# A Result About the Classification of Quantum Covariance Matrices Based on
Their Eigenspectra

Arik Avagyan<EMAIL_ADDRESS>Department of Physics, University of
Colorado, Boulder, Colorado, 80309, USA

###### Abstract

The set of valid covariance matrices of a continuous-variable quantum system
with a finite number of degrees of freedom is a strict subset of the set of
real positive-definite matrices due to Heisenberg’s uncertainty principle.
This has the implication that, in general, not every orthogonal transform of a
diagonal quantum covariance matrix produces a valid quantum covariance matrix.
A natural question thus arises, to find the set of quantum covariance matrices
consistent with a given eigenspectrum. For the special class of pure Gaussian
states the set of quantum covariance matrices with a given eigenspectrum
consists of a single orbit of the action of the orthogonal symplectic group.
The eigenspectra of the covariance matrices of this class of states are
composed of pairs that each multiply to one. We describe a larger non-trivial
class of eigenspectra with the property that the set of quantum covariance
matrices corresponding to any eigenspectrum in this class are related by
orthogonal symplectic transformations. Further, we show that all non-
degenerate eigenspectra with this property must belong to this class, and that
the set of such eigenspectra coincides with the class of non-degenerate
eigenspectra that identify the physically relevant thermal and squeezing
parameters of a Gaussian state.

##  1 Overview

A quantum system is a continuous-variable system when it is associated with an
infinite-dimensional separable Hilbert space and has observables with
continuous eigenspectra. An important area of physics that concerns itself
with the study of such systems is quantum optics. There, the quantized
electromagnetic field is usually described by a finite collection of one-
dimensional harmonic oscillators, or modes. The covariance matrix (CM) of such
a system captures the second moment of the canonical position and momentum
operators of the modes. For a classical system described by a finite set of
position and momentum variables, any real positive-definite matrix is a valid
CM of a probability distribution on phase-space. In quantum physics, however,
Heisenberg’s uncertainty principle sets a fundamental constraint on the size
of the second moment. For example, for a system occupying a single mode, where
the diagonal elements of the CM $\Gamma$ are
$2(\langle\hat{q}^{2}\rangle-\langle\hat{q}\rangle^{2})$ and
$2(\langle\hat{p}^{2}\rangle-\langle\hat{p}\rangle^{2})$, and each off-
diagonal element is
$\langle\hat{q}\hat{p}+\hat{p}\hat{q}\rangle-2\langle\hat{q}\rangle\langle\hat{p}\rangle$,
the uncertainty principle can be stated as $\det(\Gamma)\geq 1$ in appropriate
units.

The literature studying the various properties of quantum CMs is very rich.
For one, quantum CMs are intrinsically interesting as they can be used to
express such a fundamental tenet of quantum physics as the uncertainty
principle. For example, Refs. [1, 2] present several equivalent formulations
of the uncertainty-principle restrictions on the CM. Central to these
formulations is Williamson’s theorem, which states that every real positive-
definite matrix can be brought to a diagonal form by a symplectic
transformation [3]. The diagonal entries of the resulting diagonal matrix,
usually called the symplectic eigenvalues, are positive and appear in pairs.
In the case of quantum CMs, the uncertainty principle sets a lower bound on
the magnitudes of their symplectic eigenvalues. In the units adopted in this
paper this bound is equal to $1$. It should be mentioned that the symplectic
decomposition of real positive-definite matrices, in part motivated by its
relevance to quantum physics, is an active field of study in mathematics, see,
for example, Refs. [4, 5, 6]. Ref. [7] is another example of the investigation
of the relationship between the uncertainty principle and quantum CMs. There,
the quantum CMs are represented as ellipsoids in phase space, where the
uncertainty principle sets constraints on the sizes and shapes of these
ellipsoids. The existence of these constraints is closely related to Gromov’s
non-squeezing theorem, a fundamental result in symplectic geometry [8].

The interest in quantum CMs increased with the advances in quantum information
theory. In particular, quantum CMs are central to the field of Gaussian
quantum information processing. A Gaussian state is completely characterized
by its first moment and its CM, and quantum channels that preserve the
Gaussianity of the state (called Gaussian channels or processes) are fully
characterized by how they transform the first moment and the CM. For example,
a unitary Gaussian process transforms the CM by a symplectic matrix, and thus
preserves the symplectic eigenvalues. For Gaussian states, the other essential
features of quantum systems besides the uncertainty principle, such as
entanglement and separability, non-locality, as well as purity and entropy can
be quantified based on particular properties of their CMs. Similarly, the CM
is central to the evaluation of various Gaussian quantum information
processing tasks, such as quantum communication, quantum cryptography, quantum
computation, quantum teleportation and quantum state discrimination. [9, 10]

The study presented here was also inspired by a particular application in
quantum information processing. In a previous work [11] I and my coauthors
showed that the total-photon-number distribution of a Gaussian state
determines and is determined by the eigenspectrum of the CM and the absolute
values of the first moment in each eigenspace. Thus, it might be possible to
experimentally estimate these properties of an unknown Gaussian state using a
device that counts the total photon-number, such as a transition-edge-sensor
[12]. In this context, it can be important to obtain estimates of more
physically relevant properties of the Gaussian state from the measurements of
the total-photon-number, such as bounds on the symplectic eigenvalues (usually
called the thermal parameters in this field) of the CM, as well as bounds on
the squeezing parameters of the CM. The latter are associated with the
symplectic matrix in the symplectic decomposition of the CM, see Sec. 2.

Our main contribution in this paper is partially characterizing the class of
quantum CMs for which the thermal and squeezing parameters are exactly
determined by the eigenspectrum (call it property $1$). This class of CMs
includes the class of quantum CMs for which the set of CMs with the same
eigenspectrum are related by orthogonal symplectic (OS) transformations
(property $2$). It can be shown that one can construct a diagonal quantum CM
from the eigenspectrum of any quantum CM, which we call the “diagonal
representative”. The thermal and squeezing parameters of a diagonal quantum CM
are uniquely specified by its diagonal entries, or, more specifically, from
the particular assignment of pairs of entries to the modes of the system. This
implies that, for an eigenspectrum to possess property $1$, there must not
exist another valid diagonal arrangement of the eigenspectrum that pairs the
eigenvalues differently. We call this the “unique pairing condition”. Part of
our findings is that among the non-degenerate eigenspectra satisfying the
unique pairing condition, those that possess properties $1$ and $2$ are
precisely the eigenspectra with diagonal representatives where at most one of
the modes has a thermal parameter of $\geq 1$ in our units. For a system
living in $S$ modes, we say the eigenspectra with the latter property satisfy
the “$S-1$-pure unique pairing condition”. The eigenspectra of pure Gaussian
states, where the thermal parameters of all modes are equal to $1$, are a
strict subset of this class of eigenspectra. Our findings are not limited to
non-degenerate eigenspectra. We find that all eigenspectra that satisfy the
$S-1$-pure unique pairing condition possess properties $1$ and $2$. Further,
for general eigenspectra, we narrow down the set of eigenspectra that need
further study to finish the characterization of eigenspectra possessing either
or both of the properties $1$ and $2$.

Our results are presented as a theorem, a proposition and a corollary of the
proposition. Thm. 4.2 shows that, given an eigenspectrum that satisfies the
$S-1$-pure unique pairing condition, any two quantum CMs with this
eigenspectrum are related by an OS transformation. Such transformations
preserve the thermal and squeezing parameters. Prop. 4.1 shows that if an
eigenspectrum does not satisfy the $S-1$-pure unique pairing condition, and is
non-degenerate in a particular subset of eigenvalues, then there exists an
orthogonal non-symplectic (ONS) transformation that alters the symplectic
eigenvalues of the CM while keeping them $\geq 1$. Cor. 4.3 generalizes Prop.
4.1 to a larger class of degeneracies in the eigenspectrum.

The paper is organized as follows. Sec. 2 is devoted to problem formulation.
Sec. 3 introduces a set of facts about orthogonal matrices. These facts are
used in the proof of Thm. 4.2 to construct a witness for the violation of the
uncertainty principle when an ONS transformation is applied to the diagonal
representative of an eigenspectrum that satisfies the $S-1$-pure unique
pairing condition. Thm. 4.2, Prop. 4.1 and Cor. 4.3 are presented in Sec. 4.
We close with a brief discussion of the implications of our results and
proposals for future research in Sec. 5.

##  2 Problem Formulation

The background material for this section can be found in a textbook such as
Ref. [10]. Consider a continuous variable system composed of $S$ modes
characterized by canonical position and momentum operators $\hat{q}_{i}$,
$\hat{p}_{i}$ for $i=1,\ldots,S$. We choose our units such that these
operators satisfy the commutation relations $[\hat{q}_{i},\hat{q}_{j}]=0$,
$[\hat{p}_{i},\hat{p}_{j}]=0$, and $[\hat{q}_{i},\hat{p}_{j}]=i\delta_{ij}$.
These commutation relations can be expressed more compactly using the anti-
symmetric $2$-form $\Omega=\bigoplus_{i=1}^{S}\begin{pmatrix}0&1\\\
-1&0\end{pmatrix}$. In particular, $[\hat{r}_{i},\hat{r}_{j}]=i\Omega_{ij}$,
where $\hat{r}_{2k-1}=\hat{q}_{k}$ and $\hat{r}_{2k}=\hat{p}_{k}$ for
$k=1,\ldots,S$. The covariance matrix $\Gamma$ of the physical system is
defined as
$\Gamma_{ij}=\langle\hat{r}_{i}\hat{r}_{j}+\hat{r}_{j}\hat{r}_{i}\rangle-2\langle\hat{r}_{i}\rangle\langle\hat{r}_{j}\rangle$.
The notation $\langle\ldots\rangle$ denotes expectation with respect to the
state of the system. We also introduce the phase-space vector
$\vec{r}=(q_{1},p_{1},\ldots,q_{S},p_{S})$. We refer to each pair
$(q_{i},p_{i})$ as conjugate coordinates. In these coordinates the restriction
imposed on $\Gamma$ by Heisenberg’s uncertainty principle can be expressed as

$\displaystyle\Gamma+i\Omega\geq 0.$ (2.1)

Using Williamson’s theorem every positive-definite matrix, and, in particular,
every CM $\Gamma$ can be decomposed as $\Gamma=ATA^{T}$, where $A$ is
symplectic, and $T$ is diagonal and consisting of $S$ blocks of the form
$\begin{pmatrix}\nu_{i}&0\\\ 0&\nu_{i}\end{pmatrix}$. With our conventions Eq.
2.1 implies that each $\nu_{i}$ must satisfy $\nu_{i}\geq 1$ for $\Gamma$ to
be a quantum CM. $A$ satisfies $A^{T}\Omega A=\Omega$, and can be further
decomposed as $A=KQL$, where $Q$ is diagonal and consisting of $S$ blocks of
the form $\begin{pmatrix}e^{r_{i}}&0\\\ 0&e^{-r_{i}}\end{pmatrix}$, and $K$
and $L$ are OS matrices. The $\nu_{i}$ are the symplectic eigenvalues or the
thermal parameters of $\Gamma$, while the $r_{i}$ are referred to as the
squeezing parameters of $\Gamma$. We usually reserve referring to the
$\nu_{i}$ as thermal parameters when they are $\geq 1$, so that they can be
associated with a quantum CM. We refer to the group of OS matrices by
$\mathrm{SpO}(2S,\mathbb{R})$.

