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2014 Oscar winner and sensation, Lupita Nyong�o has accepted to star in a Cameroonian movie alongside some of the industry�s leading actors including Solange Yijika, Jeffery Epule, Mbufung Seikeh and Anurin Nwenembom. The Kenyan-born Hollywood actress signed a contract with a USA-based Cameroonian producer to that effect early Tuesday.
According to Lupita, she is excited to do the Cameroonian movie for two main reasons � first, Bimbia where the movie will be shot has a much to offer as regards the history of the Slave Trade. Bimbia, located near Limbe in the South West region of Cameroon was one of the transit points of the Slave Trade and scores of tourists go there annually to experience the relics of what has been described as the worse episode in human history. Secondly, the best supporting actress from 12 Years a Slave explained that the Cameroonian film industry has been making great strides in recent years and this has not escaped her attention.
Shooting of the said movie to be directed by Enah John Scott begins on Monday April 14, 2014. Lupita, the producer has revealed will be in Cameroon two days to the kick off of the project. Other crew members will include Penjo Baba (D.O.P), Yibain Emile-Aime Chah (Assistant D.O.P), Tanko Francois (Lighting Technician), Ernest Bah (Sound Technician), Musing Derick (Editor), etc. The movie will be released by July this year.
Wait a minute � what is today�s date? April 1? Ok�.APRIL FOOOOOOL��
But there maybe more to this Read Here
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Last Updated on Tuesday, 01 April 2014 11:52
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Founding director of Innergise and Australia and New Zealand School of Government educator Dr Nick Fleming GAICD gives his top 10 tips for combatting disruption and harnessing innovation from the board down.
Discuss innovation as a board. What does it mean in your business context?
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Estimate the cost of inaction. What innovation is already occurring? What aspects of the business must still improve? What value is at risk? On a risk- weighted basis, what level of investment in innovation is prudent?
Develop a list of the board's key innovation questions. Which questions are most common and important and what simple actions can be taken to address them? Beyond refreshing the board composition, actions should include discussions with technology specialists, engagement with directors operating in disruptive industries and study tours.
Ask how you would measure innovation. What does success look like and how will you know when you've achieved it?
Examine whether the incentive structures for executives help or hinder innovation.
Ask management about your customers' key unmet needs.
Ask management to explain how they've considered different options for delivery of major initiatives and their associated risk/reward profiles.
Scan the emerging operating environment to gather evidence on opportunities and risks. Move beyond uncertainty and paralysis to objective action.
Construct a portfolio of innovative initiatives to optimise your return on innovation investment while building an enduring innovation capability.
Train your people to solve problems. Problem-solving skills are not strong in most organisations, especially skills central to everyday tasks and innovation.


text:
Tseng Kwong Chi: Performing for the Camera
Sep 17 2016 - Dec 11 2016
Block Museum of Art
"Tseng's photographs address issues of popular culture, politics, cosmopolitanism, and cultural diversity. Humor and a keen observational eye combine to produce work that is both intelligent and accessible," Janet Dees, Block Museum Curator
Born in Hong Kong, educated in Vancouver and Paris, and later based in New York City, Tseng Kwong Chi (1950–1990) produced a large body of witty, playful, performance-based photography that both captured the pivotal downtown Manhattan art and club scenes of the 1980s and reflected the increasingly globalized movement of people across nations and continents. Tseng called himself an "inquisitive traveler, a witness of my time and an ambiguous ambassador." His works alternately function as witness to his life and community and as wry social commentary, raising critical questions about identity and culture.
Tseng Kwong Chi: Performing for the Camera, is the first major solo museum exhibition of the photographer's works, which have long sparked the imaginations of younger artists. The exhibition features over 80 photographs including well-known works— such as Tseng's collaborations with Keith Haring and his East Meets West and Expeditionary series—as well as examples from the artist's archive that have rarely been shown.
Whether you go or not, Tseng Kwong Chi: Performing for the Camera is the first comprehensive survey of the work of Tseng Kwong Chi, the revered photographer and performance artist of the 1980s. Reproducing more than 100 works by Tseng from the late 1970s to the late 1980s, and including archival materials from his commissions for the Soho Weekly News, the book presents Tseng's best-known self-portrait series, East Meets West, as well as lesser--known works, plus portraits of his friends Keith Haring, Jean-Michel Basquiat and Kenny Scharf, among others.
Tseng Kwong Chi

redpajama_set_name: RedPajamaCommonCrawl


text:
\section{Introduction}

The difference between two classically equivalent definitions of the mutual information as a measure of total correlations in a bipartite state of a composite quantum system
stays at the origin of the quantum discord idea \cite{OZ,HV}. Such a difference arises when actions and measurements are performed on one or more subsystems.
In the case of mixed states, quantum discord is a measure of quantumness whose relation to entanglement is not a simple one.  Evaluation of up-to-date defined degrees of correlations
is a difficult problem both analytically and numerically. To be more specific, the entanglement of formation is defined as an optimization over all pure-state decompositions  of the given state \cite{Bennett}, while the discord emerges from an optimization over the set of all local measurements.  The computational complexity of such measures of quantum correlations in the discrete-variable systems was recently analyzed in Ref. \cite{Huang1}
and found to increase exponentially with the dimension of the Hilbert space. That is why, despite all efforts,
only few analytic results could be found so far even in the simplest cases. For instance, the entropic discord originally defined in Refs. \cite{OZ,HV} was investigated for two-qubit X states  in Ref. \cite{alber} but, according to Ref. \cite{Huang2}, an analytic formula of quantum discord for general two-qubit X states
is still to be derived.

In the continuous-variable settings, evaluation of the original discord for two-mode Gaussian states (GSs) could be accomplished
by restricting the set of local measurements to the Gaussian ones to obtain what is now called the Gaussian discord \cite{PG,AD}.
Interestingly, for a large class of two-mode GSs it was recently proven that the Gaussian discord is an exact result, namely, the optimal local measurement defining the discord is a Gaussian
Positive Operator-Valued Measure (POVM) \cite{PSBCL}. A review of the increasing interest, progress, and application of classical and quantum correlations quantified by quantum discord and other measures can be found in Ref. \cite{modi1}. Among these quantifiers, an important role was recently attributed to the so-called geometric measures of quantum discord \cite{modi2,DVB,Luo2}. Let us recall that quantum properties
of the states involved in various protocols were successfully quantified using distance-type measures.
The distance from a given state having a specific property to a reference set of states not having it can be interpreted as a quantifier of that property \cite{Hil,PVK}. When using true distances such as
the trace  \cite{Hil},  Hilbert-Schmidt  \cite{Dod}, Bures  \cite{PTH02}, or Hellinger \cite{Luo1} ones ,
we speak about a geometric measure of that property.
For instance, in Refs. \cite{Hil,Dod,PTH02},  geometric measures
of non-classicality of one-mode states were defined and investigated. Alternatively, one can use other "distances" which are not true metrics, but are known to possess good distinguishability properties,
such as the relative entropy for entanglement \cite{PVK}, discord \cite{modi2}, and non-Gaussianity \cite{PT2013} and the quantum Chernoff bound for non-classicality \cite{BGPT} and polarization \cite{GBPT}.
To efficiently apply a distance-type measure to the case of discord, a primary condition was to identify
the set of zero-discord states
 \cite{PHH} according to the original definition \cite{OZ,HV}.
A geometric measure was first considered by using
the Hilbert-Schmidt metric \cite{DVB},
\begin{equation}
D_{HS} ({\hat \rho}):= \min_{\{\hat \chi\}}{[d_{HS}}({\hat \rho},\,\hat\chi)]^2,
\label{d0}
\end{equation}
where $\{\hat \chi\}$ is the set of zero-discord states.
More recently \cite{Luo2}, a geometric discord was defined  in a slightly different manner:
\begin{equation}
\bar D_{HS} ({\hat \rho}):=\min_{\{\hat \Pi^{(a)}\}}{[d_{HS}}
({\hat \rho},\,{\hat \Pi^{(a)}(\hat\rho}))]^2,
\label{d1}
\end{equation}
where the minimum is taken over all the von Neumann measurements applied
to subsystem $(a)$. The measurement operators  $\hat \Pi^{(a)}$
are orthogonal projectors of rank one. Being in terms of local measurements, definition\ (\ref{d1}) is therefore an inspiration of the original proposal \cite{OZ,HV}.
The Hilbert-Schmidt geometric discord\ (\ref{d0}) was found to have a closed expression for any two-qubit state \cite{DVB}. Moreover,  Luo and Fu proved that the results of the two definitions\ (\ref{d0}) and\ (\ref{d1}) nicely coincide for any bipartite state belonging to a finite-dimensional  Hilbert space \cite{Luo2}.
As quite severe drawbacks of the Hilbert-Schmidt geometric discord  were signalled in Ref. \cite{P},
some other distances were recently explored to soundly define geometric measures of discord.
In Ref. \cite{Luo3} explicit formulae of the  geometric  discord\ (\ref{d1}) were given for pure states,
as well as for $(2 \times n)$-dimensional states using the Hellinger distance. The same metric was shown
to arise in a more general treatment of discord-type non-classical correlations based on local quantum uncertainty \cite{adesso3}. In parallel, the Bures metric was employed
as a measure of quantum correlations  for states lying in finite-dimensional  Hilbert spaces too  \cite{SO}. When applied to two-mode Gaussian states in Refs. \cite{adesso1,adesso2},
the  definition\  (\ref{d1}) was modified by allowing  general  local  measurements  (POVMs).

In the present work, we take inspiration from a recent proposal
in Ref. \cite{Luo3} and use the Hellinger metric \cite{Luo1}
as a geometric measure of discord for two-mode Gaussian states.
The paper is organized as follows. In Sec. 2, we recapitulate
some of the beneficial features of the Hellinger distance as presented
in Refs. \cite{Luo1,Luo3}. Section 3 recalls the general structure
of a GS. Then we write the Hellinger distance between two $n$-mode GSs
by using some of our recent findings in Ref. \cite{PT2012}.
In Sec. 4, we point out that any Gaussian geometric discord suffers
from the drawback of not distinguishing between quantum and classical correlations and therefore being a Gaussian measure of the total
amount of correlations in a two-mode GS. By using the Hellinger metric,
we find analytically in Sec. 5 the nearest product GS with respect
to a given two-mode GS. This enables us to write a closed-form
expression of the Hellinger discord valid for an arbitrary two-mode GS.
In Sec. 6, we show that the obtained formula turns out to be
considerably simpler for symmetric states. Section 7 is devoted
to the Hellinger discord for two special classes of two-mode GSs which are interesting for theory and experiment as well: the squeezed thermal states and the mode-mixed thermal ones. In Sec. 8, the agreement between the Hellinger measure of all the correlations and the quantum  mutual information is illustrated within the Gaussian approach.  We conclude with a summary of our results. Finally,
in the Appendix, for some types of two-mode GSs, we express the covariance matrix of the square-root state in terms of that
of the state itself.

