Who said you can't lose weight through walking alone, and without going to the gym? Don't listen to them! You can become 20 pounds lighter in the next few months without lifting a weight or going on a hunger strike. Walking is a form of exercise that is safe and easy for everyone to execute, and you can use it to achieve your weight loss goals. This article explains how you can use walking to answer that age-old question of "how to lose weight" and maintain your weightloss without going to the gym or going on a serious diet.
Walking is a simple form of workout exercise that is often underrated in terms of health benefits, although it does have numerous health benefits. Walking will give you a stronger heart and bones, it helps lower your blood pressure and ease your stress, and help you drop more pounds than you'd imagine. The good thing about walking is that you can do it anywhere, with little effort, and it doesn't require any equipment.
How Much Weight Can You Lose By Walking?
The number of pounds you can drop by walking is determined by your body weight, walking intensity, and your pace. If you walk at a common pace of 4 miles per hour you are most likely to burn about 400 calories in one hour. But bear in mind that this is just on average, so you don't have to walk 4 miles a day. You can go 3 miles to burn 300 calories or 2 miles to burn 200 calories. And if you don't have much stamina, you can cover a shorter distance and add other exercises for a better result.
How Often Should You Walk?
The frequency of your workout should depend on your state of health and targeted goals. If you are healthy enough to engage in walking and you want to achieve optimal results, then you should be walking 3 days a week for about 20-30 minutes to start. As you are progressing, you should increase the number of days as well as the duration; like every day for 45-60 minutes.
How Can You Walk For A Longer Period Without Boredom?
To have an interesting walking routine you need to try different ways of carrying out the exercise, such as taking a different route from time to time, walking outdoors and indoors, listening to music, or even singing your favorite song out loud while you walk. You might be surprised at the distance and duration you can cover by adding some of these things to your walking.
Avoid looking at your feet while walking, look forward as this will help keep your neck in a comfortable position.
Try to brace your core and maintain good posture by pulling your belly button in the direction of your spine.
For you to be able to go a longer distance you should try tightening your glutes. This will help push you forward.
Is It Important To Monitor The Number Of Steps You Walk?
An important way to boost your chances of success is to know the number of steps you take every day. This will help maintain a particular number of steps you would need to go to get a particular result (amount of weight loss). The device that can help you with the number count is a pedometer (like this one) or a wristband (like this one). With these devices, you can know how many miles you would need to go to get to the number of pounds you intended on dropping. Once you get a pedometer, you will attach it to your clothes at your hip, or elsewhere while you walk.
What Should You Do To Lose Weight And Maintain It Through Walking?
You are definitely going to lose weight through walking alone if you continue with it. however, you can increase the time it would take to lose weight by adding resistance training to your walking routine. You can walk with dumbbell (arm) weights, or try power walking by increasing your pace some days, alternated with your normal walking days. You can even set aside a particular day in the week for just resistance exercise, such as weightlifting or jogging, in order to keep the pounds off that you will drop through walking. Also, you should eat healthily in order to maintain your weight.
If you have recently Googled "how to lose weight" and are planning on starting out with walking as an exercise routine to lose weight then I encourage you to do so, because your effort is not going to be wasted. Walking is a simple form of exercise that can result in weight loss if performed properly and consistently. You can be sure that you will lose pounds within few weeks after starting your walking routine.


text:
Afera Daily Telegraph bila je afera koja je uslijedila nakon objavljivanja 28. listopada 1908. u britanskom listu Daily Telegraph primjedbi njemačkog cara Vilima II. s ciljem poboljšanja njemačko-britanskih odnosa. Bila je to velika diplomatska zabluda koja je pogoršala odnose i jako narušila Kaiserov ugled; nakon toga igrao je mnogo manju ulogu u odlučivanju vanjske politike. Ova epizoda imala je daleko veći odjek u Njemačkoj nego u inozemstvu.

Telegraph je predstavio ono što se činilo kao intervju s carem. Zapravo su to bile prepravljene bilješke o razgovoru koji je vodio britanski časnik Edward Montagu-Stuart-Wortley s Vilimom II. 1907. Telegraph je "intervju" poslao na odobrenje Vilimu II., koji ga je pak proslijedio kancelaru Bernhardu von Bülowu, koji je kasnije izjavio da je previše zauzet za uređivanje dokumenta, iako su Bülowljevi kritičari tvrdili da ne želi proturječiti caru.  Umjesto toga Bülow je to proslijedio Ministarstvu vanjskih poslova na njihov pregled, što se očito nije dogodilo. Uključivao je neodmjerene izjave i diplomatski štetne primjedbe, od kojih je bila najzloglasnija

Wilhelm je intervju vidio kao priliku za promicanje svojih pogleda i ideja o anglo-njemačkom prijateljstvu, ali zbog svojih emocionalnih ispada tijekom intervjua, završio je s daljnjim otuđivanjem ne samo Britanaca, već i Francuza, Rusa, i Japanaca. Izravno je, između ostalog, rekao da Nijemce uopće nije briga za Britance; da su Francuzi i Rusi pokušali potaknuti Njemačku na intervenciju u Drugom burskom ratu; i da je gradnja njemačke mornarice bila usmjerena protiv Japana, a ne Britanije.

Učinak

Britansko je vodstvo već shvatilo da je Wilhelm pomalo psihički uznemiren i ovo je vidjelo kao daljnji dokaz njegove nestabilne osobnosti, nasuprot naznakama službenog njemačkog neprijateljstva.

Ova kriza duboko je ranila Wilhelmovo prethodno neokrnjeno samopouzdanje, ubrzo je pretrpio i jaku depresiju od koje se nikad nije potpuno oporavio. Izgubio je velik dio utjecaja koji je prethodno vršio u unutarnjoj i vanjskoj politici.

Učinak u Njemačkoj bio je prilično značajan; snažna sramota praćena je ozbiljnim pozivima na Kaiserovo odstupanje s vlasti. Konzervativni Junker političar Elard von Oldenburg-Januschau bio je jedini član Reichstaga koji je branio Kaisera tijekom cijele afere. Wilhelm se držao vrlodiskretno mjesecima nakon fijaska Daily Telegraph-a, potonuvši u duboku depresiju, nikad se potpuno oporavivši od poniženja. Kasnije se osvetio prisiljavajući na ostavku kancelara Bülowa, koji je pustio cara da se javno osramoti ne uredivši prijepis prije njegove njemačke objave. Bülow se u svojim memoarima podsjetio toga:

Izvori

Povijest Njemačke (1871. – 1914.)

redpajama_set_name: RedPajamaWikipedia


Today the Treasurer has announced the appointment of Michael Brennan as the new Chair of the Productivity Commission - I could not be more pleased with this appointment.
Michael has been an outstanding member of the Executive Committee seeing the Department through three Commonwealth Budgets. The Productivity Commission will be well served by his high level strategic, economic analysis and leadership qualities. I would like to personally thank Michael for his support, sound reasoning and guidance. I wish him all the best in his new and very important role.
With Michael's upcoming departure, I would also like to announce that Simon Atkinson will join Treasury as the Deputy Secretary of Fiscal Group in late August. Simon has deep experience in fiscal policy and will make an immediate impact in the Department.
Michael's last day in Treasury will be 29 August, before taking some leave to commence his new role as Chair of the Productivity Commission on 11 September.


Karen Diggs at Be Nourished offers nutritional consultations and personal chef services. The Be Nourished Web site contains useful information about the healing properties of herbs and spices.
Attaining Harmony School of T'ai Chi Ch'uan in Russia.
Our T'ai Chi School in Moscow Attaining Harmony School of T'ai Chi Ch'uan.
In addition to teaching T'ai Chi Chuan Elena Kotenkova, MS, CCHH, has her own practice of Classical homeopathy in San Francisco.
Other homeopathy related sites include Simply homeopathy in San Francisco.


Our client is an O&G refining company and a listed company. Its main business is to refine and manufacture petroleum products and it is the one of the largest refinery in Malaysia. They are looking for a Senior Legal Counsel to oversee and manage the legal issues and ensure corporate governance compliance for the Company.
Reporting to the Deputy CEO.
Provide sound risk legal advice on the business and operational matters.
Lead and support any disputes, litigation or arbitration matter involving the company.
Supporting the management team in their day-to-day business operations.
Monitor and review legislative developments and support the business in the implementation when required.
Review and advise management on legal implications of internal policies and procedures and ensure that they are in compliance with legal requirements.
Engage with key external legal stakeholders for the Company, e.g. Bursa Securities, the Companies Commission of Malaysia, Securities Commission Malaysia.
A qualified lawyer called to the English Bar and the Malaysian Bar.
Overall 20 years of working experience – a practicing lawyer for at least 4+ years; in-house legal counsel with public listed company (listed on the UK and Malaysian stock exchange) for at least 10 years.
Has around 5+ years experience in Oil and Gas related / Engineering multi-national company.
Has a Master's Degree in Commercial Law.


Home Reference Books Colorado : a liquid history & tavern guide to the highest state / Thomas J. Noel.
Colorado : a liquid history & tavern guide to the highest state / Thomas J. Noel.
Title Colorado : a liquid history & tavern guide to the highest state / Thomas J. Noel.
Publisher Golden, Colo. : Fulcrum Pub., c1999.
Physical Description xviii, 254 p. : ill., map ; 23 cm.
Subject Bars (Drinking establishments) -- Colorado -- Guidebooks. Colorado -- Social life and customs. Colorado -- History.
Notes ncludes bibliographical references (p. 249) and index.