For the important class of pure Gaussian states $T$ equals the identity, and
hence the CM of such a state has the form $\Gamma=KQ^{2}K^{T}$. It can be seen
that the eigenspectrum of the CM of a pure Gaussian state determines and is
determined by the squeezing parameters, and that the orthogonal matrix that
diagonalizes the CM is OS. Further, due to the form of $\Gamma$, the CMs of
pure Gaussian states with the same eigenspectra are related by OS
transformations, or, in other words, are in a single orbit of the action of
$\mathrm{SpO}(2S,\mathbb{R})$ on the space of $2S\times 2S$ quantum CMs. This
has the implication that any non-trivial ONS transformation of $\Gamma$
results in a CM that violates the uncertainty principle. By a “non-trivial”
ONS transformation of $\Gamma$ we mean an orthogonal transformation $O$,
acting as $O^{T}\Gamma O$, that cannot be written as $O=O^{\prime}W$, where
$W$ is OS and $O^{\prime}$ commutes with $\Gamma$. This should not be confused
with the statement about any matrix that acts non-trivially in some subspace,
by which we mean that the matrix does not necessarily act as the identity, or
trivially, on this subspace. One can prove the aforementioned fact using
Weyl’s monotonicity principle [13, Ch. 3], but we opt to show a more
complicated proof as part of the proof of Thm. 4.2. At the risk of sounding
redundant, we emphasize that, given an eigenspectrum that is composed of the
diagonal entries of a diagonal matrix of the form of $Q^{2}$, the set of all
quantum CMs with this eigenspectrum are the set of OS transforms of $Q^{2}$.

Since a general quantum CM $\Gamma$ has the decomposition
$\Gamma=KQLTL^{T}QK^{T}$, any transformation of $\Gamma$ by an OS matrix does
not affect the thermal and squeezing parameters. However, a transformation by
a non-trivial ONS matrix usually produces a CM with different thermal and
squeezing parameters. If any of the thermal parameters of the new CM are less
than $1$, the CM is not a valid quantum CM. So, in general, there are many
quantum CMs with different thermal and squeezing parameters that have the same
eigenspectrum. These facts make the broad task of classifying quantum CMs
based on their eigenspectra a challenging problem. In this paper we make
significant progress in solving two related subproblems of this general
problem. To formulate the problems, let $\Lambda=\\{\lambda_{i}\\}_{i=1}^{2S}$
denote the eigenspectrum of a quantum CM of a system living in $S$ modes.

######  Problem 2.1.

Find the full set of conditions on $\Lambda$ for which the set of quantum CMs
with this eigenspectrum consist of a single orbit under the action of
$\mathrm{SpO}(2S,\mathbb{R})$.

######  Problem 2.2.

Find the full set of conditions on $\Lambda$ for which the set of quantum CMs
with this eigenspectrum have the same thermal and squeezing parameters.

Let us denote the (so far unknown) class of eigenspectra that satisfy the
conditions of Prob. 2.1 (Prob. 2.2) by
$\mathcal{P}_{1}(S)$($\mathcal{P}_{2}(S)$).

We need the following observations. First, given an eigenspectrum
$\Lambda=(\lambda_{i})_{i=1}^{2S}$ of a quantum CM $\Gamma$, where the
$\lambda_{i}$ are in a non-ascending order, there exists a diagonal quantum CM
with this eigenspectrum. For a proof see, for example, Lem. 4.3 in Ref. [11].
This diagonal representative $D_{\Lambda}$ is composed of $S$ consecutive
$2\times 2$ blocks, with the $j$’th block having the diagonal
$(\lambda_{j},\lambda_{2S+1-j})$. Thus, if $\Lambda$ satisfies the conditions
specified in Prob. 2.1, then there exists an OS matrix $K$ such that
$\Gamma=KD_{\Lambda}K^{T}$. Additionally, notice that the symplectic
decomposition of $D_{\Lambda}$ is of the form $D_{\Lambda}=Q^{2}T$, and,
therefore, the squeezing and thermal parameters of $D_{\Lambda}$ are
identified by $\Lambda$. This shows that
$\mathcal{P}_{1}(S)\subseteq\mathcal{P}_{2}(S)$ as any two CMs that are
related by an OS transformation have the same thermal and squeezing
parameters, and the latter are identified by an eigenspectrum that belongs to
$\mathcal{P}_{1}(S)$.

The second observation is a particular consequence of Eq. 2.1 that we use in
the construction of the uncertainty-principle-violating witness in the proof
of Thm. 4.2. $\Gamma+i\Omega\geq 0$ implies that $\Gamma_{1}+i\Omega_{1}\geq
0$, where $\Gamma_{1}$ is the upper-left $2\times 2$ block of $\Gamma$ and
$\Omega_{1}=\begin{pmatrix}0&1\\\ -1&0\end{pmatrix}$. The latter condition in
turn implies that $\det(\Gamma_{1}+i\Omega_{1})\geq 0$, which can be seen to
be equivalent to the condition $\det(\Gamma_{1})\geq 1$ since $\Gamma_{1}$ is
symmetric.

##  3 On Some Properties of Orthogonal Matrices

Here we state or derive several properties of orthogonal matrices that we use
in the proof of Thm. 4.2. We denote the orthogonal group in $n$ dimensions by
$\mathrm{O}(n)$.

######  Fact 3.1.

Consider an arbitrary $O\in\mathrm{O}(n)$ and an arbitrary set of $p$ indices
$i_{1}<\ldots<i_{p}$, where $i_{p}\leq n$. Denote the submatrix of $O$ that is
produced from the columns $i_{1},\ldots,i_{p}$ and rows $j_{1},\ldots,j_{p}$
of $O$ by $O_{j_{1},\ldots,j_{p}}^{i_{1},\ldots,i_{p}}$. Then,

$\sum_{j_{1}<\ldots<j_{p}}\left[\det(O_{j_{1},\ldots,j_{p}}^{i_{1},\ldots,i_{p}})\right]^{2}=1$
(3.1)

Fact 3.1 is a straightforward application of the Cauchy-Binet formula [14,
Chap. 3] to the product of any matrix composed of mutually orthonormal columns
with its transpose. This fact implies that the squares of the determinants of
all square submatrices in any column set of $O$ can be thought of as a finite
set of weights that sum to $1$. By transposition the same is true for any row
set.

For the last several properties presented in this section we need the fact
that $\textrm{SpO}(2S,\mathbb{R})$ is isomorphic to a representation of the
unitary group in $S$ dimensions, $\mathrm{U}(S)$ [2]. In particular, the
matrix $W$ in $\textrm{SpO}(2S,\mathbb{R})$ corresponding to the unitary
$U\in\mathrm{U}(S)$ can be represented by

$\displaystyle W$
$\displaystyle=\frac{1}{2}\begin{pmatrix}U+U^{*}&\vline&i(U-U^{*})\\\
\hline\cr i(U^{*}-U)&\vline&U+U^{*}\end{pmatrix}$
$\displaystyle=\begin{pmatrix}\textrm{Re}(U)&\vline&-\textrm{Im}(U)\\\
\hline\cr\textrm{Im}(U)&\vline&\textrm{Re}(U)\end{pmatrix}$ (3.2)

Here the phase-space coordinates are ordered as
$(q_{1},\ldots,q_{S},p_{1},\ldots,p_{S})$. Eq. 3 is obtained by conjugating
$\begin{pmatrix}U&\vline&0\\\ \hline\cr 0&\vline&U^{*}\end{pmatrix}$ with the
unitary $\frac{1}{\sqrt{2}}\begin{pmatrix}I&\vline&iI\\\ \hline\cr
I&\vline&-iI\end{pmatrix}$, where $I$ is the $S\times S$ identity.

In the rest of this section we order the phase-space coordinates as
$(q_{1},p_{1},\ldots,q_{S},p_{S})$.

######  Lemma 3.1.

Let us denote the map returning the column $i$ of an arbitrary matrix by
$c_{i}[\cdot]$. Then, given an arbitrary $O\in\textrm{O}(2S)$, and for any
$1\leq i\leq S$ and $1\leq j\leq 2S$, there exists a matrix
$W\in\textrm{SpO}(2S,\mathbb{R})$ such that $c_{2i-1}[OW]$ and $c_{2i}[OW]$
are proportional by the same non-zero constant except for the row $j$.

###### Proof.

$c_{2i-1}[OW]$ and $c_{2i}[OW]$ are equal to $Oc_{2i-1}[W]$ and $Oc_{2i}[W]$,
respectively. According to the discussion in the preceding paragraph, in our
coordinate basis the columns $c_{2i-1}[W]$ and $c_{2i}[W]$ can be expressed as

$\displaystyle c_{2i-1}[W]$
$\displaystyle=(\mathrm{Re}(u_{1}),\mathrm{Im}(u_{1}),\ldots,\mathrm{Re}(u_{S}),\mathrm{Im}(u_{S}))$
$\displaystyle c_{2i}[W]$
$\displaystyle=(-\mathrm{Im}(u_{1}),\mathrm{Re}(u_{1}),\ldots,-\mathrm{Im}(u_{S}),\mathrm{Re}(u_{S}))$
(3.3)

where the $u_{i}$ are complex numbers such that
$\sum_{i=1}^{S}\absolutevalue{u_{i}}^{2}=1$.