\section{Fidelity and affinity}

The statistical overlap between two probability  distributions
$f(s)$ and $g(s)$,
\begin{equation}
{\cal A}(f,g):=\sum_{s}\sqrt{f(s) g(s)},
\label{so}
\end{equation}
is widely used in statistical physics and interpreted as a measure of classical distinguishability \cite{B}. Indeed,
Eq.\ (\ref{so}) is one of the R\'enyi overlaps  which are distinguishability measures in their own right \cite{Fuchs}. In the quantum scenario, one usually considers a general measurement (POVM), i. e., a set
of non-negative operators ${\{\hat E_b\}}$ which is complete on the Hilbert space ${\cal H}$, or,
in other words, is a resolution of the identity:
${\sum_{b} \hat E_b}=\hat I$.
The subscript $b$ indexes the possible outcomes of the measurement whose probabilities in the quantum states $\hat \rho$
and $\hat \sigma$ are, respectively,
$p_{\hat \rho}(b)={\rm Tr}(\hat \rho \hat E_b)$
and $p_{\hat \sigma}(b)={\rm Tr}(\hat \sigma \hat E_b).$

An indicator of the closeness of the two quantum states
via measurements can be built by associating to any POVM
the statistical overlap\ (\ref{so}) of the probability
distributions $p_{\hat \rho}(b)$ and $p_{\hat \sigma}(b)$:
\begin{equation}
{\cal A}_{\{\hat E_b\}}(p_{\hat \rho}, p_{\hat \sigma})
:=\sum_b \sqrt{p_{\hat \rho}(b)p_{\hat \sigma}(b)}.
\label{ca}
\end{equation}
An important theorem proven in Refs. \cite{Fuchs,FC,barn1,barn2} states that the minimal overlap\ (\ref{ca}) over all quantum measurements is equal to the square root of fidelity:
\begin{equation}
\min_{\{\hat E_b\}}[{\cal A}_{\{\hat E_b\}}(p_{\hat \rho},
p_{\hat \sigma})]=\sqrt{{\cal F}(\hat \rho, \hat \sigma)}.
\label{fi1}
\end{equation}
Nowadays a widely accepted figure of merit in quantum information science, fidelity has been introduced by Uhlmann
in a quantum-mechanical framework \cite{Uhl}. It is related to
the Bures distance \cite{Bures},
\begin{equation}
[d_{B}(\hat \rho,\hat\sigma)]^2
:=2-2\sqrt{{\cal F}(\hat\rho,\hat\sigma)},
\label{db}
\end{equation}
and to the statistical distance \cite{BC}. Uhlmann derived
the explicit expression of fidelity in terms of density operators \cite{Uhl} :
\begin{equation}
\sqrt{{\cal F}(\hat \rho, \hat \sigma)}={\rm Tr}\left[ \left(
\sqrt{\hat \sigma}\,\hat \rho\,\sqrt{\hat \sigma}\right)^{\frac{1}{2}}\right]
=||\sqrt{\hat \rho}\,\sqrt{\hat \sigma}||_1.
\label{fi2}
\end{equation}
Here $||\hat B||_1:={\rm Tr}|\hat B|$ denotes the trace norm.
A friendly review of its main properties was written
by Jozsa \cite{Jozsa}.

Why is theorem\ (\ref{fi1}) so interesting? First, it tells us
that fidelity cannot decrease under any POVM. This feature is important in defining fidelity-based distance-type measures of various quantum properties. The most prominent example is
the Bures metric\ (\ref{db}) which is therefore contractive.
Second, Eq.\ (\ref{fi1}) exhibits the square root of fidelity
as a classically defined statistical overlap\ (\ref{so})
in the quantum-measurement version.

However, an appropriate quantum analogue
of the overlap\ (\ref{so}) of two probability distributions has recently been pointed out in Refs. \cite{Luo1,Luo3}. This is the affinity of two quantum states $\hat\rho$ and $\hat\sigma$:
\begin{equation}
{\cal A}(\hat\rho,\,\hat\sigma):={\rm Tr}(\sqrt{\hat\rho}\,
\sqrt{\hat\sigma}).
\label{aff}
\end{equation}
Affinity is a non-negative functional which is symmetric with
respect to $\hat\rho$ and $\hat\sigma$ and is connected
to the Hellinger distance between these states \cite{Luo1}:
\begin{equation}
[d_H(\hat \rho,\hat \sigma)]^2:={\rm Tr}\left[ \left(
\sqrt{{\hat \rho}}-\sqrt{{\hat \sigma}}\right)^2 \right]
=2-2 {\cal A}(\hat \rho,\hat \sigma).
\label{hd}
\end{equation}
Note that we employ as Hellinger distance the square root $d_H$
of the original one introduced in Ref. \cite{Luo1} on probabilistic grounds, because $d_H$ fulfills the triangle inequality \cite{Mend} and therefore is a true metric.

Affinity can be considered a closeness measure in its own right.
Its aspect indicates a close relation with the square root of the Uhlmann fidelity\ (\ref{fi2}).
Indeed, the square root of fidelity as well as affinity possess
some convenient features that ensure both the Bures and Hellinger
metrics, Eqs.\ (\ref{db}) and\ (\ref{hd}), respectively, to be good candidates for quantifying properties such as quantum correlations.
We list them as follows \cite{Fuchs,Luo1}.
\begin{enumerate}
\item ${\cal A}(\hat\rho,\,\hat\sigma)\leqq 1$.\\
Affinity reaches unity if and only if the quantum states
$\hat\rho$ and $\hat\sigma$ coincide.
\item {Invariance under unitary transformations:}
$${\cal A}(\hat U \hat \rho \hat U^{\dag}, \hat U \hat \sigma
\hat U^{\dag})={\cal A}(\hat \rho, \hat \sigma).$$
\item {Multiplicativity:}
$${\cal A} (\hat\rho_1\otimes\hat\rho_2, \hat\sigma_1\otimes\hat\sigma_2)={\cal A}(\hat \rho_1,\hat \sigma_1)
{\cal A}(\hat \rho_2,\hat \sigma_2).$$
\item {Joint concavity with respect to both arguments:}
$${\cal A}\left( \sum_j\lambda_j\hat \rho_j,
\sum_j\lambda_j\hat \sigma_j \right) \geqq \sum_j\lambda_j
{\cal A}(\hat \rho_j,\hat \sigma_j), \qquad \left( \lambda_j>0, \quad \sum_j\lambda_j=1 \right).$$
This property obviously implies the concavity of affinity in any
of its two arguments.
\item The inequality
$${\cal A}(\hat\rho, \hat\sigma)\leqq
\sqrt{{\cal F}(\hat \rho, \hat \sigma)}$$
holds \cite{Luo1} and saturates if and only if the states commute:
$${\cal A}(\hat\rho, \hat\sigma)
=\sqrt{{\cal F}(\hat \rho, \hat \sigma)}\;\Longleftrightarrow \;
[\hat\rho,\,\hat\sigma]=\hat 0.$$
\end{enumerate}
We emphasize that affinity can be employed in an equivalent way
with another figure of merit in discriminating between two quantum states, namely, their trace distance. This property was found
long ago by Holevo, who proved the following pair of inequalities
\cite{Hol}:
\begin{equation}
1-{\cal A}(\hat\rho, \hat\sigma)\leqq T(\hat\rho, \hat\sigma)
\leqq \sqrt{1-[{\cal A}(\hat\rho, \hat\sigma)]^2}.
\label{4}
\end{equation}
Here $T(\hat\rho,\hat\sigma):=\frac{1}{2}||\hat\rho
-\hat\sigma||_1$ is the trace distance between the states
$\hat\rho$ and $\hat\sigma $. More recently, Fuchs and
van de Graaf derived similar inequalities for the square root
of fidelity \cite{FG}:
\begin{equation}
1-\sqrt{{\cal F}(\hat\rho, \hat\sigma)}\leqq T(\hat\rho,
\hat\sigma) \leqq \sqrt{1-{\cal F}(\hat\rho, \hat\sigma)}.
\label{4a}
\end{equation}
The trace metric is a particularly important distinguishability
measure owing to its connection with the probability of error
between two quantum states \cite{FG}. Recall that it was used
by Hillery to formulate the first proposal of a distance-type
measure of non-classicality in Ref. \cite{Hil}. Despite of still being considered as a sensitive distance-type measure of any quantum property \cite{Breuer}, the trace metric remains difficult to deal with analytically.
In turn,  Eqs.\ (\ref{4}) and\ (\ref{4a}) show that both the Bures metric,
$1-\sqrt{{\cal F}(\hat\rho,\hat\sigma)}$,
and the Hellinger metric, $1-{\cal A}(\hat\rho,\hat\sigma)$, compare closely to the trace metric as measures
of distinguishability. In contrast to fidelity, affinity presents the advantage of being easily computable
for many classes of states.

\section{Affinity of two $n$-mode Gaussian states}

The Gaussian states of the quantum radiation field are important resources in many quantum information protocols \cite{BL,WPGCRSL,QC3}.
In particular, the two-mode ones are experimentally quite accessible and constitute a perfect test bed for studying
all kind of correlations between modes.
It appears to the present authors that the affinity of two arbitrary
$n$-mode GSs was first derived  by Holevo in Ref.\cite{Hol}
within the formalism of the $C^*$-algebra of commutation relations.
Here we recover it by using our recent findings on the products of two Gaussian operators in Ref. \cite{PT2012}, where we have also synthesized
some of the notions and notations employed for GSs. From now on,
we drop the subscript $G$ expressing the Gaussian character of the operators we deal with.

Let us consider an arbitrary $n$-mode GS ${\hat \rho}$. We find it convenient to introduce a $2n$-dimensional row vector $(\hat u)^T$  consisting of the canonical quadrature operators of all the modes:
\begin{equation}
(\hat u)^T:=(\hat q_1,\; \hat p_1,\;\dots,\; \hat q_n,\;\hat p_n),
\label{hatu}
\end{equation}
where $\hat{q}_j,\; \hat{p}_j$ are the quadrature operators
of the $j^{\mathrm{th}}$ field mode. Further, we employ a related
vector $u \in \mathbb{R}^{2n}$ whose components are arbitrary
eigenvalues of the quadrature operators of the modes:
\begin{equation}
u^T:=(q_1,\; p_1,\;\dots,\; q_n,\; p_n).
\label{Tu}
\end{equation}
In particular, let us denote $v \in \mathbb{R}^{2n}$ the vector
consisting of the expectation values of the quadrature operators
in the chosen state ${\hat \rho}$:
$v:={\langle{\hat u}\rangle}_{\hat{\rho}}.$
The second-order moments of the deviations from the means
of the canonical quadrature operators are collected as entries
of the symmetric covariance matrix (CM)
${\mathcal V}\in M_{2n}(\mathbb{R}),$ which is positive definite: ${\mathcal V}>0$. Recall that the characteristic function
$\chi (u)$ of the GS $\hat{\rho}$ is determined only by its first- and second-order moments of the canonical quadrature operators and has a specific exponential form:
\begin{equation}
\chi (u):={\rm Tr}\left[ \hat{D}(u) \hat{\rho}\right]
=\exp\left(-\frac{1}{2}\,u^{T}{\mathcal V}\,u-iv^{T}u\right).
\label{CF}
\end{equation}
In Eq.\ (\ref{CF}), $\hat{D}(u)$ stands for an  $n$-mode Weyl displacement operator, whose factors are single-mode Weyl shifting operators:
\begin{equation}
\hat{D}(u):=\bigotimes_{j=1}^{n}\hat{D}_j(q_j,\, p_j), \qquad
\hat{D}_j(q_j,\, p_j):
=\exp \left[-i\left(q_j\hat{p}_j-p_j\hat{q}_j\right)\right].
\label{D}
\end{equation}

According to Williamson's theorem \cite{W}, the CM ${\mathcal V}$ is congruent via a symplectic matrix $S \in Sp(2n, \mathbb{R})$ to a diagonal matrix:
\begin{equation}
{\cal V}=S\left[\bigoplus_{j=1}^{n}\,\left({\kappa}_j\,\sigma_0
\right) \right]S^T, \qquad \det({\cal V})
=\prod_{j=1}^{n}({\kappa}_j)^2.
\label{W}
\end{equation}
In Eq.\ (\ref{W}), the positive numbers ${\kappa}_j, \;\;
(j=1,2,\dots, n),$ are the symplectic eigenvalues of the CM
${\cal V}$, and $\sigma_0$ denotes the $2\times 2$ identity
matrix.
We take advantage of the standard matrix $J$ of the symplectic form on $\mathbb{R}^{2n},$
\begin{eqnarray}
J:=\bigoplus_{k=1}^{n}\,J_k, \quad J_k:=\left(
\begin{array}{cc}
0  & 1\\ -1 & 0
\end{array}
\right), \quad (k=1,2,\dots, n),
\label{J}
\end{eqnarray}
to write the Robertson-Schr\"odinger uncertainty relation
in two equivalent ways:
\begin{equation}
{\cal V}+\frac{i}{2}J  \geqq 0\;\;\Longleftrightarrow \;\; {\kappa}_j \geqq \frac{1}{2}, \qquad (j=1,2,\dots, n).
\label{R&S}
\end{equation}

It is worth emphasizing the Hilbert-space counterpart of the Williamson transformation\ (\ref{W}), namely,
the statement that any $n$-mode GS ${\hat \rho}$ is unitarily similar to an $n$-mode thermal state (TS)
${\hat \rho}_T$:
\begin{equation}
{\hat \rho}=\hat D(v)\hat U(S)\hat \rho_T\hat U^{\dag}(S)\hat D^{\dag}(v).
\label{u}
\end{equation}
The mapping  $\hat U(S)$ is called the metaplectic representation of the symplectic group $Sp(2n, \mathbb{R})$.
Obviously, the density operators ${\hat \rho}$ and $\hat \rho_T$
have the same discrete spectrum consisting of non-degenerate
and strictly positive eigenvalues. These are determined
by the symplectic eigenvalues of the CM ${\cal V}$,
\begin{equation}
\kappa_j =\bar{n}_j+\frac{1}{2}, \qquad (j=1,2,\dots, n),
\label{kappaj}
\end{equation}
where $\bar{n}_j$ are the mean photon occupancies of the modes in
the TS $\hat \rho_T$.