At Highland, we've taken STEM education – which combines science, technology, engineering, and mathematics – to the next level. By adding Art to create STEAM, our students are getting an introduction to critical core concepts in a hands-on way that's more engaging and relevant to a broader range of learners. And they're getting rewarded with better comprehension and higher SAT scores.
STEM programs are designed to integrate science, technology, engineering, and mathematics in the classroom. These programs aim to teach students to think critically and have an engineering or design approach towards real-world problems while building on their math and science base. STEAM programs add art to STEM curriculum by drawing on design principles and encouraging creative and innovative solutions.
At Highland, STEAM education is a fun and meaningful way for our teachers to deliver more engaging concepts and deeply embedding them in a creative way that goes beyond typical STEM education. Why differentiate between the two? STEM is based on skills generally using the left half of the brain. They are analytical and logic-driven. Research shows that activities like the Arts, which uses the right side of the brain supports and fosters creativity, are essential to innovation. Not only that, students that focus on the arts as a part of their education perform better academically. STEAM involves the right side of the brain in a way that STEM education – which is purely left-brained – does not.
A recent University of Florida study shows that, on average, students who study the arts for four years in high school score 98 points higher on the SATs compared to those who study the same for half a year or less. Additionally, students who studied music appreciation scored 61 points higher on the verbal section and 42 points higher on the math section.
Students who study the arts as part of a STEAM-based curriculum tend to perform better academically than those who do not, but there are more reasons why the arts are an important part of an effective education. The fine and performing arts help students build confidence, develop motor skills, and hone their decision-making and problem-solving skills.


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\title{{\bf Holographic entanglement entropy for disconnected regions}}

\author{Veronika E Hubeny\footnote{veronika.hubeny@durham.ac.uk} \ and Mukund Rangamani\footnote{mukund.rangamani@durham.ac.uk}\\ \\
\small \sl   Centre for Particle Theory \& Department of
Mathematical Sciences,
\\[-1.5mm]
\small \sl Science Laboratories, South Road, Durham DH1 3LE, United Kingdom \\
}

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\begin{abstract}
We present a simple derivation  of the entanglement entropy for a region made up of a union of disjoint intervals in 1+1 dimensional quantum field theories using holographic techniques. This generalizes the results for 1+1 dimensional conformal field theories derived previously by exploiting the uniformization map.
We further comment on the generalization of our result to higher dimensional field theories.
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\section{Introduction}
 \label{intro}

Entanglement entropy is an important concept in field theory systems, with applications to condensed matter systems, quantum information, {\it etc.}. Further, given its non-extensive nature {\it i.e.}, area scaling  \cite{Srednicki:1993im,Kabat:1994vj}, one is tempted,  in the context of holography, to  think of it as providing a measure for the effective degrees of freedom associated with a given region. This interpretation is natural in light of the recent proposal  by Ryu \& Takayanagi \cite{Ryu:2006bv,Ryu:2006ef} to compute entanglement entropy in quantum field theories by finding the area of an appropriate bulk surface in Planck units. Further support for this prescription arises in the  covariant generalization of their proposal \cite{Hubeny:2007xt}, where it was argued that the entanglement entropy is related to  light-sheet constructions of the covariant entropy bound \cite{Bousso:1999xy} in the bulk spacetime.

Before proceeding to the technical aspects, let us recall the definition of entanglement entropy and the holographic prescription for computing it. We consider a quantum field theory on $\p{\cal M} = {\cal N} \times {\bf R}_t$ and focus on a region $\CA\subset {\cal N}$. Given a density matrix $\rho$ (or a pure state) on ${\cal N}$ we define the reduced density matrix $\rho_\CA = {\rm Tr}_{{\cal N} \backslash \CA} \( \rho\)$. The entanglement entropy is then simply the von Neumann entropy associated with $\rho_\CA$: $S_\CA = - \Tr{\rho_\CA \, \log \, \rho_\CA}$. The covariant holographic prescription \cite{Hubeny:2007xt} for computing this  is to consider the problem of finding a co-dimension two extremal surface $\CW_\CA$ in the bulk geometry  ${\cal M}$ (dual to the state in question), which ends on the boundary of the chosen region $\CA$ at the spacetime boundary $\p{\cal M}$. $S_\CA$ is then given by the area of $\CW_\CA$ in Planck units {\it i.e.}, $S_\CA = {{\rm Area }\(\CW_\CA\) \over 4 G_N}$. In the situation where the bulk geometry is static (and hence the dual state is time-invariant) the problem reduces to the simpler one of finding minimal area surfaces \cite{Ryu:2006bv,Ryu:2006ef} (see \cite{Fursaev:2006ih} for a proof). Recent discussions of entanglement entropy in the holographic context include braneworld scenarios \cite{Emparan:2006ni}, closed string tachyon condensation \cite{Nishioka:2006gr}, and confinement-deconfinement transitions \cite{Nishioka:2006gr, Klebanov:2007ws,Faraggi:2007fu}.

The technology to compute entanglement entropy is best developed in the case of 1+1 dimensional CFTs \cite{Calabrese:2004eu} ({\it cf.},  \cite{Holzhey:1994we} for earlier work and \cite{Cardy:2007mb} for generalizations to integrable non-conformal systems).  In such situations one can exploit the power of 1+1 dimensional conformal invariance to compute the entanglement entropy for a region composed of a disjoint union of intervals on a spatial slice.\footnote{The trick is to consider evaluating $\Tr{\rho_\CA^n}$ using a path integral prescription. One computes $S_\CA$ by analytic  continuation and taking the derivative as $n \to 1$. This replica trick allows one to relate the computation of the entanglement entropy to the  computation of twist operator correlation functions in CFTs \cite{Calabrese:2004eu}.} However, a key feature used in this derivation, the Riemann mapping theorem,  is not available for non-conformal theories or for higher dimensional examples.

On the other hand, the holographic recipe for computing entanglement entropy, which requires us to compute the area of an extremal co-dimension two spacelike surface (or a minimal surface for static states), is much more powerful, since we only need to solve a classical problem in a given bulk geometry. As an illustrative example, this holographic relation has been used recently \cite{Headrick:2007km} to give a simple and elegant proof of the  strong sub-additivity property. The proof of this statement in standard quantum field theory is quite involved, see \cite{Lieb:1973cp,Lieb:1973lr}.

Here we use the holographic prescription to derive the expression for entanglement entropy of disconnected regions in 1+1 dimensions for time independent states (or density matrices).  We  obtain the answer predicted by the field theory analysis of \cite{Calabrese:2004eu} for CFTs, but now only using geometrical arguments.  In particular,  the definition of minimal surface, which in turn guarantees the strong sub-additivity property, plays a crucial role.  We can easily generalise the calculation to $N$ disconnected regions in $1+1$ dimensional CFT (in a static state), again giving agreement with the field theoretic predictions. However, the derivation we present is true for any 1+1 dimensional quantum field theory, not necessarily one enjoying conformal symmetry, demonstrating the power of the geometric construction.
We also generalise our calculations to higher dimensions in some particular situations. Throughout, we assume a static configuration (so we can consider bulk minimal surfaces on a constant time slice), and comment on potential generalizations in the Discussion.

We begin by deriving a formula for entanglement entropy for disconnected regions in the case of 1+1 dimensional field theories in \sec{disconnect}. We then present a conjecture for the higher dimensional case in \sec{disconnectd} and conclude with a set of open questions in \sec{discussion}.

\section{Disconnected regions in $1+1$ dimensional field theories}
 \label{disconnect}

Consider a $1+1$ dimensional QFT, living on the boundary of a $2+1$ dimensional asymptotically-AdS bulk spacetime. We are interested in calculating the entanglement entropy of a region composed of disconnected intervals. We will label the points bounding the intervals as $p_i$ with $i = 1, \ldots, n$ and the region of interest will be
\begin{equation}
X = \bigcup_{m=0}^{[n/2]} \, \[p_{2m+1}, p_{2m}\] .
\Label{Xregion}
\end{equation}
Further, we will denote the simply connected region with endpoints $p_i$ and $p_j$ by $\R{ij}$, and the corresponding entanglement entropy of this region by $S(\R{ij})=\s{ij}$. By the holographic prescription of \cite{Ryu:2006bv,Ryu:2006ef}
we have
\begin{equation}
s_{ij}= {1\over 4 \, G_N^{(3)}} \, {\rm Length} \(\gamma_{ij}\)
\Label{holdefs}
\end{equation}
where $\gamma_{ij}$ is the minimal length curve (spacelike geodesic) connecting $p_i$ and $p_j$ through the bulk.\footnote{The curves under consideration should be homologous to the boundary region of interest {\it i.e.}, one must be able to deform the bulk curve back to the boundary without any obstruction.}  Note that the entanglement entropy of a region consisting of a single point vanishes, so that $\s{ii}=0$.   Furthermore, we don't require any orientation, {\it i.e.},  $\R{ij}= \R{ji}$ and correspondingly $\s{ij}=\s{ji}$.  Hence we have $n(n-1)/2$ such regions.

