text:
\begin{document}

\begin{center}

{\Large Braiding and entanglement in spin networks:}\\
{\Large a combinatorial approach to topological phases}

\vskip 1cm

{\large Z. K\'ad\'ar$^{\dag \,(1)}$,
 A Marzuoli$^{\ddagger \,(2)}$
and M Rasetti$^{\sharp\,(3)}$}

\vskip .5cm

{\small
$^{\dag}$ Institute for Scientific Interchange Foundation, Villa Gualino, Viale Settimio Severo 75, 10131 Torino (Italy)\\
$^{\ddagger}$Dipartimento di Fisica Nucleare e Teorica, Universit\`a degli Studi di Pavia and Istituto
Nazionale di Fisica Nucleare, Sezione di Pavia, via A. Bassi 6, 27100 Pavia (Italy)\\
$^{\sharp}$ Dipartimento di Fisica, Politecnico di Torino, corso Duca degli Abruzzi 24, 10129 Torino (Italy)}\\
\end{center}

\vskip 1cm

\begin{abstract}
\noindent The spin network quantum simulator relies
on  the $su(2)$ representation ring
(or its $q$--deformed counterpart at $q=$ root of unity)
and its basic features naturally include (multipartite)
entanglement and braiding.
In particular,
$q$--deformed spin network automata are able to
perform efficiently approximate calculations
of topological invarians of
knots and $3$--manifolds.
The same algebraic background is shared
by $2D$ lattice models
supporting topological phases of matter that have recently
gained much interest in condensed matter physics.
These developments are motivated by the possibility
to store quantum information fault--tolerantly
in a physical system supporting fractional statistics
since a part of the associated Hilbert space is insensitive
to local perturbations.
Most of currently addressed approaches are framed within
a `double' quantum Chern--Simons field theory, whose quantum amplitudes
represent  evolution histories of local lattice degrees of freedom.

We propose here a novel combinatorial approach based
on `state sum' models of the Turaev--Viro type associated with
$SU(2)_q$--colored triangulations of the ambient $3$--manifolds.
We argue that boundary $2D$ lattice models (as well as observables in the form of colored graphs
satisfying braiding relations) could be consistently addressed.
This is supported by the proof that
the Hamiltonian of the Levin--Wen condensed string net model in a surface
$\Sigma$ coincides with the corresponding Turaev--Viro amplitude on $\Sigma \times [0,1]$
presented in the last section.
\end{abstract}

\vspace{15pt}

\noindent {\small
{\bf PACS}:
03.67.Lx (Quantum Computation);
11.15.--q (Gauge field theories);
04.60.Kz (Lower dimensional models in Quantum Gravity);
02.10.Kn (Knot theory);
02.20.Uw (Quantum Groups)}

\vspace{10pt}

\noindent {\small
{\bf MSC}:
81P68 (Quantum Computation and Cryptography);
57R56 (Topological Quantum Field Theories);
57M27 (Invariants of knots and $3$--manifolds)}

\vspace{40pt}

\noindent
---------------------------------------\\
(1) kadar@isi.it\\
(2) annalisa.marzuoli@pv.infn.it\\
(3) mario.rasetti@polito.it\\

\vfill
\newpage

\section{Braiding and entanglement in spin networks}

By {\em spin networks} we mean here
`computational' graphs the nodes and edges of
which are labeled by dimensions of $SU(2)$ irreducible
representations (irreps) and by $SU(2)$ recoupling transformations, respectively. For this reason
spin networks can be thought of as a generalized quantum computational framework
for dealing with unitary transformations
among (entangled) many--angular momenta states.
Actually, the computational space of the {\em spin network simulator}\cite{MaRa}
encodes in its very definition the representation ring
of the Lie group $SU(2)$ given by finite--dimensional Hilbert spaces
supporting irreps of $SU(2)$ endowed
with  two  operations, tensor product  $\otimes$
and direct sum $\oplus$ (providing a ring
structure over the field $\mathbb{C}$) as well as
unitary operators relating (multiple tensor products of)
such spaces.

Unlike the usual standard
circuit model\cite{NiCh}, here it is possible to handle  eigenstates
of $N$  (pairwise coupled) angular momenta labeled by
integers and half--integers $j_1,j_2,\ldots,j_N$ (in $\hbar$ units) and
not simply $N$--qubit states for which
 $j_1=j_2= \ldots =j_N=1/2$. The (re)coupling theory of $N$ $SU(2)$
 angular momenta provides the whole class of unitary transformations that can be
 performed on many--body quantum systems described by this kind of states\cite{BiLo9}.
 Any such general transformation --expressed in terms of a $3nj$--coefficient
($N=n+1$) when a choice of basis sets is made
explicit-- can be split into a finite sequence of
two basic unitaries, called {\em twist} (or `trivial braiding')
and {\em associator} in the language of tensor categories\cite{FuSc}.
The former acts on the (ordered) tensor product
of two Hilbert spaces $V, W$ supporting irreps of $SU(2)$ by swapping them, namely
\begin{equation}\label{twist}
T_{V,W}\,:\;V\,\otimes\,W\;\rightarrow\;W\,\otimes\,V\;\;\;
\text{with}\;\;\;T_{W,V} \circ T_{V, W}\,=\text{Id}_{V \otimes W}
\end{equation}
and its action
on a binary coupled state amounts to a trivial phase transform.\\
The associator $F$ relates different binary bracketing structures
in the triple tensor product of irreps $V,U,W$ (intertwiner)
 \begin{equation}\label{assoc}
F\,:\;(V\,\otimes\,U)\,\otimes\,W\;\rightarrow\;V\,\otimes\,(U\,\otimes\,W)
\end{equation}
and is implemented on any binary coupled state as a transformation involving one Racah--Wigner
$6j$ symbol.
Notice that both \eqref{twist} and
\eqref{assoc} are isomorphisms  but the associator reflects a physically
measurable modification of the way in which intertwiner spaces are coupled.
More complicated multiple
tensor products can be related by various combinations of
braidings and associators, so that the basic isomorphisms must satisfy
compatibility conditions, a so--called pentagon identity
and two hexagon identies\cite{FuSc} (in $SU(2)$ angular momentum theory
they correspond to the Biedenharn--Elliott identity and
the Racah identity, respectively\cite{BiLo9}).

The most effective way of dealing
with non--trivial braiding operators --to be used in connection
with the study of both braid group representations and braid statistics
(typically occuring in anyonic systems)-- is to move
to the representation ring $\mathfrak{R}$ $(SU(2)_q\,)$ (modular tensor category)
of the $q$--deformed
Hopf algebra of the Lie group $SU(2)$, $SU(2)_q$ ($q$ a root of unity).
Then the operation $T$ in \eqref{twist} turns out to be substituted
by a non--trivial braiding
\begin{equation}\label{braidmor}
R_{V,W}\,:\;V\,\otimes\,W\;\rightarrow\;W\,\otimes\,V \;\;\;
 \text{with}\;\; R_{W,V} \circ R_{V, W} \neq
\text{Id}_{V \otimes W}.
\end{equation}
The $q$--{\em deformed} spin network model of
computation\cite{GaMaRa1} is modelled on the $q$--representation ring
\begin{equation}\label{qcategory}
(\,\mathfrak{R} (SU(2)_q\,)\,;R\,;F\,),
\end{equation}
where $F$ denotes from now on the $q$--counterpart
of the associator \eqref{assoc}. Once suitable basis sets are
chosen, the isomorphisms $R$ and $F$ can be made
explicit. In particular, $F$ contains the
$q$--deformed counterpart of the $6j$--symbol and, regarding it
as a unitary matrix, it is also referred to as `duality' (or
`fusion') matrix in conformal field theories\cite{GoRuSi}.

The framework outlined above was exploited
in a series of papers\cite{GaMaRa1,GaMaRa2}, where families of
$q$--deformed spin network automata were implemented
for processing efficiently classes of computationally--hard problems
in geometric topology --in particular, approximate calculations
of topological invarians of links\cite{Jon} (collections of
knots) and $3$--manifold\cite{ReTu}. A prominent role was played there by
`universal' unitary braiding operators associated with
representations of the braid group in $\mathfrak{R}$ $(SU(2)_q\,)$.
Suitable traces of matrices
of these representations provide polynomial
invariants of $SU(2)_q$--colored links, while
weighted sums of the latter give topological
invariants of $3$--manifolds presented as complements
of  knots in the $3$--sphere.  These  invariants
are  in turn recognized as partition functions and vacuum
expectation values of physical observables in $3$--dimensional
Chern--Simons--Witten (CSW)  Topological Quantum Field Theory (TQFT)\cite{Wit1}.
The CSW environment actually provides not only a physical interpretation of such
quantities but it is universal in the sense that includes,
besides the quantum group interpretation quoted above,
monodromy representations of the braid group arising in
a variety of (boundary) conformal field theories\cite{GoRuSi}
(where point--like excitations confined in $2$--dimensional
regions evolve along braided worldlines).

It is worth mentioning that the $q$--spin network
approach represents  a naturally `discretized' version of the topological
framework for quantum computation proposed a few years ago
and further improved recently\cite{Fre}. In the light of some
basic questions raised by that paper, we are going to
outline a novel approach to the whole matter of topogical phases (section 2)
providing in particular a transparent combinatorial description
of condensed strings nets (section 3).

\section{$SU(2)_q$--colored  triangulations: $3D$ invariant partition functions and
induced $2D$ lattice models}

The issue of connections between (topological) gauge theories
in $3$ spacetime dimensions and $2D$
(integrable) lattice models\cite{Wit2}
has been intensively investigated over the years
in a variety of different contexts.
The renewed interest driven by the search for a fault--tolerant
quantum computer  based on manipulations of Non--Abelian quantum Hall
states\cite{Fre,toricc} represents a major challenge on both
experimental and theoretical grounds.
Most of the currently addressed approaches are framed within
a CSW environment, but
an {\em ab initio} discretized framework
could reveal much more effective, as briefly
outlined in the following.

The Turaev--Viro (TV) `state sum' model\cite{tv}
provides   a well--defined ({\em i.e.} finite)
topological invariant for any closed $3$--manifolds $M^3$.
Given a triangulation $T^3$ of $M^3$, namely a dissection into tetraheda,
 an $SU(2)_q$--coloring is assigned to it
according to a set of admissible initial data (not made explicit here).
A state functional for $T^3$ is  built, where the $N_0$ vertices,
$N_1$ edges and $N_3$
tetrahedra are suitably weighted. Finally, a summation over all
admissible colored triangulations of $M^3$
is performed, the resulting functional being invariant under
a set of combinatorial moves ensuring that its value depends
only on the topological type of the manifold $M^3$. The TV state sum reads
\begin{equation}\label{TVstsum}
\mathbf{Z}_{\,TV}\,[M^3;q]\,=\,\sum_{\{j\}}\;\mathbf{w}^{-N_0}\,
\prod_{A=1}^{N_1} \mathbf{w}_A
\,\prod_{B=1}^{N_3} \;
\begin{Bmatrix}
j_1 & j_2 & j_3 \\
j_4 & j_5 & j_6
\end{Bmatrix}_B \,,
\end{equation}
where the summation is over all colourings $\{j\}$ labeling irreps of $SU(2)_q$
\newline
($q=\exp\{2\pi i /r\}$, with $\{j=0,1/2,1 \dots, r-1\}$); $\mathbf{w}_A\doteq$
$(-1)^{2j_A}[2j_A+1]_q$
where $[\,]_q$ denote the quantum dimension of the irrep;
$\mathbf{w}=2r/(q-q^{-1})^2$ and $\{\,:::\}_B$ represents the $q-6j$ symbol
whose entries are associated with the six edges of tetrahedron $B$.

The invariant \eqref{TVstsum} equals the
square modulus of the Reshetikhin--Turaev invariant\cite{ReTu}, which in turn represents
the Chern--Simons partition function $\mathbf{Z}_{\,CS}$ for an {\em oriented} $3$--manifold
$\mathcal{M}^3$, namely
$\mathbf{Z}_{\,TV}\,[\mathcal{M}^3;q\,]\,\longleftrightarrow$
$|\,\mathbf{Z}_{\,CS}\,[\mathcal{M}^3;k\,]\,|^2\,,
$
where $k= 2(r-1)$ is the level of Chern--Simons functional.
This feature is crucial in view of applications
to topological phases, where  a `doubling'
of the basic CS setting is needed\cite{Fre}.

The TV model can be further extended to deal with both
$3$--manifolds with $2D$ boundary component(s) and
`observables' related to embedded links and (ribbon) graphs.
The extension to a pair $(M^3, \Sigma)$ with $\Sigma \equiv \partial
M^3$ can be carried out in two different ways,
depending on whether the boundary surface $\Sigma$
inherits a fixed triangulation\cite{ks}
or a `fluctuating' one\cite{CaMa}.
Observables in the form of colored graphs,
satisfying braiding relations, can be consistently
introduced\cite{ks,db} in any oriented
triangulated compact manifold $(M^3, \Sigma)$.

On the basis of such results we are currently addressing
a combinatorial reformulation of  theoretical
foundations of topological phases\cite{KaMaRa}.
A first result is discussed in the last section.

\section{Topological spin liquids: a combinatorial description}

We will now briefly introduce the condensed string nets of Levin and Wen\cite{levinwen} in $(2+1)$ dimension and
show that the projector to the ground state of the boundary model can be obtained from the corresponding Turaev--Viro
partition function $Z_{TV}(\Sigma\times [0,1],X)$ for the cylinder $M^3 = \Sigma\times [0,1]$
($\Sigma$ is oriented and $X$ denotes a fixed
identical triangulation on both $\Sigma\times\{0\}$ and $\Sigma\times\{1\}$).