Let us denote by $\tilde{O}$ the matrix produced by removing row $j$ from $O$.
Then, the claim is that one can find an OS matrix $W$ such that
$\tilde{O}c_{2i-1}[W]=\lambda\tilde{O}c_{2i}[W]$ for some real $\lambda$. This
implies that $c_{2i-1}[W]-\lambda c_{2i}[W]$ lies in the kernel of
$\tilde{O}$. The kernel of $\tilde{O}$ is one-dimensional and, by the
arbitrariness of $O$, it can be the span of any vector in $\mathbb{R}^{2S}$.
Thus, we need to show that given any one-dimensional subspace of
$\mathbb{R}^{2S}$ there exists an OS matrix $W$ and a constant $\lambda\neq 0$
such that $c_{2i-1}[W]-\lambda c_{2i}[W]$ spans this subspace. To do this, we
substitute Eq. 3 to obtain

$\displaystyle c_{2i-1}[W]-\lambda c_{2i}[W]=$
$\displaystyle(\mathrm{Re}(u_{1})+\lambda\mathrm{Im}(u_{1}),\mathrm{Im}(u_{1})-\lambda\mathrm{Re}(u_{1}),\ldots,\mathrm{Re}(u_{S})+\lambda\mathrm{Im}(u_{S}),\mathrm{Im}(u_{S})-\lambda\mathrm{Re}(u_{S})).$
(3.4)

It can be seen that, for any $\lambda\neq 0$, $c_{2i-1}[W]-\lambda c_{2i}[W]$
is equal to an invertible matrix times $c_{2i-1}[W]$. Since $c_{2i-1}[W]$ can
be chosen to point in any direction in $\mathbb{R}^{2S}$, we can choose this
direction such that $c_{2i-1}[W]-\lambda c_{2i}[W]$ lies in the desired
subspace. ∎

This lemma implies that for any $O\in\textrm{O}(2S)$, for any two of its
columns associated with conjugate coordinates $(q_{i},p_{i})$, and for any row
$j$, there exists a matrix $O^{\prime}$ in the OS right coset of $O$, that is,
$O^{\prime}\in\\{OW\mid W\in\textrm{SpO}(2S,\mathbb{R})\\}$, such that the
determinants of all square matrices in these two columns of $O^{\prime}$,
where none of the rows is $j$, vanish. A particular case of interest occurs
when $O$ is symplectic. Then the right coset coincides with
$\textrm{SpO}(2S,\mathbb{R})$. Therefore, looking at Eq. 3, the two columns,
after removing row $j$ from each, can only be proportional if all the $u_{i}$s
except the one associated with row $j$ are zero.

######  Lemma 3.2.

Let us denote the map returning the column $i$ of a matrix by $c_{i}[\cdot]$.
Then, given an arbitrary $O\in\textrm{O}(2S)$, for any $1\leq i\leq S$, and
for any $S-1$ rows $j_{1},\ldots,j_{S-1}$ there exists a matrix
$W\in\textrm{SpO}(2S,\mathbb{R})$ such that these rows in $c_{2i-1}[OW]$ and
$c_{2i}[OW]$ are zero.

###### Proof.

As stated in the proof of the previous lemma $c_{2i-1}[OW]$ and $c_{2i}[OW]$
are equal to $Oc_{2i-1}[W]$ and $Oc_{2i}[W]$, respectively. The claim is that
there exists an OS matrix $W$ such that $(O_{l,1},\ldots,O_{l,2S})\cdot
c_{2i-1}[W]=0$ and $(O_{l,1},\ldots,O_{l,2S})\cdot c_{2i}[W]=0$ for all
$l\in\\{j_{1},\ldots,j_{S-1}\\}$. The second set of equalities are equivalent
to

$(-O_{l,2},O_{l,1},\ldots,-O_{l,2S},O_{l,2S-1})\cdot c_{2i-1}[W]=0.$ (3.5)

This can be seen from the relationship between $c_{2i}[W]$ and $c_{2i-1}[W]$
in Eq. 3. Thus the requirement on $W$ is that $c_{2i-1}[W]$ is in the kernel
of a $(2S-2)\times 2S$ matrix which is obtained by stacking the $S-1$ rows
$(O_{l,1},\ldots,O_{l,2S})$ vertically on top of the $S-1$ rows
$(-O_{l,2},O_{l,1},\ldots,-O_{l,2S},O_{l,2S-1})$. This is always possible
since $c_{2i-1}[W]$ can be chosen to be any unit vector in $\mathbb{R}^{2S}$.
∎

######  Lemma 3.3.

Consider an arbitrary $O\in\textrm{O}(2S)$ and any basis vector
$\vec{e}\in\mathbb{R}^{2S}$ in our coordinate basis. Then, there exists a
matrix $W\in\textrm{SpO}(2S,\mathbb{R})$ such that $OW$ acts trivially on
$\vec{e}$.

###### Proof.

Let $j$ denote location of $1$ in the representation of $\vec{e}$ in our
coordinate ordering. The $j$’th column of $W$ can be chosen to be any unit
vector in $\mathbb{R}^{2S}$. Any $W$ with a $j$’th column that is equal to the
$j$’th row of $O$ has the necessary properties. To see this, note that, by
orthonormality of the columns of $W$, the $j$’th row of $OW$ is guaranteed to
be equal to $\vec{e}^{T}$. And due to the requirement that the columns of $OW$
must have unit length, this implies that the $j$’th column of $OW$ is equal to
$\vec{e}$. ∎

######  Lemma 3.4.

Assume $O\in\mathrm{O}(2S)$ and let $a$ be any integer in the range $S\geq
a\geq 1$. Assume the columns $(2a-1,2a)$ of $O$ have the following form:

$\displaystyle\begin{pmatrix}\Delta(1)\lambda_{1}z_{1}&\Delta(1)\lambda_{1}z_{2}\\\
\Delta(2)\lambda_{2}z_{1}&\Delta(2)\lambda_{2}z_{2}\\\ \vdots&\vdots\\\
\Delta(j)\lambda_{j}z_{1}&\Delta(j)\lambda_{j}z_{2}\\\ x_{1}&x_{2}\\\
\Delta(j+1)\lambda_{j+1}z_{1}&\Delta(j+1)\lambda_{j+1}z_{2}\\\
\vdots&\vdots\\\
\Delta(2S-1)\lambda_{2S-1}z_{1}&\Delta(2S-1)\lambda_{2S-1}z_{2}\\\
\end{pmatrix},$ (3.6)

where $(x_{1},x_{2})$ can lie on row $1$ or row $2S$ as well, and
$\Delta(\cdot)$ can be $0$ or $1$. Further, $z_{i}\neq 0$ for $i=1,2$ and
$\lambda_{i}\neq 0$ for all $i$. Denote the row index of $(x_{1},x_{2})$ by
$k$. Denote the matrix composed of these columns by $\tilde{O}$. Then, one can
choose an OS matrix $W$, and an orthogonal matrix $R$ such that $ROW$ has the
form

$\displaystyle ROW=O_{1}\oplus\begin{pmatrix}1&0\\\ 0&1\end{pmatrix}\oplus
O_{2},$ (3.7)

where $O_{1}$ and $O_{2}$ have the dimensions $2\lfloor(k-1)/2\rfloor$ and
$2S-2\lfloor(k-1)/2\rfloor-2$, respectively. Importantly, $R$ can be chosen to
act as the identity on the subspace of coordinates associated with the zero
rows of $\tilde{O}$, with the exception of the coordinate that is conjugate to
the coordinate corresponding to row $k$.

###### Proof.

Notice that the columns of $\tilde{O}$, when the row $(x_{1},x_{2})$ is
removed, are proportional. We first consider the case where $k$ is odd. We
start by right-multiplying $O$ with an OS matrix $W_{1}$ that acts as the
identity on the subspace spanned by the set of coordinates
$\\{q_{i},p_{i}\\}_{i\neq j}$, and as
$\begin{pmatrix}\cos(\theta)&\sin(\theta)\\\
-\sin(\theta)&\cos(\theta)\end{pmatrix}$ on the subspace spanned by
$q_{j},p_{j}$. We choose $\theta$ such that
$\cos(\theta)z_{1}-\sin(\theta)z_{2}=0$. Then, $\tilde{O}$ transforms as
follows:

$\displaystyle\tilde{O}W_{1}=\begin{pmatrix}0&\Delta(1)\lambda_{1}z^{\prime}\\\
\vdots&\vdots\\\ 0&\Delta(j)\lambda_{j}z^{\prime}\\\ 1&0\\\
0&\Delta(j+1)\lambda_{j+1}z^{\prime}\\\ \vdots&\vdots\\\
0&\Delta(2S-1)\lambda_{2S-1}z^{\prime}\\\ \end{pmatrix},$ (3.8)

where $(1,0)$ lies on the $k$’th row, and
$z^{\prime}=\sin(\theta)z_{1}+\cos(\theta)z_{2}$. $(x_{1},x_{2})$ was forced
to become $(1,0)$ to preserve the orthonormality of the columns.

Next, we restrict $R$ to act trivially on row $k$. On the rest of the
coordinates $R$ can be chosen to act non-trivially only on the subspace
associated with row $k+1$ and with all rows where $\Delta(\cdot)\neq 0$. It
can be seen that even with these restrictions on the form of $R$, it can be
chosen such that it transforms the second column of $\tilde{O}$ into the basis
vector with the number $1$ lying in the $k+1$’th position. Namely,

$\displaystyle\tilde{O}W_{1}=\begin{pmatrix}0&0\\\ \vdots&\vdots\\\ 1&0\\\
0&1\\\ \vdots&\vdots\\\ 0&0\\\ \end{pmatrix}.$ (3.9)

Due to the orthonormality of the rows of $ROW_{1}$, this implies that the rest
of the elements of the rows $k$ and $k+1$ of $ROW_{1}$ are zero. Afterwards,
we apply a right-multiplication of $ROW_{1}$ with a symplectic permutation
$W_{2}$ that exchanges the column set $(2a-1,2a)$ with the column set
$(k,k+1)$. The resulting matrix can be seen to be of the desired form, and
$W=W_{1}W_{2}$.

When $k$ is even, the construction of $W_{1}$, $R$ and $W_{2}$ proceeds
analogously. In this case $\theta$ is chosen such that
$\sin(\theta)z_{1}+\cos(\theta)z_{2}=0$, in which case $\tilde{O}$ transforms
into

$\displaystyle\tilde{O}W_{1}=\begin{pmatrix}\Delta(1)\lambda_{1}z^{\prime\prime}&0\\\
\vdots&\vdots\\\ \Delta(j)\lambda_{j}z^{\prime\prime}&0\\\ 0&1\\\
\Delta(j+1)\lambda_{j+1}z^{\prime\prime}&0\\\ \vdots&\vdots\\\
\Delta(2S-1)\lambda_{2S-1}z^{\prime\prime}&0\\\ \end{pmatrix},$ (3.10)

where $z^{\prime\prime}=\cos(\theta)z_{1}-\sin(\theta)z_{2}$. $R$ is then
designed to transform the $k-1$’th row of $OW_{1}$ into $(1,0)$, and the
symplectic permutation $W_{2}$ exchanges the column set $(2a-1,2a)$ of
$ROW_{1}$ with the column set $(k-1,k)$.