The quantity of interest will be the entanglement entropy of a non-simply connected region $X$, and we will seek a universal expression for $x \equiv S(X)$ in terms of the entanglement entropies of related simply connected regions, namely the $\s{ij}$'s.

\subsection{Two disconnected regions}
 \label{disconnecttwo}

Let us now concentrate of the case of two disconnected regions, $n=4$; the case of general $n$ will be considered in the next subsection.
The six minimal surfaces\footnote{Given the linear relation between ${\rm Length}(\gamma_{ij})$ and $\s{ij}$, we will henceforth use $\s{ij}$  to denote the entanglement entropy as well as the bulk minimal surface.} for the $n=4$ case are sketched in \fig{minsurfX}.
\begin{figure}
\begin{center}

 \includegraphics[width=14cm]{minsurfX}
\caption{Minimal surfaces $\gamma_{ij}$ in the bulk, whose length gives the entanglement entropy $\s{ij}$, corresponding to the boundary intervals $\R{ij}$ bounded by points $p_i$ and $p_j$. The disconnected boundary region $X$ is indicated in red.}
\label{minsurfX}
\end{center}
\end{figure}

We can construct the disconnected region of interest $X$ either as union or equivalently as a difference of two regions:
\begin{equation}
X = \R{12} \cup \R{34} = \R{14} \, \backslash \, \R{23}
\Label{XdefR}
\end{equation}
Our main objective is to determine $x=S(X)$ in terms of $\s{ij}$.   We will achieve this by constraining $x$ using the minimal area property of each $\s{ij}$ and strong sub-additivity.

\paragraph{Minimal surface and sub-additivity inequalities:} Let us start by writing the inequalities obeyed by the various minimal surfaces  $\s{ij}$.
Consider two endpoints, $p_i$ and $p_j$ and consider any bulk `surface' (in this $1+1$ dimensional case just a curve) ending on these endpoints.  By definition of minimal surface, $\s{ij}$ must be the smallest of all such surfaces.  In particular, the minimal surface $\s{ij}$ cannot have greater area than the surface composed of two minimal surfaces $\s{ik}$ and $\s{jk}$ which meet at the same point $p_k$ on the boundary. This implies the set of inequalities of the form\footnote{Strictly speaking, in  any local quantum field theory, the entanglement entropy is divergent due to contributions from the degrees of freedom localized near the boundaries. In the holographic computation this is manifested by the divergent area of any bulk surface in asymptotically AdS spacetimes. We will assume henceforth that we are working with a fixed regulator to make sense of the inequalities. Alternately, one can imagine working with an appropriate background subtraction scheme.}
\begin{equation}
\s{ij} \le \s{ik} + \s{jk}
\Label{sijk}
\end{equation}
for $i,j,k = 1, \ldots, 4$.
Since this is non-trivial only when $i,j$ and $k$ are all distinct, and since $\s{ij}$ is symmetric, we have a-priori $12$ independent inequalities.

However, this set of inequalities can be substantially shortened by using the property of strong sub-additivity \cite{Lieb:1973cp} (see \cite{Lieb:1974qr,Araki:1970ba} for excellent discussions of entropy inequalities in classical and quantum systems).
For arbitrary regions $A$ and $B$, we have the two basic strong sub-additivity inequalities:
\begin{equation}
S(A) + S(B) \ge S(A \cup B) + S(A \cap B)
\Label{subaddgenABcup}
\end{equation}
\begin{equation}
S(A) + S(B) \ge S(A \, \backslash \, B) + S(B \, \backslash \, A)
\Label{subaddgenABmin}
\end{equation}
In \cite{Headrick:2007km} it was  shown that these inequalities also follow immediately from the definition of minimal surfaces ({\it cf.},  \cite{Hirata:2006jx} for early discussions of sub-additivity in the holographic prescription and \cite{Fursaev:2007sg} for recent discussions).  To see this, consider the minimal surfaces $\s{ij}$ shown in \fig{minsurfX}.  There are exactly two pairs of surfaces ending on distinct endpoints which don't intersect, namely $(\s{12},\s{34})$ and $(\s{23},\s{14})$, and there is one pair of intersecting surfaces, $(\s{13},\s{24})$.  We can now decompose the latter pair in two different ways, forming two non-minimal surfaces glued together at the intersection.  Using the same argument as above, the sum of the areas of these two non-minimal surfaces cannot be smaller than the sum of the corresponding minimal surfaces.  This yields the two stronger inequalities  \cite{Headrick:2007km}:
\begin{equation}
\s{14} + \s{23} \le \s{13} + \s{24}
\Label{subaddcup}
\end{equation}
\begin{equation}
\s{12} + \s{34} \le \s{13} + \s{24}
\Label{subaddmin}
\end{equation}
Note that we can obtain \req{subaddcup} and \req{subaddmin} from \req{subaddgenABcup} and \req{subaddgenABmin}, respectively, by taking $A = \R{13}$ and $B= \R{24}$.

Each of \req{subaddcup} and \req{subaddmin} combines with two equations of the form \req{sijk} to yield two of the other equations of the form \req{sijk}, rendering the latter set redundant.  For example, adding
$\s{ij} + \s{kl} \le  \s{ik} + \s{jl} $ and
$\s{ik} \le  \s{ij} + \s{jk} $ yields
$\s{kl} \le  \s{jl} + \s{jk} $.
Eliminating the redundant equations in this manner, we reduce the 12 original equations \req{sijk} to the following four:
\begin{eqnarray}
\s{13}  &\le& \s{12} + \s{23} \cr
\s{13}  &\le& \s{14} + \s{34} \cr
\s{24}  &\le& \s{12} + \s{14} \cr
\s{24}  &\le& \s{23} + \s{34}
\Label{sijkmin}
\end{eqnarray}
The set of the six inequalities \req{subaddcup}, \req{subaddmin}, and \req{sijkmin} forms the minimal set of independent constraints on the minimal surfaces $\s{ij}$.
Indeed, we can easily check that there exist examples of $\s{ij}$ satisfying any five of the six inequalities but not the sixth one.

\paragraph{Constraints on $x$:} We now turn to constraining the entanglement entropy $x = S(X)$ of the disconnected region $X$.  The strategy is to use the strong sub-additivity property \req{subaddgenABcup} and \req{subaddgenABmin} for suitable regions $A$ and $B$ such that one of the terms corresponds to $S(X)$.  There are only two nontrivial possibilities for obtaining an upper bound on $x$, which yield:
\begin{equation}
x \le \s{12} + \s{34}
\Label{xupcup}
\end{equation}
\begin{equation}
x \le \s{14} + \s{23}
\Label{xupmin}
\end{equation}
These follow from
\req{subaddgenABcup} with $A=\R{12}$ and $ B=\R{34}$  and from
\req{subaddgenABmin} with $A=\R{14}$ and $ B=\R{23}$, respectively.
The possibilities for obtaining lower bounds on $x$ are more numerous, since here we can simply let $A=X$ and consider all other regions as canditates for $B$.
A-priori, we can let $B$ be any of the six simple regions $\R{ij}$, and for each case, we have the two constraints \req{subaddgenABcup} and \req{subaddgenABmin}.  However, it turns out that four of the twelve resulting inequalities are trivial, and four more are redundant (in particular they follow from the remaining four constraints and \req{sijkmin}).
The four nontrivial independent lower bounds on $x$ which remain are the ones obtained from \req{subaddgenABcup} and \req{subaddgenABmin} with $A=X$ and $B= \R{13}, \R{24}$:
\begin{eqnarray}
\s{12} + \s{14} - \s{13}  &\le& x \cr
\s{14} + \s{34} - \s{24}  &\le& x \cr
\s{12} + \s{23} - \s{24}  &\le& x \cr
\s{23} + \s{34} - \s{13}  &\le& x
\Label{xlowbnd}
\end{eqnarray}
The six inequalities \req{xupcup}, \req{xupmin}, and \req{xlowbnd} form the minimal set of independent constraints on $x$.
As we will see, how strongly these bounds constrain the actual value of $x$ in any particular case depends on how nearly the inequalities \req{subaddcup}, \req{subaddmin}, and \req{sijkmin}  are saturated.   However, as we will argue below, it is easy to construct specific examples where two of the six inequalities in \req{subaddcup}-\req{sijkmin} are saturated, thereby forcing the lower and upper bounds on $x$ from \req{xupcup}-\req{xlowbnd} to actually coincide.

We now wish to use these six constraints on $x$, along with the six further constraints on the minimal surfaces $\s{ij}$, to determine $x$ completely in terms of $\s{ij}$.  We begin by observing that there is a natural pairing between the inequalities such that each pair contains all minimal surfaces exactly once.  For example, \req{subaddcup} is paired with \req{xupcup}, \req{subaddmin} is paired with \req{xupmin}, the first of \req{sijkmin} is paired with the second of \req{xlowbnd}, {\it etc.}.  Moreover, the relations between the coefficients of $\s{ij}$ are everywhere the same.  In particular, $\s{12}$, $\s{14}$, $\s{23}$, and $\s{34}$ appear with coefficient $+1$ while $\s{13}$ and $\s{24}$ appear with coefficient $-1$.  This is strongly suggestive of a specific pattern, which we now proceed to unearth.

