1 of 4 Heat oven to 500 degrees F. Line 13 x 9 x 2-inch baking pan with foil.
2 of 4 Arrange chicken pieces in pan.
3 of 4 Add sun-dried tomatoes, 2 tablespoons oil from tomatoes, basil, Parmesan, garlic and pepper to food processor or blender. Whirl until smooth. Spread about half (about 1/4 cup) of the pesto over chicken.
4 of 4 Bake in 500 degree F oven for 15 minutes. Remove from oven. Heat broiler. Spread remaining pesto over chicken. Broil 6 inches from heat for 3 to 5 minutes or until browned. Serve with steamed new potatoes.
Per Serving: 345 mg sodium, 13 g Fat, total, 181 mg chol., 3 g sat. fat, 384 kcal cal., 58 g pro., 7 g carb.


text:
CCB raids houses of absconding IMA boss Mansoor Khan's THREE wives in Bengaluru
Bengaluru, Jun 18: Central Crime Branch (CCB) on Monday conducted raids at three different locations at the residences of wives of I Monetary Advisory (IMA) jewels founder Mansoor Khan who is an accused perpetrating financial fraud in Bengaluru.
Raids were conducted in Shivajinagar, the residence of Khan's first wife and two different residences in Tilak Nagar related to his second and third wife.
CCB collected some important documents of investments and details of Khan's financial issues, 1.5-kilogram gold and Rs 2.7 lakh cash.
Karnataka government had constituted an 11-member Special Investigation Team (SIT) to probe the alleged fraud by the IMA, which has an estimated Rs 200 crore investment of Muslim women alone. Over 38 thousands of complaints have been filed by people who had invested in IMA.
18 complaints filed in high court to direct the state government to handover the IMA case to the Central Bureau of Investigation. Several investors also hail from Andhra Pradesh, Telangana and Tamil Nadu.
Police have also registered a case against IMA Jewels and Khan and teams have been formed to nab him. Earlier, seven directors of different entities linked to Khan had been arrested.
It is alleged that IMA Jewels, with Muslims as its prime investors, has not paid interests on investments for the last three months.
Earlier, scores of investors staged protests outside IMA Jewels' office in Sivaji Nagar, demanding the firm to return their money.

redpajama_set_name: RedPajamaCommonCrawl


text:
Poet Focus: Marianne Moore
I live in the oldest incorporated suburb of St. Louis, a town of 27, 540 called Kirkwood. It became a town in 1853, when a rail line was extended west. The project manager for the line was a surveying engineer from New York named James Pugh Kirkwood.
For the first 40 or 50 years of its life, Kirkwood was likely typical of towns in the 19th century Midwest — small town center serving both the rail line and the farms in the surrounding region. Merchants, clergymen, and others built homes near the town center, many of which survive today in good working order.
This was the town and culture where poet Marianne Moore was born in 1887. Her grandfather, the Rev. John Warner, had been a chaplain at the Battle of Gettysburg and was called to be pastor of the First Presbyterian Church of Kirkwood in 1867. Moore and her family lived with her grandparents until 1896, when Warner died and the family moved back to Pennsylvania. (The church still exists, although not the original church building.)
From Carlisle, Pennsylvania, it was Bryn Mawr College (Moore received a B.A. degree), a commercial school, and a teaching stint, until she and her mother moved to New York City in 1918. She worked at the New York Public Library and later served as acting editor of a literary magazine, The Dial. Moore lived in Brooklyn until her death in 1972. She won numerous recognitions for her poetry, including the 1952 Pulitzer Prize for her Collected Poems.
My edition of her Complete Poems is the paperback edition published by Penguin Books in 1994. The original version was published in 1967, and other editions have been subsequently published. Mine includes several poems previously unpublished, along with some of her prose as well; it contains the poems from all 11 of her published works, from 1921 to 1966. And yet the Complete Poems is still a relatively slender volume.
Through her work at The Dial, Moore got to know such poets as Wallace Stevens and William Carlos Williams, Ezra Pound, and "H.D., " a.k.a Hilda Doolittle, poets who were part of the "Imagist" movement (H.D. published Moore's first volume of poetry without Moore's knowledge). From our vantage point, we would say she was squarely in the middle of the Modernist period in American literature.
And yet, for all of her modernist associations, Moore's poetry didn't exactly fit the category. There's a richness, almost a lushness, in many of her poems that's absent from the moderns. She ranged over history and literature — Rome and Greece, Britain and Ireland, and America — as well as music and the natural world. (I've read only one poem about weasels, and Marianne Moore wrote it.)
Moore wrote story poems, too, including one entitled "Marriage" that runs to nine pages. And a poem on baseball. Her poetic eye ranged over all of life, like this one, "No Swan So Fine":
"No waters as still as the
dead fountains of Versailles." No swan,
with swart blind look askance
and gondoliering legs, so fine
as the chintz china one with fawn-
brown eyes and toothed gold
collar on to show whose bird it was.
Lodged in the Louis Fifteenth
candelabrum-tree of cockscomb-
tinted buttons, dahlias,
sea-urchins, and everlastings,
it perches on the branching foam
of polished sculptured
flowers—at ease and tall. The king is dead.
Ezra Pound and Marianne Moore in the late 1960s.
Read that one aloud (the Poetry Foundation has a reading of the poem by Melissa Severin). Her poems are best read aloud, and slowly, to hear the music and interior sense of rhyme, order, and alliteration. She was a modernist who wasn't a modernist.
While these qualities of her poetry likely come from the experiences of a lifetime, I like to think that those first years of her early life in Kirkwood were critical.
She wouldn't recognize most of the town today. But a number of the homes, a few downtown buildings, even the 1893 train station (yes, we have a functioning Amtrak station) might be familiar. And she might still recognize the spirit of the place, for it has not changed that much in the 117 years since she left.
I can imagine a little girl, dressed in her 1890s dress and with ribbons in her hair, walking unpaved streets, past her grandfather's church, stepping carefully around the ubiquitous horse droppings. She's going to the general store across from the train station, the store with its farm implements and seed, dry goods and feed, and the occasional candies offering sweetness and excitement, making the walk worthwhile. And all of this would become stored for later use by her poetic eye.
Kirkwood's streets are paved now, and the horse droppings and wagons long gone. But you can walk by the Fishback House (1867) and Mudd's Grove (1859), and many others, and still hear the sounds of her poetry in the air, if you listen closely enough.
Photograph by Fortimbras. Sourced via Flickr. Post by Glynn Young, author of the novels Dancing Priest and the recently published A Light Shining.
Buy a year of happy work mornings today, just $5.99. In June we're exploring the theme Mirror, Mirror.
Filed Under: article, Bird Poems, Blog, Literary Tour, Poems, poetry, poetry teaching resources, Poets, Swans Swallows Phoenix
Excellent review. I read it out loud before your suggestion. I read poetry out loud 90% of the time. Her poetry is rubbing off into yours Glynn. The Kirkwood muse is still hangin' round.
Mary Sayler says
Interesting article about an interesting poet, Glynn! Thanks. Moore brought flair to her poems – and also to her discussions of her work. I especially like how her poems developed musicality when read aloud even though she often wrote syllabic verse. That awakened me to the fact that rhythmic poems don't necessarily need iambic pentameter.
Maureen Doallas says
When I was in Philly last summer I stopped at the Rosenbach (a wonderful museum) and got to see some of the Moore collection there, as well as her (reconstructed) living room from her Greenwich Village apartment. Some fascinating stuff there.
L. L. Barkat says
I am quite taken with the event of her poetry being published without her knowledge. How'd that go? Was she pleased? Displeased?
Imagine someone loving your stuff that much to go and do that 🙂


text:
2517 Orma è un asteroide della fascia principale. Scoperto nel 1968, presenta un'orbita caratterizzata da un semiasse maggiore pari a 3,1845300 UA e da un'eccentricità di 0,1728338, inclinata di 2,62935° rispetto all'eclittica.

È stato così chiamato in riferimento al vocabolo italiano orma, nel senso di traccia, impronta. Va considerato inoltre che il nome è un anagramma di quello di un altro asteroide: 1257 Móra.

Note

Collegamenti esterni

Corpi celesti scoperti nel 1968
Orma

redpajama_set_name: RedPajamaWikipedia


"From sea to shining sea, the climate for guns is changing, and the progress extends beyond Starbucks. Major retailers such as Home Depot, Best Buy and Barnes & Noble apparently also are friendly to people who openly pack heat, according to the Wall Street Journal. The Brady Campaign warns businesses that allowing customers to carry guns will scare away other customers. Yet it seems pretty obvious that the businesses themselves - despite all the pressure they face from trial lawyers and bureaucrats to ban guns - are in a much better position to know what their customers want."
An interesting point. We're seeing, simultaneously, (1) an increase in gun sales to record levels, and increases in new gun owners; (2) Collapsing poll figures for public belief that guns are bad and gun restrictions are good; (3) legislatures predictably pulling away from those ideas, as well, and (4) the judiciary (and not just the Supreme Court, either) endorsing gun-related rights. It's hard to say which is cause and which is effect, or whether all are being driven by some manner of collective consciousness.
Your "collective consciousness" thought is provocative. I have the same little red flag waving in the back of my mind, too.
Suddenly, out of nowhere, AR's, ammo, and handguns are selling like hotcakes?
What event stirred that expenditure, even in the face of a serious recession?
"The Brady Campaign warns businesses that allowing customers to carry guns will scare away other customers."
Having briefly worked in a B&N store, I would suggest the customers that are scared away are not the type to buy much, anyway. They are the ones who will buy a small coffee that they nurse for hours, while they read Chomsky for free. These are not kids, either. That's my unscientific observation, at least.
More than What the Supreme Court Says, Congressional Research Services (Fall 2008) finds the SAME effect essentially saying that the U.S. Supreme Court does NOT have the final word in constitutional interpretation, not even Congress, but the American People!


Hang out with friends when you're older.
Stay indoors, you are too young.
Nobody has it easy, adjust.
Stay married, your kids will thank you later.
Divorce at this age? Are you crazy?
Stay indoors, you are old and frail.
Linking this to Writing Wednesdays and to Friday Reflections.
Nailed the sad truth that Indian women have to put up with. So few words, yet so much emotions. Bhavya, you rock!
So true… ppl n India are fkd up !! Even the famous malayalam swearing keep hitting female gender !!!


text:
Home » Articles » Articles
Booklist Editors Read for Fun 2007
American Libraries Associate Editor Dan Kraus talked to seven editor-reviewers for Booklist Magazine about the books they read in their spare time in 2007. Learn one editor's excuse for not reading Harry Potter, how another found cheer in "an intrusive and lying government, torture, and nuclear experiments," and the treasure a third found in Montana READ MORE
The Civil Rights Movement: Sites for Students and Researchers
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The Supreme Court's Brown versus Board of Education decision turned 50 in 2004. Over the next several years, many of the perennially popular research topics of the Civil Rights Movement will celebrate equally momentous anniversaries. Media attention and scholarly interest increase with each significant anniversary.Fortunately for librarians and researchers, the events, people, and places of the Civil Rights Movement are well represented online. Web sites include an excellent array of primary source materials, including papers, laws, photographs, oral histories, and speeches. These digitized collections will interest researchers of any age group or scholastic level. The digital files offer a great opportunity to listen to Martin Luther King's most galvanizing speeches in his own voice, to read Malcolm X's letters to his mother, and to look at photographs of the freedom riders. In addition to primary source documents, many federal sites provide in-depth data, history and government documents from the Civil Rights Movement. The museums dedicated to civil rights provide outstanding study guides, biographical information, and photographs online. READ MORE
Sunshine Week: Your Right to Know
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>Download Sunshine Week poster (.pdf format).What Is Sunshine Week? READ MORE
Booklist Editors' Choice: 2007
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Every year, the editors of Booklist magazine prepare a set of Editors' Choice lists. These lists contain dozens of great reads representative of the year's most outstanding books. Books included on these lists balance popular appeal with intellectual, literary, and aesthetic excellence. READ MORE
Welcoming Libraries: How Communities' Favorite Public Institutions Are Settling New Immigrants
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If you're listening to the presidential debates, you know immigration continues to be a hot issue in America. Foreign-born residents now constitute nearly 13% of the American population, a rate not seen since 1910. A new report from the Urban Libraries Council (ULC) entitled "Welcome, Stranger: Public Libraries Build the Global Village" reports on trends for the spread of immigration into new cities, and the role public libraries play in welcoming and settling new residents READ MORE
First Lady Laura Bush
American Libraries Editor Leonard Kniffel visited the White House March 19 for an exclusive interview with First Lady Laura Bush. Mrs. Bush, a teacher and librarian and the first librarian to serve as First Lady, covers a host of topics, including the role she will play in the George W. Bush Presidential Library to be built at Southern Methodist University in Dallas, how her work as a librarian influenced has influenced her own initiatives, and why young people should consider librarianship as a career. From American Libraries Focus. READ MORE
Children's Author Deedy, Oscar Nominee Breslin Named Spokespeople for School Library Media Month
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Award-winning author Carmen Agra Deedy and Oscar-nominated actor Abigail Breslin have been named national spokespeople for the 2008 School Library Media Month, celebrated in April. School Library Media Month is sponsored by the American Association of School Librarians (AASL), a division of the American Library Association (ALA), and celebrated by school library media centers around the country READ MORE
Small Change Makes a Big Difference
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If Barker residents are fond of their pennies, they had best lock them away and guard them carefully.Otherwise, a Pratt Elementary School pupil could very well snatch them up and deliver them to Barker Free Library, which is a bit squeezed for cash these days. READ MORE
Step Up to the Plate @ your library Swings into Action
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The boys of summer are stepping up to the plate, so why not join them? The American Library Association and the National Baseball Hall of Fame and Museum are getting into the swing of things by launching the fourth season of Step Up to the Plate @ your library. It could be your chance to win a trip to the National Baseball Hall of Fame! READ MORE
In Celebration of National Library Week
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One might hear this week how we celebrate libraries because they house national treasures—e.g., knowledge, history, creativity, and equal accessibility.But I celebrate libraries for very personal reasons. I grew up in northwestern New Jersey, surrounded by the patchwork earth of farmland that grew sweet corn and isolation. Boredom was my county's most plentiful crop. During the internet-less, video-game-less, and seemingly endless summers of my childhood, I could ride my bike to the Washington Borough Public Library and within one minute be transported to the world of Dr. Doolittle; The Hardy Boys; and Babe Ruth, All-American Hero. Each book was a planet with a spine. The librarian was an organizing star, keeping all those spheres in their places for future explorers to discover. The library itself was a universe—a macrocosm between paint-chipped walls, below a roof paid for by bake sales, sandwiched between a tattoo parlor and halfway house. It was the most fecund place I knew—a greenhouse for my imagination, where fluorescence had to do with my mind's branches spreading. O the joyful fire in the astronaut's skull when divagation led to apprehension READ MORE


