For each state $\rho$, we have
\begin{equation}
\Tr(E^{\psi}_{n}(\{T\}) \rho^{\otimes n}) = \Tr(P_{\psi} \rho)^{n} = \exp(-n I_{\rho}(T)),
\end{equation}
and therefore the above $(\mathrm{is}_{\psi})$-estimation scheme satisfies the large deviation principle with rate function given by
\begin{align}
\begin{split}
I_{\rho}(T) &= - \ln \Tr(P_{\psi} \rho), \\
I_{\rho}(F) &= \begin{cases} \infty & \rho \neq P_{\psi}, \\ 0 & \rho = P_{\psi}. \end{cases}
\end{split}
\end{align}
\end{exam}

\chapter{Realizability problems}
\label{chap:realizability}
\section{Realizability}

The purpose of this section is to formally define the notion of a realizability problem for a property (or collection of properties) of a class of quantum states.
Recall that a property of quantum states is considered to be a (measurable) function, $f : \s S \to X$, mapping each quantum state, $\rho \in \state$, to their corresponding property value $x = f(\rho) \in X$.
The problem of determining or characterizing which values the function $f$ can take when evaluated over the set of all quantum states in $\s S$ is called the \textit{realizability problem for $f$}.

\begin{defn}%[Direct Image and Inverse Image]
Given a measurable function $f : \state \to X$ with $(\state, \borel{\state})$ and $(X, \borel{X})$ standard Borel spaces, define
\begin{enumerate}[(i)]
\item the \defnsty{inverse image} of $f$ as $f^{-1} : \borel{X} \to \borel{\s S}$ for $\Delta \in \borel{X}$ by
\begin{equation}
f^{-1}(\Delta) = \{ \rho \in \state \mid f(\rho) \in \Delta \},
\end{equation}
\item and the \defnsty{direct image} of $f$, by an abuse of notation, as $f : \borel{\s S} \to \borel{X}$ for $S \in \borel{\state}$ by
\begin{equation}
f(S) = \{ x \in X \mid f^{-1}(x) \cap S \neq \emptyset \}.
\end{equation}
\end{enumerate}
\end{defn}
The direct image, $f$, and inverse image, $f^{-1}$, of $f$ form an adjoint pair in the sense that for all $S \in \borel{\s S}$ and $\Delta \in \borel{X}$,
\begin{equation}
\label{eq:adjoint_pair}
f(S) \subseteq \Delta \iff S \subseteq f^{-1}(\Delta).
\end{equation}
\begin{defn}
Given a function $f : \s S \to \s X$, the inverse image evaluated at a singleton $\{x\} \in \borel{X}$,
\begin{equation}
f^{-1}(\{ x \}) = \{\rho \in \s S \mid f(\rho) = x\} \subseteq \s S,
\end{equation}
is called the \defnsty{fiber over $x$}.
\end{defn}
\begin{rem}
Whenever the function $f$ under consideration has codomain belonging to the set of real numbers, $X \subseteq \mathbb R$, the fiber over $x \in \mathbb R$ may also be referred to as the \textit{level-set} at $x$.
\end{rem}
\begin{defn}%[Realizability]
Given a property $f : \s S \to X$ and an element $x \in X$, the following conditions are equivalent:
\begin{enumerate}[(i)]
\item there exists a $\rho \in \s S$ such that $f(\rho) = x$,
\item the fiber over $x$ is non-empty, i.e. $f^{-1}(\{x\}) \neq \emptyset$,
\item $x$ is an element of the direct image $f(\s S)$ of $\s S$.
\end{enumerate}
If any, and thus all, of the above conditions are satisfied, the element $x$ is said to be \defnsty{realizable}.
Otherwise, $x$ is said to be \defnsty{unrealizable}.
The subset $f(\s S) \subseteq X$ of realizable elements will be referred to as the \defnsty{realizable region} of $f$.
The \defnsty{realizability problem} for $f$ is to decide, given $x \in \s X$, whether $x$ is realizable or unrealizable.
\end{defn}
\begin{rem}
Since the fibers of any function are necessarily disjoint,
\begin{equation}
x \neq y \implies f^{-1}(\{x\}) \cap f^{-1}(\{y\}) = \emptyset,
\end{equation}
the collection of non-empty fibers forms a \textit{partition} of $\s S$ indexed by elements of the realizable region, denoted
\begin{equation}
\s S / f \coloneqq \{ f^{-1}(\{x\}) \mid x \in f(\s S) \}.
\end{equation}
Altogether, we have the following commutative diagram of maps
\begin{equation}
\begin{tikzcd}
\s S \arrow[r, "f"] \arrow[d, twoheadrightarrow, "m_f"'] & \s X \\
\s S / f \arrow[r, equal, "\simeq"] & f(\s S) \arrow[u, hookrightarrow, "\subseteq"']
\end{tikzcd}
\end{equation}
where the map $m_f : \s S \to \s S/f$ sends each $\rho \in \s S$ to the unique non-empty fiber it belongs to:
\begin{equation}
m_f(\rho) = f^{-1}(\{f(\rho)\}).
\end{equation}
\end{rem}

\begin{exam}
Consider $\s S$ to be set of density operators acting on a two-dimensional complex Hilbert space $\s H \cong \mathbb C^{2}$, which may be identified with a point in the Bloch ball, e.g., by a point $(x, y, z) \in \mathbb R^{3}$ such that $\sqrt{x^2 + y^2 + z^2} \leq 1$ via the well-known bijection
\begin{equation}
\rho \mapsto (\tr(\sigma_X\rho), \tr(\sigma_Y \rho), \tr(\sigma_Z \rho)) \qquad (x,y,z) \mapsto \frac{1}{2}(I + x \sigma_X + y \sigma_Y + z \sigma_Z),
\end{equation}
where $\sigma_X, \sigma_Y$ and $\sigma_Z$ are the usual Pauli-operators.
A familiar example of a function of density operators that one might consider is the $\sigma_Z$-expectation value of $\rho$, i.e. $\langle \sigma_Z \rangle (\rho) = \tr(\sigma_Z \rho)$.
Evidently, the set of values for $\tr(\sigma_Z \rho)$ that can be realized by density operators is equal to the closed interval $[-1, +1]$, while the corresponding fibers over $z \in [-1, +1]$, as depicted by \cref{fig:bloch_ball_fibers} (b), are closed disks of radius $\sqrt{1-z^2}$ in the $xy$-plane displaced by a height $z$.
For a second example, consider the function $\rho \mapsto \tr(\rho^{2})$, otherwise known as the \textit{purity} of $\rho$. Evaluated in $(x,y,z)$-coordinates, purity is equal to $(1+x^2 + y^2 + z^2)/2$ and thus the realizable region for purity is the closed interval $[1/2, 1]$.
The fibers of purity, parameterized by $p = \tr(\rho^{2})$, correspond to spheres of radius $\sqrt{2p-1}$ as depicted by \cref{fig:bloch_ball_fibers} (a).
\begin{figure}
\subfloat[{$(1 + r(x,y,z)^2)/2 \in [1/2,1]$}]{
\includegraphics[width=0.28\textwidth]{figures_spheres.pdf}
\quad
\subfloat[{$z \in [-1,+1]$}]{
\includegraphics[width=0.28\textwidth]{figures_pancakes.pdf}
\quad
\subfloat[{$\arccos\left(z/r(x,y,z)\right) \in [0, \pi]$}]{
\includegraphics[width=0.28\textwidth]{figures_conics.pdf}
\caption{Various functions of the coordinates $(x,y,z) \in \mathbb R^{3}$, together with a depiction of their non-empty fibers in the unit ball. Also note $r(x,y,z) = \sqrt{x^2 + y^2 + z^2}$. In subfigure (c), the origin is treated as its own fiber and must be given its own special label distinct from those in $[0, \pi]$, such as $\infty$.}
\label{fig:bloch_ball_fibers}
\end{figure}
\end{exam}

Before continuing, it will be important to elucidate the subtle, yet ultimately critical, distinction between between \textit{properties} and \textit{parameters}.
This claimed distinction is subtle simply because properties and parameters are often colloquially used to describe aspects or features of a system and in most settings, can be used interchangeably.
Ultimately, in the setting investigated by this thesis, these two concepts can be seen to be dual to each other and thus cannot be used interchangeably.
If the state or configuration of a system is described by some element of a space of possible states, $\s S$, then a \textit{parameterization} of the system is described by a function $g : \Theta \to \s S$ where the set $\Theta$ (typically taken to be $\mathbb R^{d}$ for some integer $d$) is called the \textit{parameter space} and its elements or coordinates called \textit{parameters}.
By comparison, the notion of a property emerges from considering a function $f : \s S \to X$ where the set $X$ is called the \textit{property space}.
Any meaningful distinction between a property and a parameter, of course, disappears when the parameterization $g : \Theta \to \s S$ happens to be invertible, in which case its inverse, $g^{-1} : \s S \to \Theta$, can be considered a property.
In precisely the same manner, if a property $f : \s S \to X$ happens to be invertible, then its inverse, $f^{-1} : X \to \s S$, can be considered as a parameterization.
The focus of this thesis will be on properties which encode or represent some partial or incomplete information about the system and thus are, by assumption, \textit{not} invertible.
From this point of view, properties are considered to be conceptually distinct from parameters on the basis that to every parameter, $\theta \in \Theta$, there exists the state $g(\theta) \in \s S$, whereas there might exist property values, $x \in X$, for which there does not exist a state possessing that property.
In other words, unlike parameters, not all property values are \textit{realizable}, and thus the problem of determining which property values are realizable becomes a non-trivial problem.

\subsection{Quantitative realizability}

Here we consider a variant of a realizability problem for the property $f : \state \to X$ which aims to essentially quantify the proportion of states $\rho \in \state$ that have the property value $f(\rho) = x$ where $x \in \Delta$ belongs to some measurable subset $\Delta \subseteq X$.
The answer to this question, of course, depends on a choice of normalized measure, $\mu : \borel{\state} \to [0,1]$, over the set of states $\rho \in \state$, of which there are numerous physically motivated options~\cite{wootters1990random,zyczkowski2001induced}.
Throughout this section, we assume $\s S = \s S(\s H)$ is the set of density operators on a complex finite-dimensional Hilbert space $\s H$ equipped with topology induced by the operator norm on $\End(\s H)$ and let the $\sigma$-algebra over $\s S$ generated by that topology be denoted by $\borel{\state}$.

\begin{rem}
Let $f : \state \to X$ be a property of quantum states and let $\mu : \borel{\state} \to [0,1]$ be a prior probability measure over the set of states $\state$. The pushforward measure of $\mu$ through the property $f$, is defined for $\Delta \in \borel{X}$ by
\begin{equation}
\label{eq:prior_property_pushforward_measure}
(f_{*} \mu) (\Delta) = (\mu \circ f^{-1})(\Delta) = \mu(f^{-1}(\Delta)).
\end{equation}
\end{rem}

\begin{exam}
For example, let $\s H = \mathbb C^{d}$ and let $A \in \End(\s H)$ be a Hermitian operator with non-degenerate spectra $\{\lambda_1, \ldots, \lambda_d\} \subset \mathbb R$, and let the property under consideration be the $A$-expectation value, $e_{A} : \state \to \mathbb R$ sending $\rho \in \state$ to $e_{A}(\rho) \in \mathbb R$.
Further suppose that the prior measure, $\mu$, only has support over the set of pure states, $\proj \s H$, and when $\mu$ is restricted to its support, is equal to the normalized, $\U(d)$-invariant Haar measure over $\proj \s H$.
Then the pushforward of $\mu$ through the $A$ expectation value map, $e_{A}$, has density equal to (\cref{thm:density_nondegen_exp_val}~\cite{zhang2022probability})
\begin{align}
\diff (\mu \circ e_{A}^{-1})(a)
&= (d-1)\sum_{i=1}^{d} \frac{(a-\lambda_i)^{d-2}\mathrm{H}(a-\lambda_i)}{\prod_{j\neq i}(\lambda_j-\lambda_i)} \diff a.
\end{align}
where $\mathrm{H}(a)$ is the Heaviside function.
\end{exam}

\begin{rem}
It is important to note that the support of the pushforward measure $f_{\ast} \mu : \borel{X} \to [0,1]$ is not necessarily equal to the image of the support of $\mu : \borel{\state} \to [0,1]$ through $f : \state \to X$, i.e., $f(\supp(\mu)) \neq \supp(f_{\ast} \mu)$.
For a simple example of this phenomena, consider the subset $\s P \subset \state$ of pure states and let $f : \state \to \{ T, F \}$ be the Boolean-valued property of whether a state is pure:
\begin{equation}
f(\rho) = \begin{cases}
T & \rho \in \mathcal P, \\
F & \rho \not \in \mathcal P.
\end{cases}
\end{equation}
Now consider a measure $\mu : \borel{\state} \to [0,1]$ for which the pure states are a null-set, i.e. $\mu(\mathcal P) = 0$, but nevertheless $\mu$ has full-support on $\state$ such as the Hilbert-Schmidt measure.
The pushforward measure in this case satisfies
\begin{align}
(f_{\ast}\mu)(T) &= \mu(f^{-1}(T)) = \mu(\mathcal P) = 0, \\
(f_{\ast}\mu)(F) &= \mu(f^{-1}(F)) = \mu(\state \setminus \mathcal P) = 1,
\end{align}
and thus its support is equal to $\supp(f_{\ast}\mu) = \{F\}$.
By comparison, since the support of $\mu$ contains both pure and mixed states, we have $f(\supp(\mu)) = \{T, F\}$.
Consequently, the pushforward measure $f_{\ast} \mu$ does not always faithfully capture the realizable region in general.
\end{rem}

While a direct computation of the pushforward measure $\mu \circ f^{-1}$ is certainly desirable, for a generic property $f$, this can be prohibitively difficult to do (see \cref{rem:multiple_observables}).
Fortunately, when provided with an $f$-estimation scheme, $\seq{E_n : \borel{X} \to \bound(X^{\otimes n})}$, it becomes possible to approximate $\mu \circ f^{-1}$.
Recall that applying the POVM $E_n$ to the state $\rho^{\otimes n}$ yields a probability distribution, defined for $\Delta \in \borel{X}$ by
\begin{equation}
\label{eq:estimate_rho_measure}
\xi_n^{\rho} (\Delta) \coloneqq \Tr(E_n(\Delta) \rho^{\otimes n}).
\end{equation}
When $\mu$ represents some prior measure describing the identity of $\rho \in \state$, we encounter the notion of a de Finetti state.
\begin{defn}%[De Finetti States]
Let $\mu : \borel{\state} \to [0,1]$ be a probability measure over $\state$ and $n \in \mathbb N$ and positive integer.
The \defnsty{$\mu$-de Finetti state} of degree $n$ is
\begin{equation}
D^{\mu}_{n} \coloneqq \int_{\state} \mu(\diff \rho) \rho^{\otimes n}.
\end{equation}
\end{defn}
\begin{prop}
\label{prop:estimation_weak_convergence}
Let $f : \state \to X$ be a property of quantum states, let $\seq{E_n : \borel{X} \to \bound(\s H^{\otimes n})}$ be an $f$-estimation scheme, and let $\mu : \borel{\state} \to [0,1]$ be a probability measure over the set of states $\state$.
Then the sequence of probability measures $\seq{\Xi_{n}^{\mu} : \borel{X} \to [0,1]}$, defined for all $\Delta \in \borel{X}$ and $n \in \mathbb N$ by
\begin{equation}
\Xi_{n}^{\mu}(\Delta) = \int_{\rho \in \state} \mu(\diff \rho) \Tr(E_n(\Delta) \rho^{\otimes n}) = \Tr(E_n(\Delta) D_n^{\mu}),
\end{equation}
converges weakly to the pushforward measure of $\mu$ through $f$.
In other words, for all bounded continuous functions $g : X \to \mathbb R$,
\begin{equation}
\int_{x \in X} g(x) \Tr(\diff E_n(x) D_n^{\mu}) = \int_{\rho \in \state} g(f(\rho)) \diff \mu(\rho).
\end{equation}
\end{prop}
\begin{proof}
The proof follows from \cref{lem:pushforward_continuous} which states that the pushforward operation, $f_{*}$, is continuous with respect to weak topologies and thus respects the notion of weak convergence.
\end{proof}

\begin{rem}
If the sequence of measures, $\seq{\xi_n^\rho : \borel{X} \to [0,1]}$, defined by \cref{eq:estimate_rho_measure} satisfies the large deviations principle with rate function $I_{\rho} : X \to [0,1]$, then one might expect the sequence of measures, $\seq{\Xi_n^\rho : \borel{X} \to [0,1]}$, defined by \cref{prop:estimation_weak_convergence}, to satisfy the large deviations principle with rate function $I' : X \to [0,1]$, defined for $x \in X$ by
\begin{equation}
I'(x) \coloneqq \inf_{\rho \in S_{\mu}} I_{\rho}(x)
\end{equation}
where $S_{\mu} = \supp(\mu)$ is the support of the prior measure $\mu$.
Unfortunately, to prove $I'$ is indeed a rate function requires certain continuity assumptions of the function $\rho \mapsto I_{\rho}(x)$ which may not hold in general.
Moreover, to prove that $\seq{\Xi_n^{\mu}}$ satisfies the large deviation principle requires a more careful analysis of the uniformity with which large deviations principles for $\seq{\xi_n^{\rho}}$ hold with respect to $\rho$.
See for example, \citeauthor{finnoff2002integration}'s integration theorem of sequences of probability kernels satisfying large deviation principles~\cite{finnoff2002integration}.
\end{rem}

\subsection{Inclusivity of de Finetti states}
\label{sec:inclusivity_of_de_finetti}

\begin{defn}%[Inclusivity]
\label{defn:inclusivity}
Let $\mu : \borel{\s S} \to [0,1]$ be a probability measure with support $\supp(\mu) \subseteq \s S$ and define the function $\gamma : \mathbb N \to \mathbb R_{\geq 0}$ for each $n \in \mathbb N$ as
\begin{equation*}
\gamma(n) \coloneqq \inf \{c \in \mathbb R_{\geq 0} \mid \forall \sigma \in \supp(\mu),  c D^{\mu}_{n} \geq \sigma^{\otimes n}\}.
\end{equation*}
If $\gamma$ does not increase too quickly, i.e.,
\begin{equation*}
\lim_{n \to \infty}\gamma^{\frac{1}{n}}(n) = 1,
\end{equation*}
then $\mu$ is said to be \defnsty{inclusive} and the function $\gamma$ is terms the \defnsty{inclusivity function}.
\end{defn}

\begin{defn}
\label{defn:uniform_symmetric_state}
Let $n \in \mathbb N$ and let $\s H$ be a complex $d$-dimensional Hilbert space.
Let projection operator onto the $S_n$-symmetric subspace $\mathrm{Sym}^{n}(\s H) \subset \s H^{\otimes n}$ be denoted by $\projsym_{n}$,
\begin{equation}
\projsym_{n} = \frac{1}{n!} \sum_{\pi \in S_n} T_{n}(\pi).
\end{equation}
The \textit{normalized} state over the symmetric subspace will be expressed by
\begin{equation}
\unisym_{n} = \frac{\projsym_{n}}{\Tr(\projsym_{n})},
\end{equation}
\end{defn}