\paragraph{Deriving $x(\s{ij})$:} Assuming that $x$ can be expressed in terms of the minimal surfaces $\s{ij}$, we will first write a simple ansatz for the form of $x$ and then use specific limits of the various regions to fix $x$ completely.  The bounds on $x$ suggest a linear relation between $x$ and the $\s{ij}$, namely:
\begin{equation}
x = c_{12} \, \s{12} + c_{13} \, \s{13} + c_{14} \, \s{14}
   + c_{23} \, \s{23} + c_{24} \, \s{24} + c_{34} \, \s{34}
\Label{xlinansatz}
\end{equation}
where the $c_{ij}$ are some constant coefficients to be determined.
In particular, according to our ansatz, $x$ is given by the expression \req{xlinansatz} for {\it any} allowed set of values of the $\s{ij}$.
Let us now consider specific limits where we know $x$ explicitly, namely the limits in which $X$ reduces to a simply connected region.  There are three nontrivial possibilities: we can take $p_{1} \to p_{2}$, $p_{2} \to p_{3}$, or $p_{3} \to p_{4}$.  In the first case, $\s{12} \to 0$ and $X \to \R{34}$, so that $x \to \s{34}$.  Furthermore, $\s{13} \to \s{23}$ and $\s{14} \to \s{24}$.  The expression \req{xlinansatz} then implies that $c_{34} = 1$ while $c_{13} + c_{23} = 0$ and $c_{14} + c_{24} = 0$.
Similarly, for $p_{2} \to p_{3}$, we have $x \to \s{14}$
while $\s{12} \to \s{13}$ and $\s{24} \to \s{34}$;
hence \req{xlinansatz} yields $c_{14} = 1$, $c_{12} + c_{13} = 0$, and $c_{24} + c_{34} = 0$.  Finally, for $p_{3} \to p_{4}$, we have $x \to \s{12}$ and therefore $c_{12} = 1$, $c_{13} + c_{14} = 0$, and $c_{23} + c_{24} = 0$.  Taken together, these determine all coefficients fully; in particular we find $c_{12} = c_{14} = c_{23} = c_{34} = -c_{13} = -c_{24} = 1$.  This therefore implies that $x$ is in general given by
\begin{equation}
x = \s{12} + \s{23} + \s{34} +  \s{14}  - \s{13} - \s{24} \ .
\Label{xanswer}
\end{equation}
To recap, using the linear ansatz \req{xlinansatz} we have shown that the formula for the entanglement entropy of two disconnected regions is indeed given by the expression we would have naturally derived from the CFT \cite{Calabrese:2004eu}, namely \req{xanswer}.  In fact, we can  generalise \req{xlinansatz} to a sum of arbitrary functions of one variable and still derive the same result \req{xanswer}, although a completely arbitrary function of the six $\s{ij}$'s is not uniquely fixed by three special limits at hand, as can be seen by Taylor expanding and solving for the coefficients at each order.

The preceding argument establishes the desired result by making a specific (albeit sensible) ansatz for the form of the entanglement entropy for disconnected regions. A-priori one might have hoped that the inequalities derived earlier \req{xupcup}-\req{xlowbnd}, along with the constraints \req{subaddcup}-\req{sijkmin}, would suffice to derive the form of $x$. This is generically not possible without making further assumptions.
However, if one makes the natural assumption that $x$ will be given by a unique expression (not necessarily linear) in terms of the six $\s{ij}$'s for {\it any} set of values $\s{ij}$ which satisfy the constraints \req{subaddcup}-\req{sijkmin}, then one can use a clever choice of specific `extremal' allowed values of the $\s{ij}$'s to derive the expression \req{xanswer}.
In particular, by explicitly assuming that the minimal surfaces saturate one of the inequalities \{\req{subaddcup},\req{subaddmin}\} and a corresponding one of \req{sijkmin}, we can obtain the desired expression for $x$ by showing that the upper and lower bounds on $x$ from \req{xupcup}-\req{xlowbnd} actually coincide.
For example, suppose that \req{subaddcup} and the first of \req{sijkmin} are saturated;\footnote{
Note that the system with one upper bound constraint and one lower bound constraint being saturated is perfectly self-consistent, in the sense that all of the constraints on $\s{ij}$ remain satisfied.} we can easily obtain (for arbitrary constants $a_1$ and $a_2$)
\begin{equation}
\s{14} + \s{34} - \s{24} + \( \s{12} + \s{23} - \s{13} \)  a_1
\le x \le
\s{12}  + \s{34} + \( \s{14} + \s{23} - \s{13} - \s{24} \)  a_2
\Label{xuplobnd}
\end{equation}
which (by setting $a_1=a_2=1$) yields \req{xanswer}.
While this particular choice of $a_1$ and $a_2$, as well as the specific pair constraints to be saturated, may seem somewhat ad-hoc, it turns out that {\it any} choice which leads to upper and lower bounds on $x$ coinciding in fact yields \req{xanswer} uniquely.

Hence this argument may be regarded as an alternative derivation of the desired result \req{xanswer}, which uses a different set of starting assumptions.
In particular, here we only needed to assume that $x$ is given by some universal expression in terms of the  $\s{ij}$, without assuming anything about the form of this expression.  However, in contrast to the previous argument, we required that this universal expression holds for any set of $\s{ij}$ which satisfy the constraints.\footnote{
Note that in the preceding argument, we required our linear ansatz to be satisfied by any {\it allowed} configuration of the $\s{ij}$'s; the set of such configurations is a subset of the all configurations satisfying the constraints.
Although these constraints comprised a complete set of properties derivable from the definition of a minimal surface, it is not clear whether the extremal cases used above are actually realised any physical system.}
The two arguments in conjunction therefore strengthen our derivation of \req{xanswer} by weakening the assumptions.

\subsection{Multiply disconnected regions}
 \label{disconnectmore}

Having considered the case of $X$ being composed of two disjoint intervals in the previous section, let us now turn to the more general case of non-simply connected regions, namely $m$ disjoint intervals.\footnote{
In one spatial dimension, the most general non-simply connected region consists of $m$ disjoint intervals for any $m \ge 2$ (in this discussion, we will ignore fractal regions such as Cantor sets, {\it etc.}); as we will see in the next section, in higher dimensions we have many more options.
}
In particular, these are bounded by the $n = 2m$ distinct endpoints $p_i$, which
define $n(n-1)/2$ distinct simple intervals $\R{ij}$, each associated to the entanglement entropy given by the minimal surfaces $\s{ij}$.  As a direct extension of \req{XdefR}, we have
\begin{equation}
X = \R{12} \cup \R{34} \cup \cdots \cup \R{n-1,n}
= \R{1n} \, \backslash \, \R{23} \, \backslash \, \R{45} \, \backslash \, \cdots \, \backslash \, \R{n-2,n-1}
\Label{XdefRn}
\end{equation}

Again, by definition of minimal surface, we have $\s{ij} \le \s{ik} + \s{jk}$ for $i,j,k =1, \ldots, n$; so there are a-priori $n(n-1)(n-2)/2$ independent inequalities.
Similarly, we have additional (strong sub-additivity) inequalities, which can be derived by considering any two intersecting surfaces.
For any set of four endpoints, say $p_i < p_j < p_k < p_l$, we have exactly two pairs of surfaces ending on distinct endpoints which don't intersect, namely $(\s{ij},\s{kl})$ and $(\s{jk},\s{il})$, and one pair of intersecting surfaces, $(\s{ik},\s{jl})$.  As before, we can break up the intersecting minimal surfaces into two sets of non-minimal surfaces, obtaining the inequalities of the form
\begin{eqnarray}
\s{ik}  + \s{jl} &\ge& \s{ij} + \s{kl} \cr
\s{ik}  + \s{jl} &\ge& \s{il} + \s{jk}
\Label{sijkln}
\end{eqnarray}
For $n$ boundary points, we have ${ 1 \over 12} \, n \, (n-1) \, (n-2) \, (n-3)$ of these stong sub-additivity type inequalities.  (Note that this grows as a quartic, which is more rapid than the cubic we obtained for the simple inequalities.)  In addition, we can consider still further inequalities by generalising this argument.  For example, for six distinct endpoints, $p_i < p_j < p_k < p_l < p_m < p_n$, we can consider the pairwise intersecting surfaces, and obtain inequalities of the form\footnote{These general entropy inequalities are a natural analog of strong sub-additivity for multiple disconnected regions. As far as we are aware they have not been considered in the field theory literature before.}
\begin{equation}
\s{ik} + \s{jm} + \s{ln} \ge \s{in} + \s{jk} + \s{lm} \ , \qquad {\it etc.}.
\Label{sijklmnn}
\end{equation}
Thus finding the minimal set of independent constraints on the minimal surfaces $\s{ij}$, not to mention on $x$, is a daunting prospect in general.

