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# Distinguishing Ion Dynamics from Muon diffusion in Muon Spin Relaxation

Takashi U. Ito  Advanced Science Research Center, Japan Atomic Energy Agency,
Tokai-mura, Naka-gun, Ibaraki 319-1195, Japan  Ryosuke Kadono  Muon Science
Laboratory, Institute of Materials Structure Science, High Energy Accelerator
Research Organization (KEK), Tsukuba, Ibaraki 305-0801, Japan

###### Abstract

We propose a model to describe the fluctuations in the internal magnetic field
due to ion dynamics observed in the muon spin relaxation ($\mu$SR) by an
Edwards-Anderson type autocorrelation function that separates the quasi-static
and dynamic components of the correlation by a parameter $Q$ (where $0\leq
Q\leq 1$). Our Monte Carlo simulations for this model showed that the time
evolution of muon spin polarization deviates significantly from the Kubo-
Toyabe (KT) function. To further validate the model, the results of
simulations were compared with the $\mu$SR spectra observed in a hybrid
organic-inorganic perovskite FAPbI3 [with FA referring to HC(NH${}_{2})_{2}$],
where local field fluctuations associated with the rotational motion of FA
molecules and quasi-static fields from the PbI3 lattice are presumed to
coexist. The least-squares curve fitting showed reasonable agreement with the
model with $Q=0.947(3)$, and the fluctuation frequency of the dynamical
component was obtained. This result opens the door to the possibility of
experimentally distinguishing fluctuations due to dynamics of ions around
muons from those due to self-diffusion of muons. Meanwhile, it suggests the
need to carefully consider the spin relaxation function when applying $\mu$SR
to the issue of ion dynamics.

##  I Introduction

Muon spin rotation ($\mu$SR) is an experimental method to probe magnetic
fields in matter, where spin-polarized muons ($\mu^{+}$) stopped in the sample
directly infer the magnitude of local field at the interstitial site(s) via
the frequency of their Larmor precession See  _et al._ (2021). The muon
gyromagnetic ratio $\gamma_{\mu}$ ($=2\pi\times 135.539$ MHz/T) is 3.18 times
greater than that of protons ($=2\pi\times 42.577$ MHz/T, which is the largest
among all the nuclei of stable elements) and thus it is the most sensitive
probe of the internal magnetic field upon their implantation to materials.
Muons are provided as a 100%-spin-polarized beam by proton accelerator
facilities, which enables $\mu$SR measurements in zero magnetic field. In
addition, $\mu$SR has various advantages: the muon implantation energy is high
enough ($\geq 4$ MeV, corresponding to the stopping range $\geq 0.1$ g/cm2) to
be surface-independent (bulk-sensitive), and the implanted muons (volume
concentration $\sim$$10^{5}$ cm-3 for the high-flux beams at J-PARC MLF) decay
to positrons and neutrinos with an average lifetime of 2.198 $\mu$s, so they
do not accumulate in the sample unlike other ion beam irradiation methods.
Most notably, the spatial distribution of the emitted high-energy decay
positrons has a large asymmetry (1/3 upon averaged over the positron energy)
with respect to the muon spin polarization, and the time evolution of the spin
polarization can be observed by measuring the time-dependent asymmetry (which
is called “$\mu$SR time spectrum”).

Recently, attempts have been made to observe the diffusive motion of ions in
battery materials and ionic conductors by $\mu$SR measurements under zero or
longitudinal magnetic field (ZF/LF-$\mu$SR) Sugiyama  _et al._ (2009), where
the time-dependent fluctuations of the weak random local field ${\bm{H}}(t)$
($\sim$10-4 T) exerted from the magnetic dipole moments of cation nuclei is
monitored via the spin relaxation of muons implanted into the materials of
interest. In these studies, it is often an issue whether the observed
fluctuations are caused by cation diffusion or muon self-diffusion, but it was
believed that the distinction could not be made solely from the $\mu$SR
spectra. Therefore, the attribution of the origin of the fluctuations has been
based on information from other experimental methods and, more recently, on
inferences from ab initio density functional theory calculations.

Figure 1: A schematic illustration of time-dependent fluctuation of local
field ${\bm{H}}(t)$ at muon sites for (a) muon self-diffusion and (b) ion
diffusion, where ions are located at corners on a square lattice. The
difference in the dipolar field from each ion (with nuclear magnetic moment
orientation shown by an arrow) are visualized by different colors. While their
combination at muon site changes randomly with muon diffusion in (a), the
contributions from two ions at $k$ and $l$ sites remain unchanged while those
at $i$ and $j$ sites migrate (b). (c) Structure of formamidinium lead iodate
[FAPbI3, with FA denoting HC(NH2)2] in the tetragonal phase, and associated
muon sites suggested from first-principles calculations (yellow-hatched
areas). Internal field fluctuations associated with the rotational motion of
FA molecules and static internal field from the PbI3 lattice are presumed to
coexist. (d) A schematic illustration of local fields consisting of static and
fluctuating components [see Eq. (12) in the main text].

The muon spin relaxation induced by ${\bm{H}(t)}$ under ZF/LF is considered to
be well approximated by the Kubo-Toyabe (KT) function Hayano  _et al._ (1979),
and it has been routinely used in the analysis of $\mu$SR time spectra by
curve fitting based on the least-squares method. In deriving the KT function,
the effect of the fluctuations is incorporated by a strong-collision model (an
approximation of the Markovian process), where ${\bm{H}}(t)$ is assumed to
have no correlation before and after a change. As illustrated in Fig. 1(a),
this approximation is considered to hold well in the case of jump diffusion of
muons, because the relative configuration of all nuclear magnetic moments is
swapped simultaneously in a single jump. On the other hand, in the case of ion
diffusion around muons, it is naturally expected to be a rare event that all
ions move simultaneously (especially for slow diffusion), resulting in a
strong autocorrelation in ${\bm{H}}(t)$ before and after the change [see Fig.
1(b)]. This is even more so when multiple kinds of nuclei contribute to the
internal magnetic field (e.g., 7Li and 59Co in LixCoO2) while only some of
them exhibit diffusion. Nevertheless, the KT function has traditionally been
used in the study of ion diffusion.

A similar situation has been observed in very recent $\mu$SR studies of hybrid
organic-inorganic perovskites (HOIPs) Koda  _et al._ (2022); Hiraishi  _et
al._ (2023), where thermally-activated local rotational motion of organic
cation molecules contributes to the fluctuation of ${\bm{H}}(t)$ while the
internal fields from PbI3 lattice remains quasi-static [see Fig. 1(c)].
Moreover, it has been suggested that the observed $\mu$SR time spectra may not
always be satisfactorily reproduced by the KT function upon the onset of
molecular motion. These circumstances led us to examine the corresponding
longitudinal spin relaxation function for such cases.

In this paper, we propose a model in which the fluctuations of ${\bm{H}}(t)$
are described by an Edwards-Anderson type autocorrelation function with a
parameter $Q$ to part the quasi-static and dynamical components in the
correlation. Our Monte Carlo simulations on the time evolution of muon spin
polarization for this model indicate that the relaxation lineshape could be
significantly different from that of the KT function. Such difference will
open the door to the possibility of experimentally distinguishing between muon
self-diffusion and cation dynamics solely from $\mu$SR experiment. We show
that the origin of the ${\bm{H}}(t)$ fluctuations can indeed be inferred by
comparing relaxation functions based on this model with high-precision $\mu$SR
time spectra observed in formamidinium lead iodide [FAPbI3, where FA denotes
HC(NH${}_{2})_{2}$] Hiraishi  _et al._ (2023). Furthermore, based on this
result, we will discuss the problems with the preceding studies of ion
diffusion by $\mu$SR.

##  II Model

###  II.1 Relaxation function for the static internal field

Since our goal is to investigate how different autocorrelations of
${\bm{H}}(t)$ can change in the spin relaxation function, we will introduce
the KT function in a simple classical way in the following. When muons are
initially polarized along the $z$ direction, and subjected to a homogeneous
magnetic field ${\bm{H}}=(H_{x},H_{y},H_{z})$, the time evolution of muon
polarization is described by

$\displaystyle\sigma_{z}(t)$ $\displaystyle=$
$\displaystyle\frac{H_{z}^{2}}{H^{2}}+\frac{H_{x}^{2}+H_{y}^{2}}{H^{2}}\cos(\gamma_{\mu}Ht)$
(1) $\displaystyle=$
$\displaystyle\cos^{2}\theta+\sin^{2}\theta\cos(\gamma_{\mu}Ht),$ (2)

where $H=|{\bm{H}}|$. Such single-frequency rotation is observed when muons
are subjected to a uniform external magnetic field (not parallel to $z$). In
the case of random local fields exerted from nuclear magnetic moments, $H$
varies with the muon position ${\bm{r}}_{\mu}$ so that $\mu$SR spectrum
corresponds to a random sampling of
${\bm{H}}({\bm{r}}_{\mu})=(H_{x}({\bm{r}}_{\mu}),H_{y}({\bm{r}}_{\mu}),H_{z}({\bm{r}}_{\mu}))$
which is described by a density distribution,

${\bm{n}}({\bm{H}})=\int\delta({\bm{H}}-{\bm{H}}({\bm{r}}_{\mu}))d{\bm{r}}_{\mu}.$
(3)

The time evolution of spin polarization projected to the $z$ axis is then
given by

$g_{z}(t)=\int\sigma_{z}(t){\bm{n}}({\bm{H}})d{\bm{H}}=\iiint_{-\infty}^{\infty}\sigma_{z}(t)\Pi_{\alpha}n_{\alpha}(H_{\alpha})dH_{\alpha},$
(4)

where $\alpha=x$, $y$, and $z$. Unless ${\bm{n}}({\bm{H}})$ is a delta
function, $g_{z}(t)$ accompanies relaxation due to the loss of phase coherence
in the spin precession.

Specifically, provided that the static distribution of
${\bm{H}}({\bm{r}}_{\mu})$ is approximated by an isotropic Gaussian
distribution,

$n_{\alpha}(H)=\frac{\gamma_{\mu}}{\sqrt{2\pi}\Delta}\exp\left(-\frac{\gamma_{\mu}^{2}H^{2}}{2\Delta^{2}}\right),\>\>(\alpha=x,y,z)$
(5)

the corresponding relaxation function is obtained analytically by executing
the integral in Eq. (4) to yield the static Kubo-Toyabe function Hayano  _et
al._ (1979),

$g_{z}(t)=\frac{1}{3}+\frac{2}{3}(1-\Delta^{2}t^{2})e^{-\frac{1}{2}\Delta^{2}t^{2}},$
(6)

where the term $1/3$ is derived by the spatial average of $\cos^{2}\theta$ in
Eq. (2), corresponding the probability that ${\bm{H}}({\bm{r}}_{\mu})$ is
parallel to $z$. The linewidth $\Delta$ is determined by the second moment of
the dipole field from the nuclear magnetic moments of the neighboring atoms
(including non-cationic ions),

$\Delta^{2}=\gamma_{\mu}^{2}\gamma_{I}^{2}\sum_{i}\sum_{\alpha=x,y}\sum_{\beta=x,y,z}(\hat{A}_{i}^{\alpha\beta}{\bm{I}}_{i})^{2},$
(7)

where

$(\hat{A}_{i})^{\alpha\beta}=\frac{1}{r^{3}_{i}}(\frac{3\alpha_{i}\beta_{i}}{r^{2}_{i}}-\delta_{\alpha\beta})\>\>(\alpha,\beta=x,y,z),$
(8)

is the dipole tensor for the $i$th nuclear magnetic moment
$\gamma_{I}{\bm{I}}_{i}$ situated at ${\bm{r}}_{i}$ from muon. Since the
magnetic dipolar field decays in proportion to $1/r_{i}^{3}$, the magnitude of
$\Delta$ is approximately determined by the nuclear magnetic moments nearest
neighboring (nn) to the muon. The Gaussian distribution for
${\bm{n}}({\bm{H}})$ works reasonably well when the number $N$ of the nn
nuclear magnetic moments satisfies the condition $N\geq 4$. Meanwhile, it is a
relatively poor approximation for $N\leq 3$, and it is necessary to calculate
$g_{z}(t)$ by treating the muon-nuclei cluster as a few-quantum spin system
using the density matrix method Brewer  _et al._ (1986); Celio (1986). With
recent advances in the computational environment, such calculations have been
performed for nuclear magnetic moments located farther from the muon, and good
agreement with experiment has been obtained Wilkinson and Blundell (2020).

