text:
MAGICK BROTHER
Buy GONG Music
- Early Morning -
1. Mystic Sister / Magick Brother (4:44)
2. Change the World (aka Rational Anthem) (4:10)
3. Glad To Sad To Say (3:35)
4. Chainstore Chant / Pretty Miss Titty (4:38)
5. Fredfish / Hope You Feel OK (3:18)
- The Late Night -
6. Ego (3:55)
7. Gongsong (4:05)
8. Princess Dreaming (2:55)
9. 5 & 20 Schoolgirls (4:23)
10. Cos You Got Green Hair (4:58)
- Gilli Smyth / vocals & voice (space whisper)
- Daevid Allen / guitars, bass, vocals
- Didier Malherbe / flute, soprano saxophone
- Rachid Houari / drums, tabla
- Burton Greene / piano (6)
- Dieter Gewissler / contrabass (1,7)
- Barre Phillips / contrabass (2,8)
- Earl Freeman / contrabass (6), piano & pianoharp (7)
- Tasmin Smyth / voice (1,8)
Artwork: Daevid Allen
LP Byg Records - 529 305 (1969, France)
LP Affinity - AFF 4 (1977, UK) Different cover art and tracks 2 & 3 switched order
LP Charly - CRL 5052 (1977, UK) As above
CD Decal - CD CHARLY 31 (1987, France) As above
CD Charly - SNAP 199CD (2004, UK) Remastered; with initial cover art & track running order
Buy GONG Magick Brother Music
Magick Brother (Japanese Mini LP Sleeve SHM-CD)
Belle Antique $39.99
Gong Magick Brother Decal 1986 Charly UK Import
Magick Brother Magick Sister
Limited Edition · Remastered
Jvc Japan 2001 $57.97
More places to buy GONG music online
Buy GONG & Prog Rock Digital Music online:
AmazonMP3: Search for GONG DRM-Free MP3 Downloads @ Amazon.com MP3
GONG Magick Brother ratings distribution
Collectors/fans only (14%)
GONG Magick Brother reviews
An absolutely strange UFO with very weird music well worthy of GonG but it is right now just Gong. Because if the zanyness is already present and makes for this album much of an interest , but it is not part of the Radio Gnome trilogy or linked to it. Only Malherbe , Allen and Smyth are on this album that will be present on the classic GonG.
But the main interest of the album is that I cannot think of one album that is more representative of hippydom, in fact it could epitomize it. As Daevid got thrown out of England (so called visa problems but as an Aussie and in the Commonwealth.....) , he went back to France and joined a hippy commune in Normandy and one can guess that he got those guys into this project . This album although structured reeks anarchy , but not in the same way that of Amon Djuul's first group. If this had been recorded in Germany , this would've had a chance to be classified as Polit-Rock alongside such as other left-wing activists as Kluster etc... Not that Daevid would be called left-wing but just an activist alone would fit him fine. Anyway , this album is very rough and raw to the bone (especially compared to Angel's Egg and You) but is much worth a spin but beware of your sanity. Doctor's prescription would say no more than one toke/poke a week to avoid addiction.
Posted Monday, November 15, 2004 | Review this album | Report (Review #27578)
Review by soundsweird
More proof that Psychedelic Rock was a key component/forerunner of Progressive Rock. The search for new sounds, the desire for greater artistic freedom, and the willingness to throw in the stylistic kitchen sink are things that both genres have in common. As another reviewer mentioned, this album is Gong's "Piper at the Gates of Dawn"; quite different from their later work, but magical nonetheless. I'm sure there are those who like this album a lot, and hate everything that came afterwards, just like those Syd Barrett-philes. Some of it doesn't work, but that's why CD players have a 'program' feature. Also, the sound is a little iffy, but it goes with the territory.
My God what kind of drugs makes you write this kind of music ? This is really psychadelic S... and I must admit it´s a bit too strange for me ( and this is a Zappa fan speaking).
The sound quality is a bit muddy and raw which is not a problem and as this is from 1969 I don´t expect fine audio quality. The music is very laid back even though it kind of rocks and Daevid Allen´s voice is very stoned which kind of emphazises the laid back nature of the songs. The songs are basically simple sixties rock songs with very few exceptions. This is definitely prog rock but not of the symphonic sort, it´s way too strange for that. The only band I can compare them to are The Mothers of Invention, even though they don´t really sound like them, but they have some of the same strange choice of notes.
All in all this is a bit too strange and not my taste at all, but I can´t deny the quality of the songs and the musicians playing are also pretty good. It doesn´t reach excellent in my book though so 3 stars it will be.
Posted Wednesday, January 16, 2008 | Review this album | Report (Review #158743)
The debut of a bizarre planet. Well, it didn't met my expectations -- it's nothing short of good, but certainly not as stellar as GONG trilogy will be. This is a shape of GONG to come, so to say. This hyper-dipper-ultra psychedelic music - Syd Barrett's influences are obvious. There are also a lot of tape manipulations, which is fine, but there lays the problem: there' are only a few conventional songs, the rest of the album are noodling. Which doesn't have to be bad, but in this case songs=ideas. There are melodies, passages, timbres, even lyrics that will be fully developed later. However, apart from three excellent tunes (one acid electric, one acoustic and one reggae) the rest is suffering from 'the debut syndrome': young people unable to construct the whole picture. But the pieces are fine too.
Production is even below 1969 standards, but I think these guys just didn't care for that. It's evident they were experimenting with different timbres, including heavy distortion, not only on guitar, but on the whole tape mixdown - that's why it's hissy at the moments. Moreover, amounts of phaser/flanger and similar swooshing effects is hilarious. But I like it.
This record will help you understand better the origins of GONG family - from the very Canterbury beginnings (it's the only GONG record that shows the hints of Canterbury sound IMHO) to the reggae approach in 1990's.
Posted Wednesday, September 10, 2008 | Review this album | Report (Review #182141)
1969 was a good year for ex-Soft Machine members kicking off new careers after leaving that band; along with Kevin Ayers recording his solo debut Joy of a Toy, Daevid Allen hooked up with Gilli Smyth and between them put this out this early proto-Gong album. I say proto-Gong because the Gong sound has not quite developed here yet. Several songs are psych-folk with lyrics alluding to the political hopes and dreams of the Summer of Love, but this typical hippy fair sits cheek by jowl with free jazz-like improv with Smyth's ethereal wailing getting an early workout. The different ingredients which make up Gong are all here but they've not yet mixed in together to make the rich broth which would appear on the later albums. Not a bad start, not a fantastic one either.
Posted Friday, June 3, 2011 | Review this album | Report (Review #456502)
Review by octopus-4
SPECIAL COLLABORATOR RIO/Avant/Zeuhl,Neo & Post/Math Teams
I'm Happy, I'm not happy, I'm sad....I'm not sad.... I think a lot of people remembers "The Witch" spelling these words in "Angel's Egg". Before coming to this album I didn't know that they were already been used on "Pretty Miss Titty".
So while I was expecting what somebody has called "Proto-Gong", I've been pleasantly surprised to hear this, maybe sour, full Gong album.
Yes, it has pure psychedelic moments perfectly in line with the London's summer of love, ,with sounds that can even remind to the early Barrett with a hint of Beatles, but first of all the great inventive and the jazzy weirdness of Daevid Allen.
We pass from the acid guitar and violin of "Glad To Say To Say" with the vocal line reminding to Who, Beatles and Pink Floyd at the same time, to the fantastic closer "Cos You Got Green Hair" and its flute that contains already all the kind of goods that can be found across the Radio Gnome Trilogy.
So there are two keys to access this album: a prelude to Radio Gnome and a link between the 60s late psychedelia and the Canterbury genre. The genius of Daevid Allen has already started actually to transform the hippy-trippy psychedelic standards into something new and different.
An album very innovative, with noisy unstructured avantgarde tracks like "Little Sister" followed by a country/psych song like "Change The World". Even a Floyd fan like me would regret thinking that this Gong would have probably been more appropriate to comment Zabriskie Point, apart of C'mon Number 59, of course.
This album is full of ideas, some of them abandoned later, or not developed, but this is normal in a debut. I find it excellent and deserving 4 stars
Posted Tuesday, December 6, 2011 | Review this album | Report (Review #583492)
After Daevid Allen left Soft Machine he spent a few years continuing his beatnik lifestyle traveling around Europe and getting into mischief before finally recording his first album under the GONG moniker. Despite the band name this is more of an Allen album with equal billing given to Gilly Smyth who makes her debut on this first GONG album as the trippy poet and space whisperer who graced not only the GONG albums of the Allen era but also many of the solo offerings from band's members. MAGICK BROTHER has also been released as MAGICK BROTHER, MYSTIC SISTER and was released early in 1970. The album is really an accumulation of musical ideas that had been with Allen since leaving Soft Machine and since those years took place in the tumultuous late 60s it is no surprise that this album has its sound firmly rooted in 60s psychedelia and the lyrics in that same idealist Utopian vision of the hippie movement.
This is in fact an interesting crossroads for Allen's career where he finally got to put to tape all of those latent ideas that didn't make it into Soft Machine that fully embraced the most psychedelic aspects that rock had to offer at that time and the more progressive ideas that were finding their way into the world. This album came out after all those groundbreaking releases of 1969 and the musical landscape was changing quickly but Allen was finding away to keep those space rock sounds going and evolving them into the next phase. On MAGICK BROTHER you can hear hippie peace protest songs "Rational Anthem" next to fully fueled freakouts that sound more like something on a Krautrock album and would make contemporary trippers like Pink Floyd proud. My guess is that many mind-altering substances were consumed in the making of this one. You think?!!
The album is divided into two sides. The first is "Early Morning" and the second is "Late Night" but to be honest I can't really tell any difference between the two as they both contain hippie folk type songs and freaked out bizarreness. This is one that I wasn't impressed with upon first listen coming to it after all the better releases that followed but it has grown on me and is in fact an interesting little album in its own right. I would not recommend skipping this one since it has a charm all its own and provides a glimpse of GONG's origins. The only problem I have is some of the folkier songs are a little lackluster and could have used a little more fine tuning but overall it doesn't diminish my listening experience. 3.5 rounded up
Posted Saturday, November 29, 2014 | Review this album | Report (Review #1317409)
After touring Europe with Soft Machine in August 1967, Daevid Allen was rejected to enter the United Kingdom due to overstaying his visa on previous staying. Allen settled in Paris and together with his partner, Gilli Smyth, he formed a band called Gong along with a few other side projects such as the Bananamoon Band. The two also took part in the 1968 Paris protests and later settled in Deià, Mallorca, where they had met a poet Robert Graves on their previous visit. In August 1969, they returned to Paris and recorded their debut album Magick Brother with Gong, which was released under the BYG Actuel label. At the time, the group did not have a bass player, so it was up to Allen to play the instrument. He also invited guest musicians, notably Barre Phillips, a renowned double bass player, who had previously performed with Eric Dolphy, Jimmi Giuffre, and Archie Shepp, Burton Greene, a pianist, who had played with Cecil Taylor and Albert Ayler, and a wind player Didier Malherbe, who would later become one of the members of the "classic" Gong era.
Since his very first days in Soft Machine, it was clear that Daevid Allen was a forward-looking, unorthodox, and immensely original individual. Brian Hopper, formerly of the Wilde Flowers with Robert Wyatt, Kevin Ayers, and Richard Sinclair to name a few, recalls meeting Allen for the first time: "[Daevid] showed up at Robert Wyatt's parents' house along with which he brought his own record collection, which was quite eclectic. And he was the first, I suppose, really hippie sort of person we'd met at that stage, you know, he was quite a sort of phenomenon, if you like, that arrived on the scene." Allen's musical vision comprises various diverse element, which fructify in a complex, odd, moody, tounge-in-cheek whole. Jazz influences are the most evident of all, reflected by swinging arrangements, free-jazz-like passages, and harmonic solutions. However, these only provides the basis for the music on Magick Brother. Avant-garde elements such as spoken word, storytelling, melody-less ambient parts, are introduced and play a crucial role in Gong's distinctive musical formula. Everything is topped with somewhat of a cosmic topping to it, whether it be lyrics, long reverb tails or quirky modulation effects. Most of all, the band builds on the legacy of the psychedelic boom of the sixties, with its escapist, hippie, trance-like feel.
Isn't it amazing that the first seconds of the very first Gong album are a sound of an oriental-sounding gong? "Mystic Sister/Magick Brother" is somewhat of a display of magic that one is dealing with on the album, being a hippie folk ballad opened by spacey ambient guitar parts with whistling, bird-like wind instruments somewhere in the distant background. "Glad To Say To Say" follows a rather simple psychedelic pop song pattern, featuring a catchy guitar riff and Daevid Allen's overdubbed vocals. Towards the end, the piece descends into trippy atonal mayhem. "Rational Anthem" is based around a quiet, blurry guitar motif with various ambient effects around. After the calm mood of the previous track, "Chainstore Chant/Pretty Miss Titty", a heavier psychedelic rock piece, offers more dynamically varied parts with Daevid Allen's guitar in the foreground. The song also features spoken word parts, delivered through Gilli Smyth's "space whisper". "Fredfish/Hope You Feel OK" opens with comedic spoken words, which sound as if broadcasted through a radio. Later, the track turns into a Syd Barret-era Pink Floyd-like ballad. Side two, labeled The Late Night (as opposed to side one, Early Morning), opens with "Ego", an avant-jazz statement with psychedelic rock influences, particularly on the sung parts. "Gong Song" tells a story of Pothead Pixie, a visitor from a distant planet, Gong. The concept would be continued on the following albums. Musically, it has somewhat of a Beatles-like sound, but with strong influences of jazz. "Princess Dreaming" opens with a repeating screech, an awfully unpleasant ear-sore, which later dissolves into a part spoken by Gilli Smyth. "5 & 20 Schoolgirls" is another piece kept in a psychedelic pop convention, but again with jazzy flavoring and a healthy dose of peculiarity. The album closes with a haunted ambient soundscape of "Cos You Got Green Hair", which seems to point the way towards the following works by Gong.
The Flying Teapot may yet be to take off to planet Gong, but Magick Brother, Gong's debut album, undoubtedly sets the stage for the band's next albums with its cosmic, trippy, escapist, and jazzy character. The release does have its flaws, poor studio recording being the most notable and disturbing, but is an incredibly rewarding and fascinating journey. Regardless of the style, whether it's a free-jazz workout, a spacey ambient trip or a psychedelic pop ballad, everything is played with great taste and precision. Highly recommended!
Posted Saturday, June 18, 2016 | Review this album | Report (Review #1579864)
A big surprise, to say at least. I had absolute no expectations when I started to listen to this album. I thought this was going to be a pretty horrid experience. I was proven wrong. Remove all the hype and all the weirdness surrounding Gong. Put your headphones on and listen very carefull ... (read more)
Report this review (#251998) | Posted by toroddfuglesteg | Saturday, November 21, 2009 | Review Permanlink
8.5/10 Great "Magick Brother" is really an underrated album in my opinion, and carries with it some amazing Gong moments and is overall great. This is it, Gong has entered the scene and we are given some really innovative music, great melodies and ideas, and an overall breathe of fresh air ... (read more)
Report this review (#148459) | Posted by The Lost Chord | Thursday, November 1, 2007 | Review Permanlink
Not really a prog album unless you consider looking backward whilst falling forward progress. Way to much dope was consumed and no doubt far to much acid by this proto- gong line up, What a re we left with firstly this LP sounds as though it was recoded in a bathroom not a studio, it is a horri ... (read more)
Report this review (#92830) | Posted by burgersoft777 | Sunday, October 1, 2006 | Review Permanlink
Gongs first LP is quite simply a Psychedelic masterpiece rivaled only by Floyd's Piper at the Gates of Dawn and White Noise's An Electric Storm. Allen's Gliss gutar was heavily influenced by Syd Barrets gutar work on Piper and like Syd, Allen is capable of taking you to musical dimensions others can ... (read more)
Report this review (#27577) | Posted by | Thursday, March 18, 2004 | Review Permanlink
Post a review of GONG "Magick Brother"