However, we can use the same trick as in \sec{disconnectmore} to find the general form of $x$ using the previous result \req{xanswer}.
In particular, assume that $x$ is given by an expression of the form
\begin{equation}
x = \mathop{ \sum_{{i,j=1}}^n}_{j>i} \, c_{ij} \, \s{ij}
\Label{xlinansatzn}
\end{equation}
for some fixed constants $c_{ij}$.
We require that $x$ reduce to the correct value for all the various combinations of $\s{ij}$ vanishing.  For example, if all but 4 endpoints $p_i < p_j < p_k < p_l$ coincide so that $X = \R{ij} \cup \R{kl}$, then \req{xlinansatzn} must reduce to
$x = \s{ij} + \s{jk} + \s{kl} + \s{il} - \s{ik} - \s{jl} $.  Using this constraint for all the possible combinations of 4 non-coincident endpoints, we obtain $c_{ij} = \pm 1$, depending on the positioning of $i$ and $j$.
In particular, $c_{ij} = (-1)^{i+j+1}$.
Hence the correct generalisation of the entanglement entropy \req{xanswer}, extended to $n/2$ distinct intervals is
\begin{equation}
x = \mathop{ \sum_{{i,j=1}}^n}_{j>i} \, (-1)^{i+j+1} \, \s{ij} \ .
\Label{xlinansatzn}
\end{equation}
This indeed confirms the corresponding expression found by CFT methods \cite{Calabrese:2004eu}.

\section{Disconnected regions in $2+1$ dimensional QFTs}
 \label{disconnectd}

In the previous section we have used bulk techniques to derive the expression for the entanglement entropy of a general disconnected region (composed of $m$ finite intervals) in $1+1$ dimensional QFT.  While the bulk derivation was quite simple, we could have nevertheless used an appropriate uniformization map to reduce the problem to one of computing twist operator correlation functions in a CFT \cite{Calabrese:2004eu}
and obtained the same result (at least for simple conformally invariant states).
However, these techniques are applicable only in $1+1$ dimensional CFT, and it is not presently clear what is the analogous expression for entanglement entropy of non-simply connected regions in higher dimensions.  Therefore in this section we explore what the bulk method can tell us about this situation.  For simplicity, we focus on $2+1$ dimensional CFT; so the regions and the bulk minimal surfaces in question are two dimensional.

The holographic dual of entanglement entropy (in some static state of the CFT) of a simply connected region on the boundary is the area of a bulk minimal surface anchored on the boundary of this region.  This relation is true in all dimensions, as are the strong sub-additivity properties of entanglement entropy; indeed, it is easy to see that the proof of \cite{Headrick:2007km} is valid in all dimensions.
Therefore, for situations where the interval structure of \fig{minsurfX} can be straightforwardly generalised to two-dimensional regions, the same reasoning as in \sec{disconnect} above will apply here as well.  However, a simple generalisation from intervals to regions will not hold in general; indeed, such a situation occurs only in a special, limiting, case.

\begin{figure}
\begin{center}

 \includegraphics[width=14cm]{twoDregions}
\caption{Three basic types of finite 2-dimensional non-simple regions $X$ (shaded). Note that case (b) is special, and can be deformed into either of the general cases (a) and (c).}
\label{twoDregions}
\end{center}
\end{figure}
This is illustrated in \fig{twoDregions}, where we indicate the basic types of non-simply connected regions.  In particular, generic regions whose boundary is not a single non-self-itersecting closed curve are either composed of disjoint simple regions as in case (a), or contain non-contractible curves as in case (c), or some combination thereof.  The intermediate special case which interpolates between (a) and (c) is the case (b) where two simple regions touch at two isolated points.  As we argue below, this is the case for which we can apply our formula for the entanglement entropy \req{xanswer}.   This is because the boundary of region $X$ in (b) naturally defines six minimal surfaces, in direct analogy to \fig{minsurfX}.  On the other hand, both (a) and (c) only define three possible minimal surfaces; two which are homologous to the simple regions and one which in case (a) looks like a handle connecting the shaded regions, and in case (c) is homologous to the shaded annulus.
However, since the configuration (b) can be `resolved' into either (a) or (c), we can use it to conjecture the form of entanglement entropy for the general case as well.\footnote{
Note that all three cases have the same level of complexity, despite case (c) involving only a single connected component, because in each case $X$ is bounded by two curves.
}

Let us focus on the case of the non-simple region $X$ having only two simply connected components, {\it i.e.}, the two-dimensional analog of the set-up of \sec{disconnecttwo}.
Applying the reasoning of the previous $1+1$ dimensional case requires that the boundary of the non-simple region $X$, generalising the four points $p_i$, has four components, say labeled by the four curves ${ \cal C }_i$ ({\it cf.}\ \fig{twoDregsCs}b) which enclose six simple regions $\R{ij}$ and to which we can associate the six minimal surfaces $\s{ij}$.  This in turn means that all curves are either infinite or meet at both endpoints.
In particular, for $\s{ij}$ given by the bulk minimal surface anchored on the boundary by the bounding curves ${ \cal C }_i$ and ${ \cal C }_j$ (which form a simple region $\R{ij}$), we obtain as the entanglement entropy of the region $x$ in \fig{twoDregions}b:
\begin{equation}
x = \s{12} + \s{23} + \s{34} +  \s{14}  - \s{13} - \s{24} \ .
\Label{xanswertwo}
\end{equation}

The simplest type of set-up for which we can calculate the entanglement entropy explicitly is the case of two infinite strips -- {\it i.e.}, region $X$ of \fig{minsurfX} smeared in one extra direction.  Clearly, we can employ the same reasoning as in \sec{disconnecttwo}, so the entanglement entropy density of $X$ is given by the entanglement entropy densities $\s{ij}$ by the expression \req{xanswertwo}.
 Note that even though the extra direction is translationally invariant, the individual $\s{ij}$'s are actually different from the one-dimensional case, because of the warp factor associated with the extra direction.  Therefore, the entanglement entropy formula would be extremely difficult to obtain directly in the CFT.

While equation \req{xanswertwo} provides a formula for the entanglement entropy of regions defined by four curves, as in \fig{twoDregions} case (b),  such configurations are highly non-generic in the set of non-simple regions.  The generic cases are given by only two curves, as in  \fig{twoDregions} cases (a) or (c).  The three minimal surfaces defined by these two curves are insufficient to determine the entanglement entropy, as we can see either by taking the limiting case (b) or by noting that the UV divergence for the unregulated entanglement doesn't scale appropriately for any non-trivial superposition of the areas of the minimal surfaces in question.\footnote{
The authors of \cite{Hirata:2006jx} give a very elegant proposal which circumvents the UV problem, while respecting sub-additivity.  In particular, for $X$ bounded by non-intersecting curves ${ \cal C }_{\alpha}$ and ${ \cal C }_{\beta}$, with the minimal surfaces $ s_{\alpha}$, $s_{\beta}$, and  $s_{\alpha\beta}$, their conjecture corresponds to $x = \min \{ s_{\alpha}+s_{\beta} , s_{\alpha\beta}\}$.  However, we believe that this simple prescription cannot hold in general: for example, for very elongated regions in case (a), which limit to the two infinite strips discussed above, we find disagreement from the expected answer:
$x = \s{12} + \s{23} + \s{34} +  \s{14}  - \s{13} - \s{24} \ne \min \{ \s{12} + \s{34} ,  \s{23} + \s{14} \}$.  (Although there may be special limits or special states for which the two prescriptions are equivalent, one can easily confirm the inequality for the vacuum state and general strips.)
We thank Tadashi Takayanagi for useful discussion on this point.
}  In the following subsection, we elaborate on these cases, and offer a bold conjecture for finding the entanglement entropy.

\subsection{Conjecture for generic non-simple regions}
 \label{conjecture}

Since we have a prescription for the entanglement entropy for three adjoining regions defining six simple regions and therefore six corresponding minimal surfaces as in \fig{twoDregions} case (b), let us try to express the more general two-curve cases (a) and (c) of \fig{twoDregions},  in a similar manner.
This involves joining the two curves by two more `auxiliary' curves, which simultaneously splits each of the original curves into two components with endpoints at the intersections.  We denote these as indicated in \fig{twoDregsCs}: in case (a), the two disjoint regions are bounded by $\left({ \cal C }_1 + { \cal C }_2\right)$ and $\left({ \cal C }_3 + { \cal C }_4\)$, while in case (c), the annulus is bounded by $\left({ \cal C }_1 + { \cal C }_4\right)$ and $\left({ \cal C }_2+{ \cal C }_3\right)$.  The two auxiliary curves which join $\{{ \cal C }_1,{ \cal C }_2,{ \cal C }_3,{ \cal C }_4\}$ are denoted by ${ \cal C }_5$ and ${ \cal C }_6$.  The special case (b) then corresponds to vanishing ${ \cal C }_5$ and ${ \cal C }_6$.  In general, we restrict the auxiliary curves ${ \cal C }_5$ and ${ \cal C }_6$ to have no intersections.
\begin{figure}
\begin{center}

 \includegraphics[width=16cm]{twoDregsCabc}
\caption{Three basic types of finite 2-dimensional non-simple regions $X$ (shaded). Note that case (b) is special, and can be deformed into either of the general cases (a) and (c).}
\label{twoDregsCs}
\end{center}
\end{figure}

Furthermore, to make the notation more explicit, let us denote the simple regions formed by joining the connected curves $\({ \cal C }_i+{ \cal C }_j+{ \cal C }_k+{ \cal C }_l\)$ by $\R{ijkl}$, and the corresponding entanglement entropy $S(\R{ijkl}) = \s{ijkl}$.  So in the first case,  \fig{twoDregsCs}a, we have the six simple regions
$\R{12}$, $\R{1536}$, $\R{1546}$, $\R{2536}$, $\R{2546}$, and $\R{34}$; and correspondingly, the entanglement entropies
$\s{12}$, $\s{1536}$, $\s{1546}$, $\s{2536}$, $\s{2546}$, and $\s{34}$.
Similarly, in the other case,  \fig{twoDregsCs}c, we have
$\R{1526}$, $\R{1536}$, $\R{14}$, $\R{23}$, $\R{2546}$, and $\R{3546}$; and
correspondingly, the entanglement entropies
$\s{1526}$, $\s{1536}$, $\s{14}$, $\s{23}$, $\s{2546}$, and $\s{3546}$.
Notice that in each case there is the third minimal surface anchored on the original non-simple regions; $\s{(12)(34)}$ for case (a) and $\s{(14)(23)}$ for case (c).
These are bounded by $\s{1546} + \s{2536}$ in case (a)  and $\s{1526} + \s{3546}$ in case (c).  In the special case (b) these naturally reduce to
$\s{14} + \s{23}$ and $\s{12} + \s{34}$, respectively.