Consider the string net model defined on the honeycomb lattice\cite{toricc}
with microscopic degrees of freedom associated to
the edges. The degrees of freedom are called string types and are endowed
with  $N+1$ labels and orientations. The change of
orientation of an edge is equivalent to the change of the label $i$ to its dual $i*$ and the $0$ label is
self--dual. The Hamiltonian is a sum of potential and kinetic energy. When the relative coefficient of the kinetic
piece is large, the ground state is a condensate of a dense string net. The universal long distance behaviour of these
nets is characterized by the so--called fixed--point wave functions on the space of all configurations. The
fixed--point wave functions $\Phi$ are in one--to--one correspondence with modular tensor categories \cite{Tur}
and obey the following rules (written by using the original notation\cite{levinwen},
although slightly different from the conventions in \eqref{TVstsum}):

\begin{align}
 \Phi
\bpm \includegraphics[height=0.3in]{Xi0.eps} \epm  =&
\Phi
\bpm \includegraphics[height=0.3in]{Xis0.eps} \epm
\label{topinv}
\\
 \Phi
\bpm \includegraphics[height=0.3in]{Xil0.eps} \epm  =&
d_i\Phi
\bpm \includegraphics[height=0.3in]{X0.eps} \epm
\label{clsdst}
\\
 \Phi
\bpm \includegraphics[height=0.3in]{XikljO.eps} \epm  =&
\delta_{ij}
\Phi
\bpm \includegraphics[height=0.3in]{XikliO.eps} \epm
\label{bubble}\\
 \Phi
\bpm \includegraphics[height=0.3in]{XijklmXO.eps} \epm  =&
\sum_{n}
F^{ijm}_{kln}
\Phi
\bpm \includegraphics[height=0.28in]{XijklnXO.eps} \epm
\label{fusion}
\end{align}

\noindent where the $F$ matrix and the quantum dimensions $d_i$ satisfy consistency conditions
given by

\begin{eqnarray}
F^{ijk}_{j^*i^*0} &=& \frac{v_{k}}{v_{i}v_{j}} \delta_{ijk} \nonumber \\
F^{ijm}_{kln} = F^{lkm^*}_{jin} &=& F^{jim}_{lkn^*} = F^{imj}_{k^*nl}
\frac{v_{m}v_{n}}{v_{j}v_{l}}
\nonumber \\
\sum_{n=0}^N
F^{mlq}_{kp^*n} F^{jip}_{mns^*} F^{js^*n}_{lkr^*}
&=&
F^{jip}_{q^*kr^*} F^{riq^*}_{mls^*}
\label{pent}
\end{eqnarray}
with $v_i=\sqrt{d_i}$ ($={\bf w}_i$ in TV notation) and $\delta_{ijk}=0,1$.
The latter is the branching rule: the triple $\{ijk\}$ is
admissible at a vertex when $\delta_{ijk}=1$. These rules coincide with those of Turaev and Viro\cite{tv} for
the evaluation of closed manifold invariants in three dimensions and the (first two lines of the) consistency
conditions encode the standard properties of the $q-6j$ symbol suitable normalized as
\beq
d_n \left\{\begin{array}{lll}i&j&m\\k&l&n\end{array}\right\}=F^{ijm}_{kln} ,
\label{norm}
\eeq
while the last relation in (\ref{pent}) is the Biedenharn--Elliott (pentagon) identiy ({\em cfr.}
section 1).

The Hamiltonian for the string--net model on the honeycomb lattice reads
\beq H=-\sum_I Q_I-\sum_p B_p\ , \eeq
where $I$ runs over vertices and $p$ over plaquettes.
$Q_I=\delta_{ijk}$ with $i,j,k$ being the colours of the edges
incident to the vertex $I$. The `magnetic constraints' are given by the sum $\sum_{s=0}^N a_s B_p^s$ with $B_p^s$ to
be described below. We consider the case when the coefficients are
\beq a_s=\frac{d_s}{\sum_{i=0}^N d_i^2}\ .\label{as}\eeq
This normalization prescription corresponds to the existence of smooth
continuum limit of the lattice model in one hand and gives the precise
weight of the Turaev--Viro evaluation on the other, as proved  below.

\vskip .5 cm

\subsection*{The magnetic constraints}

There is a nice intuitive way of determining the matrix elements of the magnetic constraints described in
appendix C\cite{levinwen}. In short, for each hexagon
of the lattice, one draws a smaller concentric hexagon with labels $s$ at every edge
in hexagon $p$, then {\bf (i)} connects the hexagons $p$ and $s$ with new edges
and labels them with $0$ (this does not change
$\Phi$) and {\bf(ii)} use the rules (\ref{fusion}) repeatedly until one reaches the single hexagon again.
The resulting coefficients define the matrix elements of the constraint given esplicitly by
\beq\begin{array}{l}
B^s_{p}\Big|\bmm\includegraphics[height=0.8in]{BsL0O.eps}\emm\Big\rangle=\\
{\displaystyle\sum_{g'h'i'j'k'l'}
F^{bg^*h}_{s^*h'g'^*}
F^{ch^*i}_{s^*i'h'^*}
F^{di^*j}_{s^*j'i'^*}
F^{ej^*k}_{s^*k'j'^*}
F^{fk^*l}_{s^*l'k'^*}
F^{al^*g}_{s^*g'l'^*}\Big|}
\bmm\includegraphics[height=0.8in]{BsL6O.eps}\emm\Big\rangle
\end{array}\eeq
The operators $B_p$ with the coefficients given by (\ref{as}) are projectors (as the $Q_I$'s) and all these constraint
operators commute with each other.

The image of the projector $P=\prod_q Q_q\prod_p B_p$ is the ground state of the model. We
are going to
show explicitly that
\beq P=Z_{TV}(\Sigma\times [0,1],X)\ ,\label{res} \eeq
where $\Sigma$ is the (oriented) surface on which the honeycomb lattice is defined
and X is the dual triangulation of
$\Sigma\times\{0\}\simeq\Sigma\times\{1\}\simeq\Sigma$.
To build the dual graph one places a vertex in the center
of each face and connects the new vertices in neighbouring faces.
In particular, the dual of the trivalent
honeycomb lattice is a triangular graph as shown in Fig.\ \ref{dgp}a).
\begin{figure}
\bc\includegraphics[width=3cm]{dgraph}
\includegraphics[width=7cm]{prismc}
\caption{\label{dgp} a) shows a part of the honeycomb lattice and its dual graph. In b) the `plane' of the
triangle $abc$ is the `plane' of the honeycomb lattice and there is another one with triangle
$a"b"c"$. The matrix element of the projector is interpreted as a three dimensional extrapolation between the bra
and the ket string net state given by a triangulation. The prism in the figure correspond to a part of the projector
$P$ where there is a tetrahedron for every associated $F$ symbol.}
\ec\end{figure}

Let us start constructing $\prod_p B_p$. Each operator $B_p^s$ contains six $F$ symbols corresponding to the six
vertices of a hexagon. We collect the three $F$ symbols from three different $B_p^s$ operators corresponding to a
given vertex. Their order does not matter since the $B$ operators commute. Choosing the order
$B_{p_1}^{s_1}B_{p_2}^{s_2}B_{p_3}^{s_3}$ ({\em cfr.}  Fig.\ \ref{dgp}a) we
get the contribution $F_{s_1 b' c'}^{acb}
F_{s_2 c" a'}^{b'ac'} F_{s_3 a" b"}^{c"b'a'}$. Note also that we have omitted denoting the
orientation for the sake of simplicity, but it is not difficult to incorporate it consistently. Comparing this
expression with Fig.\ \ref{dgp}b), we find that it corresponds to the triangulation of the prism with boundary
triangles $abc$ and $a"b"c"$. In this identification we have associated a $q-6j$ or $F$ symbol to every tetrahedron,
which is the prescription of the Turaev--Viro state sum (\ref{TVstsum}).

The second step consists in checking whether the gluing along the
triangulated boundary quadrilaterals can be made consistent with the algebraic pattern. Using the freedom of
multiplying the F symbols at a given vertex in arbitrary order (due to the commuting $B_p$ operators that they are
part of), the consistency can be achieved. It is clear now that we have a triangulation given by the dual graph with
edge labels $a"b"c"d"\dots$ and an identical one with labels $abcd\dots$ and these bound a $3$--dimensional
triangulation given by triangular prisms suitably glued together.

We now check whether the weights
associated to edges and vertices match that of the TV prescription.
Taking into account the normalisation (\ref{norm}) we find that there is a factor of $d_i$ associated to each internal
edge denoted by $a'b'c'd'\dots$. Because of the coefficient (\ref{as}) it is also true for the internal edges
`perpendicular' to $\Sigma$ labelled by $s_i$ in the figures (this weight comes from the numerator of $a_s$,
the denominator
will be associated to vertices bounding these edges). To the boundary edges with labels $a"b"c"d"\dots$ there is
also a $d_{a"},d_{b"}$ etc. associated, whereas to those with label $abcd\dots$ there is no non--trivial weight.
This is consistent with the TV prescription\footnote{For example we can introduce a black and white colour on
boundaries and glue only black to white boundary, that is one with $d_i$ weights to one with no weights of vertices.}.
The last type
of simplexes we have and not discussed yet are the boundary vertices. We associate the square root of the
denominator $(\sum_i d_i^2)^{-1/2}\equiv D^{-1}$
($={\bf w}^{-1}$ in TV notation) of the coefficient $a_s$ to the two boundary vertices connected by
the edge labeled by $s$. This way we covered both boundaries and found agreement with the TV prescription.

The last step is the sum over internal colouring, which has to be performed to get the partition function.
This corresponds to the summation over the labels $s_i$ and $a'b'c'd'\dots$, which
come from $\sum_i a_s B_p^s$ and the matrix
multiplication in $\prod_p B_p$. Finally, note, that the electric
constraints $\prod_q Q_q$ are taken care of by the product $\prod B_p$ as had
non--admissible labels met at a vertex, a corresponding $q-6j$ symbol gives
zero. This concludes the proof of (\ref{res}).

\vfill
\newpage


text:
\section{Vector Subspace Dimension One Less}
Tags: Linear Algebra

\begin{theorem}
Let $K$ be a [[Definition:Field (Abstract Algebra)|field]].
Let $M$ be a [[Definition:Vector Subspace|subspace]] of the [[Definition:Dimension of Vector Space|$n$-dimensional]] [[Definition:Vector Space|vector space $K^n$]].
The following statements are equivalent:
:$(1): \quad \map \dim M = n - 1$
:$(2): \quad M$ is the [[Definition:Kernel of Linear Transformation|kernel]] of a nonzero [[Definition:Linear Form|linear form]]
:$(3): \quad$ There exists a [[Definition:Sequence|sequence]] $\sequence {\alpha_n} $ of [[Definition:Scalar (Vector Space)|scalars]], not all of which are zero, such that:
:::$M = \set {\tuple {\lambda_1, \ldots, \lambda_n} \in K^n: \alpha_1 \lambda_1 + \cdots + \alpha_n \lambda_n = 0}$
Also, suppose the above hold.
Let $\sequence {\beta_n}$ be a [[Definition:Sequence|sequence]] of [[Definition:Scalar (Vector Space)|scalars]] such that:
:$M = \set {\tuple {\lambda_1, \ldots, \lambda_n} \in K^n: \beta_1 \lambda_1 + \cdots + \beta_n \lambda_n = 0}$
Then there is a non-zero [[Definition:Scalar (Vector Space)|scalar]] $\gamma$ such that:
:$\forall k \in \closedint 1 n: \beta_k = \gamma \alpha_k$
\end{theorem}

\begin{proof}
Let $M^\circ$ be the [[Definition:Annihilator on Algebraic Dual|annihilator]] of $M$.
Let $N = M^{\circ}$.
By [[Results Concerning Annihilator of Vector Subspace]], $N$ is [[Definition:Dimension of Vector Space|one-dimensional]] and $M = \map {J^{-1} } {N^\circ}$.
Let $\phi \in N: \phi \ne 0$.
Then $N$ is the set of all [[Definition:Scalar Multiplication on Vector Space|scalar multiples]] of $\phi$.
Because:
:$\map {J^{-1} } {N^\circ} = \set {x \in K^n: \forall \psi \in N: \map \psi x = 0}$
it follows that $\map {J^{-1} } {N^\circ}$ is simply the [[Definition:Kernel of Linear Transformation|kernel]] of $\phi$.
Hence $(1)$ implies $(2)$.
By [[Rank Plus Nullity Theorem]], $(2)$ also implies $(1)$.
{{qed|lemma}}
Suppose $\sequence {\alpha_n}$ is any sequence of [[Definition:Scalar (Vector Space)|scalars]].
Let $\sequence {e'_n}$ be the [[Definition:Ordered Basis|ordered basis]] of $\paren {K^n}^*$ [[Definition:Ordered Dual Basis|dual to]] the [[Definition:Standard Ordered Basis|standard ordered basis]] of $K^n$.
Let $\displaystyle \phi = \sum_{k \mathop = 1}^n \alpha_k e'_k$.
Then, by simple calculation:
:$\map \ker \phi = \set {\tuple {\lambda_1, \ldots, \lambda_n}: \alpha_1 \lambda_1 + \cdots + \alpha_n \lambda_n = 0}$
{{explain|Prove the above.}}
It follows that:
:$\phi \ne 0 \iff \exists k \in \closedint 1 n: \alpha_k \ne 0$
Thus $(2)$ and $(3)$ are equivalent.
{{qed|lemma}}
Suppose $M = \map \ker \psi$, where $\displaystyle \psi = \sum_{k \mathop = 1}^n \beta_k e'_k$.
Then $\psi = M^\circ$.
As $M^\circ$ is one-dimensional and since $\psi \ne 0$, it follows that:
:$\exists \gamma \ne 0: \psi = \gamma \phi$
Therefore:
:$\forall k \in \closedint 1 n: \beta_k = \gamma \alpha_k$
{{qed}}
\end{proof}

config: wiki


TITLE: Entropy generation in an engine
QUESTION [1 upvotes]: I'm trying to distinguish between maximum and minimum entropy generation by an engine, and ran into a conundrum.
The total entropy generation is found by applying an entropy balance to the red control volume in the figure:
$ S_{gen} = \frac{Q_L}{T_L} - \frac{Q_H}{T_H}$
Here, it looks like $S_{gen}$ is maximum when $Q_H = 0$.
Now consider another entropy balance on a control volume around the top "hot" resistance:
$(S_{gen})_H = \frac{Q_H}{T_{H'}} - \frac{Q_H}{T_H}$
where $T_{H'}$ is the temperature at the end of the resistance.
This implies that $(S_{gen})_H = 0$ if $Q_H = 0$.
This is a major conundrum of our previous result with the larger control volume, that says $(S_{gen})_H$ is maximum when $Q_H = 0$.
What is the source of my confusion?

