $|\psi_{T_{1}}\rangle$
and $|\psi_{T_{2}}\rangle$, respectively, then we can generate a new density
matrix:

$\rho_{M}=\frac{1}{2}(|\psi_{T_{1}}\rangle\langle\psi_{T_{1}}|+|\psi_{T_{2}}\rangle\langle\psi_{T_{2}}|),$
(22)

which corresponds to the point $M$, in the middle of $T_{1}$ and $T_{2}$.
Using the same optimization method to find a Hamiltonian $H^{\prime\prime}$,
for which

$\rho_{M}=\frac{e^{-H^{\prime\prime}}}{\operatorname{Tr}e^{-H^{\prime\prime}}}.$
(23)

Here, the witness is the hyperplane (line in 2-dimension space) through all
$T_{i}$s, as shown in Fig. 1.

We now summarize our algorithm in Algorithm 1 based on the loss function given
in Eq. (9).

Hermitian operator $A_{i}$s and expectation $a_{i}$.

A maximum entropy state $\rho^{\prime}=\frac{e^{-H}}{\operatorname{Tr}e^{-H}}$
satisfies $a_{i}=\operatorname{Tr}\rho^{\prime}A_{i}$

Randomly initialize coefficients $\vec{c}=\\{c_{i}\\}$.

$\rho\leftarrow\frac{e^{-H(\vec{c})}}{\operatorname{Tr}(e^{-H(\vec{c})})}$.

Compute the loss function $f(\vec{c})=\sum_{i}(a_{i}-\operatorname{Tr}\rho
A_{i})^{2}$.

while $f(\vec{c})$ has not converged do

Estimate the gradient $\nabla f(\vec{c})$ through finite differencing.

$\vec{c}=\vec{c}-\alpha\nabla f(\vec{c})$.

$\rho\leftarrow\frac{e^{-H(\vec{c})}}{\operatorname{Tr}(e^{-H(\vec{c})})}$.

Compute the loss function $f(\vec{c})$.

end while

$\rho^{\prime}\leftarrow\rho$.

Return $\vec{c}$, $f(\vec{c})$, $\rho^{\prime}$.

if $f(\vec{c})<=\epsilon$ then

desired $\rho^{\prime}$ is found

else if $\operatorname{rank}(\rho^{\prime})=1$ then

Witness is the hyperplane:
$\frac{x-a_{1}}{c_{1}}+\frac{y-a_{2}}{c_{2}}+\cdots=0$ (24)

else

Find all pure states of
$\rho^{\prime}$:$|\psi_{1}\rangle,|\psi_{2}\rangle,\cdots,|\psi_{n}\rangle$

Generate an equally mixed state
$\rho"=\frac{1}{n}\sum_{i=1}^{n}|\psi_{i}\rangle\langle\psi_{i}|,$ (25)

Find $H"$ and $\vec{c"}$ using the above method, so that
$\rho"=\frac{e^{-H"}}{\operatorname{Tr}e^{-H"}},$ (26)

Witness is the hyperplane:
$\frac{x-a_{1}}{c"_{1}}+\frac{y-a_{2}}{c"_{2}}+\cdots=0$ (27)

end if

Algorithm 1 The Maximum Entropy Method

##  III Results

We now apply our algorithm to study some concrete quantum state compatibility
problems. We start from a single case where $m=2$, i.e. only two measurement
operators $A_{1},A_{2}$. In this case, we are looking at the set of
$\\{a_{1},a_{2}\\}$. The compatible set is then given by the points
$\\{\operatorname{tr}(\rho A_{1}),\operatorname{tr}(\rho A_{2})\\}$, which is
known to be the same as the joint numerical range of $A_{1},A_{2}$ Bonsall and
Duncan (1971, 1973); Jonckheere  _et al._ (1998); Gutkin  _et al._ (2004);
Gutkin and Życzkowski (2013), given by

$W(A_{1},A_{2})=\\{(\langle x|A_{1}|x\rangle,\langle
x|A_{2}|x\rangle)||x\rangle\in\mathcal{H},\langle x|x\rangle=1\\}.$ (28)

In this case we can represent the Hamiltonian $H$ as

$H=\cos\theta A_{1}+\sin\theta A_{2}.$ (29)

The ground state of $H$ is $|g\rangle$, then the trajectory of

$(\langle g|A_{1}|g\rangle,\langle g|A_{2}|g\rangle)$ (30)

as $\theta$ goes from 0 to $2\pi$ forms the boundary of $W(A_{1},A_{2})$.

As examples, we now consider the case of two qubits, i.e. $n=2$. And we choose
$A_{1}=\sigma_{x}\otimes\sigma_{x}$ and $A_{1}=\sigma_{z}\otimes I_{2}$. For
any state $\rho$, Let $\operatorname{Tr}(\rho A_{1})$ be $x$-axis and
$\operatorname{Tr}(\rho A_{1})$ be $y$-axis. The boundary of $W(A_{1},A_{2})$
is a circle, whose center is $(0,0)$ and radius is $1$, as shown in Fig. 2.