text:
Ted Cruz says billions in assets tied to 'El Chapo' should fund the border wall
Joel Shannon
U.S. prosecutors are seeking to seize billions of dollars in assets connected to convicted drug lord Joaquín "El Chapo" Guzmán, and Sen. Ted Cruz, R-Texas, said Tuesday that he believes the money should be used to build a border wall with Mexico.
In reference to Guzmán's conviction on 10 criminal counts – including drug trafficking, weapons violations and operating a continuing criminal enterprise – Cruz tweeted: "U.S. prosecutors are seeking $14 billion in drug profits & other assets from El Chapo which should go towards funding our wall to #SecureTheBorder."
It's a nod to a previously introduced bill, CNN reports. The "EL CHAPO Act" – or "Ensuring Lawful Collection of Hidden Assets to Provide Order Act" – was first introduced in April 2017 and reintroduced in January of this year, the network reports.
"The bill would reserve any amounts forfeited to the U.S. Government as a result of the criminal prosecution of 'El Chapo' ... and other drug lords for border security assets and the completion of the wall along the U.S.-Mexico border," a January release from Cruz says.
The idea has particular relevance this week. Guzmán was convicted Tuesday and lawmakers are currently sparing on border security funding issues as another possible government shutdown looms.
Prosecutors have said Guzmán's Sinaloa Cartel generated billions of dollars by smuggling tons of cocaine, heroin, methamphetamine and other drugs.
But the EL CHAPO Act has faced skepticism before. Forbes reported in April 2017 that most of the assets sought by U.S. authorities are likely in Mexican territory.
"The lion's share of any of his assets seized, rightfully – by law and agreement – belong to Mexico. They are unlikely to find much – certainly not $14 billion. Mexico will never agree," the publication quoted Bruce M. Bagley, a University of Miami professor who has written on drug trafficking in the Americas.
The unpopularity of President Donald Trump's proposed border wall in Mexico could further complicate enforcement of the act, if it were to pass, Forbes reported.
Some Americans face a costly surprise this tax season: tiny refunds or bigger bills Supreme Court strikes blow against states that raise revenue by hefty fines, forfeitures How bad is Disney's 'Aladdin' blue Genie problem after the Will Smith reveal tanked? Police say Jussie Smollett is a suspect in setting up his own attack
Contributing: Kevin McCoy


Minted is a website that offers artists and designers across the country the opportunity to showcase their work in a way that connects with consumers. Being that it is getting closer to the holidays, I began my search for "the" perfect Christmas card to send out this year. After looking at the new designs that are being offered on the Minted site, I am having trouble deciding which one I will choose. There are so many amazing designs that this year is going to be a tough one, but I know the one I pick will be beautiful! Here are a few of the designs that are at the top of my list!
Aren't these cards gorgeous? I think they are. You can purchase the cards flat or folded, with embellishments, on different kinds of paper and with customized sayings. Minted also offers holiday cards for businesses as well. My favorites are the cards that capture a certain metropolitan city, like New York City. These cards just remind me of home!
Most designs on Minted, also offer the option of different shapes and font colors. Think about it! Your holiday cards this year won't look like anyone else's and just be thrown on a stack of mail. Your cards will become unique gifts that will be treasured by those who receive them.
But don't think for a minute that Minted only offers holiday-related items! Minted offers so much more! Whether it's cards for all different occasions, wedding invitations, baby announcements, party decorations, stationery, or gifts, Minted is your one stop shop for all of them! Here's a sample of their wedding items!
I cannot wait to order my Christmas cards this year! Don't delay! Before you know it, the season will be upon us, you will be rushing around and will end up with just an ordinary card. Why be just ordinary this year, when you can be extraordinary?
This post is sponsored by minted.com. I received a Minted credit for in exchange for my time and honest review. I only share products with you that I believe in 100%!


Australian Effects and Animation Festival is an international competition for visual effects as well as creation of animation. It will consider entries from all over the world and then screen them. These awards are the recognition of creative along with technical excellence in using visual effects and creative animation.
Technical skills in visual effects.
Visual effects along with artistry.
Level of quality in visual effects.
Story telling about the whole project.
Animation/VFX categories in their entries need to use significant amount of effects as well as animation.
All the entries must be designed using Digital Media Technology only.
Finalists will be chosen depending upon artistic and craft media by board of judges.
Winners in all categories will be determined on highest score which is aggregation of scores from all judges.
Incase if there is a tie among the winners in any categories then Festival Director will have decide and vote for the winner.
�� Entry form which is completed.
�� DVD's should be in any one of the digital media PAL, NTSC format and Region code 0 through 4.
�� Preferred format of DVD will be 1920 x 1080 pixels.
Winners and finalists of AEAF awards in all categories will be given Gold, Silver or Bronze medals respectively. Awards reel will be screened on July 30th 2014 in Sydney,�Australia at a gala event.


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The online admissions for the primary school year starting September 2019 has now closed.
For Blackpool Council Tax payers: If you have missed the deadline you should complete a paper application which is available from the Blackpool Council School Admissions Team, telephone 01253 476637. An application submitted after the closing date will be classed as late.
In Autumn we hold an open morning in school to give you the opportunity to come and have a look round. For more information contact the school.


2011 BIG Holiday Promotion Gift!
Wow! What an incredible holiday season this has already been at Passanante's Home Food Service! We want to thank all of you so very much for helping us make our 2011 BIG Holiday Promotion such an incredible success! We delivered dozens and dozens of samples to eagerly awaiting families and are so happy you all took the time to send us such wonderful referrals.
In case you were wondering, the holiday package your friends and family members received included a boneless chicken breast, 3 handmade italian sausage links, an 8 oz. marinated flatiron steak, a jumbo steak burger, and a package of organic mixed vegetables.
– No wonder they were all so excited, right?!
We are having so much fun getting to know our new and existing customers as we continue to grow Passanante's. As you know, our goal is to always provide you with delicious, healthy, and organic foods that you feel good about serving to your family. We are always available to answer any questions you may have about organic foods or hormone-free meats. Check our blog each week for useful information from what cut of meat comes from where, to delectable recipes, photos and more!
Thanks again to all of our loyal customers as well as the willing referrals we have been able to get to know over the last few weeks. We genuinely appreciate all of you!!
This entry was posted in Passanante's and tagged 2011 BIG Holiday Promotion, flatiron steak, Free, Home Food Delivery Service, italian sausage, jumbo steak burger, Organic Vegetables, Passanante's Home Food Service by Passanante's Home Food Service. Bookmark the permalink.


text:
\section{\label{sec:introduction} introduction}
Quantum entanglement is a crucial resource for quantum computation
\cite{Joz97,RB01} and other  information processing tasks,
such as quantum teleportation~\cite{BBCJPW93}, superdense coding~\cite{BW92} and quantum key distribution \cite{Eke91}.
In the past few decades, there have been tremendous efforts in understanding
various aspects of entanglement in both bipartite and multipartite settings.
One of the central issues in entanglement theory is the characterization and
quantification of multipartite entanglement \cite{PV07,HHHH09}. Among the many
approaches to the investigation of entanglement, several geometrically
motivated measures have been proposed and proved to be useful, such as
relative entropy of entanglement \cite{VPRK97,VP98}, \emph{geometric measure
of entanglement (GM)} \cite{Shi95,WG03,Bro01} and logarithmic global
robustness \cite{VT99,HN03}. Remarkably, these three measures turn out to be
related~\cite{HMM06,HMM08,WEGM04,Cav06,Wei08,ZCH10}.

Among the three measures, GM seems to be the easiest to handle and has thus
started to attract attention in recent years. Moreover, its applications have
gone beyond entanglement theory. GM is closely related to the construction of
optimal entanglement witnesses \cite{WG03}, experimental estimation of
entanglement~\cite{GRW07}, and discrimination of quantum states under local
operations and classical communications (LOCC) \cite{HMM06,HMM08,MMV07}.
Recently, GM has also been utilized to determine the universality of resource
states for one-way quantum computation \cite{NDMB07,MPMNDB10}, and to study
 generic multipartite pure states  as computational resources
\cite{GFE09,ZCH10}. In the context of condensed matter physics, GM has been
demonstrated to be useful for studying quantum many-body systems, such as
characterizing ground state properties and detecting phase transitions
\cite{WDMVG05,ODV08,Oru08,Oru08b,Wei10a,OW09,SAPFV10}.

Given the above applications of GM, it is thus interesting to
explore its connection with other important research fields, such as
quantum state estimation \cite{PR04,LR09}.  Quantum state estimation
is  a procedure of inferring the state of a quantum system by
general measurements---positive-operator-valued measures (POVMs). It
is a central issue of quantum mechanics and a cornerstone  of
various quantum-information processing tasks, such as quantum
computation, quantum communication and quantum cryptography.  In
this context, many measurement and reconstruction  methods have been
proposed to estimate the target state. Among the reconstruction
methods, an efficient one is the \emph{maximum likelihood method},
which has been widely used in experiments~\cite{Hra97,PR04,
RHJ01,RH07,JKMW01,dldg08,LR09,as10}.

Concerning measurement schemes, \emph{mutually unbiased bases
(MUBs)} \cite{Iva81,WF89,Woo04, DEBZ10, App08} and \emph{symmetric
informationally complete (SIC) POVMs} \cite{Zau99,Fuc02,RBSC04,
App05,ADF07, AFF09,SG10,Zhu10}, which stand for very efficient von
Neumann measurements and POVMs \cite{Sco06}, respectively, are two
focuses in the current research community.  Great efforts have been
directed to solving their existence problem and understanding their
structure
\cite{Iva81,WF89,Woo04,App08,Zau99,RBSC04,App05,AFF09,SG10,Zhu10};
since they are closely related to the physics in finite-dimensional
Hilbert spaces~\cite{DEBZ10,Fuc02,ADF07}.

In this article, we aim to establish a general connection between GM
and quantum state estimation theory, and MUBs, SIC~POVMs in
particular. The cross fertilizing of these geometric ideas may bring
insights to both research fields. We begin by studying the GM of
pure symmetric states related to rank-one POVMs, and establishing a
general connection with quantum state estimation theory, especially
the maximum likelihood principle \cite{Hra97,PR04,RHJ01,RH07,LR09}.
Based on this connection, we provide a method for computing the GM
of these states and demonstrate its additivity property under
certain conditions. In particular, we prove the additivity of the GM
of pure symmetric multiqubit states whose Majorana points  under
Majorana representation \cite{Maj32, BKMGLS09,MGBBB10, Mar10,AMM10}
are distributed within a half sphere, including all pure symmetric
three-qubit states. We then introduce a family of  symmetric states
that are generated from MUBs, and derive an analytical formula for
their GM. These states reduce to Dicke states in special cases,
which are useful for quantum communication and have been realized in
experiments \cite{KSTSW07,WKKMTW09,PCTPWKZ09}. Next, we compute the
GM of symmetric states generated from SIC~POVMs. This result is then
used to  characterize all inequivalent SIC~POVMs in
three-dimensional Hilbert space that are covariant with respect to
the Heisenberg--Weyl group \cite{Zau99, RBSC04, App05,Zhu10}.
Finally, we propose an experimental scheme for creating the
symmetric multiqubit states studied in this article.