###  II.2 Effect of fluctuating internal fields

In general, the effect of fluctuations in linear response theory is treated by
a Gaussian-Markov process. In this model, the autocorrelation of fluctuations
is given by

$\frac{\langle{\bm{H}}(t){\bm{H}}(0)\rangle}{\langle
H(0)^{2}\rangle}=\frac{\langle{\bm{H}}(t){\bm{H}}(0)\rangle}{\Delta^{2}/\gamma_{\mu}^{2}}=e^{-\nu
t},$ (9)

where $\langle...\rangle$ denotes the thermal average over the canonical
ensemble. (For the more rigorous treatment, see, for example, Ref. for example
_et al._ (2020).) The strong-collision model (also called random Markov
process or random phase approximation) is a simplified version of this model,
where it is assumed that the density distribution $n_{\alpha}(H)$ is constant
at any time $t$, and that ${\bm{H}}(t)$ changes with an average rate
$\nu=1/\tau$ without correlation before and after a change. The resulting
relaxation function is known to be a good approximation of a Gaussian process
except in the $\nu t\ll 1$ region Kehr  _et al._ (1978). In this case, it is
known that the relaxation function $G_{z}(t)$, which incorporates the effect
of fluctuations on the static relaxation function $g_{z}(t)$, is derived by
solving the following integral equation (a linear Volterra equation of the
second kind),

$G_{z}(t)=e^{-\nu t}g_{z}(t)+\nu\int_{0}^{t}d\tau
e^{-\nu(t-\tau)}g_{z}(t-\tau)G_{z}(\tau).$ (10)

This integral equation can be solved analytically by Laplace transform of
$g_{z}(t)$ and $G_{z}(t)$ Hayano  _et al._ (1979) or direct numerical
integration Dalmas de Reotier and Yaouanc (1992). The dynamical KT function,
$G^{\rm KT}_{z}(t;\Delta,\nu)$, obtained in this way successfully reproduced
the spin relaxation due to muon jump diffusion in a nonmagnetic metal Clawson
_et al._ (1983); Kadono  _et al._ (1989); Luke  _et al._ (1991). This
indicates that fluctuations in the internal magnetic field due to muon self-
diffusion can be well described by the strong-collision model with Eq. (9).

On the other hand, fluctuations in ${\bm{H}}$ due to ion diffusion do not
necessarily occur in its entire amplitude; let us consider the situation in
Fig. 1(b) where part of ${\bm{H}}$ is due to contributions from cations or
anions that remain stationary while other ions exhibit local motion. In this
case, it is clear that the autocorrelation of fluctuation differs from Eq.
(9), and it may be given by an equation

$\frac{\langle{\bm{H}}(0){\bm{H}}(t)\rangle}{\Delta^{2}/\gamma_{\mu}^{2}}\approx(1-Q)e^{-\nu_{\mu}t}+Qe^{-(\nu_{\rm
i}+\nu_{\mu})t},$ (11)

where $Q$ is the parameter to describe the fractional amplitude exhibiting
fluctuations with the rate $\nu_{\rm i}$, and $\nu_{\mu}$ is the fluctuation
rate due to muon diffusion. The equation with $\nu_{\rm\mu}=0$ corresponds to
a model that was introduced to describe the slowing down of magnetic
fluctuations (with a rate $\nu_{\rm i}$) in spin glasses, where $1-Q$ is
interpreted as an order parameter to describe “ordering in time” Edwards and
Anderson (1975, 1976). This model was also applied to derive relaxation
functions for analyzing the $\mu$SR spectra observed in dilute-alloy spin
glasses Uemura  _et al._ (1985).

However, the corresponding $G_{z}(t)$ has only been discussed in approximate
forms obtained as analytical expressions when $\nu_{\mu}\ll\Delta\ll\nu_{\rm
i}$ Koda  _et al._ (2022); Uemura  _et al._ (1985), and a prescription for the
general case is still awaited. Therefore, in the following, we investigate the
behavior of $G_{z}(t)$ corresponding to arbitrary $Q$ and $\nu_{i}$ by Monte
Carlo simulations.

###  II.3 Monte Carlo simulation

We performed numerical simulations of the spin relaxation function for $Q>0$
and $\nu_{\mu}=0$, which we call $G^{\rm ID}_{z}(t;\Delta,\nu_{\rm i},Q)$,
where the ion dynamics is the sole origin of fluctuations with the amplitude
$Q\Delta^{2}$ and the frequency $\nu_{\rm i}$,

$\gamma_{\mu}^{2}\langle{\bm{H}}(0){\bm{H}}(t)\rangle=[\sqrt{1-Q}\Delta]^{2}+[\sqrt{Q}\Delta]^{2}e^{-\nu_{\rm
i}t}.$ (12)

Figure 1d) schematically illustrates the situation in Eq. (12) under these
conditions. In our Monte Carlo simulation, the local field that each muon
feels was approximately expressed as a vector sum of static and dynamical
fields. These fields were randomly generated according to the Gaussian
probability distribution $n_{\alpha}(H)$ with the widths of
$\sqrt{(1-Q)}\Delta/\gamma_{\mu}$ and $\sqrt{Q}\Delta/\gamma_{\mu}$,
respectively. Note that the assignment of $1-Q$ and $Q$ is opposite to that in
Ref. Edwards and Anderson (1975, 1976), where $Q$ represents the static
component; this is so defined that the case $Q=0$ should be reduced to the
conventional KT function (with $\nu_{\mu}>0$ for muon diffusion). The
fluctuation of the dynamical component was implemented in accordance with the
strong-collision model by regenerating the dynamical field at the average rate
of $\nu_{\rm i}$. The time evolution of a muon spin, initially oriented to the
$z$ direction, in the local field was calculated and its $z$-component was
averaged for $10^{8}$ muons to obtain $G_{z}^{\rm ID}(t)$ for ZF. For finite
LFs, the calculation was performed for the vector sum of the local field and
the LF along the $z$ direction.

As shown in Fig. 2a)–c), in the case of $0<Q<1$, the $\frac{1}{3}$ term once
decays as $\nu_{\rm i}$ increases, then begins to recover with the apparent
linewidth decreasing to $\sqrt{1-Q}\Delta$. Provided that the $\mu$SR spectra
showing such temperature dependence is analyzed with the usual KT function
(Fig. 2d), the apparent fluctuation rate would exhibit an increase followed by
decrease. Thus, it is likely to lead to an incorrect interpretation that the
cause of the fluctuations is different from a simple thermal excitation
process. An important clue in avoiding this is whether the linewidth changes
at both ends of the apparent increase and decrease in $\nu$ derived from the
KT function. In Figs. 2b) and 2c), it can be seen that as $\nu_{\rm i}$
increases, the $Q\Delta^{2}$ contribution disappears due to motional
narrowing, and the relaxation rate decreases (converging to
$\sqrt{1-Q}\Delta$) while maintaining the Gaussian linshape. Such a behavior
is actually observed in HOIP Koda  _et al._ (2022); Hiraishi  _et al._ (2023),
and the change is found to correspond approximately to the case of $Q\simeq
0.9$.

Figure 2: Some examples of the simulated muon spin relaxation function $G^{\rm
ID}_{z}(t;\Delta,\nu_{\rm i},Q)$ incorporating the effect of ion dynamics by
the parameter $Q$, where $Q\Delta^{2}$ is the amplitude of fluctuation. a)–d)
under zero field with different $Q$ and $\nu_{\rm i}$, and e)–h) under a
variety of longitudinal field $B_{\rm LF}$ with $Q=0.5$ and $\Delta/\nu_{\rm
i}=0.5$. Note that $Q=1$ (and 0) corresponds to the conventional Kubo-Toyabe
function.

The LF dependence of $G^{\rm ID}_{z}(t;\Delta,\nu_{\rm i},Q)$ is indispensable
information for more accurate determination of the fluctuation frequency in
$\mu$SR measurements. Let us compare the field dependence of the time spectrum
for $Q=0.5$ and $\Delta/\nu_{\rm i}=0.5$ shown in Fig. 2e) with that of the
conventional KT function in detail. First, the KT function calculated for
$\Delta/\nu_{\rm i}=\sqrt{0.5}$ is shown in Fig. 2f); this is because the
linewidth for the fluctuations in the former case is $\sqrt{Q}\Delta$ rather
than $\Delta$. The oscillations around $\Delta t\simeq 2$ [originating from
the static component corresponding to $(1-Q)$] seen in Fig. 2e) are absent,
indicating qualitatively different behavior. Furthermore, assuming practical
situations of curve fitting by the KT function, the results with freely
varying $\nu_{\rm i}$ to reproduce the behavior of Fig. 2e) are shown in Figs.
2g) and 2h). Fig. 2g) reproduces the low-field spectrum well, while the fit is
poor on the high-field side, while Fig. 2h) shows the opposite trend. From
these comparisons, we can conclude that curve fitting with the KT function
does not accurately reproduce the field dependence of the spectrum when $Q<1$.

Now, let us discuss the behavior of the relaxation function for some extreme
cases, for which we define $\Delta_{\mu}=\sqrt{(1-Q)}\Delta$ and $\Delta_{\rm
i}=\sqrt{Q}\Delta$. When the ion dynamics is fast enough to satisfy $\nu_{\rm
i}\gg\Delta$, the spin relaxation for that part is approximated to

$G_{\rm i}(t)\simeq G_{z}^{\rm KT}(t;\Delta_{\rm i},\nu_{\rm
i})=e^{-\frac{2\Delta_{\rm i}^{2}t}{\nu_{\rm i}}}.$ (13)

Then, if the muon is quasi-static, the corresponding total relaxation function
is

$G^{\rm ID}_{z}(t)\simeq G_{z}^{\rm KT}(t;\Delta_{\mu},0)G_{\rm
i}(t)=[\frac{1}{3}+\frac{2}{3}(1-\Delta_{\mu}^{2}t^{2})e^{-\frac{1}{2}\Delta_{\mu}^{2}t^{2}}]e^{-\frac{2\Delta_{\rm
i}^{2}t}{\nu_{\rm i}}},$ (14)

indicating that the curve fit with the conventional KT function will yield a
reduced linewidth $\sqrt{(1-Q)}\Delta$.

Conversely, in the limit of slow ion dynamics ($\nu_{\rm i}\ll\Delta$), the
change in the relaxation function is independent of linewidth, and the
exponential relaxation of the $\frac{1}{3}$ component dominates;

$G^{\rm ID}_{z}(t)\simeq\frac{1}{3}e^{-Q\nu_{\rm
i}t}+\frac{2}{3}(1-\Delta^{2}t^{2})e^{-\frac{1}{2}\Delta^{2}t^{2}},$ (15)

which is valid for $Q\approx 1$. This suggests it difficult to distinguish the
ion dynamics from muon diffusion in the slow exponential damping. As
demonstrated in the case of FAPbI3 below, the difference from the conventional
KT function becomes noticeable in the intermediate region of $\nu_{\rm
i}\gtrsim\Delta$, so the spectra in such a temperature region are important in
data analysis to distinguish between ion dynamics and muon self-diffusion.

###  II.4 Comparison with experimental result

It has been inferred from the previous $\mu$SR study that muons in FAPbI3 are
exposed to random local fields originating from the nuclear magnetic moments
of both FA molecules and the PbI3 lattice Hiraishi  _et al._ (2023). Among
these, the magnetic field from the FA molecule (mainly due to the proton
magnetic moment) is expected to exhibit fluctuation due to rotational motion
induced by thermal excitation with increasing temperature. The analysis of the
$\mu$SR time spectra in the tetragonal phase of FAPbI3 by the KT function
showed that the linewidth $\Delta$ appears to decrease monotonously above
$\sim$50 K while the fluctuation frequency $\nu$ exhibits a weak peak around
100 K. Moreover, the global curve fit including the spectra under various LFs
are relatively poor in the 50–120 K range where $\Delta$ changes
significantly. These observations point to the situation that the actual
relaxation function deviates from the KT function, as demonstrated in Fig. 2.
Therefore, we focused on the temperature variation of the ZF-$\mu$SR spectrum
and investigated whether the spectral changes could be reproduced by the
appropriate $Q$, $\Delta$, and $\nu_{\rm i}$ when the data were analyzed with
the new model.

Figure 3: a) Zero-field $\mu$SR spectra observed at low temperatures in FAPbI3
fitted with a relaxation function obtained by Monte Carlo simulation (global
fit assuming that the initial asymmetry, $Q$, and $\Delta$ are common for all
temperatures). All the spectra are reproduced by $Q=0.947(3)$ and
$\Delta=0.191(1)$ MHz. b) Arrhenius plot of the internal field fluctuation
frequency $\nu_{\rm i}$ attributed to FA molecular motion obtained in this
analysis. The dashed line is the fit by the sum of the thermally activated
function and the constant. Inset: temperature dependence of the constant
background (see text for detail).

Specifically, we prepared a multi-dimensional numerical table of $G^{\rm
ID}_{z}(t;\Delta,\nu_{\rm i},Q)$ for the parameters $\Delta t$, $Q$, and
$\nu_{\rm i}$ by Monte Carlo simulation, and performed least-squares curve
fitting of the time-dependent asymmetry using the equation

$A(t)=A_{0}G^{\rm ID}_{z}(t;\Delta,\nu_{\rm i},Q)+A_{\rm b},$ (16)

where $A_{0}$ is the initial asymmetry depending on the instrument, and
$A_{\rm b}$ is the background from muons that missed the sample. In addition,
a global fit for multiple spectra was performed, where $A_{0}$, $\Delta$, and
$Q$ were common for the spectra at all temperatures. As shown in Fig. 3a), the
entire spectra were well reproduced (reduced net chi-square = 1.33), and the
values of the two common parameters were obtained as $Q=0.947(3)$ and
$\Delta=0.191(1)$ MHz, respectively. The values of $\Delta$ and
$\sqrt{1-Q}\Delta$ ($=0.044$ MHz) are in good agreement with those obtained
from the KT function analysis near 50 K and 120 K Hiraishi  _et al._ (2023).
Furthermore, as shown in Fig. 3b), $\nu_{\rm i}$ shows a temperature
dependence consistent with thermally activated process at higher temperatures,
which is also in line with the earlier suggestion Hiraishi  _et al._ (2023).
Curve fit using the Arrhenius function, $\nu_{\rm i}=\nu_{0}\exp(-E_{\rm
a}/k_{B}T)+\nu_{c}$, yields excellent agreement with $\nu_{0}=3.6(6)\times
10^{8}$ s-1, $E_{\rm a}/k_{B}=630(18)$ K, and $\nu_{c}=1.62(9)\times 10^{5}$
s-1 (see the dashed line in Fig. 3b). $A_{\rm b}$ is small ($\simeq 0.015\ll
A_{0}$) and nearly independent of temperature as expected, in support of the
present model.