REPLY [1 votes]: I think that this is an interesting set-up.  For ideal infinite reservoirs and an ideal engine, at least at steady state, all the entropy generation occurs in the two resistances (although the generated entropy in the resistances is then continually transferred to the working fluid and reservoirs).
You correctly determined that the rate of entropy generated in the hot resistance is given by $$(\dot{S}_{gen})_H=\left(\frac{1}{T_{H'}}-\frac{1}{T_H}\right)\dot{Q}_H=\frac{T_H-T_{H'}}{T_HT_{H'}}\dot{Q}_H\tag{1}$$And in the cold resistance, it is given by$$(\dot{S}_{gen})_L=\left(\frac{1}{T_{L}}-\frac{1}{T_{L'}}\right)\dot{Q}_L=\frac{T_{L'}-T_{L}}{T_LT_{L'}}\dot{Q}_L\tag{2}$$The total rate of entropy generation in the universe is obtained by summing these equations to yield:$$\dot{S}_{gen}=(\dot{S}_{gen})_H+(\dot{S}_{gen})_L=\frac{\dot{Q}_L}{T_L}-\frac{\dot{Q}_H}{T_H}\tag{3}$$
From the heat conduction equation, we also know that $$\dot{Q}_H=kA\frac{T_H-T_{H'}}{\Delta L}\tag{4}$$where $\Delta L$ is the thickness of the resistance, A is the area for heat flow, and k is the thermal conductivity.  If we substitute this into Eqn. 1, we obtain:$$(\dot{S}_{gen})_H=kA\Delta L\frac{[(T_H-T_{H'})/\Delta L]^2}{T_HT_{H'}}\tag{5}$$This shows that the rate of entropy generation is roughly proportional to the square of the temperature gradient within the resistance.  A similar relation can be written for the cold resistance.


TITLE: How to express $z^8 − 1$ as the product of two linear factors and three quadratic factors
QUESTION [2 upvotes]: Verify
$$(z-e^{i \theta} ) (z - e^{-i \theta} ) ≡ z^2 - 2\cos \theta + 1$$
Hence express $z^8 − 1$ as the product of two linear factors and three quadratic factors, where all
coefficients are real and expressed in a non-trigonometric form.

For the first part I just expanded  the LHS and showed its equal to RHS.
Roots of $z^8-1$:
$$z= e^{i \frac{\pi k}{4}}$$

Where $k=0,1,2,3,4,5,6,7$

Ok since they want non-Trignometric so :
I know two roots, which are obvious:
$$z=1,-1$$
$$z^8-1=(z-1)(z+1)(z^6+z^5+z^4+z^3+z^2+z+1)$$
They want two linear factors which I believe I have found: $(z+1)$ and $(z-1)$
How do I make
$$(z^6+z^5+z^4+z^3+z^2+z+1)$$
to three quadratic factors? And they mentioned Hence , so I have to use the identity I verified. Please help.

REPLY [1 votes]: Here is a method which make use of your identity.
$$z^8-1=(z-1)(z+1)(z^2+1)(z^4+1)$$
\begin{align*}
(z^4+1)&=(z^2+i)(z^2-i)\\
&=(z^2-e^{i-\frac{\pi}{2}})(z^2-e^{i\frac{\pi}{2}})\\&=(z+e^{-i\frac{\pi}{4}})(z-e^{-i\frac{\pi}{4}})(z+e^{i\frac{\pi}{4}})(z-e^{i\frac{\pi}{4}})\\&=(z-e^{-i\frac{5\pi}{4}})(z-e^{i\frac{5\pi}{4}})(z-e^{i\frac{\pi}{4}})(z-e^{-i\frac{\pi}{4}})\end{align*}
Then use your formula, you get $$z^4+1=(z^2+\sqrt2 z+1)(z^2-\sqrt2 z+1)$$


TITLE: minimal polynomial of $7^{th}$ root of unity
QUESTION [1 upvotes]: Let $\alpha=\zeta_7+\zeta_7^2+\zeta_7^4$. What's the minimal polynomial of $\zeta_7$ over $\mathbb{Q}(\alpha).$

I have managed to show that $[\mathbb{Q}(\alpha):\mathbb{Q}]=2$ and $[\mathbb{Q}(\zeta):\mathbb{Q}]=6$. By tower rule, the minimal polynomial has degree 3. As long as we find a polynomial with degree 3 such that $\zeta$ is a root, then we are done. But I got stuck in finding such polynomial. Any hint is appreciated.

REPLY [3 votes]: Following you're analysis of the indices, the Galois group of the extension $\mathbb{Q}(\zeta_7)\supseteq \mathbb{Q}(\alpha)$ is a (the) subgroup of order $3$ of $\text{Gal}(\mathbb{Q}(\zeta_7)/\mathbb{Q})\cong (\mathbb{Z}/7\mathbb{Z})^\times\cong\mathbb{Z}/6\mathbb{Z}$. This subgroup is the cyclic subgroup generated by the element $$
\sigma\in\text{Gal}(\mathbb{Q}(\zeta_7)/\mathbb{Q}), \,\,\,\sigma(\zeta_7)=\zeta_7^2
$$
A great trick which often helps in finding polynomials (or really any type of object in the context of galois/representation theory) which are invariant under the action of some group of transformations is taking the 'average'.
More explicitly, if we're interested in finding a polynomial in $\mathbb{Q}(\alpha)[x]=(\mathbb{Q}(\zeta_7))^{\langle \sigma \rangle}[x]$ which has $\zeta_7$ as a root, we can construct it by taking the product of the orbit of $(x-\zeta_7)$ under $\langle \sigma \rangle$: $$
p(x)=(x-\zeta_7)(x-\sigma(\zeta_7))(x-\sigma^2(\zeta_7))=(x-\zeta_7)(x-\zeta_7^2)(x-\zeta_7^4)
$$ Because of the construction, clearly $p(x)$ is invariant under the action of $\langle \sigma \rangle$ (thought of as an automorphism of the ring $\mathbb{Q}(\zeta_7)[x]$ by acting on coefficients) and hence the coefficients of the polynomial $p(x)$ must all lie in $(\mathbb{Q}(\zeta_7))^{\langle \sigma\rangle}=\mathbb{Q}(\alpha)$.
At this point, I think it's quite satisfying to verify this last statement directly:$$
p(x)=(x-\zeta_7)(x-\zeta_7^2)(x-\zeta_7^4)= $$ $$=x^3-(\zeta_7+\zeta_7^2+\zeta_7^4)x^2+(\zeta_7^3+\zeta_7^5+\zeta_7^6)x-1=x^3-\alpha x^2 + \frac{\alpha^2-\alpha}{2}x-1.
$$ And low and behold the coefficients in fact are in $\mathbb{Q}(\alpha)$.
Hope this helps :)
POST: bare in mind that this is all possible because $\mathbb{Q}(\zeta_7)\supseteq \mathbb{Q}$ is Galois.


TITLE: Effect of black coating on the inside of a hobby telescope's hull?
QUESTION [1 upvotes]: I was watching this (rather old) episode of The Making, where they show how a telescope is built in a factory. At 7:34, the cylindrical body of the telescope is shown to have a dark coating on the inside and the video explains that this prevents internal reflection of the light received from whatever you're observing.
I saw an interesting comment from Izumikawa Fukumi who asked if the telescope would become more accurate if they coated the inside with the blackest black in the world. How would such a dark coating affect the telescope's performance?

REPLY [2 votes]: Reflections from the wall striking the eyepiece would reduce the contrast in the image you are trying to observe.  It wold be like the effect of city lights reflected from the atmosphere when you are trying to observe the stars.


TITLE: Move an object a percentage every frame ("Zeno's Arrow" style)
QUESTION [0 upvotes]: Every frame, I want an object to move a percentage of the way from where it is to its target. A value of 1 means it reaches the target in exactly one frame, and a value of 0.5 means in one frame it has gone 0.5 way to its target, the next frame 0.75, etc.

(diagram stolen from this similar question)
Since we are moving at fixed frames (not infinitesimals), barring floating-point precision, the object will never reach the target.
The code for this is easy. Here it is in C# pseudocode:
void onEveryFrame() {
    currentX += (targetX - currentX) * speed;
}

The problem is, How do I make this frame-rate independent?
So someone running at 30 frames per second will see the object move at about the same speed as someone running at 60 frames per second?

REPLY [0 votes]: The answer by Carser gave me almost everything I needed. The following function returns a value from 0 (at he starting position) to 1 (at the target), with $t$ representing the current number of seconds elapsed.
$$
x = 1 - \left(\frac{1}{2} \right)^t
$$
There are few things we need to add for the equation to work in our code:
$$
x = x_0 + \Delta x * \left( 1 - \left(1-rate \right)^t \right)
$$
In the equation, $x_0$ is our starting value, $\Delta x$ is the total distance to the target (in our case targetX - startX), and $rate$ is what I called speed in my original answer, renamed for clarity.
The $rate$ variable determines how close to the target we get each second. A value of 0.5 means every second we are half as close to the target as the previous second. A value of 1 means we are "instantly teleported" to our target in that second, and a value of 0 means we never reach the target.
And for some actual code, the following should be valid in C#, and fully framerate independent (even supports reversing time, though, you might get some funny results if you go into negative time):
x = startX + (targetX - startX) * (1 - Math.Pow(1-rate, currentTime - startTime));


text:
\section{Introduction}

	This article aims to study the Hausdorff dimension of certain orbits of diagonalizable flows on homogeneous spaces with prescribed recurrence behavior.
    This problems has origins in the theory of diophantine approximation. To properly motivate our results, we briefly survey the history of the subject.

    In~\cite{Dani-Bounded,Dani-Bounded2}, Dani studied the problem of bounded orbits in two settings: orbits of diagonalizable flows on homogeneous spaces of rank $1$ Lie groups and orbits in $\mrm{SL}(m+n,\R)/\mrm{SL}(m+n,\Z)$ of the form $g_t u_Y \G$, where $\G = \mrm{SL}(m+n,\Z)$ and for $t\in \R$ and $Y\in M_{m,n}$ an $m\times n$ real matrix,
    \begin{equation} \label{Dani's thrms}
    g_t = \mrm{diag}(e^{t/m},\dots,e^{t/m},e^{-t/n},\dots,e^{-t/n}),\quad u_Y = \begin{pmatrix}
    		\mathrm{I}_m & Y \\ \mathbf{0} & \mathrm{I}_n \end{pmatrix}.
    \end{equation}
    We refer to $g_t$ as a diagonal element with weight $(1/m,\dots,1/m,1/n,\dots,1/n)$.
    It is shown that bounded orbits of diagonalizable flows on rank $1$ homogeneous spaces have full Hausdorff dimension.
    It is also shown that orbits of the form $(g_t u_Y \G)_{t\geqslant 0}$ are bounded if and only if $Y$ is badly approximable, i.e., there exists $\d>0$ such that for all $(\mbf{p},\mbf{q}) \in \Z^m\times \Z^n$, with $\mbf{q} \neq 0$,
    \begin{equation*}
    	\norm{Y \mbf{q} -\mbf{p} }^n \norm{\mbf{q}}^m > \d.
    \end{equation*}
	Using the results of Schmidt on badly approximable systems of linear forms~\cite{Schmidt-BadLinearSystems}, this implies that bounded orbits for $g_t$ as in~\eqref{Dani's thrms} have full dimension.
    These results were generalized in~\cite{KleinbockWeiss-SchmidtSLn+m,KleinbockWeiss-SchmidtExpanding} where bounded orbits of \emph{non-quasiunipotent} flows were shown to have full dimension.

    All of these results were obtained by showing that bounded orbits are winning for variants of a game invented by Schmidt in~\cite{Schmidt-Games}.
    The winning property is much stronger than having full Hausdorff dimension since it is stable under countable intersections and implies thickness, i.e., the intersection of a winning set with any non-empty open set has full dimension. We refer the reader to~\cite{KleinbockWeiss-SchmidtSLn+m} for more details on Schmidt's original game as well as a new variant introduced by the authors.
    More recently, far reaching generalizations of these results were obtained in~\cite{BadziahinEtal-SchmidtConjecture}, in particular settling an old conjecture of Schmidt on the intersection of sets of weighted badly approximable vectors with different weights.

    Dani also studied the existence and classification of divergent orbits of diagonalizable flows on homogeneous spaces in~\cite{Dani-Divergent}.
    Among the results obtained by Dani is the fact that divergent orbits on non-compact homogeneous spaces of a rank $1$ Lie group $G$ are \emph{degenerate}, i.e., can be detected using the behavior of finitely many vectors in some fixed representation of $G$.
    In particular, the set of divergent orbits consists of a countable collection of immersed submanifolds in $G/\G$.
    This result also holds for quotients of Lie groups by arithmetic lattices of rational rank $1$.
    By contrast, quotients by higher rank arithmetic lattices always admit non-degenerate divergent orbits~\cite{Dani-Divergent,Weiss-Divergent}.

    In a landmark paper, the precise Hausdorff dimension of divergent orbits under the flow induced by $g_t$ in~\eqref{Dani's thrms} was calculated when $(m,n)=(2,1)$ in~\cite{Cheung-SingularPairs}. This result was extended in~\cite{CheungChevallier} to the case when $\min(m,n)=1$.
    These results build on earlier ideas of Cheung in~\cite{Cheung-Sl2products} where the Hausdorff dimension of divergent orbits in $\mrm{SL}(2,\R)^n/\mrm{SL}(2,\Z)^n$ for $n\geq 2$ under the flow induced by a diagonal matrix in each coordinate was determined to be $3n-1/2$.
    In~\cite{KKLM-SingSystems}, a sharp upper bound on the dimension of divergent orbits for general $m$ and $n$ was obtained by different methods.
    The proof in~\cite{KKLM-SingSystems} relies on the powerful technique of systems of integral inequalities introduced in~\cite{EskinMargulisMozes} in the context of quantifying Margulis' work on the Oppenheim conjecture.