\paragraph{Conjecture:}  Since the entanglement entropies cannot depend on our choice of the auxiliary curves ${ \cal C }_5$ and ${ \cal C }_6$, we {\it minimise} over all such possible configurations, namely:
\begin{eqnarray}
 x &=& \inf_{{ \cal C }_5,{ \cal C }_6}
 \left\{ \s{12} -\s{1536} + \s{1546} + \s{2536} - \s{2546} + \s{34} \right\}
 \qquad \rm{for \ case \ (a)} \ , \nonumber \\
 x &=& \inf_{{ \cal C }_5,{ \cal C }_6}
\left\{ \s{1526} -\s{1536} + \s{14} + \s{23} - \s{2546} + \s{3546} \right\}
 \qquad \rm{for \ case \ (c)} \ .
\Label{xanswerac}
\end{eqnarray}
Note that the partition of the original two curves into the curves ${ \cal C }_i$ for $i = 1, \ldots, 4$ also depends on the choice of  ${ \cal C }_5$ and ${ \cal C }_6$, in particular their endpoints.  Thus our prescription involves choosing two pairs of points on each of the original curves and two curves connecting these points, finding the entanglement entropy using the previously-derived expression \req{xanswertwo}, and minimising over all allowed configurations.

We have chosen the notation in \fig{twoDregsCs} so as to make the limiting case of both (a) and (c) coincide with the special case (b).  The price we pay for this choice is that we have to treat (a) and (c) separately; in particular in  \req{xanswerac} we write distinct expressions for each case.  However, there is a simplifying symmetry that our system respects: in particular, by relabeling ${ \cal C }_2 \longleftrightarrow { \cal C }_4$ in case (c), the two expressions in \req{xanswerac} become identical.  Thus, we could have written only one formula instead which covers all cases; nevertheless, to help visualising how the various minimal surfaces morph into each other as the case (b) is deformed into either (a) or (c), we will adhere to the notation introduced above.

\paragraph{Support for the conjecture:}
The above conjecture, albeit reasonably natural, has been only minimally motivated. Unfortunately, at present we can furnish neither a derivation, nor an explicit test.  Instead, we offer some supporting evidence for it and leave a detailed analysis for the future.  The main consistency checks are based mostly on the observations that the ultraviolet divergences cancel out correctly, and that the expressions \req{xanswerac} reduce correctly to the known cases in specific limits.
\begin{itemize}
\item
As pointed out previously, we do not expect to obtain an expression for entanglement entropy solely in terms of the three available minimal surfaces; hence we need to furnish more information.  Obtaining the requisite number of minimal surfaces seems the most natural way to proceed.  In particular, in the limiting case (b), the six surfaces $\{\s{12}, \s{13}, \s{14}, \s{23}, \s{24}, \s{34}\}$ comprise the minimal amount of information we need to supply (without specifying the QFT state) in order to determine the entanglement entropy.  On the other hand, the answer can not depend on the extra supplied information.  The expression \req{xanswerac} is therefore the simplest, minimal ansatz.
\item
Note that the infimum ansatz of  \req{xanswerac} is not a-priori the only possible way to satisfy the above requirements of supplying enough information and the final answer independent of the extra supplied information.  For instance, we could instead consider the supremum over all configurations, or more generally, some weighted sum of all configurations.  However, we now argue that these do not appear correct.  To see that the supremum is incorrect, consider {\it e.g.}\ case (a) with ${ \cal C }_5$ and ${ \cal C }_6$ coinciding.  Then we would have $x \ge \s{12} + \s{34}$, which is in manifest contradiction with \req{xupcup}, unless $x = \s{12} + \s{34}$.  But the latter leads to the unphysical result that the entanglement entropy is independent of the separation between the regions.\footnote{It is however possible as noted in \cite{Kitaev:2005dm} that suitable linear combinations of entanglement entropy can be made insensitive to the auxiliary curves ${ \cal C }_5$ and ${ \cal C }_6$. } The possibility of expressing $x$ in terms of some weighted sum of the expression $\{ \ldots \}$ in \req{xanswerac} is more difficult to rule out; however we note that this seems to require too much information (namely the weighing factor for each possible configuration).\footnote{In fact the simplest possibility  is that the term in curly braces in \req{xanswerac} be independent of ${ \cal C }_5$ and ${ \cal C }_6$.  However, this is unlikely as can be seen using the following argument by contradiction: If the above were  true, we could in particular take ${ \cal C }_5$ and ${ \cal C }_6$ to coincide and deduce that $x = \s{12}+\s{34}$, independent of the separation between $\R{12}$ and $\R{34}$. To see this cannot happen, let ${ \cal C }_2$ and ${ \cal C }_3$ coincide. Then the region $X$ is simply connected and $x = \s{14}$ for any state (which in general is different from  $\s{12} + \s{34}$).}
\item
A more quantitative way to motivate \req{xanswerac} is to consider the ultraviolet divergences of the unregulated entanglement entropies.  The leading divergence comes from the boundary term, and therefore is proportional to length of the bounding curves.  By construction, we therefore expect that the leading divergent part of $x$ is proportional to the length of ${ \cal C }_1+{ \cal C }_2+{ \cal C }_3+{ \cal C }_4$.  This is indeed exactly reproduced by the expressions in \req{xanswerac}, as can be seen by noting that each subscript $i$ corresponds to ${ \cal C }_i$ and then counting the subscripts.   For example, the first expression in $\{ \ldots \}$ of \req{xanswerac} gives the UV divergence proportional to  $( { \cal C }_1 + { \cal C }_2 - { \cal C }_1 - { \cal C }_5 -{ \cal C }_3 -{ \cal C }_6 + { \cal C }_1 + { \cal C }_5 + { \cal C }_4 + { \cal C }_6 + { \cal C }_2 + { \cal C }_5 + { \cal C }_3 + { \cal C }_6 - { \cal C }_2 - { \cal C }_5 - { \cal C }_4 - { \cal C }_6 + { \cal C }_3 + { \cal C }_4 )
= { \cal C }_1+{ \cal C }_2+{ \cal C }_3+{ \cal C }_4$.
\item
Furthermore, as remarked above, both expressions in \req{xanswerac} reduce correctly to the limiting case (b).  Although it is manifest that by shrinking ${ \cal C }_5$ and ${ \cal C }_6$ to a point, the expressions $\{ \ldots \}$ in \req{xanswerac} coincide with \req{xanswertwo}, we still need to show that this corresponds to the infimum over all (in this limit closed) curves ${ \cal C }_5$ and ${ \cal C }_6$.  Nevertheless, the latter  follows from the observation that the minimal surfaces bounded by ${ \cal C }_5$ and ${ \cal C }_6$ cancel out in \req{xanswerac}.
\item
Since $x$ is an infimum over a set of configurations of ${ \cal C }_5$ and ${ \cal C }_6$, by evaluating specific configurations, the expression \req{xanswerac} automatically yields upper bounds for $x$.
For example, by taking coincident ${ \cal C }_5$ and ${ \cal C }_6$ in case (a), we get an upper bound which coincides precisely with the previously obtained upper bound on $x$, namely $x \le \s{12} + \s{34}$.  By the symmetry of interchanging ${ \cal C }_2$ and ${ \cal C }_4$ and case (a) with case (c), we likewise obtain the other upper bound $x \le \s{14} + \s{23}$.
\item
Finally, we can easily check that in the limit of $X$ being composed of only one simple region, we obtain the correct $x$.  Specifically, for case (a), consider {\it e.g.}\ the case of ${ \cal C }_3$ and ${ \cal C }_4$ coinciding.  (A special case of this is when ${ \cal C }_3$ and ${ \cal C }_4$ both shrink to a point; however our argument applies more generally.)  Then $\s{34}=0$, $-\s{1536}+\s{1546}=0$, and $\s{2536}-\s{2546}=0$, so \req{xanswerac} reduces to $x = \inf_{{ \cal C }_5,{ \cal C }_6} \{\s{12}\} = \s{12}$, as required.  Similarly, in case (c), coincident ${ \cal C }_2$ and ${ \cal C }_3$ gives $x = \s{14}$.
\end{itemize}

Above, we have explained the motivation for the conjecture that the entanglement entropy for a general non-simple regions of \fig{twoDregions} is given by \req{xanswerac}.  However, this implicitly assumes that the entanglement entropy does have a geometric description, in particular that this can be expressed as a function of minimal surface areas of particular regions. It would be interesting to derive this result from first principles, which we hope to return to in the future.