##  III Discussion

We have shown that when the fluctuation of ${\bm{H}}(t)$ is due to the ion
dynamics corresponding to $0<Q<1$ in Eq. (12), the analysis by the KT function
would encounter a region of $\nu$ where the $\mu$SR spectrum is not well
reproduced by the KT function. Therefore, a careful analysis of the time
spectrum in the temperature region that exhibits such $\nu$ has the potential
to distinguish the origin of fluctuation.

Historically, the possibility of applying $\mu$SR to the study of ion dynamics
was first discussed for the diffusion of Li+ ions in LixCoO2, a typical
cathode material for Li-ion batteries Sugiyama  _et al._ (2009), and then its
application to other transition metal oxides containing Li has been attempted
Baker  _et al._ (2011); Sugiyama  _et al._ (2011, 2012); Ashton  _et al._
(2014); Månsson  _et al._ (2014); Amores  _et al._ (2016); Laveda  _et al._
(2018); Benedek  _et al._ (2020); McClelland  _et al._ (2020). In these
studies, the $\mu$SR time spectra reflecting ${\bm{H}(t)}$ originating from
the 7Li and other nuclear magnetic moments (e.g., 59Co in LixCoO2) was
analyzed by the dynamical KT function, and it was commonly assumed that the
obtained fluctuation rate $\nu$ directly corresponds to the jumping frequency
of Li+ ions.

However, the interpretation of $\mu$SR result has been significantly changed
in a very recent paper reporting the results of “operando” $\mu$SR
measurements in the charging and discharging of LixCoO2 incorporated in a half
battery Ohishi  _et al._ (2022). Namely, it is argued that the fluctuation
rate $\nu$ (derived from the analysis using the KT function) reflects both
contributions of Li+ and muon jump diffusion, and that $\nu$ divided by the
square root of the Li-muon mass ratio ($\sqrt{m_{\rm Li}/m_{\mu}}\simeq 7.9$)
is the jump frequency of Li. This means that the observed fluctuations mainly
reflect the muon jump frequency ($\nu_{\mu}\gg\nu_{\rm i}$).

From the studies of muon diffusion in solids accumulated to date, it has been
established that the thermally-activated tunneling process dominates muon
diffusion at temperatures higher than about a fraction of the Debye
temperature $\Theta_{D}$ Kadono (2003); Fukai (2005). The above interpretation
is qualitatively predictable given the reported $\Theta_{D}\approx 300$ K for
LiCoO2, since muon diffusion via the thermally-activated tunneling process is
generally considered to prevail over ion diffusion in the relevant temperature
range.

Another important factor in terms of identifying the origin of fluctuation is
the temperature dependence of $\Delta$ derived from analysis by the KT
function. While the decrease in apparent $\Delta$ with increasing temperature
may indicate a contribution of ion dynamics to spin relaxation, muon diffusion
is also expected in the temperature region of interest. Our simulation results
show that when the latter is dominant ($\nu_{\mu}\gg\nu_{\rm i}$), the spin
relaxation can be well reproduced by the KT function. Therefore, when the
spectra are well reproduced by the KT function even though $\Delta$ exhibits
change, it is more likely that the cause of change in $\Delta$ is attributed
to muon site issues rather than ion dynamics; note that $\Delta$ can vary with
the muon position ${\bm{r}}_{\mu}$ in inhomogeneous materials. For example,
muons may prefer to occupy atomic vacancies or voids to reduce the zero-point
energy, which has a significant impact on the behavior of the light particles
Ito  _et al._ . Provided that $\Delta=\Delta({\bm{r}}_{\mu})$ is small for
such defect sites, the mean $\Delta$ will decrease as the probability of a
muon reaching these sites increases with increasing temperature in a
diffusion-limited process. Such behavior has been actually observed, for
example, in InGaZnO Kojima  _et al._ (2019).

In examining previous literature reporting $\mu$SR studies of Li ion
diffusion, there are many cases in which $\Delta$ decreases with increasing
temperature, but so far there has been little discussion on such behavior from
the aforementioned viewpoint. If the observed spin relaxation is well
reproduced by the KT function, it is incompatible with the ion diffusion for
which the autocorrelation function is assumed to be given by Eq. (12) with
$Q<1$. To determine whether the observed fluctuations are due to ion
diffusion, it will be necessary to examine the reproducibility of $\mu$SR
spectra by the model presented here with $Q<1$.

##  IV Conclusion

We have performed Monte Carlo simulations to investigate how the residual
correlations in the fluctuations described by an Edwards-Anderson type
parameter $Q$ affect the lineshape of the relaxation function observed in the
$\mu$SR experiment. As a result, we found a significant deviation from the
Kubo-Toyabe function for $0<Q<1$. We then compared the simulated relaxation
function, $G_{z}^{\rm ID}(t;\Delta,\nu_{\rm i},Q)$, with the $\mu$SR spectra
observed in FAPbI3 in which internal field fluctuations associated with the
rotational motion of FA molecules and quasi-static internal field from the
PbI3 lattice are presumed to coexist. The curve fits show that the model
provides proper description of the temperature-dependent lineshape with
reasonable $\Delta$, $\nu_{\rm i}$ and $Q$, which is in line with cation
molecular motion. This result paves the way for the possibility of
experimentally distinguishing fluctuations ion dynamics around muons from muon
self-diffusion, which has been considered difficult. It also calls into
question the attribution of the internal field fluctuations entirely to ion
diffusion for the $\mu$SR time spectra that are well reproduced by the Kubo-
Toyabe function, suggesting the need for a major reexamination of the results
of previous analyses.

## Acknowledgment

We thank M. Hiraishi, H. Okabe, A. Koda, and K. Fukutani for helpful
discussion during the preparation of the manuscript. The $\mu$SR results on
FAPbI3 quoted in this paper were those published in Ref. Hiraishi  _et al._
(2023). The Monte Carlo simulation was conducted with the supercomputer HPE
SGI8600 in Japan Atomic Energy Agency. This work was partially supported by
JSPS KAKENHI (Grant No. 23K11707, 21H05102, and 20H01864) and the MEXT
Program: Data Creation and Utilization Type Material Research and Development
Project (Grant No. JPMXP1122683430).

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# Monogamy of entanglement between cones

Guillaume Aubrun  Institut Camille Jordan, Université Claude Bernard Lyon 1,
43 boulevard du 11 novembre 1918, 69622 Villeurbanne cedex, France
<EMAIL_ADDRESS>,  Alexander Müller-Hermes  Department of
Mathematics, University of Oslo, P.O. box 1053, Blindern, 0316 Oslo, Norway
<EMAIL_ADDRESS>and  Martin Plávala  Naturwissenschaftlich-Technische
Fakultät, Universität Siegen, 57068 Siegen, Germany martin.plavala@uni-
siegen.de

###### Abstract.

A separable quantum state shared between parties $A$ and $B$ can be
symmetrically extended to a quantum state shared between party $A$ and parties
$B_{1},\ldots,B_{k}$ for every $k\in\mathbf{N}$. Quantum states that are not
separable, i.e., entangled, do not have this property. This phenomenon is
known as “monogamy of entanglement”. We show that monogamy is not only a
feature of quantum theory, but that it characterizes the minimal tensor
product of general pairs of convex cones $\mathsf{C}_{A}$ and
$\mathsf{C}_{B}$: The elements of the minimal tensor product
$\mathsf{C}_{A}\otimes_{\min}\mathsf{C}_{B}$ are precisely the tensors that
can be symmetrically extended to elements in the maximal tensor product
$\mathsf{C}_{A}\otimes_{\max}\mathsf{C}^{\otimes_{\max}k}_{B}$ for every
$k\in\mathbf{N}$. Equivalently, the minimal tensor product of two cones is the
intersection of the nested sets of $k$-extendible tensors. It is a natural
question when the minimal tensor product
$\mathsf{C}_{A}\otimes_{\min}\mathsf{C}_{B}$ coincides with the set of
$k$-extendible tensors for some finite $k$. We show that this is universally
the case for every cone $\mathsf{C}_{A}$ if and only if $\mathsf{C}_{B}$ is a
polyhedral cone with a base given by a product of simplices. Our proof makes
use of a new characterization of products of simplices up to affine
equivalence that we believe is of independent interest.

##  1\. Introduction

Let $V$ denote a finite-dimensional vector space over the reals and
$\mathsf{C}\subseteq V$ a convex cone. We say that the cone $\mathsf{C}$ is
_proper_ if it is closed, does not contain any line, and is not contained in
any hyperplane. Any proper cone $\mathsf{C}\subseteq V$ can be described in
terms of its dual $\mathsf{C}^{*}\subseteq V^{*}$, given by

$\mathsf{C}^{*}=\\{f\in V^{*}~{}:~{}f(x)\geqslant 0\text{ for any
}x\in\mathsf{C}\\},$

since by the bipolar theorem we have $(\mathsf{C}^{*})^{*}=\mathsf{C}$. While
any proper cone can be described from the inside by its extremal rays, the
bipolar theorem describes the cone from the outside as an intersection of
half-spaces. This duality is fundamental in the study of convex cones and it
is a basic problem in convex analysis to switch between these two
descriptions.

We will be interested in the two canonical tensor products of any pair of
proper cones. Given proper cones $\mathsf{C}_{A}\subseteq V_{A}$,
$\mathsf{C}_{B}\subseteq V_{B}$, we may form the _minimal tensor product_

$\mathsf{C}_{A}{\otimes_{\min}}\mathsf{C}_{B}=\operatorname{\mathrm{conv}}\\{x\otimes
y~{}:~{}x\in\mathsf{C}_{A},\ y\in\mathsf{C}_{B}\\}\subseteq V_{A}\otimes
V_{B},$

and the _maximal tensor product_

$\mathsf{C}_{A}{\otimes_{\max}}\mathsf{C}_{B}=(\mathsf{C}^{*}_{A}{\otimes_{\min}}\mathsf{C}^{*}_{B})^{*}=\\{z\in
V_{A}\otimes V_{B}~{}:~{}(f\otimes g)(z)\geqslant 0,\ f\in\mathsf{C}_{A}^{*},\
g\in\mathsf{C}_{B}^{*}\\}.$

For proper cones $\mathsf{C}_{A}$ and $\mathsf{C}_{B}$ both tensor products
are again proper cones. Moreover, the tensor products may be iterated and
since they are associative we may define the $k$-fold versions
$\mathsf{C}^{\otimes_{\min}k}$ and $\mathsf{C}^{\otimes_{\max}k}$ for any
proper cone $\mathsf{C}$. The two tensor products reflect the aforementioned
descriptions of convex cones: The minimal tensor product is described by
specifying generating elements and the maximal tensor product by an
intersection of half-spaces. Moreover, their discrepancy can be seen as a
general form of quantum entanglement [1].

A central feature of quantum physics is known as _monogamy of entanglement_.
In our language, it can be described using the cones of positive semidefinite
$d\times d$ matrices $\mathsf{M}^{+}_{d}$ with complex entries, which is a
proper cone inside the real vector space $\mathsf{M}_{d}^{\mathrm{sa}}$ of
Hermitian $d\times d$ matrices. A bipartite quantum state is given by a matrix
$\rho_{AB}\in\mathsf{M}_{d_{A}d_{B}}^{+}\simeq(\mathsf{M}_{d_{A}}\otimes\mathsf{M}_{d_{B}})^{+}\subsetneq\mathsf{M}_{d_{A}}^{+}{\otimes_{\max}}\mathsf{M}_{d_{B}}^{+}$
for some integers ${d_{A},d_{B}\geqslant 2}$ and it is called _entangled_ if
it does not belong to
$\mathsf{M}^{+}_{d_{A}}\otimes_{\min}\mathsf{M}^{+}_{d_{B}}$. The _monogamy
theorem_ [6, 16] asserts that a quantum state
$\rho_{AB}\in(\mathsf{M}_{d_{A}}\otimes\mathsf{M}_{d_{B}})^{+}$ is entangled
if and only if there exists an integer $k\geqslant 2$ for which no quantum
state $\sigma_{AB_{1}\cdots
B_{k}}\in(\mathsf{M}_{d_{A}}\otimes\mathsf{M}^{\otimes k}_{d_{B}})^{+}$
satisfies the equation

$\rho_{AB}=\left(\mathrm{Id}_{\displaystyle\mathsf{M}^{\mathrm{sa}}_{d_{A}}}\otimes\gamma^{\operatorname{Tr}}_{k}\right)\left(\sigma_{AB_{1}\cdots
B_{k}}\right),$

where

$\gamma^{\operatorname{Tr}}_{k}=\frac{1}{k}\sum_{j=1}^{k}\operatorname{Tr}^{\otimes(j-1)}\otimes\mathrm{Id}_{\displaystyle\mathsf{M}^{\mathrm{sa}}_{d_{B}}}\otimes\operatorname{Tr}^{\otimes(k-j)}$

corresponds to (symmetrically) discarding all but one out of the $k$ tensor
factors labelled by $B_{1},\ldots,B_{k}$. Physically, this means that a
bipartite quantum state is entangled if and only if it cannot be partially
shared with arbitrarily many parties. Our first main result establishes a
similar property for general minimal and maximal tensor products.