For further convenience, we define a square-root state as
\begin{equation}
\hat \rho_{sr}:=\left( {\rm Tr} \sqrt{\hat \rho}\right)^{-1}
\sqrt{\hat \rho}.
\label{sr}
\end{equation}
Let us consider the square-root state $(\hat \rho_T)_{sr}$,
which is itself an $n$-mode TS. Its CM has the symplectic eigenvalues specified in Ref. \cite{PT2012}:
\begin{equation}
\tilde{\kappa_j}=\kappa_j+\sqrt{{\left({\kappa}_j \right)}^2
-\frac{1}{4}}, \qquad (j=1,2,\dots, n).
\label{srsymp}
\end{equation}
Note that Eq.\ (\ref{u}) implies the transformation formula
\begin{equation}
\hat \rho_{sr}=\hat D(v)\hat U(S)(\hat \rho_T)_{sr}\,\hat U^{\dag}(S)
\hat D^{\dag}(v).
\label{usr}
\end{equation}
This means that $\hat \rho_{sr}$ is an $n$-mode GS whose CM is obtained from the CM of the state ${\hat \rho}$ just by replacing
the symplectic eigenvalues $\kappa_j$ with the corresponding ones
$\tilde{\kappa_j}$, Eq.\ (\ref{srsymp}).

The product of two Gaussian density operators,
$\hat{\mathcal B}:=\hat{\rho}^{\prime}\hat{\rho}^{\prime \prime},$
is largely investigated in Ref. \cite{PT2012} using as a main tool
the characteristic function\ (\ref{CF}) of the GS $\hat{\rho}$.
It turns out that $\hat{\mathcal B}$ is an $n$-mode Gaussian operator whose weight function in the Weyl expansion is of the type\ (\ref{CF})
up to a prefactor \cite{PT2012}. Its symmetric matrix
${\mathcal F}\in M_{2n}(\mathbb{C})$ is positive definite and is determined by the following composition rule \cite{PT2012}:
\begin{equation}
{\mathcal F}=-\frac{i}{2}J+\left({\mathcal V}^{\prime \prime}
+\frac{i}{2}J \right)
\left({\mathcal V}^{\prime}+{\mathcal V}^{\prime \prime}\right)^{-1}
\left({\mathcal V}^{\prime}+\frac{i}{2}J\right),
\label{rhorho}
\end{equation}
Further, in Ref. \cite{PT2012}, we have written the trace of the operator $\hat{\mathcal B}$ in terms of the CMs ${\mathcal V}^{\prime}$ and ${\mathcal V}^{\prime\prime}$ of the two factors, and the relative average $n$-mode displacement
$\delta v:=v^{\prime}-v^{\prime \prime}
=\langle{\hat u}\rangle_{{\hat{\rho}}^{\prime}}
-\langle{\hat u}\rangle_{{\hat{\rho}}^{\prime \prime}}$:
 \begin{eqnarray}
{\rm Tr}\,(\hat{{\mathcal B}})=\left[\det\left({\cal V}^{\prime}
+{\cal V}^{\prime\prime}\right)\right]^{-\frac{1}{2}}
\, \exp{\left[-\frac{1}{2}\left(\delta v \right)^T \left({\cal V}^{\prime}+{\cal V}^{\prime\prime}\right)^{-1}
\delta v \right]}>0.
\label{TR_HS}
\end{eqnarray}

According to Eq.\ (\ref{aff}), our evaluation of the affinity of
two multi-mode GSs involves square roots of Gaussian density operators.
 Let us denote by
 $\tilde{\mathcal V}^{\prime}$ and $\tilde{\mathcal V}^{\prime\prime}$ the CMs of the GSs $(\hat{\rho}^{\prime})_{sr}$ and
$(\hat{\rho}^{\prime\prime})_{sr}$, respectively.
Taking account of Eqs.\ (\ref{sr}),\ (\ref{usr}), and\ (\ref{TR_HS}),
we get the affinity
\begin{eqnarray}
&{\cal A}(\hat\rho^{\prime}, \hat\rho^{\prime\prime}):={\rm Tr}\left( \sqrt{\hat{\rho}^{\prime}}\,
\sqrt{\hat{\rho}^{\prime\prime}}\right)=
\frac{{\rm Tr}\left(\sqrt{\hat{\rho}^{\prime}}\right){\rm Tr}\left(\sqrt{\hat{\rho}^{\prime\prime}}\right)}
{\left[\det\left(\tilde{\mathcal V}^{\prime}
+\tilde{\mathcal V}^{\prime\prime}\right)\right]^{\frac{1}{2}}}
\nonumber\\
&\times \exp{\left[-\frac{1}{2}\left(\delta v \right)^T
\left(\tilde{\mathcal V}^{\prime}
+\tilde{\mathcal V}^{\prime\prime}\right)^{-1}\delta v \right]}.
\label{sqTR}
\end{eqnarray}
In the particular case $\hat{\rho}^{\prime}
=\hat{\rho}^{\prime\prime}=:\hat{\rho},$  Eq.\ (\ref{sqTR}) reduces
to the identity
\begin{equation}
{\rm Tr}\left(\sqrt{\hat{\rho}}\right)={\left[\det\left(2 \tilde{\mathcal V}\right)\right]^{\frac{1}{4}}}.
\label{sqTR1}
\end{equation}
Besides, Eq.\ (\ref{rhorho}) becomes a relationship between the CMs
${\mathcal V}$ and $\tilde{\mathcal V}$ of the $n$-mode GSs
$\hat{\rho}$ and $\hat{\rho}_{sr}$, respectively:
\begin{equation}
{\cal V}=\frac{1}{2}\left(\tilde{\cal V}
-\frac{1}{4}J{\tilde{\cal V}}^{-1}J \right).
\label{tildeV}
\end{equation}
 We now are ready to write down the affinity of two multi-mode GSs:
\begin{eqnarray}
{\cal A}(\hat\rho^{\prime}, \hat\rho^{\prime\prime})=\frac{2^{n}
\left[ \det(\tilde{\mathcal V}^{\prime})\,\det(\tilde{\mathcal V}^{\prime\prime})\right]^{\frac{1}{4}}}{\left[\det\left(\tilde{\mathcal V}^{\prime}
+\tilde{\mathcal V}^{\prime\prime}\right)\right]^{\frac{1}{2}}}
\, \exp{\left[ -\frac{1}{2}\left(\delta v \right)^T
\left( \tilde{\mathcal V}^{\prime}+\tilde{\mathcal V}^{\prime\prime}\right)^{-1}\delta v \right]}.
\label{sqTR2}
\end{eqnarray}
Remark that the affinity\ (\ref{sqTR2}) is written in terms of the CMs
$\tilde{\mathcal V}^{\prime}$ and $\tilde{\mathcal V}^{\prime\prime}$
of the square-root states $(\hat{\rho}^{\prime})_{sr}$ and
$(\hat{\rho}^{\prime\prime})_{sr}$, respectively.

\section{Significance of any Gaussian geometric discord}

The reason to define distance-type quantifiers of various
properties is in general given by the presumably simpler way of their evaluation. We are expecting that the results obtained
by a distance-type measure to be consistent with the original meaning of that property, usually based on complicated
extremization procedures. If not shown by explicit analytic
results, consistency could be checked by observing the behaviour
of the competing measures under some accepted  requirements formulated adequately for any quantum property.

Interestingly, when analyzing the consistency between different treatments of Gaussian discord, we are from the very beginning
in a privileged position to draw some conclusions. This happens because the originally defined discord \cite{OZ,HV}
for two-mode GSs has been calculated under the approach
of limiting  the set of all invoked one-party measurements
to the Gaussian ones \cite{PG,AD}. We were thus provided with
an analytic formula of the Gaussian discord in terms of one-mode von Neumann entropies. Moreover, according to Ref. \cite{PSBCL}, at least for the special two-mode GSs analyzed in the present paper, the Gaussian discord is {\em the} discord.
A counterintuitive result has also emerged:
it has been found that the only zero-discord GSs are the product ones \cite{PG,AD}. Even the separable GSs identified with Simon's criterion \cite{Si} do contain a good amount of quantum correlations measured by their discord. This finding has serious consequences on the interpretation of a geometric Gaussian discord defined
in Eq.\ (\ref{d0}), regardless of the distance we use.
Indeed, following Ref.\cite{DVB}, the definition of a geometric Gaussian discord for the two-mode GS
${\hat \rho}$ could be, similarly to Eq.\ (\ref{d0}) :
\begin{equation}
D_G ({\hat \rho})\sim \min_{{\hat \sigma} \in {\cal G}_P}
[{d}({\hat \rho},\,{\hat \sigma})]^2.
\label{gd}
\end{equation}
In Eq.\ (\ref{gd}),  ${\cal G}_P$ is the set of all two-mode
product GSs ${\hat \sigma}$ and $d$ is any distance having the required properties of discriminating among quantum states \cite{PVK}.
As the reference set of zero-discord states coincides with that of uncorrelated (product) states, in fact, Eq.\ (\ref{gd}) gives a measure of {\em all} the Gaussian correlations. Therefore, while according to the recent results of Ref.\cite{PSBCL} the originally defined discord could be Gaussian, all geometric Gaussian measures of discord appear to be just upper bounds for the corresponding geometric discord. They quantify all the correlations in a two-mode GS, both quantum and classical, being thus geometric analogues of the quantum mutual information,
\begin{equation}
 {\cal I}(\hat {\rho}_{AB}):={\cal S}(\hat \rho_A)+{\cal S}(\hat \rho_B)
 -{\cal S}(\hat \rho_{AB}).
\label{MI}
\end{equation}
In Eq.\ (\ref{MI}), ${\cal S}(\hat \rho)$ denotes the von Neumann entropy of the state $\hat \rho$: ${\cal S}(\hat \rho):
=-{\rm Tr}[\hat \rho \ln (\hat \rho)].$
In general, ${\cal I}(\hat {\rho}_{AB})$ is known to measure the total amount of correlations between the subsystems $A$ and $B$ in any given bipartite state $\hat \rho_{AB}$.
It appears that  we cannot separate the quantum correlations from their classical counterparts when using a distance-type Gaussian measure for the quantum discord. We have thus to admit that the closest zero-discord state to a Gaussian one is not Gaussian and this happens for {\em any} distance used as a quantifier of quantum discord.

As far as we know, a geometric discord for two-mode Gaussian states
was evaluated using the  Hilbert-Schmidt metric in Ref. \cite{adesso1}, and its {\em rescaled} version in the recent Ref. \cite{adesso2}.
Nevertheless, we find it instructive to take advantage of the beneficial properties of affinity and define a Gaussian geometric discord
of the type\ (\ref{d0}) in terms of the Hellinger distance between
an arbitrary two-mode GS $\hat \rho $ and the whole set ${\cal G}_P$
of the zero-discord GSs, i. e., the product ones:
\begin{equation}
D_H ({\hat \rho}):=\min_{{\hat \sigma}
\in {\cal G}_P}\frac{1}{2}[d_H ({\hat \rho},{\hat \sigma})]^2
= 1-\max_{{\hat \sigma} \in {\cal G}_P}{\cal A}({\hat \rho},
{\hat \sigma}).
\label{gdh}
\end{equation}

To perform the optimization\ (\ref{gdh}), we shall use
the relationship between the CMs
$\tilde{\mathcal V}_p$ and ${\mathcal V}_p$ in the case of a product state. This is considered in the Appendix together with a more general example.  We there write the entries of the CM $\tilde{\mathcal V}$ corresponding to a scaled standard-form CM ${\mathcal V}$ and find it
to be in a scaled standard form as well.