    Parallel to these developments and motivated by problems in diophantine approximation, the study of the evolution of curves on homogeneous spaces under diagonal flows attracted a lot of interest.
    In~\cite{KleinbockMargulis}, Kleinbock and Margulis showed that the push-forward of certain ``non-degenerate" smooth curves in the group $\set{u_Y:Y \in M_{1,n}}$ by diagonal elements similar to $g_t$ in~\eqref{Dani's thrms} do not diverge in $\mrm{SL}(n+1,\R)/\mrm{SL}(n+1,\Z)$.
    This allowed them to settle a conjecture due to Baker and Sprind\v{z}uk showing that the Lebesgue measure of very well approximable vectors belonging to such curves is $0$.
    This result has been generalized in numerous directions, cf.~\cite{KleinbockLindenstraussWeiss,BeresnevichKleinbock-WeakNonPlanar,Aka-ExtremalCriterion} for notable examples.

    In~\cite{Shah-Inventiones,Shah-JAMS}, using Ratner's theorems and the linearization technique, Shah extended the results of Kleinbock and Margulis by showing that the push-forwards of the parameter measure on these curves, in fact, become equidistributed towards the Haar measure on $G/\G$.
    These results build on earlier work of Shah in~\cite{Shah-Duke1,Shah-Duke2} where the push-forwards of certain smooth curves on the unit tangent bundle of hyperbolic manifolds by the geodesic flow were shown to be equidistributed towards the Haar measure.

    On the other hand, the problem of determining the Hausdorff dimension of bounded and divergent orbits restricted to curves as above is far less understood.
    In a breakthrough article, Beresnevich showed in~\cite{Beresnevich-BadCurves} that the Hausdorff dimension of finite intersections of weighted badly approximable vectors on non-degenerate analytic curves in $M_{1,n}$ is full.
	By means of Dani's correspondence, this implies that bounded orbits of diagonal elements similar to $g_t$ in~\eqref{Dani's thrms} with more general weights than $(1,1/n,\dots,1/n)$ starting from points on curves on the group $\set{u_Y:Y \in M_{1,n}}$ is equal to $1$.
    We refer the reader to~\cite{Beresnevich-BadCurves} for more on the history of this problem and to~\cite{AnBeresnevichVelani-WinningPlanarCurves} where these bounded orbits were shown to be in fact winning in the sense of Schmidt for planar curves.

    The dimension of bounded orbits starting from curves on homogeneous spaces other than $\mrm{SL}(n,\R)/\mrm{SL}(n,\Z)$ was studied in~\cite{Aravinda} in rank $1$ homogeneous spaces and in~\cite{Einsiedler-CurvesNumfields} in quotients of $\mrm{SL}(2,\R)^r\times \mrm{SL}(2,\C)^s$ by irreducible lattices.
	To the best of our knowledge, the problem of the dimension of divergent orbits starting from curves has not been previously addressed in the literature.

    The purpose of this article is to provide an abstract framework to study the recurrence of orbits of diagonalizable flows starting from curves on homogeneous spaces.
    We verify our hypotheses in the setting of homogeneous spaces of products of real rank $1$ Lie groups (Theorem~\ref{thrm: rank 1 DOA}), a more general class of curves on homogeneous spaces of products of $\mrm{SO}(n,1)$ (Theorem~\ref{thrm: products of sl2} and~\ref{thrm: products of so(n,1)}), and actions of $\mrm{SL}(2,\R)$ on any homogeneous space of finite volume (Theorem~\ref{thrm: sl2 actions}).
    Curves on more general arithmetic homogeneous spaces are studied in Section~\ref{section: CH higher rank}.

    In all these cases, we provide an upper bound on the dimension of divergent on average trajectories (Definition~\ref{defn: divergence on average}) and show that bounded orbits are winning for a Schmidt game on intervals of the real line (see Section~\ref{section: schmidt games} for detailed definitions).
    Moreover, we establish, in a quantitative form, the non-divergence of expanding push-forwards of shrinking curve segments (Proposition~\ref{propn: non-divergence of shrinking curves}).

	In Section~\ref{section: applications}, we present applications of our results to problems in intrinsic diophantine approximation on spheres (Corollary~\ref{cor: dioph approx on Sn}), diophantine approximation by number fields (Corollary~\ref{cor: dioph approx in num fields}), and diophantine properties of lines in the space of square systems of linear forms $M_{n,n}$ (Corollary~\ref{cor: dioph approx of linear forms}).

config: arxiv


text:
\begin{document}
\title{A Szeg\H o type theorem and distribution of symplectic
eigenvalues}
\author{Rajendra Bhatia}
\address{Department of Mathematics, Ashoka
University, Rajiv Gandhi Education City, P.O.Rai, Sonepat, Haryana 131 029, India.}
\email[Rajendra Bhatia]{\href{mailto:rajendra.bhatia@ashoka.edu.in}{rajendra.bhatia@ashoka.edu.in}}
\author{Tanvi Jain}
\address{Theoretical Statistics and Mathematics Unit, Indian
Statistical Institute, Delhi Centre, 7 S J S Sansanwal Marg,
New Delhi 110 016, India.}
\email[Tanvi Jain]{\href{mailto:tanvi@isid.ac.in}{tanvi@isid.ac.in}}
\author{Ritabrata Sengupta}
\address{Department of Mathematical Science, Indian Institute of
Science, Education, \& Research (IISER), Berhampur, Transit campus -
Govt. ITI, NH 59, Berhampur 760 010, Ganjam, Odisha, India.}
\email[Ritabrata
Sengupta]{\href{mailto:rb@iiserbpr.ac.in}{rb@iiserbpr.ac.in}}

\begin{abstract}
\par We study the properties of stationary
G-chains in terms of their generating functions. In
particular, we prove an analogue of the Szeg\H o limit theorem for symplectic eigenvalues,
derive an expression for the entropy rate of stationary quantum Gaussian processes,
and study the distribution of symplectic eigenvalues of truncated block Toeplitz matrices.
We also introduce a concept of symplectic numerical range, analogous to that of numerical range,
and study some of its basic properties, mainly in the context of block Toeplitz operators.

\smallskip
\noindent \textbf{Keywords.} Symplectic eigenvalue,
symplectic numerical range, Szeg\H o limit theorem,
Gaussian state, stationary Gaussian chain, entropy rate.

\smallskip

\noindent \textbf{Mathematics Subject Classification
(2010):} 81P45, 94A15, 94A17, 54C70.
\end{abstract}
\thanks{The work of RB is supported by a Bhatnagar Fellowship of the CSIR.
TJ acknowledges financial support from {\sf SERB
MATRICS grant number MTR/2018/000554}. RS acknowledges
financial support from {\sf SERB MATRICS grant number
MTR/2017/000431} and {\sf DST/ICPS/QuST/Theme-2/2019/General Project number Q-90}.
}
\maketitle

\maketitle

\section{Introduction}
\par A quantum  state $\rho$  in a bosonic
Fock space $\Gamma(\mathbb{C}^k)$ is a positive semidefinite
operator with trace one. Let  $q_1, p_1;
\ldots;q_k, p_k$ be $k$ pairs of position-momentum
observables of a quantum system with $k$ degrees of
freedom satisfying the canonical commutation relations. We
introduce the observables $(X_1, X_2, \ldots, X_{2k-1},
X_{2k}) = (q_1, p_1, \ldots, q_k, p_k)$. Then if $\rho$ has
finite second moments, we write the covariance
matrix of $\rho$ as $A=[\mathrm{Cov}_\rho (X_i, X_j) ]_{i,j=1}^{2k},$ where
\[\mathrm{Cov}_\rho (X_i,X_j)
= \tr \frac{1}{2} (X_1X_2+X_2X_1)\rho -(\tr X_1\rho)(\tr X_2
\rho).\]
The complete Heisenberg uncertainty principle for
all the position and momentum observables assumes the form of
the following matrix inequality:
\begin{equation}\label{eqb1}
A+\frac{\imath}{2}J_{2k} \ge0,
\end{equation}
where $J_{2k}= {\underset{k\,
\textrm{times}}{\underbrace{J_2\oplus\cdots\oplus J_2}}}$
with $J_2=\begin{bmatrix}0 & 1\\ -1 & 0\end{bmatrix}.$

\par Following the terminology in \cite{krprb2}, we call
a real $2k \times 2k$ positive definite matrix $A$ satisfying inequality
\eqref{eqb1} a \emph{G-matrix}.
A standard result in quantum theory states that a $k$-mode,
mean zero, Gaussian quantum state  is uniquely represented by
its covariance matrix, which is a G-matrix. Conversely any
$2k \times 2k$ G-matrix is the
covariance matrix of a unique (up to permutation) $k$-mode mean zero quantum
Gaussian state in the Fock space $\Gamma(\mathbb{C}^k)$;  \cite{krpg,
MR2986302}. Finite mode quantum Gaussian states and quantum
Gaussian processes have been extensively studied in quantum
optics, quantum probability, and quantum information -  both
in theory as well as in experiments.  A comprehensive
survey of Gaussian states and their properties can be found in the two
books of Holevo \cite{holevo1, MR2986302}. For their applications to
quantum information theory the reader is referred to the
survey article by Weedbrook et al \cite{RevModPhys.84.621},
Holevo's book \cite{MR2986302}, and the new book of
Serafini \cite{Sarafini}.

\par In the present paper, our concern is with a stationary
quantum Gaussian process. This is a chain of
finite mode  ($k$ mode) quantum Gaussian states exhibiting
stationarity. Let $\{\rho_n\}$ be a chain of
quantum Gaussian states with covariance matrices $\{T_n\}$.
The stationarity property means that each $T_n$ is a
positive definite block Toeplitz matrix such that $T_n$ is
the leading principal sub-matrix of $T_{n+1}$. This
sequence $\{T_n\}$ gives rise to an infinite block Toeplitz
matrix $\Sigma$. We call this chain $\{\rho_n\}$ of quantum
Gaussian states a \emph{stationary} quantum Gaussian process
and the infinite matrix $\Sigma$  a \emph{G-chain}
\cite{krprb2}. Thus a G-chain $\Sigma$ is an infinite block Toeplitz matrix.
The classical
version of such objects has been well studied in
probability theory. (See for instance \cite{MR543837}.) A study
of the quantum version has been initiated in
\cite{krprb2, krprb4}. In order to study G-chains, we need to study properties of infinite block Toeplitz matrices with blocks of size $2k \times 2k$.

Every leading $n \times n$ principal block sub-matrix gives a
covariance matrix of an $nk$-mode quantum Gaussian state.
Toeplitz matrices play an
important part in the study of stationary processes in
classical probability theory as well. See, e.g.,
Grenander and Szeg\H o \cite{grenander-szego}.

\par  Among real positive definite matrices, G-matrices are characterised by a simple property of their symplectic eigenvalues.
{\it Williamson's theorem} \cite{zbMATH03035616} tells us that
for every $2k\times 2k$ real positive definite matrix $A,$
there exists a symplectic matrix $M$ such that
\[MAM^T= d_1(A) I_2 \oplus \cdots \oplus d_k(A) I_2,\]
where $d_1(A)\le \cdots\le d_k(A)$ are positive numbers uniquely determined by $A$.
These are uniquely determined by $A.$
We call these numbers the {\it symplectic eigenvalues}
of $A.$
 We can see that a matrix $A$ is a
G-matrix if and only if all its symplectic eigenvalues
$d_j(A)\ge \frac{1}{2}.$  There has recently been
considerable interest in the study of various properties of
symplectic eigenvalues (see for instance
\cite{bhatia-jain,  gosson, MR2391195, PhysRevA.73.012330}), due to
their close connection with quantum optics and
thermodynamics \cite{MR2986302, mandel_wolf}.
\par Given a $k$-mode quantum Gaussian state with covariance
matrix $A$, the {\it von Neumann entropy} of the state is
given by
\begin{equation}\label{eqn1}
S(A)= \sum_{j=1}^k \frac{2d_j(A)+1}{2} H\left( \frac{2d_j(A)
-1}{2d_j(A)+1} \right),
\end{equation}
where $H$ is the Shannon entropy function given by $H(t) =
-t \log t - (1-t) \log (1-t), \, 0 \le t \le 1,$ and $H(0)
=H(1) =0.$ See \cite{MR3800794, krp9}, or \cite{Sarafini} pages 61 -- 62.
Let $T_n$ be the covariance matrix of a
$k$-mode stationary quantum Gaussian process, truncated at
level $n.$ The \emph{entropy rate} of the process is defined as
\[\lim_{n \to \infty} \frac{S(T_n)}{n}.\]
An important problem in information theory has been the
study of the entropy rate of any given stationary process. This
can be very complicated \cite{coverthomas, gray-book}.
The entropy rate for a certain type of stationary quantum Gaussian
process was
calculated in \cite{krprb2}.
We compute the entropy rate for a more general class, namely, the class of bounded partially symmetric stationary quantum Gaussian processes.
Let $\Sigma=\begin{bmatrix}A_{i-j}\end{bmatrix}$ be a G-chain corresponding to a stationary quantum Gaussian process.
We
 call this process {\it bounded} if $\Sigma$ is a bounded operator on $l^2_{2k}$ (the space of square summable sequences of elements of $\mathbb{C}^{2k}$).
In this case $\Sigma$ is a Toeplitz operator generated by a matrix symbol $\wt{A}$ in $L^\infty_{2k\times 2k}.$
The process is {\it partially symmetric} if $A_{-n}=A_n$ for all $n\in\mathbb{N}.$
We show that a stationary quantum Gaussian process is partially symmetric and bounded if and only if
its corresponding G-chain is generated by an $\wt{A}$ in $L^\infty_{2k\times 2k}$ such that $\wt{A}(\theta)$ is a G-matrix for almost all $\theta.$
The computation of the entropy rate requires a study of the
distribution of symplectic eigenvalues of block Toeplitz
matrices. To achieve this  we
prove a symplectic analogue of a fundamental
theorem for the distribution of eigenvalues of
Toeplitz matrices, well-known as the Szeg\H o limit theorem
\cite{grenander-szego, MR0059482}. The classical Szeg\H o theorem can
be stated as follows: \emph{ Suppose $\varphi: (-\pi, \pi)
\to \mathbb{R}$ is an essentially bounded function, and
$(T_n)$ is
the sequence of Hermitian Toeplitz matrices generated by
$\varphi.$
Then for every function $f,$ continuous on the interval}
$[\textrm{essinf}~ \varphi,\, \textrm{esssup} ~\varphi],$ \emph{one has
\begin{equation}\label{eqn2}
\lim_{n \to \infty} \frac{1}{n}
\sum_{j=1}^n f(\lambda_j(T_n)) = \frac{1}{2\pi}
\int_{-\pi}^\pi f(\varphi(x)) \dd x,
\end{equation}
where $\lambda_j(T_n), \, j = 1, 2, \cdots, n,$ are the
eigenvalues of $T_n.$ }  Many different versions and  proofs
of this theorem are available in the
literature \cite{bs, MR0270218,  simon-ortho-1, tilli-1, tilli-2, MR761763,
Tyrtyshnikov, MR0409511, MR0409512}.