######  Theorem 1.

Consider proper cones $\mathsf{C}_{A}\subseteq V_{A}$ and
$\mathsf{C}_{B}\subseteq V_{B}$ and a linear form $\phi$ in the interior of
$\mathsf{C}_{B}^{*}$. Then, we have

(1) $\mathsf{C}_{A}{\otimes_{\min}}\mathsf{C}_{B}=\bigcap_{k\geqslant
1}\left(\mathrm{Id}_{V_{A}}\otimes\gamma_{k}^{\phi}\right)\left(\mathsf{C}_{A}{\otimes_{\max}}\mathsf{C}_{B}^{{\otimes_{\max}}k}\right),$

where, for an integer $k\geqslant 1$, the _$k$ th reduction map_
$\gamma_{k}^{\phi}:V_{B}^{\otimes k}\to V_{B}$ is defined as

$\gamma_{k}^{\phi}=\frac{1}{k}\sum_{j=1}^{k}\phi^{\otimes(j-1)}\otimes\mathrm{Id}_{V_{B}}\otimes\phi^{\otimes(k-j)}.$

Our Theorem 1 shows that the monogamy property of quantum entanglement is
actually a common feature of the most general forms of entanglement for any
pair of cones. The question whether Theorem 1 holds has been asked in [11,
§2.4.1]. Even for cones of positive semidefinite matrices our results are new:
They show that entanglement cannot be partially shared with arbitrarily many
parties even if we allow for unphysical quantum states represented in the
maximal tensor product.

Equation (1) produces a decreasing sequence of outer approximations to the
minimal tensor product, called the _extendability hierarchy_. A natural
question is whether this sequence stops after finitely many steps. We answer
this question in terms of combinatorial properties of the base
$K_{\phi}=\mathsf{C}_{B}\cap\phi^{-1}(1)$. We have the following theorem:

######  Theorem 2.

Consider a proper cone $\mathsf{C}_{B}\subseteq V_{B}$ and a linear form
$\phi$ in the interior of $\mathsf{C}_{B}^{*}$. Then, for every integer
$k\geqslant 1$, the following are equivalent:

  1. (i)

For every proper cone $\mathsf{C}_{A}\subseteq V_{A}$, we have the identity

$\mathsf{C}_{A}{\otimes_{\min}}\mathsf{C}_{B}=\left(\mathrm{Id}_{V_{A}}\otimes\gamma_{k}^{\phi}\right)\left(\mathsf{C}_{A}{\otimes_{\max}}\mathsf{C}_{B}^{{\otimes_{\max}}k}\right);$

  2. (ii)

The base $K_{\phi}$ is affinely equivalent to a Cartesian product of
$\leqslant k$ simplices.

As a byproduct of our proof, we obtain a characterization of products of
simplices as the only polytopes (up to affine equivalence) for which the
operations “intersection” and “affine hull” commute when applied to the face
lattice (see Figure 1 for an illustration). We denote by
$\operatorname{\mathrm{aff}}(X)$ the affine hull of a subset $X$ of a vector
space, using the convention that
$\operatorname{\mathrm{aff}}(\emptyset)=\emptyset$.

######  Theorem 3.

Let $P$ be a polytope. The following are equivalent:

  1. (i)

The polytope $P$ is affinely equivalent to a Cartesian product of simplices;

  2. (ii)

For every faces $(F_{i})_{i\in I}$ of $P$, we have

(2) $\operatorname{\mathrm{aff}}\left(\bigcap_{i\in
I}F_{i}\right)=\bigcap_{i\in I}\operatorname{\mathrm{aff}}(F_{i}).$

In the case of dimension $3$, Theorem 3 means that the only polyhedra without
stellation are the simplex, the cube and the triangular prism, extending an
observation by Wenninger [15, p.35].

Figure 1. The planar case of Theorem 3: if a polygon is neither a triangle nor
a parallelogram, it has disjoint edges whose affine hulls intersect.

### Conventions and preliminaries

Given an integer $n\geqslant 1$, we denote by $[n]$ the set $\\{1,\dots,n\\}$.
All the vector spaces we consider are implicitly assumed to be finite-
dimensional. If $V$ is a vector space, the action of $f\in V^{*}$ on $x\in V$
is denoted as $f(x)$, $\langle f,x\rangle$ or $\langle x,f\rangle$. We always
identify the double dual $V^{**}$ with $V$ itself. Similarly, for an integer
$k\geqslant 2$ we always identify $(V^{\otimes k})^{*}$ with $(V^{*})^{\otimes
k}$.

Given vector spaces $V_{1},V_{2}$, the adjoint of a linear map $\Phi:V_{1}\to
V_{2}$ is denoted as $\Phi^{*}:V_{2}^{*}\to V_{1}^{*}$. If $X\subseteq V$, the
affine hull $\operatorname{\mathrm{aff}}(X)$ is the set of elements of the
form $\lambda_{1}x_{1}+\dots+\lambda_{n}x_{n}$ for an integer $n\geqslant 1$,
elements $x_{1},\dots,x_{n}$ in $X$ and real numbers
$\lambda_{1},\dots,\lambda_{n}$ such that $\lambda_{1}+\dots+\lambda_{n}=1$.
An element $y\in V$ belongs to $\operatorname{\mathrm{aff}}(X)$ if and only if
every affine function $f:V\to\mathbf{R}$ which vanishes on $X$ vanishes at
$y$.

### Structure of the paper

After reviewing the basic properties of the symmetric subspace, we discuss the
basic properties of the generalized extendibility hierarchy (Section 2.1), a
useful dual formulation (Section 2.2) and some examples (Section 2.3)
including the case of quantum theory. In Section 3 we prove Theorem 1 by
making use of a generalization of the de Finetti theorem due to Barrett and
Leifer [3]. The proof of Theorem 2 is presented in Section 4. It relies on
Theorem 3 which we prove in Section 5.

##  2\. Basic properties of the extendibility hierarchy and examples

Let $V$ be a finite-dimensional real vector space and $k\geqslant 1$ an
integer. The symmetric group $\mathfrak{S}_{k}$ acts naturally on $V^{\otimes
k}$: if $\sigma\in\mathfrak{S}_{k}$, we define $U_{\sigma}:V^{\otimes k}\to
V^{\otimes k}$ as the linear map such that

$U_{\sigma}(x_{1}\otimes\dots\otimes
x_{k})=x_{\sigma^{-1}(1)}\otimes\dots\otimes x_{\sigma^{-1}(k)}$

for every $x_{1},\dots,x_{k}\in V$. We define the _symmetric subspace_
$\mathrm{Sym}_{k}(V)$ as the subspace of $V^{\otimes k}$ which is invariant
for this action

$\mathrm{Sym}_{k}(V)=\\{x\in V^{\otimes k}\ :\ U_{\sigma}x=x\textnormal{ for
every }\sigma\in\mathfrak{S}_{k}\\}.$

The _symmetric projection_ is the operator $P_{\mathrm{Sym}_{k}(V)}:V^{\otimes
k}\to V^{\otimes k}$ defined as

$P_{\mathrm{Sym}_{k}(V)}=\frac{1}{k!}\sum_{\sigma\in\mathfrak{S}_{k}}U_{\sigma}.$

It is a projection whose range equals $\mathrm{Sym}_{k}(V)$. If we identify
$(V^{\otimes k})^{*}$ with $(V^{*})^{\otimes k}$, we have the relation

$P_{\mathrm{Sym}_{k}(V)}^{*}=P_{\mathrm{Sym}_{k}(V^{*})}.$

Moreover, if $\mathsf{C}\subseteq V$ is a proper cone, then

$P_{\mathrm{Sym}_{k}(V)}(\mathsf{C}^{{\otimes_{\min}}k})=\mathsf{C}^{{\otimes_{\min}}k}\cap{\mathrm{Sym}_{k}(V)},$
$P_{\mathrm{Sym}_{k}(V)}(\mathsf{C}^{{\otimes_{\max}}k})=\mathsf{C}^{{\otimes_{\max}}k}\cap{\mathrm{Sym}_{k}(V)}.$

###  2.1. The extendibility hierarchy

Consider proper cones $\mathsf{C}_{A}\subseteq V_{A}$,
$\mathsf{C}_{B}\subseteq V_{B}$ and
$\phi\in\operatorname{int}(\mathsf{C}_{B}^{*})$. For an integer $k\geqslant
1$, denote by

(3)
$\mathsf{Ext}_{k}(\mathsf{C}_{A},\mathsf{C}_{B},\phi)=(\mathrm{Id}_{V_{A}}\otimes\gamma_{k}^{\phi})\left(\mathsf{C}_{A}{\otimes_{\max}}\mathsf{C}_{B}^{{\otimes_{\max}}k}\right)$

the cone appearing in Theorem 1. Since
$\gamma_{k}^{\phi}=(\mathrm{Id}_{V_{B}}\otimes\phi^{\otimes(k-1)})\circ
P_{\mathrm{Sym}_{k}(V_{B})}$, an element $x\in V_{A}\otimes V_{B}$ belongs to
$\mathsf{Ext}_{k}(\mathsf{C}_{A},\mathsf{C}_{B},\phi)$ if and only if

$\exists y_{k}\in\left(\mathrm{Id}_{V_{A}}\otimes
P_{\mathrm{Sym}_{k}(V_{B})}\right)\left(\mathsf{C}_{A}{\otimes_{\max}}\mathsf{C}_{B}^{{\otimes_{\max}}k}\right)\
:\
x=\left(\mathrm{Id}_{V_{A}}\otimes\mathrm{Id}_{V_{B}}\otimes\phi^{\otimes(k-1)}\right)(y_{k}).$

Such an element $y_{k}$ is called a _$k$ -extension_ of $x$ and
$\mathsf{Ext}_{k}(\mathsf{C}_{A},\mathsf{C}_{B},\phi)$ is called the cone of
_$k$ -extendible elements_. Observe that

$\left(\mathrm{Id}_{V_{A}}\otimes
P_{\mathrm{Sym}_{k}(V_{B})}\right)\left(\mathsf{C}_{A}{\otimes_{\max}}\mathsf{C}_{B}^{{\otimes_{\max}}k}\right)=\left(V_{A}\otimes\mathrm{Sym}_{k}(V_{B})\right)\cap\left(\mathsf{C}_{A}{\otimes_{\max}}\mathsf{C}_{B}^{{\otimes_{\max}}k}\right)$

and therefore, whenever $y_{k}$ is a $k$-extension of $x$, then
$(\mathrm{Id}_{V_{A}}\otimes\mathrm{Id}_{V_{B}}^{\otimes(k-1)}\otimes\phi)(y_{k})$
is a $(k-1)$-extension of $x$. It follows that the sequence
$\mathsf{Ext}_{k}(\mathsf{C}_{A},\mathsf{C}_{B},\phi)$ is a decreasing
sequence of cones. We also define

$\mathsf{Ext}_{\infty}(\mathsf{C}_{A},\mathsf{C}_{B},\phi)=\bigcap_{k\geqslant
1}\mathsf{Ext}_{k}(\mathsf{C}_{A},\mathsf{C}_{B},\phi).$

The first level of this hierarchy is the maximal tensor product itself, i.e.,
we have
$\mathsf{Ext}_{1}(\mathsf{C}_{A},\mathsf{C}_{B},\phi)=\mathsf{C}_{A}{\otimes_{\max}}\mathsf{C}_{B}$,
and Theorem 1 asserts that
$\mathsf{Ext}_{\infty}(\mathsf{C}_{A},\mathsf{C}_{B},\phi)=\mathsf{C}_{A}{\otimes_{\min}}\mathsf{C}_{B}$.

###  2.2. The dual of the extendibility hierarchy

The adjoint of the $k$th reduction map $\gamma_{k}^{\phi}$ is the map
$(\gamma_{k}^{\phi})^{*}:V_{B}^{*}\to(V_{B}^{*})^{\otimes k}$ which for
$\psi\in V_{B}^{*}$ reads as

$(\gamma_{k}^{\phi})^{*}(\psi)=P_{\mathrm{Sym}_{k}(V_{B}^{*})}(\psi\otimes\phi^{\otimes(k-1)}).$

Let $\zeta\in V_{A}^{*}\otimes V_{B}^{*}$. It follows immediately from (3)
that

(4)
$\zeta\in\mathsf{Ext}_{k}(\mathsf{C}_{A},\mathsf{C}_{B},\phi)^{*}\iff(\mathrm{Id}_{V_{A}}\otimes(\gamma_{k}^{\phi})^{*})(\zeta)\in\mathsf{C}_{A}^{*}{\otimes_{\min}}(\mathsf{C}_{B}^{*})^{{\otimes_{\min}}k}.$

In particular, we can write down a statement which is dual to Theorem 1.