text:
Home » Natural Heritage » Chatham Island black robin
Chatham Island black robin
by Dave Houston|Published June 7, 2012
Status (2012):
Population (2013):
Trend: Nationally critical
298 mature individuals
Restricted to Little Mangere Island for over 80 years, the deteriorating condition of the small area of forest available to the birds and their resulting population decline saw the seven remaining birds transferred to a larger patch of bush on adjacent Mangere Island in 1976-77. After reaching a low of just 5 birds in 1980, intensive and innovative management combined with the extraordinary efforts of 'Old Blue', the only breeding female, saw the population increase to around 80 birds in 1989 with robins also having been established on Rangatira Island in 1983. At that time intensive management stopped, but intensive monitoring continued to monitor the growth of the population up until 1999, by which time the population had reached around 200 mature individuals. Seemingly secure on the pest-free island homes, monitoring was scaled back to allow work on other species.
Translocation failure
The growing population allowed the transfer of forty birds to be moved from Rangatira and Mangere islands to a predator-proof enclosure within the Ellen Elizabeth Preece Conservation Covenant (a.k.a. Caravan Bush) on Pitt Island between 2001 and 2005. Despite two breeding attempts the robins failed to thrive and the last bird 'Spike' disappeared late in 2007. The reasons for failure are not well understood, however the abundant mouse population within the covenant may have seriously depleted the robins' insect and spider food source.
Colour-banded robin. Photo: Dave Houston
Understanding population trends
Throughout the 2000's population estimates were made using a sampling method which over time appeared to show a decline in the population on both Mangere and Rangatira Islands. As concern grew over the apparent population trends, researchers from Canterbury University fortuitously started working black robins and other species Rangatira in 2007 and this enabled full population counts to be resumed. Initially, the low census counts obtained indicated that the population had declined on Rangatira, but no obvious reason cound be found for a decline aside from seasonal changes in productivity and survival. Over subsequent seasons the population on Rangatira has grown from 120 mature individuals in 2008 to 210 in 2012. While the robins have certainly had a couple of good years in recent times, it appears that the largest factor in the 'growth' of the population is increased observer effort.
Counting black robins is a tricky business. Those who have encountered them will remember robins boldly approaching on the lookout for food, either invertebrates you have disturbed or perhaps a mealworm proffered in attempt to lure the bird into the perfect photo position. If you're in the business of undertaking a distance sampling type population estimate, having robins chase you about the forest invalidates the method. However, if you're trying to count every robin on the island then having the robins come to you when you enter their territory is rather handy. It seems that in the period of low monitoring effort, the robins became progressively harder to detect as they encountered humans and their mealworm handouts less frequently. Once the researchers arrived robins were again offered handouts in order to keep them friendly the detectability of the robins increased. Along with increased effort put into searching infrequently visited areas of the island, the 'feed and count' methodology has again provided us with a robust population estimate. Unfortunately, this method is quite labour intensive and it takes two or three years to make the population sufficiently friendly.
Mangere Island has also had its ups and downs but probably for different reasons. The mid-90's saw 50+ birds on the island before declining to the mid-30's by 2005 and then increasing again to 46 birds in 2012. Rather than varying observer effort influencing population estimates it is likely that the carrying capacity of the small area of forest has been reached and the population is significantly influenced by seasonal conditions.
Keeping black robins secure on their island homes is the highest priority, hence the Department of Conservation maintains strict access and biosecurity restrictions on the islands to prevent incursions of rodents or other pests. The next highest priority is providing more habitat for the robins to occupy. On Mangere providing more habitat has meant the planting of over 100,000 trees and shrubs over the last 30 years. Unfortunately, trees take a long time to grow on windswept islands like Mangere and still have a bit of growing to do before the robins can move in.
Little Mangere from Mangere summit. Photo: Dave Houston
Over on Rangatira a 'hands off' policy has relied on natural regeneration of the formerly farmed island to fill the gaps between forest remnants. Again the process of growing has been slow and consideration is being given to transplanting some of the natural seedlings along track margins into areas of muehlenbeckia and bracken fern in order to speed up the joining of forest fragments. Undertaking an all-out planting assault like on Mangere would be inappropriate on this island given the number of seabird burrows present and the biosecurity risks of bringing large numbers of plants onto the island.
Little Mangere – difficult to access. Photo: Don Merton
Moving black robins back to Pitt Island is desirable, but keeping a sufficiently large area of forest mouse-free presents a significant challenge and the current predator-proof fence would require extensive modifications. Increasing the density of seabirds within forested areas like Caravan Bush may also increase improve the robins outlook as the increased fertility the seabirds bring benefits invertebrates.
Habitat quality on Little Mangere Island has improved markedly over the last 30 years and while it would do little to boost the population (there is room for only 6-10 pairs), it would complete the circle and return robins to their source. Complicating a return is the very difficult access to the privately owned island. Regular monitoring is required to ensure the long-term success of the transfer and the lack of a helicopter in the Chathams currently makes this impractical.
Restoring black robins to Chatham Island presents an even larger challenge. Again a predator-proof enclosure could be used, but again it needs to be around a sufficiently large area of forest to support a self-sustaining population, but ideally be isolated from adjacent forested areas that birds could disperse into. There are few such areas on Chatham Island and currently the Henga Scenic Reserve leads the list of possibly suitable candidates.
The black robin population currently stands at 298 mature indivduals and is moderately secure on predator-free islands. Population growth is limited by habitat availability and a third breeding site capable of supporting at least 50 mature individuals is very desirable.
black robin
rangatira
Dave Houston
Visiting the Chatham Islands since 1996, Dave works for the Department of Conservation providing advice on the management of threatened species.
New Endemic Plants for the Chatham Islands
As progress toward the preparation of a Chatham Islands Flora continues, the number of endemic vascular plants accepted for the Chatham Islands has now increased from 38 to 42 with the formal recognition of three new scurvy grasses and one new hook sedge (oft known as bastard grass) from the islands.
Apparently extinct lichen discovered on the Chatham Islands
A common woodland lichen of eastern Australia that has been recorded only twice from New Zealand in 1934 and 1976, and then only from the far north of the North Island has turned up on the Chatham Islands. The lichen, Heterodea muelleri is a leafy species that in Australia grows in moderately open woodland habitats. In New Zealand, until it was recognised from the Chatham Islands it had only ever been recorded from dune slacks somewhere on the Ninety Mile Beach and from damp sandstone ridges in light scrub near Puheke, Karikari Peninsula.
The Chatham gatherings came from the north-western end of Ocean Bay, Chatham Island and from the top of Hakepa Hill (Walkemup), Pitt Island. At Ocean Bay the lichen grew on sandy peat and clay above schist on the margin of salt and wind blasted vegetation. In this habitat it was associated with the lichens Cladia aggregata and C. retipora, and sedge Lepidosperma australe. On Hakepa Hill specimens were gathered from amongst the dense drifts of Cladonia lichens that grow between the low, windswept fernland that covers most of that trachyte peaks summit.
New Checklist of the Chatham Islands Plants Published
As progress toward the preparation of the first flora of the Chatham Islands since 1864 Department of Conservation and Landcare Research scientists […]
Previous post Chatham Island oystercatchers
Next post Chatham petrel population growing