Figure 2: Result of the random operators (a) The given point is inside the
numerical range. (b) The given point is outside the numerical range without
degeneracy. (c) The given point is outside the numerical range with
degeneracy.

Now, we have a state $\rho_{1}$ which satisfy $a_{1}=0.5$ and $a_{2}=0.5$.
Obviously, the point $S(0.5,0.5)$ is inside $W(A_{1},A_{2})$, as shown in Fig.
2(a). The optimization will give a Hamiltonian

$H_{1}^{\prime}=-2.4929A_{1}-2.4929A_{2},$ (31)

corresponding the maximum entropy state

$\rho_{1}^{\prime}=\begin{pmatrix}0.3750&0&0&0.1250\\\ 0&0.3750&0.1250&0\\\
0&0.1250&0.1250&0\\\ 0.1250&0&0&0.1250\end{pmatrix},$ (32)

and one can readily check that $\operatorname{Tr}(\rho_{1}^{\prime}A_{1})=0.5$
and $\operatorname{Tr}(\rho_{1}^{\prime}A_{2})=0.5$, which is shown as point
$P$ in Fig. 2(a). That is we found a maximum entropy state. You can see that
$P$ and $S$ are almost the same points. Also, the rank of $\rho_{1}^{\prime}$
is 4, which means $\rho_{1}^{\prime}$ is a mixed state. The eigenvalues of
$H_{1}^{\prime}$ are $\\{-0.8814,-0.8814,0.8814,0.8814\\}$, and the energy gap
between the ground state and the first excited states is $\thicksim 1.8$.

The state $\rho_{2}$, which satisfies $a_{1}=2$ and $a_{2}=2$, is obviously
outside $W(A_{1},A_{2})$, as point $S$ shown in Fig. 2(b). The optimization
will give a Hamiltonian $H_{2}^{\prime}=-28.7177A_{1}-28.7177A_{2}$ and the
corresponding maximum entropy state is

$\rho_{2}^{\prime}=\begin{pmatrix}0.4268&0&0&0.1768\\\ 0&0.4268&0.1768&0\\\
0&0.1768&0.0732&0\\\ 0.1768&0&0&0.0732\end{pmatrix},$ (33)

and one can readily check that
$\operatorname{Tr}(\rho_{2}^{\prime}A_{1})=0.7071$ and
$\operatorname{Tr}(\rho_{2}^{\prime}A_{2})=0.7071$, which is shown as point
$P$ in Fig. 2(a). One thing should be noticed here is that
$\operatorname{rank}{\rho_{2}^{\prime}}$ is 2, which means $\rho_{2}^{\prime}$
is a mixed state of two degenerate ground states. These two ground states has
the form:

$|\psi_{1}\rangle=\begin{pmatrix}0.9239\\\ 0\\\ 0\\\
0.3827\end{pmatrix},|\psi_{2}\rangle=\begin{pmatrix}0\\\ 0.9239\\\ 0.3827\\\
0\end{pmatrix}.$ (34)

and

$\rho_{2}^{\prime}=0.5|\psi_{1}\rangle\langle\psi_{1}|+0.5|\psi_{2}\rangle\langle\psi_{2}|$
(35)

However, both $|\psi_{1}\rangle$ and $|\psi_{2}\rangle$ satisfy:

$\displaystyle\langle\psi_{1}|A_{1}|\psi_{1}\rangle=\langle\psi_{2}|A_{1}|\psi_{2}\rangle=0.7071,$
(36)
$\displaystyle\langle\psi_{1}|A_{2}|\psi_{1}\rangle=\langle\psi_{2}|A_{2}|\psi_{2}\rangle=0.7071.$
(37)

Also, one can readily verify that
$\operatorname{Tr}(\rho_{2}^{\prime}A_{1})=0.7071$ and
$\operatorname{Tr}(\rho_{2}^{\prime}A_{1})=0.7071$. This is exactly the point
on $W(A_{1},A_{2})$ which has the smallest Euclidean distance to the point
$(2,2)$, as the point $P$ shown in Fig. 2(b). Therefore, although
$\rho_{2}^{\prime}$ is degenerate, it is already an equally superpositioned
state. Moreover, both the expectation of $|\psi_{1}\rangle$ and
$|\psi_{2}\rangle$ corresponding to the same point, the point $P$, as shown
inFig. 2(b). The four eigenvalues of $H_{2}^{\prime}$ is
$\\{-10.1532,-10.1532,10.1532,10.1532\\}$ and the energy gap between the
ground state and the first excited state is $\thicksim 20.3$, much greater
than 1. Compared with the energy gap of $H_{1}^{\prime}$ in Eq. 31, the energy
gap of $H_{2}^{\prime}$ is much greater, which guarantees that
$\rho_{2}^{\prime}$ is a mixed ground state.