######  Theorem 4.

Let $\mathsf{C}_{A}\subseteq V_{A}$ and $\mathsf{C}_{B}\subseteq V_{B}$ be
proper cones. For every
$x\in\operatorname{int}(\mathsf{C}_{A}{\otimes_{\max}}\mathsf{C}_{B})$ and
$y\in\operatorname{int}(\mathsf{C}_{B})$, there is an integer $k\geqslant 1$
such that

$\left(\mathrm{Id}_{V_{A}}\otimes P_{\mathrm{Sym}_{k}(V_{B})}\right)(x\otimes
y^{\otimes(k-1)})\in\mathsf{C}_{A}{\otimes_{\min}}\mathsf{C}_{B}^{{\otimes_{\min}}k}.$

###### Proof.

By Theorem 1 applied to $\mathsf{C}_{A}^{*}$ and $\mathsf{C}_{B}^{*}$, we have

$\mathsf{C}_{A}^{*}{\otimes_{\min}}\mathsf{C}_{B}^{*}=\mathsf{Ext}_{\infty}(\mathsf{C}_{A}^{*},\mathsf{C}_{B}^{*},y)=\bigcap_{k\in\mathbf{N}}\mathsf{Ext}_{k}(\mathsf{C}_{A}^{*},\mathsf{C}_{B}^{*},y).$

Using the bipolar theorem in the form
$(\mathsf{C}^{*})^{*}=\overline{\mathsf{C}}$ whenever $\mathsf{C}$ is a not-
necessarily-closed convex cone, this is equivalent to the dual equation

$\mathsf{C}_{A}{\otimes_{\max}}\mathsf{C}_{B}=\left(\bigcap_{k\in\mathbf{N}}\mathsf{Ext}_{k}(\mathsf{C}_{A}^{*},\mathsf{C}_{B}^{*},y)\right)^{*}=\overline{\bigcup_{k\in\mathbf{N}}\mathsf{Ext}_{k}(\mathsf{C}_{A}^{*},\mathsf{C}_{B}^{*},y)^{*}}.$

Since $x\in\operatorname{int}(\mathsf{C}_{A}{\otimes_{\max}}\mathsf{C}_{B})$,
there is an integer $k\geqslant 1$ such that
$x\in\mathsf{Ext}_{k}(\mathsf{C}_{A}^{*},\mathsf{C}_{B}^{*},y)^{*}$. By (4),
this means that
$(\mathrm{Id}_{V_{A}}\otimes(\gamma_{k}^{y})^{*})(x)\in\mathsf{C}_{A}{\otimes_{\min}}\mathsf{C}_{B}^{{\otimes_{\min}}k}$,
which is the desired conclusion. ∎

###  2.3. Examples

#### Simplicial cones

We say that a cone $\mathsf{C}\subseteq V$ is _simplicial_ (or _classical_) if
any of its bases is a simplex. Since $\gamma_{1}^{\phi}$ is the identity
operator for any nonzero linear form $\phi$, the case $k=1$ of Theorem 2 says
that a cone $\mathsf{C}_{B}$ is simplicial if and only if
$\mathsf{C}_{A}{\otimes_{\min}}\mathsf{C}_{B}=\mathsf{C}_{A}{\otimes_{\max}}\mathsf{C}_{B}$
for every proper cone $\mathsf{C}_{A}$. This result can be traced back to [12]
and is much simpler to prove than the case $k>1$ of Theorem 2. A strengthening
of the $k=1$ case was recently obtained [1]: given two proper cones
$\mathsf{C}_{A}$, $\mathsf{C}_{B}$, we have
$\mathsf{C}_{A}{\otimes_{\min}}\mathsf{C}_{B}=\mathsf{C}_{A}{\otimes_{\max}}\mathsf{C}_{B}$
if and only if either $\mathsf{C}_{A}$ or $\mathsf{C}_{B}$ is simplicial.

#### Cone over a square

Consider the following vectors in $\mathbf{R}^{3}$

(5) $x_{+0}=(1,1,0),\ x_{-0}=(1,-1,0),\ x_{0+}=(1,0,1),\ x_{0-}=(1,0,-1)$

and let $\mathsf{C}_{B}$ be the cone they generate (see Figure 2).

$\bullet$$0$$x_{-0}$$x_{+0}$$x_{0-}$$x_{0+}$ Figure 2. The cone
$\mathsf{C}_{B}$ over a square

The dual cone $\mathsf{C}^{*}_{B}$ is generated by the following linear forms,
identified with vectors in $\mathbf{R}^{3}$

$\psi_{++}=\frac{1}{2}(1,1,1),\ \psi_{+-}=\frac{1}{2}(1,1,-1),\
\psi_{-+}=\frac{1}{2}(1,-1,1),\ \psi_{--}=\frac{1}{2}(1,-1,-1).$

Consider the linear form $\phi=(1,0,0)$. The corresponding base $K_{\phi}$ of
$\mathsf{C}_{B}$ is the square whose vertices are the vectors (5). The second
reduction map satisfies $\gamma^{\phi}_{2}(u\otimes v)=\frac{u+v}{2}$ for
every $u,v\in K_{\phi}$. One can check that it can be rewritten, for
$y\in\mathbf{R}^{3}\otimes\mathbf{R}^{3}$, as

$\begin{split}2\gamma^{\phi}_{2}(y)=\
&(\psi_{++}\otimes\psi_{+-}+\psi_{+-}\otimes\psi_{++})(y)\cdot x_{+0}\\\
&+(\psi_{-+}\otimes\psi_{--}+\psi_{--}\otimes\psi_{-+})(y)\cdot x_{-0}\\\
&+(\psi_{++}\otimes\psi_{-+}+\psi_{-+}\otimes\psi_{++})(y)\cdot x_{0+}\\\
&+(\psi_{+-}\otimes\psi_{--}+\psi_{--}\otimes\psi_{+-})(y)\cdot
x_{0-}.\end{split}$

Let $\mathsf{C}_{A}\subseteq V_{A}$ be a arbitrary cone. For any
$z\in\mathsf{C}_{A}{\otimes_{\max}}\mathsf{C}_{B}{\otimes_{\max}}\mathsf{C}_{B}$,
this formula shows that $(\mathrm{Id}_{A}\otimes\gamma_{2}^{\phi})(z)$ belongs
to $\mathsf{C}_{A}{\otimes_{\min}}\mathsf{C}_{B}$, since it produces a
decomposition into a sum of $4$ terms (with nonnegative weights) of the form
$x_{A}\otimes x_{B}$ with $x_{A}\in\mathsf{C}_{A}$ and $x_{B}$ one of the
vectors in (5).

We also observe that the choice of the linear form $\phi$ is crucial. If we
choose $\phi^{\prime}\in\operatorname{int}(\mathsf{C}_{B}^{*})$ which is not
proportional to $\phi$, then the corresponding base $K_{\phi^{\prime}}$ is a
quadrilateral which is not a parallelogram. Theorem 2 then implies that the
extendibility hierarchy for $\phi^{\prime}$ does not stop after finitely many
steps.

#### The quantum case

Consider an operator
$\rho_{AB}\in(\mathsf{M}_{d_{1}}\otimes\mathsf{M}_{d_{2}})^{+}$ and an integer
$k\geqslant 1$. We say that $\rho_{AB}$ is $k$-max-extendible if there exists
an operator $\sigma_{AB_{1}\cdots
B_{k}}\in\mathsf{M}_{d_{1}}^{+}{\otimes_{\max}}\left(\mathsf{M}_{d_{2}}^{+}\right)^{{\otimes_{\max}}k}$
such that

$\rho_{AB}=\left(\mathrm{Id}_{\displaystyle\mathsf{M}^{\mathrm{sa}}_{d_{1}}}\otimes\gamma^{\operatorname{Tr}}_{k}\right)\left(\sigma_{AB_{1}\cdots
B_{k}}\right).$

If moreover the operator $\sigma_{AB_{1}\cdots B_{k}}$ can be chosen to be
positive semidefinite, we say that $\rho_{AB}$ is $k$-PSD-extendible. An
operator which is $k$-PSD-extendible is also $k$-max-extendible, and the
following proposition shows that the converse is false.

######  Proposition 5 (Proved in Appendix 5).

There is an operator $\rho_{AB}\in(\mathsf{M}_{3}\otimes\mathsf{M}_{3})^{+}$
which is $2$-max-extendible but not $2$-PSD-extendible.

The usual monogamy theorem [6, 16] states that an operator which is $k$-PSD-
extendible for every integer $k\geqslant 1$ is separable; the stronger version
given by Theorem 1 is that an operator which is $k$-max-extendible for every
$k\geqslant 1$ is separable.

Finally, let us mention the following consequence of Theorem 4. To our
knowledge, this result does not appear in the quantum information literature.

######  Corollary 6.

Let $\rho_{AB}\in(\mathsf{M}_{d_{1}}\otimes\mathsf{M}_{d_{2}})^{+}$ be of full
rank. Then there exists an integer $k\geqslant 1$ such that the operator on
$\mathsf{M}_{d_{1}}\otimes\mathsf{M}_{d_{2}}^{\otimes k}$ defined as

$\frac{1}{k}\sum_{i=1}^{k}\rho_{AB_{i}}=\left(\mathrm{Id}_{\displaystyle\mathsf{M}_{d_{1}}^{\mathrm{sa}}}\otimes
P_{\mathrm{Sym}_{k}(\mathsf{M}_{d_{2}}^{\mathrm{sa}})}\right)\left(\rho_{AB}\otimes\mathbf{1}^{\otimes(k-1)}_{d_{2}}\right)$

is fully separable, i.e., belongs to
$\mathsf{M}_{d_{1}}^{+}{\otimes_{\min}}\left(\mathsf{M}_{d_{2}}^{+}\right)^{{\otimes_{\min}}k}$.

##  3\. Proof of Theorem 1

One direction is easy: if $x\in\mathsf{C}_{A}{\otimes_{\min}}\mathsf{C}_{B}$,
we may write $x=\sum_{i}a_{i}\otimes b_{i}$ with $a_{i}\in\mathsf{C}_{A}$ and
$b_{i}\in\mathsf{C}_{B}$. We may assume by rescaling $a_{i}$ that
$\phi(b_{i})=1$. The formula $y_{k}=\sum_{i}a_{i}\otimes b_{i}^{\otimes k}$
shows that $x$ is $k$-extendible for every integer $k$.

The other direction of the proof relies on a specific choice of extensions. We
say that $(y_{k})_{k\geqslant 0}$ is a _compatible sequence_ (for
$(\mathsf{C}_{A},\mathsf{C}_{B},\phi)$) if the following conditions hold:

  1. (i)

We have $y_{0}\in\mathsf{C}_{A}\setminus\\{0\\}$;

  2. (ii)

For every $k\geqslant 1$, we have
$y_{k}\in\Big{(}V_{A}\otimes\mathrm{Sym}_{k}(V_{B})\Big{)}\cap\left(\mathsf{C}_{A}{\otimes_{\max}}\mathsf{C}_{B}^{{\otimes_{\max}}k}\right)$;

  3. (iii)

For every $k\geqslant 1$, we have

$y_{k-1}=\left(\mathrm{Id}_{V_{A}}\otimes\mathrm{Id}_{V_{B}}^{\otimes(k-1)}\otimes\phi\right)(y_{k}).$

Iterating the last property shows that whenever $(y_{k})_{k\geqslant 0}$ is a
compatible sequence, then $y_{k}$ is a $k$-extension of $y_{1}$ for every
$k\geqslant 1$.

Our proof strategy is to first show in Lemma 7 that for every
$x\in\mathsf{Ext}_{\infty}(\mathsf{C}_{A},\mathsf{C}_{B},\phi)$ the extensions
can be chosen such that they form a compatible sequence. Then we use a
representation of compatible sequences given by Proposition 8 to finish the
proof.

######  Lemma 7.

Let $x\in\mathsf{Ext}_{\infty}(\mathsf{C}_{A},\mathsf{C}_{B},\phi)$. Then
there exists a compatible sequence $(y_{k})_{k\geqslant 0}$ such that
$y_{1}=x$.

###### Proof.

We necessarily have $y_{1}=x$ and $y_{0}=(\mathrm{Id}_{V_{A}}\otimes\phi)(x)$.
For every $k\geqslant 2$, let $x_{k}$ be an arbitrary $k$-extension of $x$.
The compatibility can be enforced by a compactness argument. For $k\leqslant
n$, the vector

$y_{k,n}=(\mathrm{Id}_{V_{A}}\otimes\mathrm{Id}_{V_{B}^{\otimes
k}}\otimes\phi^{\otimes(n-k)})(x_{n})$

is a $k$-extension of $x$. Since for every integer $k$ the set of
$k$-extensions of $x$ is compact, by a diagonal extraction process (see, e.g.,
[13, Theorem I.24]), we may find an increasing function
$g:\mathbf{N}\to\mathbf{N}$ such that the limit
$y_{k}=\lim_{n\to\infty}y_{k,g(n)}$ exists for every $k\geqslant 2$. The
sequence $(y_{k})_{k\geqslant 0}$ is compatible, as needed. ∎

######  Proposition 8.

Let $\mathsf{C}_{A}\subseteq V_{A}$, $\mathsf{C}_{B}\subseteq V_{B}$ be proper
cones and $\phi\in\operatorname{int}(\mathsf{C}_{B}^{*})$. Let
$(y_{k})_{k\geqslant 0}$ be a compatible sequence. Then there exists a couple
$(\pi,\alpha)$, where

  1. (i)

$\pi$ is a Borel probability measure on $K_{\phi}$,

  2. (ii)

$\alpha:K_{\phi}\to V_{A}$ is a Borel map such that
$\pi(\\{\alpha\in\mathsf{C}_{A}\\})=1$,

  3. (iii)

for every integer $k\geqslant 0$,

(6) $y_{k}=\int_{K_{\phi}}\alpha(\omega)\otimes\omega^{\otimes
k}\,\mathrm{d}\pi(\omega).$

Moreover, if $(\pi^{\prime},\alpha^{\prime})$ is another couple satisfying
these three conditions, then $\pi=\pi^{\prime}$ and
$\pi(\\{\alpha=\alpha^{\prime}\\})=1$.