\section{Gaussian discord with the Hellinger distance}

\subsection{The closest Gaussian product state}

According to definition\ (\ref{gdh}), we now address
the following question: which  product state belonging
to the set ${\cal G}_P$ has the maximal affinity with respect to
a given two-mode GS $\hat \rho$ ?  Let us choose a state $\hat \rho$ whose CM ${\mathcal V}$ has a scaled standard form,
\begin{eqnarray}
{\mathcal V}=\left(
\begin{array} {cc}
\mathcal V_{1}       & {\mathcal C}\\
{\mathcal C} & \mathcal V_{2}
\end{array}
\right),
\label{st1}
\end{eqnarray}
i. e., it is partitioned into the following $2\times 2$ scaled
diagonal submatrices:
\begin{eqnarray}
&\mathcal V_{j}=\left(
\begin{array}  {cc}
b_j s_j & 0\\
0 & b_js^{-1}_j
\end{array}
\right),
\quad {\mathcal C}=\left(
\begin{array}  {cc}
c\sqrt{s_1s_2} & 0\\
0 & d (s_1s_2)^{-\frac{1}{2}}
\end{array}
\right), \nonumber\\
&\left(j=1,2,\;\; b_{j}\geqq \frac{1}{2},
\;\; c\geqq |d|\right).
\label{st2}
\end{eqnarray}
In Eq.\ (\ref{st2}),  $s_j$ are one-mode squeeze factors.
It has been proven in the Appendix that the CM
$\tilde{\mathcal V}$ of the square-root state\ (\ref{sr}) has
a similar scaled standard form\ (\ref{st1})--\ (\ref{st2}),
with entries specified by Eqs.\ (\ref{tildeb})
and\ (\ref{tildeb1}).
Let us examine the affinity\ (\ref{sqTR2}) between the given GS
$\hat \rho$ whose CM is $\mathcal V$,
Eqs.\ (\ref{st1})--\ (\ref{st2}), and a product state
$\hat\sigma$ whose CM is
${\mathcal V}_p$, Eqs.\ (\ref{st3})--\ (\ref{prod}).
It is obvious that the closest product state, hereafter denoted
by $\hat\sigma_*$, has to maximize the exponential
in Eq.\ (\ref{sqTR2}), so that it should have the same displacement parameters as $\hat \rho$. We are thus left to perform the maximization of the undisplaced affinity
\begin{eqnarray}
&{\cal A}_1(\hat\rho, \hat\sigma)
=\frac{4\left[\det(\tilde{\mathcal V})
\det(\tilde{\mathcal V}_p)\right]^{\frac{1}{4}}}{\left[ \det\left(
\tilde{\mathcal V}+\tilde{\mathcal V}_p\right)\right]^{\frac{1}{2}}},
\label{a1}
\end{eqnarray}
 with respect to the one-mode squeeze parameters $\varphi_j,r_j$,
and the symplectic eigenvalues $\eta_j$. We routinely find that
$\det(\tilde{\mathcal V}+\tilde{\mathcal V}_p)$ is minimal
at $\varphi_j=0,\; (j=1,2)$, so that our problem
is reformulated as the maximization of the function
\begin{eqnarray}
&{\cal A}_2(\hat\rho, \hat\sigma):=4\sqrt{\frac{{\tilde \kappa}_1 {\tilde \kappa}_2 {\tilde \eta}_1{\tilde \eta}_2}{\delta_1\delta_2}},
\label{a2}
\end{eqnarray}
with respect to the variables $r_j,\,\tilde \eta_j$.
In Eq.\ (\ref{a2}), use is made of the resolution into factors
$\,\det(\tilde{\mathcal V}+\tilde{\mathcal V}_p)=\delta_1 \delta_2,$
with:
\begin{eqnarray}
&\delta_1:=(\tilde b_1 \tilde s_1+\tilde \eta_1 {\rm e}^{2 r_1})(\tilde b_2 \tilde s_2+\tilde \eta_2 {\rm e}^{2 r_2})-{\tilde c}^2 \tilde s_1 \tilde s_2,\nonumber \\
&\delta_2:=\left( \frac{\tilde b_1}{\tilde s_1}+\tilde \eta_1
{\rm e}^{-2 r_1}\right) \left(\frac{\tilde b_2}{ \tilde s_2}
+\tilde \eta_2 {\rm e}^{-2 r_2}\right) -\frac{{\tilde d}^2}
{\tilde s_1 \tilde s_2}.
\label{del}
\end{eqnarray}
We finally list four conditions to be  fulfilled by the variables
$r_1, r_2, \tilde \eta_1, \tilde \eta_2$ at the maximum point
of the affinity\ (\ref{a2}):
\begin{eqnarray}
&(\tilde b_1 \tilde s_1+\tilde \eta_1 {\rm e}^{2 r_1})(\tilde b_2 \tilde s_2-\tilde \eta_2 {\rm e}^{2 r_2})-{\tilde c}^2 \tilde s_1 \tilde s_2=0,
\nonumber \\
&(\tilde b_1 \tilde s_1-\tilde \eta_1 {\rm e}^{2 r_1})(\tilde b_2 \tilde s_2+\tilde \eta_2 {\rm e}^{2 r_2})-{\tilde c}^2 \tilde s_1 \tilde s_2=0,
\nonumber \\
&\left( \frac{\tilde b_1}{\tilde s_1}+\tilde \eta_1 {\rm e}^{-2 r_1}\right) \left( \frac{\tilde b_2}{ \tilde s_2}-\tilde \eta_2 {\rm e}^{-2 r_2}\right) -\frac{{\tilde d}^2}{\tilde s_1 \tilde s_2}=0,\nonumber \\
&\left(\frac{\tilde b_1}{\tilde s_1}-\tilde \eta_1 {\rm e}^{-2 r_1}
\right) \left( \frac{\tilde b_2}{ \tilde s_2}+\tilde \eta_2
{\rm e}^{-2 r_2}\right) -\frac{{\tilde d}^2}{\tilde s_1 \tilde s_2}=0.
\label{max2}
\end{eqnarray}
The closest product two-mode state $\hat \sigma_* $ to the given state $\hat \rho$ is determined by the following parameters
arising from  Eq.\ (\ref{max2}):
\begin{eqnarray}
&(\tilde \eta_1)_*=\sqrt{\frac{\tilde b_1}{\tilde b_2}\tilde \kappa_1 \tilde \kappa_2}, \quad
(\tilde \eta_2)_*=\sqrt{\frac{\tilde b_2}{\tilde b_1}\tilde \kappa_1 \tilde \kappa_2},\nonumber\\
&{\rm e}^{2(r_j)_*}=\tilde s_j\left[\frac{\tilde b_1\tilde b_2-{\tilde c}^2}{\tilde b_1\tilde b_2-{\tilde d}^2}\right]^{\frac{1}{4}},
\quad (\varphi_j)_*=0,\quad (j=1,2).
\label{max3}
\end{eqnarray}
Although  $\hat \sigma_* $ is the nearest product state
to the state $\hat \rho$ when using the Hellinger metric, we find it
convenient to determine it in terms of the parameters
of the square-root state $\hat \rho_{sr}$, Eq.\ (\ref{sr}).

Some remarks concerning our results\ (\ref{max3}) are now at hand.
\begin{enumerate}
\item Equations\ (\ref{max3}) and\ (\ref{sr}) show that the GSs
$\hat \sigma_*$ and $\hat \rho_{sr}$ have the same purity:
\begin{equation}
{\rm Tr}\left[ (\hat \sigma_*)^2 \right]
={\rm Tr}\left[ (\hat \rho_{sr})^2 \right]
=(4\tilde \kappa_1 \tilde \kappa_2)^{-1}.
\end{equation}
\item The closest GS  $\hat \sigma_* $ is a product of one-mode squeezed thermal states for an arbitrary  given undisplaced GS
$\hat \rho$, except for any state with $|d|=c,$ whose closest product state is a two-mode thermal one.
\item For any undisplaced and scaled pure GS, the closest state
$\hat \sigma_*$ is also a pure state, namely, the product of
one-mode squeezed vacuum states  having the squeeze parameters $(r_j)_*=\ln s_j,\; (j=1,2),$ specified by Eq.\ (\ref{max3}).
Recall that, apart from the vacuum state, the only unscaled and
unshifted pure two-mode GSs are the two-mode squeezed vacuum states.
They are characterized by the following properties of their
standard-form parameters \cite{PTH2003,PT2008}:
\begin{equation}
b_1=b_2=:b>\frac{1}{2},\quad c=-d>0,\quad
b^2-c^2=\frac{1}{4}.
\label{sfpSVS}
\end{equation}
\end{enumerate}

\subsection{Explicit result}

The maximal affinity can straightforwardly be written in terms
of the parameters of the square-root state $\hat \rho_{sr}$,
Eq.\ (\ref{sr}), via Eqs.\ (\ref{max3}). We get first
\begin{eqnarray}
&(\delta_1)_*=2 \tilde s_1 \tilde s_2 \sqrt{\tilde b_1 \tilde b_2
-{\tilde c}^2}\left[\sqrt{\tilde b_1\tilde b_2}
+\sqrt{\tilde b_1\tilde b_2-{\tilde c}^2}\right],\nonumber\\
&(\delta_2)_*=2 \frac{1}{\tilde s_1 \tilde s_2} \sqrt{\tilde b_1\tilde b_2-{\tilde d}^2}\left[\sqrt{\tilde b_1\tilde b_2}
+\sqrt{\tilde b_1\tilde b_2-{\tilde d}^2}\right],
\end{eqnarray}
which yields, via Eq.\ (\ref{a2}), the maximal affinity
\begin{eqnarray}
&{\mathcal A}(\hat \rho):={\mathcal A}(\hat \rho,\, \hat \sigma_*)
\nonumber\\
&=\left[\frac{4\sqrt{\det \tilde{\mathcal V}}}{\left(\sqrt{\tilde b_1\tilde b_2}+\sqrt{\tilde b_1\tilde b_2
-{\tilde c}^2}\right) \left(\sqrt{\tilde b_1\tilde b_2}
+\sqrt{\tilde b_1\tilde b_2-{\tilde d}^2}\right)}\right]
^{\frac{1}{2}}.
\label{MA}
\end{eqnarray}
As expected from the properties of the Hellinger distance,
the Gaussian discord\ (\ref{gdh}) evaluated via Eq.\ (\ref{MA}) does not depend on the local squeezing factors. It has a remarkably simple analytic aspect in terms of the entries of the CM
$\tilde {\cal V}$. This nice formula hides a rather complicated structure when expressed with the entries of the CM ${\cal V}$.
For instance, it does not display a dependence of the sign of the parameter $\tilde d$.  However, as we shall see and comment later,  application of the transformation
relations\ (\ref{tildeb})--\ (\ref{tildeb4})  gives us distinct
expressions for the maximal affinity of the states whose CMs
differ only by the sign of $d$.

A tedious calculation of $\delta_{1*}$ and $\delta_{2*}$ by inserting
the correspondence rules\ (\ref{tildeb}) leads
to a convenient form of the maximal affinity\ (\ref{MA}):
\begin{eqnarray}
\mathcal A (\hat \rho)=\frac{2 {\cal K}(\det {\cal V})^{\frac{1}{4}}}
{\left[{\cal K}^2 \sqrt{\det {\cal V}}
+{\cal K}(\det {\cal V})^{\frac{1}{4}} \sqrt{\cal Q}
+\sqrt{{\cal B}_1 {\cal B}_2}\right]^{\frac{1}{2}}}.
\label{MA1}
\end{eqnarray}
Here we have denoted:
\begin{eqnarray}
{\cal B}_{1,2}:= b_1 b_2\, {\cal K}^2+\frac{1}{4}\left( b_{1,2}\,c
+b_{2,1}\,d \right)^2,
\label{B12}
\end{eqnarray}
\begin{eqnarray}
{\cal Q}:=&\left(\sqrt{b_1 b_2-c^2}+\sqrt{b_1 b_2-d^2}\right)^2
\left[b_1 b_2 \left(\sqrt{{\cal M}_1}+\sqrt{{\cal M}_2}\right)^2\right.\nonumber\\
&-\left.\frac{1}{4}(b_1-b_2)^2\right]-\left(\sqrt{{\cal B}_1}
-\sqrt{{\cal B}_2}\right)^2.
\label{C}
\end{eqnarray}
The notations ${\cal K}, {\cal M}_1, {\cal M}_2$ are introduced in Eqs.\ (\ref{tildeb1}) and\ (\ref{M}).