The uniform symmetric state defined above constitutes the motivating example of an inclusive probability measure where the prior measure $\mu$ is taken to be the $\U(d)$-invariant Fubini-Study or Haar measure on $\s S$ with support over the pure states, i.e., $\supp(\mu) = \{ \rho \in \s S | \mathrm{rank}(\rho) = 1\}$.
In this case, the corresponding inclusivity function is given by the dimension of the $n$th symmetric subspace, i.e., $\gamma(n) = \binom{n+d-1}{n}$ where $d = \dim(\s H)$.
Of course, this example can also be used to directly prove the inclusivity of the Hilbert-Schmidt measure with support over the entire set of density operators $\s S$ where the corresponding inclusivity measure is $\gamma(n) = \binom{n+d^2-1}{n}$.
Moreover, it is conjectured that essentially all commonly encountered probability measures over the space of quantum states, such as those considered in~\cite{zyczkowski2001induced}, are inclusive with respect to definition \cref{defn:inclusivity}.
Perhaps the simplest examples of inclusive probability measures over quantum states are the discrete probability measures.
For example, consider a discrete probability measure, $\mu$, with binary support $\supp(\mu) = \{\rho, \sigma\}$ of the form
\begin{equation}
\mu = \lambda \delta_{\rho} + (1-\lambda) \delta_{\sigma}.
\end{equation}
In this case, the inclusivity function, $\gamma(n) = \max\{\frac{1}{\lambda}, \frac{1}{1- \lambda}\}$, is independent of $n$.

\begin{thm}
\label{thm:symmetric_inclusivity}
Let $\s H$ be a $d$-dimensional complex Hilbert space, let $p \in \mathbb N$ be a positive integer and let
\begin{equation}
\seq{X_n \in \bound(\s H^{\otimes pn})}
\end{equation}
be a sequence of positive semidefinite operators.
Let $\unisym_{np}$ be the uniform state on the symmetric subspace $\mathrm{Sym}^{np}(\s H) \subseteq \s H^{\otimes np}$.
Further assume that the following limit exists for all $\psi \in \proj \s H$,
\begin{equation}
C(\psi) \coloneqq \limsup_{n\to\infty} \Tr(X_n P_{\psi}^{\otimes pn})^{\frac{1}{n}}.
\end{equation}
Then the supremum of $C(\psi)$ over all $\psi \in \proj \s H$ equals
\begin{align}
\sup_{\psi \in \proj \s H} C(\psi)
&= \limsup_{n\to\infty} \Tr(X_n \unisym_{pn})^{\frac{1}{n}} \\
\end{align}
\end{thm}
\begin{proof}
Since the ray $\psi^{\otimes pn}$ is a subspace of $\mathrm{Sym}^{pn}(\s H)$ and $X_n$ is positive semidefinite, we have
\begin{align}
\sup_{\psi \in \proj \s H} \Tr(X_n P_{\psi}^{\otimes pn})
&\leq \Tr(X_n \projsym_{pn}).
\end{align}
Furthermore, since $\mathrm{Sym}^{pn}(\s H)$ supports an irreducible representation of $\SU(\s H)$, we have by Schur's lemma
\begin{align}
\Tr(X_n \projsym_{pn}) = \dim(\mathrm{Sym}^{pn}(\s H)) \int_{\phi \in \proj \s H} \diff \mu(\phi) \Tr(X_n P_{\phi}^{\otimes pn}),
\end{align}
where $\mu$ is the normalized Haar measure for $\SU(\s H)$.
Moreover, since $\mu$ is a probability measure,
\begin{align}
\label{eq:probability_upper_bound_mean}
\int_{\phi \in \proj \s H} \diff \mu(\phi) \Tr(X_n P_{\phi}^{\otimes pn})
\leq \sup_{\psi \in \proj \s H} \Tr(X_n P_{\psi}^{\otimes pn}).
\end{align}
Finally, since $\dim(\mathrm{Sym}^{pn}(\s H)) = \binom{pn + d - 1}{pn}$ grows at polynomial rate with increasing $n$,
\begin{equation}
\label{eq:dimsym_irrelavent}
\lim_{n \to \infty} \dim(\mathrm{Sym}^{pn}(\s H))^{\frac{1}{pn}} = \lim_{n \to \infty} \dim(\mathrm{Sym}^{pn}(\s H))^{-\frac{1}{pn}} = 1.
\end{equation}
Taking the supremum limit as $n \to \infty$ then yields
\begin{equation}
\sup_{\psi \in \proj \s H} C(\psi) \leq \limsup_{n\to\infty} \Tr(X_n \unisym_{pn})^{\frac{1}{n}} \leq \sup_{\psi \in \proj \s H} C(\psi),
\end{equation}
which proves the claim.
\end{proof}

\subsection{Moment polytopes}
\label{sec:moment_polytope}

Consider a representation $\grep : G \to \GL(\s H)$ with moment map $\momap_\grep : \proj \s H \to i \mathfrak k^{*}$.
For which values of $\omega = i \mathfrak k^{*}$ does there exist a ray $\psi \in \proj \s H$ such that $\momap_\grep (\psi) = \omega$?

To answer this question, our strategy is to make use of the correspondence, provided by \cref{prop:deform_cap_properties}, between the $\omega$-capacity of a ray and the moment map of that ray equaling $\omega$.
In this setting, for any given $\psi$, we have
\begin{equation}
\projcapacity_{\grep}^{\omega}(\psi) = 1 \quad \Longleftrightarrow \quad \momap_{\grep}(\psi) = \omega,
\end{equation}
which, in turn, implies
\begin{equation}
\sup_{\psi \in \proj \s H} \projcapacity_{\grep}^{\omega}(\psi) = 1 \quad \Longleftrightarrow \quad \exists \psi \in \proj \s H :  \momap_{\grep}(\psi) = \omega.
\end{equation}
Therefore, the problem of determining which $\omega$ belong to the image of the moment map $\momap_{\grep}(\proj \s H)$, is equivalent to the problem of determining the maximum value of $\projcapacity_{\grep}^{\omega}(\psi)$.
Fortunately, the deformed strong duality result of \cref{thm:deformed_strong_duality} offers an avenue for computing $\projcapacity_{\grep}^{\omega}(\psi)$ at least when $\omega \in i \mathfrak k^{*}$ has a rational coadjoint orbit (where $\omega = \Ad^{*}(h)(\omega_+)$ and $\ell \omega_+$ is dominant, analytically integral for some $\ell$).
Given an $\omega \in i \mathfrak k^{*}$ of this form, \cref{thm:deformed_strong_duality} yields,
\begin{equation}
\projcapacity_{\grep}^{\omega}(\psi) = \limsup_{n\to\infty} \norm{\ratsub{\omega}{\grep^{\otimes n}} P_{\psi}^{\otimes n}}^{\frac{1}{n}},
\end{equation}
where we have defined $\ratsub{\omega}{\grep^{\otimes n}}$ as the projection operator onto the subspace of highest weight vectors for $\grep^{\otimes n}$ with weight $n \omega_+$ rotated by $h \in K$, in accordance with \cref{defn:rational_projectors},
\begin{equation}
\label{eq:Q_defn}
\ratsub{\omega}{\grep^{\otimes n}} = \grep^{\otimes n}(h)\hwsub{n\omega_+}{\grep^{\otimes n}} \grep^{\otimes n}(h^{-1}).
\end{equation}
In order to compute the supremum over all $\psi \in \proj \s H$ we can make use of \cref{thm:symmetric_inclusivity} (for the special case where $p = 1$ and $X_n = \ratsub{\omega}{\grep^{\otimes n}}$) to obtain the following result.
\begin{prop}
\label{prop:realizability_deformed_single}
Let $\momap_{\grep} : \proj \s H \to i \mathfrak k^*$ be the moment map for a representation $\grep : G \to \GL(\s H)$, and let $\omega \in i \mathfrak k^{*}$ have a rational coadjoint orbit. Then
\begin{equation}
\limsup_{n\to\infty} \Tr(\ratsub{\omega}{\grep^{\otimes n}} \unisym_n)^{\frac{1}{n}} = \sup_{\psi \in \proj \s H} \projcapacity_{\grep}^{\omega}(\psi),
\end{equation}
where $\unisym_n$ is the uniform state on the $S_n$-symmetric subspace $\mathrm{Sym}^{n}(\s H) \subseteq \s H^{\otimes n}$.
Therefore, there exists $\psi \in \proj \s H$ such that $\momap_{\grep}(\psi) = \omega$ if and only if
\begin{equation}
\label{eq:asymptotic_momap_realizability}
\limsup_{n\to\infty} \Tr(\ratsub{\omega}{\grep^{\otimes n}} \unisym_n)^{\frac{1}{n}} = 1.
\end{equation}
\end{prop}

In other words, the asymptotics of the quantity $\Tr(\ratsub{\omega}{\grep^{\otimes n}} \unisym_n)$ completely determines the realizability of $\omega \in i \mathfrak k^{*}$ as the moment map of some state $\psi \in \proj \s H$.
Using the symmetries of operator $\unisym_n$, one can simplify the result of \cref{prop:realizability_deformed_single}.
\begin{cor}
\label{cor:only_fundamental}
There exists $\psi \in \proj \s H$ such that $\momap_{\grep}(\psi) = \omega$ if and only if
\begin{equation}
\limsup_{n\to\infty} \Tr(\hwsub{n\omega_+}{\grep^{\otimes n}} \unisym_n)^{\frac{1}{n}} = 1,
\end{equation}
where $\hwsub{n\omega_+}{\grep^{\otimes n}}$ is the projection operator onto the subspace of highest weight vectors for $\grep^{\otimes n}$ with weight $n \omega_+$. In other words, the realizability of a rational $\omega$ depends only on where its coadjoint intersects the fundamental Weyl chamber, $\omega_+ \in C_+$.
\end{cor}
\begin{proof}
Since symmetric subspace supports an irreducible representation of $\U(\s H)$, $\grep$ restricted to the compact subgroup $K$ is a unitary representation, we have for all $k \in K$,
\begin{equation}
\label{eq:commuting_rep_unisym}
[\grep^{\otimes n}(k), \unisym_n] = 0,
\end{equation}
and therefore by \cref{eq:Q_defn} we conclude
\begin{equation}
\Tr(\ratsub{\omega}{\grep^{\otimes n}} \unisym_n) = \Tr(\grep^{\otimes n}(h)\hwsub{n\omega_+}{\grep^{\otimes n}} \grep^{\otimes n}(h^{-1}) \unisym_n) = \Tr(\hwsub{n\omega_+}{\grep^{\otimes n}} \unisym_n).
\end{equation}
The result then follows from \cref{eq:asymptotic_momap_realizability} from \cref{cor:only_fundamental}.
\end{proof}
Note that the statement of \cref{cor:only_fundamental} could have been anticipated in light of the fact that the moment map is $K$-equivariant (\cref{lem:momap_equivariance}).
Specifically, the realizability of $\omega$ implies the realizability of $\Ad^{*}(k)(\omega)$ because
\begin{equation}
\momap_{\grep}(\psi) = \omega \implies \momap_{\grep}(k \cdot \psi) = \Ad^{*}(k)(\omega).
\end{equation}
An even more useful simplification arises from noticing the actions of $S_n$ and $\grep^{\otimes n}$ on $\s H^{\otimes n}$ commute with each other.
Therefore, $[\hwsub{n\omega_+}{\grep^{\otimes n}}, \projsym_n] = 0$, and therefore, we have
\begin{equation}
\Tr(\hwsub{n\omega_+}{\grep^{\vee n}}) = \Tr(\projsym_n \hwsub{n\omega_+}{\grep^{\otimes n}}),
\end{equation}
where $\grep^{\vee n}$ is the $n$th symmetric power representation of $G$ on $\mathrm{Sym}^{n}(\s H)$ from \cref{exam:nth_tensor_power} and $\hwsub{n\omega_+}{\grep^{\vee n}}$ is the projection operator onto subspace of highest weight vectors with weight $n \omega_+$ which are also $S_n$-symmetric (otherwise known as \textit{covariants} of weight $n\omega_+$~\cite{walter2013entanglement}).
From this observation, one obtains a slightly more surprising result.
\begin{thm}
\label{thm:moment_polytope}
Let $\Delta_{\grep}$ denote the intersection of the image of the moment map with the positive Weyl chamber $i \mathfrak t_+^{*}$,
\begin{equation}
\Delta_{\grep} \coloneqq \{ \omega_+ \in i \mathfrak t_+^{*} \mid \exists \psi : \momap_{\grep}(\psi) = \omega_+ \}.
\end{equation}
Then $\Delta_{\grep}$ is a convex polytope, known as the \defnsty{moment polytope} for $\grep$.
\end{thm}
\begin{proof}
To prove $\Delta_{\grep}$ is a convex set, let $\alpha_+, \beta_+ \in \Delta_{\grep}$ be rational such that there exists $\psi, \psi' \in \proj \s H$ satisfying
\begin{equation}
\label{eq:alpha_beta_realized}
\momap_{\grep}(\psi) = \alpha_+, \qquad \momap_{\grep}(\psi') = \beta_+.
\end{equation}
Then for each $a, b \in \mathbb N$ consider the projection operators $\hwsub{a \alpha_+}{\grep^{\vee a}}$ and $\hwsub{b \beta_+}{\grep^{\vee b}}$.
From \cref{cor:only_fundamental}, \cref{eq:alpha_beta_realized} and \cref{eq:dimsym_irrelavent}, we conclude
\begin{align}
\label{eq:alpha_beta_asym}
\begin{split}
\limsup_{a\to\infty} \Tr(\hwsub{a \alpha_+}{\grep^{\vee a}})^{\frac{1}{a}} &= 1, \\
\limsup_{b\to\infty} \Tr(\hwsub{b \beta_+}{\grep^{\vee b}})^{\frac{1}{b}} &= 1.
\end{split}
\end{align}
More importantly, since $\hwsub{a \alpha_+}{\grep^{\vee a}}$ and $\hwsub{b \beta_+}{\grep^{\vee b}}$ are projection operators, their trace must be a non-negative integer, and therefore, \cref{eq:alpha_beta_asym} implies that there exists arbitrarily large $a, b \in \mathbb N$ such that $\Tr(\hwsub{a \alpha_+}{\grep^{\vee a}}) \geq 1$ and $\Tr(\hwsub{b \alpha_+}{\grep^{\vee b}}) \geq 1$.
Given a covariant of degree $a$ with weight $a \alpha_+$ and a covariant of degree $b$ with weight $b \beta_+$, the symmetrization of their tensor product is necessarily a non-zero covariant of degree $a+b$ with weight $a \alpha_+ + b \beta_+$ and therefore,
there exists arbitrarily large $a, b \in \mathbb N$ such that
\begin{equation}
\label{eq:covariants_non_empty}
\Tr(\hwsub{a \alpha_+ + b \beta_+}{\grep^{\vee (a+b)}}) \geq 1,
\end{equation}
and similarly, for all $n \in \mathbb N$,
\begin{equation}
\label{eq:covariants_non_empty_powers}
\Tr(\hwsub{n a \alpha_+ + n b \beta_+}{\grep^{\vee n(a+b)}}) \geq 1.
\end{equation}
Let $\gamma_+$ be the convex combination of $\alpha_+$ and $\beta_+$,
\begin{equation}
\gamma_+ = \frac{a \alpha_+ + b \beta_+}{a+b}.
\end{equation}
Then \cref{eq:covariants_non_empty_powers} implies
\begin{align}
\limsup_{n\to\infty} \Tr(\hwsub{n\gamma_+}{\grep^{\otimes n}})^{\frac{1}{n}} = 1,
\end{align}
which means, by \cref{cor:only_fundamental}, that $\gamma_+ = \momap_{\grep}(\psi'')$ is the moment map of some $\psi'' \in \proj \s H$ and thus $\gamma_+ \in \Delta_{\grep}$.
To further prove that $\Delta_{\grep}$ is a polytope requires a result which states the algebra of symmetric highest weight vectors, viewed as covariant polynomial functions in $\s H$, is finitely generated~\cite{walter2014multipartite,brion1987image}.
\end{proof}

\begin{rem}
The moment polytope for $\grep$ provides a solution to the realizability problem for a given moment map.
In particular, $\omega \in i \mathfrak k^{*}$ is realizable as the moment map of some ray $\psi \in \proj \s H$ if and only if the coadjoint orbit of $\omega$, denoted by $\Ad^{*}(K)(\omega) \subseteq i \mathfrak k^{*}$, which intersects the positive Weyl chamber uniquely at $\omega_+ \in i \mathfrak t_+^*$, is such that $\omega_+$ lies inside the moment polytope for $\grep$:
\begin{equation}
\exists \psi \in \proj \s H : \momap_{\grep}(\psi) = \omega \Longleftrightarrow \Ad^{*}(K)(\omega) \cap \Delta_{\grep} \neq \emptyset \Longleftrightarrow \omega_+ \in \Delta_{\grep}.
\end{equation}
\end{rem}

\subsection{Beyond moment polytopes}
\label{sec:beyond_polytopes}

The purpose of this section is to consider realizability problems which go beyond the realizability problem associated to a single moment map.
Our approach is to consider the generalization of \cref{prop:realizability_deformed_single} to the case where one is interested in characterizing the image of a moment map $\momap_{\grep_{(p)}} : \proj (\s H^{\otimes p}) \to i \mathfrak k^{*}$ with respect to a representations of the form $\grep_{(p)} : G \to \GL(\s H^{\otimes p})$.

\begin{thm}
\label{thm:poly_realizability}
Let $p \in \mathbb N$ be a positive integer and let $\grep_{(p)} : G \to \GL(\s H^{\otimes p})$ be a representation of a complex reductive group $G$ on $\s H^{\otimes p}$.
Let $\omega = \Ad^{*}(h)(\omega_+) \in i \mathfrak k^{*}$ have rational coadjoint orbit and let $\ratsub{\omega}{\grep^{\otimes n}}$ be the projection operator onto the subspace $(\s H^{\otimes p})^{\otimes n}$ defined by
\begin{equation}
\ratsub{\omega}{\grep^{\otimes n}} = \grep_{(p)}^{\otimes n}(h) \hwsub{n\omega_+}{\grep_{(p)}^{\otimes n}}\grep_{(p)}^{\otimes n}(h^{-1})
\end{equation}
and where $\hwsub{n\omega_+}{\grep_{(p)}^{\otimes n}}$ is the projection operator onto the highest weight subspace of $(\s H^{\otimes p})^{\otimes n}$ with weight $n \omega_+$.
Then there exists a ray $\psi \in \proj \s H$ such that
\begin{equation}
\momap_{\grep_{(p)}}(\psi^{\otimes p}) = \omega \in i \mathfrak k^{*},
\end{equation}
if and only if
\begin{equation}
\limsup_{n\to\infty} \Tr(\ratsub{\omega}{\grep^{\otimes n}} \unisym_{pn})^{\frac{1}{n}}= 1.
\end{equation}
\end{thm}
\begin{proof}
From \cref{thm:deformed_strong_duality}, we have for all $\Psi \in \proj \s H^{\otimes p}$,
\begin{equation}
\projcapacity_{\grep_{(p)}}^{\omega}(\Psi) = \limsup_{n\to\infty} \Tr(\ratsub{\omega}{\grep^{\otimes n}} \Psi^{\otimes n})^{\frac{1}{n}}.
\end{equation}
Which implies for all $\psi \in \proj \s H$,
\begin{equation}
\projcapacity_{\grep_{(p)}}^{\omega}(\psi^{\otimes p}) = \limsup_{n\to\infty} \Tr(\ratsub{\omega}{\grep^{\otimes n}} \psi^{\otimes p})^{\frac{1}{n}}.
\end{equation}
Now applying \cref{thm:symmetric_inclusivity} to the case where $X_n = \ratsub{\omega}{\grep^{\otimes n}}$ yields
\begin{equation}
\sup_{\psi\in\proj \s H} \projcapacity_{\grep_{(p)}}^{\omega}(\psi^{\otimes p}) = \limsup_{n\to\infty} \Tr(\ratsub{\omega}{\grep^{\otimes n}} \unisym_{pn})^{\frac{1}{n}}.
\end{equation}
But $\sup_{\psi\in\proj \s H} \projcapacity_{\grep_{(p)}}^{\omega}(\psi^{\otimes p}) = 1$ if and only if there exists a $\psi \in \proj \s H$ such that $\momap_{\grep_{(p)}}(\psi^{\otimes p}) = \omega$, which concludes the proof.
\end{proof}
\begin{rem}
In the special case where $p = 1$, \cref{thm:poly_realizability} reduces to \cref{prop:realizability_deformed_single}.
Unlike \cref{prop:realizability_deformed_single}, however, the $S_{np}$-symmetric subspace is not necessarily invariant under the action of $\grep_{(p)}^{\otimes n}$ and therefore we generically have
\begin{equation}
[\grep_{(p)}^{\otimes n}(k), \unisym_{np}] \neq 0.
\end{equation}
Because of this lack of symmetry, we do not have a generalization of \cref{cor:only_fundamental} for $p > 1$ and therefore no generalization of \cref{thm:moment_polytope}.
\end{rem}