The result of Proposition 8 is very similar to (and could be deduced from) a
theorem by Barrett and Leifer [3] which is a version of the de Finetti theorem
for arbitrary cones (see also [5] for related results). This extends the
argument given in [4] for the quantum case. For the reader’s convenience, we
will present a new simpler proof of Proposition 8 using directly the classical
de Finetti theorem.

Let us recall the statement of the de Finetti theorem. Let $F$ be a finite set
and $(Y_{n})_{n\geqslant 1}$ be a sequence of $F$-valued random variables
which is _exchangeable_ (i.e., for any integer $n\geqslant 1$ and any
permutation $\sigma\in\mathfrak{S}_{n}$, the tuples $(Y_{1},\dots,Y_{n})$ and
$(Y_{\sigma(1)},\dots,Y_{\sigma(n)})$ have the same distribution). The de
Finetti theorem asserts that there is a unique Borel probability measure $\pi$
on the set $\operatorname{Prob}(F)$ of probability measures on $F$ such that
for every $n\geqslant 1$ and $x_{1},\dots,x_{n}\in F^{n}$,

$\mathbf{P}(X_{1}=x_{1},\dots,X_{n}=x_{n})=\int_{\operatorname{Prob}(F)}p(x_{1})\dots
p(x_{n})\,\mathrm{d}\pi(p).$

We need a small extension of this statement. Assume that $W$ is an additional
random variable taking values in a finite set $E$ and such that the sequence
$(X_{n})$ is exchangeable conditionally on $W$. Then there is a unique family
$(\pi_{i})_{i\in E}$ of positive Borel measures on $\operatorname{Prob}(F)$
such that, for $i\in E$

(7)
$\mathbf{P}(W=i,X_{1}=x_{1},\dots,X_{n}=x_{n})=\int_{\operatorname{Prob}(F)}p(x_{1})\dots
p(x_{n})\,\mathrm{d}\pi_{i}(p).$

We necessarily have $\pi=\sum_{i\in E}\pi_{i}$. For $i\in E$, denote by
$\alpha_{i}$ the Radon–Nikodym derivative of $\pi_{i}$ with respect to $\pi$.
We can rewrite (7) as

(8)
$\mathbf{P}(W=i,X_{1}=x_{1},\dots,X_{n}=x_{n})=\int_{\operatorname{Prob}(F)}\alpha_{i}(p)p(x_{1})\dots
p(x_{n})\,\mathrm{d}\pi(p).$

We also use the following simple lemma.

######  Lemma 9.

Let $K\subseteq\mathbf{R}^{n}$ be a convex body, i.e., a compact convex set
with non-empty interior. There is a sequence $(\Delta_{k})_{k\geqslant 1}$ of
simplices such that $\bigcap\Delta_{k}=K$.

###### Proof.

Without loss of generality, assume that $0\in K$. It suffices to show that for
every $x\in\mathbf{R}^{n}\setminus K$, there is a simplex $\Delta$ containing
$K$ but not $x$. This can be seen as follows: by the Hahn–Banach separation
theorem, there is a linear form $f_{1}$ such that $f_{1}(x)>B:=\max_{K}f_{1}$.
We may find linear forms $f_{2},\dots,f_{n+1}$ such that
$0\in\operatorname{int}\operatorname{\mathrm{conv}}(f_{1},\dots,f_{n+1})$. For
$A>0$, the set

$\Large\\{x\in\mathbf{R}^{n}\ :\ f_{1}(x)\leqslant B,\ f_{i}(x)\leqslant
A\textnormal{ for }i\in\\{2,\dots,n+1\\}\Large\\}$

is a simplex which does contain $x$; for $A$ large enough, it contains $K$ as
needed. ∎

###### Proof of Proposition 8.

Suppose first that the cones $\mathsf{C}_{A}$ and $\mathsf{C}_{B}$ are
simplicial. Without loss of generality, we may then identify $\mathsf{C}_{A}$
and $\mathsf{C}_{B}$ with the cone of positive measures on some finite sets
$E$ and $F$, and $K_{\phi}$ with the set of probability measures on $F$. Up to
rescaling, we may also assume that $y_{0}$ is a probability measure on $E$. In
that case, the conclusion is equivalent to the conditional de Finetti theorem
as in (8).

Consider now the general case. We use repeatedly the fact that the sequence
$(y_{k})_{k\geqslant 0}$ is a compatible sequence for
$(\mathsf{C},\mathsf{C}^{\prime},\phi)$ whenever $\mathsf{C}$ and
$\mathsf{C}^{\prime}$ are proper cones containing respectively
$\mathsf{C}_{A}$ and $\mathsf{C}_{B}$, and such that
$\phi\in\operatorname{int}(\mathsf{C}^{\prime*})$. In particular, choosing
$\mathsf{C}$ and $\mathsf{C}^{\prime}$ to be simplicial cones, we obtain the
existence of a positive Borel measure
$\pi=\pi_{\mathsf{C},\mathsf{C}^{\prime}}$ on
$\Delta:=\mathsf{C}^{\prime}\cap\phi^{-1}(1)$ and a Borel map
$\alpha=\alpha_{\mathsf{C},\mathsf{C}^{\prime}}$ such that for every
$k\geqslant 0$

$y_{k}=\int_{\Delta}\alpha(\omega)\otimes\omega^{\otimes
k}\,\mathrm{d}\pi(\omega).$

To finish the proof, it suffices to show that (i) the measure $\pi$ is
supported on $K_{\phi}$ and (ii) the function $\alpha$ is $\pi$-almost surely
$\mathsf{C}_{A}$-valued; the uniqueness property for
$(\mathsf{C}_{A},\mathsf{C}_{B})$ then follows from the uniqueness property
for $(\mathsf{C},\mathsf{C}^{\prime})$.

Let $\Delta_{1}$ be another simplex in the affine hyperplane $\phi^{-1}(1)$
such that $K_{\phi}\subseteq\Delta_{1}$ and $\Delta_{2}$ a third simplex such
that $\Delta\cup\Delta_{1}\subseteq\Delta_{2}$. By the uniqueness property for
$(\mathsf{C},\operatorname{\mathrm{cone}}(\Delta_{2}))$, it follows that
$\pi=\pi_{\mathsf{C},\operatorname{\mathrm{cone}}(\Delta_{1})}=\pi_{\mathsf{C},\operatorname{\mathrm{cone}}(\Delta_{2})}$.
In particular, the measure $\pi$ is supported in $\Delta\cap\Delta_{1}$.
Applying Lemma 9 proves that $\pi$ is supported on $K_{\phi}$.

Similarly, let $\mathsf{C}_{1}$ be another simplicial cone such that
$\mathsf{C}_{A}\subseteq\mathsf{C}_{1}$ and $\mathsf{C}_{2}$ a third
simplicial cone such that
$\mathsf{C}\cup\mathsf{C}_{1}\subseteq\mathsf{C}_{2}$. By the uniqueness
property for $(\mathsf{C}_{2},\mathsf{C}^{\prime})$, we have
$\alpha=\alpha_{\mathsf{C}_{1},\mathsf{C}^{\prime}}=\alpha_{\mathsf{C}_{2},\mathsf{C}^{\prime}}$
$\pi$-a.s. It follows that $\alpha$ takes $\pi$-a.s. values in
$\mathsf{C}\cap\mathsf{C}_{1}$. Applying Lemma 9 proves that $\alpha$ takes
$\pi$-a.s. values in $\mathsf{C}_{A}$. ∎

###### Proof of Theorem 1.

By Lemma 7, given an arbitrary element
$x\in\mathsf{Ext}_{\infty}(\mathsf{C}_{A},\mathsf{C}_{B},\phi)$, there is a
compatible sequence $(y_{k})_{k\geqslant 0}$ with $y_{1}=x$. Equation (6)
applied with $k=1$ shows that
$x=y_{1}\in\mathsf{C}_{A}{\otimes_{\min}}\mathsf{C}_{B}$. ∎

######  Remark 1.

Theorem 1 can be extended to the case of more than two factors. Given proper
cones $\mathsf{C}_{A},\mathsf{C}_{B_{1}},\dots,\mathsf{C}_{B_{d}}$ and for
every $i\in[d]$ a linear form
$\phi_{i}\in\operatorname{int}(\mathsf{C}_{B_{i}}^{*})$, the minimal tensor
product
$\mathsf{C}_{A}{\otimes_{\min}}\mathsf{C}_{B_{1}}{\otimes_{\min}}\dots\otimes_{\min}\mathsf{C}_{B_{d}}$
is equal to

$\bigcap_{k_{1},\dots k_{d}\geqslant
1}\left(\mathrm{Id}_{V_{A}}\otimes\gamma_{k_{1}}^{\phi_{1}}\otimes\dots\otimes\gamma_{k_{d}}^{\phi_{d}}\right)\left(\mathsf{C}_{A}{\otimes_{\max}}\mathsf{C}_{B_{1}}^{{\otimes_{\max}}k_{1}}{\otimes_{\max}}\dots\otimes_{\max}\mathsf{C}_{B_{d}}^{{\otimes_{\max}}k_{d}}\right).$

The proof of this result is similar to the proof of Theorem 1. The
corresponding classical de Finetti theorem is the following. Suppose that
$(X_{n}^{i})_{n\in\mathbf{N},i\in[d]}$ is a family of random variables such
that, for every finitely supported permutations $\sigma_{1},\dots,\sigma_{d}$
of the integers, $(X_{\sigma_{i}(n)}^{i})_{n\in\mathbf{N},i\in[d]}$ has the
same distribution as $(X_{n}^{i})_{n\in\mathbf{N},i\in[d]}$. Then
$(X^{i}_{n})$ is distributed as a mixture of i.i.d. variables, each element in
the mixture being distributed as an independent tuple $(Y_{i})_{i\in[d]}$. We
omit details.

##  4\. Proof of Theorem 2

Fix a proper cone $\mathsf{C}\subseteq V$ and
$\phi\in\operatorname{int}(\mathsf{C}^{*})$. The $k$th reduction map
$\gamma_{k}^{\phi}:V^{\otimes k}\to V$ can be viewed as a tensor, namely the
unique element of $(V^{*})^{\otimes k}\otimes V$ such that

(9) $\langle\gamma_{k}^{\phi},x_{1}\otimes\dots\otimes
x_{k}\otimes\psi\rangle=\frac{\psi(x_{1})+\dots+\psi(x_{k})}{k}.$

for every $x_{1},\dots,x_{k}\in K_{\phi}$ and $\psi\in V^{*}$.

We say that the $k$th reduction map $\gamma_{k}^{\phi}$ is _entanglement-
breaking_ if, as a tensor, it belongs to
$(\mathsf{C}^{*})^{{\otimes_{\min}}k}{\otimes_{\min}}\mathsf{C}$. We use the
following basic property of entanglement-breaking maps (see for instance [2,
Proposition 2.2]). Given proper cones $\mathsf{C}_{1}\subseteq V_{1}$ and
$\mathsf{C}_{2}\subseteq V_{2}$ and a linear map $\Phi:V_{1}\to V_{2}$, the
following are equivalent:

  1. (i)

The map $\Phi$, as a tensor in $V_{1}^{*}\otimes V_{2}$, belongs to
$\mathsf{C}_{1}^{*}{\otimes_{\min}}\mathsf{C}_{2}$;

  2. (ii)

For every proper cone $\mathsf{C}_{A}\subseteq V_{A}$, we have

$(\mathrm{Id}_{V_{A}}\otimes\Phi)(\mathsf{C}_{A}{\otimes_{\max}}\mathsf{C}_{1})\subseteq\mathsf{C}_{A}{\otimes_{\min}}\mathsf{C}_{2}.$

Using this equivalence for the cones
$\mathsf{C}_{1}=\mathsf{C}^{{\otimes_{\max}}k}$, $\mathsf{C}_{2}=\mathsf{C}$
and the map $\Phi=\gamma_{k}^{\phi}$, we may reformulate Theorem 2 as follows.

######  Theorem 2’.

Let $\mathsf{C}\subseteq V$ be a proper cone,
$\phi\in\operatorname{int}(\mathsf{C}^{*})$ and $k\geqslant 1$ an integer. Set
$K_{\phi}=\mathsf{C}\cap\phi^{-1}(1)$. The following are equivalent:

  1. (i)

The $k$th reduction map $\gamma_{k}^{\phi}$ is entanglement-breaking;

  2. (ii)

The base $K_{\phi}$ is affinely isomorphic to the Cartesian product of
$\leqslant k$ simplices.