∎

##  4 Classification of Eigenspectra

In this section we present the main contributions of this paper. To state
them, we first give a more precise description of the “$t$-pure unique pairing
condition”. See, for example, Ref. [15] for the relevant background in graph
theory. In this section we order the phase-space coordinates as
$(q_{1},p_{1},\ldots,q_{S},p_{S})$.

######  Definition 4.1.

Consider the eigenspectrum $\Lambda=(\lambda_{i})_{i=1}^{2S}$ of a quantum CM
$\Gamma$, where the $\lambda_{i}$ are in a non-ascending order. Let
$G(\Lambda)$ be the graph with vertices $\\{1,\ldots,2S\\}$ and edges
$\\{\\{i,j\\}:i<j,\lambda_{i}\lambda_{j}\geq 1\\}$. An edge $\\{i,j\\}$ of
$G(\Lambda)$ is called pure if $\lambda_{i}\lambda_{j}=1$. Then $\Lambda$
satisfies the unique pairing condition if $G(\Lambda)$ has a unique perfect
matching up to the permutations of any set of vertices associated with
degenerate eigenvalues. $\Lambda$ satisfies the $t$-pure unique pairing
condition if at least $t$ of the edges in the perfect matching are pure.

Thm. 4.2 is about $\Lambda$ that satisfy the $S-1$-pure unique pairing
condition. To build an intuition and a different representation of such
$\Lambda$, consider its diagonal representative $D_{\Lambda}$. Using the fact
that the symplectic decomposition of $D_{\Lambda}$ is of the form $Q^{2}T$, we
can assign a set of thermal and squeezing parameters to $\Lambda$. Then, $S-1$
of the thermal parameters are equal to $1$. We can apply a symplectic
permutation to $D_{\Lambda}=Q^{2}T$ to obtain a quantum CM of the form

$\displaystyle D=\textrm{diag}[\nu e^{-2r_{1}},\nu
e^{2r_{1}},e^{-2r_{2}},e^{2r_{2}},\ldots,e^{-2r_{S}},e^{2r_{S}}],$ (4.1)

where $\nu\geq 1$, $r_{i}\geq 0$ for all $i$, and $r_{i}\geq r_{j}$ when
$i>j>1$. Of course, the diagonal entries of $D$ are composed of the elements
of $\Lambda$. When $\nu=1$, $D$ is a CM of a pure Gaussian state. When
$\nu>1$, the $S-1$-pure unique pairing condition requires that, for any
$j\in\\{2,\ldots,S\\}$, either $\nu e^{2r_{1}-2r_{j}}\leq 1$ or $\nu
e^{-2r_{1}+2r_{j}}\leq 1$. This is because, otherwise, the $4$-vertex subgraph
composed of the vertices associated with the eigenvalues $\nu e^{-2r_{1}},\nu
e^{2r_{1}},e^{-2r_{j}}$ and $e^{2r_{j}}$ would have another perfect matching:
the two edges connecting $\nu e^{-2r_{1}}$ with $e^{2r_{j}}$ and $\nu
e^{2r_{1}}$ with $e^{-2r_{j}}$, and both of the edges are not pure. In
contrast, when, for example, $\nu e^{2r_{1}-2r_{j}}=1$, then the vertices
associated with the eigenvalues $\nu e^{2r_{1}}$ and $e^{-2r_{j}}$ can be
exchanged to obtain the original perfect matching. Similarly, this restriction
does not arise when $\nu=1$, since one can have $r_{1}=r_{j}$, in which case
the second perfect matching is equal to the original when the vertex
associated with $e^{-2r_{j}}$ ($e^{2r_{j}}$) is exchanged with the vertex
associated with $e^{-2r_{1}}$ ($e^{2r_{1}}$).

With these observations we find it useful to divide the set of eigenspectra
satisfying the $S-1$-pure unique pairing condition into three classes. The
first class is composed of the eigenspectra with $\nu=1$. In the second class,
$\nu>1$ and $\nu e^{2r_{1}-2r_{2}}\leq 1$, which implies that $\nu
e^{2r_{1}-2r_{j}}\leq 1$ for all $j=2,\ldots,S$. Finally, for the eigenspectra
in the third class, $\nu>1$ and $\nu e^{-2r_{1}+2r_{2}}\leq 1$.

######  Theorem 4.2.

Let $\Gamma$ and $\Gamma^{\prime}$ be two quantum CMs with the same
eigenspectrum $\Lambda=\\{\lambda_{i}\\}_{i=1}^{2S}$. Suppose that $\Lambda$
satisfies the $S-1$-pure unique pairing condition. Then $\Gamma^{\prime}$ is
an OS transform of $\Gamma$.

###### Proof.

It is sufficient to show that any non-trivial ONS transform of the diagonal
representative $D_{\Lambda}$ (or of any other OS transform of $D_{\Lambda}$
such as $D$ in Eq. 4.1) results in a CM that violates the uncertainty
principle. If this is not evident, see the discussion after the problem
statements in Sec. 2. We show that, given a non-trivial ONS matrix $O$ acting
on $D$ as $O^{T}DO$, we can find an OS matrix $W_{f}$ such that the upper-left
$2\times 2$ block of $W_{f}^{T}O^{T}DOW_{f}$ has a determinant that is $<1$.
The latter implies that $W_{f}^{T}O^{T}DOW_{f}$, and therefore $O^{T}DO$ as
well, violate the uncertainty principle, according to the discussion at the
end of Sec. 2.

Before moving on, let us give some examples of what a non-trivial ONS matrix
for $D$ looks like. If there are no degeneracies in the eigenspectrum, then
every ONS matrix is non-trivial. “Trivial” ONS matrices arise when any two
diagonal entries of $D$ that belong to different modes are equal. To clarify,
by belonging to different modes we mean associated with different pairs of
conjugate coordinates $(q_{i},p_{i})$. Then, any non-trivial orthogonal
transformation that acts as identity on the rest of the entries, is an ONS
transformation and does not affect $D$. Thus, the action of the right OS coset
of such a transformation on $D$ is equal to the action of
$\textrm{SpO}(2S,\mathbb{R})$, and thus preserves the thermal and squeezing
parameters.

Consider an arbitrary orthogonal matrix $O^{\prime}$. We derive an expression
for the determinant of the upper-left $2\times 2$ block of $O^{\prime
T}DO^{\prime}$. We use the superscript $[2\times 2]$ to refer to this block of
any matrix. Now,

$\displaystyle(O^{\prime T}DO^{\prime})^{[2\times 2]}_{ij}$
$\displaystyle=\sum_{k=1}^{2S}\sum_{l=1}^{2S}(O^{\prime
T})_{ik}D_{kl}O^{\prime}_{lj}.$ (4.2)

Notice that the decomposition of $(O^{\prime T}DO^{\prime})^{[2\times 2]}$ in
Eq. 4.2 can be represented as a product of a $2\times 2S$ matrix
$\tilde{O}^{T}$ with a $2S\times 2$ matrix $D\tilde{O}$. Here, $\tilde{O}$ is
the matrix composed of the first two columns of $O^{\prime}$. We can thus use
the Cauchy-Binet formula to obtain

$\displaystyle\det((O^{\prime T}DO^{\prime})^{[2\times 2]})$
$\displaystyle=\sum_{k>l}\det\begin{pmatrix}\tilde{O}^{T}_{1l}&\tilde{O}^{T}_{1k}\\\
\tilde{O}^{T}_{2l}&\tilde{O}^{T}_{2k}\end{pmatrix}\det\begin{pmatrix}(D\tilde{O})_{l1}&(D\tilde{O})_{l2}\\\
(D\tilde{O})_{k1}&(D\tilde{O})_{k2}\end{pmatrix}$
$\displaystyle=\sum_{k>l}\det\begin{pmatrix}\tilde{O}_{l1}&\tilde{O}_{l2}\\\
\tilde{O}_{k1}&\tilde{O}_{k2}\end{pmatrix}\det\begin{pmatrix}(D\tilde{O})_{l1}&(D\tilde{O})_{l2}\\\
(D\tilde{O})_{k1}&(D\tilde{O})_{k2}\end{pmatrix},$ (4.3)

where the sum is over all $k>l$, for $l=1,\ldots,2S-1$. We can use the fact
that $D$ is diagonal to express
$\begin{pmatrix}(D\tilde{O})_{l1}&(D\tilde{O})_{l2}\\\
(D\tilde{O})_{k1}&(D\tilde{O})_{k2}\end{pmatrix}=\begin{pmatrix}D_{ll}\tilde{O}_{l1}&D_{ll}\tilde{O}_{l2}\\\
D_{kk}\tilde{O}_{k1}&D_{kk}\tilde{O}_{k2}\end{pmatrix}$. Then,

$\displaystyle\det((O^{\prime T}DO^{\prime})^{[2\times 2]})$
$\displaystyle=\sum_{k>l}\det\begin{pmatrix}\tilde{O}_{l1}&\tilde{O}_{l2}\\\
\tilde{O}_{k1}&\tilde{O}_{k2}\end{pmatrix}\det\begin{pmatrix}D_{ll}\tilde{O}_{l1}&D_{ll}\tilde{O}_{l2}\\\
D_{kk}\tilde{O}_{k1}&D_{kk}\tilde{O}_{k2}\end{pmatrix}$
$\displaystyle=\sum_{k>l}\left(\det\begin{pmatrix}\tilde{O}_{l1}&\tilde{O}_{l2}\\\
\tilde{O}_{k1}&\tilde{O}_{k2}\end{pmatrix}\right)^{2}D_{ll}D_{kk}$
$\displaystyle=\sum_{k>l}\left(\det\begin{pmatrix}O^{\prime}_{l1}&O^{\prime}_{l2}\\\
O^{\prime}_{k1}&O^{\prime}_{k2}\end{pmatrix}\right)^{2}D_{ll}D_{kk}.$ (4.4)

According to Fact 3.1, the set of squares of the determinants appearing in the
last line sum to $1$. We can thus think of them as forming a discrete
probability distribution, and denote
$\left(\det\begin{pmatrix}O^{\prime}_{l1}&O^{\prime}_{l2}\\\
O^{\prime}_{k1}&O^{\prime}_{k2}\end{pmatrix}\right)^{2}$ by $p_{lk}$ for
brevity. We also sometimes refer to a particular $p_{lk}$ by $p_{kl}$ for
convenience. It should be clear from the context which orthogonal matrix the
$p_{lk}$ are associated with. Substituting the $D_{ii}$ with their
corresponding entries in Eq. 4.1, we obtain

$\displaystyle\det((O^{\prime T}DO^{\prime})^{[2\times
2]})=\sum_{l=1}^{2S-1}\sum_{k=l+1}^{2S}p_{lk}D_{kk}D_{ll}$
$\displaystyle=p_{1,2}\nu^{2}+\sum_{i=2}^{S}p_{2i-1,2i}$
$\displaystyle+\sum_{j=2}^{S}\left[p_{1,2j-1}\nu
e^{-2r_{1}-2r_{j}}+p_{1,2j}\nu e^{-2r_{1}+2r_{j}}+p_{2,2j-1}\nu
e^{2r_{1}-2r_{j}}+p_{2,2j}\nu e^{2r_{1}+2r_{j}}\right]$
$\displaystyle+\sum_{i=2}^{S-1}\sum_{j=i+1}^{S}\left[p_{2i-1,2j-1}e^{-2r_{i}-2r_{j}}+p_{2i-1,2j}e^{-2r_{i}+2r_{j}}+p_{2i,2j-1}e^{2r_{i}-2r_{j}}+p_{2i,2j}e^{2r_{i}+2r_{j}}\right]$
(4.5)

We consider the class of eigenspectra with $\nu=1$ first. This class of
eigenspectra corresponds to that of pure Gaussian states. As mentioned
earlier, the theorem statement for this class of eigenspectra can be proven
using a different, and a significantly shorter, method. But we opt to show
this version of the proof in detail as the main features of the proof are also
used for the more complicated classes of eigenspectra later on. Hopefully,
this makes it easier to convey the proofs for the other two classes.