Our main application of \cref{thm:poly_realizability} is to consider the case where $\psi \in \proj \s H$
\begin{cor}
\label{thm:birealizable_occasional}
Consider two complex reductive groups, $G_1$ and $G_2$, with representations acting on the same Hilbert space,
\begin{equation}
\grep_1 : G_1 \to \GL(\s H), \qquad \grep_2 : G_2 \to \GL(\s H),
\end{equation}
and assume $\omega_1 \in i \mathfrak k_1^*$ and $\omega_2 \in i \mathfrak k_2^*$ have rational coadjoint orbits.
Then there exists a ray $\psi \in \proj \s H$ satisfying
\begin{equation}
\momap_{\grep_1}(\psi) = \omega_1, \qquad \momap_{\grep_2}(\psi) = \omega_2,
\end{equation}
if and only if
\begin{equation}
\limsup_{n\to\infty} \Tr( \unisym_{2n} ( \ratsub{\omega_1}{\grep_1^{\otimes n}} \otimes \ratsub{\omega_2}{\grep_2^{\otimes n}}))^{\frac{1}{2n}}= 1.
\end{equation}
\end{cor}
\begin{proof}
The proof follows from \cref{thm:poly_realizability} by setting $p = 2$, $G \coloneqq G_1 \times G_2$ and $\grep_{(2)} = \grep_1 \boxtimes \grep_2$, where
\begin{equation}
(\grep_1 \boxtimes \grep_2)(g_1, g_2) = \grep_1(g_1) \boxtimes \grep_2(g_2).
\end{equation}
Then by \cref{lem:moment_map_ext}, we have
\begin{equation}
\momap_{\grep_1 \boxtimes \grep_2}(\psi^{\otimes 2}) = \momap_{\grep_1}(\psi) \oplus \momap_{\grep_2}(\psi)
\end{equation}
and moreover for any $n \in \mathbb N$,
\begin{equation}
\ratsub{\omega_1\oplus \omega_2}{(\grep_1\boxtimes\grep_2)^{\otimes n}} = \ratsub{\omega_1}{\grep_1^{\otimes n}} \otimes \ratsub{\omega_2}{\grep_2^{\otimes n}}.
\end{equation}
\end{proof}
The proof technique from \cref{cor:joint_realizable_occasional} trivially generalizes to all $p \in \mathbb N$.
\begin{cor}
\label{cor:joint_realizable_occasional}
Let $[p] = \{1, \ldots, p\}$ be an index set and for each index $j \in [p]$, let $\grep_j : G_j \to \GL(\s H)$ be a representation of a complex reductive group $G_j$, let $\momap_{\grep_j} : \proj \s H \to (i \mathfrak k_j)^{*}$ be the moment map for $\grep_j$, and let $\omega_j \in i \mathfrak k_j^{*}$ have a rational coadjoint orbit.
Then there exists a ray $\psi \in \proj \s H$, simultaneously satisfying
\begin{equation}
\forall j \in [p] : \momap_{\grep_j}(\psi) = \omega_j,
\end{equation}
if and only if
\begin{equation}
\limsup_{n\to\infty} \Tr(\unisym_{np} ( \ratsub{\omega_1}{\grep_{1}^{\otimes n}} \otimes \cdots \otimes \ratsub{\omega_p}{\grep_{p}^{\otimes n}}))^{\frac{1}{np}}= 1.
\end{equation}
\end{cor}

%    \label{lem:asymptotic_test_property}
%    Suppose $\grep : G \to \GL(\s H)$ is a representation of $G$ and let $x \in (i\mathfrak k)^{*}$ be rational in the sense that there exists a positive integer $\ell \in \mathbb N$ such that $\lambda \coloneqq \ell x$ is analytically integral.
%    Then the ray $\psi \in \proj \s H$ has moment map $\momap_{\grep}(\psi) = x$ if and only if
%    \begin{equation}
%        \limsup_{n\to\infty} \Tr[(P_{\psi}^{\otimes \ell n} \otimes P_{\psi_{\lambda}}^{\otimes n}) \Pi_n^{0}] = 1.
%    \end{equation}
%    where $\psi_{\lambda}$ is a highest weight ray for the irreducible representation $\grep_{\lambda} : G \to \GL(\s H_{\lambda})$ and where $\Pi_n^{0} \in \End((\s H^{\otimes \ell} \otimes \s H_{\lambda})^{\otimes n})$ is the projection operator onto the subspace fixed by the action of $\grep^{\otimes \ell} \otimes \grep_{\lambda}$.

\input{imports_biriffle}

\section{Examples}

\subsection{Absolutely maximally entangled states}
\label{sec:absolutely_maximally_entangled}

In this section we cover the problem of determining the existence of so-called absolutely maximally entangled states previously discussed in the introduction.
Absolutely maximally entangled states appear in a variety of contexts in quantum information theory, including quantum error correction and quantum secret sharing~\cite{huber2017quantum,scott2004multipartite,helwig2013absolutely}.

To begin we consider a fairly general setup.
Let $[p] = \{1, \ldots, p\}$ be a set of $p \in \mathbb N$ indices and let $\s H \cong \bigotimes_{i \in [p]} \s H_{i}$ be a finite-dimensional $p$-partite Hilbert space.
Let the dimension of the local Hilbert spaces be expressed as $d_{i} = \dim(\s H_{i})$, while the dimension of the whole Hilbert space is expressed as $D = \dim(\s H) = \prod_{i\in[p]} d_i$.
Now fix a subset $S \subseteq [p]$ of indices, and consider the $\card{S}$-partite subsystem $\s H_S \coloneqq \bigotimes_{i\in S} \s H_{i}$ with dimension $d_S = \dim(\s H_S)$, alongside a complementary subsystem $\s H_{\neg S} \coloneqq \bigotimes_{j \in [p] \setminus S} \s H_{j}$ with dimension $d_{\neg S} = \dim(\s H_{\neg S})$ such that $D = d_{S} d_{\neg S}$.
Given a pure quantum state, described by a ray $\psi \in \proj \s H$ over the whole Hilbert space, the quantum state describing the subsystem $S$ can be obtained from $\psi$ by applying the partial trace over the complementary subsystem $\s H_{\neg S}$:
\begin{equation}
\rho_{S}^{\psi} = \Tr_{\s H_{\neg S}} (P_{\psi}).
\end{equation}
Alternatively, one can obtain the reduced state $\rho_{S}^{\psi}$ by first considering a unitary representation, $\grep_{S} : \SU(d_{S}) \to \SU(D)$ of $\SU(d_{S})$ acting locally on the subsystem $\s H_S$ by sending the $d_{S}$-dimensional unitary $U \in \SU(d_{S})$ to the $D$-dimensional unitary,
\begin{equation}
\label{eq:local_action}
\grep_{S}(U) \coloneqq U \otimes \ident_{\s H_{\neg S}}.
\end{equation}
Then, the moment map of the representation $\grep_{S}$ is a function of the form
\begin{equation}
\momap_{S} : \proj \s H \to (i\mathfrak{su}(d))^{*}.
\end{equation}
Using the moment map, we note a correspondence between $\rho_{S}^{\psi}$ and $\momap_{S}(\psi)$, captured by the following expression, which holds for all traceless Hermitian operators $X \in i\mathfrak{su}(d)$:
\begin{equation}
\Tr(\rho_{S}^{\psi} X) = \momap_{S}(\psi)(X).
\end{equation}
In other words, the moment map for the representation $\grep_{S}$ applied to $\psi$ determines the reduced state of $\psi$ on subsystem $S$ up to normalization.

Now a pure state, $\psi \in \proj \s H$, is said to be \textit{$S$-uniform} if its reduced state onto subsystem $\s H_{S}$ is proportional to the identity operator, meaning
\begin{equation}
\rho_{S}^{\psi} = \frac{\ident_{\s H_{S}}}{d_{S}},
\end{equation}
or equivalently, the moment map associated to the representation $\grep_S$ vanishes, i.e.,
\begin{equation}
\momap_{S}(\psi) = 0.
\end{equation}
A pure state, $\psi \in \proj \s H$, is said to \textit{absolutely maximally entangled}, if for all subsystems $S \subseteq [p]$ of size
\begin{equation}
\abs{S} \leq \left\lfloor\frac{D}{2}\right\rfloor,
\end{equation}
the state $\psi$ is $S$-uniform.

\begin{exam}
For example, consider the case where $p = 2$, $d_1 = d_2 = 2$, and $D=4$.
In this case, the familiar two-qubit Bell state $\psi \in \proj (\mathbb C^{2} \otimes \mathbb C^{2})$ associated to the unit vector
\begin{equation}
v \coloneqq \frac{1}{\sqrt{2}}\left(e_0 \otimes e_0 + e_1 \otimes e_1\right) \in \mathbb C^{2} \otimes \mathbb C^{2},
\end{equation}
is absolutely maximally entangled because both of its reduced states (on either the first or second subsystem) are proportional to identity matrix on $\mathbb C^{2}$, or equivalently, for all $2$-dimensional traceless Hermitian operators $X \in i \mathfrak {su}(2)$,
\begin{align}
\begin{split}
\Tr(\rho_1 X) &= \momap_1(\psi)(X) = \braket{v, (X \otimes \ident) v} = 0, \\
\Tr(\rho_2 X) &= \momap_2(\psi)(X) = \braket{v, (\ident \otimes X) v} = 0.
\end{split}
\end{align}
\end{exam}

Using the strong-duality result from \cref{sec:strong_duality}, we see that $\psi \in \proj \s H$ is $S$-uniform, i.e. $\momap_{S}(\psi) = 0$, if and only if
\begin{equation}
\limsup_{n\to\infty} \Tr(P_{\psi}^{\otimes n} \fsub{\grep_S^{\otimes n}})^{\frac{1}{n}} = 1,
\end{equation}
where, for each $n \in \mathbb N$, $\fsub{\grep_S^{\otimes n}}$ is the projection operator onto the subspace of vectors in $(\s H_S \otimes \s H_{\neg S})^{\otimes n}$ which are invariant under the action of $\SU(d_S)$ through the representation $\grep_S$.

Combining this observation with the results from \cref{sec:beyond_polytopes}, we obtain the following result which may be used to determine, for any given collection of subsystems $S_1, \ldots, S_m \subseteq [p]$, whether or not there exists a state which is $S_j$-uniform for all $j \in \{1, \ldots, m\}$.
\begin{cor}
\label{cor:absolutely_maximally_entangled}
Let $S_1, \ldots, S_m \subseteq [p]$ be a collection of subsystems indexed by $j \in [m] \in \{1, \ldots, m\}$.
Then, for each $n \in \mathbb N$ and $j \in [m]$, let $\fsub{\grep_{S_j}^{\otimes n}}$ be the aforementioned projection operator onto the subspace of vectors in $(\mathbb C^{D})^{\otimes n}$ invariant under the action of $\grep_{S_j}^{\otimes n} : \SU(d_j) \to \SU(D^{n})$ defined by
\begin{equation}
\grep_{S_j}(U_{S_j}) \coloneqq U_{S_j}^{\otimes n} \otimes \ident_{\neg S_j}^{\otimes n}.
\end{equation}
Then there exists a ray $\psi \in \proj (\mathbb C^{D})$ which is $S_j$-uniform for all $j \in [m]$ if and only if
\begin{equation}
\limsup_{n \to \infty} \Tr(\unisym_{nm}( \fsub{\grep_{S_1}^{\otimes n}} \otimes \cdots \otimes \fsub{\grep_{S_m}^{\otimes n}} )) = 1
\end{equation}
where $\unisym_{nm}$ is defined in \cref{defn:uniform_symmetric_state}.
\end{cor}
This result thus constitutes a first step toward, at least partially, generalizing the result of \citeauthor{bryan2018existence}~\cite{bryan2018existence} for the complete solution to the existence or non-existence of \textit{locally} maximally entangled states (i.e., the case where $\card{S_j} = 1$ for all $j$) as a function of local dimensions.

\subsection{Quantum marginals}
\label{sec:quantum_marginals_early_example}

The quantum marginals problem can be understood as a non-uniform generalization of the problem considered in \cref{sec:absolutely_maximally_entangled}.
In fact, by using the marginal representations defined by \cref{eq:local_action}, we observe that \cref{cor:joint_realizable_occasional} already provides an asymptotic solution to the quantum marginals problem, at least for marginal states with rational spectra.
In \cref{chap:qmp}, we will revisit the quantum marginals problem in greater detail and will encounter an alternative, but ultimately related, strategy for asymptotically characterizing which collections of marginals are jointly realizable.

%\subsection{Euclidean geometry from representation theory}
%If $K$ is a compact matrix Lie group and $\grep : K \to \U(\s H)$ is a unitary representation of $K$ on a Hilbert space $\s H$, by the complete reducibility of $K$, $\s H$ admits of a decomposition into irreducible invariant subspaces,
%    \s H = \bigoplus_{\lambda} \s H_{\lambda} \otimes \mult_{\lambda}(\s H)
%where $\mult_{\lambda}(\s H) = \Hom_{K}(\s H_{\lambda}, \s H)$ is the multiplicity space for $\s H_{\lambda}$ in $\s H$ and $\grep$ admits of a decomposition into irreducible subrepresentations,
%    \grep(g) = \bigoplus_{\lambda} \grep_{\lambda}(g) \otimes I_{\mult_{\lambda}(\s H)}.
%For example if $K = \SU(2)$, $\s H = (\mathbb C^{2})^{\otimes n}$ and $\grep$ is the $n$-fold tensor-power representation of the defining repesentation of $\SU(2)$ acting on $\mathbb C^2$, then
%    \grep(g) = \bigoplus_{\ell} \grep_{\ell}(g) \otimes I_{\mult_{\ell}^{(n)}}.
%where $\ell \in \mathbb Z_{\geq 0}$, $\grep_{\ell} : \SU(2) \to \U(\mathbb C^{\ell + 1})$ is the well-known, unitary spin-$\ell/2$ representation of $\SU(2)$ on with dimension $\ell + 1$ and $\mult_{\ell}^{(n)} \coloneqq \mult_{\ell}((\mathbb C^{2})^{\otimes n})$ is non-zero if and only if $\ell \in \{0, 2, 4, \ldots, n\}$ when $n$ is even and $\ell \in \{1, 3, \ldots, n\}$ when $n$ is odd.

%\subsection{Subspace intersections}
%    \braket{v,P_1v} = \braket{v, P_2v} = 1
%    \braket{v,P_1v}^{n_1} \braket{v, P_2v}^{n_2} = \braket{v^{\otimes (n_1 + n_2)}, (P_1^{\otimes n_1} \otimes P_2^{\otimes n_2}) v^{\otimes (n_1 + n_2)}} = 1
%    s_{d}(n) \coloneqq \binom{n+d-1}{n} = \dim(\mathrm{Sym}^{n}(\mathbb C^d)) \leq (n+1)^{d-1}.
%Now let $q_{n}(a) \coloneqq \frac{\binom{n}{a}^{2}}{\binom{2n}{n}}$ (where $0 \leq q_{n}(a) \leq 1$).
%    &\Tr(\unisym_{2n}( P_1^{\otimes n} \otimes P_2^{\otimes n})) \\
%    &\qquad = \Tr(\unisym_{2n}(P_{\mathrm{Sym}^{n}(\s V_1)} \otimes P_{\mathrm{Sym}^{n}(\s V_2)})) \\
%    &\qquad = \frac{1}{s_{d}(2n)} \frac{1}{(2n)!} \sum_{\pi \in S_{2n}} \Tr[T_{2n}(\pi) (P_{\mathrm{Sym}^{n}(\s V_1)} \otimes P_{\mathrm{Sym}^{n}(\s V_2)})] \\
%    &\qquad = \frac{1}{s_{d}(2n)} \sum_{a = 0}^{n} q_{n}(a) \Tr[(P_{\mathrm{Sym}^{n}(\s V_1)} \otimes I_{\s H}^{\otimes (n-a)}) (I_{\s H}^{\otimes (n-a)} \otimes P_{\mathrm{Sym}^{n}(\s V_2)})] \\
%    &\qquad = \frac{s_{d_1}(n)s_{d_1}(n)}{s_{d}(2n)} \sum_{a = 0}^{n} \frac{q_{n}(a)}{s_{d_1}(a)s_{d_2}(a)} \Tr[P_{\mathrm{Sym}^{a}(\s V_1)}P_{\mathrm{Sym}^{a}(\s V_2)}] \\
%    &\qquad = \frac{s_{d_1}(n)s_{d_1}(n)}{s_{d}(2n)} \sum_{a = 0}^{n} \frac{q_{n}(a)}{s_{d_1}(a)s_{d_2}(a)} \Tr(\unisym_{a}((P_1P_2)^{\otimes a})) \\
%    &\qquad \leq \frac{s_{d_1}(n)s_{d_1}(n)}{s_{d}(2n)} \sum_{a = 0}^{n}\frac{q_{n}(a)s_{d}(a)}{s_{d_1}(a)s_{d_2}(a)} \lambda_{\max}^{a}
%Now let $g_{d_1,d_2}(n)$ be
%    g_{d_1,d_2}(n) \coloneqq \frac{s_{d_1}(n)}{s_{d_1}(a)}\frac{s_{d_2}(n)}{s_{d_2}(a)} s_{d}(a)
%and pick a threshold value $\tau \in \mathbb N$ such that $0 < \tau < \frac{n}{2}$.
%Then for all $a \leq \tau$, $\binom{n}{a} \leq \binom{n}{\tau}$ and thus $q_{n}(a) \leq q_{n}(\tau)$ and for all $a \geq \tau$, $\lambda_{\max}^{a} \leq \lambda_{\max}^{\tau}$ because $\lambda_{\max} \leq 1$.
%    \sum_{a = 0}^{n} q_{n}(a) \lambda_{\max}^{a} = \left(\sum_{a = 0}^{\tau-1} + \sum_{a = \tau}^{n}\right) q_{n}(a) \lambda_{\max}^{a} \leq \tau q_{n}(\tau) + \lambda_{\max}^{\tau}

%    Let $\seq{E_n : \borel{X} \to \bound(\s H^{\otimes n})}$ be an $f$-estimation scheme such that for each $\rho \in \state$, $\seq{\xi_n^{\rho} : \borel{X} \to [0,1]}$ satisfies the large deviation principle with rate function $I_{\rho} : X \to [0,1]$.
%    and let $\mu : \borel{\state} \to [0,1]$ be a probability measure over the set of states $\state$.
%    Then the sequence of probability measures $\seq{\Xi_{n}^{\mu}}$, given by \cref{eq:estimate_de_finetti_measure}, satisfies the large deviation principle with rate function $I : X \to [0,1]$ defined for $x \in X$ by
%    \begin{equation}
%        I(x) = \inf_{\rho \in \supp(\mu)} I_{\rho}(x).
%    \end{equation}

%The following theorem demonstrates that $\Tr(D_{n}^{\mu} p_n(x))^{\frac{1}{n}}$ can only be close to unity if there exist some state $\sigma$ in the support of $\mu$ such that $I_{\sigma}(x) \approx 0$,
%    Let $\mu : \borel{\state} \to [0,1]$ be probability measure with support $\supp(\mu) = S_{\mu} \subseteq \state$.
%    Now suppose that $\seq{g_n : X \to \s B(\s H^{\otimes n})}$ is a sequence of functions such that $g_n(x) \geq 0$ is positive semidefinite for all $x \in X$ and $n \in \mathbb N$.
%    Further suppose there exists a function $(\rho, x) \mapsto I_{\rho}(x) \in [0, \infty]$ such that for all $n \in \mathbb N$,
%    \begin{equation}
%        \Tr(\rho^{\otimes n} g_n(x)) \leq \exp(-n I_{\rho}(x)).
%    \end{equation}
%    If for some $\epsilon \geq 0$,
%    \begin{equation}
%        \label{eq:de_Finetti_quasitypical}
%        \Tr(D_{n}^{\mu} g_n(x))^{\frac{1}{n}} \geq 1 - \epsilon,
%    \end{equation}
%    where $D_n^{\mu}$ is the $\mu$ de Finetti state of order $n$, then there exists $\sigma \in S_{\nu}$ satisfying
%    \begin{equation}
%        \label{eq:rate_upper_bound}
%        I_{\sigma}(x) \leq \ln\left(\frac{1}{1-\epsilon}\right).
%    \end{equation}
%    Given that $\mu$ is a probability distribution, we have for any positive semidefinite operator $g_n(x) \in \End(\s H^{\otimes n})$, the inequality
%    \begin{equation}
%        \Tr(D_{n}^{\mu} g_n(x)) = \int_{\rho \in S_{\mu}}\mu(\diff \rho) \Tr(\rho^{\otimes n} F_n) \leq \sup_{\sigma \in S_{\mu}} \Tr(\sigma^{\otimes n} F_n) \leq \exp(-n \inf_{\sigma \in S_{\mu}} I_{\sigma}(x))
%    \end{equation}
%    Combining this upper-bound on $\Tr(D_{n}^{\mu} g_{n}(x))^{\frac{1}{n}}$ with the lower-bound offered by \cref{eq:de_Finetti_quasitypical} yields
%     \begin{equation}
%        1- \epsilon \leq \exp(-n \inf_{\rho \in S_\mu} I_{\sigma}(x)).
%    \end{equation}
%    Rearranging for $I_{\sigma}(x)$ and noting that the infimum of a rate function over closed set $S_\mu = \supp(\mu)$, is necessarily attained by some $\sigma \in S_\mu$ yields \cref{eq:rate_upper_bound}.