If $P$ is a polytope, we denote by $\mathcal{V}(P)$ the set of its vertices
and by $\mathcal{F}(P)$ the set of its facets (=faces of codimension $1$). We
will start by proving the easy direction:

###### Proof that $(ii)$ implies $(i)$ in Theorem 2’.

A basic observation is that given two polytopes $P$ and $Q$, the faces of the
Cartesian product $P\times Q$ have the form $F\times G$, where $F$ is face of
$P$ and $G$ is face if $Q$. Consider simplices $\Delta_{1},\dots,\Delta_{k}$,
possibly $0$-dimensional, and the polytope
$\Pi=\Delta_{1}\times\dots\times\Delta_{k}$. Its vertices have the form
$(v_{1},\dots,v_{k})$ for $v_{i}\in\mathcal{V}(\Delta_{i})$ and its facets
have the form

(10) $\Delta_{1}\times\dots\times\Delta_{i-1}\times
F\times\Delta_{i+1}\times\dots\times\Delta_{k},$

for $i\in[k]$ and $F\in\mathcal{F}(\Delta_{i})$. Assume that $K_{\phi}$ is
affinely isomorphic to $\Pi$. We may label the vertices of $K_{\phi}$ as

$\\{x_{v_{1},\dots,v_{k}}\ :\ v_{i}\in\mathcal{V}(\Delta_{i})\\}$

and define, for every $i\in[k]$ and $v\in\mathcal{V}(\Delta_{i})$, an affine
map $\psi_{i}^{v}:\operatorname{\mathrm{aff}}(K_{\phi})\to\mathbf{R}$ by

$\psi_{i}^{v}(x_{v_{1},\dots,v_{k}})=\begin{cases}1&\textnormal{ if
}v_{i}=v\\\ 0&\textnormal{otherwise.}\end{cases}$

We may extend $\psi_{i}^{v}$ to a linear map defined on $V$, which we still
denote by $\psi_{i}^{v}$, and which is an element of $\mathsf{C}^{*}$. We
observe that for every $j\in[k]$,

(11) $\phi=\sum_{v\in\mathcal{V}(\Delta_{j})}\psi_{j}^{v}$

since both sides evaluate to $1$ on any vertex of $K_{\phi}$. Let $i\in[k]$,
$v\in\mathcal{V}(\Delta_{i})$ and $z_{1},\dots,z_{k}\in K_{\phi}$. We compute,
using the fact that $P_{\mathrm{Sym}_{k}(V)}^{*}=P_{\mathrm{Sym}_{k}(V^{*})}$
and (11) for every $j\neq i$,

$\displaystyle\mathllap{\left\langle\sum_{v_{1}\in\mathcal{V}(\Delta_{1})}\dots\sum_{v_{k}\in\mathcal{V}(\Delta_{k})}P_{\mathrm{Sym}_{k}(V^{*})}(\psi_{1}^{v_{1}}\otimes\dots\otimes\psi_{k}^{v_{k}})\otimes
x_{v_{1},\dots,v_{k}},z_{1}\otimes\dots\otimes
z_{k}\otimes\psi_{i}^{v}\right\rangle}$ $\displaystyle\begin{aligned}
&=\sum_{v_{1}\in\mathcal{V}(\Delta_{1})}\dots\sum_{v_{k}\in\mathcal{V}(\Delta_{k})}{\bf
1}_{\\{v_{i}=v\\}}\langle\psi_{1}^{v_{1}}\otimes\dots\otimes\psi_{k}^{v_{k}},P_{\mathrm{Sym}_{k}(V)}(z_{1}\otimes\dots\otimes
z_{k})\rangle\\\
&=\langle\phi^{\otimes(i-1)}\otimes\psi_{i}^{v}\otimes\phi^{\otimes(k-i)},P_{\mathrm{Sym}_{k}(V)}(z_{1}\otimes\dots\otimes
z_{k})\rangle\\\ &=\frac{\psi_{i}^{v}(z_{1})+\dots+\psi_{i}^{v}(z_{k})}{k}\\\
&=\langle\gamma_{k}^{\phi},z_{1}\otimes\dots\otimes
z_{k}\otimes\psi_{i}^{v}\rangle.\end{aligned}$

Since elements of the form $z_{1}\otimes\dots\otimes z_{k}\otimes\psi_{i}^{v}$
span the space $V^{\otimes k}\otimes V^{*}$, we conclude that

$\gamma_{k}^{\phi}=\sum_{v_{1}\in\mathcal{V}(\Delta_{1})}\dots\sum_{v_{k}\in\mathcal{V}(\Delta_{k})}P_{\mathrm{Sym}_{k}(V^{*})}(\psi_{1}^{v_{1}}\otimes\dots\otimes\psi_{k}^{v_{k}})\otimes
x_{v_{1},\dots,v_{k}}$

and therefore the map $\gamma_{k}^{\phi}$ is entanglement-breaking. ∎

The second half of the proof of Theorem 2’ requires a couple of preparatory
lemmas and some terminology about polytopes. Let $P$ denote a polytope. The
_avoiding set_ of a vertex $x\in\mathcal{V}(P)$ is given by

$\operatorname{Av}(x)=\\{F\in\mathcal{F}(P)\ :\ x\not\in F\\},$

i.e., the set of facets not containing $x$. We say that a tuple
$(F_{1},\dots,F_{k},x)\in\mathcal{F}(P)^{k}\times\mathcal{V}(P)$ is
_admissible_ if $\operatorname{Av}(x)\subseteq\\{F_{1},\dots,F_{k}\\}$.

######  Lemma 10.

If $\gamma_{k}^{\phi}$ is entanglement-breaking, then $K_{\phi}$ is a
polytope.

###### Proof.

By assumption, there exist an integer $N\geqslant 1$, elements
$x_{1},\dots,x_{N}$ in $K_{\phi}$ and $(f_{i}^{j})_{i\in[N],j\in[k]}$ in
$\mathsf{C}^{*}$ such that

(12) $\gamma_{k}^{\phi}=\sum_{i=1}^{N}f_{i}^{1}\otimes\dots\otimes
f_{i}^{k}\otimes x_{i}.$

For every $y\in K_{\phi}$, we have

$y=\gamma_{k}^{\phi}\left(y^{\otimes
k}\right)=\sum_{i=1}^{N}\left(\prod_{j=1}^{k}f_{i}^{j}(y)\right)x_{i}.$

It follows that
$K_{\phi}\subseteq\operatorname{\mathrm{conv}}\\{x_{1},\dots,x_{N}\\}$ and
therefore $K_{\phi}=\operatorname{\mathrm{conv}}\\{x_{1},\dots,x_{N}\\}$. In
particular, $K_{\phi}$ is a polytope. ∎

We now assume that $\gamma^{\phi}_{k}$ is entanglement-breaking and hence that
$K_{\phi}$ is a polytope. To every facet $F\in\mathcal{F}(K_{\phi})$ we
associate an element $\psi_{F}\in\mathsf{C}^{*}$ with the property that
$F=K_{\phi}\cap\ker\psi_{F}$ (this determines $\psi_{F}$ up to a positive
scalar).

######  Lemma 11.

If $\gamma_{k}^{\phi}$ is entanglement-breaking, then $\gamma_{k}^{\phi}$
belongs to the cone generated by elements of the form
$\psi_{F_{1}}\otimes\cdots\otimes\psi_{F_{k}}\otimes x$, where
$(F_{1},\ldots,F_{k},x)\in\mathcal{F}(K_{\phi})^{k}\times\mathcal{V}(K_{\phi})$
is admissible.

###### Proof.

For every facet $F\in\mathcal{F}(K_{\phi})$, pick an arbitrary element $x_{F}$
in the relative interior of $F$. With this we define the _vertex-facet tensor_
as

(13) $\omega_{k}=\sum_{F\in\mathcal{F}(K_{\phi})}x_{F}^{\otimes
k}\otimes\psi_{F}\in V^{\otimes k}\otimes V^{*}.$

Since $\psi_{F}(x_{F})=0$, it follows from (9) that
$\langle\gamma_{k}^{\phi},\omega_{k}\rangle=0$.

Given a tuple
$(F_{1},\dots,F_{k},x)\in\mathcal{F}(K_{\phi})^{k}\times\mathcal{V}(K_{\phi})$,
we have

$\langle\psi_{F_{1}}\otimes\dots\otimes\psi_{F_{k}}\otimes
x,\omega_{k}\rangle=\sum_{F\in\mathcal{F}(K_{\phi})}\psi_{F}(x)\prod_{j=1}^{k}\psi_{F_{j}}(x_{F})=\sum_{F\in\operatorname{Av}(x)}\psi_{F}(x)\prod_{j=1}^{k}\psi_{F_{j}}(x_{F}).$

This quantity is nonnegative and vanishes if and only if, for every
$F\in\operatorname{Av}(x)$, there is $j\in[k]$ such that
$\psi_{F_{j}}(x_{F})=0$. By definition of $x_{F}$, we have
$\psi_{F_{j}}(x_{F})=0$ if and only if $F=F_{j}$. We conclude that

(14) $\langle\psi_{F_{1}}\otimes\dots\otimes\psi_{F_{k}}\otimes
x,\omega_{k}\rangle\geqslant 0$

with equality if and only if $(F_{1},\dots,F_{k},x)$ is admissible.

Since $\mathsf{C}^{*}=\operatorname{\mathrm{cone}}\\{\psi_{F}\ :\
F\in\mathcal{F}(K_{\phi})\\}$, we may expand the decomposition (12) in the
form

$\gamma_{k}^{\phi}=\sum_{F_{1}\in\mathcal{F}(K_{\phi})}\dots\sum_{F_{k}\in\mathcal{F}(K_{\phi})}\sum_{x\in\mathcal{V}(K_{\phi})}\lambda_{F_{1},\dots,F_{k},x}\psi_{F_{1}}\otimes\dots\otimes\psi_{F_{k}}\otimes
x$

for some $\lambda_{F_{1},\dots,F_{k},x}\geqslant 0$. Since
$\langle\gamma_{k}^{\phi},\omega_{k}\rangle=0$, we conclude from (14) that
$\lambda_{F_{1},\dots,F_{k},x}=0$ whenever $(F_{1},\dots,F_{k},x)$ is not
admissible. This finishes the proof. ∎

###### Proof that $(i)$ implies $(ii)$ in Theorem 2’.

Assume that the map $\gamma_{k}^{\phi}$ is entanglement-breaking. By Lemma 11
the base $K_{\phi}$ is a polytope. We now show that

(15)
$\operatorname{\mathrm{aff}}\left(\bigcap_{j=1}^{n}F_{j}\right)=\bigcap_{j=1}^{n}\operatorname{\mathrm{aff}}(F_{j}),$

for any integer $n\geqslant 1$ and
$F_{1},\dots,F_{n}\in\mathcal{F}(K_{\phi})$. Since any face is an intersection
of facets, (15) implies that $K_{\phi}$ satisfies the condition (ii) from
Theorem 3 and therefore is affinely equivalent to a product of simplices.

Note that $\operatorname{\mathrm{aff}}\left(\bigcap
F_{j}\right)\subseteq\bigcap\operatorname{\mathrm{aff}}(F_{j})$. Conversely,
consider $a\in\bigcap\operatorname{\mathrm{aff}}(F_{j})$ and any $h\in V^{*}$
which vanishes on $\bigcap F_{j}$. For every $j\in[n]$, we have
$\psi_{F_{j}}(a)=0$ since $a\in\operatorname{\mathrm{aff}}(F_{j})$. For any
admissible
$(G_{1},\dots,G_{k},y)\in\mathcal{F}(K_{\phi})^{k}\times\mathcal{V}(K_{\phi})$,
we have

(16) $\langle\psi_{G_{1}}\otimes\dots\otimes\psi_{G_{k}}\otimes y,a^{\otimes
k}\otimes h\rangle=\psi_{G_{1}}(a)\dots\psi_{G_{k}}(a)h(y)=0.$

To prove (16), observe that if $h(y)\neq 0$, then $y\not\in\bigcap F_{j}$ and
hence there is an index $j\in[n]$ such that $y\not\in F_{j}$. This means that
$F_{j}\in\operatorname{Av}(y)$ and therefore $F_{j}=G_{i}$ for some $i\in[k]$.
We obtain that $\psi_{G_{i}}(a)=\psi_{F_{j}}(a)=0$, proving (16).

By Lemma 11, it follows that $\langle\gamma_{k}^{\phi},a^{\otimes k}\otimes
h\rangle=0$. By (9), we have $h(a)=\langle\gamma_{k}^{\phi},a^{\otimes
k}\otimes h\rangle$ and therefore $h(a)=0$. Since every $h\in V^{*}$ which
vanishes on $\bigcap F_{j}$ vanishes at $a$, we have
$a\in\operatorname{\mathrm{span}}(\bigcap F_{j})$. We conclude the proof of
(15) by observing that
$a\in\operatorname{\mathrm{aff}}(K_{\phi})\cap\operatorname{\mathrm{span}}(\bigcap
F_{j})=\operatorname{\mathrm{aff}}(\bigcap F_{j})$.