With $H_{2}^{\prime}$, we can draw a line through the point of
$(0.7071,0.7071)$ with equation

$\frac{x-0.7071}{28.7177}+\frac{y-0.7071}{28.7177}=0.$ (38)

This line is a tangent line through point $(0.7071,0.7071)$. In Fig. 2(b), the
function Eq. 38 represents the blue tangent line of the circle $W$. Obviously,
the points of $W(A_{1},A_{2})$ and the point $(2,2)$ lie at different sides of
the line. Therefore, this line is a witness.

Now let us look at a case with flat boundary (corresponding to the degenerate
ground states of some $H$). Suppose we have $3$ qubits and the subsystems are
$\\{1,2\\}$ and $\\{2,3\\}$. Choose $A_{1}$ and $A_{2}$ random Hermitian
operators acting on subsystem $\\{1,2\\}$ and $\\{2,3\\}$, respectively. For
Hamiltonian

$H=\sin\theta A_{1}+\cos\theta A_{2},$ (39)

the locality would guarantee that the ground state would be degenerate for
some certain $\theta$, as shown in Fig. 2(c). Unlike the circle shown in Fig.
2(b), some parts of the boundary are flat, meaning that the ground state here
is degenerate. Now, suppose we have a point $S(0,1.5)$, which is obviously
outside $W$. The nearest point $P$, given by the maximum entropy state is two-
degenerate: The rank of the density matrix $\rho_{P}$ corresponding $P$ is 2,
which means there are two components in mixed state $\rho_{P}$:

$\rho_{P}=p_{1}|\psi_{1}\rangle\langle\psi_{1}|+p_{2}|\psi_{2}\rangle\langle\psi_{2}|,$
(40)

where $|\psi_{1}\rangle$ and $|\psi_{2}\rangle$ corresponds to point $T_{1}$
and $T_{2}$, respectively. Using the maximum entropy method again, we can find
$T_{1}$ and $T_{2}$, as shown in Fig. 2(c). Draw a line through $T_{1}$ and
$T_{2}$, this line is obviously a witness.

##  IV The quantum marginal problem

In this section, we apply our method to study a special case of quantum
compatibility problem, namely the quantum marginal problem. The quantum
marginal problem asks whether there exists a global quantum state $\rho_{N}$,
whose reduced density matrices
$\tilde{\rho}_{S_{i}}=\operatorname{Tr}_{(S_{i})^{c}}\rho_{N}$ on subsystem
$S_{i}\subset\\{1,2,\cdots,N\\}$Klyachko (2006) coincide with given
set$\\{\rho_{S_{i}}\\}$. The related problem in fermionic (bosonic) systems is
the so-called $N$-representability problem Coleman (1963). It is worth
mentioning that, the quantum marginal problem and the $N$-representative
problem have been shown to be in the complexity class QMA, even for the
relatively simple case in which all the given marginals are two-particle
states Liu (2006); Liu  _et al._ (2007); Wei  _et al._ (2010). The MaxExt then
states that there exists a compatible $\rho_{N}$ if and only if there exists
the $\bar{\rho}_{N}$ which has the maximum entropy among all the states that
are compatible with the RDMs.

To be more precise, the quantum marginal problem could be defined as follows.
Given a set of density matrices on several $2$-local systems $S_{j}$, such as
$\\{1,2\\},\\{2,3\\},\\{3,4\\}\cdots$, which are $\rho_{S_{j}}$ respectively.
Quantum marginal problem asks whether these density matrices are compatible,
i.e. whether there exists a global state $\rho_{N}$, such that,

$\operatorname{Tr}_{(S_{j})^{c}}\rho_{N}=\rho_{S_{j}},\qquad\forall
S_{j}\subset\\{1,2,\cdots,N\\}.$ (41)

Denote $\operatorname{Tr}_{(S_{j})^{c}}\rho_{N}$ by $\rho^{\prime}_{S_{j}}$.
If all $A_{i}$s in Eq. 8 forms a complete basis of Hermitian operator on all
$S_{j}$s, then Eq. 41 is established if and only if

$\operatorname{Tr}(A_{i}\rho^{\prime}_{S_{j}})=\operatorname{Tr}(A_{i}\rho_{S_{j}})=a_{i},\,\forall
A_{i}s.$ (42)

Therefore, the problem is transformed into the compatibility problem: for all
$A_{i}$s, find a maximum entropy state

$\rho^{\prime}=\frac{e^{-H}}{\operatorname{Tr}e^{-H}},$ (43)

where

$H^{\prime}=\sum c_{i}A_{i},$ (44)

so that

$\operatorname{Tr}(A_{i}\rho^{\prime}_{S_{j}})=\operatorname{Tr}(A_{i}\rho_{S_{j}})=a_{i},\,\forall
A_{i}s.$ (45)

The maximum entropy method could be directly applied.