######  Case 1.

The proof for eigenspectra with $\nu=1$.

For this case we can assume $D$ is of the form

$\displaystyle
D=\textrm{diag}[e^{-2r_{1}},e^{2r_{1}},e^{-2r_{2}},e^{2r_{2}},\ldots,e^{-2r_{S}},e^{2r_{S}}],$
(4.6)

where $r_{i}\geq 0$ for all $i$, and $r_{i}\geq r_{j}$ when $i>j\geq 1$,
without loss of generality. We exclude the instance where $D$ is the identity,
as in that case there are no non-trivial ONS matrices. Then, it should be
clear that each of the terms of the form $e^{-2r_{S}-2r_{j}}$ in the last line
of Eq. 4 are $<1$, and each of the terms of the form $e^{-2r_{S}+2r_{j}}$ are
$\leq 1$. According to Lem. 3.1, we can find an OS matrix $W$ such that the
first two columns of $OW$ are proportional except for one of the rows of our
choosing. Let us denote $OW$ by $O^{\prime}$, and by $\tilde{O}$ the matrix
composed of its first two columns. Let us choose this row to be the $2S-1$th
row. Then, all determinants corresponding to the weights $p_{lk}$ where $l\neq
2S-1$, $k\neq 2S-1$, vanish. Thus, Eq. 4 reduces to

$\displaystyle\det((O^{\prime T}DO^{\prime})^{[2\times
2]})=p_{2S-1,2S}+\sum_{j=1}^{S-1}\left[p_{2j-1,2S-1}e^{-2r_{S}-2r_{j}}+p_{2j,2S-1}e^{-2r_{S}+2r_{j}}\right].$
(4.7)

The remaining weights must, of course, sum to $1$. Let us consider what values
the remaining weights can take. Since we have designed the first two columns
of $O^{\prime}$ such that $O^{\prime}_{l2}=\lambda O^{\prime}_{l1}$ for all
$l\neq 2S-1$ for some $\lambda\neq 0$, each of the the remaining weights
$p_{l,2S-1}$ is either $0$, when the corresponding
$O^{\prime}_{l1}=O^{\prime}_{l2}=0$, or equal to the same value as the other
nonzero weights. If, for some $j\neq S$, $O^{\prime}_{2j-1,1}\neq 0$, then the
corresponding $p_{2S-1,2j-1}\neq 0$. In this case $\det((O^{\prime
T}DO^{\prime})^{[2\times 2]})<1$ as it is a uniformly weighed combination of
terms that are $\leq 1$ and at least one other term that is $<1$.

We need to do more work when $p_{2j-1,2S-1}=0$ for all $j\neq S$. In this
case, $O^{\prime}_{l1}=O^{\prime}_{l2}=0$ for $l=1,3,\ldots,2S-3$. If, for
some $j\neq 0$, $r_{j}<r_{S}$ and the corresponding $p_{2j,2S-1}\neq 0$, then
it can be seen that $\det((O^{\prime T}DO^{\prime})^{[2\times 2]})<1$. A more
complicated situation arises when for all $j$ such that $r_{j}<r_{S}$, the
corresponding $p_{2j,2S-1}=0$. Note that such $j$ might not exist, namely,
when all $r_{j}$ are equal to each other. We can capture all possible
instances with the following representation of the first two columns of
$O^{\prime}$:

$\displaystyle\tilde{O}=\begin{pmatrix}0&0\\\ \Theta(1)y&\Theta(1)\lambda y\\\
0&0\\\ \Theta(2)y&\Theta(2)\lambda y\\\ \vdots&\vdots\\\ 0&0\\\
\Theta(S-1)y&\Theta(S-1)\lambda y\\\ x_{1}&x_{2}\\\
\Theta(S)y&\Theta(S)\lambda y\end{pmatrix}.$ (4.8)

Here $\Theta(j)=0$ for all $j$ such that $r_{j}<r_{S}$, and is either $0$ or
$1$ for $j$ where $r_{j}=r_{S}$. Note that $\Theta(j)=1$ for at least one
value of $j$ since the columns are orthonormal. For the same reason, $x_{1}$
and $x_{2}$ are nonzero. $\tilde{O}$ has the form of Eq. 3.6 in Lem. 3.4, and
thus we can apply the transformation in the lemma statement to turn $O$ into a
matrix of the form

$\displaystyle O^{(1)}=\begin{pmatrix}O_{S-1}&\vline&\begin{matrix}0&0\\\
\vdots&\vdots\\\ 0&0\end{matrix}\\\ \hline\cr\begin{matrix}0&\ldots&0\\\
0&\ldots&0\end{matrix}&\vline&\begin{matrix}1&0\\\
0&1\end{matrix}\end{pmatrix}.$ (4.9)

To see that this transformation is allowed, that is, it results in an OS
transform of $O^{\prime T}DO^{\prime}$, we note that the orthogonal matrix $R$
used for this transformation can be chosen to act as the identity on all rows,
except row $2S$ of course, where $\Theta(j)=0$. The rows on which $R$ acts
non-trivially correspond to identical entries in $D$, and thus $R$ is a
trivial ONS matrix with respect to $D$.

Let us step back and recap what we have accomplished. We have shown that,
starting with $D$ in the form of Eq. 4.6, and given a non-trivial ONS
transformation $O$, we can right-multiply $O$ with an OS matrix $W$ so that
$\det((W^{T}O^{T}DOW)^{[2\times 2]})$ has the form in Eq. 4.7. Then, either
the dereminant is $<1$, in which case we do not need to proceed further, or
the first two columns of $O^{\prime}=OW$ have the form in Eq. 4.8. If the
latter is true, then there exists a trivial ONS matrix $R$ and an OS matrix
$W^{\prime}$, such that $RO^{\prime}W^{\prime}=O^{(1)}$ has the form in Eq.
4.9. In this case, the crucial fact to notice is that we have effectively
removed the mode $S$ from the problem. More specifically, the upper-left
$2\times 2$ block of $O^{(1)T}DO^{(1)}$ does not contain the last two diagonal
entries of $D$, that is, $e^{2r_{S}}$ and $e^{-2r_{S}}$. Thus, the determinant
of $(O^{(1)T}DO^{(1)})^{[2\times 2]}$ has the same form as Eq. 4 but with the
indices of the weights $p_{kl}$ running from $1$ to $2S-2$. In the rest of the
proof, when saying “we have removed the mode $i$ from the problem” by
transforming $O$ onto some $O^{(n)}$, we mean that (1) the entries of $D$
associated with mode $i$ do not appear in $(O^{(n)T}DO^{(n)})^{[2\times 2]}$,
(2) $O^{(n)}$ acts trivially on the subspace associated with $(q_{i},p_{i})$,
and (3) we can restrict all consequent transformations of $O^{(n)}$ to act
trivially on this subspace as well.

It can be seen that the set of steps described in the previous three
paragraphs can be performed on the modes $1,\ldots,S-1$. For example, the
application of Lem. 3.1 to the first two columns of $O^{(1)}$ can be performed
such that the last two rows remain zero. This is because the OS matrix can be
chosen to act non-trivially on the first $S-1$ modes of $O^{(1)}$, and as
identity on mode $S$. If at some point in the sequence of steps the
determinant of the upper-left $2\times 2$ block falls below zero, then the
looked-for OS matrix has been found. Conversely, the sequence of steps will
result in transforming $O^{(1)}$ into a matrix of the form

$\displaystyle O^{(2)}=\begin{pmatrix}O_{S-2}&\vline&\begin{matrix}0&0&0&0\\\
\vdots&\vdots&\vdots&\vdots\\\ 0&0&0&0\end{matrix}\\\
\hline\cr\begin{matrix}0&\ldots&0\\\ 0&\ldots&0\\\ 0&\ldots&0\\\
0&\ldots&0\end{matrix}&\vline&\begin{matrix}1&0&0&0\\\ 0&1&0&0\\\ 0&0&1&0\\\
0&0&0&1\end{matrix}\end{pmatrix},$ (4.10)

in which case the mode $S-1$ has been removed from the problem as well.

Repeating this procedure we are guaranteed to transform $O$ onto some $O_{f}$,
such that $\det((O_{f}^{T}DO_{f})^{[2\times 2]})<1$, by a sequence of right-
multiplications with OS matrices and a sequence of left-multiplications with
trivial ONS matrices. Otherwise, we would have shown that $O$ can be
transformed into the identity by such a set of sequences. This would imply
that $O$ was either a trivial ONS matrix, or an OS matrix, contradicting the
original assumption.

######  Case 2.

The proof for eigenspectra with $\nu>1$ and $\nu e^{2r_{1}-2r_{j}}\leq 1$ for
all $j=2,\ldots,S$.