We prove an analogue of this
theorem for symplectic eigenvalues,
and apply this to compute the entropy rate and to
study the distribution of symplectic eigenvalues of block Toeplitz matrices.

In particular we prove that the union of the set of all
symplectic eigenvalues of truncated $n \times n$ block
Toeplitz matrices $T_n(\wt{A})$ is dense in the set of all
symplectic eigenvalues of $\wt{A}(\theta)$ where
 $\wt{A}(\theta)$   varies over  the essential range of $\wt{A}.$

\par In classical operator theory, the numerical range is
an important and useful concept. We introduce an analogous notion
of the symplectic numerical range  and study its basic properties.

We show that the closure of the symplectic numerical range of an operator is convex and contains the symplectic spectrum.
We give a relationship between the symplectic numerical
ranges of truncated block Toeplitz matrices and their
symbol. This, in turn, helps us to have a better understanding
of the distribution of symplectic eigenvalues of the
truncated block Toeplitz matrices.

\par The paper is organised as follows:
We give some basic notations and results in Section
\ref{S2}, introduce the notion of symplectic numerical range in
Section \ref{S3}, and study some of its basic properties,
especially in the context of block Toeplitz operators.
In Section \ref{S4} we prove a symplectic analogue of
Szeg\H o limit theorem, and give its applications.

\section{Preliminaries} \label{S2}

\par We begin with some basic facts about
Toeplitz operators. For proofs and other
details, the reader may refer to the book of B\"ottcher and
Silbermann \cite{bs}.

\par Let $L^\infty_{k\times k}$ denote the set of all
functions $\wt{A}=\begin{bmatrix}\wt{a}_{ij}\end{bmatrix}$ from $[-\pi,\pi]$ to the set of all $k\times k$ complex matrices, with
$\wt{A}(-\pi)=\wt{A}(\pi)$ and $\wt{a}_{ij}$ essentially bounded for all $i,j=1,\ldots,k.$

For an $\wt{A}$
in $L^\infty_{k\times k},$ we define
\begin{equation}\label{eqn3}
\|\wt{A}\|={\underset{\theta\in
[-\pi,\pi]}{\textrm{esssup}}}\|\wt{A}(\theta)\|,
\end{equation}
where $\|\wt{A}(\theta)\|$ denotes the operator norm of
$\wt{A}(\theta).$ It is easy to see that $\|\cdot\|$ is a
norm on $L^\infty_{k\times k}.$
The space $L^\infty_{k\times k}$ is a $C^*$-algebra with the usual operations.

Let $L^2_k$ be the set of all functions
$\wt{x}=(\wt{x}_1,\ldots,\wt{x}_k)$ from $[-\pi,\pi]$ to $\mathbb{C}^k$ with $\wt{x}(-\pi)=\wt{x}(\pi)$ and $\wt{x}\in L^2$ for all $i=1,\ldots,k.$

The space $L^2_k$ is a Hilbert space with the inner product
\begin{equation}\label{eqn4}
\langle \wt{x},\wt{y}\rangle = \frac{1}{2\pi}
\int\limits_{-\pi}^{\pi}\langle
\wt{x}(\theta),\wt{y}(\theta)\rangle\dd \theta = \frac{1}{2\pi}
\int\limits_{-\pi}^{\pi} \sum\limits_{i=1}^{k}
\overline{\wt{x}_i(\theta)}\wt{y}_i(\theta)\dd\theta.
\end{equation}
With each $\wt{A}$ in $L^\infty_{k\times k},$ we can
associate the multiplication map $M_{\wt{A}}$ on $L^2_k$
defined as
\begin{equation}\label{eqn5}
M_{\wt{A}}(\wt{x})(\theta)=\wt{A}(\theta)\wt{x}(\theta),
\end{equation}
where $\tilde{x}$ is here understood as a column vector.
It can be verified that $M_{\wt{A}}$ is a bounded
linear operator on $L^2_k,$

and $\|M_{\wt{A}}\|\le \|\wt{A}\|.$
The space $\{M_{\wt{A}}:\wt{A}\in L^{\infty}_{k\times k}\}$ is a $C^*$-algebra, and the map $\wt{A}\mapsto M_{\wt{A}}$ is a surjective isomorphism.
This implies that
\[\|M_{\wt{A}}\|=\|\wt{A}\|\]
for all $\wt{A}$ in $L^\infty_{k\times k}.$

\par Next let $l^2_k$ be the set of all sequences of vectors
$\hat{x}=(x_0,x_1,x_2,\ldots),$ $x_i\in\mathbb{C}^k$ such that
$\sum\limits_{i=0}^{\infty} \|x_i\|^2 < \infty.$
Here $\|x_i\|$ is the Euclidean norm of
$x_i=(x_i^{(1)},\ldots,x_i^{(k)})$.

The
space $l^2_k$ is a Hilbert space with the inner product
given by
\begin{equation}\label{eq1}
\langle \hat{x},\hat{y}\rangle = \sum\limits_{i=0}^{\infty} \langle x_i,y_i\rangle.
\end{equation}
Clearly this inner product induces the $l^2$ norm on $l^2_k.$
We denote this norm by $\|\cdot\|_2.$
In a similar way, $l^2_k(\mathbb{Z})$ is the Hilbert space of all square summable doubly infinite sequences $\hat{x}$ of vectors with the $l^2$ norm.

\par Throughout this paper, we denote the elements of $L^\infty_{k\times k}$($L^2_k$) by
$\wt{A},\wt{B},\ldots$ ($\wt{u},\wt{x},\ldots$), the elements of $l^2_k$ by $\hat{u},\hat{x},\ldots,$
and the usual matrices (vectors) by $A,B,\ldots$ ($u,x,\ldots$),
unless we mention otherwise.
\par Let $\wt{A}\in L^\infty_{k\times k}.$
For each $n\in\mathbb{Z},$ let $A_n$ be the $n\textrm{th}$ Fourier coefficient of $\wt{A}$  given by
$$A_n=\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}\wt{A}(\theta)\e^{-\imath n\theta}\dd\theta.$$

 Suppose that $L(\wt{A})$ is the doubly infinite
$k\times k$ block Toeplitz matrix
$\begin{bmatrix}A_{i-j}\end{bmatrix}_{i,j=-\infty}^{\infty}.$
Since $l^2_k(\mathbb{Z})$ and $L^2_k$ are isomorphic Hilbert spaces, we
can identify $L(\wt{A})$ with the linear operator $M_{\wt{A}}$ defined
in \eqref{eqn5}.
Let  $T(\wt{A})$ be the infinite block Toeplitz matrix
$\begin{bmatrix}A_{i-j}\end{bmatrix}_{i,j=0}^{\infty}.$ This
is a principal submatrix of $L(\wt{A}).$ If for
$n\in\mathbb{N},$  $\overline{P}_n$ is the projection
operator on $l^2_k(\mathbb{Z})$ defined as
\[\overline{P}_n(\ldots,x_{-n},x_{-(n-1)},\ldots,x_0,\ldots,x_n,\ldots)
= (\ldots,0,0,x_{-(n-1)},\ldots,x_0,\ldots,x_n,\ldots),\]
then $\overline{P}_nL(\wt{A})\overline{P}_n$ converges strongly to
$L(\wt{A}),$ and for every $n,$
$\overline{P}_nL(\wt{A})\overline{P}_n=T(\wt{A}).$

For $\wt{A}$ in $L^\infty_{k\times k},$
we say $T(\wt{A})$ is the infinite block Toeplitz matrix generated by $\wt{A}$
and  $\wt{A}$ is the \emph{symbol} of the block Toeplitz operator $T(\wt{A}).$

\begin{prop}\label{prop1}
Let $\Sigma=\begin{bmatrix}A_{i-j}\end{bmatrix}_{i,j=0}^{\infty}$ be an infinite block Toeplitz matrix.
Then $\Sigma$ is a bounded linear operator on $l^2_k$ if and only if $\Sigma=T(\wt{A})$ for some $\wt{A}$ in $L^\infty_{k\times k}.$
In this case $\|\Sigma\|=\|\wt{A}\|.$
For every $\hat{x}\in l^2_k,$
\begin{equation}\label{eqn6}
\langle \hat{x},T(\wt{A})\hat{x}\rangle = \frac{1}{2\pi}
\int\limits_{-\pi}^{\pi} \langle \wt{x}(\theta),
\wt{A}(\theta)\wt{x}(\theta)\rangle \dd\theta,
\end{equation}
where $\wt{x}$ is the element of $L^2_k$ defined as
$\wt{x}(\theta)=\sum\limits_{n=0}^{\infty}x_n\e^{\imath
n\theta}.$
\end{prop}

\par A Hermitian operator $T$ on a Hilbert space $\mathcal{H}$ is said to be \emph{positive semidefinite} if $\langle x,Tx \rangle \ge 0$ for all $x$ in $\mathcal{H}$. If equality here holds only for the null vector, then $T$ is said to be \emph{positive definite}. If the space $\mathcal{H}$ is finite-dimensional, a positive semidefinite operator is positive definite if and only if it is invertible. This is not the case when $\mathcal{H}$ is infinite-dimensional (consider, e.g., the operator $T= \mathrm{diag}(1, 1/2, 1/3,.....)$ on the space $l_2$). So, we will use the term \emph{positive invertible} for an operator that is positive definite and invertible.

Let $T_n(\wt{A})$ be the truncated $n\times n$ block
Toeplitz matrix $\begin{bmatrix} A_{i-j}
\end{bmatrix}_{i,j=0}^{n-1}.$
The operator $T(\wt{A})$ is positive semidefinite if and only if all $T_n(\wt{A})$ are positive semidefinite.

\par The \emph{essential range} of $\wt{A}$ is given by the set
of all $k \times k$ matrices $B$ such
that for every $\epsilon >0,\, m(\{t: \|\wt{A}(t) -B\| <
\epsilon\})>0.$
Here $m(\cdot)$ denotes the Lebesgue measure.
We denote the essential range of $\wt{A}$ by
$\mathcal{R}(\wt{A}).$ Clearly the essential range of $\wt{A}$ is closed in the space of $k\times k$ matrices and
is contained in the closure of the range of $\wt{A}.$
So, if $\wt{A}\in L^\infty_{k\times k},$ then
$\mathcal{R}(\wt{A})$ is compact. Also if $X\subseteq
[-\pi,\pi]$ is any set such that $\wt{A}(X)\cap
\mathcal{R}(\wt{A})=\emptyset,$ then $m(X)=0.$

\begin{prop}\label{prop3}
Let $\wt{A}\in L^\infty_{k\times k}.$
Then $T(\wt{A})$ is a positive semidefinite operator on $l^2_k$
if and only if all matrices $\wt{A}(\theta)$ in $\mathcal{R}(\wt{A})$ are positive semidefinite.
Consequently the matrices $T_n(\wt{A})$ are positive semidefinite for all $n$ if and only if all matrices $\wt{A}(\theta)$ in $\mathcal{R}(\wt{A})$ are positive semidefinite.

If $T(\wt{A})$ is positive invertible,
then all matrices $\wt{A}(\theta)$ in $\mathcal{R}(\wt{A})$ are positive definite.
\end{prop}

\par The following proposition gives an equivalent condition for $T_n(\wt{A})$ to be positive definite for each $n.$
See \cite{Miranda-Tilli}.

\begin{prop}\label{propf5}
For every $\wt{A}\in L^\infty_{k\times k},$ $T_n(\wt{A})$ is positive definite for every $n$ if and only if
all matrices $\wt{A}(\theta)$ in $\mathcal{R}(\wt{A})$ are positive semidefinite, and
$\wt{A}(\theta)$ are positive definite for all $\theta$ in some subset of $[-\pi,\pi]$ that has positive measure
\end{prop}

\par We call a Toeplitz operator $\Sigma=\begin{bmatrix}A_{i-j}\end{bmatrix}$  {\it partially symmetric} if each
$A_n$ is a real matrix and $A_{-n}=A_n$ for all $n\in\mathbb{N}.$
An element $\wt{A}$ of $L^\infty_{k\times k}$ is \emph{even} if $\wt{A}(-\theta)=\wt{A}(\theta)$ for almost all $\theta\in [-\pi,\pi].$

\begin{prop}\label{propps}
For any $\wt{A}$ in $L^\infty_{k\times k},$ the following statements are equivalent.
\begin{itemize}
\item[(i)] $T(\wt{A})$ is partially symmetric.
\item[(ii)] $\wt{A}$ is even and every matrix $\wt{A}(\theta)$ in $\mathcal{R}(\wt{A})$ is real.
\item[(iii)] The infinite matrix $T(\wt{A})$ is real and every matrix $\wt{A}(\theta)$ in $\mathcal{R}(\wt{A})$ is real.
\end{itemize}
\end{prop}

\par Here we point out that symplectic eigenvalues of $T_n(\wt{A})$ and $\wt{A}(\theta)$
 are defined only when $T(\wt{A})$ is a partially symmetric operator on $l_{2k}^2$, and $T_n(\wt{A})$ and $\wt{A}(\theta)$ are positive definite.