The rest of the article is organized as follows. In
Sec.~\ref{sec:GenConnection}, we establish a general connection
between the GM of pure symmetric states and the maximum likelihood
principle in quantum state estimation theory. In
Sec.~\ref{sec:AddGM},  we prove the additivity of the GM of pure
symmetric multiqubit states whose Majorana points are distributed
within a half sphere, including all pure symmetric three-qubit
states. In Sec.~\ref{sec:MUBGM}, we introduce a family of symmetric
states generated from MUBs, and derive an analytical formula for
their GM. In Sec.~\ref{sec:SICGM}, we derive the GM of symmetric
states generated from SIC~POVMs and use it to characterize
inequivalent SIC~POVMs in three-dimensional Hilbert space. We
discuss experimental methods for realizing the symmetrized states of
this article in Sec.~\ref{sec:creation}. We conclude with a summary.

\section{\label{sec:GenConnection} GM of pure symmetric states: a connection with quantum state estimation}
In this section, we study the GM of pure symmetric states related to
rank-one POVMs, and  establish a general connection with quantum
state estimation theory, especially the maximum likelihood
principle. Based on this connection, the GM of many pure symmetric
states can be computed analytically, and its additivity property  be
demonstrated. In the next section, this connection will be used to
show the additivity of the GM of pure symmetric multiqubit states
whose Majorana points are distributed within a half sphere,
including all pure symmetric three-qubit states. Additional examples
related to MUBs and SIC~POVMs will be presented in the following
sections to illustrate this general idea.

\subsection{ Preliminary}

The geometric measure  of entanglement  measures  the maximum overlap between
a given state and the set of separable states, or equivalently, the set of
pure product states, and is defined as \cite{WG03,HMM06}
\begin{eqnarray}
  \Lambda^2(\rho)&:=&\max_{\sigma\in \mathrm{SEP}} \mathrm{tr}(\rho\sigma) =
\max_{|\Ph\rangle\in \mathrm{PRO}}    \label{eq:overlap}
  \langle\Ph|\rho|\Ph\rangle,\\
  G(\rho) &:=& -2~\log \Lambda(\rho).    \label{eq:GM}
\end{eqnarray}
Here ``SEP" denotes the set of separable states, and ``PRO" the set
of   pure product states that fully factorize; ``log" has base 2
throughout this article. Any pure product state maximizing
Eq.~(\ref{eq:overlap}) is a \emph{closest product state} of $\rho$.

For symmetric states, the computation of GM can be greatly
simplified due to a result in Refs.~\cite{HKW09,
ZCH10,HMMOV09,WS10}:
\begin{proposition}\label{pro:symmetric}
  The closest product state to any  pure or mixed symmetric
  $N$-partite state $\rho$ can be chosen to be symmetric; it is  necessarily symmetric if
  $N\geq3$:
   \bea
  \label{ea:puresymmetricGM}
  \L^2(\rho)
  &=& \max_{\ket{\ph_1}, \ldots, \ket{\ph_N}}
\bigg( \bigotimes^N_{j=1} \bra{\ph_j} \bigg)\rho \bigg(
\bigotimes^N_{j=1} \ket{\ph_j} \bigg)\nonumber\\
  &=& \max_{\ket{\ph}}
 \bra{\ph}^{\ox N}\rho  \ket{\ph}^{\ox N}.
\eea
\end{proposition}
Recently, the GM of pure symmetric  three-qubit states have been
derived based on this observation \cite{CXZ10}, and a class of
maximally entangled three-qubit states has also been
obtained~\cite{TWP09,CXZ10}. The problem is still open for more
general situations, although progress has been made with respect to
the connection between the singular values of a hypermatrix and the
GM~\cite{HS10}. Below, we shall provide many examples where
analytical solutions can be obtained.

Another key ingredient in our investigation is the maximum
likelihood (ML) principle of quantum state estimation
\cite{Hra97,PR04,RHJ01,RH07,LR09}. Consider state estimation using a
rank-one POVM composed of $M$ outcomes that are represented by
subnormalized pure projectors $\Pi_j=|\psi_j\rangle\langle\psi_j|$
such that $\sum_j\Pi_j=I$. If we are given $N$ copies of an unknown
input state and perform $N$ measurements independently, then the
outcome statistics  obey a multinomial distribution. Suppose outcome
$j$ occurs $n_j$ times for $j=1,2\ldots,M$ ($\sum_j n_j=N$); then
the frequency of obtaining outcome $j$ is $f_j=n_j/N$. In the
standard state reconstruction, the estimator is obtained by solving
the following set of equations:
\begin{eqnarray}\label{eq:standard}
\langle\psi_j|\rho|\psi_j\rangle=f_j \quad\forall j.
\end{eqnarray}
However, such a solution does not always exist.  The ML principle
consists in choosing a state $\r_{\mathrm{ML}}$ that maximizes the
likelihood functional $\mathcal{L}(\rho)$ as an estimator of the
true state \cite{PR04,LR09,Hra97,RHJ01,RH07},
\begin{eqnarray}\label{eq:likelihood}
\mathcal{L}(\rho)=\prod_{j=1}^M p_j^{n_j},
\end{eqnarray}
where $p_j=\langle\psi_j|\rho|\psi_j\rangle$ is the probability of
obtaining outcome $j$ given the input state $\rho$. If there exists
a state $\rho_{\mathrm{s}}$ that satisfies Eq.~(\ref{eq:standard}),
that is, the probabilities $p_j$ derived from this state coincide
with the frequencies $f_j$, then $\rho_{\mathrm{s}}$ is also the
maximum point of the likelihood functional. In general, there is an
efficient iterative algorithm for finding the ML estimator if the
POVM is informationally complete (IC)
\cite{PR04,LR09,Hra97,RHJ01,RH07}.  A POVM is IC if we can
reconstruct any input state according to the measurement statistics.
If the POVM is not IC,   the maximum of the likelihood functional
can still be computed efficiently, but the ML estimators are
generally not unique. In addition, the ML principle is also
applicable when the ($|\psi_j\rangle\langle\psi_j|$)'s form an
incomplete POVM, that is $\sum_j|\psi_j\rangle\langle\psi_j|=\Pi\leq
I$ \cite{PR04,LR09,Hra97,RHJ01,RH07} (be sure to distinguish
``complete" and ``informationally complete").

\subsection{ Connection}

We are now ready to show the connection between the GM of pure
symmetric states and quantum state estimation theory. Following the
above notation, assume $\sum_j|\psi_j\rangle\langle\psi_j|=\Pi\leq
I$, and the largest eigenvalue of $\Pi$ is 1. Define
$\Psi(\{n_j\})\rangle$ as the symmetrized state of the product state
$\bigotimes_{j=1}^M|\psi_j\rangle^{\otimes n_j}$,
\begin{eqnarray}
  \label{ea:symmetrizedstate}
  |\Psi(\{n_j\})\rangle=c P_\mathrm{sym}
  \bigotimes_{j=1}^M|\psi_j\rangle^{\otimes n_j},
  \end{eqnarray}
where  $P_\mathrm{sym}$ is the projector onto the symmetric
subspace, and $c$ is a normalization constant, which can be assumed
to be positive without loss of generality. The effect of the
projector $P_\mathrm{sym}$ is determined by its action on pure
product states. The action of  $P_\mathrm{sym}$ on the tensor
product of $N$ single-particle kets $|a_k\rangle$ is given by
\begin{eqnarray}
P_\mathrm{sym} \bigotimes_{k=1}^N
|a_k\rangle=\frac{1}{N!}\sum_{\sigma\in S_N} \bigotimes_{k=1}^N
|a_{\sigma(k)}\rangle,
\end{eqnarray}
where $S_N$ is the symmetry group of $N$ letters. Now suppose
$|a_k\rangle$ is the $k$-th member of the multiset consisting of
$n_1$ copies of $|\psi_1\rangle$, $n_2$ copies of $|\psi_2\rangle$
and so on. Define $A$ as the Gram matrix of the kets $|a_k\rangle$,
i.e.,
\begin{eqnarray}
A_{jk}=\langle a_j|a_k\rangle, \quad j,k=1,2,\ldots,N.
\end{eqnarray}
The dependence of $A$ on $|\psi_j\rangle$ and $n_j$ has been
suppressed to simplify the notation. The normalization constant $c$
is thus given by $c=\sqrt{N!/\mathrm{perm}(A)}$~\cite{Wei10b}, where
$\mathrm{perm}(A)$ denotes the permanent of the matrix $A$ (for
other connections to permanent, see Ref.~\cite{WS10}).

According to Proposition~\ref{pro:symmetric}, the GM of
$|\Psi(\{n_{j}\})\rangle$ reads
\begin{eqnarray}
 \Lambda^2(|\Psi(\{n_{j}\})\rangle)&=&\max_{|\varphi\rangle}
 |\langle\varphi|^{\otimes N}|\Psi(\{n_j\})\rangle|^2\nonumber\\
 &=&\frac{N!}{\mathrm{perm}(A)}\max_{|\varphi\rangle}\prod_{j=1}^M|\langle\varphi|\psi_j\rangle|^{2n_{j}}\nonumber\\
 &\le&\frac{N!}{\mathrm{perm}(A)}\max_{\r}\cL(\r)\nonumber\\
 &\leq&\frac{N!}{\mathrm{perm}(A)}\max_{ 0\leq p_{j}\leq 1,\; \sum_{j}p_{j}=1}\;\prod_{j=1}^M p_{j}^{n_{j}}\nonumber\\
 &=&\frac{N!}{\mathrm{perm}(A)}\prod_{j=1}^M f_{j}^{n_{j}}
 \label{ea:upperboundforlikelihood}.
\end{eqnarray}
Note that, apart from a constant factor, the functional to be
maximized in the second line of the above equation is exactly the
likelihood functional $\mathcal{L}(\rho)$ associated with the POVM
$\{\Pi_j\}$ if we take $f_j$ as ``frequency" [see
Eq.~(\ref{eq:likelihood})], except that $\rho$ is restricted to pure
states here; this is the reason for the first inequality in
Eq.~(\ref{ea:upperboundforlikelihood}). The ``frequencies" $f_j$ are
called \emph{compatible} if there exists a normalized pure state
$|\varphi_{\mathrm{s}}\rangle$ such that
$|\langle\varphi_{\mathrm{s}}|\psi_j\rangle|^2=f_{j}\;\forall j$;
that is, the $f_j$'s  can coincide with the probabilities derived
from a pure state. Note that such a pure state  is necessarily
contained in the eigenspace to the largest eigenvalue of $\Pi$. The
maximum in the last line of Eq.~(\ref{ea:upperboundforlikelihood})
can be obtained if and only if the $f_j$'s are compatible.  In that
case, the likelihood functional $\cL(\r)$ is maximized at the pure
state $|\varphi_{\mathrm{s}}\rangle$, and
$|\varphi_\mathrm{s}\rangle^{\otimes N}$ is a closest product state
to $|\Psi(\{n_{j}\})\rangle$.