The proof in this case is very similar to the proof for Case $1$. Therefore,
we find it satisfactory to show how the two cases are similar, and leave out
the details of the steps that are identical to those used in the proof for
Case $1$. The basic idea is to again find a set of OS transformations of
$O^{T}DO$ such that all weights in Eq. 4 associated with the coefficients that
are $>1$ vanish, and there is at least one weight associated with a
coefficient that is $<1$ that does not vanish. We start by applying Lem. 3.1
to the first two columns of $O$, and again choosing the row for which the
entries of the two columns are not proportional to be the row $2S-1$. Then,
Eq. 4 reduces to

$\displaystyle\det((W^{T}O^{T}DOW)^{[2\times 2]})$
$\displaystyle=p_{2S-1,2S}+p_{1,2S-1}\nu e^{-2r_{1}-2r_{S}}+p_{2,2S-1}\nu
e^{2r_{1}-2r_{S}}$
$\displaystyle+\sum_{j=2}^{S-1}\left[p_{2j-1,2S-1}e^{-2r_{S}-2r_{j}}+p_{2j,2S-1}e^{-2r_{S}+2r_{j}}\right].$
(4.11)

Notice that, if $\nu e^{-2r_{1}-2r_{S}}<1$, the set of weights associated with
the coefficients that are $\leq 1$, and the set of weights associated with the
coefficients that are $<1$ is the same as the corresponding sets of weights in
Eq. 4.7. These weights only depend on $OW$. If any of the weights associated
with a coefficient that is $<1$ is nonzero, then we do not need to proceed
further. Otherwise, the same set of steps as in Case $1$ can be performed to
transform $O$ into a matrix $O^{(1)}$ of the form in Eq. 4.9.

If $\det((W^{T}O^{T}DOW)^{[2\times 2]})=1$ and $\nu e^{-2r_{1}-2r_{S}}=1$,
which can only occur if $r_{1}=0$ and $e^{2r_{S}}=\nu$, the first row in Eq.
3.8 is not necessarily empty. To represent all possible instances of
degeneracies in the entries of $D$, we substitute the first row of Eq. 3.8
with $(\Theta(0)y,\Theta(0)\lambda y)$, where we have redefined the function
$\Theta(i)$ to also include the value $0$ as argument. $\Theta(0)$ can be
either $0$ or $1$ depending on $O$ and the OS matrix $W$ used in the
application of Lem. 3.1. It can be seen that Lem. 3.4 can be applied here as
well to obtain a matrix $O^{(1)}$ of the form in Eq. 4.9. We have thus reduced
the number of modes in the problem by removing mode $S$.

Then, we repeat this procedure for the modes $1,\ldots,S-1$. In particular, we
apply Lem. 3.1 and choose the OS matrix $W^{(1)}$ such that it acts trivially
on the mode $S$, and such that the first two columns of $O^{(1)}W^{(1)}$ are
proportional except for the row $2S-3$. Then,

$\displaystyle\det((W^{(1)T}O^{(1)T}DO^{(1)}W^{(1)})^{[2\times 2]})$
$\displaystyle=p_{2S-3,2S-2}+p_{1,2S-3}\nu e^{-2r_{1}-2r_{S-1}}+p_{2,2S-3}\nu
e^{2r_{1}-2r_{S-1}}$
$\displaystyle+\sum_{j=2}^{S-2}\left[p_{2j-1,2S-3}e^{-2r_{S-1}-2r_{j}}+p_{2j,2S-3}e^{-2r_{S-1}+2r_{j}}\right].$
(4.12)

Since $\nu e^{2r_{1}-2r_{j}}\leq 1$ for all $j=2,\ldots,S-1$, and $r_{S-1}\geq
r_{j}$ for all $j=2,\ldots,S-2$, if
$\det((W^{(1)T}O^{(1)T}DO^{(1)}W^{(1)})^{[2\times 2]})=1$, we can apply Lem.
3.4 and reduce the problem by removing the mode $S-1$. As argued at the end of
the proof for Case $1$, by repeating this procedure for each consecutive mode,
we are guaranteed to find a transform of $O$ onto some $O_{f}$ such that
$\det((O_{f}^{T}DO_{f})^{[2\times 2]})<1$.

######  Case 3.

The proof for eigenspectra with $\nu>1$ and $\nu e^{-2r_{1}+2r_{2}}\leq 1$.

The steps of this proof are somewhat more involved than the proofs for the
previous two cases. The main reason is that this class of eigenspectra covers
a larger variety of subclasses. To see this, note that, since we have ordered
the diagonal entries of $D$ such that $r_{i}\geq r_{j}$ for $r>j>1$, given the
fact that we only require $\nu e^{-2r_{1}+2r_{2}}\leq 1$, there might exist
some $j$ such that $\nu e^{-2r_{1}+2r_{i}}>1$ for all $i>j$, and this $j$ can
be any number between $2$ and $S-1$. As remarked earlier, if such a $j$
exists, then the $S-1$-pure unique pairing condition implies that $\nu
e^{2r_{1}-2r_{i}}\leq 1$ for all $i>j$. Let us use the special character $\xi$
to denote such a $j$. We assume $\xi$ can take values from $2$ to $S$, where,
if $\xi=S$, then $\nu e^{-2r_{1}+2r_{i}}\leq 1$ for all $i\geq 2$. We find
that we need Lem. 3.2 in addition to Lem. 3.1 to develop a procedure of
finding $W_{f}$ that can be applied to every eigenspectrum in this class.

Before describing the general procedure, we want to exclude from it a
particular subclass of eigenspectra for which every ONS transform is a trivial
ONS transform. This subclass is represented by $D$ of the form

$\displaystyle D=\textrm{diag}[\nu e^{2r_{1}},1,1,1,\ldots,1,1].$ (4.13)

It can be seen that such eigenspectra satisfy the $S-1$-pure unique pairing
condition and fall within the class considered here. Since $\nu
e^{-2r_{1}}=1$, we could also write $\nu^{2}$ in place of $\nu e^{2r_{1}}$ in
Eq. 4.13. The fact that for such $D$ every ONS transform is trivial follows
from Lem 3.3. According to that lemma, given an arbitrary orthogonal matrix
$O^{\prime}$, we can bring it into the form $O^{\prime}\mapsto 1\oplus X$ by a
right-multiplication with an OS matrix. Clearly, any orthogonal matrix of the
form $1\oplus X$ commutes with $D$ in Eq. 4.13.

We now consider the eigenspectra in the rest of this class. Notice that, since
we are working with $D$ of the form in Eq. 4.1, such eigenspectra must satisfy
either $\nu e^{-2r_{1}}<1$ or $e^{-2r_{S}}<1$, or both conditions at the same
time. Let us denote the subclass where $\nu e^{-2r_{1}+2r_{i}}\leq 1$ for all
$i\geq 2$ by $\mathcal{C}(S)$. We first show that, if $\xi<S$, then we can
remove the modes $\xi+1,\ldots,S$ from the problem and thus reduce the
effective eigenspectrum to one that belongs to $\mathcal{C}(\xi)$. Depending
on $O$, it is possible that in the process of removing these modes we stumble
upon $W_{f}$, in which case we do not need to proceed further.

When $\xi<S$, then $\nu e^{2r_{1}-2r_{S}}\leq 1$. This implies that
$e^{-2r_{S}}<1$. Thus, if we use Lem. 3.1, and pick row $2S-1$ of the first
two columns of $OW$ to be the row where the entries are not proportional by
the same constant as the rest of the rows, then we end up with Eq. 4. Further,
the set of coefficients that are $<1$ and the set of coefficients that are
$\leq 1$ are the same as in Case $2$. Thus, if any of the weights associated
with these coefficients is nonzero, we do not need to proceed further.
Otherwise, we can perform the same steps of transformations on $OW$ as in Case
$2$ to remove the mode $S$ from the problem. We can repeat this process for
the modes $\xi+1,\ldots,S$, and if, during this process, we do not transform
$O$ into some $O_{f}$ such that $\det((O_{f}^{T}DO_{f})^{[2\times 2]})<1$,
then we have transformed $O$ into a matrix that acts non-trivially only on
modes $1,\ldots,\xi$. In this case Eq. 4 reduces to an equation of the same
form but with $\xi$ substituted in the place of $S$.

Since $S$ is arbitrary, to finish the proof, it suffices to consider the
eigenspectra in $\mathcal{C}(S)$.

######  Subcase 3.1.

The proof for eigenspectra in $\mathcal{C}(S)$.

According to Lem. 3.2, for any set of $S-1$ rows, there exists an OS matrix
$W$ such that these rows in the first two columns of $OW$ vanish. We can
choose $W$ such that the row $2$ and and the rows $2j$, for al $j\geq 3$,
vanish. Let us denote the first two columns of the resulting matrix by
$\tilde{O}$. In this case the weights $p_{kl}$ and $p_{lk}$, where $l=2$ or
$l=2j$ for $j\geq 3$, are zero. Then, Eq. 4 becomes:

$\displaystyle\det((W^{T}O^{T}DOW)^{[2\times 2]})$
$\displaystyle=p_{3,4}+p_{1,4}\nu
e^{-2r_{1}+2r_{2}}+\sum_{j=2}^{S}p_{1,2j-1}\nu e^{-2r_{1}-2r_{j}}$
$\displaystyle+\sum_{i=2}^{S-1}\sum_{j=i+1}^{S}p_{2i-1,2j-1}e^{-2r_{i}-2r_{j}}+\sum_{j=3}^{S}p_{4,2j-1}e^{2r_{2}-2r_{j}}.$
(4.14)

It can be seen that the coefficient of every weight in Eq. 4 is $\leq 1$.
Further, notice that for the eigenspectra in this subclass $\nu
e^{-2r_{1}}<1$. This is because, otherwise, the condition that $\nu
e^{-2r_{1}+2r_{i}}\leq 1$ for all $i\geq 2$ would imply that $D$ is of the
form in Eq. 4.13. Thus, the coefficients $\nu e^{-2r_{1}-2r_{j}}$ are $<1$,
and if any of the corresponding weights is nonzero, we do not need to proceed
further. Otherwise, the set of weights $\\{p_{1,2j-1}\\}_{j=2}^{S}$ can vanish
only when either the first row is zero or all nonzero rows except row $4$ are
proportional to each other. Let us consider the $\tilde{O}$ of the latter kind
first.