\par A stationary G-chain $\Sigma=\begin{bmatrix}A_{i-j}\end{bmatrix}$
is \emph{bounded} if it is
bounded as a linear operator on $l^2_{2k},$
and is \emph{partially symmetric} if it is a partially symmetric linear operator.

The following theorem gives a characterisation of a
partially symmetric bounded stationary G-chain
in terms of its symbol.

\begin{thm}\label{thm4}
Let $\Sigma$ be an infinite real matrix.
Then $\Sigma$ is a partially symmetric bounded stationary G-chain if
and only if it is generated by an $\wt{A}$ in $L^\infty_{2k\times 2k}$ such that $\wt{A}(\theta)$ is a G-matrix for all
$\wt{A}(\theta)$ in $\mathcal{R}(\wt{A}).$
\end{thm}

\begin{proof}
By Proposition \ref{prop1} $\Sigma$ is a bounded linear operator on $l^2_{2k}$ if and only if $\Sigma=T(\wt{A})$ for some $\wt{A}$ in $L^\infty_{2k\times 2k}.$

By Propositions \ref{prop3} and \ref{propps} we know that $\wt{A}(\theta)$ are real positive semidefinite
matrices for all $\wt{A}(\theta)\in\mathcal{R}(\wt{A})$ if and only if
$T(\wt{A})$ is partially symmetric and positive semidefinite.
Let $\Sigma_0=T(\wt{A})+\frac{\imath}{2}J_\infty,$
where $J_\infty$ is the infinite block diagonal matrix $\oplus_{\mathbb{N}}J_2.$
Clearly $\Sigma_0$ is the infinite $2k\times 2k$ block Toeplitz matrix
corresponding to the sequence $\langle
B_n\rangle_{n\in\mathbb{Z}},$ where
\[B_n=\begin{cases}
A_n & n\ne 0,\\
A_0+\frac{\imath}{2}J_{2k} & n=0.
\end{cases}\]
One can see that $\Sigma_0$ is generated by the function
$\wt{B}=\wt{A}+\frac{\imath}{2}J_{2k}.$ Now $\wt{A}(\theta)$ is a G-matrix
if and only if $\wt{B}(\theta)$ is positive semidefinite.
Similarly, $T(\wt{A})$ is a G-chain if and only if $T_n(\wt{B})$
is positive semidefinite for every $n\in\mathbb{N}.$ Hence we obtain the theorem by
using Proposition \ref{prop3}.
\end{proof}

\section{Symplectic numerical range} \label{S3}

\par Let $\mathcal{H}$ be a real separable Hilbert space. We denote
the direct sum $\mathcal{H} \oplus \mathcal{H}$ by $\wt{\mathcal{H}}.$ It is easy to
see that the space $\wt{\mathcal{H}}$ is isomorphic to
$\bigoplus_\mathbb{N} \mathcal{K}$ where $\mathcal{K}$ is a two dimensional real
Hilbert space and the operator $J = \begin{bmatrix} 0
& I \\ -I & 0 \end{bmatrix}$ on $\wt{\mathcal{H}}$ is orthogonally
equivalent to $\bigoplus_{\mathbb{N}} J_2.$
Henceforth we will identify $\wt{\mathcal{H}}$  with  $\bigoplus_\mathbb{N} \mathcal{K}$
and the operator $J$ with $\bigoplus_{\mathbb{N}} J_2.$

\begin{defin}
Let $A$ be a positive definite operator on
$\wt{\mathcal{H}}.$ We define the symplectic numerical range
of $A$ to be the set
\[W_s(A) = \left\{ \frac{1}{2} ( \langle u, A u\rangle +
\langle v, Av \rangle): \langle u,Jv\rangle=1,\, u, \, v \in \wt{\mathcal{H}}\right\}.\]
\end{defin}

This is a subset of $(0,\infty)$. It is unbounded as the set of vectors $(u,v)$  with $\langle u, J v \rangle =1$ is unbounded.

An infinite dimensional version of Williamson's theorem
was proved in \cite{tiju}:
for any
positive invertible operator $A$ on $\wt{\mathcal{H}}$ there exists a positive invertible
operator $P$ on  $\mathcal{H}$
and a symplectic transformation $L: \wt{\mathcal{H}} \to \wt{\mathcal{H}}$ such that
\[ A = L \begin{bmatrix} P & 0 \\ 0 & P \end{bmatrix} L^T.\]
The \emph{symplectic spectrum} of $A$ is the spectrum of the positive
invertible operator  $P.$
If $A$ is a $2n\times 2n$ real positive definite matrix, then
its symplectic spectrum is the set of its symplectic eigenvalues
$\{d_1(A),\ldots,d_n(A)\}\subseteq (0,\infty).$

We denote by
$\sigma_s(A)$ the symplectic spectrum of $A.$

\begin{prop} \label{prop4}
Let $\mathcal{H}$ be a real separable Hilbert space and $A$ a
bounded positive invertible operator on $\wt{\mathcal{H}}.$ Then
\begin{enumerate}[(i)]
\item $W_s(A) = W_s(MAM^T)$ for every symplectic transformation $M.$
\item $\sigma_s(A)\subseteq\overline{W_s(A)}=[\inf \sigma_s(A),\infty).$
\item If $\mathcal{H}$ is finite-dimensional,
then $W_s(A)$ is the closed set $[d_1(A),\infty),$
where $d_1(A)$ is the minimum symplectic eigenvalue of $A.$
\end{enumerate}
\end{prop}

\begin{proof}
Part (i)  follows from the fact that
$\langle Mu,JMv\rangle=\langle u,Jv\rangle$
for every symplectic transformation $M.$

\par Let $u,v\in\wt{\mathcal{H}}$ be such that $\langle u,Jv\rangle =1.$
Let $\alpha=\frac{1}{2}\left(\langle u,Au\rangle+\langle v,Av\rangle\right),$ $\alpha_1=\langle u,Au\rangle/2$ and $\alpha_2=\langle v,Av\rangle/2.$
For any $t>0,$ $\langle tu,Jv/t\rangle=1.$
Let
\begin{equation*}
\alpha(t)=\frac{\langle tu,A(tu)\rangle+\langle v/t,A(v/t)\rangle}{2}
= t^2\alpha_1+\frac{1}{t^2}\alpha_2.
\end{equation*}
Clearly $\alpha(t)$ is continuous in $t$ and $\alpha(1)=\alpha.$
Since $\lim_{t\to\infty}\alpha(t)=\infty,$
by the intermediate value theorem, $[\alpha,\infty)\subseteq W_s(A).$
Thus $\overline{W_s(A)}=[\inf W_s(A),\infty).$
Now, let $P$ be the positive invertible operator on $\mathcal{H}$ such that
$$A=L\hat{P}L^T,$$
where $L$ is a symplectic transformation on $\wt{\mathcal{H}}$ and $\hat{P}=\begin{bmatrix}P & O\\O & P\end{bmatrix}.$
We know that $\sigma_s(A)=\sigma(P)=\sigma(\hat{P}).$
Thus, we only need to show that
$\inf W_s(A)=\inf \sigma(\hat{P}).$
Let $u,v$ be two distinct unit vectors in $\wt{\mathcal{H}}.$
Without loss of generality, we can assume that
$\langle u,Jv\rangle>0.$
Clearly $\langle u,Jv\rangle\le 1.$
Let $u_0=u/\sqrt{\langle u,Jv\rangle}$ and $v_0=v/\sqrt{\langle u,Jv\rangle}.$
Then $\langle u_0,Jv_0\rangle=1,$ and
\begin{equation}
\frac{\langle u_0,Au_0\rangle+\langle v_0,Av_0\rangle}{2}=\frac{\langle u,Au\rangle+\langle v,Av\rangle}{2\langle u,Jv\rangle}
\ge\frac{\langle u,Au\rangle+\langle v,Av\rangle}{2}.\label{eqntt1}
\end{equation}
Since the left-hand side of \eqref{eqntt1} belongs to $W_s(\hat{P})$
and the right-hand side to $W(\hat{P}),$ it follows that
\begin{equation}
\inf W_s(\hat{P})\ge\inf W(\hat{P}).\label{eqnt2}
\end{equation}
Now let $x$ be any unit vector in $\mathcal{H},$
and let $u=\frac{1}{\sqrt{2}}(x\oplus x)$ and $v=\frac{1}{\sqrt{2}}(-x\oplus x)$.
Then $\langle u,Jv\rangle=1.$
We see that
$$\langle x,Px\rangle=\frac{\langle u,\hat{P}u\rangle+\langle v,\hat{P}v\rangle}{2}.$$
This implies that
\begin{equation}
\inf W(P)\ge \inf W_s(\hat{P}).\label{eqnt3}
\end{equation}
Combining \eqref{eqnt2} and \eqref{eqnt3}, and using the fact that
$\inf W(\hat{P})=\inf W(P)=\inf \sigma(P),$
we obtain (ii).

\par When $\mathcal{H}$ is finite-dimensional,  we have
$$d_1(A)={\underset{\underset{\langle u,Jv\rangle=1}{u,v\in\mathcal{H}}}{\min}}\frac{\langle u,Au\rangle+\langle v,Av\rangle}{2}.$$
(See Theorem 5 of \cite{bhatia-jain}.)
This gives part (iii).
\end{proof}

\par Let $\wt{A}\in L^\infty_{2k\times 2k}$ be such that all matrices $\wt{A}(\theta)$ in $\mathcal{R}(\wt{A})$ are real positive definite. Then the symplectic numerical range of
$\wt{A}$ is the set
\begin{multline*}
W_s(\wt{A}) = \left\{ \frac{1}{2\pi} \int_{-\pi}^\pi
\frac{\langle \wt{u}(\theta), \wt{A}(\theta)\wt{u}(\theta)\rangle +
\langle \wt{v}(\theta), \wt{A}(\theta)\wt{v}(\theta)\rangle}{2} \dd \theta
: \right.\\
\left. \frac{1}{2\pi}
\int_{-\pi}^\pi \langle \wt{u}(\theta),
J\wt{v}(\theta)\rangle \dd \theta
=1,\;\wt{u}, \wt{v} \in L_{2k}^2\right\}.
\end{multline*}

\par We next give a relationship between
$W_s(\wt{A})$ and $W_s(\wt{A}(\theta))$ for $\wt{A}(\theta)\in\mathcal{R}(\wt{A}).$

\begin{thm}\label{the1}
Let $\wt{A}$ be an element of $L^\infty_{2k\times 2k}$ such that all matrices $\wt{A}(\theta)$ in $\mathcal{R}(\wt{A})$ are real positive definite.
The set $\overline{W_s(\wt{A})}$ is the same as the closed convex hull of $\bigcup_{\wt{A}(\theta)\in \mathcal{R}(\wt{A})}
W_s(\wt{A}(\theta)).$
\end{thm}

\begin{proof}
Let $B=\wt{A}(\theta)\in \mathcal{R}(\wt{A})$ and $\mu \in W_s(B).$ Then there
exists a pair $(u,v)$ in $\mathbb{R}^{2k}\times\mathbb{R}^{2k}$ such that $\langle u, J_{2k}v\rangle =1$ and $\mu = \frac{1}{2} (\langle u, Bu\rangle +
\langle v, Bv\rangle).$
For $n \in \mathbb{N},$  let $S_n$ be the set

\[S_n= \left\{ t: \|\wt{A}(t) -B\| <\frac{1}{n(\|u\|^2
+\|v\|^2)}\right\},\]
and let $m_n =m(S_n)$, the measure of $S_n$.
Since $B\in\mathcal{R}(\wt{A}),$ $m_n>0$ for every $n.$
Define the vector functions $\wt{u}_n$ and $\wt{v}_n$ on $[-\pi,\pi]$ as
\[ \wt{u}_n(t) = \begin{cases}
\sqrt{\frac{2\pi}{m_n}}u & t\in S_n\\
0 & \text{otherwise,}
\end{cases}
\text{\qquad and \qquad}
\wt{v}_n(t) = \begin{cases}
\sqrt{\frac{2\pi}{m_n}}v & t\in S_n\\
0 & \text{otherwise.}
\end{cases}\]
Clearly $\wt{u}_n$ and $\wt{v}_n$ are in $L_{2k}^2,$
and $\langle\wt{u}_n,J_{2k}\wt{v}_n\rangle=1.$
Let
$\mu_n = \frac{1}{2} ( \langle \wt{u}_n, \wt{A} \wt{u}_n
\rangle + \langle \wt{v}_n, \wt{A} \wt{v}_n \rangle).$
Then $\mu_n \in W_s(\wt{A}),$  and we have
\begin{align*}
|\mu_n - \mu| &=
\left|\frac{1}{2\pi}\int_{-\pi}^{\pi}\frac{1}{2}\left(\langle
\wt{u}_n(t),\wt{A}(t)\wt{u}_n(t)\rangle+\langle\wt{v}_n(t),\wt{A}(t)\wt{v}_n(t)\rangle\right)
\dd t  -  \frac{1}{2}\left(\langle u,Bu\rangle+\langle v,Bv\rangle\right)\right|\\

&= \left|\frac{1}{m_n}\int_{S_n}\frac{1}{2}\left(u,\wt{A}(t)u\rangle+\langle v,\wt{A}(t)v\rangle\right)\dd t  - \frac{1}{m_n}\int_{S_n}\frac{1}{2}\left(\langle u,Bu\rangle+\langle v,Bv\rangle\right)\dd t\right|\\

&= \left| \frac{1}{m_n} \int_{S_n} \frac{1}{2}
\left(\langle u, (\wt{A}(t)-B) u \rangle + \langle v, (\wt{A}(t)-B) v \rangle \right) \dd
t \right|\\
&\le \frac{1}{2m_n} \left(\|u\|^2 + \|v\|^2 \right) \int_{S_n} \left\|\wt{A}(t) -
B \right\| \dd t\\
&\le \frac{1}{2n}.
\end{align*}
This proves $\mu_n\to \mu.$ Hence $W_s(B) \subseteq \overline{W_s(\wt{A})}.$
 Since $W_s(\wt{A})$
is convex, the closed convex hull of
$\bigcup_{B \in \mathcal{R}(\wt{A})} W_s(B)$ is contained in $\overline{W_s(\wt{A})}.$