Moreover, if the maximum of $\mathcal{L}(\rho)$ can be obtained at a
pure state, which is true if the $f_j$'s are compatible, then the GM
of $|\Psi(\{n_{j}\})\rangle$ and any pure symmetric $N$-partite
state $|\Phi\rangle$ is additive; that is,
$G(|\Psi(\{n_{j}\})\rangle \ox
\ket{\Ph})=G(|\Psi(\{n_{j}\})\rangle)+G(\ket{\Ph})$ (see
Ref.~\cite{ZCH10} for a detailed discussion on the additivity
property of GM). We prove this statement in Appendix A. In that
case, the GM of $|\Psi(\{n_{j}\})\rangle$ is equal to its asymptotic
regularized quantity and gives a lower bound for asymptotic relative
entropy of entanglement and asymptotic logarithmic global robustness
\cite{ZCH10}. For convenience,  we summarize the above observations
as the following theorem.

\begin{theorem}\label{thm:POVMstateGM}
Suppose  there are $M$ kets $|\psi_j\rangle$  (not necessarily
normalized) that satisfy
$\sum_{j=1}^M|\psi_j\rangle\langle\psi_j|=\Pi$, and the largest
eigenvalue of $\Pi$ is 1; $n_1, n_2,\ldots, n_M$ are non-negative
integers with sum $N=\sum_j n_j$ and $f_j=n_j/N$. Define $A$ as the
Gram matrix of the multiset consisting of $n_1$ copies of
$|\psi_1\rangle$, $n_2$ copies of $|\psi_2\rangle$ and so on; and
$|\Psi(\{n_j\})\rangle=
\sqrt{N!/\mathrm{perm}(A)}P_{\mathrm{sym}}\bigotimes_{j=1}^M|\psi_j\rangle^{\otimes
n_j}$. Then the  GM of  $|\Psi(\{n_j\})\rangle$ is lower bounded by
\begin{eqnarray}\label{eq:POVMstateGM}
G(|\Psi(\{n_{j}\})\rangle) &\geq
&-\log\biggl(\frac{N!}{\mathrm{perm}(A)}\prod_{j=1}^M
f_{j}^{n_{j}}\biggr);
\end{eqnarray}
the bound is saturated if and only if  the $f_j$'s are compatible,
that is, there exists a normalized pure state
$|\varphi_{\mathrm{s}}\rangle$ such that
\begin{eqnarray}\label{eq:compatible}
|\langle\varphi_{\mathrm{s}}|\psi_j\rangle|^2=f_{j}\quad\forall j.
\end{eqnarray}
Moreover, if the maximum of $\mathcal{L}(\rho)$ [see
Eq.~(\ref{eq:likelihood}) and (\ref{ea:upperboundforlikelihood})]
can be obtained at a pure state, which is true if the $f_j$'s are
compatible, then the GM of $|\Psi(\{n_{j}\})\rangle$ and  any pure
symmetric $N$-partite state $|\Phi\rangle$ is  additive; that is,
\begin{eqnarray}\label{eq:POVMstateGMadd}
G(|\Psi(\{n_{j}\})\rangle \ox
\ket{\Ph})=G(|\Psi(\{n_{j}\})\rangle)+G(\ket{\Ph}).
\end{eqnarray}
\end{theorem}

No matter whether Eq.~(\ref{eq:compatible}) can be satisfied or not,
the maximum of the likelihood functional $\cL(\r)$ in
Eq.~(\ref{ea:upperboundforlikelihood}) can be computed efficiently
with an iterative algorithm \cite{Hra97,PR04,RHJ01,RH07,LR09}. It
gives an upper bound for $\Lambda^2(|\Psi(\{n_j\})\rangle)$ and thus
a lower bound for $G(|\Psi(\{n_j\})\rangle)$. On the other hand,
when Eq.~(\ref{eq:compatible}) is not satisfied, it is still
possible that the maximum of the likelihood functional is obtainable
at a pure state. Then the $N$-fold tensor product of the pure state
is a closest product state to $|\Psi(\{n_j\})\rangle$, and the GM of
$|\Psi(\{n_{j}\})\rangle$ and any pure symmetric $N$-partite  state
is additive (see Sec.~\ref{sec:AddGM} for examples). In other words,
the compatibility condition on the $f_j$'s is sufficient but not
necessary for Eq.~(\ref{eq:POVMstateGMadd}) to hold. It is really
remarkable that we can derive the GM of these symmetric states and
its additivity property from the property of the likelihood
functional.

Besides, the  condition on the largest eigenvalue of $\Pi$ in
Theorem~\ref{thm:POVMstateGM} is not essential; it is adopted mainly
for a closer connection with state estimation theory. If instead the
largest eigenvalue of $\Pi$  is $g>0$, then
Theorem~\ref{thm:POVMstateGM} is still applicable as long as $f_j$
is replaced by $gf_j$ in Eqs.~(\ref{eq:POVMstateGM}) and
(\ref{eq:compatible}). One advantage of this alternative convention
is that $|\psi_j\rangle$ can now be chosen to be normalized. A
simple example in the case of qubits is as follows. Suppose there
are $M$ normalized kets $|\psi_j\rangle$  whose Bloch vectors are
distributed on a circular cone around the $z$ axis, such that
$\sum_{j=1}^M|\psi_j\rangle\langle\psi_j|=M(I+r \sigma_z)/2$ with
$0< r <1$; and
$|\Psi\rangle=\sqrt{M!/\mathrm{perm}(A)}P_{\mathrm{sym}}\bigotimes_{j=1}^M|\psi_j\rangle$.
Then $|\Psi\rangle$ has a unique closest product state
$|0\rangle^{\otimes M}$ and its  GM  is
\begin{eqnarray}
G(|\Psi\rangle)
&=&-\log\biggl[\frac{M!}{\mathrm{perm}(A)}\Bigl(\frac{1+r}{2}\Bigr)^M\biggr].
\end{eqnarray}

\section{\label{sec:AddGM} Additivity of GM of pure symmetric three-qubit states}
Every pure symmetric $N$-qubit state $|\Psi\rangle$ can be written
in the form in Eq.~(\ref{ea:symmetrizedstate}), that is,
\begin{eqnarray}
|\Psi\rangle\propto
P_{\mathrm{sym}}\bigotimes_{j=1}^M|\psi_j\rangle^{\otimes n_j};
\end{eqnarray}
and this representation is  unique up to permutations of the
$|\psi_j\rangle$'s and some phase factors. This  is well-known as
the Majorana representation \cite{Maj32}. Under this representation,
each pure symmetric $N$-qubit state corresponds to $N$ points on the
Bloch sphere.  Following Refs.~\cite{Mar10, AMM10}, the
$|\psi_j\rangle$'s are called Majorana points  of $|\Psi\rangle$.
Recently, Majorana representation has found many applications in the
study of multipartite entanglement, such as classification of
entanglement of pure symmetric multiqubit states under stochastic
LOCC (SLOCC) \cite{BKMGLS09}, investigation  of the GM of these
states and maximally entangled states among them
\cite{Mar10,MGBBB10,AMM10}.

As  an important application of the theory developed  in the
previous section and the Majorana representation, in this section,
we prove the additivity of the GM of pure symmetric multiqubit
states whose Majorana points are distributed within a half sphere,
which is true for all pure symmetric three-qubit states. As we shall
see shortly, the structure of the state space and its boundary plays
a crucial role in proving this additivity property.
\begin{theorem}\label{thm:SymMulQubit}
Suppose $|\Psi\rangle$ is a pure symmetric $N$-qubit state whose
Majorana points are distributed within a half sphere under the
Majorana representation; then the GM of $|\Psi\rangle$ and any pure
symmetric $N$-partite state is additive.
\end{theorem}
According to Theorem~\ref{thm:POVMstateGM}, to prove this theorem,
it suffices to show that the maximum of the following functional can
be obtained at a pure state,
\begin{eqnarray}
\mathcal{L}(\rho)=\prod_{j=1}^M
(\langle\psi_j|\rho|\psi_j\rangle)^{n_j}.
\end{eqnarray}
Without loss of generality, we can assume that the Majorana points
of $|\Psi\rangle$ lie within a half sphere with $z\geq0$, and the
Bloch vector of $\rho$ is $\bm{r}=(x,y,z)$ with $x^2+y^2+z^2\leq 1$.
Then it is straightforward to verify that $\mathcal{L}(\rho)$ is
nondecreasing with $z$, and thus its maximum can be obtained at the
boundary of the Bloch sphere, that is, at a pure state. This
completes the proof of Theorem~\ref{thm:SymMulQubit}.

For any pure symmetric three-qubit state, the three Majorana points
 lie within some half sphere;  hence, Theorem~\ref{thm:SymMulQubit} is applicable.
 The same is true for any
pure symmetric multiqubit state that has at most three distinct
Majorana points.
\begin{corollary}\label{cor:sym3Qubit}
The GM of any pure symmetric $N$-qubit  state that has at most three
distinct Majorana points and  any pure symmetric $N$-partite state
is additive. In particular, the GM of any pure symmetric three-qubit
state and any pure symmetric tripartite state is additive .
\end{corollary}

\begin{corollary}\label{cor:sym3points}
Suppose $|\Psi\rangle$ is a pure symmetric $N$-partite state which
can be written in the form
\begin{eqnarray}
|\Psi\rangle\propto
P_{\mathrm{sym}}\bigotimes_{j=1}^3|\psi_j\rangle^{\otimes n_j},
\end{eqnarray}
where $n_1+n_2+n_3=N$; then the GM of $|\Psi\rangle$ and any pure
symmetric $N$-partite  state is additive.
\end{corollary}
Without loss of generality, we can assume the three states
$|\psi_j\rangle$ belong to a three-dimensional Hilbert space. As in
the case of qubit, they  can be seen as three extremal points of the
eight-dimensional state space whose origin is the completely mixed
state; however, the boundary of the state space is no longer a
sphere, and the states on the boundary are not necessarily pure. In
addition, we can find  a suitable hyperplane passing through the
origin such that the three points are on the same side of the
hyperplane (or on the hyperplane). According to a similar reasoning
that leads to Theorem~\ref{thm:SymMulQubit} and
Corollary~\ref{cor:sym3Qubit}, the maximum of the following
functional
\begin{eqnarray}
\mathcal{L}(\rho)=\prod_{j=1}^3
(\langle\psi_j|\rho|\psi_j\rangle)^{n_j}
\end{eqnarray}
can be obtained at a state $\rho_{\mathrm{ML}}$ on the boundary of
the state space, whose rank is at most two. If $\rho_{\mathrm{ML}}$
is pure, then we are done. Otherwise, when $\rho$ is restricted to
the support of $\rho_{\mathrm{ML}}$, we have
\begin{eqnarray}
\mathcal{L}(\rho)=\prod_{j=1}^3
(\langle\psi_j^\prime|\rho|\psi_j^\prime\rangle)^{n_j}.
\end{eqnarray}
where $|\psi_j^\prime\rangle$ is the projection of $|\psi_j\rangle$
onto the support of $\rho_{\mathrm{ML}}$. Now applying the same
reasoning that leads to Corollary~\ref{cor:sym3Qubit} shows that the
maximum of $\mathcal{L}(\rho)$ can be obtained at a pure state.
Therefore, the Corollary follows from Theorem~\ref{thm:POVMstateGM}.

Theorem~\ref{thm:SymMulQubit} and Corollaries~\ref{cor:sym3Qubit}
and \ref{cor:sym3points} provide a method for computing the
asymptotic GM, which in turn provides a lower bound for the
asymptotic relative entropy of entanglement and the asymptotic
logarithmic global robustness \cite{ZCH10}. They are also useful in
the  study of multipartite pure states  as computational resources,
since  GM and its additivity property  are closely related to
whether these states are universal for quantum computation
\cite{NDMB07, MPMNDB10,GFE09,ZCH10}. Corollary~\ref{cor:sym3Qubit}
also implies the multiplicativity of the output purity of the
quantum channels associated with pure symmetric three-qubit states
according to the Werner-Holevo recipe \cite{WH02,ZCH10}.

\section{\label{sec:MUBGM} symmetric states generated from mutually unbiased bases}

To illustrate the general idea presented in
Sec.~\ref{sec:GenConnection}, in this section we consider the
situation where the POVM can be decomposed into a set of von Neumann
measurements, in particular the scenario where the  bases of the von
Neumann measurements are mutually unbiased \cite{DEBZ10}. We first
consider pure symmetric states generated from two bases of the qubit
Hilbert space, which reduce to Dicke states as special cases. We
then generalize the idea to higher-dimensional Hilbert spaces and
point out the role played by mutually unbiasedness.