redpajama_set_name: RedPajamaCommonCrawl


text:
\section{Introduction}

The implementation of positive operator-valued measures (POVMs)
for photonic quantum state signals is important for a variety of
quantum information protocols, in particular, for quantum
communication schemes such as quantum teleportation
\cite{Teleportation}, quantum key distribution
\cite{QuantumCrypto}, and collective decoding in quantum channel
coding \cite{Hausladen96,HSW,Fujiwara03}. Unlike conventional
optical detection technologies, POVMs for optical quantum
information protocols generally include a projection onto
superposition states or entangled states. In order to implement
such measurements, normally a nonlinear interaction of the signal
states (described by a Hamiltonian at least cubic in the optical
mode operators \cite{LloydBraunstein}) is needed. At present,
however, these nonlinear processes are hard to realize on the
level of single photons.

One possibility for inducing a nonlinear element is to exploit the
effective nonlinearity associated with a measurement.
In particular, for photonic-qubit states, universal gating
operations and hence any POVM for these states can be realized
asymptotically by using linear optics, photon counting, highly
entangled auxiliary states of $n$ photons, and conditional
dynamics (feedforward). Here, conditional dynamics means the
successive application of linear transformations on the remaining
modes conditioned upon the detection of a subset of modes
\cite{KLM01}. In the special case of a projection measurement,
corresponding to the discrimination of an orthogonal set of
states, perfect distinguishability is achieved in the asymptotic
limit of large $n$. However, with current technology, it is hard
to generate the entangled auxiliary states even for modest $n$.
More recent investigations, therefore, have focused on the
question whether one can implement a given measurement, or more
generally a set of universal quantum gates, via cheaper and/or
finite resources. For example, by applying the cluster-state model
of quantum computation \cite{RaussendorfBriegel} to linear optics
\cite{Nielsen}, the cost of the extra entangled resources may be
significantly reduced \cite{BrowneRudolph,Nielsen}.

In this paper, we address the following question: is it possible
to implement a given projection measurement in the asymptotic
limit of infinitely many, arbitrarily fast conditional-dynamics
steps without using any entangled auxiliary states at all?
Thereby, the intermediate detections upon which the conditional
dynamics relies are not supposed to be finite either, but they
shall be arbitrarily weak. In other words, instead of using
arbitrarily expensive auxiliary states of arbitrarily many photons
and a finite number of finite measurements plus feedforward
\cite{KLM01}, we employ infinitely many steps of feedforward
alone, based upon arbitrarily weak measurements. In fact, a nice
example for the latter approach was already given by Dolinar
\cite{Dolinar73} in the field of
quantum communication and detection theory, namely,
for the minimum error discrimination of binary coherent signals.

As an extension of the conventional signal detection theory,
quantum detection theory has been studied to give an optimal
signal decision strategy for noncommutative quantum signals
\cite{Helstrom_QDET}. It is motivated by fundamental interest, but
it also aims at investigating ultimate performance of optical
communication systems where information is usually carried by
coherent-state signals. Now, in this context, the simplest
scenario would be a communication scheme that is based on
the discrimination of the binary phase-shift keyed (BPSK) coherent
signals $\{ |\alpha\rangle,\,|\!-\! \alpha\rangle \}$ (in the
following, simply called the binary coherent signals). The optimal
POVM that discriminates these nonorthogonal signal states with the
minimum average error is described by a projection onto the
orthonormal basis consisting of superposition states of
$|\alpha\rangle$ and $|\!-\!\alpha\rangle$. This scheme is
sometimes called the Helstrom measurement \cite{Helstrom_QDET}.
Kennedy \cite{Kennedy73} first showed a simple physical model that
achieves the near-optimal measurement: the binary signal is
displaced to $\{|2\alpha\rangle,\,|0\rangle\}$ and then measured
by a photodetector that discriminates whether the signal contains
photons or not. Dolinar \cite{Dolinar73} (see also
\cite{Helstrom_QDET}) extended the Kennedy scheme demonstrating
that the perfect implementation of the Helstrom measurement is
possible by using linear optics, photon counting, and {\it
infinitesimally fast} feedforward. Other related proposals on the
implementation of the (near-)optimal measurement of binary
coherent states were given in
Refs.\cite{Bondurant93,Sasaki96_1,Sasaki96_2}.