Before proceeding to apply the formula\ (\ref{MA1}) to several notorious GSs, we recall the Peres-Simon theorem \cite{Si}: Preservation of non-negativity of the density matrix under
partial transposition (PT) is a necessary and sufficient
condition for the separability of any two-mode GS.
The standard form\ (\ref{st1})--\ (\ref{st2}) of the CM
of a separable two-mode GS $\hat \rho$ transforms into that
of its positive partial transpose ${\hat\rho}^{PT}$ just
by changing the sign of the parameter $d$: $d\rightarrow -d$.
In view of a lemma proven by Simon \cite{Si}, the two-mode GSs
with positive $d$ are separable. However, their discord does not vanish \cite{PG,AD} and can be evaluated within our present approach.

\section{Symmetric two-mode Gaussian states}

As a first application of the formula\ (\ref{MA1}), we now consider
the geometric discord for the noteworthy class of the symmetric
two-mode GSs. These are defined as states possessing equal marginal purities, i. e., with $b_1=b_2=:b.$
Notice that definition\ (\ref{B12}) implies the equality ${\cal B}_{1}={\cal B}_{2}=:{\cal B}$. The explicit symplectic eigenvalues of the CM of a symmetric two-mode GS \cite{PT2008},
\begin{equation}
\kappa_{1}=\sqrt{(b+c)(b+d)},\quad \kappa_{2}=\sqrt{(b-c)(b-d)},
\label{ksym}
\end{equation}
allow us first to write $\kappa_{1}^2-\kappa_{2}^2=2 b(c+d),$ and then
Eq.\ (\ref{B12}) becomes via Eqs.\ (\ref{tildeb2})--\ (\ref{N}):
\begin{eqnarray}
{\cal B}={\cal K}^2\left[b^2+\frac{1}{4}\left(\sqrt{{\cal M}_1}
-\sqrt{{\cal M}_2}\right)^2\left(\sqrt{{\cal N}_1}
-\sqrt{{\cal N}_2}\right)^2\right].
\label{ks}
\end{eqnarray}
After a routine algebra, the Gaussian Hellinger discord,
Eq.\ (\ref{gdh}), simplifies to:
\begin{eqnarray}
& D_H({\hat \rho}_{sym})=1-\mathcal A (\hat \rho_{sym})
\nonumber \\
& =1-\frac{4(\det {\cal V})^{\frac{1}{4}}}{ \kappa_{1}^{PT}
+\kappa_{2}^{PT}+2(\det {\cal V})^{\frac{1}{4}}
\left( \sqrt{{\cal N}_1}-\sqrt{{\cal N}_2}\right)}.
\label{s1}
\end{eqnarray}
Here we have employed the symplectic eigenvalues of the CM
of the PT symmetric two-mode GS:
\begin{equation}
\kappa_{1}^{PT}=\sqrt{(b+c)(b-d)}, \quad \kappa_{2}^{PT}
=\sqrt{(b-c)(b+d)}.
\label{ksympt}
\end{equation}
The quantities  ${\cal N}_1,\; {\cal N}_2$
are defined in Eq.\ (\ref{N}). Equation\ (\ref{s1})
encompasses both cases $d=\pm|d|$.

It is remarkable that the obtained maximal affinity
of the symmetric GSs depends only on the symplectic eigenvalues of the state and of its partial transpose (PT).
 Moreover, for the symmetric GSs with $d=-|d|$ we can compare this dependence with some results for the degree of entanglement.
 Recall that the symmetric GSs are entangled when the condition
$\kappa_{2}^{PT}=\sqrt{(b-c)(b-|d|)}<\frac{1}{2}$
is met. Their entanglement of formation could be exactly evaluated
\cite{G1} and has been found to depend only of the smallest symplectic eigenvalue $\kappa_{2}^{PT}$ of the CM
of the PT symmetric GS. A similar $\kappa_{2}^{PT}$-dependence
has been found by the present authors for a Gaussian distance-type
degree of entanglement defined with the Bures metric \cite{PT2008}.

\section{States with $|d|=c$}

Owing to their experimental applications, the two-mode GSs
identified by the relation $|d|=c$ are particularly relevant
when evaluating quantum correlations.  The sign of the parameter
$d$ splits this case in so far that we discuss two distinct classes
of states.

For negative $d=-c$, we deal with the familiar {\em two-mode squeezed
thermal states} (STSs) \cite{PTH2003}.  According to Simon's separability criterion \cite{Si}, a STS is separable when it meets the condition
\begin{equation}
\left( b_1-\frac{1}{2}\right) \left( b_2-\frac{1}{2}\right)
-c^2\geqq 0,
\label{sep}
\end{equation}
 and entangled in the opposite case \cite{PTH2001,PTH2003}.

On the contrary, all the GSs with positive $d=c$ are separable. Specifically, such a state is the partial transpose of a separable STS. Since states with $d=c>0$ can be obtained by mixing two input modes in distinct TSs in a lossless beam splitter, we call them
{\em mode-mixed thermal states} (MTSs).

According to Eqs.\ (\ref{gdh}) and\ (\ref{MA}), the Gaussian geometric discord simplifies to
\begin{eqnarray}
 D_H ({\hat \rho}_{|d|=c})=\frac{\sqrt{\tilde b_1\tilde b_2}
-\sqrt{\tilde b_1\tilde b_2
-{\tilde c}^2}}{\sqrt{\tilde b_1\tilde b_2}
+\sqrt{\tilde b_1\tilde b_2-{\tilde c}^2}}.
\label{D-sts}
\end{eqnarray}
Our aim is to express the Gaussian discord
$D_H ({\hat \rho}_{|d|=c})$ in terms of the entries of the CM
$\cal V$ of the state $\hat \rho$ for both classes of states presented above. We can do this by using our general
equations\ (\ref{tildeb})--\ (\ref{tildeb4}) and specializing
the maximal affinity\ (\ref{MA}). The current parametrizations of these states enable us to obtain more insightful formulae.

\subsection{Case $d=-c<0:$ Two-mode squeezed thermal states}

A STS $\hat \rho_{ST}$ is the result of the action of a two-mode
squeeze operator  \cite{SC1985},
\begin{equation}
\hat S_{12}(r,\phi):=\exp{\left[ \,r \left( {\rm e}^{i\phi}
\hat a^{\dag}_1 \hat a^{\dag}_2-{\rm e}^{-i\phi} \hat a_1
\hat a_2 \right) \right] },\nonumber\\
 \left( r>0,\;\;
\phi\in (-\pi,\pi] \right),
\label{S_12}
\end{equation}
on a product-thermal state \cite{PTH2003}:
\begin{equation}
 \hat \rho_{ST}=\hat S_{12}(r,\phi)\hat \rho_T(\kappa_1,\kappa_2)
 \hat S^{\dag}_{12}(r,\phi),
\label{STS}
\end{equation}
The symplectic eigenvalues of its CM are determined
by the thermal mean occupancies $\bar n_j$ of the two-mode TS
$\hat \rho_T(\kappa_1,\kappa_2): \, \kappa_j=\bar n_j+\frac{1}{2},\;
(j=1,2)$. We find the  entries of the standard-form CM ${\cal V}$ depending on the squeeze parameter $r$ as follows \cite{PTH2003}:
\begin{eqnarray}
&b_{1,2}=\kappa_{1,2}[\cosh(r)]^2
+\kappa_{2,1}[\sinh(r)]^2,\nonumber \\
&c=-d=(\kappa_{1}+\kappa_{2})\cosh(r)\sinh(r).
\label{sts1}
\end{eqnarray}
Insertion of Eq.\ (\ref{sts1}) into Eq.\ (\ref{sep}) allows one
to write explicitly the separability threshold $r_s$ of the STSs
\ (\ref{STS}) \cite{PTH2003}:
\begin{equation}
[\sinh(r)]^2 \leqq [\sinh(r_s)]^2:=\frac{\bar n_1 \bar n_2}
{\bar n_1+\bar n_2+1}.
\label{r_s}
\end{equation}
Any separable two-mode STS fulfils the condition $r \leqq r_s,$
Eq.\ (\ref{r_s}), and, moreover, is  classical, i. e., it has
a well-behaved Glauber-Sudarshan $P$ representation.

From the general equations\ (\ref{tildeb})--\ (\ref{tildeb4})
we get the maximal affinity
\begin{eqnarray}
\mathcal A (\hat \rho_{ST})=\frac{2\left( \sqrt{{\cal N}_1}
+\sqrt{{\cal N}_2}\right)}{\sqrt{{\cal N}_1}+\sqrt{{\cal N}_2}
+\left [\left( \sqrt{{\cal N}_1}+\sqrt{{\cal N}_2}\right)^2
+4c^2\right]^{\frac{1}{2}}}.
\label{maxaff2}
\end{eqnarray}
Then, taking into account the definitions\ (\ref{N}),
Eq.\ (\ref{maxaff2}) simplifies to
\begin{eqnarray}
\mathcal A (\hat \rho_{ST})=\frac{2 (\kappa_1+\kappa_2)}{\kappa_1
+\kappa_2+[(\kappa_1+\kappa_2)^2
+8c^2(\kappa_1\kappa_2+1/4-\sqrt{\cal D})]^{\frac{1}{2}}},
\label{maxaff2-}
\end{eqnarray}
where the symplectic eigenvalues $\kappa_{1,2}$ of a STS are expressed
in terms of the standard-form parameters as it follows
from Eq.\ (\ref{sts1}):
\begin{equation}
\kappa_{1,2}=\frac{1}{2}\left[ \sqrt{(b_1+b_2)^2-4 c^2}
\pm (b_1-b_2)\right].
\label{kappaSTS}
\end{equation}
The maximal affinity\ (\ref{maxaff2-}) becomes
\begin{equation}
\mathcal A (\hat \rho_{ST})=\frac{2}{\sqrt{\cal X}+1}, \quad
{\cal X}:=1+2\left(\sqrt{\det{\cal V}}+\frac{1}{4}
-\sqrt{{\cal D}}\right)[\sinh(2r)]^2,
\label{X}
\end{equation}
where the symplectic invariant ${\cal D}$ is written
in Eq.\ (\ref{tildeb4}). Alternatively, Eq.\ (\ref{X}) can be obtained
by making direct use of the standard-form CM of the square-root state $(\hat\rho_{ST})_{sr}$, Eq.\ (\ref{sr}), whose entries are:
\begin{eqnarray}
&\tilde b_{1,2}=\tilde\kappa_{1,2}[\cosh(r)]^2+\tilde\kappa_{2,1}
[\sinh(r)]^2,\nonumber \\
&\tilde c=-\tilde d=(\tilde\kappa_{1}
+\tilde\kappa_{2})\cosh(r) \sinh(r),
\label{sts2}
\end{eqnarray}
with $\tilde\kappa_{1,2}$ given by Eq.\ (\ref{srsymp}).
After some algebra we recover the result\ (\ref{X})
that we now specialize to a couple of interesting cases.