\chapter{Quantum marginal problems}
\label{chap:qmp}
The purpose of this chapter is to explore a class of realizability problems known as quantum marginal problems.
From the perspective of \cref{chap:realizability}, and for the purposes of this chapter, a quantum marginal problem can be understood as a realizability problem where the properties under investigation are \textit{localized} in the sense that they pertain to subsystems of a multipartite Hilbert space.
Although a solution to the quantum marginal problem was already presented in \cref{sec:quantum_marginals_early_example}, this chapter provides an alternative, albeit related, solution in the form of \cref{thm:main} which has the advantage of being much easier to articulate.
The contents of this chapter are taken directly from my paper on the quantum marginals problem~\cite{fraser2022sufficient}, with only minor modifications.

\section{Introduction}

The quantum marginal problem (QMP) is interested in characterizing the space of reduced/marginal states of a multipartite quantum state, and is widely regarded as being an important, albeit challenging problem to solve.

Connections and applications of the QMP to other topics in quantum theory (and beyond) include multipartite entanglement and separability~\cite{coffman2000distributed,walter2013entanglement}, quantum error correction~\cite{ocko2011quantum,huber2017quantum,yu2021complete}, entropic constraints~\cite{majenz2018constraints,christandl2018recoupling,kim2020entropy,osborne2008entropic}, other forms of quantum realizability problems~\cite{heinosaari2016invitation,haapasalo2021quantum,doherty2008quantum}, the asymptotics of representations of the symmetric groups~\cite{daftuar2005quantum,christandl2006spectra,christandl2018recoupling}, the asymptotic restriction problem for tensors~\cite{christandl2018universal}, random matrix theory~\cite{collins2021projections,christandl2014eigenvalue}, and many-body physics~\cite{coleman2001reduced,schilling2015quantum}.

The quantum marginal problem, and its associated terminology, is derived from an analogous problem in probability theory called the \textit{classical} marginal problem. The classical marginal problem is concerned with characterizing the relationships between the various marginal distributions of a joint, multivariate distribution~\cite{fritz2012entropic,vorob1962consistent,malvestuto1988existence}. As a joint probability distribution and its marginals can always be faithfully represented by the eigenvalues of a joint quantum state and its marginals using a product eigenbasis, the quantum marginal problem subsumes the classical marginal problem, and consequently, any of its applications~\cite{fritz2012entropic,liang2011specker,abramsky2011sheaf,fraser2018causal}.

One of the earliest formulations of the problem dates back to the early 1960s, when, for the purposes of simplifying calculations involving the atomic and molecular structure, quantum chemists became interested in characterizing the possible reduced density matrices of a system of $N$ interacting fermions~\cite{coulson1960present,coleman1963structure}. This version of the problem, referred to as the $N$-representability problem, has a long history~\cite{coleman2000reduced,coleman2001reduced,lude2013functional,borland1972conditions,ruskai2007connecting,klyachko2009pauli} that continues to evolve~\cite{mazziotti2012structure,mazziotti2012significant,klyachko2006quantum,castillo2021effective}.

Now the QMP comes in a variety of flavours which can be broadly organized by considering any additional assumptions or constraints that are imposed on either i) the properties of the joint state, and/or ii) the properties of the set of candidate density operators~\cite{tyc2015quantum}.

When focusing on the joint state, specializations of the QMP exist where the joint state is assumed to be fermionic~\cite{coleman2000reduced,schilling2013pinning}, bosonic~\cite{wei2010interacting}, Gaussian~\cite{eisert2008gaussian,vlach2015quantum}, separable~\cite{navascues2021entanglement}, or having symmetric eigenvectors~\cite{aloy2020quantum}. Generally speaking, the QMP is difficult in the sense that it is a QMA-complete problem~\cite{liu2006consistency,liu2007quantum,wei2010interacting,bookatz2012qma}. For the purposes of this section, the only restriction imposed on joint states will be that they live in a finite-dimensional Hilbert space, with a finite and fixed number of subsystems.

Regarding the list of candidate density operators, e.g., $(\rho_{AB}, \rho_{BC}, \rho_{AC})$, the list of subsystems they correspond to, e.g., $(AB, BC, AC)$, is known as the \textit{marginal scenario}, while the individual elements, e.g., $AB$, $BC$, or $AC$, will be referred to as \textit{marginal contexts}. A key consideration for understanding previous work on the QMP is whether the marginal contexts are disjoint. When the marginal contexts are disjoint, a complete solution to the QMP is known~\cite{klyachko2004quantum,klyachko2006general}. For a given specification of Hilbert space dimensions, this solution takes the form of a finite list of linear inequality constraints on the spectra of the candidate density operators. These solutions furthermore recover earlier results pertaining to low dimensional Hilbert-spaces~\cite{higuchi2003one,higuchi2003qutrit,bravyi2003requirements,han2005compatibility}.

In contrast, when the marginal scenario involves overlapping marginal contexts, existing results are comparatively more sporadic and typically weaker, being only applicable to low-dimensional systems, small numbers of parties, or only yielding necessary but insufficient constraints~\cite{chen2014symmetric,carlen2013extension,butterley2006compatibility,hall2007compatibility,chen2016detecting}. One promising approach, developed in~\cite{christandl2018recoupling} for relating marginal spectra to the recoupling theory of the symmetric group, appears limited to situations where the marginal contexts do not overlap too much.

Whenever a candidate set of density operators is explicitly given, one strategy to decide their realizability is to use convex optimization techniques, e.g., semidefinite programming~\cite{vandenberghe1996semidefinite}. If the joint state is not necessarily pure, realizability can be decided with a single semidefinite program~\cite{hall2007compatibility}. Additionally, when the joint state is assumed pure, realizability can still be decided by an infinite hierarchy of semidefinite programs~\cite{yu2021complete}. In either case, analytic inequality constraints that serve as witnesses for unrealizability can be extracted from the outputs of such semidefinite programs~\cite{hall2007compatibility}.

The objective of this chapter is to improve our understanding of the QMP, in particular for the case of overlapping marginal contexts, by i) deriving inequality constraints that are necessarily satisfied by all realizable collections of density operators, and ii) proving that if a collection of density operators satisfies these inequalities, then they are realizable. This chapter begins by formally defining the QMP and then reformulating it from a different perspective. The primary advantage of this reformulation of the QMP is that it exposes an implicit symmetry of the problem which is helpful in deriving our main result.

% ========= Setting up the supplemental material reference =======
%The supplementary material for this letter provides additional proofs and explores some of the consequences of the the main theorem presented here~\footnote{See Supplemental Material at %
%[URL to be inserted by publisher] %
%which references~\cite{procesi2007lie,hall2015lie,harrow2013church,fritz2012entropic,cvitanovic2008group,penrose1971applications,selinger2012finite,coecke2010compositional,wood2011tensor,biamonte2017tensor,tyc2015quantum,klyachko2004quantum,keyl2001estimating,christandl2006spectra,sra2016inequalities,sagan2013symmetric,littlewood1934group,fulton2000eigenvalues,pak2019largest,howe1987gl,walter2014multipartite,reid2009generalised,klyachko2002coherent,keyl2006quantum,odonnell2016efficient,fisk2005very,horn1985matrix,kullback1997information,zhelobenko1973compact} and contains proofs of \cref{lem:de_finetti_realizable} (\cref{sec:proof_finetti_lemma}), and \cref{eq:unrealizability_measure} (\cref{sec:keyl_exclusive}, \cref{rem:state_discrim}).}.
%\newcounter{suppcounter}\setcounter{suppcounter}{\thefootnote} % store the value of the current footnote
%\newcommand{\suppkey}{Note\thesuppcounter} % the key associated to the footnote for the supplemental material

\section{A Reformulation of the QMP}

This section introduces some notation and terminology that is sufficient to formally define both the QMP and an equivalent reformulation that is better suited for the techniques developed in subsequent sections.

First and foremost, every Hilbert space considered will be complex, finite-dimensional, and labeled by some subscript $X$, e.g., $\s H_{X}$. For each labeled Hilbert space, $\s H_{X}$, we will implicitly assume there exists some canonical orthonormal basis such that $\s H_{X} \cong \mathbb C^{d_X}$ where $d_X = \dim(\s H_{X})$. The corresponding set of linear operators, density operators (states), and pure states of $\s H_{X}$ are respectively denoted $\s L(\s H_{X})$, $\dens(\s H_{X})$, and $\proj(\s H_{X})$. The identity operator on $\s H_{X}$ is denoted $\ident_{X}$.

Given a list of labels, $S = (X_1, \ldots, X_k)$, which we identify with the concatenated string, $S \simeq X_1\cdots X_k$, the composite Hilbert space $\s H_{X_1} \otimes \cdots \otimes \s H_{X_k}$, will be labeled by $S$ itself and thus denoted $\s H_{S}$. For instance, if $S = ABC$, then $\s H_{ABC} = \s H_{A} \otimes \s H_{B} \otimes \s H_{C}$. Additionally, for any positive integer $n$ and label $X$, the list of labels consisting of $n$ copies of $X$ will be abbreviated by
\begin{equation}
nX \coloneqq (X, \stackrel{n}{\ldots}, X) \simeq X\stackrel{n}{\cdots}X
\end{equation}
Using this notational convention, the $n$th tensor-power of a Hilbert space $\s H_{X}$ can be written as $\s H_{X}^{\otimes n} = \s H_{nX} = \s H_{X\stackrel{n}{\cdots}X}$.

Associated to any instance of the QMP, is a \textit{joint} (or \textit{global}) Hilbert space $\s H_{J}$ where $J = (X_1, \ldots, X_p)$ is a given finite list of labels called the \textit{joint context}. Every non-empty sublist $S$ of $J$ will be called a \textit{marginal context}. For each $S \subseteq J$, the partial trace from $\s H_{J}$ onto $\s H_{S}$ will be denoted by
\begin{equation}
\label{eq:partial_trace}
\Tr_{J \setminus S} : \s L(\s H_{J}) \to \s L(\s H_{S}).
\end{equation}
A finite, non-empty tuple of marginal contexts,
\begin{equation}
\s M = (S_1, \ldots, S_m),
\end{equation}
is called a \textit{marginal scenario}. The cardinality of a marginal scenario will always be denoted by $m = \abs{\s M}$.
\begin{prob}[QMP]
\label{prob:qmp}
Given a marginal scenario, $\s M = (S_1, \ldots, S_m)$, and list of states $(\rho_{S_1}, \ldots, \rho_{S_m})$ where $\rho_{S_i} \in \dens(\s H_{S_i})$ for each $i \in \{1, \ldots, m\}$, decide if there exists a joint pure state $\psi_{J} \in \proj(\s H_J)$ such that
\begin{equation}
\label{eq:qmp}
\forall S_i \in \s M : \rho_{S_i} = \Tr_{J\setminus S_i}(\psi_{J}).
\end{equation}
\end{prob}
The assumption above of purity for the joint state is made without loss of generality as shown in~\cref{sec:pure_vs_mixed_qmp}.

Whenever such a pure state $\psi_{J}$ exists, the $m$-tuple of states $(\rho_{S_1}, \ldots, \rho_{S_m})$ is said to be \textit{realizable} and otherwise they are \textit{unrealizable}.

For the sake of brevity, unless otherwise specified, a joint Hilbert space $\s H_{J}$ with joint context $J$ will always be implicitly given together with a particular marginal scenario $\s M = (S_1, \ldots, S_m)$ of length $m$.

Our first step is to reinterpret the $m$ linear constraints imposed on the joint state $\psi_{J}$ by \cref{eq:qmp} as a single linear constraint on the $m$th tensor-power state, $\psi_{J}^{\otimes m}$. Specifically, the $m$-tuple of states $(\rho_{S_1}, \ldots, \rho_{S_m})$ satisfies \cref{eq:qmp} if and only if
\begin{equation}
\label{eq:parallel}
\begin{split}
\rho_{S_1} \otimes \cdots \otimes \rho_{S_m}
&= \Tr_{J \setminus S_1}(\psi_{J}) \otimes \cdots \otimes \Tr_{J \setminus S_m}(\psi_{J}), \\
&= (\Tr_{J \setminus S_1} \otimes \cdots \otimes \Tr_{J \setminus S_m})(\psi_{J}^{\otimes m}).
\end{split}
\end{equation}
This observation motivates the following definitions.
\begin{defn}
The \textit{Hilbert space on $\s M$}, denoted $\s H_{\s M}$, is
\begin{equation}
\s H_{\s M} \coloneqq \s H_{S_1} \otimes \cdots \otimes \s H_{S_m}.
\end{equation}
The \textit{partial trace from $mJ$ onto $\s M$}, denoted $\Tr_{mJ\setminus \s M}$, is
\begin{equation}
\Tr_{mJ\setminus \s M} \coloneqq \Tr_{J\setminus S_1} \otimes \cdots \otimes \Tr_{J\setminus S_m}.
\end{equation}
\end{defn}
Note that the partial trace from $mJ$ onto $\s M$, $\Tr_{mJ\setminus \s M}$, is simply the partial trace operation mapping elements of $\s L(\s H_{mJ}) = \s L(\s H_{J}^{\otimes m})$ to elements of $\s L(\s H_{\s M}) = \s L(\s H_{S_1} \otimes \cdots \otimes \s H_{S_m})$.
\begin{defn}
An \textit{$\s M$-product state} is any state, $\rho_{\s M} \in \dens(\s H_{\s M})$, of the form
\begin{equation}
\rho_{\s M} = \rho_{S_1} \otimes \cdots \otimes \rho_{S_m}
\end{equation}
where each component, $\rho_{S_i}$, is a state in $\dens(\s H_{S_i})$.
\end{defn}
Since there is a bijection between $m$-tuples of states $(\rho_{S_1}, \ldots, \rho_{S_m})$ and $\s M$-product states $\rho_{S_1} \otimes \cdots \otimes \rho_{S_m}$, the QMP can be equivalently restated entirely in terms of $\rho_{\s M}$.
\begin{prob}
\label{prob:cqmp}
Given an $\s M$-product state, $\rho_{\s M}$, determine whether or not there exists a pure state $\psi_{J}$ such that
\begin{equation}
\label{eq:cqmp}
\rho_{\s M} = \Tr_{mJ\setminus \s M}(\psi_{J}^{\otimes m}).
\end{equation}
\end{prob}
The equivalence between \cref{prob:qmp} and \cref{prob:cqmp} follows from \cref{eq:parallel}; moreover, whenever such a pure state $\psi_{J}$ exists in either formulation of the QMP, it satisfies both \cref{eq:qmp} and \cref{eq:cqmp}. Pursuant to this equivalence, an $\s M$-product state, $\rho_{S_1} \otimes \cdots \otimes \rho_{S_m}$, is said to be \textit{realizable} whenever the $m$-tuple of states $(\rho_{S_1}, \ldots, \rho_{S_m})$ is realizable (and \textit{unrealizable} otherwise). The set of all realizable $\s M$-product states will be denoted $\s C_{\s M}$.

\section{Necessary and Sufficient Inequality Constraints}

In this section, we construct inequalities that are necessarily satisfied by \textit{all} realizable $\s M$-product states, $\rho_{\s M}$.
These inequalities, therefore, can be used to answer the QMP in the negative; if an $\s M$-product state violates any of the forthcoming inequalities, then it is unrealizable.
In addition, it will be shown that if an $\s M$-product state, $\rho_{\s M}$, satisfies all of the forthcoming inequalities, then it must be realizable.

These inequalities emerge from considering the permutation symmetry of the $k$th tensor power, $\psi_J^{\otimes k}$, of a pure state $\psi_J \in \proj(\s H_{J})$. For each $k \in \mathbb N$, let $\sym_{k}$ be the symmetric group on $k$ symbols, and let $T_{J} : \sym_{k} \to \s L(\s H_{J}^{\otimes k})$ be the representation of $\sym_k$ acting on $\s H_{J}^{\otimes k}$ by permutation of its $k$ factors.
\begin{defn}
\label{defn:symsub}
The $k$th symmetric subspace $\symsub^{k}\s H_{J} \subseteq \s H_{J}^{\otimes k}$ is defined as
\begin{equation}
\symsub^{k}\s H_{J} = \{ \ket{\phi} \in \s H_{J}^{\otimes k} \mid \forall \pi \in \sym_k, T_{J}(\pi) \ket{\phi} = \ket{\phi} \}.
\end{equation}
The orthogonal projection operator onto $\symsub^{k}\s H_{J}$ will be denoted by $\Pi_{J}^{(k)}$.
\end{defn}
Given any vector $\ket{\psi_{J}} \in \s H_{J}$, it is straightforward to verify that $\ket{\psi_{J}}^{\otimes k}$ is an element of the $k$th symmetric subspace $\symsub^{k}\s H_{J} \subseteq \s H_{J}^{\otimes k}$.
\begin{prop}
\label{prop:copy_sym}
Let $\psi_{J} = \ket{\psi_J}\bra{\psi_J}$ be a pure state, let $k \in \mathbb N$. Then
\begin{equation}
\label{eq:kth_inclusion}
\psi_{J}^{\otimes k} \leq \Pi_{J}^{(k)},
\end{equation}
where $\Pi_{J}^{(k)}$ is defined in \cref{defn:symsub}.
\end{prop}
Throughout this chapter, an inequality $A \geq B$ between Hermitian operators $A$ and $B$ always indicates that $A - B$ is positive semidefinite, i.e. $A - B \geq 0$. See \cite[Section V]{bhatia1997matrix}.