\subsection{\label{sec:gDickeGM} Generalization of Dicke states}

Given $\bm{r}_0=(0,0,1)$ and  $\bm{r}_1=(\sin\theta,0,\cos\theta)$,
let $|0\rangle$, $|1\rangle$ denote the eigenbasis of
$\bm{r}_0\cdot\bm{\sigma}=\sigma_z$, and
$|\theta_+\rangle=\cos(\theta/2)|0\rangle+\sin(\theta/2)|1\rangle$,
$|\theta_-\rangle=\sin(\theta/2)|0\rangle-\cos(\theta/2)|1\rangle$
the eigenbasis of $\bm{r}_1\cdot\bm{\sigma}$.  Given four
nonnegative integers $\{n_{jk}\}=\{n_{00},n_{01};n_{10},n_{11}\}$,
let $N=n_{00}+n_{01}+n_{10}+n_{11}$,  $N_j=n_{j0}+n_{j1}$,
$f_{jk}=n_{jk}/N_j$ (assuming $N_j\neq 0$). Define $A$ as the Gram
matrix of the multiset consisting of  $n_{00}, n_{01}, n_{10},
n_{11}$ copies of $|0\rangle, |1\rangle, |\theta_+\rangle,
|\theta_-\rangle$, respectively; and
\begin{eqnarray}\label{eq:gDicke}
\bigl|\Psi\bigl(\theta,\{n_{jk}\}\bigr)\bigr\rangle:&=&\sqrt{\frac{N!}{\mathrm{perm}(A)}}P_\mathrm{sym}
\Bigl(|0\rangle^{\otimes n_{00}}\otimes|1\rangle^{\otimes
n_{01}}\nonumber\\
&&\otimes|\theta_+\rangle^{\otimes
n_{10}}\otimes|\theta_-\rangle^{\otimes n_{11}}\Bigr).
\end{eqnarray}
Here  the permanent $\mathrm{perm}(A)$ can be computed efficiently,
see Appendix B. Note that $|\Psi(\theta,\{n_{jk} \})\rangle$ can be
seen as a generalization of the Dicke state; it reduces to the Dicke
state when $\theta=0,\pi$ or $N_0=0$ or $N_1=0$. The GM of the Dicke
state has been derived in Ref.~\cite{WG03}, and its additivity
property has been demonstrated in Ref.~\cite{ZCH10}. When
$N_0,N_1\neq 0$, we have the following theorem.
 \begin{theorem}
 \label{thm:DickeGM}The GM of $|\Psi(\theta,\{n_{jk}\})\rangle$ is
lower bounded by
\begin{eqnarray}\label{eq:gDickeGM}
&&G(|\Psi(\theta,\{n_{jk}\})\rangle)
\geq-\mathrm{log}\Bigl(\frac{N!}{\mathrm{perm}(A)}f_{00}^{n_{00}}f_{01}^{n_{01}}f_{10}^{n_{10}}f_{11}^{n_{11}}\Bigr);\nonumber\\
\end{eqnarray}
the bound is saturated  if and only if there exists a qubit state
$|\varphi\rangle$ such that
\begin{eqnarray}\label{eq:CPsConstraint2}
|\langle\varphi|0\rangle|^2=f_{00},
\quad|\langle\varphi|1\rangle|^2=f_{01},\nonumber\\
|\langle\varphi|\theta_+\rangle|^2=f_{10},\quad
|\langle\varphi|\theta_-\rangle|^2=f_{11}.
\end{eqnarray}
When $\theta\neq0,\pi$, this condition  is equivalent to the
following one:
\begin{eqnarray}\label{eq:constraint}
|h_0\bm{s}_0+h_1\bm{s}_1|\leq 1,
\end{eqnarray}
where $h_j=f_{j0}-f_{j1}$, and $\bm{s}_0=(-\cot\theta,0,1)$,
$\bm{s}_1=(\csc\theta,0,0)$ is the dual basis of $\bm{r}_0,\bm{r}_1$
in the $x$-$z$ plane.

The GM of $|\Psi(\theta,\{n_{jk}\})\rangle$  and any pure symmetric
$N$-partite state is  additive, irrespective whether the condition
Eq.~(\ref{eq:CPsConstraint2}) is satisfied or not.
 \end{theorem}

We can briefly show Theorem~\ref{thm:DickeGM} as follows. First
Eqs.~(\ref{eq:gDickeGM}) and (\ref{eq:CPsConstraint2}) can be
derived according to a similar reasoning that leads to
Theorem~\ref{thm:POVMstateGM}.  Equation~(\ref{eq:constraint}) has a
nice geometric interpretation under the Bloch sphere representation.
Suppose the Bloch vector of $|\ph\rangle$ is $\bm{s}$. The first two
equations in Eq.~(\ref{eq:CPsConstraint2}) restrict $\bm{s}$ to a
plane satisfying $\bm{s}\cdot\bm{r}_0=h_0$, which is perpendicular
to $\bm{r}_0$, and the last two restrict $\bm{s}$ to a plane
satisfying $\bm{s}\cdot\bm{r}_1=h_1$, which is perpendicular to
$\bm{r}_1$. There exists a pure state satisfying
Eq.~(\ref{eq:CPsConstraint2}) if and only if the intersection of the
two planes passes through the Bloch sphere; that is, the
intersection of the two planes and the $x$-$z$ plane lies within the
unit circle centered at the origin on the $x$-$z$ plane. This is
exactly what  Eq.~(\ref{eq:constraint}) means. In the special case
$\theta=\frac{\pi}{2}, \frac{3\pi}{2}$, where the two bases
$|0\rangle,|1\rangle$ and $|\theta_\pm\rangle=|\pm\rangle$ are
mutually unbiased, Eq.~(\ref{eq:constraint}) simplifies to
$h_0^2+h_1^2\leq1$.  The additivity property of the GM of
$|\Psi(\theta,\{n_{jk}\})\rangle$ follows from
Theorem~\ref{thm:SymMulQubit}. This completes the proof of
Theorem~\ref{thm:DickeGM}.

When $n_{01}=n_{00}$ and $n_{11}=n_{10}$,
Eq.~(\ref{eq:CPsConstraint2}) is trivial to satisfy; and  the
$N$-fold tensor product of the two eigenstates of $\sigma_y$,
respectively, are  exactly two closest product states to
$|\Psi\bigl(\theta,\{n_{jk}\}\bigr)\bigr\rangle$. Besides,
Eq.~(\ref{eq:gDickeGM}) reduces to
\begin{eqnarray}
G(|\Psi(\theta,\{n_{jk}\})\rangle)&=&-\log\Bigl(\frac{N!}{2^N\mathrm{perm}(A)}\Bigr).
\end{eqnarray}
Remarkably, the GM is completely determined by the number of qubits
$N$ and $\mathrm{perm}(A)$. An interesting example is the state
$\bigl|\Psi\bigl(\theta,\{1,1;1,1\}\bigr)\bigr\rangle$ obtained when
$n_{00}=n_{01}=n_{10}=n_{11}=1$. It is a balanced four-qubit Dicke
state when $\theta=0,\pi$, and is equivalent to four-qubit GHZ state
under a suitable local unitary transformation when
$\theta=\frac{\pi}{2}, \frac{3\pi}{2}$, according to
Refs.~\cite{BKMGLS09,MGBBB10, Mar10,AMM10,Bas09}. Calculation shows
that $\mathrm{perm}(A)=[7+\cos(2\theta)]/2$, hence,
\begin{eqnarray}
G(|\Psi(\theta,\{1,1;1,1\})\rangle)&=&\log\frac{7+\cos(2\theta)}{3}.
\end{eqnarray}
The maximum is obtained at $\theta=0,\pi$, where the two bases
coincide, and the minimum at $\theta=\frac{\pi}{2},\frac{3\pi}{2}$,
where the two bases are mutually unbiased.

\subsection{A connection with MUBs}

To generalize the above idea to higher dimension, let
$|e^j_k\rangle$ for $j=0,1,\ldots,b-1, k=0,1,\ldots,d-1$ be $bd$
single-particle states such that the $d$ states for given $j$ form
an orthonormal basis \cite{DEBZ10}. Let $n_{jk}$ be a $b\times d$
matrix composed of nonnegative integers, $N=\sum_{j,k} n_{jk}$,
$N_j=\sum_{k}n_{jk}$, and $f_{jk}=n_{jk}/N_j$ (assuming $N_j\neq
0$). Define
\begin{eqnarray*}
\bigl|\Psi\bigl(\{n_{jk}\}\bigr)\bigr\rangle:=\sqrt{\frac{N!}{\mathrm{perm}(A)}}P_\mathrm{sym}
\bigotimes_{j,k} |e^j_k\rangle^{\otimes n_{jk}}.
\end{eqnarray*}
This state  reduces to a generalized Dicke state when there is only
one basis, that is $b=1$. According to the same reasoning as before,
the GM of $\bigl|\Psi\bigl(\{n_{jk}\}\bigr)\bigr\rangle$ is lower
bounded by
\begin{eqnarray*}
G(|\Psi(\{n_{jk}\})\rangle)
&\geq&-\log\biggl(\frac{N!}{\mathrm{perm}(A)}\prod_{j,k}f_{jk}^{n_{jk}}\biggr);
\end{eqnarray*}
the bound is saturated if and only if  there exists a pure state
$|\varphi\rangle$ such that
\begin{eqnarray}\label{eq:mubcond1}
|\langle\varphi|e^j_k\rangle|^2=f_{jk}\quad \forall j,k.
\end{eqnarray}
In that case, the GM of
$\bigl|\Psi\bigl(\{n_{jk}\}\bigr)\bigr\rangle$ and  any pure
symmetric $N$-partite state is additive .

In general, it is not easy to determine whether such a state exists
or not; the structure of the state space plays a crucial role here.
Here we are content to point out a connection with MUBs. Suppose the
$b$ bases $|e^j_k\rangle$ are mutually unbiased, and $n_{jk}=n_{j},
f_{jk}=1/d\; \forall j,k$. Then satisfying the set of constraints in
Eq.~(\ref{eq:mubcond1}) amounts to the existence of a pure state
that is mutually unbiased with all states $|e^j_k\rangle$.

\bt
   \label{thm:mubexistence}
   Suppose $\ket{e^j_k}$ for $j=0,1,\ldots,b-1 (b \le d+1)$
   and $k=0,1,\ldots,d-1$ are $b$
   MUBs, i.e.,
   $\abs{\braket{e^j_k}{e^l_m}}^2 = \frac1d(1-\d_{j,l})+\d_{j,l}
   \d_{k,m}$.  Then the $N$-partite $(N=d \sum^b_{j=1} n_j,~n_j\ge1)$ symmetric state
   $\ket{\Psi(d,\{n_j\})}
   := \sqrt{N!/\mathrm{perm}(A)}P_\mathrm{sym}
   \bigl(\bigotimes^b_{j=1} \bigotimes^d_{k=1} \ket{e^j_k}^{\otimes n_{j}}\bigr)$
   has GM lower bounded by
\begin{eqnarray}\label{eq:MUBGm}
   G(\ket{\Psi(d,\{n_j\})})\geq
   -\log\frac{N!}{d^N\mathrm{perm}(A)};
\end{eqnarray}
the bound is saturated if and only if  there exists a pure state
that is mutually unbiased to all states $\ket{e^j_k}$. If such a
state exists, then  the GM of $\ket{\Psi(d,\{n_j\})}$ and any pure
symmetric $N$-partite state $\ket{\Phi}$ is additive; that is,
\begin{eqnarray}
\label{eq:MUBGmAdd}
  G(\ket{\Psi(d,\{n_j\})} \ox \ket{\Phi})
   &=&G(\ket{\Psi(d,\{n_j\})})
  +
   G(\ket{\Phi}).  \nonumber\\
\end{eqnarray}
\et

For example, the inequality in Eqs.~(\ref{eq:MUBGm}) is saturated
and (\ref{eq:MUBGmAdd}) is  applicable when $b=2$ and the
$|e^{0,1}_k\rangle$'s are the eigenbases of $Z$ and $X$,
respectively, where $Z$ and  $X$ are phase operator and cyclic shift
operator, respectively. They  are defined according to their action
on the computational basis $|e_k\rangle=|e^0_k\rangle$,
\begin{eqnarray}\label{eq:XZ}
Z|e_k\rangle&=&\omega^k|e_k\rangle, \quad \omega=\mathrm{e}^{2\pi\mathrm{i}/d}\nonumber\\
X|e_k\rangle&=&|e_{(k+1)\, {\rm mod}\, d}\rangle,
\end{eqnarray}
where we use ``mod'' to denote the modulo operation. Since the
eigenbases of $X, Z$ and $XZ$ are mutually unbiased \cite{DEBZ10},
the $N$-fold tensor product of any eigenstate of $XZ$ is a closest
product state to $\ket{\Psi(d,\{n_j\})}$.

If $d$ is a prime power, there exists a complete set of $d+1$ MUBs
\cite{WF89,DEBZ10}. The inequality in Eqs.~(\ref{eq:MUBGm}) is
saturated and (\ref{eq:MUBGmAdd}) is  applicable if
$|e^j_k\rangle$'s are chosen from $b$ bases with $1\leq b\leq d$
from the complete set. However,  this is not the case if $b=d+1$,
since there is no pure state that is mutually unbiased to all states
in a complete set of MUBs \cite{WF89}.

\section{\label{sec:SICGM} symmetric states generated from  SIC~POVMs}

In a $d$-dimensional Hilbert space, a SIC~POVM \cite{Zau99, RBSC04,
App05,SG10} consists of $d^2$ outcomes that are  subnormalized
projectors onto pure states
$\Pi_j=\frac{1}{d}|\psi_j\rangle\langle\psi_j|$ for
$j=1,\ldots,d^2$, such that
\begin{eqnarray}\label{eq:SIC}
|\langle\psi_j|\psi_k\rangle|^2=\frac{1+d\delta_{jk}}{d+1}.
\end{eqnarray}
The condition  $\sum_j\Pi_j=I$ is already implied by the above
equation and need not  be imposed separately. Most known SIC~POVMs
are generated from a \emph{fiducial state} under the action of the
Heisenberg--Weyl (HW) group, which is generated by the two operators
$X,Z$ defined in Eq.~(\ref{eq:XZ}). A fiducial state $|\psi\rangle$
of the HW group obeys the following equations,
\begin{eqnarray}
|\langle\psi|X^{k_1}Z^{k_2}|\psi\rangle|=\frac{1}{\sqrt{d+1}}\quad
\end{eqnarray}
for all $(k_1,k_2)\neq (0,0)\mod d$. If $|\psi\rangle$ is a fiducial
state, then the $d^2$ states $X^{k_1}Z^{k_2}|\psi\rangle$ for
$k_1,k_2=0,1,\ldots,d-1$ form a SIC~POVM that is covariant with
respect to the HW group. The Clifford group is the normalizer of the
HW group that consists of unitary operators. Likewise, the extended
Clifford group is the larger group that contains also anti-unitary
operators. For any operator $U$ in the extended Clifford group,
$U|\psi\rangle$ is a fiducial state whenever $|\psi\rangle$ is.
Fiducial states and SIC~POVMs form disjoint orbits under the action
of the extended Clifford group. SIC~POVMs on the same orbit of the
extended Clifford group are \emph{equivalent} in the sense that they
can be transformed into each other with unitary or antiunitary
operations \cite{App05,Zhu10}.