For quantum marginal problem, if the RDMs are not compatible, the final
Hamiltonian $H^{\prime}$ is a witness. Let $E_{g}$ be the lowest eigenvalue of
$H^{\prime}$, the ground state energy. For any global state $\rho$, we have

$\operatorname{Tr}(\rho H^{\prime})\geq E_{g}.$ (46)

If the given RDMs $\\{\rho_{S_{i}}\\}$s are not compatible, then the
Hamiltonian $H^{\prime}$ obtained should satisfy

$\sum_{i}H^{\prime}_{i}\rho_{S_{i}}<E_{g}.$ (47)

We now consider some examples, starting from $n=3$.

###  IV.1 3-qubits

It is known that for 3 qubit states, nearly all pure states are uniquely
determined by their 2-reduced density matrices among all states except the GHZ
type states $a|000\rangle+b|111\rangle$ Linden  _et al._ (2002).

First, let us look at the $W$-state:

$|\psi_{W}\rangle=\frac{1}{\sqrt{3}}(|100\rangle+|010\rangle+|001\rangle$ (48)

By using the maximum entropy method, we will recover the result:

$\rho^{\prime}_{W}=\frac{e^{-H^{\prime}_{W}}}{\operatorname{Tr}e^{-H^{\prime}_{W}}}=\begin{pmatrix}0&0&0&0&0&0&0&0\\\
0&0.3333&0.3333&0&0.3333&0&0&0\\\ 0&0.3333&0.3333&0&0.3333&0&0&0\\\
0&0&0&0&0&0&0&0\\\ 0&0.3333&0.3333&0&0.3333&0&0&0\\\ 0&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0\end{pmatrix},$ (49)

which is the density matrix of $W$-state. The lowest two eigenvalues of the
corresponding Hamiltonian $H^{\prime}_{W}$ are $-28.8105$ and $-18.3106$,
which guarantees that the rank of the density matrix $\rho^{\prime}_{W}$ is 1,
hence a pure ground state.

Now, suppose we have the GHZ-state:

$|\psi_{\text{GHZ}}\rangle=\frac{1}{\sqrt{2}}(|000\rangle+|111\rangle).$ (50)

This time, the maximum entropy state will give a density matrix

$\rho^{\prime}_{\text{GHZ}}=0.5(|000\rangle\langle 000|+|111\rangle\langle
111|).$ (51)

It can be easily verified that $\rho^{\prime}_{\text{GHZ}}$ and
$|\psi_{\text{GHZ}}\rangle\langle\psi_{\text{GHZ}}|$ have the same 2-body
reduced density matrix. Obviously, the populations of the two components -
$|000\rangle$ and $|111\rangle$ \- in state $\rho^{\prime}_{\text{GHZ}}$ are
both 0.5. For a maximum entropy state, or a thermal state, the only
possibility that two components have the same population is that these two
components are degenerate. Let the corresponding Hamiltonian of
$\rho^{\prime}_{\text{GHZ}}$ be $H^{\prime}_{\text{GHZ}}$, we have

$\displaystyle H^{\prime}_{\text{GHZ}}|000\rangle$
$\displaystyle=-32.2662|000\rangle,$ (52) $\displaystyle
H^{\prime}_{\text{GHZ}}|111\rangle$ $\displaystyle=-32.2662|111\rangle.$ (53)

Both $|000\rangle$ and $|111\rangle$ have the same eigen-energy, which means
they are both ground state of the Hamiltonian $H^{\prime}_{\text{GHZ}}$.

###  IV.2 4-qubit chain

Now suppose we have a system with 4 qubits labelled as 1,2,3,4, and the
subsystems are

$\displaystyle S_{1}$ $\displaystyle=\\{1,2\\};$ (54) $\displaystyle S_{2}$
$\displaystyle=\\{2,3\\};$ (55) $\displaystyle S_{3}$
$\displaystyle=\\{3,4\\},$ (56)

Now the problem is that given the reduced density matrix $\rho_{12},\rho_{23}$
and $\rho_{34}$, determine whether there exist a global density matrix, of
which $\rho_{12},\rho_{23}$ and $\rho_{34}$ are the reduced density matrices.

Suppose we have a bell state

$|\psi_{B}\rangle=\frac{1}{\sqrt{2}}(|00\rangle+|11\rangle),$ (57)

with the density matrix

$\rho_{B}=|\psi_{B}\rangle\langle\psi_{B}|.$ (58)

Suppose $\rho_{12}=\rho_{23}=\rho_{34}=\rho_{B}$. Because of entanglement
monogamy, we know that there does NOT exist a global $\rho$, of which the
reduced density matrices are $\rho_{12},\rho_{23}$ and $\rho_{34}$.