By comparing \cref{eq:cqmp} with \cref{eq:kth_inclusion}, and recalling that partial traces are positive channels, it becomes possible to \textit{eliminate} $\psi_{J}$ from \cref{eq:cqmp}. For example, when $k=m$, the partial trace $\Tr_{mJ \setminus \s M}$ applied to \cref{eq:kth_inclusion} yields $\Tr_{mJ\setminus\s M}(\psi_{J}^{\otimes m}) \leq \Tr_{mJ\setminus\s M}(\Pi_{J}^{(m)})$ and thus we obtain the following corollary.
\begin{cor}
\label{cor:n1}
If $\rho_{\s M}$ is a realizable $\s M$-product state, then
\begin{equation}
\label{eq:n1}
\rho_{\s M} \leq \Tr_{mJ\setminus\s M}(\Pi_{J}^{(m)}).
\end{equation}
\end{cor}
In \cref{sec:n1}, it is shown that the utility of this constraint is quite sensitive to the marginal scenario under consideration. For certain marginal scenarios, \cref{eq:n1} happens to be satisfied by all $\s M$-product states, and thus is useless for the purposes of the QMP. Nevertheless, for other marginal scenarios, \cref{eq:n1} happens to be violated by some $\s M$-product states, and thus it serves as a non-trivial condition for the realizability of an $\s M$-product state $\rho_{\s M}$.

Analogous reasoning can be used to construct stronger inequality constraints for the QMP. When $k$ is a multiple of $m$, $k=nm$, one can apply the $n$th tensor power of $\Tr_{mJ\setminus\s M}$ to both sides of \cref{eq:kth_inclusion}. While it is clear from the preceding discussion that this will yield inequality constraints necessarily satisfied by all $\s M$-product states, we will additionally show that their satisfaction for all $n \in \mathbb N$ is \textit{sufficient} to conclude that $\rho_{\s M}$ is a realizable $\s M$-product state.
\begin{thm}
\label{thm:main}
An $\s M$-product state, $\rho_{\s M}$, is realizable if and only if for all $n \in \mathbb N$,
\begin{equation}
\label{eq:nth_order}
\rho_{\s M}^{\otimes n} \leq \Tr_{mJ\setminus\s M}^{\otimes n}(\Pi^{(nm)}_{J}).
\end{equation}
\end{thm}
Note that $\Tr_{mJ\setminus\s M}^{\otimes n}$ is the partial trace operation taking elements of $\s L(\s H_{J}^{\otimes nm})$ to elements of $\s L(\s H_{\s M}^{\otimes n})$.

To prove \cref{thm:main}, we first need the following lemma which states that the upper-bound in \cref{eq:nth_order} represents the expected value of $\sigma_{\s M}^{\otimes n}$, up to normalization, when $\sigma_{\s M}$ is sampled according to a probability measure, $\nu_{\s M}$, whose support is precisely the set of realizable $\s M$-product states, denoted $\s C_{\s M}$.
\begin{lem}
\label{lem:de_finetti_realizable}
There exists a probability measure, $\nu_{\s M}$, over $\dens(\s H_{\s M})$, with support $\s C_{\s M}$, such that for all $n \in \mathbb N$,
\begin{equation}
\Tr_{mJ\setminus\s M}^{\otimes n}(\Pi^{(nm)}_J) = \tbinom{nm+d_{J}-1}{nm} \int_{\s C_{\s M}} \hspace{-1em}\nu_{\s M}(\diff \sigma_{\s M}) \sigma_{\s M}^{\otimes n},
\end{equation}
where $d_J = \dim(\s H_J)$.
\end{lem}
\begin{proof}
See \cref{sec:proof_finetti_lemma}.
\end{proof}
Now consider a measurement effect, $E_{n}$, acting on $\s H_{\s M}^{\otimes n}$, i.e. a Hermitian operator $E_n \in \s L(\s H_{\s M}^{\otimes n})$ such that $0 \leq E_n \leq \ident_{\s M}^{\otimes n}$. If \cref{eq:nth_order} is satisfied by some state $\rho_{\s M} \in \dens(\s H_{\s M})$, and $\Tr(E_n\rho_{\s M}^{\otimes n}) \neq 0$, then \cref{lem:de_finetti_realizable} implies
\begin{equation}
\label{eq:prob_ratio_lower}
\sup_{\sigma_{\s M} \in \s C_{\s M}}\frac{\Tr(E_n\sigma_{\s M}^{\otimes n})}{\Tr(E_n\rho_{\s M}^{\otimes n})} \geq \tbinom{nm+d_{J}-1}{nm}^{-1}.
\end{equation}
Therefore, to show that every $\s M$-product state, $\rho_{\s M}$, eventually violates \cref{eq:nth_order}, and thus prove \cref{thm:main}, it suffices to prove the existence of a sequence of measurement effects, $n \mapsto E_n$, such that the above ratio of probabilities, as a function of increasing $n$, approaches zero faster than $\tbinom{nm+d_{J}-1}{nm}^{-1}$, thus violating \cref{eq:prob_ratio_lower}.

The particular problem of finding a sequence of measurements such that $\Tr(E_n\rho_{\s M}^{\otimes n})$ stays reasonably large while simultaneously minimizing $\Tr(E_n\sigma_{\s M}^{\otimes n})$ for all $\sigma_{\s M}$ distinct from $\rho_{\s M}$ is related to the problems of asymmetric quantum state discrimination and quantum hypothesis testing~\cite{hiai1991proper,ogawa2005strong,pereira2022analytical,hayashi2001asymptotics,hayashi2002two}. Broadly speaking, existing results in these fields are sufficiently strong to establish the claimed violation of \cref{eq:prob_ratio_lower} for each unrealizable $\rho_{\s M} \not \in \s C_{\s M}$.

Our specific approach, largely inspired by applications of the spectral estimation technique~\cite{keyl2001estimating} to the spectral QMP~\cite{christandl2006spectra,christandl2007nonzero,christandl2018recoupling}, relies on ideas developed by \citeauthor{keyl2006quantum}~\cite{keyl2006quantum} for the purposes of quantum state estimation.
In the interest of being constructive and non-asymptotic, in \cref{sec:keyl_exclusive} it is shown how to construct, for each $\rho_{\s M}$, an explicit sequence of projection operators, $n \mapsto E_n$, such that for all $n \in \mathbb N$,
\begin{equation}
\label{eq:unrealizability_measure}
\sup_{\sigma_{\s M} \in \s C_{\s M}}\frac{\Tr(E_{n} \sigma_{\s M}^{\otimes n})}{\Tr(E_{n} \rho_{\s M}^{\otimes n})} \leq \exp( - (n-d_{\s M}^2) \infkeyl{\rho_{\s M}} + c(\rho_{\s M}))
\end{equation}
where $c(\rho_{\s M}) \geq 0$ and $\infkeyl{\rho_{\s M}} \geq 0$ are quantities independent of $n$ (but dependent on $\rho_{\s M}$) and $d_{\s M} = \dim(\s H_{\s M})$. Additionally, it is shown that the exponential rate, $\infkeyl{\rho_{\s M}}$, vanishes if and only if $\rho_{\s M}$ is realizable and thus its positivity can serve as a witness of the unrealizability of $\rho_{\s M}$.

\begin{proof}[Proof of \cref{thm:main}]
The discussion preceding \cref{thm:main} has already established the ``only if'' portion of \cref{thm:main}: applying $\Tr_{mJ\setminus\s M}^{\otimes n}$ to \cref{eq:kth_inclusion} when $k = nm$ yields
\begin{align}
\rho_{\s M}^{\otimes n}
= \Tr_{mJ\setminus\s M}^{\otimes n}(\psi_J^{\otimes nm})
\leq \Tr_{mJ\setminus\s M}^{\otimes n}(\Pi^{(nm)}_{J}).
\label{eq:nth_disco}
\end{align}
Therefore, all that remains is to prove the ``if'' portion of \cref{thm:main}.
Suppose $\rho_{\s M} \in \dens(\s H_{\s M})$ is a state that satisfies \cref{eq:nth_order} for some particular value of $n$. By combining \cref{eq:unrealizability_measure} with \cref{eq:prob_ratio_lower}, we conclude
\begin{align}
\label{eq:unrealizability_upper_bound}
\infkeyl{\rho_{\s M}}
&\leq \frac{\ln \tbinom{nm + d_{J} - 1}{nm} + c(\rho_{\s M})}{n-d_{\s M}^{2}}.
\end{align}
Since $\tbinom{nm + d_{J} - 1}{nm} \in O(n^{d_{J}-1})$ is polynomial of degree $d_{J}-1$ in $n$, the upper bound above approaches zero in the limit as $n \to \infty$. For finite $n$, the inequality in \cref{eq:unrealizability_upper_bound} merely implies that $\infkeyl{\rho_{\s M}}$ must be small. For the purposes of \cref{thm:main}, if a state $\rho_{\s M}$ satisfies \cref{eq:nth_order} for all $n$, \cref{eq:unrealizability_upper_bound} implies that $\infkeyl{\rho_{\s M}} = 0$ and thus $\rho_{\s M} \in \s C_{\s M}$ must be a realizable $\s M$-product state.
\end{proof}

\section{Conclusion}

This chapter makes progress toward an analytic solution to the quantum marginal problem (QMP) by constructing a countably infinite family of necessary operator inequalities whose satisfaction by a given tuple of density operators is sufficient to conclude their realizability.
The primary advantage of this approach is its generality: for any finite Hilbert space dimension(s) and any number of subsystems with arbitrary overlap, the corresponding family of necessary inequalities is shown to be sufficient.
The results of this chapter, therefore, constitute the first solution to the QMP for overlapping marginal contexts that is free of existential quantifiers.
However, the characterization of realizable density operators produced by this approach is not finite, and thus inherently more challenging to compute.
Evidently, further insights will be required to produce a finite set of necessary and sufficient conditions for the overlapping QMP.

\section{Supporting results}

\subsection{Schur-Weyl Decompositions}
\label{sec:schur_weyl}

Given any representation of the symmetric group $\sym_n$ over a finite-dimensional complex space, such as the aforementioned tensor-permutation representation $T : \sym_n \to \s L((\mathbb C^d)^{\otimes n})$, Maschke's Theorem guarantees the representation is \textit{completely reducible} and therefore decomposes into irreducible subrepresentations~\cite[Theorem 1.5.3]{sagan2013symmetric}. Furthermore, the complete set of non-isomorphic irreducible representations of $\sym_n$ is isomorphic to the set of conjugacy classes of $\sym_n$~\cite[Proposition 1.10.1]{sagan2013symmetric} which itself is isomorphic to the set of partitions of $n$.
\begin{defn}
A \textit{partition of $n$}, $\lambda = (\lambda_1, \ldots, \lambda_\ell)$, is a sequence of non-increasing ($\lambda_i \geq \lambda_{i+1}$) positive integers ($\lambda_i \in \mathbb N$) whose total sum is $n$ (${\sum}_i \lambda_i = n$). The length of $\lambda$ is denoted by $\ell = \ell(\lambda)$. The \textit{set of all partitions of $n$} will be denoted $\yf_n$.
\end{defn}
For each partition $\lambda \in \yf_n$, let $\lilspecht{\lambda} : \sym_n \to \specht{\lambda}$ denote the corresponding irreducible representation of $\sym_n$, otherwise known as the Specht module for $\lambda$~\cite[Section 2.3]{sagan2013symmetric}. Using this notation, the Maschke decomposition of the tensor-permutation representation $T : \sym_n \to \s L((\mathbb C^d)^{\otimes n})$ yields a decomposition of $(\mathbb C^d)^{\otimes n}$,
\begin{align}
(\mathbb C^{d})^{\otimes n} \cong {\bigoplus}_{\lambda \in \yf_n} \specht{\lambda} \otimes \schur{\lambda}^d,
\label{eq:maschkes_theorem}
\end{align}
where the $\schur{\lambda}^{d}$ denotes the \textit{multiplicity space}, whose dimension counts the number of isomorphic copies of $\specht{\lambda}$ in $(\mathbb C^{d})^{\otimes n}$. It is also worth noting that $\dim(\schur{\lambda}^d) > 0$ if and only if $\ell(\lambda) \leq d$~\cite{sagan2013symmetric} and therefore the above summands over $\lambda$ are implicitly restricted to the subset $\yf^d_n \subseteq \yf_n$ of partitions of $n$ with length at most $d$. Another result, referred to as Schur-Weyl duality \cite[Chapter 9]{procesi2007lie}, implies that the multiplicity space $\schur{\lambda}^d$ itself supports an irreducible representation of $\GL(d)$, denoted $\lilschur{\lambda} : \GL(d) \to \s L(\schur{\lambda}^d)$. Let $\ket{\phi_{\lambda}} \in \schur{\lambda}^{d}$ denote the unique highest weight vector of $\schur{\lambda}^{d}$ characterized by the property that
\begin{equation}
\lilschur{\lambda}(\diag(x_1, \ldots, x_d)) \ket{\phi_{\lambda}} = \prod_{i=1}^{d} x_i^{\lambda_i} \ket{\phi_{\lambda}}
\end{equation}
for all $\diag(x_1, \ldots, x_d) \in \mathrm{GL}(d)$.
Furthermore, for each partition $\lambda \in \yf^d_n$, let
\begin{align}
\label{eq:iota_iso}
\iota_{\lambda} : \specht{\lambda} \otimes \schur{\lambda}^d \xhookrightarrow{} (\mathbb C^d)^{\otimes n}
\end{align}
be the $\sym_n \times \GL(d)$-intertwining isometry from the isotypic subspace $\specht{\lambda} \otimes \schur{\lambda}^d$ into $(\mathbb C^d)^{\otimes n}$ associated to $\lambda$. Furthermore, let
\begin{equation}
\Pi_{d}^{\lambda} \coloneqq (\iota_{\lambda})(\iota_{\lambda})^{\dagger}
\end{equation}
denote the corresponding orthogonal projection operator acting on $(\mathbb C^d)^{\otimes n}$.
\begin{prop}
\label{prop:decomp_sym_operator}
Let $Q \in \s L((\mathbb C^d)^{\otimes n})$ be an $\sym_n$-invariant operator in the sense that
\begin{equation}
\forall g \in \sym_n : T(g) Q T^{\dagger}(g) = Q.
\end{equation}
Then $Q$ admits of the following decomposition,
\begin{equation}
\label{eq:Q_decomp}
Q = \bigoplus_{\lambda \in \yf^d_n} \ident_{\specht{\lambda}} \otimes \tau_{\lambda}(Q),
\end{equation}
where the $\lambda$-component of $Q$, $\tau_{\lambda}(Q) \in \s L(\schur{\lambda}^{d})$, is defined as
\begin{equation}
\tau_\lambda(Q) = \frac{\Tr_{\specht{\lambda}}(\iota_{\lambda}^{\dagger} Q \iota_\lambda)}{\dim(\specht{\lambda})}.
\end{equation}
\end{prop}
\begin{proof}
This follows from an application of Schur's lemma~\cite[Theorem 4.29]{hall2015lie}.
\end{proof}

\begin{defn}
\label{defn:twirled}
Let $\lilschur{\lambda} : \GL(d) \to \schur{\lambda}^d$ be the irreducible representation of $\GL(d)$ with highest weight vector $\ket{\phi_\lambda} \in \schur{\lambda}^d$ where $\lambda \in \yf_n^d$. For each unitary operator $U \in \U(d) \subseteq \GL(d)$, let the \textit{twirled highest weight vector} be defined as $\ket{\phi^{U}_\lambda} \coloneqq \lilschur{\lambda}(U) \ket{\phi_\lambda} \in \schur{\lambda}^{d}$.
\end{defn}
Consider the quantity $\bra{\phi_\lambda^{U}} \tau_\lambda(\rho^{\otimes n}) \ket{\phi_{\lambda}^{U}}$, which depends only on $\lambda \in \yf_n^d$ and $U^{\dagger} \rho U \in \dens(\mathbb C^d)$.
In \cref{sec:keyl_exclusive}, specifically \cref{prop:hwv_lpm}, we will show that this quantity can be extended to a formula that remains well-defined even when $\lambda$ is permitted to be a non-increasing sequence of \textit{non-negative real numbers}.

\subsection{Spectra \& Partitions}
\label{sec:spectra_partitions}

The purpose of this subsection is to develop a connection between i) partitions $\lambda \in \yf_n^d$ with length at most $d$, and ii) the possible eigenvalues of density operators $\rho \in \dens(\mathbb C^d)$.
\begin{defn}
A subset $C \subseteq \mathbb R^d$ is called a \textit{convex cone} if it is closed under
\begin{enumerate}[i)]
\item addition: for any $x, y \in C$, $x+y \in C$, and
\item multiplication: for any $x \in C$, and $a \geq 0$, $a x \in C$.
\end{enumerate}
\end{defn}
Two convex cones that are relevant here will be the cone of non-negative real numbers,
\begin{equation}
\mathbb R_{\geq 0}^{d} = \{ (x_1, \ldots, x_d) \in \mathbb R^d \mid \forall i : x_i \geq 0\},
\end{equation}
and the subset of non-increasing non-negative real numbers,
\begin{equation}
\mathbb R_{\geq 0}^{d;\downarrow} = \{ (x_1, \ldots, x_d) \in \mathbb R^d \mid x_1 \geq \cdots \geq x_d \geq 0\}.
\end{equation}
While there is a natural surjective map from $\mathbb R_{\geq 0}^{d}$ to $\mathbb R_{\geq 0}^{d;\downarrow}$ which sorts the elements of $(x_1, \ldots, x_d)$ in a non-increasing order, there is also a \textit{bijective} linear map $\gamma : \mathbb R_{\geq 0}^{d} \to \mathbb R_{\geq 0}^{d;\downarrow}$ which takes partial sums. Specifically, $\gamma$ maps $y = (y_1, \ldots, y_{d}) \in \mathbb R_{\geq 0}^{d}$ to $\gamma(y) = (\gamma_1(y), \ldots, \gamma_d(y))$ where
\begin{equation}
\label{eq:cumm}
\gamma_i(y) = y_i + y_{i+1} + \cdots + y_d.
\end{equation}
The inverse of $\gamma$, henceforth denoted $\fd : \mathbb R_{\geq 0}^{d;\downarrow} \to \mathbb R_{\geq 0}^{d}$, takes finite differences; specifically, $\fd$ maps $x = (x_1, \ldots, x_{d}) \in \mathbb R_{\geq 0}^{d;\downarrow}$ to $\fd(x) = (\fd_1(x), \ldots, \fd_d(x))$, where
\begin{equation}
\label{eq:diffs}
\fd_i(x) = \fd_i(x_1, \ldots, x_{k}) = \begin{cases} x_i - x_{i+1} & 1 \leq i < d \\ x_d & i = k \end{cases}.
\end{equation}
\begin{defn}
For each $x = (x_1, \ldots, x_d) \in \mathbb R_{\geq 0}^{d}$, the \textit{size of $x$}, $\abs{x}$, is the sum of its elements
\begin{equation}
\abs{x} = x_1 + \cdots + x_d,
\end{equation}
If $x$ is not equal to all-zero $d$-tuple, $x \neq (0, \ldots, 0)$, then $\abs{x} > 0$, and the \textit{normalization of $x$} is defined as
\begin{equation}
\frac{x}{\abs{x}} = \left(\frac{x_1}{\abs{x}}, \ldots, \frac{x_d}{\abs{x}}\right).
\end{equation}
\end{defn}
Two subsets of $\mathbb R_{\geq 0}^{d;\downarrow}$ will be crucial to the results of \cref{sec:keyl_exclusive}. The first subset was already discussed in \cref{sec:schur_weyl}, namely partitions of $n$ with length at most $d$: $\yf^d_n \subseteq \mathbb R_{\geq 0}^{d;\downarrow}$. If the length, $\ell$, of $\lambda = (\lambda_1, \ldots, \lambda_\ell)$ is strictly less than $d$, then it can be viewed as a element of  $\mathbb R_{\geq 0}^{d;\downarrow}$ by padding $\lambda$ with $d-\ell$ zeros, i.e. $\lambda \cong (\lambda_1, \ldots, \lambda_\ell, 0, \ldots, 0)$. The second subset corresponds to the set of possible eigenvalues, or spectra, of density operators $\dens(\mathbb C^d)$.
\begin{defn}
The set of \textit{spectra}, or sorted probability distributions, is
\begin{equation}
\spectra^{d} = \{ s \in \mathbb R_{\geq 0}^{d;\downarrow} \mid  {\sum}_{i=1}^{d} s_i = \abs{s} = 1 \}.
\end{equation}
\end{defn}
While the normalization of any partition, $\lambda \in \yf^d_n$, is a spectrum, $\frac{\lambda}{n} \in \spectra^d$, multiplying a spectrum, $s \in \spectra^d$, by $n \in \mathbb N$ does not necessarily produce a partition because the entries of $ns$, $(ns_1, \ldots, ns_d)$, may not be integer-valued. Nevertheless, $ns \in \mathbb R_{\geq 0}^{d;\downarrow}$ can always be approximated by a partition, $\lambda \in \yf_n^d$, so that $\abs{\lambda_i - ns_i} \leq 1$ for all $i \in \{1, \ldots, d\}$ \footnote{An explicit scheme for accomplishing such an approximation is to let $t = n - {\sum}_{i}\lfloor n s_i \rfloor$ and define $\lambda_i = \lfloor n s_i \rfloor + 1$ whenever $i \leq t$ and $\lambda_i = \lfloor n s_i \rfloor$ whenever $i > t$.}. In \cref{sec:keyl_exclusive}, it will be useful to consider approximating $n s$ with a partition, $\lambda$, in a different manner, where i) degeneracies of $s$ are preserved, i.e., $\delta_i(s) = 0 \implies \delta_i(\lambda) = 0$, and ii) non-degeneracies of $s$ are adequately represented, e.g., $\delta_i(\lambda) \geq \delta_i(ns)$. The next lemma shows that this can always be accomplished by partitions, $\lambda$, whose size is approximately $n$.