In this section we study the GM of symmetric states generated from
SIC~POVMs,
 \be
   \ket{\Ps_d^{\mathrm{SIC}}} :=\sqrt{\frac{d^2!}{\mathrm{perm}(A)}}
   P_\mathrm{sym}(\ket{\ps_1}\ox\cdots\ox\ket{\ps_{d^2}}).
 \ee
Since the completely mixed  state is the only state that satisfies
$\langle\psi_j|\rho|\psi_j\rangle =1/d$ for $j=1, 2, \ldots, d^2$,
the GM of $\ket{\Ps_d^{\mathrm{SIC}}}$ cannot be computed according
to Theorem~\ref{thm:POVMstateGM}. In Appendix C, we derive the GM of
$\ket{\Ps_d^{\mathrm{SIC}}}$ by virtue of the special properties of
SIC~POVMs; the result is  summarized in the following theorem. \bt
   \label{thm:SIC-GM}
   Suppose $d^2$ normalized states $|\psi_j\rangle$ for $j=1,2,\ldots, d^2$ in a
   $d$-dimensional Hilbert space  satisfy $\sum^{d^2}_{j=1}\proj{\ps_j} = d I$, and $A$ is the Gram matrix of the $|\psi_j\rangle$'s; define
   $\ket{\Ps} = \sqrt{d^2!/\mathrm{perm}(A)}P_\mathrm{sym}(\ket{\ps_1}\ox\cdots\ox\ket{\ps_{d^2}})$. Then
   the following four statements are equivalent.
\begin{enumerate}
\item
   \label{item:sic}
   The $d^2$ states  $|\psi_j\rangle$ form a SIC~POVM, that is, they satisfy Eq.~(\ref{eq:SIC});

\item
   \label{item:closeproductstate}
   Up to global phases, the $d^2$ states $\ket{\ps_j}^{\ox d^2}$ for $j=1, \cdots,
   d^2$ are the only closest product states to
   $\ket{\Ps}$; and
\be
   \L^2(\ket{\Ps})= {d^2! \over {(d+1)^{d^2-1}}\mathrm{perm}(A)};
\ee

\item
   \label{item:innerproduct}
  The  $d^2$  states
   $\ket{\ps_j}^{\ox d^2}$ for $j=1,\cdots,d^2$ satisfy
\be
  \abs{ \bra{\ps_j}^{\ox d^2} \ket{\Ps} }^2
    ={d^2! \over {(d+1)^{d^2-1}}\mathrm{perm}(A)};
\ee
\item
   \label{item:closestSICPOVM}
  There exists a SIC~POVM consisting of  $d^2$ states $|\varphi_j\rangle$ for $j=1,2,\ldots,d^2$ such that
  $|\varphi_j\rangle^{\otimes d^2}$ satisfies
\be
   \abs{ \bra{\ph_j}^{\ox d^2} \ket{\Ps} }^2= {d^2! \over {(d+1)^{d^2-1}}\mathrm{perm}(A)}.
\ee
\end{enumerate}
\et

In dimension two, there is only one orbit of fiducial states, one of
them is given by \cite{Zau99,RBSC04,App05}
\begin{eqnarray}
\bigl|\psi^\mathrm{f}_2\bigr\rangle=\sqrt{(3+\sqrt{3})/6}\,|e_0\rangle+e^{\mathrm{i}\pi/4}\sqrt{(3-\sqrt{3})/6}\,|e_1\rangle.
\end{eqnarray}
Moreover, all SIC~POVMs are equivalent to the one thus generated.
The four states of each SIC~POVM form a regular tetrahedron when
represented on the Bloch sphere. Calculation shows that
$\mathrm{perm}(A)=\frac{8}{3}$, which, together with
Theorem~\ref{thm:SIC-GM}, implies that
\begin{eqnarray}
\L^2\bigl(\bigl|\Ps_2^{\mathrm{SIC}}\bigr\rangle\bigr)=\frac{1}{3},
\quad G\bigl(\bigl|\Ps_2^{\mathrm{SIC}}\bigr\rangle\bigr)=\log 3.
\end{eqnarray}
This result coincides with that obtained  by Martin et al.
\cite{MGBBB10} and Aulbach et al. \cite{AMM10}. They also showed by
numerics that the state generated from the SIC~POVM is the maximally
entangled pure symmetric four-qubit  state with respect to GM.

On the other hand, there is a one-parameter family of  fiducial
states in dimension three \cite{Zau99, RBSC04, App05,Zhu10},
\begin{eqnarray}
\label{eq:fiducial3}
 \bigl|\psi_3^\mathrm{f}(t)\bigr\rangle=\frac{1}{\sqrt{2}}\bigl(|e_1\rangle
   - \mathrm{e}^{\mathrm{i} t}|e_2\rangle\bigr);
\end{eqnarray}
and there is a one-to-one correspondence between orbits of the
extended Clifford group and the parameter $t$ for
$t\in[0,\frac{\pi}{3}]$. There are three kinds of orbits, two
exceptional orbits corresponding to the endpoints $t=0$ and
$t=\frac{\pi}{3}$, respectively, and infinitely many generic orbits
corresponding to $0<t<\frac{\pi}{3}$ . In addition, there is a
one-to-one correspondence between inequivalent SIC~POVMs and the
parameter $t$ for $0\leq t\leq \frac{\pi}{9}$; the SIC~POVMs on the
three orbits $t, \frac{2\pi}{9}-t, \frac{2\pi}{9}+t$, respectively,
are equivalent under unitary transformations. All these inequivalent
SIC~POVMs can be classified  in terms of  geometric phases
associated with fiducial states  \cite{Zhu10}. Here we provide an
alternative characterization in terms of GM.

\begin{figure}
  \includegraphics[width=7cm]{SIC3dGM}\\
  \caption{\label{fig:SIC3dGM} The GM of symmetric states $\bigl|\Ps_3^{\mathrm{SIC}}(t)\bigr\rangle$
   for $0\leq t\leq \frac{\pi}{3}$ generated from
  HW covariant SIC~POVMs in three-dimensional Hilbert space.
     Each SIC~POVM is uniquely specified by $G(t)$ up to the equivalence class; recall that
     there is a one-to-one correspondence between inequivalent SIC~POVMs and the parameter $t$ for $0\leq t\leq \frac{\pi}{9}$ \cite{Zhu10}. }
\end{figure}

Let $\bigl|\Ps_3^{\mathrm{SIC}}(t)\bigr\rangle$ denote the symmetric
state related to the SIC~POVM generated from the fiducial state
$\bigl|\psi_3^\mathrm{f}(t)\bigr\rangle$ under  the HW group, $A(t)$
the corresponding Gram matrix. Surprisingly, the permanent of $A(t)$
can be given by a simple formula:
$\mathrm{perm}(A(t))=\frac{27}{32}\bigl[61-\cos(9t)\bigr]$.
According to Theorem~\ref{thm:SIC-GM} and Eq.~(\ref{eq:GM}),
\begin{eqnarray}
G\bigl(\bigl|\Ps_3^{\mathrm{SIC}}(t)\bigr\rangle\bigr)&=&\log\frac{16[61-\cos(9t)]}{105}.
\end{eqnarray}
Figure~\ref{fig:SIC3dGM} shows  $G(t):=
G\bigl(\bigl|\Ps_3^{\mathrm{SIC}}(t)\bigr\rangle\bigr)$ for $0\leq
t\leq \frac{\pi}{3}$. Up to the equivalence class, each SIC~POVM is
uniquely specified by
$G\bigl(\bigl|\Ps_3^{\mathrm{SIC}}(t)\bigr\rangle\bigr)$. There is
only a small difference among the GM of symmetric states generated
from different SIC~POVMs. The minimum and the maximum of the GM are
obtained at the exceptional orbits $t=0$ and $t=\frac{\pi}{3}$,
respectively. Note that the SIC~POVMs on the orbits
$t=\frac{2\pi}{9}$ and $t=\frac{\pi}{9}$ are equivalent to the ones
on the orbits $t=0$ and $t=\frac{\pi}{3}$, respectively
\cite{Zhu10}.

\section{\label{sec:creation} Creating symmetric multi-qubit states}
In this section, we describe how to create symmetric multi-qubit
states mentioned in this article, following the approach by Bastin
et al.~\cite{Bas09}. For non-qubit systems, a generalization could
be made, but we do not know whether suitable physical systems exist.

Imagine we have an atom or trapped ion initially in the excited
state $|e\rangle$, and it has two stable ground states, labeled by
$|0\rangle$ and $|1\rangle$, which are  connected to the excited
state via emitting right-circularly ($R$) and left-circularly ($L$)
polarized light, respectively. If the detection of the emitted
photon is $R$ ($L$) then we know the atom  is now in the state
$|0\rangle$ ($|1\rangle$). The photon polarization carries the
which-way information. If we place a polarizer in $(|R\rangle
+|L\rangle)/\sqrt{2}$ to erase the which-way information and if the
detector behind the polarizer clicks, the atom is then projected to
an equal superposition of $|0\rangle$ and $|1\rangle$. In general,
if the polarizer is in $\alpha|R\rangle + \beta|L\rangle$, after
detecting the photon, the atom is in the state $\alpha|0\rangle +
\beta|1\rangle$.

Now suppose we have $N$ excited atoms and $N$ photon detectors. A
simultaneous detection of $N$ photons will project the $N$-atom
state, depending on the polarizers' settings. Suppose the $k$-th
polarizer is in $\alpha_k|R\rangle +\beta_k|L\rangle$. If the $k$-th
polarizer and detector sit close to the $k$-th atom and detect the
emitted photon by the nearby atom, and if  each detector-atom system
is far away from each other, then after detection of a photon at
every detector, the $N$-atom state is $\otimes_k (\alpha_k
|0\rangle_k +\beta_k |1\rangle_k)$. Bastin et al.~\cite{Bas09}
proposed the setup such that the $N$ detectors are placed in the
far-field, roughly at an equal distance to all atoms. Because of
multipath quantum interference (which-path information is erased),
after the detection, the atoms should be in a state that is
invariant under permutations, namely,
\begin{equation}
|\Psi\rangle \propto P_{\rm sym}\bigotimes_k (\alpha_k |0\rangle_k
+\beta_k |1\rangle_k).
\end{equation}
Thus, using the scheme by Bastin et al.~\cite{Bas09}, the symmetric
multi-qubit states discussed in this article can be realized in
principle, not just mathematical objects.

For non-qubit systems, one needs to first identify a physical system
that contains an excited state coupled to $d$ longlived sublevels
and that the decay into each level is associated with a
distinguishable which-way information. Deleting the which-way
information from the decay and the which-atom information will
enable the creation of symmetric multiqudit states considered in
this article. But we have not yet identified such a system.