We can represent any $\tilde{O}$ of this form by

$\displaystyle\tilde{O}=\begin{pmatrix}y_{1}&y_{2}\\\ 0&0\\\
\Theta^{\prime}(2)\lambda_{2}y_{1}&\Theta^{\prime}(2)\lambda_{2}y_{2}\\\
x_{1}&x_{2}\\\
\Theta^{\prime}(3)\lambda_{3}y_{1}&\Theta^{\prime}(3)\lambda_{3}y_{2}\\\
0&0\\\ \vdots&\vdots\\\
\Theta^{\prime}(S)\lambda_{S}y_{1}&\Theta^{\prime}(S)\lambda_{S}y_{2}\\\
0&0\end{pmatrix},$ (4.15)

where $\Theta^{\prime}(\cdot)$ assumes values of $0$ or $1$, and the
$\lambda_{i}$ are real numbers. It can be seen that Eq. 4 reduces to

$\displaystyle\det((W^{T}O^{T}DOW)^{[2\times 2]})$
$\displaystyle=p_{3,4}+p_{1,4}\nu
e^{-2r_{1}+2r_{2}}+\sum_{j=3}^{S}p_{4,2j-1}e^{2r_{2}-2r_{j}},$ (4.16)

where $p_{1,4}\neq 0$. Clearly, if $\nu e^{-2r_{1}+2r_{2}}<1$ then
$\det((W^{T}O^{T}DOW)^{[2\times 2]})<1$. Otherwise, $e^{-2r_{2}}=\nu
e^{-2r_{1}}<1$, which implies that $r_{i}\neq 0$ for all $i>1$. If there
exists an $r_{j}>r_{2}$ such that $\Theta^{\prime}(j)=1$, then
$\det((W^{T}O^{T}DOW)^{[2\times 2]})<1$. Otherwise, $\Theta^{\prime}(j)=1$
only for $j$ such that $r_{j}=r_{2}$. In this case every nonzero row besides
row $4$ corresponds to an entry in $D$ that is equal to $e^{-2r_{2}}$. We can
thus use Lem. 3.4, where we choose the trivial ONS matrix $R$ such that it
acts non-trivially on the subspace composed of the coordinates corresponding
to the rows $1$, $3$, and all $2j-1$ with $\Theta^{\prime}(j)=1$. It can be
seen that this results in removing the mode $2$ from the problem.

We now consider the $\tilde{O}$ where the first row is zero. In this case
$p_{1,4}=0$, so Eq. 4 reduces to

$\displaystyle\det((W^{T}O^{T}DOW)^{[2\times 2]})$
$\displaystyle=p_{3,4}+\sum_{i=2}^{S-1}\sum_{j=i+1}^{S}p_{2i-1,2j-1}e^{-2r_{i}-2r_{j}}+\sum_{j=3}^{S}p_{4,2j-1}e^{2r_{2}-2r_{j}}.$
(4.17)

Let us start with discussing the simplest instance, where $r_{i}=0$ for all
$i\geq 2$. In this case the diagonal entries of $D$ corresponding to the modes
$2,\ldots,S$ are all $1$. It should be clear that we can left multiply
$\tilde{O}$ with a trivial ONS matrix to bring it into the form

$\displaystyle\tilde{O}\mapsto\begin{pmatrix}0&0\\\ 0&0\\\ 1&0\\\ 0&1\\\
0&0\\\ \vdots&\vdots\\\ 0&0\end{pmatrix},$ (4.18)

and then remove mode $2$ from the problem by right-multiplying with a
symplectic permutation matrix that exchanges the column sets associated with
modes $1$ and $2$.

For the more general instances, the crucial observation is that, if there
exists a smallest $j>2$ such that $r_{j}>r_{2}\geq 0$, then the set of weights
$p_{2i-1,2k-1}$ for all $i\geq 2$ and $k\geq j$, and the set of weights
$p_{4,2k-1}$ for all $k\geq j$ must be zero, else
$\det((W^{T}O^{T}DOW)^{[2\times 2]})<1$. This is possible only when the rows
$2k-1$ for all $k\geq j$ are zero. By the assumption on $j$, the remaining
nonzero odd rows all correspond to the eigenvalue $e^{-2r_{2}}$. If $r_{2}=0$,
then the remaining nonzero rows all correspond to the eigenvalue of $1$, and,
therefore, we can left multiply $\tilde{O}$ with a trivial ONS matrix to bring
it into the form of Eq. 4.18. If $r_{2}>0$, then the weights $p_{2i-1,2k-1}$
for all $2\leq i<k<j$ must be zero, else $\det((W^{T}O^{T}DOW)^{[2\times
2]})<1$. This is only possible if either all nonzero odd rows are proportional
to each other, or if all rows besides $3$ and $4$ are zero. In either case, we
can bring $\tilde{O}$ into the form of Eq. 4.18 by a left-multiplication with
a trivial ONS matrix. If a value of $j$ such that $r_{j}>r_{2}$ does not
exist, then $r_{i}=r_{2}$ for all $i>2$. Since $S$ is arbitrary, the same
argument can be applied here as well to show that we can bring $\tilde{O}$
into the form of Eq. 4.18, assuming, of course, that all weights associated
with coefficients that are $<1$ are zero.

We have thus shown that, if $\det((W^{T}O^{T}DOW)^{[2\times 2]})=1$, we can
right-multiply $OW$ with an OS matrix and left-multiply with a trivial ONS
matrix, such that the resulting matrix $O^{(1)}$ acts trivially on mode $2$.
Then, applying Lem. 3.2 to the modes $1,3,\ldots,S$, we can right-multiply
$O^{(1)}$ with an OS matrix $W^{(1)}$, that acts as identity on mode $2$, such
that the rows $2$ and $2j$, for $j\geq 4$, of the first two columns of
$O^{(1)}W^{(1)}$ vanish. Then, Eq. 4 reduces to

$\displaystyle\det((W^{(1)T}O^{(1)T}DO^{(1)}W^{(1)})^{[2\times 2]})$
$\displaystyle=p_{5,6}+p_{1,6}\nu
e^{-2r_{1}+2r_{3}}+\sum_{j=3}^{S}p_{1,2j-1}\nu e^{-2r_{1}-2r_{j}}$
$\displaystyle+\sum_{i=3}^{S-1}\sum_{j=i+1}^{S}p_{2i-1,2j-1}e^{-2r_{i}-2r_{j}}+\sum_{j=3}^{S}p_{6,2j-1}e^{2r_{3}-2r_{j}}.$
(4.19)

If $\det((W^{(1)T}O^{(1)T}DO^{(1)}W^{(1)})^{[2\times 2]})=1$, we can repeat
the procedure described in this subcase to remove mode $3$ from the problem.
Repeating this process we are guaranteed to transform $O$ onto some $O_{f}$
such that $\det((O_{f}^{T}DO_{f})^{[2\times 2]})<1$.

∎

It follows from Thm. 4.2 that if an eigenspectrum $\Lambda$ satisfies the
$S-1$-pure unique pairing condition, then all corresponding quantum CMs have
the same thermal and squeezing parameters. This is a direct consequence of the
fact that the set of quantum CMs with such a $\Lambda$ form a single orbit
under the action of $\mathrm{SpO}(2S,\mathbb{R})$. We can find examples of
eigenspectra that do not satisfy the $S-1$-pure unique pairing condition, but
nevertheless belong to both $\mathcal{P}_{1}(S)$ and $\mathcal{P}_{2}(S)$. For
any $S$, any $\Lambda$ that is completely degenerate and composed of elements
that are $>1$, clearly falls in this category. For such eigenspectra, the
orbit under the action of $\mathrm{SpO}(2S,\mathbb{R})$ is composed of a
single CM. But we can find less trivial examples for $S=2$ modes. Consider an
eigenspectrum with a diagonal representative
$D_{\Lambda}=\textrm{diag}[\nu_{1},\nu_{1},\nu_{2}e^{2r_{2}},\nu_{2}e^{-2r_{2}}]$,
where $\nu_{2}e^{2r_{2}}=\nu_{1}$, $r_{2}>0$, and both thermal parameters are
$>1$. Then, it can be seen, by applying Lem. 3.3, that every ONS matrix $O$ is
trivial since we can write $O=O^{\prime}W$, where $O^{\prime}$ acts trivially
on the last coordinate and $W$ is OS. Hence, any quantum CM with such an
eigenspectrum is an OS transform of $D_{\Lambda}$.

Both sets of examples of eigenspectra we gave above have degeneracies, and
satisfy the unique pairing condition. It is easy to see that, if an
eigenspectrum $\Lambda$ does not satisfy the unique pairing condition, then
there are at least two sets of thermal and squeezing parameters compatible
with $\Lambda$. This is because the thermal and squeezing parameters
corresponding to a given pair of eigenvalues determine and are determined by
the pair. The set of quantum CMs with such a $\Lambda$ cannot be in a single
orbit under the action of $\mathrm{SpO}(2S,\mathbb{R})$, since the permutation
matrix taking the diagonal representative to a diagonal matrix corresponding
to the different pairing is not OS.

Thus, $\mathcal{P}_{1}(S)$ and $\mathcal{P}_{2}(S)$ must necessarily each be a
subset of the class of eigenspectra that satisfy the unique pairing condition,
but be strictly larger than the class of eigenspectra that satisfy the
$S-1$-pure unique pairing condition. We can represent any such eigenspectrum
that does not satisfy the $S-1$-pure unique pairing condition by the diagonal
CM

$\displaystyle
D=\textrm{diag}[\nu_{1}e^{-2r_{1}},\nu_{1}e^{2r_{1}},\ldots,\nu_{S}e^{-2r_{S}},\nu_{S}e^{2r_{S}}],$
(4.20)

where $\nu_{1}\geq\nu_{2}\geq...\geq\nu_{S}$, $\nu_{2}>1$, and for all pairs
$(i,j)$ in $1\leq i<j\leq S$, either $\nu_{i}\nu_{j}e^{-2r_{i}+2r_{j}}\leq 1$
or $\nu_{i}\nu_{j}e^{2r_{i}-2r_{j}}\leq 1$, without loss of generality. The
next result derived in this paper concerns a subset of such $\Lambda$ where
the ordered $4$-tuple
$(\nu_{1}e^{-2r_{1}},\nu_{1}e^{2r_{1}},\nu_{2}e^{-2r_{2}},\nu_{2}e^{2r_{2}})$
has at most a $2$-fold degeneracy. By the latter we mean that we exclude those
$4$-tuples where $3$ or $4$ of the elements are equal.

######  Proposition 4.1.

Let an eigenspectrum $\Lambda$ of a quantum CM satisfy the unique pairing
condition, but not the $S-1$-pure unique pairing condition. Further, let the
$4$-tuple
$(\nu_{1}e^{-2r_{1}},\nu_{1}e^{2r_{1}},\nu_{2}e^{-2r_{2}},\nu_{2}e^{2r_{2}})$
in the representation $D$ of $\Lambda$ in Eq. 4.20 have at most a $2$-fold
degeneracy. Then, (1) the set of thermal and squeezing parameters consistent
with $\Lambda$ is not unique, and (2) there exist two quantum CMs with
$\Lambda$ that are related by a non-trivial ONS transformation.