\par To prove the reverse inclusion, we use the fact that every element of $W_s(\wt{A})$ is a
limit of  finite sums of the form
\[ \sum_j \frac{\alpha_j}{4\pi} ( \langle u_j, \wt{A}(\theta_j) u_j
\rangle + \langle v_j, \wt{A}(\theta_j) v_j \rangle ),\]
where $\wt{A}(\theta_j) \in \mathcal{R}(\wt{A})$, and $\alpha_j \ge 0$
are such that $\sum_j \alpha_j \langle u_j, J v_j \rangle =1.$
Let $\beta_j = \langle u_j, J v_j \rangle .$ Without loss of
generality we may assume that $\beta_j \ge 0$ for all $j.$
Replacing $u_j$ by $\sqrt{\beta_j} u_j$ and $v_j$ by
$\sqrt{\beta_j} v_j$ we can take $\langle u_j, J_{2k} v_j \rangle
=1$ for every $j,$ and $\sum \alpha_j =1.$
This shows that every element of $W_s(\wt{A})$ is a limit of
convex combinations of elements of $\bigcup_{\wt{A}(\theta)\in\mathcal{R}(\wt{A})} W_s(\wt{A}(\theta)).$

\end{proof}

\par Let $\hat{u}, \, \hat{v} \in l_{2k}^2,$  $\hat{u}=(u_1, u_2, \ldots)$
and $\hat{v}=(v_1, v_2, \ldots).$
Define $\wt{u}(\theta)=
\sum u_n e^{\imath n \theta}$ and $\wt{v}(\theta)=
\sum v_n e^{\imath n \theta}.$ Clearly
\[\langle \hat{u}, J \hat{v} \rangle = \frac{1}{2\pi} \int_{-\pi}^\pi
\langle \wt{u}(\theta), J_{2k} \wt{v}(\theta) \rangle \dd
\theta. \]
Using \eqref{eqn6} we see that
\[W_s\left(T(\wt{A})\right) \subseteq W_s(\wt{A})\]
for every partially symmetric, bounded, positive invertible operator $T(\wt{A})$ on $l^2_{2k}.$
Since
$T_n(\wt{A})$ is a principal submatrix of $T(\wt{A}),$ we have
\begin{equation}
W_s\left(T_n(\wt{A})\right) \subseteq W_s\left(T_{n+1}(\wt{A}) \right)\subseteq W_s\left(T(\wt{A})\right) \subseteq W_s(\wt{A}).
\end{equation}

Let $\wt{A}\in L^\infty_{2k\times 2k}$ be such that all matrices $\wt{A}(\theta)$ in $\mathcal{R}(\wt{A})$ are real positive definite.
Let
\begin{align}
\mathfrak{m}_{\wt{A}} &={\underset{\theta \in[-\pi,\pi]}{\textrm{essinf}}}
d_1(\wt{A}(\theta)).\label{eqnt1}

\end{align}

Using Theorem \ref{the1} we
see that
\begin{equation}
\mathfrak{m}_{\wt{A}} = \inf W_s(\wt{A}).\label{eqt1}
\end{equation}

\begin{thm}\label{the2}
Let $T(\wt{A})$ be a partially symmetric, bounded, positive invertible operator on $l^2_{2k}.$
Let $n\in\mathbb{N},$ and let $d$ be a symplectic eigenvalue of $T_n(\wt{A}).$
Then $d\ge \mathfrak{m}_{\wt A}.$
If $d=\mathfrak{m}_{\wt{A}},$ then $d_1(\wt{A}(\theta))$ is the constant $\mathfrak{m}_{\wt{A}}$ for almost all $\theta.$
\end{thm}

\begin{proof}
Let $d$ be a symplectic eigenvalue of $T_n(\wt{A}).$ Then there
exist vectors $\hat{u} = (u_1, \cdots , u_n), \, \hat{v}= (v_1,
\cdots, v_n),$ $u_j, \, v_j \in \mathbb{R}^{2k}$
with  $\langle \hat{u}, J \hat{v} \rangle =1$ and
\[d= \frac{1}{2} (\langle \hat{u}, T_n(\wt{A}) \hat{u} \rangle + \langle \hat{v},
T_n(\wt{A}) \hat{v} \rangle).\]
Let $\wt{u}$ and $\wt{v}$ be the elements of $L_{2k}^2$ given by
$\wt{u}(\theta) = \sum u_j e^{\imath j \theta}, \,
\wt{v}(\theta) = \sum v_j e^{\imath j \theta}.$
Then
\begin{align*}
d - \mathfrak{m}_{\wt{A}} &= \frac{1}{2} (\langle \hat{u}, T_n(\wt{A}) \hat{u}
\rangle + \langle \hat{v}, T_n(\wt{A}) \hat{v} \rangle) -
\mathfrak{m}_{\wt{A}}\\
& =  \frac{1}{2\pi} \int_{-\pi}^\pi \left[ \frac{1}{2} \left(\left\langle
\wt{u}(\theta), \wt{A}(\theta) \wt{u}(\theta) \right\rangle + \left\langle \wt{v}(\theta),
\wt{A}(\theta) \wt{v}(\theta) \right\rangle\right)  -
\mathfrak{m}_{\wt{A}}\left\langle \wt{u}(\theta),J_{2k}\wt{v}(\theta)\right\rangle  \right) \dd
\theta.
\end{align*}
Using \eqref{eqt1}, we know that the above integrand is nonnegative almost everywhere.
Hence, we have $d \ge \mathfrak{m}_{\wt{A}}.$
Also $d=\mathfrak{m}_{\wt{A}}$ if and only if
\[\frac{1}{2}
\left(\langle
\wt{u}(\theta),\wt{A}(\theta)\wt{u}(\theta)\rangle+\langle\wt{v}(\theta),\wt{A}(\theta)\wt{v}(\theta)\rangle
\right)=\mathfrak{m}_{\wt{A}}\langle \wt{u}(\theta),J_{2k}\wt{v}(\theta)\rangle\]
for almost all $\theta.$
So the last statement of the theorem by using Proposition \ref{prop4}(iii) and \eqref{eqnt1}.
\end{proof}

\section{A Szeg\H o type theorem for symplectic eigenvalues
and applications} \label{S4}

\par We first recall some basic facts about symplectic
eigenvalues. See \cite{bhatia-jain, gosson}, and \cite{jain-h} for 		details.
A positive number $d$ is a symplectic eigenvalue of a
$2k\times 2k$ positive definite matrix $A$ if and only if
$\pm d$ are the eigenvalues of the (non-Hermitian) matrix $\imath
J_{2k}A.$

Thus each symplectic eigenvalue
$d_i$ of $A$ lies in the interval $[0,\|A\|].$
Let $T(\wt{A})$ be a bounded, partially symmetric, positive invertible operator on $l^2_{2k}$ generated by $\wt{A}.$
Let $d_1^{(n)}\le\cdots\le d_{nk}^{(n)}$ denote the symplectic eigenvalues of $T_n(\wt{A})$ arranged in increasing order.
 Since $\wt{A}\in L^\infty_{2k\times 2k},$
each $d_i^{(n)}\le \|T_n(\wt{A})\|\le\|\wt{A}\|.$

\begin{thm}\label{thm_main}
Let $\Sigma$ be a partially symmetric, bounded positive invertible operator on $l^2_{2k}$
generated by $\wt{A}.$
Let $d_1^{(n)} \le \cdots \le d_{nk}^{(n)}$
denote the symplectic eigenvalues of $T_n(\wt{A}).$ Then for
every function $f$ continuous on $[0,\|\wt{A}\|],$
\begin{equation}\label{eqn_main}
\lim\limits_{n\to\infty} \frac{1}{n} \sum_{j=1}^{nk} f(d_j^{(n)}) =
\frac{1}{2\pi} \int_{-\pi}^{\pi} \sum_{j=1}^{k}
f\left(d_j(\wt{A}(\theta))\right) \dd\theta.
\end{equation}
\end{thm}

\begin{proof}
By Theorem 6.24 of \cite{bs}, we know that if $\wt{B}\in
L^\infty_{2k\times 2k},$ and $\lambda_1^{(n)}, \ldots,
\lambda_{2nk}^{(n)}$ are the eigenvalues of the $n\times n$
truncated block Toeplitz matrix $T_n(\wt{B}),$ then
\begin{equation}\label{eqna1}
\lim\limits_{n\to\infty} \frac{1}{n} \sum\limits_{j=1}^{2nk}
\left(\lambda_j^{(n)}\right)^{m} = \frac{1}{2\pi}
\int_{-\pi}^{\pi} \sum\limits_{j=1}^{2k}
\left(\lambda_j(\wt{B}(\theta))\right)^{m} \dd\theta,
\end{equation}
for every nonnegative integer $m.$
Suppose $\wt{B}=\imath J_{2k}\wt{A}.$
Then $T_n(\wt{B})=\imath J_{2nk}T_n(\wt{A}),$ and the
eigenvalues $\lambda_1^{(n)}, \ldots, \lambda_{2nk}^{(n)}$
of $T_n(\wt{B})$ are $\pm d_1^{(n)},\ldots,\pm
d_{nk}^{(n)}.$
Also the eigenvalues $\lambda_j(\wt{B}(\theta))$ of $\wt{B}(\theta)$ are $\pm d_j(\wt{A}(\theta)).$
Hence for every non-negative integer $m,$
\begin{equation}\label{eqna2}
\lim_{n\to\infty} \frac{1}{n} \sum_{j=1}^{nk}
\left(d_j^{(n)}\right)^{2m} = \frac{1}{2\pi}
\int_{-\pi}^{\pi} \sum_{j=1}^{k}
\left(d_j(\wt{A}(\theta))\right)^{2m} \dd\theta.
\end{equation}
By linearity, we can extend \eqref{eqna2} to polynomials in
$\left(d_j^{(n)}\right)^2,$ i.e.,
\begin{equation}\label{eqna3}
\lim_{n\to\infty} \frac{1}{n} \sum_{j=1}^{nk} p
\left({d_j^{(n)}}^2\right) = \frac{1}{2\pi}
\int_{-\pi}^{\pi} \sum_{j=1}^{k} p \left(d_j^2(\wt{A}(\theta))\right) \dd\theta
\end{equation}
for every polynomial $p.$ Each $d_j^{(n)}\in
\left[0,\|\wt{A}\|\right],$  and hence $\left(d_j^{(n)}\right)^2\in
\left[0,\|\wt{A}\|^2\right].$
Let $q(x)$ be any polynomial and let $s(x)=q(\sqrt{x}).$
Clearly $s$  is continuous on $\left[0,\|\wt{A}\|^2\right].$
For a given $\epsilon>0,$ we can find a polynomial $p$ such that
\[\sup_{x\in \left[0,\|\wt{A}\|^2\right]} |s(x)-p(x)|
<\epsilon.\]
Since $d_j(\wt{A}(\theta)) \le \|\wt{A}(\theta)\|$ and
$\|\wt{A}(\theta)\|
\le \|\wt{A}\|$ for almost all $\theta,$

\begin{multline}\label{eqna4}
\left\vert \frac{1}{2\pi} \int_{-\pi}^{\pi}
s\left(d_j^2(\wt{A}(\theta))\right) \dd\theta - \frac{1}{2\pi}
\int_{-\pi}^{\pi} p\left(d_j^2(\wt{A}(\theta))\right) \dd\theta
\right\vert \\ \le \frac{1}{2\pi} \int_{-\pi}^{\pi}
\left\|s\left(d_j^2(\wt{A}(\theta))\right) -
p\left(d_j^2(\wt{A}(\theta))\right) \right\| \dd\theta \; \le \epsilon,
\end{multline}

holds for all $j=1,\ldots,k.$
Similarly,\begin{equation}\label{eqna5}
\left|s\left({d_j^{(n)}}^2\right) -
p\left({d_j^{(n)}}^2\right)\right| < \epsilon
\end{equation}
for all $j=1,\ldots,nk.$
Combining the relations \eqref{eqna3}, \eqref{eqna4}, and
\eqref{eqna5}, we see that
\[\lim_{n\to\infty} \frac{1}{n} \sum_{j=1}^{nk}
s\left({d_j^{(n)}}^2\right) = \frac{1}{2\pi}
\int_{-\pi}^{\pi} s\left(d_j^2(\wt{A}(\theta))\right) \dd\theta.\]
Since $s(x^2)=q(x),$ we have
\begin{equation}\label{eqna6}
\lim_{n\to\infty} \frac{1}{n} \sum_{j=1}^{nk}
q\left(d_j^{(n)}\right) = \frac{1}{2\pi} \int_{-\pi}^{\pi}
q\left(d_j(\wt{A}(\theta))\right) \dd\theta.
\end{equation}
Now let $f$ be any continuous function on $\left[0,
\|\wt{A}\|\right].$ Then by using the Weierstrass
approximation theorem and arguing as above, we can show that
\eqref{eqn_main} holds for $f.$
\end{proof}

\begin{rem}
\emph {
We know that $d$ is a symplectic eigenvalue of a positive definite matrix $A$ if and only if $\pm d$ are eigenvalues of the non-Hermitian matrix $iJA$. In \cite{tilli-2} Tilli has proved a very general version of Szeg\H o's limit theorem for non-Hermitian block Toeplitz matrices. It is possible to derive Theorem \ref{thm_main} from Tilli's general results. The proofs of the general version are, naturally, more intricate. We have given a short self-contained presentation for the special case we need.
}