\section{Discussion}
 \label{discussion}

We have presented a simple derivation for the entanglement entropy of disconnected regions in 1+1 dimensional quantum field theories, by exploiting the holographic prescription for computing the same. The result, \req{xanswer}, is consistent with the formulae known for 1+1 CFTs and generalizes them to the non-conformal realm. In deriving this result, we crucially exploited the definition of minimal surfaces and the strong sub-additivity property of entanglement entropy (which also follows from the minimal surface prescription \cite{Headrick:2007km}).  Our derivation consisted of two separate arguments, both yielding the same answer; one employed a natural ansatz and obtained the requisite formula by requiring that it reduce to correct expression in specific limits, while the other relied on self-consistency and used the constraints to obtain coinciding upper and lower bounds on the desired expression.
One intriguing aspect of our derivation are the entropy inequalities \req{sijklmnn}, which appear when we have more than two disconnected regions. These have not hitherto been discussed in the field theory literature and appear to be natural analogs of strong sub-additivity in the general situation.

We then proposed a conjecture for the formula in higher dimensions, concentrating specifically on the 2+1 dimensional case.   When the regions in question touch at two isolated points, our proposal \req{xanswertwo} is a natural generalization of the 1+1 dimensional result. However, as discussed in \sec{disconnectd} one can have more complicated set-up; we conjecture that the entanglement entropy in such circumstances is obtained by adding `virtual regions' to reduce the problem to the previous case, and then minimizing over the space of possibilities for such regions; see \req{xanswerac}. While we have not provided any concrete evidence for our conjecture, we have shown that it is consistent with known data. In particular, we have argued that it correctly captures the leading divergence of the entanglement entropy (the area law)  and reduces to the known results when some components are collapsed to obtain connected regions. It would be very interesting to prove this result from first principles.

We have noted in \sec{disconnectd} that while \cite{Hirata:2006jx} propose a far simpler expression for the entanglement entropy of a region bounded by two curves (namely cases (a) and (c) of \fig{twoDregions}), involving only the three naturally defined minimal surfaces, this doesn't seem to reduce to the requisite result for the infinite strip case.  However, one of the motivations of \cite{Hirata:2006jx} was based on the close analogy with Wilson loop computations. Recall that Wilson loop expectation values in the AdS/CFT context are also given by the regulated area of a bulk minimal surface \cite{Maldacena:1998im,Rey:1998ik} which corresponds to a string world-sheet.  As discussed in  \cite{Gross:1998gk}, the correlation function of two Wilson loops has a transition in the dominant world-sheet from the connected surface ending on the two loops to a disconnected surface ending on each individual loop. However, one important distinction in this situation is the fact that string world-sheeets are oriented (for unitary gauge group). It is therefore plausible that the Wilson loop calculation receives contributions from only two distinct configurations, while the entanglement entropy is sensitive to the correlations across various regions as given by \req{xanswerac}.

So far, in all of our discussion, we have assumed that given a simple region $\R{ij}$ on the boundary, there exists a corresponding bulk minimal surface $\s{ij}$ homologous to $\R{ij}$.  While this is true for the vacuum state (and should be true for any pure state which is sufficiently near the vacuum state, essentially because we expect the minimal surface to exist close to the AdS boundary), $\s{ij}$ is not a-priori guaranteed to exist for a general state. In this context it is useful to recall the examples discussed in \cite{Nishioka:2006gr, Klebanov:2007ws,Faraggi:2007fu}, where the bulk surface contributing to the entanglement entropy changes discontinuously as the length of the boundary region is varied (in a fashion similar to phase transitions involving Wilson loops in thermal AdS/CFT \cite{Witten:1998zw}). However, the local existence of the minimal surface,  coupled with the requirement that we obtain the correct dependence on the UV cut-off for local QFTs,  indicates that the formulae we give are correct, but allow the possibility of the minimal surfaces jumping discontinuously under continuous deformations of the boundary regions.

In fact, if we allow for the minimal surfaces to be topologically nontrivial, our conjecture \req{xanswerac} naturally contains the apparently mysterious minimal surfaces anchored on both of the original curves and nowhere else, denoted $\s{(12)(34)}$ and $\s{(14)(23)}$ above.  For example, the former may arise from $\s{1546}$ when ${ \cal C }_5$ and ${ \cal C }_6$ coincide {\it and} this surface has smaller area than the more natural $\s{12}+\s{34}$ configuration.  In other words, as ${ \cal C }_5$ and ${ \cal C }_6$ are brought close enough together, the minimal surface $\s{1546}$ may develop a handle.  It would be interesting to explore the topological restrictions, in particular the requirement of the surfaces being homologous to the corresponding boundary regions, in this context.

An important open problem that we have not discussed in the present work is to generalize the entanglement entropy formulae to the case with time dependence. As discussed in \cite{Hubeny:2007xt}, for general non-static situations one needs to replace the minimal surface by an extremal surface in the bulk.\footnote{This is also necessary for stationary spacetimes, as the timelike Killing field at infinity is not hypersurface orthogonal.} Unfortunately, the arguments given above (or indeed those used in \cite{Headrick:2007km}) do not carry over to the time dependent case, since the extremal surfaces of interest do not lie on a single spacelike surface in the bulk.
This prevents direct comparison of the areas of the surfaces; indeed even the simple inequalities \req{sijk} are harder to obtain in this case owing to the possibility that one can reduce the area of surfaces by wiggling them in the timelike direction. It would be interesting to establish sub-additivity explicitly from the geometrical viewpoint (thereby lending further credence to the proposal of \cite{Hubeny:2007xt}) and derive the formulae for disconnected regions.

\section*{Acknowledgements}
 \label{acks}
It is a pleasure to thank Joan Camps and Tadashi Takayanagi for discussions and comments on the manuscript. This work is supported in part by STFC.

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Outskirts – Platform
King Midas Sound
Banana Oil
Freelove + Eilidh Rogers
Outskirts is an annual festival held at Platform, an arts centre in Easterhouse, whose very name evokes both the venue's location, on the geographical margins of Glasgow, and the porous nature of the (often) multidisciplinary work on offer. Everything does have one foot in performance – whether music, theatre or performance art – though the festival takes pleasure in mixing it all up.
Jessica Higgins' dedications is a much more productive and enjoyable approach to performance, a willfully absurdist collection of nonsense and non sequiturs derived from radio announcements– with half a dozen performers scampering about a stage giving a collection of shout outs, often to the weather, one particularly absurd routine involving a mouldy jumper affected by said weather. A major aspect of why this is so compelling is the fluid musical backing of Banana Oil, occasionally referencing cover versions, a three piece jazz band who could make reading a telephone directory a thing of joy. After the performance art is over, the trio snake back into the venue – Sun Ra Arkestra style – with toy instruments and play a gig proper.
While may have earned a reputation for being very serious – academic even – Banana Oil always look as if they're having as much fun as their audience are. It's always a pleasure to watch the interaction between Joe Howe, looking like a mad scientist as he alternates between alto sax and keyboards, Niall Morris, who can't stop moving to his own supple bass lines, and the endlessly inventive, propulsive drumming of Laurie Pitt – so propulsive that he knocks his hi hat off his kit, not once, but twice.
King Midas Sound are a very different proposition, being one of Kevin Martin's long running investigations into black and electronic music. For this project, he works with English/Trinidadian poet Roger Robinson, who perform their new album, Solitude. Essentially a long spoken word piece with a sparse soundscape backing from Martin, it fits the Outskirts bill perfectly.
The stage is filled with clouds of dry ice, subsiding to reveal Martin playing synths, and occasional bells and gongs, against an array of dimly throbbing lights. Robinson emerges through the haze to deliver what is essentially an extraordinary, extended monologue on the aftermath of a relationship, and the obsessive behavior of the abandoned lover. The effect is hypnotic, the restrained intensity of Robinson's measured delivery matched to eerie precision by Martin's backing. This may be the bleakest Saturday night entertainment I've ever witnessed, but it's incredibly powerful, and even deeply moving.
Light relief comes courtesy of David Sherry, a Northern Irish performance artist working in Glasgow who's been responsible for some of the wackiest, actual LOL inducing work I've seen in the field. Tonight he embarks on a stream of consciousness monologue crossing from bookshops to Brexit to the state of the economy, aided by some of his trademark absurdist props, such as a Spiral of Downward Mobility and a Brexit map of Europe wind instrument. Halfway through, he sticks his head in a crash helmet while a guy comes over of the audience and beats him on the head with a bat. As you do. It culminates with Sherry smashing out a poem on a keyboard with a hammer, shards of plastic flying into the audience. The poem is gibberish; the routine is brilliant.
Outskirts always builds up to Easterhouse Conversations, where musicians are invited to a residency with the local community to create new music. This year's crop are Free Love (the artists formerly known as Happy Meals) and Eilidh Rogers, the drummer from Sacred Paws. Free Love's set is all new material. Though the music is their characteristically intoxicating brew of psychedelic Italo-inflected sound, only strengthened by Rogers' live percussion. This is also unusual in being presented in a seated theatre format, Free Love being a band who thrive in a club setting, acknowledged by singer Suzi Rodden, where she tells the audience, "Treat it as immersive". Which it certainly is, as we all writhe along in our seats, before Rodden gives a shout out to the people of Easterhouse who inspired it.
Brian Beadie.