\begin{prop}
\label{prop:crit_approx}
Let $s = (s_1, \ldots, s_k) \in \spectra_{d}$ be a spectrum and $n \in \mathbb N$.
\begin{equation}
\lambda = (\lambda_1, \ldots, \lambda_d) \in \mathbb N_{\geq 0}^{d;\downarrow}
\end{equation}
be defined by
\begin{equation}
\lambda_i = \lceil n (s_i - s_{i+1}) \rceil + \cdots + \lceil n (s_{d-1} - s_{d}) \rceil + \lceil n s_d \rceil,
\end{equation}
such that $\fd_i(\lambda) = \lceil \fd_i(ns) \rceil$ holds. Then $\lambda$ is a partition of size $\abs{\lambda}$ where
\begin{equation}
\label{eq:crit_approx_3}
n \leq \abs{\lambda} \leq n + \tbinom{d+1}{2} - 1.
\end{equation}
\end{prop}
\begin{proof}
First note that for all $1 \leq i \leq d$,
\begin{equation}
\epsilon_i \coloneqq \lceil \fd_i(ns) \rceil - \fd_i(ns),
\end{equation}
is upper and lower bounded by $0 \leq \epsilon_i < 1$.
From this observation, it will be shown that $\lambda$ approximates $ns$, specifically,
\begin{equation}
\label{eq:crit_approx_2}
0 \leq \lambda_i - n s_i < d - i + 1,
\end{equation}
To prove \cref{eq:crit_approx_2}, we use (reverse) induction starting from the base case of $i = d$.
Since $\lambda_d = \fd_d(\lambda) = \lceil \fd_d(ns) \rceil = \lceil n s_d \rceil = n s_d + \epsilon_d$, we have $\lambda_d - n s_d = \epsilon_d$ and thus \cref{eq:crit_approx_2} holds when $i = d$. Then, assuming \cref{eq:crit_approx_2} holds for $i = j+1$, we prove it holds for $i = j$. Since
\begin{align}
\fd_j(\lambda)
&= \lambda_{j} - \lambda_{j+1} = \lceil \fd_j(ns) \rceil \\
&= \fd_j(ns) + \epsilon_j = n s_j - n s_{j+1} + \epsilon_j,
\end{align}
we conclude that $\lambda_{j} - n s_j = \lambda_{j+1} - n s_{j+1} + \epsilon_j$ and thus $0 \leq \lambda_{j} - ns_{j} < d - j + 1$ which is \cref{eq:crit_approx_2} for $i = j$.
Finally, \cref{eq:crit_approx_3} follows from \cref{eq:crit_approx_2} by summing over all $i$:
\begin{equation}
0 \leq \abs{\lambda} - n \abs{s} < \sum_{i=1}^{d} (d-i+1) = \tbinom{d+1}{2}.
\end{equation}
Since $\abs{\lambda}$ is necessarily an integer, $\abs{s} = 1$, and the upper bound above is strict, \cref{eq:crit_approx_2} holds.
\end{proof}
The bounds proven above are also tight for every $d$: if $s = \tbinom{d+1}{2}^{-1}(d, d-1, \ldots, 1)$, then $n = 1$ or $n = \tbinom{d+1}{2}$ yields $\lambda = (d, d-1, \ldots, 1)$ with size $\abs{\lambda} = \tbinom{d+1}{2}$ which achieves the upper bound when $n = 1$ and the lower bound when $n = \tbinom{d+1}{2}$.

\subsection{Proof of \texorpdfstring{\cref{lem:de_finetti_realizable}}{a realizability lemma}}
\label{sec:proof_finetti_lemma}

\begin{proof}[Proof of \cref{lem:de_finetti_realizable}]
Let $d_{J} = \dim(\s H_J)$, let $\mu_{J}$ be the $U(d_{J})$-invariant Haar probability measure over the space of pure states $\proj(\s H_J)$,
For any $k \in \mathbb N$, the orthogonal projection operator, $\Pi_J^{(k)}$, onto the symmetric subspace, $\symsub^{k} \s H_J$, is proportional to the expected value of $\psi_J^{\otimes k}$ when $\psi_J$ is sampled according to the probability measure $\mu_{J}$:
\begin{equation}
\label{eq:haar_sym}
\Pi_{J}^{(k)} = \tbinom{k+d_{J}-1}{k}\int_{\proj(\s H_{J})} \mu_{J}(\diff \psi_{J}) \psi_{J}^{\otimes k},
\end{equation}
where the normalization factor is simply $\Tr[\Pi_{J}^{(k)}] = \tbinom{k+d_{J}-1}{k}$. The proof of \cref{eq:haar_sym} follows from Schur's lemma (see \cite[Proposition 6]{harrow2013church}).

Next, define the map $\tau_{\s M} : \proj(\s H_J) \to \dens(\s H_{\s M})$ by
\begin{equation}
\tau_{\s M}(\psi_J) = \Tr_{mJ\setminus\s M}(\psi_J^{\otimes m}).
\end{equation}
Let $\nu_{\s M}$ be the push-forward measure of $\mu_{J}$ through $\tau_{\s M}$, i.e. $\nu_{\s M} = \mu_{J} \circ \tau_{\s M}^{-1}$.

Next note that the coefficients $\tau_{\s M}(\psi_J)$ are homogeneous polynomials of degree $m$ in the coefficients of $\psi_{J}$, and thus $\tau_{\s M}$ is continuous and measurable. Additionally, by construction, the image of $\tau_{\s M}$ is precisely the set of realizable $\s M$-product states $\s C_{\s M}$. Therefore, since $\proj(\s H_J)$ is compact (as $\s H_J$ is finite-dimensional), $\s C_{\s M}$ is also compact (and thus closed).

Moreover, the support of the pushforward measure, $\nu_{\s M} = \mu_J \circ \tau_{\s M}^{-1}$ is equal to $\s C_{\s M}$. This is because, by the closure of $\s C_{\s M}$, $\rho_{\s M} \not \in \s C_{\s M}$ implies there exists an open set, $O$ containing $\rho_{\s M}$, such that $O \cap \s C_{\s M} = \emptyset$ which implies $\nu_{\s M}(O) = \mu_J(\tau_{\s M}^{-1}(O)) = \mu_J(\emptyset) = 0$, i.e. $\rho_{\s M}$ is not in support of $\nu_{\s M}$. Moreover, if $\sigma_{\s M} \in \s C_{\s M}$ and $O'$ is any open set containing $\sigma_{\s M}$, $\tau_{\s M}^{-1}(O')$ is non-empty and open (by continuity of $\tau_{\s M}$) in $\proj(\s H_J)$ and thus $\nu_{\s M}(O') = \mu_J(\tau_{\s M}^{-1}(O')) > 0$. Therefore, because $\nu_{\s M}(O') > 0$ for all open sets containing $\sigma_{\s M}$, $\sigma_{\s M}$ is in the support $\nu_{\s M}$.

Finally, using \cref{eq:haar_sym}, linearity of $\Tr_{mJ\setminus\s M}$, and a change of variables,
\begin{align}
&\Tr_{mJ\setminus\s M}^{\otimes n}(\Pi^{(nm)}_J) \nonumber \\
&\quad\propto \int_{\proj(\s H_J)} \mu_{J}(\diff \psi_J) (\Tr_{mJ\setminus\s M}(\psi_J^{\otimes m}))^{\otimes n}, \\
&\quad= \int_{\proj(\s H_J)} \mu_{J}(\diff \psi_J) (\tau(\psi_J))^{\otimes n}, \\
&\quad= \int_{\s C_{\s M}} \nu_{\s M}(\diff \sigma_{\s M}) \sigma_{\s M}^{\otimes n}.
\end{align}
\end{proof}

\subsection{Keyl Divergence \& State Discrimination}
\label{sec:keyl_exclusive}

The purpose of this subsection is to prove \cref{thm:constructive_state_discrim} which can be interpreted as an explicit strategy for asymmetric quantum state discrimination. While \cref{thm:constructive_state_discrim} is exclusively used by this paper in the proof of \cref{thm:main}, it may be of independent interest. Many of the results of this subsection come directly from \citeauthor{keyl2006quantum}'s work on a large-deviation-theoretic approach to quantum state estimation~\cite{keyl2006quantum}. The only additional insight not taken from \cite{keyl2006quantum} is the use of \cref{prop:crit_approx} in the proof of \cref{thm:constructive_state_discrim}. Both \cref{sec:schur_weyl} and \cref{sec:spectra_partitions} are considered prerequisites for this subsection.

\begin{defn}
Let $x \in \mathbb R_{\geq 0}^{d;\downarrow}$ and let $\rho \in \dens(\mathbb C^{d})$, define
\begin{equation}
\label{eq:gpf}
\Delta_{x}(\rho) = {\prod}_{i=1}^{d} \lpm_{i}(\rho)^{\fd_i(x)},
\end{equation}
where $\lpm_i(\rho)$ is the $i$th (leading) principal minor of $\rho$, i.e. the determinant of the upper-left $i\times i$-submatrix of $\rho$ with respect to some fixed, computational basis $\{ \ket{0}, \ldots, \ket{d}\}$. \footnote{If it happens that $\lpm_i(\rho) = 0$ and $\fd_i(x) = 0$ for some index $i$, then the indeterminant expression $0^0$ is taken to be equal to $1$.}
\end{defn}
The function defined in \cref{eq:gpf} is also referred to as the \textit{generalized power function}~\cite[Notation 4.1]{odonnell2016efficient}. Note however, in \cite{odonnell2016efficient}, $x$ is restricted to be a partition with length at most $d$, while $\rho$ is permitted to be any $d \times d$ complex-valued matrix.

\begin{prop}
Let $s = (s_1, \ldots, s_d) \in \spectra^d$ be the spectrum of $\sigma \in \dens(\mathbb C^d)$. For all $x = (x_1, \ldots, x_d) \in \mathbb R_{\geq 0}^{k; \downarrow}$,
\begin{equation}
\Delta_{x}(\sigma) \leq \Delta_{x}(\diag(s_1, \ldots, s_d)) = {\prod}_{i=1}^{d} s_i^{x_i},
\end{equation}
with equality holding if and only if $\sigma = \diag(s_1,\ldots, s_d)$.
\end{prop}
\begin{proof}
Consider any $i \in \{1, \ldots, d\}$. Let $(s^{(i)}_{1}, \ldots, s^{(i)}_i)$ with $s^{(i)}_1 \geq \cdots \geq s^{(i)}_i$ denote the eigenvalues of the $i\times i$ leading principal submatrix of $\sigma$ so that $\lpm_i(\sigma) = s^{(i)}_1 \cdots s^{(i)}_i$.
According to Cauchy's interlacing theorem (see \cite{fisk2005very} or \cite[Thm. 4.3.17]{horn1985matrix} noting the reversed ordering of labels), for all $1 < i \leq d$,
\begin{equation}
s^{(i)}_{1} \geq s^{(i-1)}_1 \geq s^{(i)}_{2} \geq \cdots \geq s^{(i)}_{i-1} \geq s^{(i-1)}_{i-1} \geq s^{(i)}_{i}.
\end{equation}
Therefore, for any $k$ and $i$ such that $1 \leq k \leq i \leq d$, $s_k = s^{(d)}_k \geq s^{(i)}_k \geq s^{(k)}_k$. Therefore, for all $i \in \{1, \ldots, d\}$,
\begin{equation}
\label{eq:maximimal_principal_minor}
\lpm_i(\sigma) \leq s_1 \cdots s_i,
\end{equation}
with equality holding (for all $i$) only if $\sigma = \diag(s_1,\ldots, s_d)$.
\end{proof}
\begin{prop}
\label{prop:shannon}
Let $s = (s_1, \ldots, s_d) \in \spectra^d \subseteq \mathbb R_{\geq 0}^{d;\downarrow}$. Then
\begin{equation}
\Delta_s(\mathrm{diag}(s)) = {\prod}_{i=1}^{d} s_i^{s_i} = \exp(- H(s)) > 0.
\end{equation}
where $H(s) = -{\sum}_{i=1}^{d} s_i \ln s_i$ is the Shannon entropy of $s$.
\end{prop}
\begin{cor}
\label{cor:keyl_div}
Let $\rho \in \dens(\mathbb C^{d})$ have spectrum $s = (s_1, \ldots, s_d) \in \spectra^d$ and let $U \in \U(d)$ be a unitary such that $\rho = U \diag(s_1, \ldots, s_d) U^{\dagger}$ and let $\sigma \in \dens(\mathbb C^{d})$. Then
\begin{equation}
\frac{\Delta_{s}(U^{\dagger} \sigma U)}{\Delta_{s}(\diag(s))} = \exp(-\keyl{\rho}{\sigma})
\end{equation}
where $\keyl{\rho}{\sigma}$ is defined as
\begin{equation}
\keyl{\rho}{\sigma} = {\sum}_{i=1}^{d} s_i \ln s_i - \fd_i(s) \ln \lpm_i(U^{\dagger} \sigma U),
\end{equation}
where $\keyl{\rho}{\sigma} \in [0, \infty]$ and $\keyl{\rho}{\sigma} = 0$ if and only if $\sigma = \rho$.
\end{cor}
Notice that if $\rho$ and $\sigma$ are simultaneously diagonalized by $U$ so that
\begin{equation}
U^{\dagger} \sigma U = \diag(t_1, \ldots, t_d),
\end{equation}
then the quantity $\keyl{\rho}{\sigma}$ simplifies to the classical relative entropy, $\kl{s}{t} = \sum_{i=1}^{d} s_i (\ln s_i - \ln t_i)$,
also known as Kullback-Liebler divergence~\cite{kullback1997information}. Also note that, in general, $\keyl{\rho}{\sigma}$ does not equal the quantum relative entropy $\qrl{\rho}{\sigma} = \Tr(\rho (\ln \rho - \ln \sigma))$, but is nevertheless bounded by $\keyl{\rho}{\sigma} \leq \qrl{\rho}{\sigma}$~\cite{keyl2006quantum}. For these reasons, we refer to the quantity $\keyl{\rho}{\sigma}$ as \textit{Keyl-divergence}.