A brief description of Dolinar's original proposal is the
following \cite{Dolinar73}. As shown in Fig.~\ref{fig:Dolinar},
the coherent signal field $s_i(t)$ ($i=0,1$) within the time
interval $T$ is displaced by one of the two local oscillators
(LOs) $\beta_0(t)$ and $\beta_1(t)$ and then incident into a
photodetector. The photodetector is assumed to have infinitesimal
time resolution and the functions $\beta_0(t)$ and $\beta_1(t)$
are appropriately chosen to minimize the error probability. The
choice of $\beta_0(t)$ or $\beta_1(t)$ at time $t'$ ($0\le t' <T$)
depends on whether the totally detected number of photons during
$[0,t')$ is even or odd. Therefore, once a signal photon triggers
a detector click, the current LO immediately has to be switched to
the other setting. After detecting the whole signal, one can infer
whether the signal was $s_0(t)$ or $s_1(t)$ by looking at the parity
of the number of the detected photons.
This scheme achieves the minimum error probability
after making an optimization of the likelyhood ratio of the binary signal
with optimal control theory \cite{Dolinar73,Geremia04}.
In the original paper \cite{Dolinar73}, the analysis was semiclassical
and later a fully quantum mechanical description was given \cite{Holevo82}.
More recently, the system performance under the realistic situation
including delay of feedforward, finite bandwidth and imperfect detection
has been investigated \cite{Yamazaki91,Geremia04}.

Compared to previous works on the Dolinar scheme, our contribution
here contains basically four new aspects. First, we revisit the
derivation of a Dolinar-type measurement scheme. However, instead
of focusing on the original signals $\{
|\alpha\rangle,\,|-\alpha\rangle \}$, we simply consider the
projection measurement onto the basis $\{
|\omega_0\rangle,\,|\omega_1\rangle \}$ that corresponds to the
minimum error discrimination of the signal states. We can then ask
whether one can discriminate the orthogonal states $\{
|\omega_0\rangle,\,|\omega_1\rangle \}$ via linear optics and
infinitely weak detections. In such a scheme, during the entire
measurement, the conditional states in every detection and
feedforward step must remain orthogonal. Using this constraint,
the derivation of the Dolinar protocol becomes simpler and more
transparent. Further, no complicated optimization procedures are
needed. In order to discuss the Dolinar receiver in the context of
linear-optics quantum information processing, we translate
Dolinar's original scheme from the time domain to the spatial
domain. In other words, infinitesimally fast feedforward is
replaced by an infinite use of spatial resources.

\begin{figure}
\begin{center}
\includegraphics[width=80mm]{dolinar.eps}   %
\caption{\label{fig:Dolinar}The Dolinar receiver. The signal
$s_i(t)$ and the local oscillator are combined at a highly
transmissive beamsplitter such that the signal is displaced to
$s_i(t)+\beta_n(t)$. The appropriate choice of $\beta_n(t)$ at
$t'$ is determined by the number of the detected photons during
$[0, t')$ (For details, see the text). }
\end{center}
\end{figure}

Secondly, we prove that there is no finite detection scheme that
attains the Helstrom bound for the minimum error discrimination of
binary coherent states. Such a no-go statement can be made by
using a set of criteria for the {\it exact} discrimination of
orthogonal states in a projection measurement \cite{vanLoock03}.
These criteria express in a simple way the requirement that the
orthogonal states must remain orthogonal after a linear-optics
transformation followed by the detection of a first mode. For
instance, in a Bell measurement for polarization-encoded photonic
qubits, this requirement can never be met and hence the Bell
measurement cannot be implemented with linear optics including
photon counting, finite steps of conditional dynamics, and
arbitrary auxiliary photon states
\cite{Luetkenhaus99,Calsamiglia01,vanLoock03}. Asymptotic schemes
\cite{KLM01} are not included in this approach. For the example of
binary coherent states investigated here, finite feedforward means
that the linear-optics transformation of the signal and auxiliary
modes before the first detection step involves a nonzero mixing
between the signal mode and the mode to be detected. Of course, in
the case of zero mixing, the orthogonality can be trivially
preserved. Here we will show how to accomplish the Helstrom
measurement based on feedforward using infinitely weak detections.

A third new aspect of our work is to demonstrate that the Dolinar
approach can be applied not only to the Helstrom measurement of
binary coherent states, but also to other measurements. In
particular, we apply the Dolinar approach to a projection
measurement in the regime of photonic-qubit signals. For a
particular example, for which any finite linear-optics scheme must
fail, we show that the Dolinar approach succeeds.

Last but not least, our analysis of the Dolinar scheme in the view
of the recent developments in linear-optics quantum information
processing demonstrates that by using infinitely many steps of
feedforward instead of arbitrarily expensive entangled photon
states, an asymptotically perfect efficiency of a projection
measurement is possible. Related to this, we note that there is
another proposal for an asymptotic linear-optics implementation of
a quantum measurement in which suboptimal unambiguous state
discrimination (USD) of $N$ symmetric coherent states for $N \ge
3$ is achieved without expensive auxiliary resources
\cite{vanEnk02}. For $N=2$, the optimal USD can be easily done
non-asymptotically by using a $50/50$ beamsplitter and photon
counting \cite{Huttner95}.

\section{Continuous Measurement via Linear Optics and Conditional Displacements}
In this section, we describe a linear-optics circuit in which a
sequence of arbitrarily weak measurements are asymptotically
combined to a continuous measurement. In particular, we discuss
the linear-optics circuit that corresponds to the Dolinar receiver
in the spatial domain. A fully quantum-mechanical description of
the Dolinar receiver was described in Ref.~\cite{Holevo82} by
applying a continuous photon-counting measurement based on the
quantum Markov process model \cite{Srinivas81,Srinivas82}.

In the following, we describe the continuous measurement based on
photon counting with a local oscillator by a sequence of linear
optics and photodetection. The equivalence between continuous
photon counting process and a sequence of beamsplitters and
photodetectors was firstly pointed out in Ref.~\cite{Ban94}. A
schematic of our measurement model is shown in
Fig.~\ref{fig:ContinuousMeasurement}(b). The scheme consists of a
sequence of weak photodetection steps. In each measurement step, a
small fraction of the signal state is reflected by a beamsplitter
and measured by a photodetector after a displacement operation
[Fig.~\ref{fig:ContinuousMeasurement}(a)]. The photodetector
counts the photon number of the signal.

\begin{figure}
\begin{center}
\includegraphics[width=80mm]{cont_meas.eps}   %
\caption{\label{fig:ContinuousMeasurement} Continuous photon
counting with displacement operations. (a) Photodetection with a
displacement operation. The beamsplitter is parametrized by
$\theta$. (b) A sequential photodetection with displacement
operations in the limit of small $\theta$ (small reflectance) and
a large number of measurement steps. }
\end{center}
\end{figure}

After one measurement step, the output state conditioned upon the
number of detected photons $k$ is described by
\begin{equation}
\label{eq:in-out}
\hat{\rho}_{\rm out}^{(k)} =
\frac{ {\rm Tr_{\it B}}\, \left[
( \hat{I}^A \otimes \hat{\Pi}_k^B ) \, \hat{\rho}'_{\rm out} \right]
}{
{\rm Tr_{\it A,B}}\, \left[
( \hat{I}^A \otimes \hat{\Pi}_k^B ) \, \hat{\rho}'_{\rm out} \right] } ,
\end{equation}
where
\begin{eqnarray}
\hat{\rho}'_{\rm out} & = &
\hat{D}^B (\beta \sin\theta)
\hat{B}^{AB} (\theta)
\left( \hat{\rho}_{\rm in}^A \otimes |0^B\rangle\langle0^B|
\right)
\nonumber\\
& & \times \hat{B}^{\dagger AB} (\theta)
\hat{D}^{\dagger B} (\beta \sin\theta) ,
\end{eqnarray}
and $\hat{\rho}_{\rm in}$ is the input state. Tr$_A$, Tr$_{A,B}$
denote the trace operations over the mode $A$ and the modes $A$
and $B$, respectively. $\hat{B}^{AB} (\phi)= \exp[
\phi(\hat{a}^{\dagger} \hat{b} - \hat{a} \hat{b}^{\dagger}) ]$ and
$\hat{D}^B (\alpha)= \exp( \alpha \hat{b}^{\dagger} - \alpha^*
\hat{b} )$ are the operators for a beamsplitter and a displacement
of mode $B$, where $\{ \hat{a},\,\hat{a}^{\dagger} \}$ and $\{
\hat{b},\,\hat{b}^{\dagger} \}$ are the annihilation and creation
operators for modes $A$ and $B$, respectively. Although
photodetection is described by a set of projection operators on
the Fock basis $\{ \hat{\Pi}_k = |k \rangle\langle k| \}$, we
assume that the parameter $\theta$ is sufficiently small such that
the probabilities of detecting more than two photons are
negligible. The complex number $\beta$ in the displacement
operator will be appropriately determined later. As it is well
known, this displacement operation can be realized using a
beamsplitter $\hat{B}(\theta)$ and a local oscillator of amplitude
$\beta$. For a pure state input $\hat{\rho}_{\rm in} = |\psi_{\rm
in}\rangle\langle\psi_{\rm in}|$, $\hat{\rho}'_{\rm
out}=|\psi'_{\rm out}\rangle\langle\psi'_{\rm out}|$ is given by
\begin{eqnarray}
\label{eq:out_dash1} |\psi'_{\rm out}\rangle & = & \hat{D}^B
(\beta \sin\theta) \hat{B}^{AB} (\theta) |\psi_{\rm in}^A\rangle
|0^B\rangle
\\
& = & e^{-\frac{1}{2}|\beta|^2 \sin^2\theta}
e^{\hat{b}^{\dagger}\beta \sin\theta} e^{- \hat{b}\,\beta^*
\sin\theta}
e^{-\hat{a}\hat{b}^{\dagger}\tan\theta} \nonumber\\
&& \times \, e^{\hat{a}^{\dagger} \hat{a}\ln\cos\theta}
e^{-\hat{b}^{\dagger} \hat{b}\ln\cos\theta}
e^{\hat{a}^{\dagger}\hat{b}\tan\theta} |\psi_{\rm in}^A\rangle
|0^B\rangle ,\nonumber
\end{eqnarray}
where we applied the Baker-Campbell-Hausdorff (BCH) formula and
for $\hat{B}^{AB}(\theta)$ an anti-normally ordered decomposition
formula such that $e^{\theta (\hat{a}^{\dagger} \hat{b} - \hat{a}
\hat{b}^{\dagger})} = e^{-\hat{a}\hat{b}^{\dagger}\tan\theta}
e^{\ln\cos\theta \, (\hat{a}^{\dagger}\hat{a} -
\hat{b}^{\dagger}\hat{b})}
e^{\hat{a}^{\dagger}\hat{b}\tan\theta}$. Throughout, we use the
notation $|\psi_{\rm in}^A\rangle|0^B\rangle \equiv |\psi_{\rm
in}^A\rangle\otimes|0^B\rangle$ etc.