\end{rem}

\par The entropy rate of a stationary quantum Gaussian process with the associated G-chain $T(\wt{A})$ is given by the formula
\[\mathfrak{S}(T(\wt{A}))=\lim_{n \to \infty} \frac{S(T_n(\wt{A}))}{n},\]
where $S(T_n(\wt{A}))$ denotes the entropy of the quantum Gaussian state with the corresponding G-matrix $T_n(\wt{A}).$
As a consequence of Theorem \ref{thm_main}, we obtain a closed expression for the
entropy rate of a partially symmetric, bounded stationary quantum Gaussian process in terms of the entropies
of the Gaussian states with G-matrices $\wt{A}(\theta).$

\begin{cor}\label{cor4.2}
Let $T(\wt{A})$ be a partially symmetric, bounded stationary G-chain generated by $\wt{A}.$
The entropy rate $\mathfrak{S}\left(T(\wt{A})\right)$ of the corresponding stationary Gaussian process is
\begin{equation}\label{eq:h}
\mathfrak{S}\left(T(\wt{A})\right) = \frac{1}{2\pi} \int_{-\pi}^{\pi}
S\left(\wt{A}(\theta)\right) \dd\theta.
\end{equation}
\end{cor}

\begin{proof}
Define the function $f:[0,\|\wt{A}\|]\to\mathbb{R}$ as
$$f(x)=\begin{cases}
\left(x+\frac{1}{2}\right)\log\left(x+\frac{1}{2}\right)-\left(x-\frac{1}{2}\right)\log\left(x-\frac{1}{2}\right) & \quad \text{ if }\frac{1}{2}<x\le\|\wt{A}\|,\\
0 & \quad \text{ if } 0\le x\le \frac{1}{2}.
\end{cases}$$
Using \eqref{eqn1}we can see that the entropy of any G-matrix $B$ can be written as
\begin{equation}
S(B)=\sum\limits_{d\in\sigma_s(B)}f(d).\label{eqer1}
\end{equation}
Hence the entropy rate $\mathfrak{S}\left(T(\wt{A})\right)$ of $T(\wt{A})$ is given by
\begin{equation}
\mathfrak{S}\left(T(\wt{A})\right)=\lim\limits_{n\to\infty}\frac{1}{n}\sum\limits_{j=1}^{nk}f\left(d_j(T_n(\wt{A}))\right).\label{eqer2}
\end{equation}
Since $f$ is continuous and $\wt{A}\in L^\infty_{2k\times 2k},$ we can apply Theorem \ref{thm_main} to get
$$\mathfrak{S}\left(T(\wt{A})\right)=\frac{1}{2\pi}\int\limits_{-\pi}^{\pi}\frac{1}{k}\sum\limits_{j=1}^{k}f\left(d_j(\wt{A}(\theta))\right)\dd\theta.$$
Using the formula \eqref{eqer1} for the sum inside this integral, we obtain \eqref{eq:h}.

\end{proof}

\par The entropy rate of a special kind of stationary quantum Gaussian process has been computed in the paper \cite{krprb2}. There the authors considered a block Toeplitz matrix $T_n(\wt{A})$ given by
\[T_n(\wt{A}) = \begin{cases}
A & n=0,\\
p_{|n|} B & \text{otherwise};
\end{cases}\]
where $A$ and  $B$ are $2k \times 2k$ real symmetric matrices such that $A + tB$ is a G-matrix for each $t \in [-2,2]$, and $\{p_1, p_2, \ldots\} $ is a probability distribution over $\{1,2, \ldots\}$. In this case $\wt{A}(\theta)$ takes the form $\left( A + \sum_{j \in \mathbb{Z}\backslash\{0\}} p_{|j|} B e^{\imath j \theta}\right),\, \theta \in [-\pi, \pi].$  Our Corollary \ref{cor4.2} gives a much more general result.

\par In the rest of the paper, we use Theorem \ref{thm_main} to study the distribution of symplectic eigenvalues of truncated block Toeplitz matrices.
Henceforth $T(\wt{A})$ is a partially symmetric, bounded, positive invertible operator on $l^2_{2k}$ generated by $\wt{A}.$ Recall the definition of $\mathfrak{m}_{\wt{A}}$ given in \eqref{eqnt1}.

\begin{thm}\label{the22}
For each $m\in\mathbb{N},$ $\lim_{n \to \infty} d_m^{(n)} = \mathfrak{m}_{\wt{A}}.$
Consequently
\begin{equation}\label{s4}
\overline{\bigcup_{n \in \mathbb{N}} W_s(T_n(\wt{A})) } = \overline{W_s(\wt{A})}.
\end{equation}
\end{thm}
\begin{proof}
For every $n\in\mathbb{N},$ $T_n(\wt{A})$ is a principal submatrix of $T_{n+1}(\wt{A}).$
Hence from the relation (42) of \cite{bhatia-jain} we see that
\[0\le d_m^{(n+1)} \le d_m^{(n)}\ \ \textrm{ for all }n\ge m.\]
 Hence $\lim\limits_{n\to\infty}d_m^{(n)}$ exists. Suppose this equals
$ r .$ By definition $r \ge \mathfrak{m}_{\wt{A}}$.  Suppose  $ r  >
\mathfrak{m}_{\wt{A}}.$ Define $f$ on $[0, \|\wt{A}\|]$ as
\[f(x) = \begin{cases}
0 & x>  r  \text{ or } x< \mathfrak{m}_{\wt{A}} -1,\\
x - \mathfrak{m}_{\wt{A}} +1 & \mathfrak{m}_{\wt{A}} -1 \le x \le
\mathfrak{m}_{\wt{A}}, \\
\frac{x-  r }{\mathfrak{m}_{\wt{A}} -  r }  &
\mathfrak{m}_{\wt{A}} \le x \le  r .
\end{cases} \]
Clearly $f$ is continuous on $[0, \|\wt{A}\|],$ and the formula
\eqref{eqn_main} holds for $f.$
But $f(d_j^{(n)}) =0$ for all
$j=m, \cdots, nk$ and for all $n \ge m.$
So the left hand side of \eqref{eqn_main} is zero.
But since $\mathfrak{m}_{\wt{A}} = \textrm{essinf}\, d_1(\wt{A}(\theta))$
and $f$ is positive for $[\mathfrak{m}_{\wt{A}},  r ],$ we
have $f\left( d_j(\wt{A}(\theta))\right) > 0$ for $\theta$ in a
set of positive measure. Hence the right hand side of \eqref{eqn_main} is strictly positive. This is a contradiction.

\par By Proposition \ref{prop4} (iii), we know that
$W_s(T_n(\wt{A})) =[d_1^{(n)}, \infty).$ Hence
\[ \overline{\bigcup_{n \in \mathbb{N}} W_s(T_n(\wt{A}))} = [\lim_{n \to \infty}
d_1^{(n)}, \infty) = [\mathfrak{m}_{\wt{A}}, \infty).\]
Since $\mathfrak{m}_{\wt{A}} = \inf W_s(\wt{A})$ and $W_s(\wt{A})$ is convex,
this proves \eqref{s4}

\end{proof}

\begin{lemma}\label{ls5}
Let $K$ be a compact subset of $\mathbb{R},$ and let
$c_n(K)$ be the cardinality of the set $\{j: d_j^{(n)} \in
K\}.$ Then
\begin{equation}\label{s5}
\lim_{n \to \infty} \frac{c_n(K)}{n} = \frac{1}{2\pi}\sum\limits_{j=1}^k
m\{ \theta \in [-\pi, \pi]: d_j\left(\wt{A}(\theta)\right) \in
K\}.
\end{equation}
\end{lemma}
\begin{proof}
Let $g$ be the distance function defined as $$g(x) = |x -K| := \inf \{|x -t|: t \in K\}.$$
For any $\epsilon>0$ define the function $f_\epsilon : [0 , \|\wt{A}\|] \to  \mathbb{R}$ as
$f_\epsilon(x) = \e^{-\frac{g(x)}{\epsilon}}.$
Clearly $f_\epsilon$ is continuous and $f_\epsilon(x)=1$ if
and only if $x \in K.$ We can see that as $\epsilon \to 0,$
$f_\epsilon$ converges to the characteristic function
$\chi_K$ in the $L^1$ norm. Hence
\begin{align*}
\lim_{\epsilon \to 0} \frac{1}{2\pi} \int_{-\pi}^\pi
\sum_{j=1}^k  f_\epsilon (d_j(\wt{A}(\theta))) \dd \theta  &=
\frac{1}{2\pi} \int_{-\pi}^\pi \sum_j \chi_K
(d_j(\wt{A}(\theta))) \dd \theta \\
&= \frac{1}{2\pi} \sum_j m(\{\theta:d_j(\wt{A}(\theta)) \in
K\}).
\end{align*}
Also
\begin{align*}
\lim_{\epsilon \to 0} \lim_{n \to \infty} \frac{1}{n}
\sum_{j=1}^{nk} f_\epsilon(d_j^{(n)})  =  \lim_{n \to
\infty} \sum_j \chi_K(d_j^{(n)}) = \lim_{n \to \infty} \frac{c_n(K)}{n}.
\end{align*}
Applying Theorem \ref{thm_main} with $f=f_\epsilon$ and taking $\epsilon
\to 0$ we get \eqref{s5}.
\end{proof}

\par We know that the map $d_j$ that takes a positive definite
matrix $B$ to its $j$th minimum symplectic eigenvalue
$d_j(B)$ is continuous \cite{bhatia-jain}. Since
$\theta \mapsto \wt{A}(\theta)$ is measurable on $[-\pi, \pi],$
the composite map $\theta \mapsto d_j(\theta) =
d_j(\wt{A}(\theta))$ is also measurable.

\par Let $\mathcal{R}_j$ denote the essential range of the
map $d_j(\theta)$ and let $\mathcal{R}= \bigcup_{j=1}^{k}
\mathcal{R}_j.$ Since $\wt{A} \in L^\infty_{2k \times 2k},$
 the set $\mathcal{R}$ is compact.

\begin{lemma}\label{lemt1}
 For every $\wt{A}(\theta)$ in $\mathcal{R}(\wt{A})$ and $1\le j\le k,$
$d_j(A(\theta))$is in $\mathcal{R}_j.$

\end{lemma}

\begin{proof}
 Let $B=\wt{A}(\theta)$ be any element of $\mathcal{R}(\wt{A}).$
We show that $d_j(B)\in\mathcal{R}_j.$
Let $\epsilon>0.$
Since the map $d_j$ is continuous on positive definite matrices, we can find a $\delta>0$ such that
\begin{equation}
\|\wt{A}(t)-B\|<\delta\implies |d_j(\wt{A}(t))-d_j(B)|<\epsilon.\label{eqt2}
\end{equation}
By the definition of the essential range of $\wt{A},$
the set $S=\{t:\|\wt{A}(t)-B\|<\delta\}$
has positive measure.
Let $T$ be the set
$$T=\{t:|d_j(\wt{A}(t))-d_j(B)|<\epsilon\}.$$
By \eqref{eqt2}
we see that $S\subseteq T.$
Hence $T$ also has positive measure.
This shows that $d_j(B)\in\mathcal{R}_j.$
\end{proof}

\par For any subset $X$ of $\mathbb{R}$ let $\mathcal{B}(X, \delta)$ be its $\delta$-neighbourhood:
\[\mathcal{B}(X, \delta) = \{ x \in \mathbb{R}: |x -s | <
\delta \text{ for some } s \in X\}.\]
\par Let $\mathcal{D}_n$ be the set of
symplectic eigenvalues of $T_n(\wt{A}).$ Let $\mathcal{D}=
\bigcup_n \mathcal{D}_n$
\begin{thm}\label{thm4.4}
The set $\mathcal{D}$ is dense in $\mathcal{R}.$ Further for each
$\delta>0$ let $X_\delta$ be the set
\begin{equation}
X_\delta =[\mathfrak{m}_{\wt{A}}, \|\wt{A}\|] \backslash
\mathcal{B}(\mathcal{R},\delta).
\end{equation}
Then
\begin{equation}
\lim\limits_{n \to \infty} \frac{c_n(X_\delta)}{n} =0.\label{eqt3}
\end{equation}
\end{thm}
\begin{proof}
Since $\mathcal{R}$ is compact, we can apply Lemma \ref{ls5}
to get
\begin{equation}\label{eq4.10}
\lim\limits_{n \to \infty} \frac{c_n(\mathcal{R})}{n} =
\frac{1}{2\pi} \sum\limits_{j=1}^k m (\{ \theta: d_j(\wt{A}(\theta)) \in
\mathcal{R}\}).
\end{equation}
\par Suppose $\mathcal{D}$ is not a dense subset of $\mathcal{R}$. Then there exist $j \in \{1, 2, \ldots,k\}$ $x \in \mathcal{R}_j$ and $\epsilon >0$ such that
$\mathcal{B}(x, \epsilon) \cap \mathcal{D} =
\emptyset.$ Since  $x \in \mathcal{R}_j$ the
set
\[S =\{ t: |x- d_j(\wt{A}(t)) |< \epsilon\}\]
has a positive measure. Let $Y = \mathcal{R} \backslash
\mathcal{B}(x, \epsilon).$ The set $Y$ is compact, hence by \eqref{s5} we have
\begin{equation}
\lim\limits_{n\to\infty}\frac{c_n(Y)}{n}=\frac{1}{2\pi}\sum\limits_{j=1}^{k}m\{t:d_j(\wt{A}(t))\in Y\}.\label{eq4.11}
\end{equation}
Since $S$ has positive measure,
$$m\{t:d_j(\wt{A}(t))\in\mathcal{R}\}>m\{t:d_j(\wt{A}(t))\in Y\}.$$
This shows that the right hand side of \eqref{eq4.10} is strictly greater than the right hand side of \eqref{eq4.11}.
But since $\mathcal{D}\cap \mathcal{B}(x,\epsilon)=\emptyset,$
 $c_n(Y)=
c_n(\mathcal{R})$ for all $n.$
So,  the left hand sides of
 \eqref{eq4.10} and \eqref{eq4.11} are equal.
This is a contradiction.
Hence $\mathcal{D}$ must be dense in $\mathcal{R}.$

\par The set $X_\delta$ is compact, and hence \eqref{s5} holds when $K=X_\delta.$
Since $X_\delta\cap\mathcal{R}_j=\emptyset$ for every $j,$
$$m\{\theta:d_j(\wt{A}(\theta))\in X_\delta\}=0.$$
This proves \eqref{eqt3}.
\end{proof}

\bibliographystyle{acm}
\bibliography{biblio}

\end{document}