###### Proof.

We start by elucidating the condition on the $4$-tuple in the proposition
statement. Let us denote the eigenvalues in the $4$-tuple by $\gamma_{i}$, so
that the $4$-tuple can be written as
$(\gamma_{1},\gamma_{2},\gamma_{3},\gamma_{4})$. With this notation,
$\nu_{1}=\sqrt{\gamma_{1}\gamma_{2}}$ and
$\nu_{2}=\sqrt{\gamma_{3}\gamma_{4}}$. It is not hard to see that the
condition implies that there exists a permutation $\sigma(\cdot)$ of the
locations of the $\gamma_{i}$, such that the symplectic eigenvalues associated
with
$(\gamma_{\sigma(1)},\gamma_{\sigma(2)},\gamma_{\sigma(3)},\gamma_{\sigma(4)})$
are different from the $\nu_{i}$. Namely, without losing generality, we can
choose $\sigma(\cdot)$ so that
$\sqrt{\gamma_{\sigma(1)}\gamma_{\sigma(2)}}\neq\nu_{i}$ for $i=1$ or $2$. And
since $\Lambda$ satisfies the unique pairing condition, either
$\sqrt{\gamma_{\sigma(1)}\gamma_{\sigma(2)}}<1$ or
$\sqrt{\gamma_{\sigma(3)}\gamma_{\sigma(4)}}<1$. Note that the product of
symplectic eigenvalues is invariant under orthogonal (as well as symplectic)
transformations, so both cannot be $<1$ simultaneously.

To prove the proposition, it suffices to show that there exists a non-trivial
ONS transform $O$ of the diagonal representative $D$, such that $O^{T}DO$ is a
quantum CM and has a different set of thermal and squeezing parameters. We
choose $O$ to act as the identity on the modes $3,\ldots,S$ in the mode
ordering corresponding to $D$. $O$ does not affect the subsets of thermal and
squeezing parameters, $\nu_{3},\ldots,\nu_{S}$ and $r_{3},\ldots,r_{S}$,
respectively. We show the action of $O$ on the first two modes can be chosen
such that the corresponding duo of symplectic eigenvalues of the resulting
matrix are different from $\\{\nu_{1},\nu_{2}\\}$ and $\geq 1$. We can
restrict our attention to the transformations of the upper-left $4\times 4$
block of $D$, which we denote by $D_{1}$. Let us further denote the
corresponding block of $O$ by $O_{1}$.

The symplectic eigenvalues of $O_{1}^{T}D_{1}O_{1}$ are given by the absolute
values of the eigenvalues of $\Omega_{2}O_{1}^{T}D_{1}O_{1}$, where
$\Omega_{2}=\Omega_{1}\oplus\Omega_{1}$ [9]. Consider the map $\tau$ from the
cone of real positive-definite $4\times 4$ matrices
$\mathbb{P}_{4}(\mathbb{R})$ to the cone $\mathbb{K}$ in $\mathbb{R}^{2}$
consisting of all tuples $(\nu_{1},\nu_{2})$, where $\nu_{1}\geq\nu_{2}>0$,
which takes any $A\in\mathbb{P}_{4}(\mathbb{R})$ to its symplectic eigenvalues
arranged in a non-ascending order. It is not hard to show that this map is
continuous in the standard topology of both spaces, see, for example, Thm. 7
in [4].

Next, note that the subset $\\{O_{1}^{T}D_{1}O_{1}\mid
O_{1}\in\mathrm{O}(4)\\}\subset\mathbb{P}_{4}(\mathbb{R})$ is a path connected
region. Thus, for any $O_{1}$, there exists a continuous curve
$O_{1}(t)D_{1}O_{1}(t)$, $t\in[0,t_{f}]$, where $O_{1}(0)=I$ and
$O_{1}(t_{f})=O_{1}$. This curve produces a continuous curve
$(\nu_{1}(t),\nu_{2}(t))$ in $\mathbb{K}$ by the continuity of $\tau$. Thus,
if there exists an $O_{1}$ such that $\nu_{2}(t_{f})<1$, then, since we
assumed $\nu_{1}(0)>1$ and $\nu_{2}(0)>1$, there exists a $t_{p}\in[0,t_{f}]$
such that $\nu_{1}(t_{p})\geq 1$, $\nu_{2}(t_{p})\geq 1$ and
$\nu_{i}(t_{p})\neq\nu_{j}$ for $i,j=1,2$. $O_{1}$ can be chosen to be the
perturbation matrix that pairs the elements of $D_{1}$ such that one of the
symplectic eigenvalues is $<1$. Such a matrix is guaranteed to exist by the
assumptions of the proposition statement, as discussed at the beginning of the
proof.

∎

Prop. 4.1 has an important consequence that we point out now.

######  Corollary 4.3.

Let an eigenspectrum $\Lambda$ of a quantum CM satisfy the unique pairing
condition, but not the $S-1$-pure unique pairing condition. Let $\zeta$ be the
index, if it exists, such that $\nu_{i}=1$ for $i>\zeta$ in the representation
$D$ of $\Lambda$ in Eq. 4.20. If all $\nu_{i}>1$, then set $\zeta=S$. Assume
the ordered $2\zeta$-tuple
$(\nu_{1}e^{-2r_{1}},\nu_{1}e^{2r_{1}},\ldots,\nu_{\zeta}e^{-2r_{\zeta}},\nu_{\zeta}e^{2r_{\zeta}})$
has at most $2\zeta-2$-fold degeneracy. Then, (1) the set of thermal and
squeezing parameters consistent with $\Lambda$ is not unique, and (2) there
exist two quantum CMs with $\Lambda$ that are related by a non-trivial ONS
transformation.

###### Proof.

Let us write the $2\zeta$-tuple in the corollary statement as
$(\gamma_{1},\gamma_{2},\ldots,\gamma_{2\zeta-1},\gamma_{2\zeta})$. The
assumption is that there exist at least two indices $i<j$ such that the
corresponding $\gamma_{i},\gamma_{j}$ are each different from the rest of the
elements of the $2\zeta$-tuple. If $\gamma_{i}$ and $\gamma_{j}$ are
associated with the same mode, then the $4$-tuple composed of
$(\gamma_{i},\gamma_{j})$ and the elements of another mode in the
$2\zeta$-tuple is at most $2$-fold degenerate. Similarly, if $\gamma_{i}$ and
$\gamma_{j}$ are associated with two different modes, then the $4$-tuple
composed of $\gamma_{i}$, $\gamma_{j}$ and of the two remaining elements in
these two modes is at most $2$-fold degenerate. Thus, it can be seen that in
either case Prop. 4.1 can be applied.

∎

We finish this section by summarizing our results about $\mathcal{P}_{1}(S)$
and $\mathcal{P}_{2}(S)$ in several bullet points. More specifically, an
eigenspectrum $\Lambda$, that satisfies the unique pairing condition,

  1. 1.

belongs to $\mathcal{P}_{1}(S)$ and $\mathcal{P}_{2}(S)$ if $\Lambda$
satisfies the $S-1$-pure unique pairing condition.

  2. 2.

belongs to $\mathcal{P}_{1}(S)$ and $\mathcal{P}_{2}(S)$ if $\Lambda$ does not
satisfy the $1$-pure unique pairing condition, and at least $2S-1$ of the
eigenvalues are identical.

  3. 3.

does not belong to $\mathcal{P}_{1}(S)$ or $\mathcal{P}_{2}(S)$ if $\Lambda$
does not satisfy the $S-1$-pure unique pairing condition, and at least $2$ of
the eigenvalues among the subset of not-pure pairs in the perfect matching of
$G(\Lambda)$ are each different from the rest of the eigenvalues in this
subset.

Notice that, for $S=2$, we have found $\mathcal{P}_{1}(2)$ and
$\mathcal{P}_{2}(2)$. For general $S$, to finish the classification problem at
hand, the only class of eigenspectra that needs further study is composed of
all $\Lambda$ that: (1) do not satisfy the $S-1$-pure unique pairing
condition, (2) satisfy the $1$-pure unique pairing condition, (3) if $\zeta$
of the edges in the perfect matching of $G(\Lambda)$ are not-pure, then at
least $2\zeta-1$ of the eigenvalues associated with these edges are identical.

##  5 Discussion

In this study we attempted to answer two related questions about the
eigenspectra of quantum CMs: (1) to characterize the eigenspectra with the
property that the corresponding set of quantum CMs form a single orbit under
the action of $\mathrm{SpO}(2S,\mathbb{R})$ (Prob. 2.1), and (2) to
characterize the eigenspectra with the property that the corresponding set of
quantum CMs have the same set of thermal and squeezing parameters (Prob. 2.2).
Our main result was finding a non-trivial class of eigenspectra that satisfy
both properties (Thm. 4.2). There is only one way to pair the elements of the
eigenspectra in this class such that the product of each pair is $\geq 1$, up
to swapping of identical elements. Further, in this pairing, the product of
only one of the pairs can be strictly greater than $1$. We named this set of
pairing conditions on the eigenspectra as the $S-1$-pure unique pairing
condition. It is easy to show that the class of eigenspectra satisfying both
properties is larger than the class we found, and we gave several examples.
However, the class we found is the largest class of eigenspectra satisfying
both properties when certain non-degeneracy conditions are set on the
eigenvalues (Prop. 4.1 and Cor. 4.3).

So, we could say that we have found the answers to Probs. 2.1 and 2.2 for the
set of non-degenerate eigenspectra, which consists of “almost all”
eigenspectra, and the answers are identical. Further work is needed to fill
the gaps in our findings, namely, to find the conditions on the eigenspectra
that violate the non-degeneracy conditions of Prop. 4.1 and Cor. 4.3, and that
do not satisfy the $S-1$-pure unique pairing condition, but possess either one
or both of the properties in the posed problems. It is likely that the general
proof methodology used to derive our results can also be successfully applied
to the remaining class of eigenspectra. We conjecture that the complete
answers to Probs. 2.1 and 2.2 are identical.

This study was in part inspired by the observation that, for a Gaussian state,
the only features of its CM that that affect its total-photon-number
distribution are the eigenvalues. This is a consequence of the fact that, in
the phase-space formulation of quantum mechanics, the quasi-probability
distribution, e.g. the Wigner function, of the total-photon-number operator is
spherically symmetric. More generally, the eigenspectrum of the quantum CM
captures the rotation-invariant features of the “spread” of the Wigner
function, while the symplectic eigenvalues capture the amount of spread in the
subspaces associated with conjugate coordinates. The latter, for physical
states, must be above certain thresholds due to Heisenberg’s uncertainty
principle. Informally speaking, we can view this study as an investigation of
what the rotation-invariant spread of the Wigner function can tell us about
the amount of spread as well as its uniformity, in the subspaces associated
with conjugate coordinates. Future research directions can include considering
much broader classes of eigenspectra of quantum CMs and attempting to
determine the ranges in which the corresponding thermal and squeezing
parameters must fall.

###### Acknowledgements.

The author acknowledges support from the Professional Research Experience
Program (PREP) operated jointly by NIST and the University of Colorado.
Discussions with Emanuel Knill contributed to thinking about the problem and
to improving the formulation of the main statements. I would also like to
thank Scott Glancy for helpful questions and suggestions to improve the
presentation.

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