Equation~(\ref{eq:out_dash1}) can be further simplified to
\begin{eqnarray}
\label{eq:out_dash} |\psi'_{\rm out}\rangle & = &
e^{-\frac{1}{2}|\beta|^2 \sin^2\theta} e^{(\beta \sin\theta -
\hat{a}\, \tan\theta) \hat{b}^{\dagger}} \nonumber\\ & & \times \,
e^{\hat{a}\, \beta^*  \frac{\sin^2\theta}{\cos\theta} }
e^{\hat{a}^{\dagger} \hat{a} \ln\cos\theta} |\psi_{\rm
in}^A\rangle|0^B\rangle .
\end{eqnarray}
Here we applied again the BCH formula such that $e^{-
\hat{b}\,\beta^* \sin\theta}
e^{-\hat{a}\hat{b}^{\dagger}\tan\theta} =
e^{-\hat{a}\hat{b}^{\dagger}\tan\theta} e^{\hat{a}\, \beta^*
\frac{\sin^2\theta}{\cos\theta} } e^{- \hat{b}\,\beta^*
\sin\theta}$, and we used $\hat{b}^{\dagger}\hat{b}\,|0^B\rangle =
\hat{b}\,|0^B\rangle = 0$.

Let us assume that the photodetector detects at most one photon.
We will now consider the two possible outcomes, i.e.
$\hat{\rho}'_{\rm out}$ is projected on $\hat{\Pi}_0$ or
$\hat{\Pi}_1$. Since these projectors are rank one operators, the
conditional output for each operator can be described by
\begin{equation}
\label{eq:two_outcomes}
\hat{\rho}_{\rm out}^{(k)} = \frac{
\hat{M}^{(k)} \hat{\rho}_{\rm in} \hat{M}^{(k)\dagger} }{
{\rm Tr_A}\, [
\hat{M}^{(k)} \hat{\rho}_{\rm in} \hat{M}^{(k)\dagger} ] }
\quad (k=0,1) ,
\end{equation}
where $\hat{M}^{(k)}$ is a Kraus operator for the $k$-photon
detection event. When no photon is detected at the photodetector,
the conditional output is given by
\begin{equation}
\label{eq:no-count-process} \hat{M}^{(0)} |\psi_{\rm in}\rangle =
e^{- \frac{1}{2} |\beta|^2 \sin^2\theta} e^{\hat{a}\, \beta^*
\frac{\sin^2\theta}{\cos\theta} } e^{\hat{a}^{\dagger} \hat{a} \ln
\cos \theta } |\psi_{\rm in}\rangle .
\end{equation}
Here we consider $N$ successive operations of
Eq.~(\ref{eq:no-count-process}) with the displacement
$\hat{D}(\beta_n \sin\theta)$ representing the $n$-th operation.
For example, for $N=2$, we have
\begin{widetext}
\begin{eqnarray}
\label{eq:twooperations} \hat{M}^{(0)}_2 \hat{M}^{(0)}_1
|\psi_{\rm in}\rangle
 =  \exp \left[- \frac{1}{2} (|\beta_1|^2 + |\beta_2|^2)
 \sin^2\theta \right]
\exp \left[\hat{a}\, \beta_2^* \frac{\sin^2\theta}{\cos\theta}
\right] \exp \left[\hat{a}\, \beta_1^*
\frac{\sin^2\theta}{\cos\theta} \,e^{-\ln \cos \theta}\right] \exp
\left[\hat{a}^{\dagger} \hat{a} \ln \cos^2 \theta \right]
|\psi_{\rm in}\rangle .\nonumber\\
\end{eqnarray}
Here we inserted the identity $e^{-\hat{a}^{\dagger} \hat{a} \ln
\cos \theta }e^{\hat{a}^{\dagger} \hat{a} \ln \cos \theta }$ and
then used $e^{\phi\hat{a}^{\dagger} \hat{a}} \,\hat{a}\,
e^{-\phi\hat{a}^{\dagger} \hat{a}}=\hat{a} e^{-\phi}$. For $N$
successive operations, the output is then given by
\begin{eqnarray}
\label{eq:N_times_no_count} \hat{M}^{(0)}_N \hat{M}^{(0)}_{N-1}
\cdots \hat{M}^{(0)}_1 |\psi_{\rm in}\rangle = \exp \left[
-\frac{1}{2} \sum^N_{n=1} |\beta_n|^2 \sin^2\theta \right] \exp
\left[ \hat{a} \sum^N_{n=1} \beta_n^* \cos^n \theta
\frac{\sin^2\theta}{\cos^{N+1} \theta} \right] \exp \left[
\hat{a}^{\dagger} \hat{a} \ln \cos^N \theta \right] |\psi_{\rm
in}\rangle .
\end{eqnarray}
Let us define the constant parameter
$L=N \theta^2$ and take the limit $\theta^2 \to 0$ and
$N \to \infty$. The displacement operations may now be described
in a continuous way via
\begin{eqnarray}
\label{eq:continuous_displacement}
& & \sum^N_{n=1} |\beta_n|^2 \sin^2\theta \approx
\sum^N_{n=1} |\beta_n|^2 \frac{L}{N} \to
\int^L_0 {\rm d}l \, |\beta (l)|^2 ,
\nonumber\\ & &
\sum^N_{n=1} \beta^*_n \cos^n \theta  \frac{\sin^2\theta}{\cos\theta} \approx
\sum^N_{n=1} \beta^*_n (\cos^N \theta)^{\frac{n}{N}} \theta^2 =
\sum^N_{n=1} \beta^*_n e^{-\frac{nL}{2N}} \frac{L}{N} \to
\int^L_0 {\rm d}l \, \beta^* (l) e^{-\frac{l}{2}} .
\end{eqnarray}
Using these relations, we obtain the output state
\begin{eqnarray}
\label{eq:0_L_no_count}
\hat{M}^{(0)}_N \hat{M}^{(0)}_{N-1} \cdots \hat{M}^{(0)}_1
|\psi_{\rm in}\rangle & = &
\exp \left[ -\frac{1}{2} \int^L_0 {\rm d}l \, |\beta(l)|^2 \right]
\exp \left[ \hat{a} e^{\frac{L}{2}}
\int^L_0 {\rm d}l \, \beta^*(l) e^{-\frac{l}{2}} \right]
\exp \left[ -\frac{L}{2} \hat{a}^{\dagger} \hat{a} \right]
|\psi_{\rm in}\rangle .
\end{eqnarray}
More generally, when no photon is detected from the $L_0$-th
detector to the $L_1$-th detector, we obtain the output state
\begin{eqnarray}
\label{eq:output_S_L1-L0}
\hat{\rho}_{\rm out} = \frac{
\hat{M}^{(0)}_{L_1} \hat{M}^{(0)}_{L_1-1} \cdots \hat{M}^{(0)}_{L_0}
\hat{\rho}_{\rm in}
\hat{M}^{(0)\dagger}_{L_0} \cdots
\hat{M}^{(0)\dagger}_{L_1-1} \hat{M}^{(0)\dagger}_{L_1} }{
{\rm Tr}\, [
\hat{M}^{(0)}_{L_1} \hat{M}^{(0)}_{L_1-1} \cdots \hat{M}^{(0)}_{L_0}
\hat{\rho}_{\rm in}
\hat{M}^{(0)\dagger}_{L_0} \cdots
\hat{M}^{(0)\dagger}_{L_1-1} \hat{M}^{(0)\dagger}_{L_1} ] }
= \frac{
\hat{S}_{L_1-L_0} \hat{\rho}_{\rm in} \hat{S}_{L_1-L_0}^{\dagger} }{
{\rm Tr}\, [
\hat{S}_{L_1-L_0} \hat{\rho}_{\rm in} \hat{S}_{L_1-L_0}^{\dagger} ] },
\end{eqnarray}
where
\begin{equation}
\label{eq:S_L1-L0}
\hat{S}_{L_1-L_0} =
\exp \left[ \hat{a} e^{\frac{L_1-L_0}{2}}
\int^{L_1}_{L_0} {\rm d}l \, \beta^*(l) e^{-\frac{l}{2}} \right]
\exp \left[ -\frac{L_1-L_0}{2} \hat{a}^{\dagger} \hat{a} \right] .
\end{equation}