Next, we discuss a possible measurement of the permanent of
$A_{ij}=\langle \psi_i|\psi_j\rangle$. Suppose one can prepare the
following symmetric $N$-particle  state
\begin{equation}
|\Psi\rangle= c P_{\rm sym} \mathop{\bigotimes}_{j=1}^N |\psi_j\rangle,
\end{equation}
with the normalization $c=\sqrt{N!/{\rm perm}(A)}$. If one chooses a
von Neumann measurement at each particle in the basis consisting of
$|\ph\rangle$ and the remaining orthonormal states, the simultaneous
detection of outcome $|\ph\rangle$ at all sites has the probability
\begin{equation}
\lambda(\ph)^2=\frac{N!}{{\rm perm}(A)} \prod_{j=1}^N
|\langle\ph|\psi_j\rangle|^2.
\end{equation}
Since  all $|\langle\ph|\psi_j\rangle|$'s are known and $\lambda(\ph)^2$ is
obtained from the measurement statistics, ${{\rm perm}(A)}$ can be inferred.
Ideally, one prefers $\lambda(\ph)^2$ to be as large as possible, as it
represents the probability of obtaining the desired outcome. One can also vary
$|\ph\rangle$; the probability $\lambda(\ph)^2$ is maximized when
$|\ph\rangle^{\otimes N}$ is the closest product state that results in the
geometric measure. According to a recent work by Martin et al.~\cite{MGBBB10},
the maximal overlap $\Lambda^2$ for highly entangled symmetric states decays
only inversely proportional to the number of qubits, so the proposed scheme
seems to be feasible even for medium size of systems that have been achieved
in various physical implementations.
\section{\label{sec:summary} summary}

We have studied the GM of pure symmetric states related to rank-one
POVMs and established its  connection with the maximum likelihood
principle in quantum state estimation theory. Based on this
connection, we provided a method for computing the GM of these
states and demonstrated its additivity property under certain
conditions. In particular, we proved the additivity of the GM of
pure symmetric multiqubit states whose Majorana points are
distributed within a half sphere, including all pure symmetric
three-qubit states. We then introduced a family of symmetric states
that are generated from MUBs and derived an analytical formula for
their GM. We also derived the GM of symmetric states generated from
SIC~POVMs and used it to characterize all inequivalent HW covariant
SIC~POVMs in three-dimensional Hilbert space. A scheme for creating
the symmetric multiqubit states studied in this article was also
proposed. Our studies promise a broad perspective of integrating two
important research areas in quantum information science, namely,
entanglement characterization and quantum state estimation.

\section*{Acknowledgment}
We thank Markus Grassl for discussions on the symmetric states
generated from SIC~POVMs and for his valuable  comments on the
manuscript. We also thank Daniel Greenberger for discussions on the
connection between the GM of symmetric states and the maximum
likelihood principle. H.Z. thanks Berthold-Georg Englert, Yong Siah
Teo and  Hui Khoon Ng for discussions on the maximum likelihood
methods. The Centre for Quantum Technologies is funded by the
Singapore Ministry of Education and the National Research Foundation
as part of the Research Centres of Excellence programme. T.-C.W.
acknowledges support from NSERC and MITACS.

\section*{Appendix A: Additivity of the GM of $|\Psi(\{n_{j}\})\rangle$}
In this appendix, we prove Eq.~(\ref{eq:POVMstateGMadd}) when the
maximum of the likelihood functional $\mathcal{L}(\rho)$ can be
obtained at a pure state. Equation~(\ref{eq:POVMstateGMadd}) is
equivalent to
 \be
 \label{appendixAeq:add}
\L^2(|\Psi(\{n_{j}\})\rangle \ox \ket{\Ph}) =
\L^2(|\Psi(\{n_{j}\})\rangle)\L^2(\ket{\Ph}).
 \ee
According  to the definition, the l.h.s is never smaller than the
r.h.s; so it suffices to show that $\L^2(|\Psi(\{n_{j}\})\rangle \ox
\ket{\Ph})\leq\L^2(|\Psi(\{n_{j}\})\rangle)\L^2(\ket{\Ph})$. Suppose
$|\Psi(\{n_{j}\})\rangle$ is shared over the parties $A_1,
A_2,\ldots, A_N$, and $|\Phi\rangle$ over the parties $B_1,
B_2,\ldots, B_N$; suppose $|a_k\rangle$ is the $k$-th member of the
multiset consisting of $n_1$ copies of $|\psi_1\rangle$, $n_2$
copies of $|\psi_2\rangle$ and so on. Then according to
Proposition~\ref{pro:symmetric},
 \bea
     &&
    \L^2(|\Psi(\{n_{j}\})\rangle \ox \ket{\Ph})\nonumber\\
  &=&\max_{\ket{\ph}} \bigl| \bra{\ph}^{\ox N}|\bigl(|\Psi(\{n_{j}\})\rangle
\ox \ket{\Ph}\bigr) \bigr|^2
    \nonumber\\
    &=& \frac{N!}{\mathrm{perm}(A)}~\max_{\ket{\ph}}
    \bigg| \biggl(\bigotimes^{N}_{k=1} \bra{\ph}_{A_k,B_k}\ket{a_k}_{A_k}\biggr)
     \ket{\Ph}_{B_1,\ldots,B_{N}} \bigg|^2
    \nonumber\\
    &=& \frac{N!}{\mathrm{perm}(A)}~
    \max_{\ket{\ph}}
    \Bigg[\biggl(
    \bigotimes^{N}_{k=1}
    \bigl|\bra{\ph}_{A_k,B_k}\ket{a_k}_{A_k}\bigr|^2
    \bigg)\nonumber\\
    &&\times
    \biggl|\bigg(
    \bigotimes^{N}_{k=1}
    {\bra{\ph}_{A_k,B_k}\ket{a_k}_{A_k}
    \over\bigl|\bra{\ph}_{A_k,B_k}\ket{a_k}_{A_k}\bigr|}
    \bigg)
    \ket{\Ph}_{B_1,\ldots,B_{N}}
    \bigg|^2\Biggr]
    \nonumber\\
    &\leq&\L^2(\ket{\Ph}) \frac{N!}{\mathrm{perm}(A)}
    \max_{\r}
    \prod^{N}_{k=1}
    \big(
    \bra{a_k}_{A_k}\,\r_{A_k}\!\ket{a_k}_{A_k}
    \big)        \nonumber\\
    &=&\L^2(\ket{\Ph}) \frac{N!}{\mathrm{perm}(A)}
    \max_{\r}\mathcal{L}(\rho)\nonumber\\
    &=& \L^2(|\Psi(\{n_{j}\})\rangle)\L^2(\ket{\Ph}).
 \eea
Here the last equality is due to our assumption that the maximum of
the likelihood functional $\mathcal{L}(\rho)$ can be obtained at a
pure state.

\section*{Appendix B: perm(A) in Eq.~(\ref{eq:gDicke})}

Note that the entries of the Gram matrix $A$ in
Eq.~(\ref{eq:gDicke}) only take on five different values
$0,1,\sin(\theta/2), \pm\cos(\theta/2)$. In contrast with the
computation of the permanent of a generic matrix,  the permanent of
$A$ can be computed efficiently as follows:
\begin{eqnarray}
&&\mathrm{perm}(A)= \Biggl(\prod_{j,k=0}^1
n_{jk}!\Biggr)\sum_{\{a,b,c,f,g\}}(-1)^{g-a}\Bigl(\cos\frac{\theta}{2}\Bigr)^{2f+2g}\nonumber\\
&&\times\Bigl(\sin\frac{\theta}{2}\Bigr)^{2a+2b+2c-2g}{n_{00}\choose
a,
b, n_{00}-a-b}\nonumber\\
&&\times{n_{01}\choose c,f,n_{01}-c-f}{n_{10}\choose g, a+c-g,
n_{10}-a-c}\nonumber\\
&&\times{n_{11}\choose a+b-g, f+g-a, n_{11}-b-f},
\end{eqnarray}
where the summation is restricted to the set of nonnegative integers
$\{a,b,c,f,g\}$ satisfying the following constraints,
\begin{eqnarray}
&&a+b\leq n_{00},\;\; c+f\leq n_{01},\;\; a+c\leq
n_{10},\;\; b+f\leq n_{11},\nonumber\\
&&g\leq a+b, \quad g\leq a+c,\quad f+g\geq a.
\end{eqnarray}

\section*{Appendix C: Proof of Theorem~\ref{thm:SIC-GM}}

   We begin by proving the implication \ref{item:sic}$\Ra$\ref{item:closeproductstate}.
   According to Proposition~\ref{pro:symmetric} and the definition of
   a SIC~POVM,
\bea
    \L^2(\ket{\Ps})&=&
    \L^2(\ket{\Ps_d^{\mathrm{SIC}}})
    = \max_{\ket{\ph}} \bigl| \bra{\ph}^{\ox d^2} \ket{\Ps_d^{\mathrm{SIC}}}
    \bigr|^2\nonumber\\
    &=&\frac{d^2!}{\mathrm{perm}(A)}~\max_{\ket{\ph}} \prod^{d^2}_{j=1} \abs{ \braket{\ph}{\ps_j}
    }^2.
\eea
  Here the maximization is subjected to the completeness condition $\sum_j \abs{ \braket{\ph}{\ps_j} }^2 =
  d$ and the following condition
\bea
    \label{ea:symmetricoperator}
    \sum_j \abs{ \braket{\ph}{\ps_j} }^4
    =  \frac{2d}{d+1},
\eea since a SIC~POVM is a 2-design \cite{RBSC04}: $\sum_j
|\psi_j\rangle\langle\psi_j|\otimes|\psi_j\rangle\langle\psi_j|=[2d/(d+1)]\Pi_{\rm
sym}$, where $\Pi_{\rm sym}$ is the projector onto the bipartite
symmetric subspace. Hence,

\bea
    \label{eqa:maximumSICPOVM}
    \L^2(\ket{\Ps_d^{\mathrm{SIC}}})
    &\leq& \frac{d^2!}{\mathrm{perm}(A)}~\max_{0\leq p_j\leq 1,\; \sum_j p_j = d, \atop\sum_j p_j^2 = \frac{2d}{d+1}}
    \prod^{d^2}_{j=1} p_j\nonumber\\
    &=&{d^2! \over (d+1)^{d^2-1}\mathrm{perm}(A)}.
\eea
  The maximum in the above equation is obtained if all $p_j$'s are
  equal to $1/(d+1)$ except one of them, which is equal to 1. A state can  satisfy these conditions if and only it  belongs to the
  SIC~POVM, hence, the implication \ref{item:sic} $\Ra$
\ref{item:closeproductstate} follows.

The implication  \ref{item:closeproductstate}
$\Ra$\ref{item:innerproduct} is obvious.  The implication
\ref{item:innerproduct}$\Ra$\ref{item:sic} and
\ref{item:closestSICPOVM} can be shown as follows, \bea
  \label{ea:innerproduct}
    \abs{ \bra{\ps_j}^{\ox d^2} \ket{\Ps} }^2
       &=&\frac{d^2!}{\mathrm{perm}(A)} \prod^{d^2}_{k=1} \abs{ \braket{\ps_j}{\ps_k} }^2
    \nonumber\\
    &\leq&\frac{d^2!}{\mathrm{perm}(A)}
    \bigg(
    {{\sum^{d^2}_{k=1,k\ne j} \abs{ \braket{\ps_j}{\ps_k} }^2 }
    \over
    {d^2-1}}
    \bigg)^{d^2-1}
    \nonumber\\
    &=&{d^2! \over {(d+1)^{d^2-1}}\mathrm{perm}(A)}.
\eea
 The inequality is saturated  if and only if $|\langle\psi_j|\psi_k\rangle|^2=1/(d+1),~\forall j,k$ and $j\neq
 k$; that is, the $\ket{\ps_j}$'s  form a SIC
  POVM, which implies \ref{item:sic} and \ref{item:closestSICPOVM}.

It remains to show the implication
\ref{item:closestSICPOVM}$\Ra$\ref{item:sic},
 \bea
|\langle\varphi_j|^{\otimes d^2}\Psi\rangle|^2&=&
    \frac{d^2!}{\mathrm{perm}(A)} \prod^{d^2}_{k=1} \abs{ \braket{\ph_j}{\ps_k} }^2
    \nonumber\\
    &=&
    \frac{d^2!}{\mathrm{perm}(A)}
    \prod^{d^2}_{k=1}
    \bigg(
    \prod^{d^2}_{j=1} \abs{ \braket{\ph_j}{\ps_k} }^2
    \bigg)
    ^{\frac{1}{d^2}}
    \nonumber\\
    &\le&
    \frac{d^2!}{\mathrm{perm}(A)}
    \max_{|\ps\rangle}
    \prod^{d^2}_{j=1}
    \abs{ \braket{\ph_j}{\ps} }^2
    \nonumber\\
    &\leq&
    {d^2! \over {(d+1)^{d^2-1}}\mathrm{perm}(A)}.
\eea
  Here the second inequality follows from Eq.~(\ref{eqa:maximumSICPOVM}), recall that the $|\ph_j\rangle$'s form a SIC
  POVM.  The inequalities are saturated if and only if  the $\ket{\ps_j}$'s are
 the same as the  $\ket{\ph_j}$'s up to some permutation and phase factors;  which implies 1.