\begin{prop}
\label{prop:hwv_lpm}
Let $\lambda \in \yf_n^d$ be a partition of $n \in \mathbb N$ and let $\ket{\phi_\lambda^U} \in \schur{\lambda}^d$ be the twirled highest weight vector for $U \in \U(d)$ (see \cref{defn:twirled}). Then for all $\sigma \in \dens(\mathbb C^d)$,
\begin{equation}
\bra{\phi^U_\lambda} \tau_{\lambda}(\sigma^{\otimes \abs{\lambda}}) \ket{\phi^U_\lambda} = \Delta_{\lambda}(U^{\dagger}\sigma U).
\end{equation}
\end{prop}
\begin{proof}
This result is noted by \citeauthor{keyl2006quantum} as \cite[Eqs. (141) \& (151)]{keyl2006quantum} with reference to \cite[Sec. 49]{zhelobenko1973compact}.
\end{proof}
Henceforth, define the projection operator $\Phi_{\lambda}^{U} \in \s L(H^{\otimes \abs{\lambda}})$ by
\begin{equation}
\Phi_{\lambda}^{U} \coloneqq \iota_{\lambda} (\ket{\phi_{\lambda}^{U}} \bra{\phi^{U}_{\lambda}} \otimes \ident_{\specht{\lambda}}) \iota_{\lambda}^{\dagger},
\end{equation}
so that,
\begin{equation}
\label{eq:defn_of_big_phi}
\Tr(\Phi_{\lambda}^{U} \rho^{\otimes \abs{\lambda}}) = \dim(\specht{\lambda})\Delta_{\lambda}(U^{\dagger} \rho U).
\end{equation}
\begin{cor}
\label{cor:rational_state_discrim}
Let $\rho, \sigma \in \dens(\mathbb C^d)$ and let $U \in \U(d)$ diagonalize $\rho$,
\begin{equation}
\rho = U\diag(s_1, \ldots, s_d)U^{\dagger}.
\end{equation}
If $\rho$ has \textit{rational} spectra $s = (s_1, \ldots, s_d) \in \spectra^{d}$, i.e. there exists a $q \in \mathbb N$ such that $qs \in \yf_{q}^{d}$ is a partition of $q$, then for all $n \in \mathbb N$,
\begin{equation}
\label{eq:rational_state_discrim}
\frac{\Tr(\Phi^{U}_{nqs} \sigma^{\otimes nq})}{\Tr(\Phi^{U}_{nqs}\rho^{\otimes nq})} = \exp (-nq\keyl{\rho}{\sigma}).
\end{equation}
\end{cor}
\begin{proof}
The proof follows from \cref{eq:defn_of_big_phi} and \cref{cor:keyl_div}. When \cref{eq:defn_of_big_phi} is applied to the numerator and denominator on the left-hand-side of \cref{eq:rational_state_discrim}, the common factor of $\dim(\specht{nqs}) > 0$ cancels out.
\end{proof}
A result similar to \cref{cor:rational_state_discrim} holds for arbitrary states $\rho \in \dens(\mathbb C^d)$, e.g., for states that do not have rational spectra.
\begin{thm}
\label{thm:constructive_state_discrim}
Let $\rho, \sigma \in \dens(\mathbb C^d)$, let $s = (s_1, \ldots, s_d) \in \spectra^{d}$ be the spectrum of $\rho$, and let $U \in \U(d)$ diagonalize $\rho$, such that $\rho = U\diag(s_1, \ldots, s_d)U^{\dagger}$.
Then there exists a sequence, $n \mapsto \lambda^{n} \in \yf_{n}^{d}$ of partitions such that for all $n \in \mathbb N$,
\begin{equation}
\label{eq:constructive_state_discrim}
\frac{\Tr(\Phi^{U}_{\lambda^{n}}\sigma^{\otimes n})}{\Tr(\Phi^{U}_{\lambda^{n}}\rho^{\otimes n})} \leq D(s)\exp (-(n-\tbinom{d+1}{2}+1) \keyl{\rho}{\sigma}).
\end{equation}
where $D(s)$ is a constant depending only on $s$.
\end{thm}
\begin{proof}
The proof relies on an explicit construction of a sequence, $n \mapsto \lambda^{n}$, that satisfies the claim.
For each non-negative integer $k \in \mathbb N_{\geq 0}$, let $\mu^{k}$ be the partition characterized by $\delta_i(\mu^k) = \lceil\fd_i(ks)\rceil$ (see \cref{prop:crit_approx}). Then for any state $\eta \in \dens(\mathbb C^d)$, we claim
\begin{equation}
\label{eq:eta_bounds}
\Delta_{ks}(\eta) \Delta_{\mu^1}(\eta) \leq \Delta_{\mu^k}(\eta) \leq \Delta_{ks}(\eta).
\end{equation}
To see the upper bound, note that for all $i \in \{1, \ldots, d\}$, $\fd_i(\mu^k) \geq \fd_i(k s) > 0$ so $\lpm_i(\eta)^{\fd_{i}(\mu^k)} \leq \lpm_i(\eta)^{\fd_{i}(ks)}$ since $\lpm_i(\eta) < 1$ (noting \cref{eq:maximimal_principal_minor}).
For the lower bound, note that $\fd_i(\mu^k) = \lceil \fd_i(ks) \rceil \leq \fd_i(ks) + \lceil \fd_i(s) \rceil = \fd_i(ks) + \fd_i(\mu^1)$, so $\Delta_{\mu^k}(\eta) \geq \Delta_{ks}(\eta) \Delta_{\mu^1}(\eta)$ holds.
Next, apply \cref{cor:keyl_div} and \cref{eq:eta_bounds} (the upper bound when $\eta = U^{\dagger} \sigma U$, and the lower bound when $\eta = U^{\dagger} \rho U = \diag(s_1, \ldots s_d)$) to obtain
\begin{equation}
\label{eq:constructive_state_discrim_k}
\frac{\Tr(\Phi_{\mu^{k}}^{U}\sigma^{\otimes \abs{\mu^k}})}{\Tr(\Phi_{\mu^{k}}^{U}\rho^{\otimes \abs{\mu^k}})} \leq \frac{\exp(-k \keyl{\rho}{\sigma})}{\Delta_{\mu^{1}}(\diag(s_1, \ldots s_d))}.
\end{equation}
Now, notice that \cref{eq:constructive_state_discrim_k} is almost in the form of \cref{eq:constructive_state_discrim}.
The main obstacle remaining is simply that the size of $\mu^{k}$ needs to be decoupled from the spectra of $\rho$.
Fortunately, \cref{prop:crit_approx} guarantees that $\abs{\mu^{k}}$ is, at least, approximately equal to $k$ because $k \leq \abs{\mu^{k}} \leq k + \tbinom{d+1}{2} - 1$. Moreover, since $\abs{\mu^{k+1}} \geq \abs{\mu^{k}}$, there always exists at least one value of $k$ such that $\mu^{k}$ has size approximately $n$ for any $n \in \mathbb N$; specifically, there exists a $k \in \mathbb N$ such that
\begin{equation}
\label{eq:optimal_k}
n - \tbinom{d+1}{2} + 1 \leq \abs{\mu^{k}} \leq n.
\end{equation}
Now simply define $\lambda^{n} \in \yf_{n}^{d}$ by
\begin{equation}
\lambda^{n} = (\mu^{k}_1 + n - \abs{\mu^{k}}, \mu^{k}_{2}, \ldots, \mu_{d}^{k}),
\end{equation}
where $k$ is the largest such that $\mu_k$ satisfies \cref{eq:optimal_k}. Note that when $n$ is small ($n < \tbinom{d+1}{2} - 1$), is entirely possible for $k = 0$ and $\mu_0 = (0,0,\ldots,0)$, in which case, $\lambda^n = (n, 0, \ldots, 0)$.
This definition ensures
\begin{align}
\Tr(\Phi^{U}_{\lambda^{n}} \rho^{\otimes n}) &= s_1^{n-\abs{\mu^{k}}} \Tr(\Phi^{U}_{\mu^k} \rho^{\otimes \abs{\mu^{k}}}), \\
\Tr(\Phi^{U}_{\lambda^{n}} \sigma^{\otimes n}) &\leq \Tr(\Phi^{U}_{\mu^k} \sigma^{\otimes \abs{\mu^{k}}}).
\end{align}
Therefore, from \cref{eq:constructive_state_discrim_k,eq:optimal_k}, we conclude \cref{eq:constructive_state_discrim} where $D(s)$ is the constant
\begin{align}
&= s_1^{1-\tbinom{d+1}{2}} \left(\Delta_{\mu^{1}}(\diag(s_1, \ldots s_d))\right)^{-1} \\
&= s_1^{1-\tbinom{d+1}{2}}\prod_{i=1}^{d} (s_1s_2 \cdots s_i)^{-\lceil \fd_i(s) \rceil}.
\end{align}
\end{proof}
\begin{rem}
\label{rem:state_discrim}
To derive the inequality in \cref{eq:unrealizability_measure} from the result of \cref{thm:constructive_state_discrim}, note that $\tbinom{d+1}{2}-1 \leq d^2$ and substitute
\begin{enumerate}[i)]
\item $E_n = \Phi_{\lambda^{n}}^{U}$,
\item $\infkeyl{\rho_{\s M}} = \inf_{\sigma_{\s M} \in \s C_{\s M}} \keyl{\rho_{\s M}}{\sigma_{\s M}}$,
\item $c(\rho_{\s M}) = \ln D(\mathrm{spec}(\rho_{\s M}))$, and
\item $d = d_{\s M} = \dim(\s H_{\s M})$.
\end{enumerate}
The claim that $\infkeyl{\rho_{\s M}}$ vanishes if and only if $\rho_{\s M} \in \s C_{\s M}$ follows from the compactness of $\s C_{\s M}$ and the following corollary.
\end{rem}
\begin{cor}
\label{cor:keyl_div_from_R}
Let $\s C \subseteq \dens(\mathbb C^d)$ be compact. Define
\begin{equation}
\infkeyl{\rho} \coloneqq \inf_{\sigma \in \s C}\keyl{\rho}{\sigma}.
\end{equation}
Then $\infkeyl{\rho} = 0$ if and only if $\rho \in \s C$.
\end{cor}
\begin{proof}
If $\rho \in \s C$, then \cref{cor:keyl_div} implies $\infkeyl{\rho} = \keyl{\rho}{\rho} = 0$. Otherwise if $\rho \not \in \s C$, then consider, for each fixed $x \in \mathbb R_{\geq 0}^{d;\downarrow}$ and $U \in \U(d)$, that the function $\sigma \mapsto \Delta_x(U^{\dagger} \sigma U) \in [0,1]$ is continuous as $\lpm_i(U^{\dagger} \sigma U)$ is a polynomial in the coefficients of $\sigma$. The compactness of $\s C$ guarantees (using the extreme value theorem) the supremum is attained by some $\sigma_{x}^{U} \in \s C$:
\begin{equation}
\label{eq:least_divergent_sigma}
\Delta_x(U^{\dagger} \sigma_x^{U} U) = \sup_{\sigma \in \s C} \Delta_x(U^{\dagger} \sigma U).
\end{equation}
Since $\infkeyl{\rho} = \keyl{\rho}{\sigma_x^{U}}$ and $\rho \neq \sigma_x^{U}$, we conclude, from \cref{cor:keyl_div}, that $\infkeyl{\rho} \neq 0$.
\end{proof}

\section{Special Cases}

\subsection{Pure vs. Full QMP}
\label{sec:pure_vs_mixed_qmp}
One might wonder why the version of the QMP considered in this paper (\cref{prob:qmp}) seems to be exclusively interested in the existence of joint states that are \textit{pure}, $\psi_J \in \proj(\s H_{J})$, instead of the more general \textit{density operator}, $\rho_{J} \in \dens(\s H_{J})$. In order to distinguish between these two types of QMP, the former is sometimes called the \textit{pure} QMP, while the latter is sometimes called the \textit{mixed} QMP. Of these two variants, the mixed QMP is arguably a much closer analogy to the classical marginals problem~\cite{fritz2012entropic}.

The distinction between these two variants is strongest when the marginal scenario under consideration, $\s M = (S_1, \ldots, S_m)$, has disjoint marginal contexts, i.e. $S_i \cap S_j = \emptyset$ for all $i \neq j$, or equivalently $\s H_{\s M} = \s H_{S_1}\otimes \cdots \otimes \s H_{S_m} \cong \s H_J$. Under the assumption of disjoint marginal contexts, the mixed QMP becomes trivial; every collection of density operators $(\rho_{S_1}, \ldots, \rho_{S_m})$ are the $\s M$-marginals of the density operator $\rho_J = \rho_{S_1} \otimes \cdots \otimes \rho_{S_m}$. On the other hand, under this assumption, the pure QMP remains non-trivial.

At the level of generality considered in this paper, wherein the marginal scenario $\s M = (S_1, \ldots, S_m)$ is permitted to contain overlapping marginal contexts, e.g., $S_i \cap S_j \neq \emptyset$, the distinction becomes less important because the marginals of a mixed state can equivalently be viewed as the marginals of any of its purifications. Specifically, there exists a density operator $\sigma_{J} \in \dens(\s H_{J})$ with marginals $\rho_{S} = \Tr_{J\setminus S}(\sigma_{J})$ for all $S \in \s M$ if and only if there exists a joint pure state $\psi_{JJ'} \in \proj(\s H_{J} \otimes \s H_{J'})$ (where $\s H_{J'} \cong \s H_{J}$) such that $(\Tr_{J\setminus S}\otimes \Tr_{J'})(\sigma_{J}) = \rho_{S}$ for all $S \in \s M$. Consequently, the techniques developed in this paper, which directly apply to the pure QMP, can also be applied to any instance of the mixed QMP without substantial modification.

\subsection{Diagrammatics}
\label{sec:diagrammatics}

The purpose of this subsection is to briefly introduce some diagrammatic notation that will be useful for performing a few calculations in \cref{sec:n1}. The particular notations involving symmetrization and antisymmetrization used here (\cref{eq:sym_2_X} and onward), are taken from \citeauthor{cvitanovic2008group}'s excellent textbook~\cite{cvitanovic2008group} on diagrammatic calculations of invariants of Lie groups, and are essentially the same those used by \citeauthor{penrose1971applications}~\cite{penrose1971applications}. For a categorical justification of this notation, see~\cite{selinger2012finite}. For further applications within quantum theory, see~\cite{coecke2010compositional,wood2011tensor,biamonte2017tensor}.

The essential idea is to depict linear operators, $L : \s H_X \to \s H_Y$, by pictures with corresponding inputs and outputs:
\begin{equation}
\strdia{generic_l_x_y}.
\end{equation}
Of course, two linear operators can be combined in at least three different ways; specifically, by addition $+$, tensor product $\otimes$, and sequential composition $\circ$. These operations are depicted respectively as
\begin{align}
\strdia{composition_addition_LHS} &= \strdia{composition_addition_RHSone} + \strdia{composition_addition_RHStwo},\\
\strdia{composition_tensor_LHS} &= \strdia{composition_tensor_RHS}, \\
\strdia{composition_circ_LHS} &= \strdia{composition_circ_RHS}.
\end{align}
Important special cases of this notation include the identity operator $\ident_X : \s H_X \to \s H_X$,
\begin{equation}
\strdia{identity_X_X_explicit} = \strdia{identity_X_X},
\end{equation}
and vectors $\ket{\psi_X} : \mathbb C \to  \s H_{X}$ (and their conjugates $\bra{\psi_X} : \s H_{X} \to \mathbb C$) as
\begin{equation}
\strdia{generic_ket}, \quad \text{and} \quad \strdia{generic_bra}.
\end{equation}
This notation is especially elegant for depicting two concepts frequently encountered in this paper: the partial trace and permutations.

First, given a bipartite operator $L : \s H_{X} \otimes \s H_{Y} \to \s H_{X} \otimes \s H_{Y}$, the partial trace $\Tr_{Y}$ over $Y$, is depicted as
\begin{equation}
\strdia{generic_partial_trace_verbose} = \strdia{generic_partial_trace}.
\end{equation}
The trace over the identity operator $\ident_{X} : \s H_{X} \to \s H_{X}$, which is equal to the dimension of $\s H_X$, is therefore depicted as a closed loop
\begin{equation}
d_X = \dim(\s H_X) = \strdia{identity_trace_explicit} = \strdia{identity_trace} = \strdia{dimension_loop}.
\end{equation}
Second, the tensor permutation representation, $T_{X} : \sym_{m} \to \s L(\s H_{X}^{\otimes m})$, of the symmetric group, $\sym_{m}$, has elements depicted naturally as follows. When $m=2$, $\sym_{2} = \{ e, (12) \}$, and the identity $T_{X}(e)$ and swap $T_{X}((12))$ are depicted respectively by
\begin{equation}
\strdia{Stwo_e_X}, \quad \text{and} \quad \strdia{Stwo_onetwo_X}.
\end{equation}
Analogously, for $m=3$, the $3!=6$ permutations in $\sym_{3}$ are depicted by
\begin{align}
\begin{split}
\strdia{Sthree_e_X}, &\qquad \strdia{Sthree_onetwo_X}, \qquad \strdia{Sthree_twothree_X},\\
\strdia{Sthree_onethree_X}, &\qquad \strdia{Sthree_onetwothree_X}, \qquad \strdia{Sthree_threetwoone_X}.
\end{split}
\end{align}
The orthogonal projection operator, $\Pi^{(2)}_{X}$, onto the symmetric subspace $\symsub^{2}\s H_{X} \subseteq \s H^{\otimes 2}_X$, referred to as the \textit{symmetrization operator}, is given the following unique notation:
\begin{equation}
\label{eq:sym_2_X}
\strdia{symmetrization_two_X}
\coloneqq
\strdia{symmetrization_two_X_explicit}
\frac{1}{2}\strdia{Stwo_e_X}
\frac{1}{2}\strdia{Stwo_onetwo_X}.
\end{equation}
Similarly, the orthogonal projection operator onto the antisymmetric subspace $\antisymsub^{2}\s H_{X} \subseteq \s H^{\otimes 2}$, denoted by $\Pi_{X}^{(1,1)}$ and referred to as the \textit{antisymmetrization operator}, is depicted in a complementary manner:
\begin{equation}
\strdia{antisymmetrization_two_X}
\coloneqq
\strdia{antisymmetrization_two_X_explicit}
\frac{1}{2}\strdia{Stwo_e_X}
\frac{1}{2}\strdia{Stwo_onetwo_X}.
\end{equation}
Generalizing this notation for the orthogonal projection operators onto the symmetric and antisymmetric subspaces of $\s H_X^{\otimes m}$ for $m > 2$ can be done recursively as follows. For the sake of clarity, the Hilbert space label, $X$, can often be omitted without introducing ambiguity.
\begin{align}
\strdia{symmetrization_n}
\frac{1}{m} \left(
\strdia{symmetrization_n_recursive_term_one}
+ (m-1)
\strdia{symmetrization_n_recursive_term_two}
\right),\\
\strdia{antisymmetrization_n}
\frac{1}{m} \left(
\strdia{antisymmetrization_n_recursive_term_one}
- (m-1)
\strdia{antisymmetrization_n_recursive_term_two}
\right).
\end{align}
Finally, in order to generalize the above symmetrization and antisymmetrization notation to the case of multipartite Hilbert spaces, e.g., $\s H_{XY} = \s H_X \otimes \s H_Y$, we introduce the following notational definition for the joint symmetrization $\Pi^{(2)}_{XY}$:
\begin{equation}
\label{eq:parallel_sym_XY}
\strdia{parallel_symmetrization_two_X_Y}
\frac{1}{2}
\strdia{parallel_e_X_Y}
\frac{1}{2}
\strdia{parallel_onetwo_X_Y}.
\end{equation}

\subsection{The degree one case}
\label{sec:n1}

This subsection explores the strength of the constraint imposed by \cref{eq:nth_order} for the special case when $n=1$ (equivalently \cref{eq:n1}) for the purposes of detecting unrealizable $\s M$-product states. For marginal scenarios involving disjoint marginal contexts, it will be shown that \cref{eq:n1} happens to be satisfied by \textit{all} $\s M$-product states, and therefore is useless for the QMP. For at least some marginal scenarios involving non-disjoint marginal contexts, it will shown that \cref{eq:n1} is already capable of witnessing the unrealizability of certain $\s M$-product states. Finally, it is shown that for some (admittedly degenerate) marginal scenarios, the constraint imposed by \cref{eq:n1} is also sufficient for the corresponding QMP.

Throughout this subsection, unipartite subsystems are labeled alphabetically, e.g., $A$, $B$, $C\ldots$, and their respective dimensions denoted by lower-case letters, e.g., $a = d_A$, $b = d_B$, etc.

Consider the marginal scenario $\s M = (A, B)$ ($m=2$) for the joint context $J = AB$. In this scenario, the QMP is already fully solved: $\rho_{A}$ and $\rho_{B}$ are the marginals of some pure state $\psi_{AB}$ if and only if $\spec(\rho_{A}) = \spec(\rho_{B})$ (see \cref{sec:bipartite}). To what extent, if any, does \cref{eq:n1} reproduce this known solution? If $\rho_{A}$ and $\rho_{B}$ are the marginals of some pure state $\psi_{AB}$, \cref{eq:n1} implies
\begin{equation}
\label{eq:n1_A_B}
\rho_{A} \otimes \rho_{B} \leq (\Tr_{B}\otimes \Tr_{A})(\Pi^{(2)}_{AB}).
\end{equation}
To calculate the right-hand side of the above inequality, it will be convenient to use the diagrammatic notation introduced in \cref{sec:diagrammatics}. Specifically, $\Pi^{(2)}_{AB}$ can be depicted using \cref{eq:parallel_sym_XY} (with $X,Y$ substituted by $A,B$), and thus $(\Tr_{B}\otimes \Tr_{A})(\Pi^{(2)}_{AB})$ is equal to
\begin{align}
\strdia{parallel_sym_AB_partial}
\frac{1}{2}
\strdia{parallel_e_AB_partial}
\frac{1}{2}
\strdia{parallel_onetwo_AB_partial},
\\
\frac{ab}{2}
\strdia{parallel_AB_ident}
\frac{1}{2}
\strdia{parallel_AB_ident},\\
&= \frac{1+ab}{2} \strdia{parallel_AB_ident}.
\end{align}
Therefore, \cref{eq:n1_A_B} is equivalent to $\rho_{A} \otimes \rho_{B} \leq \frac{1}{2}(1+ab) \ident_{A} \otimes \ident_{B}$, i.e.
\begin{equation}
\strdia{tensor_rA_rB} \leq \frac{1+ab}{2} \strdia{parallel_AB_ident_label_adjust},
\end{equation}
which is an inequality satisfied by all $(A, B)$-product states $\rho_{A} \otimes \rho_{B}$ because $\rho_X \leq \ident_X$ already holds for all $\rho_X \in \dens(\s H_X)$ and $ab \geq 1$. In fact, it is not too difficult to show that when $\s M = (X_1, \ldots, X_k)$ contains disjoint contexts, i.e. $X_i \cap X_j = \emptyset$ for $i \neq j$, the inequality in \cref{eq:n1} is always trivial because $\Tr_{mJ\setminus\s M}(T_{J}(\pi)) \geq \ident_{\s M}$ holds for all $\pi \in \sym_k$ and thus $\Tr_{mJ\setminus\s M}(\Pi^{(m)}_{J}) \geq \ident_{\s M}$ also. Fortunately, the same is not necessarily true for non-disjoint marginal scenarios.