However, when a photon is detected at the $L$-th photodetector,
the conditional operation is described by
\begin{equation}
\label{eq:count-process} \hat{M}^{(1)}_L |\psi_{\rm in}\rangle =
\exp \left[ -\frac{1}{2} |\beta_L|^2 \sin^2\theta \right] (
\beta_L \sin\theta - \hat{a} \tan\theta ) \exp \left[ \hat{a}
\beta_L^* \frac{\sin^2\theta}{\cos\theta} \right] \exp \left[
\hat{a}^{\dagger} \hat{a} \ln \cos \theta \right] |\psi_{\rm
in}\rangle .
\end{equation}
\end{widetext}
Upon taking the limit $\theta^2 \to 0$ and using the continuous
representation given in Eq.~(\ref{eq:continuous_displacement}),
all the exponential terms in Eq.~(\ref{eq:count-process}) approach
unity. The output now becomes
\begin{equation}
\label{eq:output_J_L}
\hat{\rho}_{\rm out} = \frac{
\hat{M}^{(1)}_L \hat{\rho}_{\rm in} \hat{M}^{(1)\dagger}_L }{
{\rm Tr}\, [
\hat{M}^{(1)}_L \hat{\rho}_{\rm in} \hat{M}^{(1)\dagger}_L ] }
= \frac{
\hat{J}_L \hat{\rho}_{\rm in} \hat{J}^{\dagger}_L }{
{\rm Tr}\, [
\hat{J}_L \hat{\rho}_{\rm in} \hat{J}^{\dagger}_L ] } ,
\end{equation}
where $ \hat{M}_L^{(1)} = \theta \hat{J}_L  $ and
\begin{equation}
\label{eq:J_L}
\hat{J}_L = \beta(L) - \hat{a} .
\end{equation}

Using these expressions, a continuous measurement based on
beamsplitting, displacement operations, and photon counting is
described via the operator
\begin{equation}
\label{eq:SJSJ_}
\hat{S}_{L_{n}-L_{n-1}} \hat{J}_{L_{n-1}} \hat{S}_{L_{n-1}-L_{n-2}}
\cdots \hat{J}_{L_1} \hat{S}_{L_1-0} .
\end{equation}

This expression can also be obtained
as a solution of the master equation of the system and
this kind of conditional dynamics is called quantum jump process
in terms of quantum statistics theory \cite{QuantumNoise}.
Although analytical solutions can be obtained in our case,
it should be noted when the system has complicated quantum jump e.g.
the measurement has continuous outcome, one effective approach is
the stochastic unraveling of the master equation.
One of the successful applications of this approach is shown in
Ref.~\cite{Wiseman93}.

In the next section, we will show how a given projective
measurement can be implemented via the apparatus discussed here.
Finally, we note that when $L$ is replaced by a time parameter
$T$, our formulation is equivalent to the conventional time domain
continuous measurement model including displacement operations
\cite{Holevo82}.

\section{Projective measurements via continuous measurement}

Projective measurements represent an important special case among
the generalized measurements. In this section, we show that the
Dolinar receiver can be systematically derived from an
orthogonality condition, similar to that used for analyzing the
exact distinguishability of orthogonal states in a projection
measurement \cite{vanLoock03}. First, we will apply this approach
to the original Dolinar receiver, that is, the discrimination of
binary coherent states. In a further example, we
examine a binary projection measurement onto a photonic-qubit
basis, which otherwise cannot be implemented with linear optics
including photon counting, finite steps of conditional dynamics,
and arbitrary auxiliary photon states.

Before discussing particular examples, we briefly summarize our
approach. The problem of implementing a complete projection
measurement $\{ \Pi_i = |\pi_i\rangle\langle\pi_i| \}$ can be
regarded as the problem of an exact discrimination of the
orthogonal signal states $|\pi_i\rangle$ \cite{vanLoock03}. In
order to achieve an exact discrimination, these signal states,
when conditionally transformed via partial measurements and
feedforward, must remain orthogonal after each step of the
intermediate measurements. In our approach, these measurements are
assumed to be arbitrarily weak, asymptotically corresponding to a
continuous measurement. Via the orthogonality constraint, we can
infer the input signal by counting the total number of detected
photons $N_{\rm tot}$. In the limit of infinitely many photodetection steps,
the final state must be in a vacuum state $|0\rangle$ due to the
energy loss at each step \cite{energyloss}.
Combining this fact with the condition
that the signals always remain mutually orthogonal, we know that
every possible result of the $N_{\rm tot}$-photon detection can be triggered
only by one of the two signal states with nonzero probability.
Eventually, we can infer the input signal state perfectly by
counting $N_{\rm tot}$.

\subsection{Minimum error discrimination
of binary coherent states}

The minimum error detection of the binary coherent states $\{
|\alpha\rangle, \, |-\alpha\rangle \}$ is achieved via the
projection operators corresponding to the orthogonal states
\begin{eqnarray}
\label{eq:Helstrom_measurement}
|\omega_0\rangle & = & \sqrt{\frac{1-P_e}{1-\kappa^2}} |\alpha\rangle
                 - \sqrt{\frac{P_e}{1-\kappa^2}} |\!-\!\alpha\rangle , \\
|\omega_1\rangle & = & \sqrt{\frac{P_e}{1-\kappa^2}} |\alpha\rangle
                 - \sqrt{\frac{1-P_e}{1-\kappa^2}} |\!-\!\alpha\rangle ,
\end{eqnarray}
where $\kappa=|\langle\alpha| \! - \! \alpha\rangle|$ and
\begin{equation}
\label{eq:min_err_probability}
P_e = \frac{1}{2} \left( 1-\sqrt{1-\kappa^2} \right) ,
\end{equation}
is the minimum error probability. For the sake of simplicity, we
assume that the a priori probabilities for the signals are equal.

Let us first prove that the projection onto
$\{|\omega_0\rangle,\,|\omega_1\rangle\}$ cannot be implemented
using linear optics, finite steps of conditional
dynamics, and arbitrary auxiliary states.
As for the detection mechanisms, we may restrict ourselves
to photon counting since homodyne detection with linear optics
never leads to non-Gaussian operation,
as required for our projection measurement.
Now, using the criteria for exact discrimination of orthogonal states
\cite{vanLoock03}, one finds that already the detection of a first
output mode, after mixing the signal mode with the auxiliary
modes, inevitably destroys the orthogonality. Defining an
arbitrary auxiliary state $|A\rangle$ with arbitrarily many modes,
the necessary conditions for preserving the orthogonality after
such a first detection (and hence for potentially enabling one to
exactly discriminate the states via further detections and
conditional transformations) are \cite{vanLoock03}
\begin{eqnarray}\label{criteria}
\langle A|\langle\omega_0| (\hat c^\dagger)^n\hat c^n
|\omega_1\rangle|A\rangle = 0,\quad \forall n=0,1,2,...\nonumber\\
\end{eqnarray}
Here, the 0th order ($n=0$) just corresponds to the orthogonality
of the signal states.
The annihilation operator
$\hat c$ represents the first mode being detected after the
linear-optics transformation. This output mode may be decomposed
into two parts of which one refers to the signal mode and the
other one to the auxiliary modes,
\begin{eqnarray}\label{decompose}
\hat c = \nu_1\hat a_1 + b_{\rm aux} \hat c_{\rm aux} + \gamma\,.
\end{eqnarray}
Here, $\nu_1\equiv U_{j1}$ is the complex entry of the unitary
matrix $U$ for describing the linear-optics mixing of the mode $j$
to be detected with the signal mode $1$. The mixing of mode $j$
with the auxiliary modes due to linear optics is described by the
annihilation operator $\hat c_{\rm aux}$, where $b_{\rm aux}$ is a
real parameter. The complex parameter $\gamma$ enables us to
include the possibility of phase-space displacements before the
detection \cite{vanLoock04inprep}. Note that without including
displacements and for signal states with a fixed number of
photons, arbitrary auxiliary states $|A\rangle$ cannot help to
provide nontrivial solutions to the conditions in
Eq.~(\ref{criteria}), if there are only trivial solutions without
an extra state $|A\rangle$ \cite{vanLoock03}. For the projection
onto $\{|\omega_0\rangle,\,|\omega_1\rangle\}$, however, the
signal states have an unfixed photon number. Thus, using an
auxiliary state $|A\rangle$ having unfixed number too
\cite{vanLoock03}, or employing a nonzero phase-space displacement
$\gamma\neq 0$ may indeed help.