Our maximum entropy method gives a density matrix
$\rho^{\prime}=e^{-H^{\prime}}/\operatorname{Tr}(e^{-H^{\prime}})$, of which
the rank is 1. This means $\rho^{\prime}$ is a pure state. It can be verified
that $\rho^{\prime}$ is the eigenstate with the lowest eigen-energy of
$H^{\prime}$ \- the ground state. The lowest energy of $H^{\prime}$ is
$-81.1733$. Therefore, for any global state $\rho$, we have

$\operatorname{Tr}(\rho H^{\prime})\geq-81.1733.$ (59)

Also, we know that $H^{\prime}$ is a local Hamiltonian:

$H^{\prime}=H^{\prime g}_{12}+H^{\prime g}_{23}+H^{\prime g}_{34},$ (60)

where the superscript $g$ means the local interaction Hamiltonian in global
form. Now, we can verify that

$\operatorname{Tr}(\rho_{12}H^{\prime}_{12})+\operatorname{Tr}(\rho_{23}H^{\prime}_{23})+\operatorname{Tr}(\rho_{34}H^{\prime}_{34})=-121.0888,$
(61)

which is smaller than the ground state energy of $H^{\prime}$. Therefore,
$H^{\prime}$ is a witness.

The above case gives a very example that $H^{\prime}$ is not degenerate.
However, if you randomly choose $\rho_{12},\rho_{23}$ and $\rho_{34}$, the
solution is mostly likely to be degenerate. For example, we randomly generate
the three reduced density matrices as follows

$\displaystyle\rho_{12}$
$\displaystyle=\begin{pmatrix}0.2408&0.1717-0.1312i&-0.1304-0.0459i&0.0306-0.0962i\\\
0.1717+0.1312i&0.3359&-0.0536-0.1599i&0.1028-0.0836i\\\
-0.1304+0.0459i&-0.0536+0.1599i&0.3336&0.0695+0.1169i\\\
0.0306+0.0962i&0.1028+0.0836i&0.0695-0.1169i&0.0896+0i\end{pmatrix};$ (62)
$\displaystyle\rho_{23}$
$\displaystyle=\begin{pmatrix}0.2534&-0.0662+0.0248i&0.1068-0.0440i&0.0746-0.1027i\\\
-0.0662-0.0248i&0.2168&0.0307-0.0383i&-0.1085+0.2051i\\\
0.1068+0.0440i&0.0307+0.0383i&0.2114&0.0309-0.0164i\\\
0.0746+0.1027i&-0.1085-0.2051i&0.0309+0.0164i&0.3183\end{pmatrix};$ (63)
$\displaystyle\rho_{34}$
$\displaystyle=\begin{pmatrix}0.1175&-0.0402-0.0453i&-0.0265+0.0143i&0.0747+0.0688i\\\
-0.0402+0.0453i&0.5037&-0.1095+0.0187i&-0.0620-0.1296i\\\
-0.0265-0.0143i&-0.1095-0.0187i&0.0883&-0.0247-0.0074i\\\
0.0747-0.0688i&-0.0620+0.1296i&-0.0247+0.0074i&0.2905\end{pmatrix}.$ (64)

The maximum entropy method will give a $\rho^{\prime}$, of which the rank is
8, meaning that $\rho$ is a mixture of 8 pure states. As mentioned earlier,
for states outside $\mathcal{B}$ shown in Fig. 1, the nearest points is on the
boundary, which is composed of ground states. Therefore, the 8 pure states are
all ground states of some Hamiltonian $H^{\prime}$, and $H^{\prime}$ is
degenerate: all the 8 pure states are ground states and have the same energy.

To find the Hamiltonian $H^{\prime}$, we generate a new density matrix:

$\rho_{g}=\sum_{i}\frac{1}{n}|\psi_{i}\rangle\langle\psi_{i}|,$ (65)

where $n$ is the degeneracy and $|\psi_{i}\rangle$ is the $i$-th ground states
in $\rho^{\prime\prime}$. By using the maximum entropy method, we can find the
Hamiltonian $H^{\prime\prime}$, of which

$\rho^{\prime\prime}=\frac{e^{-H^{\prime\prime}}}{\operatorname{Tr}e^{-H^{\prime\prime}}}.$
(66)

In our example, the lowest eigen-energy of $H^{\prime\prime}$ is around -7.2:
because of numerical precision, there are slightly difference between these
energies. The first exited states have the energy of 1.7, which makes the gap
between the ground state be about 8.9. This gap makes the state
$\rho^{\prime\prime}$ a ground state.

Similarly, we could calculate the energy of
$\\{\rho_{12},\rho_{23},\rho_{34}\\}$ based on $H"$:

$\operatorname{Tr}(\rho_{12}H^{\prime\prime}_{12}+\operatorname{Tr}(\rho_{23}H^{\prime\prime}_{23})+\operatorname{Tr}(\rho_{34}H^{\prime\prime}_{34})=-7.5,$
(67)

which is smaller than the ground state energy of $H^{\prime\prime}$.
Therefore, $H^{\prime\prime}$ is a witness.