Let us first write the geometric discord for {\em symmetric} STSs $(b_1=b_2=:b)$. Equation\ (\ref{X}) leads to an expression similar to Eq.\ (\ref{D-sts}):
\begin{equation}
 D_H({\hat \rho}_{ST})=\frac{b-\sqrt{b^2-{c}^2}}{b+\sqrt{b^2
 -{c}^2}},\qquad (b_1=b_2=:b).
\label{D-sts2}
\end{equation}
In addition, by applying Eq.\ (\ref{sts1}), we find that
the geometric discord quantified by Hellinger distance for all
the symmetric STSs is independent of the degree of mixing and
has the simple expression
\begin{eqnarray}
 D_H({\hat \rho}_{ST})=[\tanh(r)]^2.
\label{sym}
\end{eqnarray}
Note that this set of states includes the two-mode squeezed vacuum ones \cite{SC1985}:
\begin{equation}
 \hat \rho_{SV}=\hat S_{12}(r,\phi)|0,0\rangle \langle 0,0|
 \hat S^{\dag}_{12}(r,\phi).
\label{SVS}
\end{equation}
In view of Eqs.\ (\ref{kappaSTS}) and\ (\ref{sts1}), respectively,
the characterization\ (\ref{sfpSVS}) of these pure states gives
the minimal symplectic eigenvalues $\kappa_1=\kappa_2=\frac{1}{2}$
and then the standard-form parameters $b=\frac{1}{2}\cosh(2r),\,
c=\frac{1}{2}\sinh(2r).$

Another simplification arises for any STS whose CM has a single minimal symplectic eigenvalue: $\kappa_2=\frac{1}{2}$. Equation\ (\ref{X})
then gives the Hellinger discord
\begin{equation}
D_H({\hat \rho}_{ST})=\frac{\{(\bar n_1+1)[\sinh(2r)]^2+1\}^{1/2}-1}
{\{(\bar n_1+1)[\sinh(2r)]^2 +1\}^{1/2}+1},
\quad \left( \kappa_2=\frac{1}{2}\right).
\end{equation}

As noticed earlier, the original geometric discord defined
with the Hilbert-Schmidt metric proved to display
some inconveniences \cite{P} related to its property
of non-contractivity.  Quite recently, a modification
of the Hilbert-Schmidt norm for continuous-variable systems was addressed in Ref. \cite{adesso2} in order to prevent
its dependence on the global purity of the states involved. The result reported in Eq. (B.6) of Ref. \cite{adesso2} for two-mode STSs can now
be compared to our Eq.\ (\ref{D-sts}). The two measures share the same functional form but with the significant difference of having the parameters $\tilde {b}_1, \tilde{b}_2,  \tilde{c}
= |\tilde{d}|$  replaced in Ref. \cite{adesso2} by the corresponding entries of the CM ${\cal V}$. Only for symmetric STSs
our Eq.\ (\ref{X}) and Eq. (B.6) of Ref.\cite{adesso2} give
the same result, namely, Eq.\ (\ref{sym}). For asymmetrical STSs,
the difference between the two expressions of the geometric discord
are determined by the asymmetry of the modes and, in general,
we found them very close. This can be seen in Fig. 1,
where the monotonic behaviour of two geometric measures of discord with the degree of squeezing is displayed by a non-symmetric example.

\begin{figure}[t,h]
\center
\includegraphics[width=6cm]{fig1.eps}
\caption{(Color online) Hellinger discord (purple dotted plot),
Eq.\ (\ref{D-sts}), and the rescaled Hilbert-Schmidt one (magenta line plot), Eq.(B.6) of Ref.\cite{adesso2}, versus the squeeze parameter $r$
for a non-symmetric STS  with the symplectic eigenvalues
$\kappa_1=0.5$ and $\kappa_2=20.5$.}
\end{figure}

\subsection{Case $d=c>0:$ Mode-mixed thermal states}

It is well known that a beam splitter mixes two incident modes
to produce two outgoing ones. Recall that the optical interference
of two modes in a reversible, lossless beam splitter is described
by a mode-mixing operator \cite{Bonny}:
\begin{equation}
\hat M_{12}(\theta,\phi):=\exp{\left[ \frac{\theta}{2}
\left( {\rm e}^{i\phi} \hat a_1  \hat a^{\dag}_2-{\rm e}^{-i\phi}
\hat a^{\dag}_1  \hat a_2 \right) \right] }.
\label{M_12}
\end{equation}
As a matter of fact, this is a $SU(2)$ displacement operator
\cite{Radcliffe, Arecchi} written employing the Jordan-Schwinger
two-mode bosonic realization of angular momentum \cite{BS, Ulf}.
Its parameters are the spherical polar angles $\theta$ and
$\phi: \theta\in [0,\pi],\; \phi\in (-\pi,\pi]$.  The co-latitude
$\theta$ determines the intensity transmission and reflection coefficients of the device, which are
$T=\left[ \cos \left( \frac{\theta}{2}\right) \right]^2$
and, respectively,
$R=\left[ \sin \left( \frac{\theta}{2}\right) \right]^2.$

When choosing an asymmetrical two-mode TS as input to a beam splitter,
then we get an emerging MTS as output:
\begin{equation}
\hat \rho_{MT}=\hat M_{12}(\theta,\phi)\hat \rho_T(\kappa_1,\kappa_2)
\hat M^{\dag}_{12}(\theta,\phi), \quad (\kappa_1 >\kappa_2).
\label{MTS}
\end{equation}
The standard-form entries of the output CM are found to be:
\begin{eqnarray}
&b_{1,2}=\kappa_{1,2}\left[ \cos \left( \frac{\theta}{2}\right)
\right]^2+\kappa_{2,1}\left[ \sin \left( \frac{\theta}{2}\right) \right]^2,\nonumber \\
&c=d=(\kappa_{1}-\kappa_{2})\cos \left( \frac{\theta}{2}\right)
\sin \left(\frac{\theta}{2}\right).
\label{m1}
\end{eqnarray}
As already mentioned, any MTS $\hat \rho_{MT}$  is a separable two-mode GSs with $d=c>0$ and, consequently, it is related to a separable STS
by partial transposition \cite{Si}. Furthermore, any MTS
$\hat \rho_{MT}$ is a classical state, i. e., it possesses a well-behaved
Glauber-Sudarshan $P$ representation.
The general equations\ (\ref{tildeb})--\ (\ref{tildeb4}) yield
the maximal affinity
\begin{eqnarray}
\mathcal A (\hat \rho_{MT})=\frac{2\left( \sqrt{{\cal M}_1}
+\sqrt{{\cal M}_2}\right)}{\sqrt{{\cal M}_1}+\sqrt{{\cal M}_2}
+\left[ \left(\sqrt{{\cal M}_1}+\sqrt{{\cal M}_2}\right)^2
+4c^2\right]^{\frac{1}{2}}}.
\label{maxaff1}
\end{eqnarray}
By use of Eq.\ (\ref{M}), Eq.\ (\ref{maxaff1}) simplifies to
\begin{eqnarray}
\mathcal A (\hat \rho_{MT}) =\frac{2(\kappa_1-\kappa_2)}
{\kappa_1-\kappa_2+\left[ (\kappa_1-\kappa_2)^2
+8c^2\left(\kappa_1\kappa_2-\frac{1}{4}-\sqrt{\cal D}\right)
\right]^{\frac{1}{2}}}.
\label{maxaff1+}
\end{eqnarray}
From Eq.\ (\ref{m1}) we get the symplectic eigenvalues $\kappa_{1,2}$
of the CM of a MTS expressed in terms of its standard-form entries:
\begin{equation}
\kappa_{1,2}=\frac{1}{2}\left[ b_1+b_2 \pm  \sqrt{(b_1-b_2)^2
+4 c^2}\right].
\label{kappaMTS}
\end{equation}
The maximal affinity\ (\ref{maxaff1+}) becomes
\begin{equation}
\mathcal A (\hat \rho_{MT})=\frac{2}{\sqrt{\cal Y}+1}, \quad
{\cal Y}:=1+2\left(\sqrt{\det{\cal V}}-\frac{1}{4}
-\sqrt{{\cal D}}\right)[\sin(\theta)]^2.
\label{Y}
\end{equation}

\section{Discussion and conclusions}

In view of the analysis in Section 4, an examination of the consistency between the Gaussian discord\ (\ref{gdh}) evaluated via Eq.\ (\ref{MA})  and the original exact one \cite{PG,AD} becomes a necessity.
To make our presentation as simple as possible, we choose to deal with symmetric two-mode GSs ($b_1=b_2=:b$) having $|d|=c$.
In this particular case, the standard-form parameter $b$ is related
to the total mean photon number $\langle \hat N \rangle$ of the state:
\begin{equation}
b=\frac{1}{2}\left(\langle \hat N \rangle +1 \right).
\label{b}
\end{equation}
 By specializing the Gaussian discord
of Refs.\cite{PG,AD}, we get the formula:
\begin{equation}
{\mathcal D}(\hat \rho_{|d|=c})=h(b)-h(\kappa_1)-h(\kappa_2)+h(y).
\label{PA}
\end{equation}
Here $h(x)$ is the entropic function
\begin{equation}
h(x):=\left( x+\frac{1}{2}\right)
\ln\left (x+\frac{1}{2}\right)
-\left( x-\frac{1}{2}\right)
\ln\left (x-\frac{1}{2}\right), \quad \left( x\geqq \frac{1}{2}\right),
\label{h}
\end{equation}
$\kappa_{1,2}$ are the symplectic eigenvalues of the CM,
and  $\;y:=b-\frac{c^2}{b+\frac{1}{2}}$.
The states being symmetric,  the quantum mutual information\ (\ref{MI}) specializes to
\begin{equation}
{\mathcal I}(\hat \rho_{|d|=c})=2 h(b)-h(\kappa_1)-h(\kappa_2).
\label{MI1}
\end{equation}
As seen from Eq.\ (\ref{PA}),  states whose CMs differ only by the sign of the parameter $d$ have different amounts of quantum correlations because they have different symplectic eigenvalues. However, they have the same amount of classical correlations:
\begin{equation}
{\mathcal C}(\hat \rho_{|d|=c}):={\mathcal I}(\hat \rho_{|d|=c})-{\mathcal D}(\hat \rho_{|d|=c})=
h(b)-h(y).
\label{CC}\end{equation}
According to Eq.\ (\ref{sep}), for $b-c\geqq\frac{1}{2},$
all the symmetric states with $d=\pm c$ are separable.
In the absence of any entanglement, their geometric Gaussian discord measures all the other Gaussian correlations (classical and quantum). Correlations of such states are described in Fig. 2 for a fixed value
of their common purity,
$${\rm Tr} \left[(\hat \rho)^2 \right]
=\left[ \det(2{\cal V})\right]^{-\frac{1}{2}}
=\frac{1}{4\left( b^2-c^2 \right)}.$$

\begin{figure*}[h]
\center
Separable states with $|d|=c$

\includegraphics[width=7cm]{fig2a.eps}
\includegraphics[width=7cm]{fig2b.eps}
\caption{ (Color online) We compare the aspect of the geometric Hellinger discord, Eq.\ (\ref{s1}), (right panel), and two measures of Gaussian correlations (left panel), when describing the same sets of separable states. The full line curves represent MTSs,
$(d=c>0)$, corresponding to  $b^2-c^2=6.25$ (fixed purity). Separable STSs,  $(d=-c<0),$ are shown by purple
dashed curves at the same value of the purity. The lower curves in the left panel represent the original Gaussian discord, Eq.\ (\ref{PA}), the upper ones are the mutual information, Eq.\ (\ref{MI1}), while the full black plot represents the classical correlations, Eq.\ (\ref{CC}). Note that symmetric STSs and MTSs possess the same amount of classical correlations at the same purity.}
\end{figure*}

It is rather intriguing that the Gaussian discord of the states with $d=c>0$, i. e., MTSs, turns out to be larger than that of the corresponding ones with $d=-c<0$, which are separable STSs. This feature is common to both Gaussian measures we have used: the geometric
Hellinger discord, Eq.\ (\ref{s1}), (right panel) and the original one (lower curves on the left panel). Remark that in this case the graphs
of the quantum mutual information (upper curves on the left panel) are just shifted with respect to the discord plots by the same amount
of classical correlations, Eq.\ (\ref{CC}).
Figure 2 exhibits another important fact too: the two Gaussian
discords\ (\ref{s1}) and\ (\ref{PA}) have a similar behaviour
with respect to the parameter $b$.  This can be interpreted as an expression of their consistency.

\begin{figure}[h]
\center
\includegraphics[width=7cm]{fig3.eps}
\caption{(Color online) Measures of Gaussian correlations for a symmetric STS with $\sqrt{\det {\cal V}}=b^2-c^2=6.25$ versus the parameter $b$. The dashed purple curve represents the quantum mutual information, Eq.\ (\ref{MI1}) with $\kappa_1=\kappa_2
=\sqrt{b^2-c^2}$. The black full line plot represents the original Gaussian discord, Eq.\ (\ref{PA}), the blue dotted line is the Hellinger discord, Eq.\ (\ref{D-sts2}), and
the dashed-dotted green curve is the entanglement of formation, Eq.\ (\ref{EF}).}
\end{figure}

We now examine the influence of entanglement on both Gaussian
discords. Let us thus consider a symmetric STS. It is separable
for $\kappa^{PT}=b-c\geqq\frac{1}{2}$ and entangled
for $\kappa^{PT}<\frac{1}{2}$.
In Fig. 3 we plot both discords\ (\ref{s1}) and\ (\ref{PA})
for a STS with $b^2-c^2=6.25$ as depending of the parameter $b$.
The state is entangled for $b>6.5$ and its exact entanglement
of formation \cite{G1},
\begin{equation}
E_f(\hat \rho_{ST})=h(z),\qquad z:=\frac{(b-c)^2
+\frac{1}{4}}{2(b-c)},
\label{EF}
\end{equation}
is also shown in Fig. 3. We first notice the smooth monotonic increasing of the two measures of Gaussian discord,
Eqs.\ (\ref{PA}) and\ (\ref{D-sts2}). Both plots are consistent
and seem to be totally insensitive to the absence or presence
of entanglement.