After some algebra, for the first-order condition $n=1$ from
Eq.~(\ref{criteria}) using Eq.~(\ref{decompose}), we obtain
\begin{equation}\label{finalcondition}
|\nu_1|^2\,|\alpha|^2 = i \sqrt{1-\kappa^2}\,{\rm Im}\,\delta\,,
\end{equation}
where $\alpha\neq 0$ is the complex amplitude given by the signal
states, the parameter $\kappa=|\langle\alpha| \! - \!
\alpha\rangle|$ is also defined through the signal states, and
${\rm Im}\,\delta$ is the imaginary part of
\begin{equation}
\delta\equiv \nu_1^*\alpha^* (\gamma + \langle A|b_{\rm aux} \hat
c_{\rm aux}|A\rangle)\,.
\end{equation}
Since $\kappa^2 < 1$ for $\alpha\neq 0$, one can easily see that
the only solution to the condition in Eq.~(\ref{finalcondition})
is trivial, $\nu_1 = 0$. In other words, only if there is no
mixing at all between the mode $j$ to be detected and the signal
mode $1$, the orthogonality is (trivially) preserved. For any
finite mixing between the modes $j$ and $1$, exact discrimination
of the signal states is no longer possible. In our approach based
upon continuous measurement, every single conditional-dynamics
step is supposed to be arbitrarily weak corresponding to the limit
$\nu_1\to 0$. Thus, in this limit, the orthogonality condition may
be satisfied. However, typically, such a scheme does not provide
any information about the input signal states. Yet in the
following, we demonstrate that by combining infinitely many
arbitrarily weak detections of the signal mode, corresponding to a
continuous measurement of the signal mode, eventually perfect
state discrimination can be accomplished.

Suppose that the states $\{|\omega_0\rangle,\,|\omega_1\rangle\}$
are sent into the continuous measurement apparatus discussed in
the preceding section. When no photon is counted during
$[0,\,L_1)$, the signals evolve as
\begin{eqnarray}
\label{eq:no-count_Opt1}
\hat{S}_{L_1-0} |\omega_0\rangle & = &
\sqrt{\frac{1-P_e}{1-\kappa^2}} \,
\exp\left[ B \right]
\, |\alpha e^{-\frac{L_1}{2}}\rangle
\nonumber\\ & &
- \sqrt{\frac{P_e}{1-\kappa^2}} \,
\exp\left[ -B \right]
\, |\!-\!\alpha e^{-\frac{L_1}{2}}\rangle ,
\nonumber\\
\\
\label{eq:no-count_Opt2}
\hat{S}_{L_1-0} |\omega_1\rangle & = &
\sqrt{\frac{P_e}{1-\kappa^2}} \,
\exp\left[ B \right]
\, |\alpha e^{-\frac{L_1}{2}}\rangle
\nonumber\\ & &
- \sqrt{\frac{1-P_e}{1-\kappa^2}} \,
\exp\left[ -B \right]
\, |\!-\!\alpha e^{-\frac{L_1}{2}}\rangle ,
\nonumber\\
\end{eqnarray}
where
\begin{equation}
\label{eq:B}
B = \alpha \int^{L_1}_0 {\rm d}l \, \beta^*(l) e^{-\frac{l}{2}} .
\end{equation}
We note that these states are unnormalized. By calculating the
inner product between Eqs.~(\ref{eq:no-count_Opt1}) and
(\ref{eq:no-count_Opt2}), we can find the condition that these
signals are orthogonal,
\begin{eqnarray}
\label{eq:Dolinar_inner_product}
& & \sqrt{P_e(1-P_e)} \left\{
\exp\left[ B+B^* \right]
+\exp\left[ -(B+B^*) \right]
\right\} \nonumber\\
& & - \left\{ P_e \exp\left[ B-B^* \right]
+ (1-P_e) \exp\left[ -(B-B^*) \right] \right\}
\nonumber\\ & &
\times \exp \left[ -2 |\alpha|^2 e^{-L_1} \right] = 0 .
\end{eqnarray}
From the orthogonality condition
Eq.~(\ref{eq:Dolinar_inner_product}) with Eq.~(\ref{eq:B}), we
obtain the function for the displacement operation
\begin{equation}
\label{eq:LO}
\beta(l) = \pm \frac{ \alpha e^{-\frac{l}{2}} }{
\sqrt{ 1 - \exp[-4|\alpha|^2 (1-e^{-l})] } } .
\end{equation}
By defining the two solutions in Eq.~(\ref{eq:LO}) as
$\beta_+(l),\,\beta_-(l)$, respectively, we can derive the relations
\begin{eqnarray}
\label{eq:beta+}
\exp\left[ \alpha \int^{L_{j+1}}_{L_j} {\rm d}l \,
\beta_+(l) e^{-\frac{l}{2}}
\right] & = &
\frac{e^{-|\alpha|^2 e^{-L_{j+1}}}}{e^{-|\alpha|^2 e^{-L_{j}}}}
\frac{\sqrt{1-P_e^{L_{j+1}}}}{\sqrt{1-P_e^{L_{j}}}} ,
\nonumber\\
\end{eqnarray}
\begin{eqnarray}
\label{eq:beta-}
\exp\left[ \alpha \int^{L_{j+1}}_{L_j} {\rm d}l \,
\beta_-(l) e^{-\frac{l}{2}}
\right] & = &
\frac{e^{-|\alpha|^2 e^{-L_{j+1}}}}{e^{-|\alpha|^2 e^{-L_{j}}}}
\frac{\sqrt{P_e^{L_{j+1}}}}{\sqrt{P_e^{L_{j}}}} ,
\nonumber\\
\end{eqnarray}
where
\begin{equation}
P_e^{L} =
\frac{1}{2} \left( 1-\sqrt{1-\exp[-4|\alpha|^2 (1-e^{-L})]} \right) .
\end{equation}