##  V The hybrid quantum-classical algorithm

In our algorithm, we need to repeatedly calculate density matrices of the form

$\rho=\frac{e^{-H}}{\operatorname{Tr}(e^{-H})}.$ (68)

Calculation of the exponential of a matrix using a classical computer is a
difficult task, because it is not scalable. To address this issue, we will
turn to some hybrid quantum-classical algorithm. Recently, there have been
some proposed hybrid quantum-classical algorithms Zeng  _et al._ (2021);
Verdon  _et al._ (2019); Wu and Hsieh (2019); Wang  _et al._ (2021); Chowdhury
_et al._ ; Zoufal  _et al._ (2021) to prepare the thermal state
$\rho_{\beta}=\frac{e^{-\beta H}}{\operatorname{Tr}(e^{-\beta H})}$. All of
these can be used to help scale our method with the existence of near-term
quantum computers.

As an example, we have used the method in Ref. Zeng  _et al._ (2021) to
calculate the thermal state density matrix, which can take advantage of the
small correlation length of the system in small $\beta$. When setting
$\beta=1$, we can obtain the density matrix in Eq. 68. To prepare the thermal
state $\rho_{\beta}$, one can apply an imaginary time evolution with a
thermofield double state. We first prepare initial state
$|\phi_{0}\rangle=\frac{1}{\sqrt{2^{n}}}\sum_{i=1}^{2^{n}}|i\rangle|i\rangle$
with $2n$ qubits. The imaginary time evolution $e^{-\beta H/2}$ on
$|\phi_{0}\rangle$ returns the state
$|\text{TFD}(\beta)\rangle=\sqrt{\frac{2^{n}}{\operatorname{Tr}(e^{-\beta
H})}}e^{-\beta H/2}|\phi_{0}\rangle$ and $\rho_{\beta}$ will be then obtained
on the first $n$ qubits by tracing out the last $n$ qubits,
$\rho_{\beta}=\operatorname{Tr}_{n+1,\ldots,2n}|\text{TFD}(\beta)\rangle\langle\text{TFD}(\beta)|$.

To implement $e^{-\beta H/2}$ on a quantum computer, where $H=\sum_{m}h[m]$
acts nontrivially only on at most $K$ neighbouring qubits, one could use a
quantum imaginary time evolution (QITE) algorithm, which is NISQ friendly
Motta  _et al._ (2020); Cao  _et al._ (2022b). The basic idea of QITE is to
approximate the $K$-local non-unitary transformation with a $D$-local unitary
transformation in each step after the Trotter decomposition,

$\displaystyle|{\psi}^{\prime}\rangle=\frac{1}{\sqrt{c}}e^{-\Delta\tau
h[m]}|\psi\rangle\approx e^{-i\Delta\tau A[m]}|\psi\rangle,$ (69)

where $c=\langle\psi|e^{-2\Delta\tau h[m]}|\psi\rangle$ is the normalization
factor. Each $h[m]$ is a $K$-local operator and $A[m]$ is Hermitian, which can
be expanded in terms of Pauli basis on $D$ qubits,

$\displaystyle
A[m]=\sum_{i_{1}i_{2}...i_{D}}a[m]_{i_{1}i_{2}...i_{D}}\sigma_{i_{1}}\sigma_{i_{1}}...\sigma_{i_{D}}=\sum_{I}a[m]_{I}\sigma_{I},$
(70)

where $a[m]_{I}$ is the coefficient of combining Pauli operator $\sigma_{I}$
and the index $I$ is a combination of qubit indexes
$\\{i_{1},i_{2},...,i_{D}\\}$ corresponding to nontrivial Pauli operators. To
find coefficients $a[m]_{I}$ of $A[m]$, we minimize the square norm of the
difference between the two quantum states

$\displaystyle\parallel|{\psi}^{\prime}\rangle-(1-i\Delta\tau
A[m])|{\psi}\rangle\parallel^{2},$ (71)

where the $(1-i\Delta\tau A[m])|{\psi}\rangle$ is the first two terms of the
taylor series of $e^{-i\Delta\tau A[m]}|\psi\rangle$. The solution of the
minimization is subject to the linear equation,

$\displaystyle(\bm{S}+\bm{S}^{T})\bm{a}[m]=-\bm{b},$ (72)

where the matrix $\bm{S}$ and vector $\bm{b}$ can be obtained by $D$ local
measurements on the quantum state $|{\psi}\rangle$,

$\displaystyle S_{IJ}$ $\displaystyle=$
$\displaystyle\langle\psi|\sigma_{I}^{\dagger}\sigma_{J}|\psi\rangle,$
$\displaystyle b_{I}$ $\displaystyle=$
$\displaystyle-2\text{Im}\left[\frac{1}{\sqrt{c}}\langle\psi|\sigma_{I}^{\dagger}h[m]|\psi\rangle\right],$
(73)

where Im[] denotes the imaginary part of the variable inside. We can obtain
the $\bm{a}[m]$ by solving the linear equation on a classical computer, and
then construct a quantum circuit to implement the unitary transformation
$e^{-i\Delta\tau A[m]}|\psi\rangle$ on an NISQ quantum devices.