To sum up, in this paper we have considered the Hellinger
metric as a measure of quantum discord for two-mode Gaussian
states. As any other distance used to define a geometric Gaussian discord, the Hellinger one is a measure of the total amount of correlations, providing just an upper bound of the exact geometric discord. We have reviewed some useful properties of the affinity,
which is a close relative of the Uhlmann fidelity. It is interesting
to note that almost simultaneously with launching fidelity
as a new tool in quantum information processing
in Ref. \cite{Jozsa}, affinity was analyzed as a possible quantifier
of the information flow in and out of a black hole \cite{Al}. It was
its clear meaning  expressed in our Eq.\ (\ref{fi1}) that gave prominence to fidelity  as a measure of closeness between quantum states. However, we here have shown that the recently reconsidered affinity \cite{Luo1,Luo3} and the more popular fidelity share
some  useful properties for defining distance-type measures
of quantum properties such as correlations.
In this paper we have defined and exactly evaluated a geometric discord based on the maximal affinity between a given two-mode GS
and the whole set of two-mode product GSs. The general analytic formula we have derived has easily been specialized to symmetric
GSs. A detailed analysis is devoted to a pair of classes
of two-mode GSs that have insightful parametrizations and at the same time can readily be prepared: the squeezed thermal states
and the mode-mixed thermal ones. The latter are separable states
and have been compared with the appropriate separable STSs,
in order to evaluate their total amounts of correlations in the absence of entanglement by means of the Hellinger discord. We have chosen symmetric states
from both classes and examined the variation of their Gaussian discords with the mean total number of photons at a fixed global purity.
Comparison of the Hellinger discord\ (\ref{s1}) with the originally defined one\ (\ref{PA}) indicates consistency by inducing the same ordering of all the Gaussian correlations.

\ack{This work was supported  by the Romanian National Authority for Scientific Research (CNCS – UEFISCDI) through Grant No.~PN-II-ID-PCE-2011-3-1012 and by the University of Bucharest through Grant No. UB 898/2013.}

redpajama_set_name: RedPajamaArXiv


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Last fall during the political season, everyone running for the Montana Legislature touted their credentials as being for public access.
But what does that really mean? For Montana's hunters, anglers, hikers and other outdoors enthusiasts, it means the ability to get to our public lands to enjoy them.
The access to the outdoors that makes Montana such a great place to live also makes it tempting for some people to try to close off public roads. By putting up gates on public roads that lead to public land, they can turn vast swaths of our national forests, Bureau of Land Management and other public lands into private playgrounds.
Everybody loses when this happens — except for the one person who claims exclusive access to that public land. It harms not only the public who enjoys public land, but also wildlife biologists with the Montana Department of Fish, Wildlife and Parks. The agency relies on public hunters to manage our public wildlife, and getting hunters on the ground helps get the harvest of deer, elk and antelope needed to meet their management objectives. When someone illegally gates a public road and blocks access for hunting, they are ultimately worsening wildlife concentration problems for neighboring farmers and ranchers.
Under current state law, the fine for gating a county road is $10 per day. That fine might have had teeth when it was set decades ago. But today, it's not a deterrent to keep people from gating these roads in the first place. Texans and Californians who can afford multi-million-dollar vacation homes in Montana would think nothing of paying $10 a day to block public access to public land.
In comparison, the fine for failing to shovel the sidewalk in Billings is initially $300, then $500 per day thereafter. And the fine for letting your sprinkler hit the sidewalk in Helena is $500.
This lack of any teeth in state law results in cases like the Hughes Creek Road in Ravalli County. That public road has been illegally gated for decades, cutting off access to the Bitterroot National Forest. The fact that it's a county road is indisputable: in 1983 the very same people who blocked it off requested the county abandon the road. Recently the Montana Supreme Court ruled that it's a county road yet it remains illegally gated.
A simple step toward getting these public county roads opened to the public would be to have a real, meaningful fine in state law. Lawmakers need to increase the fine to up to $500 per day, with no minimum.
This would give county attorneys a powerful tool to address blocked roads, create a strong disincentive to gate roads in the first place, and ultimately improve public access to our public lands.
Every legislator says they're for public access. Increasing the fine for illegally blocking a public road gives them a chance to walk the walk.
Bill Geer is president of the Montana Wildlife Federation.


text:
I caught a few folks on Facebook recently who posted something to the effect of: "Tell me things that would get your geek card revoked. Here are some of mine…" They would then proceed to rattle off some bullet points about how they don't like The Lord of the Rings and the Marvel Cinematic Universe, or how they think certain popular creators are criminally over-rated, or whatever opinions they have that would be generally considered unpopular to the point of heretical in "typical" geek circles.
I put "typical" in quotes because there of course is no typical geek. The type of person who is drawn to the work of, say, Joss Whedon is not the same type of person who is drawn to the X-Files. There's probably some overlap between them, but we're still talking about two different and fairly distinct groups. So lumping them all together, as often happens at comic cons, is a bit misleading as it suggests their fandoms are the same.
To be fair, in many respects, these fandoms are all the same. Their most devoted members express the same passion, and the primary difference is in where and how it's directed. But because fans are drawn to different things for different reasons, the traits that tend to surface most frequently in a fandom vary based on what the fans are responding to. The sense of hope and eternal optimism in the face of overwhelming odds might be more prevalent among Star Wars fans doesn't necessarily translate very well to the quirky irreverence in Doctor Who. That's not to say you can't be a fan of both, but in the company of other Doctor Who fans, the traits more closely associated with the Doctor will be more prominent, where in the company of other Star Wars fans, the traits more often found in those movies will be on greater display.
Based on the ideals and mores shown in the source material, fans will often take those concepts and hold them up as ideals of the fan group. Then it becomes an unwritten rule that being a fan in the group "requires" one to strive for that ideal. In psychological terms, it's called a prototype: an idealized form of what a "true" fan should be within that group. The closer an individual comes to embodying that prototype, the "greater" of a fan they're often seen as.
Being in different fandoms, though, means that you'd be striving to match multiple prototypes simultaneously, even if some of those ideals are in contrast with one another. Hence, emphasizing different traits among different groups. It's essentially a variation on code-switching, which you may have heard about in some news reports in recent years. There's nothing wrong with it, and it's perfectly natural; we are all individuals and come at everything with our own individual preferences, baggage, and biases.
What that means, though, is that even within a single fandom, there really is no one who perfectly embodies the prototype's features. And bringing together the collection of all geeks, I doubt it's even possible to reconcile all of the different prototypes to form a single "geek" prototype. And with no geek prototype to try to mold yourself after, who's to say whether you're geek enough for the rest of the crowd? No one issues geek cards. So like what you like, and don't worry about conforming to some standard that doesn't even exist!
Doctor WhoJoss WhedonLord of the RingsMarvel Cinematic UniverseStar WarsX-Files
Senior Editor, Comics & Lifestyle
Sean Kleefeld is an independent researcher whose work has been used by the likes of Marvel Entertainment, Titan Books and 20th Century Fox. He writes the ongoing "Incidental Iconography" column for The Jack Kirby Collector and had weekly "Kleefeld on Webcomics" and "Kleefeld's Fanthropology" columns for MTV Geek. He's also contributed to Alter Ego, Back Issue and Comic Book Resources. Kleefeld's 2009 book, Comic Book Fanthropology, addresses the questions of who and what comic fans are. He blogs daily at KleefeldOnComics.com.
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MONEYBALL: Miller's latest is set against a backdrop of wealth and privilege, but doesn't have much to say about either.
"I hate Bennett Miller." "Everything I've ever said positive about the movie I take back. I hate it. i hate it. i hate it. i hate it. i hate it. i hate it. i hate it."
No, those aren't the sentiments of a reviewer underwhelmed by Miller's bafflingly pointless new film. They're sentiments tweeted last month by one of the men it's about, Olympic wrestling champion Mark Schultz.
"You crossed the line Miller," Schultz continued. "You're [sic] career is over." "You think I can't take you down coz ur a director. Watch me Bennett."
Perhaps most significant was the tweet that sought to direct readers to the memoir Schultz published in November chronicling the events dramatized in Miller's film. "For the true story (not the one the movie is based on ... actually I don't know what story the movie is based on) read my book."
In Foxcatcher, Channing Tatum plays decorated athlete Schultz — now 54 — as an inarticulate lug seething with undefined bitterness and given to wild mood swings. Anyone who reads Schultz's tweets and sees the film is likely to feel the actor captured something of his essence. Did I mention that Schultz raved about Foxcatcher after its Cannes premiere? And that he's an associate producer?
I can't account for his change of heart. Members of the Hollywood press, however, have postulated that, after multiple viewings, Schultz suddenly realized that screenwriters E. Max Frye and Dan Futterman were hinting at a sexual undercurrent in the strange relationship between the wrestler and his sponsor, multimillionaire ornithologist, cokehead and diagnosed paranoid schizophrenic John du Pont. And he freaked.
Perhaps you're picking up on the fact that this is a movie about disturbed dudes. That much is true. But almost everything else in Miller's latest is not. Events depicted in the film are either completely fictionalized to serve the director's theme or chronologically rearranged or compressed. The bottom line: Any given thing you see in the picture probably never happened.
Steve Carell delivers a super-creepy performance as du Pont, with the assistance of a prosthetic beak so impressive it almost merits a costar credit. The film suggests that du Pont built a wrestling training facility in the 1980s ostensibly as an act of patriotism, but in reality as a means of dealing with monumental loneliness, insecurity and mommy issues. He invited Mark to live there and, later, his older brother, Dave, also a gold medalist. Mark Ruffalo portrays Dave as the sole voice of reason in an increasingly crowded madhouse.
The movie takes 135 minutes to say absolutely nothing of consequence about America, entitlement, the power of privilege, the haves and have-nots, or what-have-you. It offers little more than the portrait of an unstable hobbyist behaving in unstable ways — a spoiled brat with a gun collection.
The considerable talents of the cast make the film mildly interesting at times, but Miller never quite gets around to having a point. Watching Foxcatcher meander dourly into narrative dead ends and leave one storyline after another dangling, it's difficult to believe the same director gave us Moneyball and Capote.
Something terrible occurs at the end of the film, but it's nothing the movie builds toward. The picture isn't in any sense about the tragedy and makes no effort to explain it. It comes out of nowhere, the random act of a deranged man. The filmmaker merely uses it as a punctuation mark, a futile grasp at gravitas calculated to move Foxcatcher up in weight class.
Foxcatcher is not showing in any theaters in the area.