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Tag Archives: Helpless
The Colorful World of Rachael Sage
No, I haven't gone mad – the title refers to the title of a new song by American singer songwriter Rachael Sage and features on the deluxe edition of her recent album, Blue Roses. The song is Coloring Book and is to be released in November – more of that to come later. Of course, if you're one of my American readers, this spelling will be completely normal!
I first became aware of Rachael Sage when she was featured on the Angry Baby blog earlier this year and then happened to catch her on BBC Radio 2's Good Morning Sunday performing another song from Blue Roses. The song was a cover of Neil Young's Helpless and features legendary singer songwriter Judy Collins, with whom she is soon to support on Collins' UK tour.
Ahead of the tour, I recently caught up with Rachael and amongst other things we talked about how the collaboration with Judy originally started. It turns out that five or six years ago, Sage performed at a music industry conference in France (she also runs her own record label) and was approached by Judy Collins's manager who had noticed the similarities between the two (at the time, Collins' was running her own label, Wildflower). One thing led to another and Sage ended up meeting Collins and performing at her Wildflower Festival. Of the experience, Sage says that Collins was "very warm and welcoming, she listened to my album and gave me feedback. We became friends and then she had me open for her at a lot of shows around the country and even in England and Scotland. Eventually, I felt comfortable enough to ask her to sing with me on my album and amazingly she said yes." Endearingly, Sage recounts this with wonder, as if she can't believe her luck.
The resulting song is a beautiful interpretation of the original, perfectly suiting both voices. A cover of another artist's song is a rarity for Sage, her long back catalogue is mostly self penned but this was a great choice. Sage says that the recording process was "wonderful, Judy was so much fun, she's hilarious. People don't realise but she brings a lot of… almost comedy to the live shows."
Collins is set to perform a few shows in across the UK in October, starting with Bristol on 20 October and ending a few days later in Cardiff (click here for more dates), with Sage in support. At present, the two will perform separate sets but when asked if they will collaborate on stage, Sage tantalises with a "maybe – her accompanist is always keen to do something, but she's so busy it's difficult to get time to sort anything out".
Collins may be busy but the same can also be said of Sage – in addition to writing and performing, she runs her own record label. To the layman, running the business side of a record label at the same time as creating your own music would seem to require two completely different mindsets (hence the comment from Sage that one is "definitely work"). Sage says not signing to another label was a conscious decision after college, when she was discovering that the music she was creating was something that "I needed to self develop, to do it on my own. If I signed to a major label based on a particular sound and songwriting ability, I'm going to get hurt as they'll tell me what to do". Showing impressive self awareness at a young age, Sage decided that in order to have creative freedom, the only way was to go it alone, which she did, collaborating with the people who had helped make her first teenage demos.
As an interviewee, Sage was very giving and I had a lot of material so look out for more later in the year. She was also a lot of fun and loved the fact that our skype interview (a first for both of us) ended with a host of bubbles from my 9yo's bubble wand floating across the screen – we ended with the quote "every interview should end with bubbles".
Rachael Sage will be supporting Judy Collins in the UK from October 20 – 26. Click here for details.
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This entry was posted in Music and tagged Blue Roses, Helpless, Judy Collins, Rachael Sage on 21st August 2015 by thoughtsofjustafan.


Tell us more about the solo exhibition that you are presenting, how did the concept come about?
Dia: Watch This Space depicts the mundane experiences, day after day, that build our character and develops our personality. My body of work is a reflective process, capturing energy in the transition from one experience to another. It is my personal belief that these veiled experiences are most profound. As an artist I work with the 'constructed image' primarily public spaces, cornering you to a contemplative experience limiting us to observe, reflect and experience the basic realities.
What was the thought behind using newspapers and magazines to build set-ups and what was the design process like?
Dia: Although it may seem mundane, the mundane is our day to day. For me, it's really the small things that build up to each day… it is all connected. And my practice works in the same way – with each tile, each pattern, is designed to precision. I am building towards the everyday. I spend a lot of time thinking about what the image will be and yet, a lot of stuff happens by accident and as much as I plan there is still so much that I can't control.
What would you want the viewers to take home?
What is the evolution of your art along these years?


A bold and uncompromising feminist manifesto that shows women and girls how to defy, disrupt, and destroy the patriarchy by embracing the qualities they've been trained to avoid.
Seizing upon the energy of the #MeToo movement, feminist activist Mona Eltahawy advocates a muscular, out-loud approach to teaching women and girls to harness their power through what she calls the "seven necessary sins" that women and girls are not supposed to commit: to be angry, ambitious, profane, violent, attention-seeking, lustful, and powerful. All the necessary "sins" that women and girls require to erupt.
Eltahawy knows that the patriarchy is alive and well, and she is fed the hell up: Sexually assaulted during hajj at the age of fifteen. Groped on the dance floor of a night club in Montreal at fifty. Countless other injustices in the years between. Illuminating her call to action are stories of activists and ordinary women around the world--from South Africa to China, Nigeria to India, Bosnia to Egypt--who are tapping into their inner fury and crossing the lines of race, class, faith, and gender that make it so hard for marginalized women to be heard. Rather than teaching women and girls to survive the poisonous system they have found themselves in, Eltahawy arms them to dismantle it.
Brilliant, bold, and energetic, The Seven Necessary Sins for Women and Girls is a manifesto for all feminists in the fight against patriarchy.
Mona Eltahawy is an author, activist, and commentator. Her essays and op-eds have appeared in the New York Times, Washington Post, Guardian, Time, NBCNews.com, and New York Magazine, among many others. Follow her on Twitter at @monaeltahawy.


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LHC forwards 'The Legend of Maula Jatt' to censor board for further deliberation
The Lahore High Court on Tuesday forwarded the appeal for the release of The Legend of Maula Jatt to the censor board, reported Express News. According to the provincial high court, the board can decide whether character names and dialogues can be used in the film or not. The plea against the characters' names and dialogues used in the Bilal Lashari […]
Gap between IT firms, corporate sector hinders growth
KARACHI: The communication gap between tech companies and the corporate sector is undermining the potential of IT ecosystem in Pakistan, despite some IT firms working for US clients. "There are so many IT companies and entrepreneurs that are doing business of product development with companies in the US and Europe, however, people in Pakistan are not […]
Chief Secretary wants to step up tree plantation
LAHORE – Punjab Chief Secretary Yousaf Naseem Khokhar has directed that the Clean and Green Pakistan campaign be continued in the province in a vigorous manner, saying that special attention should be paid to improving cleanliness, besides massive tree plantation in cities. He issued these directions while presiding over a meeting at Commissioner's Office during […]
PBA opposes issuance of licences for satellite broadcast stations
Broadcasting, Media and Governance, Television
KARACHI: The Pakistan Broadcasters Association (PBA) has criticised a decision by Pakistan Electronic Media Regulatory Authority to conduct auction proceedings for the issuance of licences for satellite TV broadcast stations without taking up a petition filed by PBA against the proposal. The association appealed to the prime minister to intervene and stop the process. According […]
Press freedom under serious threat from govts worldwide: IPI
Digital Media, Freedom of Expression, Front Page, Media Safety
KARACHI: Governments are increasingly clamping down on press freedom as new laws are being enacted to curtail media independence around the world, while 55 journalists were killed since May last year, according to the International Press Institute (IPI). In a press statement issued on Wednesday, Barbara Trionfi, executive director of the Vienna-based IPI — a […]
Murder of Journalists: Blood money lets culprits go scot-free
Freedom of Expression, Media Safety
Blood money is considered a stumbling block in the way of bringing to justice the culprits of slain journalists in Sindh southern province of Pakistan. The murder of Amir Bux Brohi and Shahid Hussain Soomro were wound up after out of court settlements were agreed upon blood money between the families of deceased journalists and […]
World Press Freedom Day and Pakistan
Three days ahead of the World Press Freedom Day, 2109, a reporter has been as killed in Dera Ismail Khan's Parowa area. Malik Amanullah Khan, who had been the chairman of the Parowa Press Club, was a target in a drive-by shooting near Landa Sharif Adda on Multan-Dera Road. He died at the scene of […]


Every year Reporters Without Borders releases the annual World Press Freedom Index. The Index ranks countries around the world based on the amount of freedom they grant their media.
The results of the 2015 World Press Freedom Index can be viewed on a Leaflet powered map. Countries on the map are colored based on their press freedom rankings. I hate to make spurious correlations - but basically the colder a country is then the more likely it is to have press freedom (except Russia!).
The map includes the option to also view the 2013 & 2014 Press Freedom Index scores for each country. Alternatively you can check-out the the 2014 World Press Freedom Index map. Last year's map included an interesting 3d view which visualizes the press freedom index as a topographical map.