The locality of $D$ qubits local unitary transformation is important. The
unitary local operator $A[m]$ should at least act nontrivially on the
$(i,i+n)$ qubit pair if its counterpart $h[m]$ act nontrivially on $i$-th
qubit Zeng  _et al._ (2021). Given a $K$-local Hamiltonian, the QITE algorithm
is capable of capturing the correlation of the original Hamiltonian only if
$D\geq 2K$. In our case it is a $2$-local Hamiltonian, so we set $D=4$. The
trotter step size $\Delta\tau$ can also affect the accuracy of the thermal
state density matrix, therefore we here choose a small value
$\Delta\tau=0.05$. Then the number of Trotter steps is
$N=\frac{\beta/2}{\Delta\tau}=10$. As described in Ref. Zeng  _et al._ (2021)
the time complexity of the QITE and the depth of the circuit construct by the
QITE algorithm depend on $N\times\mathcal{O}(e^{D})$. The density matrix in
Eq. 68 thus can be calculated efficiently on NISQ devices.

In previous section we showed the results in the 3-qubit case, where for GHZ
state, the variation method returns the density matrix

$\rho_{\text{GHZ}}=0.5(|000\rangle\langle 000|+|111\rangle\langle 111|).$ (74)

and for W state, it returns exactly the target density matrix, which means
that our variational method works as desired.

##  VI Conclusion

In this work, we applied the maximum entropy method to solve the quantum state
compatibility problem. Compared to the traditional SDP method, there are
several advantages of the method. At first, only a small number of parameters
(the same as the number of measurement operators) are required to parametrize
to obtain the desired density matrix. Secondly, the hybrid quantum-classical
algorithms replace some subroutines with help of quantum devices may give
further advantages with the incoming NISQ era. Thirdly, in case the solution
does not exist, our method will further produce a witness which is the
supporting hyperplane of the compatible set; making use of these ”byproducts”
might be able to provide us some insight in further understanding about the
global features of the compatible sets in future studies.

##  VII Acknowledgement

S-Y.Hou is supported by National Natural Science Foundation of China under
Grant No. 12105195. YN-Li is supported by National Natural Science Foundation
of China under Grant No. 12005295. Z-P. Wu, C.-F. Cao, B.Zeng. are supported
by GRF16305121.