redpajama_set_name: RedPajamaArXiv


With the days getting colder and winter surely on its way, we thought it would be fitting to share with you some of our top book recommendations for those autumn evenings best spent on the couch with a cup of tea in one hand and a book in the other. Whether it be a tale of coming-of-age, an exploration of identity and self, or a comedic look at our feminist hypocrisies, these charismatic stories provide insight into experiences of femininity around the world – and should most definitely make it onto your reading list.
'We Need New Names' is a coming-of-age novel recounting the journey of a young girl out of Mugabe's Zimbabwe and towards America's promise of a reinvented-self. The New York Times describes the protagonist as 'A Child of Two Lands', as the novel explores those challenges faced by immigrants in the U.S. – identity, cultural adaptation, and the ever-looming threat of deportation. Bulawayo's literary debut is topical given Mugabe's recent removal from power, and has earned her the distinction of becoming the first woman African writer to be shortlisted for the Man Booker Prize; two great reasons to add this to your 'to read' list this season!
'Milk and Honey' is a collection of poetry and prose by Indian-Canadian writer Rupi Kaur, which takes a modern look at femininity and how experiences of violence, love and loss affect how women relate to the world, and to men. Kaur's minimalist style and delicate, accompanying illustrations reflect life's bittersweet take on hurt, hope, and healing – a universal and captivating tri-set of themes which is sure to inspire you!
Nepali novelist Manjushree Thapa probes themes of gender, displacement and humanitarianism, through her captivating account of an unlikely group of strangers from across the world and the surprising ways in which their lives and ambitions overlap. Intensely perceptive and honest, this fiction novel takes a feminist look at the world of international aid, and the interconnectedness of those within it.
Titled by Trump's viral labelling of Hilary Clinton as a 'Nasty Woman', this collection of essays provides insight into the experiences of individuals from all walks of life, as they discuss the ever-normalising set of misogynies that shape what it means to be a woman today. In a book Margaret Atwood has described as 'an essential window into many of the hazard-strewn worlds younger women are living in right now', this collection delves into issues of reproductive rights, cruelty towards those with disabilities, and life as a queer black woman in America. Punk, pressure, politics and personal - this diverse and relatable collection is sure to ignite your feminist fury and inspire you to be the nastiest, most defiant version of yourself!
Makumbi's multi-generational drama, in which the interwoven stories of five different members of the cursed Kintu Clan are told, explores how family legacies and old folktale are kept alive within a modernising Uganda. This fictional novel delves deep into the nation's rich culture and history as Makumbi expresses an image of the real fabric of Ugandan society – making this novel both an entertaining and enlightening read!
In this book, Nigerian-British journalist Reni Eddo-Lodge opens up a dialogue on structural racism, white privilege, and the disconnect with reality which often runs through accounts of British history. This Top 5 Sunday Times Bestseller is Eddo-Lodge's call to end the silence which continues to echo the topic of race in the UK, challenging us all to listen, to talk, and to act. Whatever your background, the frustrations and fears discussed in this book are sure to challenge your assumptions and open your eyes to insidious racial prejudice.
Fate, obsession, and the deceitful nature of memories are the central themes explored in Gappah's novel 'The Book of Memory'. Convicted of the murder of a white man and confined to Chikueubi Maximum Security Prison, protagonist Memory narrates the series of events leading up to her arrest as she remembers them. Her story reveals the dichotomy between what is real and what is assumed, told against the backdrop of a racially divided Zimbabwe in which white occupation complicates Memory's quest for justice further.
Following her hit "50-million download" comedy podcast, comedian Deborah Frances-White released in September her literary debut exploring the feminist musings and concerns we find ourselves coming back to again and again. From the diet industry to toxic masculinity and including interviews from feminist entertainers and activists alike, this comedic look at what modern feminism really means has been described as passionate, humorous and intelligent, and is certainly privileged on our reading lists this autumn.
The diversity of these books demonstrates that when it comes to being a woman and to being a feminist – anything goes! Through absorbing as many perspectives of life as possible, fictional or not, we can better understand other women and how to best celebrate them. Happy reading!


She's been walking since 2013, and I thought it was time to bring her home. With this piece I tried to outdo anything I've done before, and I can only hope I succeeded.
First it was just Twilight and the open sky. Then it felt lonely and barren, so I gave her a photo to hold. Then she needed something to walk on, some solid ground. Oh, and where did the tree come from? Long story short, it was a case of just adding onto things until it felt right.
It's been a long few years following along, and we're gonna do it one more time with season 9. I can't express just how much these colorful cartoon horses mean to me, but this is a shot in the dark to at least -try-. Dear Princess Twilight, thank you for showing us how to be the best person we can be.
Now this does put a smile on my face. I'm glad to see other people love Twilight as much as I do. I discovered my little pony at a time in my life when i had been let down by friend after friend after friend and i had decided that only my family mattered to me and no one else was worth it. then Twilight came and taught me the magic of friendship and i met the best friend ever. Thank you Princess Twilight Sparkle the Lavender Unicorn, You changed my heart.
Right! That was the sentiment I was trying to get across, and I'm glad it worked. Glad you're doing better these days and that these little ponies have changed a few people for the better!
Hey, thanks! I really appreciate the nice words! I was definitely going for nostalgic, so that's great to hear. Cheers!
This looks totally amazing! Compared to the versions from previews years the step up seems insane!
I must say, Twilight looks really really good on this one.


text:
90 Day Fiance: The Other Way fans bash Steven Johnston for hitchhiking with Alina
Tue Sep 28, 2021 at 10:46am ET
By Mona Wexler
When Steven Johnston hitchhiked with Alina in Turkey, 90 Day Fiance fans slammed his decision. Pic credit: TLC
Fans of 90 Day Fiance: The Other Way are not happy with Steven Johnston's decision to hitchhike a ride in Turkey with his fiance, Alina.
90 Day Fiance: The Other Way viewers have already come to know "Steven with a V" as quirky and even "creepy."
A devout Mormon from Salt Lake City, Utah, Steven, 25, met his love interest Alina, a 20-year-old Russian student and artist, on a language exchange site.
Due to border closures amid the pandemic, Steven and Alina decided to meet up in Turkey and get to know each other before tying the knot and moving to Russia together.
During the latest episode of The Other Way, Steven and Alina met with an attorney to discuss their marriage options while staying in Turkey.
At the time, Steven and Alina didn't have a car and it was extremely hot outside as they began to walk to the lawyer's office.
So, what better way to safely get you and your fiance to an appointment with an attorney than to hitchhike, right?
A post shared by 90DayFianceNow (@90dayfiancenow)
That's exactly what Steven did and it had 90 Day Fiance fans (and Alina) wondering what he was thinking.
Steven Johnston hitchhikes with Alina in Turkey
During the scene, Steven stuck out his thumb and started waving down passing cars, hoping they would stop and give him and Alina a ride.
"No one is going to stop," Alina embarrassingly tells Steven.
Steven admitted that he has hitchhiked in "at least five countries," and didn't think much of it, calling it a "cheap mode of transportation."
"I think it's a little bit dangerous, don't you think so?" Alina asked Steven.
"Trust me, it'll be fine. We just gotta get someone," Steven said, trying to reassure Alina that hitchhiking in a foreign country was safe.
Steven explained his technique to the cameras during his and Alina's confessional and said that he typically tries to make eye contact with drivers and wave them down, in hopes that someone will eventually stop.
Someone did eventually stop and give Steven and Alina a ride and during their ride, Steven had some shocking advice for his fiancee, who felt uneasy about the situation.
"If I get a bad feeling I'll let you know and you can jump out [of] the car," Steven told Alina, suggesting that she just "roll out" of the door or roll down the window and escape that way.
https://www.instagram.com/p/CTzYuXHLIbU/?utm_source=ig_web_copy_link
90 Day Fiance: The Other Way fans bash Steven for hitchhiking
A 90 Day Fiance fan page on Instagram, 90 Day Fiance Now, shared the clip from The Other Way, and fans of the show blasted Steven for hitchhiking with Alina.
One fan of 90 Day Fiance: The Other Way felt that Steven could have put out the money to catch a cab or an Uber rather than hitchhike and risk his and Alina's safety.
Their comment read, "Is he [too] cheap to get a cab? Turkey is not Utah. Glad that's not my daughter."
Pic credit: @90dayfiancenow/Instagram
One fan of 90 Day Fiance brought up the safety factor and thought that Alina would be safer alone rather than in Steven's care.
"I thought she wanted him to stay with her so she was safe!!!!! Hahahaaaa you'll be safer by yourself hun xxx," the commenter wrote.
Another confused and concerned fan of The Other Way commented, "This was a bit dangerous. They could have just gotten a taxi or took a bus."
Steven and Alina seemed an unlikely couple from the start, and their recent behavior on social media has fans of the show wondering if their relationship ended before they made it to the altar.
Steven was recently spotted with a mystery woman while spending some time with Chantel and Pedro Jimeno, also from the 90 Day Fiance franchise.
Fans of 90 Day Fiance: The Other Way will have to tune in every week to follow Steven and Alina's story and see if their romance stood the test of time.
90 Day Fiance: The Other Way airs on Sundays at 8/7c on TLC and Discovery+.
More: 90 Day Fiance: The Other Way, Steven Johnston


text:
Tag Archives: Shredder
April O'Neil is investigating the rise of the Foot Clan who have been causing trouble in the city of New York. She hopes to find out who is behind the clan so that she can leave her currently uninspired news reporting at Channel 6, which sees her doing elaborate bird themed work outs in front of the camera. Her intuition soon brings her to the Foot Clan suspiciously unloading cargo from the docks. As she videos the crime taking place, she witnesses unidentifiable vigilantes stopping the Foot Clan in their tracks. This won't be the last time she sees them.
Michael Bay brings another childhood staple to the big screen but this time from the producers chair. Although he may as well have directed the TMNT reboot as Jonathan Liebesman clearly takes notes from the Transformers director. Everything from the glossy CGI to the Bayhem that ensues, it would be hard not to draw similarities. At least this time Megan Fox isn't just there for her looks, although it's hard not to notice her in that bright yellow jacket.
Despite having watched the cartoons at a very young age, I wasn't very enthused about the release of this film. Maybe because I'd forgotten a lot about the franchise or after watching Transformers: Age of Extinction a month prior, I just couldn't stomach another Michael Bay film. That said, I decided to pass on the ticket price and waited for it to come out on video. Although I'm glad I didn't pay the $10 to watch this film in the cinema, I can't say that I would have been disappointed if I had. Teenage Mutant Ninja Turtles from the outset is clearly a child friendly film. Once I got in that mindset, it was easier to just sit back and enjoy it. It was nice falling in love with these characters again and remembering the old days coming back from school and watching these cartoons. For hard out fans it may not be the same pleasant case of nostalgia as I'm aware they've completely butchered the character of shredder, literally and figuratively. He looked like a walking Swiss army knife. The turtles and Splinter aren't too pretty either, unluckily for splinter he doesn't have a mask to hide it. That being said I don't think this is a completely bad film. It's a decent origin story and first installment in the franchise, and for me personally has aroused enough interest for me to be more enthusiastic about the sequel which is already planned for 2016.
TMNT doesn't have much going for it in terms of story. It's incredibly generic and worn out. Bad guy wants to release a biological weapon on the city, corrupt business man makes tonnes of money from curing everyone. Being child friendly isn't a good enough excuse for a lack of originality. Although The Lego Movie pretty much had that same plot, except it had a nice underlying message behind it. TMNT is quite simply shallow mindless fun which is forgivable if it succeeds in being fun. In my opinion it succeeds thanks to the facetious nature of the turtles and the cartoony action. Although it is let down by a script caked in exposition. There's a huge portion in the middle of the film which consists of flashbacks and explanations, which ultimately ruins the whole flow of the film.
I welcomed the prospect of seeing a Megan Fox led film, partly because she's gorgeous but also because I think she has more to give. After seeing her in This is 40, it's clear that she is funny and has a sense of comedic timing which makes a film like TMNT a place she can thrive. I was disappointed not to see a lot of comedic moments where she could showcase that, and when she did have that opportunity, she was let down by the script. That being said I didn't connect with her character as much as the turtles. Overall there weren't any standout performances from the cast; solid but comfortable.
Teenage Mutant Ninja Turtles may not be the best film but thanks to some charismatic turtles and a few exciting action sequences, it prevents itself from being a bad one. It's a film that kids are sure to enjoy but the script is neither exciting or funny enough for the whole family to enjoy together. For me personally, I'm glad that a sequel is being made so that it has another shot at success.
Posted in Roze-Reviews	| Tagged April O'Neil, Donatello, Film, Johnny Knoxville, Leonardo, Megan Fox, Michaelangelo, Movie, Raphael, Rating, Review, Shredder, Splinter, Teenage Mutant Ninja Turtles, The Popcorn Bandit, TMNT, Will Arnett, William Fitchner	| Leave a reply


Welcome to the US SNES Games Section. All of these SNES Games are US NTSC Super Nintendo games. To run these NTSC Super NES Games you will need either a US console, a modified European Super Nintendo or a SNES Import Adapter Cartridge. I can supply any of these, and they can be found in the Super Nintendo Hardware or Super Nintendo Consoles section.
If you are a European or Australian customer you should be looking in the Super Nintendo Games section for PAL SNES Games.
Please enquire beforehand if you are unsure about which games run on a normal console. We have a great selection of Imported Japanese Super Nintendo Games (Super Famicom) in the Super Famicom Games section. All games have been cleaned as described in the Care of Second Hand Games section.
Notes: Complete with all maps and in excellent condition.