text:
Houston's choice
Unless Houston changes fundamentally over the next decade it is going to become Detroit.Houston's choice is to grow into new markets or die. Right now solar energy is about as popular a topic at the Petroleum Club as a gay mayor.
Written by Dana Blankenhorn, Inactive on Dec. 27, 2009
I have a special place in my heart for Houston, Texas.
I went to college there. I met my wife there. I got my first job there.
Houston is a great city and not at all stuck-up. It is getting a new mayor next month, and all the media wants to say about her is that she's gay. (I wanted to talk about having gone to college with her, but no one cared.)
She comes to office at a key time. There are more important choices facing her than whether to play Melissa Etheridge or the Indigo Girls on the office iPod.
Unless Houston changes fundamentally over the next decade it is going to become Detroit.
Houston's wealth is based entirely on hydrocarbons. Money comes from the ground. The petrochemical plants along the Houston Ship Channel are just like Detroit's auto plants. Their pollution is the perfume of money being made.
Just like in Detroit, its economy is a dinosaur. It's a zombie. It needs to eat brains.
Houston's oil executives must become convinced to put at least as much capital to work in solar, wind, tidal, and hydrogen technologies as they do on oil and gas and coal, or Houston's future as a city is dire.
All sorts of breakthroughs are being made in this area, just not in Houston. QuantumSphere, for instance, has discovered a way to store hydrogen reliably for use in portable power applications. The work was done in Tampa and in Alabama. It was not done in Houston.
Right now only one of the biggest wind turbine companies is American. Only one of the top 10 solar cell producers is American, and they are based in Phoenix. Brazil dominates in biofuels. Some 2.3 million people are now in some form of renewable energy, investment is nearing $80 billion per year, and Houston is nowhere.
This is just what Detroit did in response to the energy crisis. Nothing. GM, Ford and Chrysler made a few tweaks here and there, but on the whole they watched passively as their industry was taken over, and Detroit has suffered grievously from that.
The opportunity Houston's business leaders have now to transform themselves and dominate these new industries won't last long. For the next few years they will retain the capital necessary to buy out the market's current leaders and bring those jobs to Houston.
But the window on that opportunity is closing fast. Houston's choice is to grow into new markets or die. Right now solar energy is about as popular a topic at the Petroleum Club as a gay mayor. But both are realities that can no longer be denied.
This post was originally published on Smartplanet.com
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BERTUZZI, Vincent Joseph
PINKHAM, Milton H.
Home Local Questions grow over Bellosguardo Foundation
Questions grow over Bellosguardo Foundation
by Robert Eringer October 28, 2021 0 comment
Editor's note: The state Attorney General's Office has called for an independent audit of the Bellosguardo Foundation for its last three tax years after the nonprofit's failure to complete financial records with the state Registry of Charitable Trusts. The Attorney General's Office gave the foundation until Nov. 29 to complete the audit. Stay tuned to the News-Press for updates on that. In the meantime, columnist Robert Eringer has raised another question concerning the foundation.
THE INVESTIGATOR – ROBERT ERINGER
When then Mayor Helene Schneider and Jeremy Lindaman began the process of selecting trustees for Bellosguardo's board of directors, The Investigator has learned, they appear to have been soliciting financial contributions for Ms. Schneider's 2016 campaign for U.S. Congress.
One prospective trustee told us: "The first people to be invited to the board of directors were people who would also be asked to donate to Helene's congressional campaign. She probably thought we would feel obligated to help fund her campaign. It was never stated but I felt implied we should feel compelled to give her a donation for her campaign."
This may prove to be a huge problem for Jeremy Lindaman, who managed Ms. Schneider's congressional campaign and was appointed by her — despite his total lack of foundation experience — to be president of the Bellosguardo Foundation at an annual salary now reaching $150,000.
Why might this be a huge problem?
Because the IRS specifically prohibits campaign solicitations in connection with a 501(c)(3) tax exempt nonprofit.
About such an entity, IRS regulations are very clear: "It may not be an action organization i.e. it may not participate in any campaign activity for or against political candidates."
Soliciting campaign donations in the same breath as holding out trustee positions would be a direct violation of strict IRS rules forbidding such use of a 501(c)(3).
With this in mind, we then cross-checked the list of political contributors to Ms. Schneider's 2016 campaign with Bellosguardo's board of directors. We discovered that no fewer than nine of the 14 listed trustees ("as of 2019") made financial contributions to the Schneider campaign during the 2016 election cycle, including board chairman Dick Wolf, who was one of Ms. Schneider's largest donors, weighing in at $9,100. (Mr. Wolf, who lives in Montecito, is well-known as the creator of the "Law & Order" and "FBI" TV franchises.)
Add Ann Towbes, who has since resigned her trusteeship (she donated $2,700) and Morris Jurkowitz (also resigned, but donated $2,000) and that means 11 of 16 or an astounding two-thirds of the board contributed to the Schneider campaign — a further indication that prospective board members may have been impelled to contribute to her congressional campaign as an "implied" condition for being selected to the board.
The Investigator reached out to both Mr. Lindaman and Ms. Schneider to ask how it was possible that so many trustees happened to contribute to the Schneider campaign as managed by Mr. Lindaman during the period when potential trustees were being consulted, cultivated and decided upon.
In an email, Ms. Schneider responded thus: "During my 2016 congressional campaign, I received contributions from several hundred individuals. Many of these individuals also gave to other candidates in the same campaign, including members who also happened to serve on the Bellosguardo Board of Directors. Of the 19 people who were on the Bellosguardo Board of Directors at that time, 3 contributed solely to my campaign, and the others either made no contributions, or contributed to both me and other candidates simultaneously, or contributed solely to my opponents. The breakdown per FEC reports are: 7 contributed to me and Salud Carbajal; 1 contributed to me, Salud Carbajal and Justin Fareed; 1 gave only to Justin Fareed; 1 gave only to Salud Carbajal; 1 gave to me and Justin Fareed; 1 gave to Salud Carbajal and Justin Fareed, and 4 made no campaign contributions."
We received no response from Mr. Lindaman.
Robert Eringer is a longtime Montecito author with vast experience in investigative journalism. He welcomes questions or comments at reringer@gmail.com.
Robert Eringer
NEWS-PRESS ENDORSEMENT


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Introducing Lyon
Sitting on the banks of the Rhone and Saone rivers, the historic city of Lyon is located in eastern France, about 70 miles west of Geneva. While package holidays in Lyon often include visits to the Alps, there is plenty to see in the city itself, which was established in 43 BC by the Romans and is now the third-largest city in France.
What to know before visiting Lyon
Although French is the main language in Lyon, many people will speak at least a few words of English. Lyon is one hour ahead of the UK, and the currency is the euro.
Although not a traditional destination for city breaks, Lyon is a major business centre, which means there are plenty of flight options. Flights from the UK to Lyon depart from several regional airports, including Manchester and London's Heathrow and London City.
When is the best time to visit Lyon?
Summers in Lyon are warm, sometimes reaching 30°C. However, being in the foothills of the Alps, the city also experiences more rain then you might expect, even during July and August. Whereas winters in Lyon are wet and cool, there are plenty ways to enjoy a weekend in Lyon at this time of year.
Each year, in early December, the city hosts the fête des lumières (Festival of Lights), a religious celebration that sees residents place a candle outside their homes. The festival has now expanded to include light-based art installations and performances.
What to do in Lyon
Explore Lyon and the surrounding area on foot, or by other forms of transport:
One of the best ways to get your bearings on city breaks in Lyon is to take a relaxing boat trip down one of the city's two rivers, the Rhone and the Saône.
If you prefer to get about on your own two feet, take a stroll around the Vieux Quartier (Old Quarter) to see some of the oldest buildings in Lyon.
The 19th-century Parc de la Tete d'Or is another great place to explore by foot. This large park has a boating lake open in the summer, a small zoo, mini-golf course and horseriding facilities.
The Rhone Valley is well known for its scenery, along with plenty of hiking routes and cycling trails. It's also home to dozens of vineyards and wineries that produce famous vallée du Rhône wine. A visit to a winery is a must on package holidays in Lyon.
What places should you visit in Lyon?
The impressive 12th-century Lyon Cathedral, which took over four centuries to build, is well worth a visit, though its exterior is much more impressive than its interior. The area of Fourvière, one of the city's oldest, is home to the Basilica of Notre-Dame de Fourvière. The structure, which boasts a golden statue of the Virgin Mary, is where the Festival of Lights begins each December. You can also visit the ruins of Roman baths and a Roman theatre, and ride along the world's oldest and active funicular lines.
Pay homage to the Lumière brothers, the inventors of the cinematograph and fathers of modern cinema, at the Institut Lumière. The institute houses a museum dedicated to the brothers, who lived in Lyon for most of their adult lives, and hosts regular cinema screenings. If you're a fan of movies, package holidays in Lyon should also include a visit to the intriguing Musée Miniature et Cinéma, which features exhibits on cinema props and special effects.
What to eat in Lyon
Lyon is well known for its food, with pâté, potatoes and pork among the many ingredients for you to try on a weekend in Lyon. The city even has a salad named after it, salade Lyonnaise, which is served warm with lardons (pieces of bacon), poached eggs and fried potatoes. For dessert, try bugnes, thin crispy doughnuts traditionally eaten on Mardi Gras (Shrove Tuesday), or matefaim, a cross between a cake and a crêpe, served with fruit and sugar.
Souvenirs from Lyon
As well as a bottle of local wine, you might also want to take home some typical Lyonnais sweets, such as Coussin de Lyon marzipan. Antoine de Saint-Exupéry's Le Petit Prince is another Lyonnais creation, and many shops sell souvenirs related to the beloved book. Lyon is also the home of Guignol puppets, the French version of Punch and Judy. The puppets make perfect souvenirs.
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Das Erzbistum Milwaukee (, ) ist eine Diözese der römisch-katholischen Kirche in den Vereinigten Staaten mit Sitz in Milwaukee, Wisconsin.

Gebiet des Erzbistums
Das Gebiet des Erzbistums umfasst die Stadt Milwaukee und Milwaukee County sowie die Gebiete von Dodge County, Fond du Lac County, Kenosha County, Ozaukee County, Racine County, Sheboygan County, Walworth County, Washington County und Waukesha County im Bundesstaat Wisconsin.

Geschichte
Das Bistum wurde am 28. November 1843 von Papst Gregor XVI. errichtet. Am 12. Februar 1875 wurde das Bistum von Papst Pius IX. zum Erzbistum erhoben. Die Kathedrale St. Johannes der Evangelist ist die Bischofskirche des Erzbistums.

Am Ende des 19. Jahrhunderts zählte das Erzbistum 250.000 Katholiken, in der Stadt Milwaukee gab es 27 katholische Gemeinden, elf deutsche, sieben englische, sieben polnische und zwei böhmische Gemeinden.

Anfang Januar 2011 wurde bekannt gegeben, dass die Erzdiözese Milwaukee auf Grund vorausgegangener Entschädigungszahlungen an Missbrauchsopfer insolvent sei.

Bedeutende Kirchengebäude
Neben der Kathedrale befindet sich in Milwaukee die Basilica of St. Josaphat. Die Basilika Maria Hilfe der Christen auf dem Holy Hill ist eine vielbesuchte Wallfahrtsstätte in Erin, Wisconsin.

Bischof von Milwaukee
 John Martin Henni (1844–1875)

Erzbischöfe von Milwaukee
 John Martin Henni (1875–1881)
 Michael Heiß (1881–1890)
 Frederick Xavier Katzer (1890–1903)
 Sebastian Gebhard Messmer (1903–1930)
 Samuel Alphonse Stritch (1930–1940)
 Moses Elias Kiley (1940–1953)
 Albert Meyer (1953–1958)
 William Edward Cousins (1959–1977)
 Rembert Weakland OSB (1977–2002)
 Timothy Dolan (2002–2009)
 Jerome Listecki (seit 2009)

Katholische Höhere Schulen
 Catholic Central High School, Burlington
 Catholic Memorial High School, Waukesha
 Divine Savior Holy Angels High School, Milwaukee
 Dominican High School, Whitefish Bay
 Marquette University High School, Milwaukee
 Messmer High School, Milwaukee
 Pius XI High School, Milwaukee
 St. Catherine's High School, Racine
 St. Joan Antida High School, Milwaukee
 St. Joseph High School, Kenosha
 St. Lawrence Seminary High School, Mt. Calvary
 St. Mary Springs High School, Fond du Lac
 Thomas More High School, Milwaukee

Siehe auch
 Liste der römisch-katholischen Diözesen
 Römisch-katholische Kirche in den Vereinigten Staaten

Weblinks

 Offizielle Website (englisch)
  (englisch)

Einzelnachweise

Milwaukee
Organisation (Milwaukee)
Christentum (Wisconsin)
Gegründet 1843

redpajama_set_name: RedPajamaWikipedia