## References

  * Pathria (2016) R. K. Pathria,  _Statistical mechanics_ (Elsevier, 2016).
  * Boltzmann (1877) L. Boltzmann,  _Über die Beziehung zwischen dem zweiten Hauptsatze des mechanischen Wärmetheorie und der Wahrscheinlichkeitsrechnung, respective den Sätzen über das Wärmegleichgewicht_ (Kk Hof-und Staatsdruckerei, 1877).
  * Shannon (1948) C. E. Shannon, The Bell system technical journal 27, 379 (1948).
  * Jaynes (1957a) E. T. Jaynes, Physical review 106, 620 (1957a).
  * Jaynes (1957b) E. T. Jaynes, Physical review 108, 171 (1957b).
  * Csiszár  _et al._ (2004) I. Csiszár, P. C. Shields, _et al._ , Foundations and Trends® in Communications and Information Theory 1, 417 (2004).
  * Palmieri and Ciuonzo (2013) F. A. Palmieri and D. Ciuonzo, Information Fusion 14, 186 (2013).
  * Botev and Kroese (2011) Z. I. Botev and D. P. Kroese, Methodology and Computing in Applied Probability 13, 1 (2011).
  * Ratnaparkhi (1998) A. Ratnaparkhi,  _Maximum entropy models for natural language ambiguity resolution_ (University of Pennsylvania, 1998).
  * Malouf (2002) R. Malouf, in  _COLING-02: The 6th Conference on Natural Language Learning 2002 (CoNLL-2002)_ (2002).
  * Holevo (1973) A. S. Holevo, Problemy Peredachi Informatsii 9, 3 (1973).
  * Bužek  _et al._ (1998) V. Bužek, R. Derka, G. Adam, and P. Knight, Annals of Physics 266, 454 (1998).
  * Řeháček  _et al._ (2001) J. Řeháček, Z. Hradil, and M. Ježek, Physical Review A 63, 040303 (2001).
  * Zhou (2008) D. Zhou, Physical review letters 101, 180505 (2008).
  * Niekamp  _et al._ (2013) S. Niekamp, T. Galla, M. Kleinmann, and O. Gühne, Journal of Physics A: Mathematical and Theoretical 46, 125301 (2013).
  * Cao  _et al._ (2020) C. Cao, S.-Y. Hou, N. Cao, and B. Zeng, Journal of Physics: Condensed Matter 33, 064002 (2020).
  * Zuo  _et al._ (2022) Y. Zuo, C. Cao, N. Cao, X. Lai, B. Zeng, and S. Du, Advanced Photonics 4, 1 (2022).
  * Amin  _et al._ (2018) M. H. Amin, E. Andriyash, J. Rolfe, B. Kulchytskyy, and R. Melko, Physical Review X 8, 021050 (2018).
  * Preskill (2018) J. Preskill, Quantum 2, 79 (2018).
  * Zoufal  _et al._ (2021) C. Zoufal, A. Lucchi, and S. Woerner, Quantum Machine Intelligence 3, 1 (2021).
  * Shingu  _et al._ (2021) Y. Shingu, Y. Seki, S. Watabe, S. Endo, Y. Matsuzaki, S. Kawabata, T. Nikuni, and H. Hakoshima, Physical Review A 104, 032413 (2021).
  * Cao  _et al._ (2022a) C. Cao, C. Zhang, Z. Wu, M. Grassl, and B. Zeng, arXiv preprint arXiv:2204.03560 (2022a).
  * Zeng  _et al._ (2021) J. Zeng, C. Cao, C. Zhang, P. Xu, and B. Zeng, Quantum Science and Technology 6, 045009 (2021).
  * Verdon  _et al._ (2019) G. Verdon, J. Marks, S. Nanda, S. Leichenauer, and J. Hidary, arXiv:1910.02071 (2019).
  * Wu and Hsieh (2019) J. Wu and T. H. Hsieh, Phys. Rev. Lett. 123, 220502 (2019).
  * Wang  _et al._ (2021) Y. Wang, G. Li, and X. Wang, Physical Review Applied 16, 054035 (2021).
  * (27) A. N. Chowdhury, G. Hao Low, and N. Wiebe, arXiv:2002.00055 .
  * BROYDEN (1970) C. G. BROYDEN, IMA Journal of Applied Mathematics 6, 76 (1970).
  * Fletcher (1970) R. Fletcher, The Computer Journal 13, 317 (1970).
  * Goldfarb (1970) D. Goldfarb, Mathematics of Computing 24, 23 (1970).
  * Shanno (1970) D. F. Shanno, Mathematics of Computing 24, 647 (1970).
  * Hestenes and Stiefel (1952) M. R. Hestenes and E. Stiefel, Journal of research of the National Bureau of Standards 49, 409 (1952).
  * Salazar  _et al._ (2012) R. Salazar, D. Goyeneche, A. Delgado, and C. Saavedra, Physics Letters A 376, 325 (2012).
  * Bonsall and Duncan (1971) F. F. Bonsall and J. Duncan,  _Numerical Ranges of Operators on Normed Spaces and of Elements of Normed Algebras_, London Mathematical Society Lecture Note Series (Cambridge University Press, 1971).
  * Bonsall and Duncan (1973) F. F. Bonsall and J. Duncan,  _Numerical ranges II_ , Vol. 10 (Cambridge university press, 1973).
  * Jonckheere  _et al._ (1998) E. A. Jonckheere, F. Ahmad, and E. Gutkin, Linear Algebra and its Applications 279, 227 (1998).
  * Gutkin  _et al._ (2004) E. Gutkin, E. A. Jonckheere, and M. Karow, Linear Algebra and its Applications 376, 143 (2004).
  * Gutkin and Życzkowski (2013) E. Gutkin and K. Życzkowski, Linear Algebra and its Applications 438, 2394 (2013).
  * Klyachko (2006) A. A. Klyachko, in  _Journal of Physics: Conference Series_ , Vol. 36 (IOP Publishing, 2006) p. 014.
  * Coleman (1963) A. J. Coleman, Reviews of modern Physics 35, 668 (1963).
  * Liu (2006) Y.-K. Liu, in  _Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques_ , Lecture Notes in Computer Science, Vol. 4110, edited by J. Diaz, K. Jansen, J. D. Rolim, and U. Zwick (Springer Berlin Heidelberg, 2006) pp. 438–449.
  * Liu  _et al._ (2007) Y.-K. Liu, M. Christandl, and F. Verstraete, Phys. Rev. Lett. 98, 110503 (2007).
  * Wei  _et al._ (2010) T.-C. Wei, M. Mosca, and A. Nayak, Phys. Rev. Lett. 104, 040501 (2010).
  * Linden  _et al._ (2002) N. Linden, S. Popescu, and W. Wootters, Physical review letters 89, 207901 (2002).
  * Motta  _et al._ (2020) M. Motta, C. Sun, A. T. Tan, M. J. O’Rourke, E. Ye, A. J. Minnich, F. G. Brandão, and G. K.-L. Chan, Nature Physics 16, 205 (2020).
  * Cao  _et al._ (2022b) C. Cao, Z. An, S.-Y. Hou, D. Zhou, and B. Zeng, Communications Physics 5, 1 (2022b).