Theorem 3 tells us that $K_{\phi}$ is affinely isomorphic to
$\Pi=\Delta_{1}\times\dots\times\Delta_{l}$ for some integer $l\geqslant 1$
and nontrivial simplices $\Delta_{1},\dots,\Delta_{l}$; we still need to show
that $l\leqslant k$. By Lemma 11, there is at least one admissible tuple
$(F_{1},\dots,F_{k},x)\in\mathcal{F}(K_{\phi})^{k}\times\mathcal{V}(K_{\phi})$.
In particular, $\operatorname{\mathrm{card}}\operatorname{Av}(x)\leqslant k$.
On the other hand, we observe from (10) that the polytope $\Pi$ has the
property that the avoiding set of any vertex contains exactly $l$ elements. It
follows that $l\leqslant k$. ∎

### Further remarks:

We finish this section with two remarks about concepts appearing in the
previous proof.

(a) It is easy to see that if $K$ is the product of $k$ nontrivial simplices,
then $\operatorname{\mathrm{card}}(\operatorname{Av}\left(x\right))=k$ for
every vertex $x\in\mathcal{V}(K)$. For $k=1$ the converse holds as well. It
would be interesting to determine which polytopes $K$ have the property that
$\operatorname{\mathrm{card}}(\operatorname{Av}\left(x\right))=2$ for every
vertex $x\in\mathcal{V}(K)$. It was observed by Martin Winter [9] that this
class of polytopes is closed under taking free joins. Therefore, it contains
examples that are not products of simplices, e.g., the free join of two
squares.

(b) The vertex-facet tensor $\omega_{k}$ from (13) is an interesting object
encoding certain combinatorial properties of the polytope $K_{\phi}$. For
example, we have the following proposition:

######  Proposition 12.

Consider a polyhedral cone $\mathsf{C}$ with base
$K_{\phi}=\phi^{-1}(1)\cap\mathsf{C}$ for some functional
$\phi\in\operatorname{int}(\mathsf{C}^{*})$ and an integer $k\geqslant 1$.
Then, the following are equivalent:

  1. (i)

We have

$\omega_{k}\in\operatorname{int}\left(\mathsf{C}^{\otimes_{\max}k}\otimes_{\max}\mathsf{C}^{*}\right);$

  2. (ii)

For every vertex $x\in\mathcal{V}(K_{\phi})$, we have
$\operatorname{\mathrm{card}}(\operatorname{Av}\left(x\right))>k$.

###### Proof.

For any proper cone $\mathsf{C}_{0}\subseteq V_{0}$, the interior of
$\mathsf{C}_{0}^{*}$ is the set of linear forms $f\in V_{0}^{*}$ such that
$f(x)>0$ for every $x\in\mathsf{C}_{0}\setminus\\{0\\}$. Using this
observation for the cone
$\mathsf{C}_{0}=\left(\mathsf{C}^{*}\right)^{{\otimes_{\min}}k}{\otimes_{\min}}\mathsf{C}$
shows that (i) is equivalent to the statement

$\forall F_{1},\dots,F_{k}\in\mathcal{F}(K_{\phi}),\ \forall
x\in\mathcal{V}(K_{\phi}),\ \exists
F\in\mathcal{F}(K_{\phi})\setminus\\{F_{1},\dots,F_{k}\\}\ :\ \psi_{F}(x)>0,$

which is equivalent to (ii). ∎

Since $\langle\gamma_{k}^{\phi},\omega_{k}\rangle=0$ always holds, Proposition
12 implies directly that $\gamma_{k}^{\phi}$ is not entanglement-breaking
whenever $\operatorname{\mathrm{card}}(\operatorname{Av}\left(x\right))>k$ for
every vertex $x\in\mathcal{V}(K_{\phi})$.

##  5\. Proof of Theorem 3

Before proving Theorem 3, we recall some facts about polytopes. Given a
polytope $P$ and a vertex $v\in\mathcal{V}(P)$, we denote by $P/v$ the _vertex
figure_ of $P$ at $v$ (see [8, Chapter 2.1] for definition). A $d$-polytope
$P$ is said to be _simple_ if any vertex is contained in exactly $d$ facets,
or equivalently if all its vertex figures are $(d-1)$-simplices. A polytope
$P$ is said to be $2$-_level_ if for every facet $F\in\mathcal{F}(P)$, all
vertices not in $F$ lie in the same translate of
$\operatorname{\mathrm{aff}}(F)$. Since the complement of any facet in the
graph of $P$ is connected [14, p. 475], $P$ is $2$-level if and only if any
face disjoint from any facet $F\in\mathcal{F}(P)$ lies in a translate of
$\operatorname{\mathrm{aff}}(F)$.

We use the following characterization of products of simplices. For
characterizations of products of simplices up to combinatorial equivalence,
see [17].

######  Theorem 13 (Kaibel–Wolff, [10]).

A polytope is affinely equivalent to a product of simplices if and only if it
is simple and $2$-level.

A polytope $P\subseteq\mathbf{R}^{n}$ is said to be a _0/1-polytope_ if its
vertices are a subset of $\\{0,1\\}^{n}$. Kaibel and Wolff proved in [10] that
a simple 0/1-polytope is a product of simplices. Since any $2$-level polytope
is affinely equivalent to a 0/1-polytope (see [7]), this implies Theorem 13.

###### Proof of Theorem 3.

The fact that a simplex satisfies (ii) is a consequence of the following
observation: if $(x_{i})_{i\in I}$ are affinely independent, then for every
$J_{1}$, $J_{2}\subseteq I$, we have

$\operatorname{\mathrm{aff}}(\\{x_{i}\ :\ i\in J_{1}\cap
J_{2}\\})=\operatorname{\mathrm{aff}}(\\{x_{i}\ :\ i\in
J_{1}\\})\cap\operatorname{\mathrm{aff}}(\\{x_{i}\ :\ i\in J_{2}\\}).$

By induction, to prove (i) $\Longrightarrow$ (ii), it suffices to prove that
the class of polytopes satisfying (ii) is closed under products. Let $P$ and
$Q$ be polytopes satisfying (ii) and consider a family $(F_{i})_{i\in I}$ of
faces of $P\times Q$. There are faces $(G_{i})_{i\in I}$ of $P$ and
$(H_{i})_{i\in I}$ of $Q$ such that $F_{i}=G_{i}\times H_{i}$ for every $i\in
I$. Using the relation $\operatorname{\mathrm{aff}}(A\times
B)=\operatorname{\mathrm{aff}}(A)\times\operatorname{\mathrm{aff}}(B)$
whenever $A$, $B$ are subsets of vector spaces, we obtain

$\operatorname{\mathrm{aff}}\left(\bigcap_{i\in
I}F_{i}\right)=\operatorname{\mathrm{aff}}\left(\bigcap_{i\in
I}G_{i}\times\bigcap_{i\in
I}H_{i}\right)=\operatorname{\mathrm{aff}}\left(\bigcap_{i\in
I}G_{i}\right)\times\left(\bigcap_{i\in I}H_{i}\right)\\\ =\left(\bigcap_{i\in
I}\operatorname{\mathrm{aff}}(G_{i})\right)\times\left(\bigcap_{i\in
I}\operatorname{\mathrm{aff}}(H_{i})\right)=\bigcap_{i\in
I}\operatorname{\mathrm{aff}}(G_{i})\times\operatorname{\mathrm{aff}}(H_{i})=\bigcap_{i\in
I}\operatorname{\mathrm{aff}}(F_{i})$

and the implication (i) $\Longrightarrow$ (ii) follows.

Conversely, let $P$ be an $n$-polytope satisfying (ii). We first show that $P$
is simple. Consider a vertex $x\in\mathcal{V}(P)$ and facets
$F_{1},\dots,F_{n-1}\in\mathcal{F}(P)$ containing $x$. The set
$\bigcap\operatorname{\mathrm{aff}}(F_{i})$ is the nonempty intersection of
$n-1$ affine hyperplanes, hence has dimension $\geqslant 1$. By (2), the face
$\bigcap F_{i}$ has dimension $\geqslant 1$. By the 1-1 correspondence between
faces of $P$ containing $x$ and faces of $P/x$ (see [8, Proposition 2.4]), the
$(n-1)$-polytope $P/x$ is dual-$(n-1)$-neighbourly, i.e., any $n-1$ facets
have a common point. Since a $d$-polytope which is dual-$k$-neighbourly for
$k>\lfloor d/2\rfloor$ is a simplex (see [8, p. 123]), it follows that $P/x$
is a simplex for every $x\in\mathcal{V}(P)$ and therefore $P$ is simple.

Let $F\in\mathcal{F}(P)$ and $G$ be a face of $P$ such that $F\cap
G=\emptyset$. By (2), we have
$\operatorname{\mathrm{aff}}(F)\cap\operatorname{\mathrm{aff}}(G)=\emptyset$
and therefore $G$ lies in a translate of $\operatorname{\mathrm{aff}}(F)$.
This means that $P$ is $2$-level. By Theorem 13, it follows that $P$ is
affinely equivalent to a product of simplices. ∎

### Acknowledgements

GA was supported in part by ANR (France) under the grant ESQuisses
(ANR-20-CE47-0014-01). AMH acknowledges funding from the European Union’s
Horizon 2020 research and innovation programme under the Marie Skłodowska-
Curie Action TIPTOP (grant no. 843414). MP was supported by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation, project numbers
447948357 and 440958198), the Sino-German Center for Research Promotion
(Project M-0294), the ERC (Consolidator Grant 683107/TempoQ) and the Alexander
von Humboldt Foundation. We thank Ludovico Lami and Christophe Sabot for
useful discussions.

##  Appendix A Proof of Proposition 5

We denote by $(|1\rangle,|2\rangle,|3\rangle)$ the canonical basis of
$\mathbf{C}^{3}$. As common in the quantum information theory literature, we
write $|ij\rangle$ as a shortcut for $|i\rangle\otimes|j\rangle$. Consider the
operators

$\displaystyle X_{\alpha,\beta,\gamma}$ $\displaystyle=$
$\displaystyle\alpha\sum_{i}|ii\rangle\langle ii|+\beta\sum_{i\neq
j}|ij\rangle\langle ij|+\gamma\sum_{i\neq j}|ii\rangle\langle jj|$
$\displaystyle=$
$\displaystyle{\footnotesize\begin{pmatrix}\alpha&&&&\gamma&&&&\gamma\\\
&\beta&&&&&&&\\\ &&\beta&&&&&&\\\ &&&\beta&&&&&\\\
\gamma&&&&\alpha&&&&\gamma\\\ &&&&&\beta&&&\\\ &&&&&&\beta&&\\\
&&&&&&&\beta&\\\ \gamma&&&&\gamma&&&&\alpha\end{pmatrix}.}$

with parameters $\alpha$, $\beta$, $\gamma\in\mathbf{R}$. This operator is
positive semidefinite if $\beta\geqslant 0$ and
$\alpha\geqslant\gamma\geqslant 0$. Set $\eta=1-\sqrt{2}/2$. Consider the
operator

$Y=X_{1,\eta,1}=\frac{1}{4}X_{4,1,2\sqrt{2}+1}+\frac{3-2\sqrt{2}}{4}X_{0,1,1}\in(\mathsf{M}_{3}\otimes\mathsf{M}_{3})^{+}$

Observe the following facts.

  * •

The operator $X_{4,1,2\sqrt{2}+1}$ is $2$-PSD-extendible, an extension being
given by
$|\psi_{1}\rangle\langle\psi_{1}|+|\psi_{2}\rangle\langle\psi_{2}|+|\psi_{3}\rangle\langle\psi_{3}|$,
with

$\displaystyle|\psi_{1}\rangle=\sqrt{2}|111\rangle+|212\rangle+|221\rangle+|313\rangle+|331\rangle,$
$\displaystyle|\psi_{2}\rangle=\sqrt{2}|222\rangle+|323\rangle+|332\rangle+|121\rangle+|112\rangle,$
$\displaystyle|\psi_{3}\rangle=\sqrt{2}|333\rangle+|131\rangle+|113\rangle+|212\rangle+|221\rangle.$

  * •

The operator $X_{0,1,1}$ is $2$-max-extendible, since its partial transpose is
$2$-PSD-extendible, an extension being given by $|\psi\rangle\langle\psi|$,
with

$|\psi\rangle=|123\rangle+|132\rangle+|213\rangle+|231\rangle+|312\rangle+|321\rangle.$

  * •

By the two previous points, the operator $Y$ is $2$-max-extendible as a
positive combination of $2$-max-extendible operators.

  * •

To show that $Y$ is not 2-PSD-extendible, consider the operator
$W=X_{1,\eta,-2\eta}$. It is easy to check numerically that the operator

$W_{2}=\left(\mathrm{Id}_{\mathsf{M}_{3}}\otimes
P_{\mathrm{Sym}_{2}(\mathsf{M}_{3})}\right)(W\otimes{\bf 1}_{3})$

is positive definite. Assume that $Y$ is $2$-PSD-extendible, with extension
$Y_{2}\in\left(\mathsf{M}_{3}^{\otimes 3}\right)^{+}$. We would then have

$\operatorname{Tr}(Y_{2}W_{2})=\operatorname{Tr}(YW)=3\cdot
1+6\cdot\eta^{2}+6\cdot(-2\eta)=0,$

showing that $Y_{2}=0$, leading to a contradiction.

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