For an example of a non-trivial instance of \cref{eq:n1}, consider the marginal scenario $\s M = (AB, AC, BC)$ for the joint context $J = ABC$. In this scenario, the aforementioned operator inequality becomes
\begin{equation}
\label{eq:AB_AC_BC_ex}
\rho_{AB} \otimes \rho_{AC} \otimes \rho_{BC} \leq (\Tr_{C} \otimes \Tr_{B} \otimes \Tr_{A})(\Pi_{ABC}^{(3)}).
\end{equation}
The projector $\Pi_{ABC}^{(3)}$ onto $\symsub^{3}(\s H_{A} \otimes \s H_{B} \otimes \s H_{C})$ can be expressed as
\begin{align}
\frac{1}{3!}
\bigg[
\strdia{ABC_e}
+\strdia{ABC_onetwo}+ \\
\bigg]. \nonumber
\end{align}
Therefore, $(\Tr_{C} \otimes \Tr_{B} \otimes \Tr_{A})(\Pi^{(3)}_{ABC})$ becomes
\begin{align}
&\strdia{parallel_sym_ABC_partial} =
\frac{1}{3!}
\bigg[
abc\strdia{ABC_eee} + \\
\bigg]. \nonumber
\end{align}
To show that \cref{eq:AB_AC_BC_ex} is a non-trivial constraint, we consider consider the case of three qubits, i.e. $a=b=c=2$. For a given pair of qubits, the unique antisymmetric pure state (also called the singlet state), $\Phi = \ket{\Phi}\bra{\Phi} \in \proj(\mathbb C^2 \otimes \mathbb C^2)$, can be identified with $\ket{\Phi} = \frac{1}{\sqrt{2}} (\ket{0 1} - \ket{1 0})$ and depicted as follows
\begin{equation}
\strdia{levi_civita_two_named} = \strdia{levi_civita_two}, \quad \text{s.t.} \quad \strdia{levi_civita_two_swap} = - \strdia{levi_civita_two}, \quad \strdia{levi_civita_two_norm} = 1.
\end{equation}
When applied to $(\Tr_{C} \otimes \Tr_{B} \otimes \Tr_{A})(\Pi^{(3)}_{ABC})$ (assuming $a = b = c = 2$) we obtain the identity
\begin{equation}
\strdia{parallel_sym_ABC_partial_levi} = \frac{1}{3!}(abc-a-b-c-2) = 0,
\end{equation}
which, when combined with \cref{eq:AB_AC_BC_ex} proves that the $(AB, AC, BC)$ marginals of a three-qubit pure state $\psi_{ABC}$ always satisfy
\begin{equation}
\label{eq:AB_AC_BC_equality}
\strdia{AB_AC_BC_equal_three_qubits} = 0.
\end{equation}
An example of an unrealizable triple of states $(\rho_{AB}, \rho_{AC}, \rho_{BC})$ whose unrealizability is witnessed by the above equality constraint is the triple of anti-correlated states, $\rho_{AB} = \rho_{AC} = \rho_{BC} = \frac{1}{2}(\ket{01}\bra{01}+\ket{10}\bra{10})$, where the left-hand side evaluates to $2^{-5}$. Other examples includes the triple of singlets $\rho_{AB} = \rho_{AC} = \rho_{BC} = \Phi$ (with value $2^{-4}$), or the triple of maximally mixed states $\rho_{AB} = \rho_{AC} = \rho_{BC} = \frac{I}{2} \otimes \frac{I}{2}$ (with value $2^{-6}$). An example of an inconsistent triple of states for which \cref{eq:AB_AC_BC_equality} happens to be satisfied is $\rho_{AB} = \rho_{BC} = \ket{00}\bra{00}$ and $\rho_{AC} = \ket{11}\bra{11}$.

To conclude, consider the rather non-standard marginal scenario $\s M = (X, X)$ for the joint context $J = X$. Taken literally, the QMP for this marginal scenario is to determine, for any given pair of states $\rho_X, \sigma_X \in \dens(\s H_X)$, whether or not there exists a pure state $\psi_X \in \proj(\s H_X)$ such that $\rho_X = \psi_X$ and $\sigma_X = \psi_X$. This marginal scenario can be regarded as ``non-standard'' for at least two reasons: (i) the marginal context $X$ is repeated twice in $\s M$, and (ii) since $X$ is not a proper subset of $X$, $\rho_X$ and $\sigma_X$ are not proper marginals of $\psi_X$. Taken together, the QMP for this scenario has a simple solution: $\rho_X$ and $\sigma_X$ are realizable if and only if they are both pure states and equal to each other. Nevertheless, in this scenario \cref{eq:n1} is a valid constraint; in particular, it simplifies to $\rho_{X} \otimes \sigma_X \leq \Pi^{(2)}_{X}$, or diagrammatically
\begin{equation}
\label{eq:XX_n1}
\strdia{tensor_rX_sX} \leq \strdia{symmetrization_two_X}.
\end{equation}
The above inequality implies that $\Tr(\sigma_X \rho_X) = 1$ since
\begin{equation}
0 \leq \strdia{tensor_rX_sX_antisym} \leq \strdia{sym_antisym_trace} = 0,
\end{equation}
\begin{align}
\strdia{tensor_rX_sX_antisym}
&= \frac{1}{2} \strdia{tensor_rX_sX_antisym_e} - \frac{1}{2}\strdia{tensor_rX_sX_antisym_onetwo}, \\
&= \frac{1}{2} \left(1 - \strdia{tensor_rX_sX_trace}\right).
\end{align}
Since $\Tr(\sigma_X \rho_X) = 1$ holds if and only if $\sigma_X = \rho_X = \psi_X$ for some pure state $\psi_X$, we see that \cref{eq:n1}, which becomes \cref{eq:XX_n1}, is both necessary and \textit{sufficient} for the $\s M = (X, X)$ instance of the QMP.

\subsection{The Bipartite QMP}
\label{sec:bipartite}

This subsection considers the bipartite marginal scenario, $\s M = (A,B)$, for the joint context $J = AB$. For this scenario, the QMP is fully solved and admits of a simple solution: $\rho_{A}$ and $\rho_{B}$ are the marginals of a joint pure state $\psi_{AB} \in \proj(\s H_{AB})$ if and only if they have the same spectrum~\cite{tyc2015quantum,klyachko2004quantum}. A natural question arises: how does \cref{thm:main} recover this well-known result?

To answer this question, first let $a = \dim(\s H_A)$, $b = \dim(\s H_B)$, and let $\ell = \min(a,b)$. For this scenario, \cref{eq:nth_order} becomes
\begin{equation}
\label{eq:bipartite_nth_order}
(\rho_{A} \otimes \rho_{B})^{\otimes n} \leq (\Tr_{B}\otimes\Tr_{A})^{\otimes n}(\Pi_{AB}^{(2n)}).
\end{equation}
Now let $s_{A} \in \spectra^{a}$ and $s_{B} \in \spectra^{b}$ be the spectra of $\rho_{A}$ and $\rho_{B}$. Using the results of \cref{sec:keyl_exclusive} (or essentially the spectral estimation theorem~\cite{keyl2001estimating,christandl2006spectra}), together with \cref{eq:haar_sym}, it is possible to show that the exponential factor in \cref{eq:unrealizability_measure}, $\Omega(\rho_{A}\otimes\rho_{B})$, depends only on $r_A$ and $r_B$ and is equal to:
\begin{align}
\Omega(\rho_{A} \otimes \rho_{B}) = \inf_{r \in \spectra^{\ell}}(\kl{s_{A}}{r}+\kl{s_{B}}{r}),
\end{align}
where $\kl{p}{q}$ is the relative entropy $\kl{p}{q} = \sum_{i} p_i (\ln p_i - \ln q_i)$. Since $\kl{p}{q}$ only vanishes if $p = q$, $\Omega(\rho_{A} \otimes \rho_{B})$ only vanishes if $s_{A} = s_{B}$. Therefore, we conclude that $\rho_{A}$ and $\rho_{B}$ are the $(A,B)$-marginals of a pure state $\psi_{AB} \in \proj(\mathbb C^{a} \otimes \mathbb C^{b})$ if and only if they have equal spectra.
Additionally, using Pinsker's inequality~\cite{reid2009generalised}, $\norm{p-q}_1^{2} \leq 2\kl{p}{q}$, and the triangle inequality for $\norm{\cdot}_1$, we obtain:
\begin{align}
\norm{s_A - s_B}_1^2
&\leq 3(\norm{s_A - r}_1^2 + \norm{s_B - r}_1^2)\\
&\leq 6 (\kl{s_A}{r}+\kl{s_B}{r}).
\end{align}
Therefore, $\Omega(\rho_{A} \otimes \rho_{B}) \geq \norm{s_A - s_B}_1^2/6$.

A more direct consequence of \cref{eq:bipartite_nth_order} is the following proposition.
\begin{prop}
Let $n \in \mathbb N$ and let $\alpha \in \yf_{n}^{a}$ and $\beta \in \yf_{n}^{b}$ be partitions. If $\rho_{A}\otimes \rho_{B}$ satisfies \cref{eq:bipartite_nth_order}, then
\begin{equation}
\label{eq:bipartite_result}
s_{\alpha}(r_{A})s_{\beta}(r_{B}) \leq \sum_{\lambda \in \yf_{2n}^{\ell}} c^{\lambda}_{\alpha\beta} \frac{\dim(\schur{\lambda}^{a})\dim(\schur{\lambda}^{b})}{\dim(\specht{\lambda})},
\end{equation}
where $r_{A}$ and $r_{B}$ are the spectra of $\rho_{A}$ and $\rho_{B}$ respectively. Additionally, $s_{\alpha}$ and $s_{\beta}$ are Schur functions~\cite{sra2016inequalities,sagan2013symmetric} and $c^{\lambda}_{\alpha\beta}$ is the Littlewood-Richardson coefficient~\cite{littlewood1934group,fulton2000eigenvalues,pak2019largest}.
\end{prop}
\begin{proof}
One of the most powerful tools for decomposing bipartite Hilbert spaces, specifically the symmetric subspace of a bipartite system $\symsub^{k}(\s H_{A} \otimes \s H_{B}) \cong V^{ab}_{(k)}$, is known as $\mathrm{GL}(a)\times\mathrm{GL}(b)$-duality~\cite{howe1987gl} (see also~\cite[Eq. (2.25)]{walter2014multipartite}):
\begin{equation}
V^{ab}_{(k)} \cong \bigoplus_{\lambda \in \yf_{k}^{\ell}} \schur{\lambda}^{a} \otimes \schur{\lambda}^{b},
\end{equation}
where $\ell = \min(a,b)$.
Using this result, and applying $\Pi_{A}^{\alpha} \otimes \Pi_{B}^{\beta}$ to the right-hand-side of \cref{eq:nth_order}, we obtain
\begin{align}
\begin{split}
&\Tr_{AB}^{\otimes n}\{(\Pi_{A}^{\alpha} \otimes \Pi_{B}^{\beta})(\Tr_{B}^{\otimes n}\otimes\Tr_{A}^{\otimes n})(\Pi_{AB}^{(2n)})\}\\
%&\quad=\Tr_{AB}^{\otimes 2n}\{((\Pi_{A}^{\alpha} \otimes \ident_{B}^{\otimes n}) \otimes (\ident_{A}^{\otimes n} \otimes \Pi_{B}^{\beta}))\Pi_{AB}^{(2n)}\},\\
%&\quad=\Tr_{AB}^{\otimes 2n}\{((\Pi_{A}^{\alpha} \otimes \ident_{B}^{\otimes n}) \otimes (\Pi_{A}^{\beta} \otimes \ident_{B}^{\otimes n})\Pi_{AB}^{(2n)}\},\\
%&\quad=\Tr_{A}^{\otimes 2n}\{(\Pi_{A}^{\alpha} \otimes \Pi_{A}^{\beta})\Tr_{B}^{\otimes 2n}(\Pi_{AB}^{(2n)})\}, \\
%&\quad=\sum_{\lambda \in \yf^{\ell}_{2n}}\Tr_{A}^{\otimes 2n}\{(\Pi_{A}^{\alpha} \otimes \Pi_{A}^{\beta})\Pi_{A}^{\lambda}\}\frac{\dim(\schur{\lambda}^{b})}{\dim(\specht{\lambda})},\\
&\quad=\dim(\specht{\alpha}) \dim(\specht{\beta}) \hspace{-0.5em}\sum_{\lambda \in \yf^{\ell}_{2n}}\hspace{-0.5em}c^{\lambda}_{\alpha\beta}\frac{\dim(\schur{\lambda}^{a})\dim(\schur{\lambda}^{b})}{\dim(\specht{\lambda})},
\end{split}
\end{align}
where $c^{\lambda}_{\alpha\beta}$ counts the multiplicity of the $\sym_n\times \sym_n$ irreducible representation space $\specht{\alpha} \otimes \specht{\beta}$ inside $\specht{\lambda}$ under the restriction of $\sym_{2n}$ to $\sym_{n} \times \sym_{n}$. By comparison, applying $\Pi_{A}^{\alpha} \otimes \Pi_{B}^{\beta}$ to the left-hand-side of \cref{eq:nth_order} yields
\begin{align}
&\Tr((\Pi_{A}^{\alpha} \otimes \Pi_{B}^{\beta})(\rho_{A}^{\otimes n} \otimes \rho_{B}^{\otimes n}))\\
&\quad=\Tr(\Pi_{A}^{\alpha}\rho_{A}^{\otimes n})\Tr(\Pi_{B}^{\beta}\rho_{B}^{\otimes n}),\\
&\quad= s_{\alpha}(r_{A})\dim(\specht{\alpha})s_{\beta}(r_{B})\dim(\specht{\beta}).
\end{align}
Therefore, \cref{eq:bipartite_nth_order} implies \cref{eq:bipartite_result} and thus the claim holds.
\end{proof}

\subsection{Fermionic \& Bosonic QMP}
\label{sec:ferm_bos}

Our sufficient family of necessary inequality constraints can be modified to handle the fermionic and bosonic variants of the QMP. Recall that a state describing a system of $p$ fermions (resp. bosons) with $f$ internal degrees of freedom, is typically modeled by an element of the antisymmetric subspace $\antisymsub^{p} \mathbb C^{f}$ (resp. the symmetric subspace $\symsub^{p}\mathbb C^{f}$). Since $\antisymsub^{p}\mathbb C^{f}$ (resp. $\symsub^{p}\mathbb C^{f}$) can be viewed as a subspace of a $p$-partite composite Hilbert space $(\mathbb C^f)^{\otimes p}$, and $\symsub^{n}\antisymsub^{p}\mathbb C^{f}$ (resp. $\symsub^{n}\symsub^{p}\mathbb C^{f}$) serves as the respresentation space for an irreducible representation of $SU(\antisymsub^{p}\mathbb C^{f}) \cong SU(\dim(\antisymsub^{p}\mathbb C^{f})) \cong SU(\tbinom{f}{p})$ (resp. $SU(\tbinom{p+f-1}{p})$), the analogue of \cref{eq:haar_sym} holds and thus an analogue of \cref{lem:de_finetti_realizable} also holds. Altogether, a generalization of \cref{thm:main} holds:
\begin{cor}
Let $\s H_{V} \subseteq \s H_{J}$ be a subspace of a joint Hilbert space, $\s H_{J}$. An $\s M$-product state, $\rho_{\s M} = \rho_{S_1} \otimes \cdots \otimes \rho_{S_m}$, is realizable by a joint pure state $\psi_{V} \in \s H_V \subseteq \s H_{J}$ in the subspace $\s H_{V}$ if and only if for all $n \in \mathbb N$,
\begin{equation}
\rho_{\s M}^{\otimes n} \leq \Tr_{m J\setminus \s M}^{\otimes n}(\Pi^{(nm)}_{V}),
\end{equation}
where $\Pi^{(nm)}_{V}$ is the projection operator onto the $nm$-symmetric subspace $\symsub^{nm}\s H_V \subseteq \s H_J^{\otimes nm}$.
\end{cor}

\subsection{Counting Solutions to the QMP}
\label{sec:ortho_sol}

Whenever a given $\s M$-product state, $\rho_{\s M} = \rho_{S_1}\otimes \cdots \otimes \rho_{S_m}$, is shown to be realizable, a natural follow-up problem is to determine whether or not the joint state, $\psi_{J}$, satisfying \cref{eq:cqmp} is unique. For the bipartite marginal scenario, $\s M = (\s A, \s B)$, if the common spectrum of $\rho_{A}$ and $\rho_{B}$ is $s = (s_1, \ldots, s_r, 0, \ldots, 0)$, with positive values distinct, i.e. $s_1 > \cdots > s_r > 0$, then the \textit{unique} pure state, $\psi_{AB} \in \proj(\s H_J)$, satisfying \cref{eq:cqmp} is $\psi_{AB} = \ket{\psi_{AB}}\bra{\psi_{AB}}$ where
\begin{equation}
\ket{\psi_{AB}} = \sum_{i=1}^{r} \sqrt{s_i} \ket{\phi_{A}^{(i)}} \otimes \ket{\phi_{B}^{(j)}},
\end{equation}
where $\{\phi_{A}^{(i)}\}_{i=1}^{r}$ and $\{\phi_{B}^{(i)}\}_{i=1}^{r}$ are the eigenvectors of $\rho_{A}$ and $\rho_{B}$. If, however, the common spectrum, $(s_1, \ldots, s_r, 0, \ldots, 0)$, is degenerate in the sense that some of its values are identical, then the solution to \cref{eq:cqmp} may not be unique. A familiar example of this phenomenon, for the two-qubit Hilbert space $\s H_J \cong \mathbb C^{2} \otimes \mathbb C^{2}$, are the four Bell states all sharing the same pair of maximally-mixed, single-qubit marginals, $(\frac{\ident}{2}, \frac{\ident}{2})$.

The following result generalizes \cref{eq:nth_order} by considering the possibility that an $\s M$-product may be realizable by multiple, orthogonal, joint states.
\begin{cor}
Let $\rho_{\s M} = \rho_{S_1} \otimes \cdots \otimes \rho_{S_m}$ be an $\s M$-product state and $\{ \psi_{J}^{(1)}, \ldots, \psi_{J}^{(v)}\}$ be a set of joint pure states, satisfying i) for all $1 \leq j,k \leq v$,
\begin{equation}
\Tr(\psi_{J}^{(j)}\psi_{J}^{(k)}) = |\braket{\psi_{J}^{(j)}|{\psi_{J}^{(k)}}}|^{2} = \delta_{j,k},
\end{equation}
and ii) for all $1 \leq i \leq m$, and $1 \leq k \leq v$,
\begin{equation}
\rho_{S_i} = \Tr_{J\setminus S_i}(\psi_{J}^{(k)}).
\end{equation}
Then, the following inequality holds:
\begin{equation}
\label{eq:ortho_ineq}
v^{nm}\rho_{\s M}^{\otimes n} \leq \sum_{\lambda \in \yf_{nm}^{v}}\Tr_{mJ\setminus\s M}^{\otimes n}(\Pi_{J}^{\lambda}).
\end{equation}
\end{cor}
\begin{proof}
Let $P_V$ be the orthogonal projection operator onto the subspace of $\s H_J$ spanned by $\{\psi_{J}^{(k)}\}_{k=1}^{v}$, i.e.,
\begin{equation}
P_V = \sum_{k=1}^{v} \psi_{J}^{(k)}.
\end{equation}
Since each $\psi_{J}^{(k)}$ has marginals $(\rho_{S_1}, \ldots, \rho_{S_m})$, we conclude
\begin{equation}
\label{eq:ortho_equality}