In the following, we choose $\beta_+(l)$ as a displacement
function for $[0,\,L_1)$ ($\beta_-(l)$ leads to the same
conclusions) and assume that a photon is detected at the $L_1$-th
detector. The equations (\ref{eq:no-count_Opt1}),
(\ref{eq:no-count_Opt2}), (\ref{eq:beta+}), and (\ref{eq:beta-}) yield
\begin{eqnarray}
\label{eq:no-count_omega1(2)}
\hat{S}_{L_1-0} |\omega_0\rangle & \propto &
\sqrt{1-P_e} \frac{\sqrt{1-P_e^{L_1}}}{\sqrt{1-P_e^{0}}}
| \alpha e^{-\frac{L_1}{2}}\rangle
\nonumber\\
& &
- \sqrt{P_e} \frac{\sqrt{P_e^{L_1}}}{\sqrt{P_e^{0}}}
| \!-\! \alpha e^{-\frac{L_1}{2}}\rangle ,
\\
\label{eq:no-count_omega2(2)}
\hat{S}_{L_1-0} |\omega_1\rangle & \propto &
\sqrt{P_e} \frac{\sqrt{1-P_e^{L_1}}}{\sqrt{1-P_e^{0}}}
|\alpha e^{-\frac{L_1}{2}}\rangle
\nonumber\\
& &
- \sqrt{1-P_e} \frac{\sqrt{P_e^{L_1}}}{\sqrt{P_e^{0}}}
| \!-\! \alpha e^{-\frac{L_1}{2}}\rangle ,
\end{eqnarray}
where unimportant global coefficients are omitted.
As for the states after detecting a photon,
by using the relations
\begin{eqnarray}
\label{eq:J1}
\beta_+(L_1) - \alpha e^{-\frac{L_1}{2}} =
\frac{2 P_e^{L_1} \alpha e^{-\frac{L_1}{2} }}{
\sqrt{1-\exp[-4|\alpha|^2 (1-e^{-L_1})]}} ,
\nonumber\\
\end{eqnarray}
\begin{eqnarray}
\label{eq:J2}
\beta_+(L_1) + \alpha e^{-\frac{L_1}{2}} =
\frac{2 (1-P_e^{L_1}) \alpha e^{-\frac{L_1}{2} }}{
\sqrt{1-\exp[-4|\alpha|^2 (1-e^{-L_1})]}} ,
\nonumber\\
\end{eqnarray}
we find
\begin{eqnarray}
\label{eq:JS0}
\hat{J}_{L_1} \hat{S}_{L_1-0} |\omega_0\rangle
& \propto &
\sqrt{1-P_e} \frac{\sqrt{1-P_e^{L_1}}}{\sqrt{1-P_e^{0}}} P_e^{L_1}
             |\alpha e^{-\frac{L_1}{2}}\rangle
\nonumber\\
& &
- \sqrt{P_e} \frac{\sqrt{P_e^{L_1}}}{\sqrt{P_e^{0}}} (1-P_e^{L_1})
             |\!-\!\alpha e^{-\frac{L_1}{2}}\rangle
\nonumber\\
& \propto &
\sqrt{1-P_e} \frac{\sqrt{P_e^{L_1}}}{\sqrt{1-P_e^{0}}}
             |\alpha e^{-\frac{L_1}{2}}\rangle
\nonumber\\
& &
- \sqrt{P_e} \frac{\sqrt{1-P_e^{L_1}}}{\sqrt{P_e^{0}}}
             |\!-\!\alpha e^{-\frac{L_1}{2}}\rangle ,
\end{eqnarray}
and similarly,
\begin{eqnarray}
\label{eq:JS1}
\hat{J}_{L_1} \hat{S}_{L_1-0} |\omega_1\rangle
& \propto &
\sqrt{P_e} \frac{\sqrt{P_e^{L_1}}}{\sqrt{1-P_e^{0}}}
             |\alpha e^{-\frac{L_1}{2}}\rangle
\nonumber\\
& &
- \sqrt{1-P_e} \frac{\sqrt{1-P_e^{L_1}}}{\sqrt{P_e^{0}}}
             |\!-\!\alpha e^{-\frac{L_1}{2}}\rangle .
\nonumber\\
\end{eqnarray}
Comparing them with Eqs.~(\ref{eq:no-count_omega1(2)}) and
(\ref{eq:no-count_omega2(2)}), we can easily see that these states
are still orthogonal to each other.
Also, we can derive the relations,
\begin{eqnarray}
\label{eq:JS_a+_b+}
& &\hat{J}_{L_{j+1}} \hat{S}_{L_{j+1}-L_j}
|\alpha e^{-\frac{L_j}{2}} \rangle
\nonumber\\
& &= \frac{1}{2}e^{-4|\alpha|^2 (1-e^{-L_{j+1}})}
\frac{\sqrt{P_e^{L_{j+1}}}}{\sqrt{1-P_e^{L_j}}} |\alpha
e^{-\frac{L_{j+1}}{2}} \rangle ,
\\
& &\label{eq:JS_a-_b+}
\hat{J}_{L_{j+1}} \hat{S}_{L_{j+1}-L_j}
|\!-\!\alpha e^{-\frac{L_j}{2}} \rangle
\nonumber\\
& &= \frac{1}{2}e^{-4|\alpha|^2 (1-e^{-L_{j+1}})}
\frac{\sqrt{1-P_e^{L_{j+1}}}}{\sqrt{P_e^{L_j}}}
|\!-\!\alpha e^{-\frac{L_{j+1}}{2}} \rangle ,
\nonumber\\
\end{eqnarray}
for the measurement process
with the displacement function $\beta_+ (l)$, and
\begin{eqnarray}
\label{eq:JS_a+_b-}
& &\hat{J}_{L_{j+1}} \hat{S}_{L_{j+1}-L_j}
|\alpha e^{-\frac{L_j}{2}} \rangle
\\
& &= \frac{1}{2}e^{-4|\alpha|^2 (1-e^{-L_{j+1}})}
\frac{\sqrt{1-P_e^{L_{j+1}}}}{\sqrt{P_e^{L_j}}}
|\alpha e^{-\frac{L_{j+1}}{2}} \rangle ,
\nonumber\\
\label{eq:JS_a-_b-}
& &\hat{J}_{L_{j+1}} \hat{S}_{L_{j+1}-L_j} |\!-\!\alpha e^{-L_j} \rangle
\\
& &= \frac{1}{2}e^{-4|\alpha|^2 (1-e^{-L_{j+1}})}
\frac{\sqrt{P_e^{L_{j+1}}}}{\sqrt{1-P_e^{L_j}}} |\!-\!\alpha
e^{-\frac{L_{j+1}}{2}} \rangle , \nonumber
\end{eqnarray}
for the measurement process
with the displacement function $\beta_- (l)$,
where $(P_e^{L_{j+1}} (1-P_e^{L_{j+1}}))^{1/2}=
e^{-4|\alpha|^2 (1-e^{-L_{j+1}})}/2$

From Eqs. (\ref{eq:beta+}), (\ref{eq:beta-}), (\ref{eq:JS0}) and
(\ref{eq:JS1}), we can determine the local oscillator function
$\beta(l)$ for the next zero photon counting process
$\hat{S}_{L_2-L_1}$. In our case, the orthogonality condition
requires using $\beta_-(l)$ for $\hat{S}_{L_2-L_1}$, as the output
states $\hat{S}_{L_2-L_1}\hat{J}_{L_1}\hat{S}_{L_1-0}
|\omega_i\rangle$ ($i=0,1$) have the similar structure as in
Eqs.~(\ref{eq:JS0}) and (\ref{eq:JS1}), respectively, except that
$L_1$ is replaced by $L_2$. Eventually, one can preserve the
orthogonality by switching the sign of the displacement function
$\beta_{\pm}(l)$ each time a photon is detected. After applying
many of these operations, the signals evolve as
$\hat{S}_{L-L_{N_{\rm tot}}}
\hat{J}_{L_{N_{\rm tot}}}
\hat{S}_{L_{N_{\rm tot}}-L_{{N_{\rm tot}}-1}} \cdots \hat{J}_{L_1}
\hat{S}_{L_1-0} |\omega_i\rangle$, $(i=0,1)$, where $N_{\rm tot}$ is the
total number of detected photons $( N_{\rm tot}=0,1 \cdots ,\infty)$
\cite{finiteN_tot} and we will take an infinitely large $L$.
The final states are now derived with the help of
Eqs.~(\ref{eq:JS_a+_b+}), (\ref{eq:JS_a-_b+}), (\ref{eq:JS_a+_b-}),
and (\ref{eq:JS_a-_b-}),
and classified via the parity of $N_{\rm tot}$.
When $N_{\rm tot}$ is zero or even, the output state for the input
$|\omega_0\rangle$ is given by
\begin{widetext}
\begin{eqnarray}
\label{eq:omega_0_odd}
|\omega_0^L\rangle & = &
\hat{S}_{L-L_{N_{\rm tot}}} \hat{J}_{L_{N_{\rm tot}}}
\hat{S}_{L_{N_{\rm tot}}-L_{{N_{\rm tot}}-1}} \cdots
\hat{J}_{L_1} \hat{S}_{L_1-0}
|\omega_0\rangle
\nonumber\\ & \propto &
\sqrt{1-P_e} \frac{\sqrt{P_e^{L_1}}}{\sqrt{1-P_e^{0}}}
\frac{\sqrt{1-P_e^{L_2}}}{\sqrt{P_e^{L_1}}}
\cdots \frac{\sqrt{1-P_e^L}}{\sqrt{1-P_e^{L_{N_{\rm tot}}}}}
|\alpha e^{-\frac{L}{2}} \rangle
- \sqrt{P_e} \frac{\sqrt{1-P_e^{L_1}}}{\sqrt{P_e^{0}}}
\frac{\sqrt{P_e^{L_2}}}{\sqrt{1-P_e^{L_1}}}
\cdots \frac{\sqrt{P_e^L}}{\sqrt{P_e^{L_{N_{\rm tot}}}}}
|\!-\!\alpha e^{-\frac{L}{2}} \rangle
\nonumber\\ & \propto &
\sqrt{1-P_e} \sqrt{1-P_e^L} |\alpha e^{-\frac{L}{2}} \rangle
- \sqrt{P_e} \sqrt{P_e^L} |\!-\!\alpha e^{-\frac{L}{2}} \rangle
\nonumber\\ & = &
e^{ -\frac{1}{2} |\alpha|^2 e^{-L} }
\sum^{\infty}_{m=0} \frac{1}{\sqrt{m!}} \left\{
\begin{array}{ll}
  \left( \sqrt{1-P_e}\sqrt{1-P_e^L} - \sqrt{P_e}\sqrt{P_e^L} \right)
  \left( \alpha e^{-\frac{L}{2}} \right)^m |m\rangle &
  \qquad \mbox{$m$ is zero or even} \\
  \left( \sqrt{1-P_e}\sqrt{1-P_e^L} + \sqrt{P_e}\sqrt{P_e^L} \right)
  \left( \alpha e^{-\frac{L}{2}} \right)^m |m\rangle &
  \qquad \mbox{$m$ is odd}
\end{array}
\right. \nonumber\\ & = &
\left( \sqrt{1-e^{-4|\alpha|^2}} +
\frac{|\alpha|^2 e^{-4|\alpha|^2}}{ \sqrt{1-e^{-4|\alpha|^2}} } e^{-L}
\cdots \right) |0\rangle + (1+\cdots) \alpha e^{-\frac{L}{2}} |1\rangle
+ \frac{1}{\sqrt{2}} \left( \sqrt{1-e^{-4|\alpha|^2}} + \cdots \right)
\alpha^2 e^{-L} |2\rangle + \cdots, \nonumber\\
\end{eqnarray}
where $|m\rangle$ is the $m$-photon number state and note that
$P_e^0 = 1/2$ and $\lim_{L \to \infty} P_e^L = P_e$. In the last
line, the terms only up to the order of $e^{-L}$ are given.
Similarly, for the input $|\omega_1\rangle$, we obtain
\begin{eqnarray}
\label{eq:omega_1_odd}
|\omega_1^L\rangle & = &
\hat{S}_{L-L_{N_{\rm tot}}} \hat{J}_{L_{N_{\rm tot}}}
\hat{S}_{L_{N_{\rm tot}}-L_{{N_{\rm tot}}-1}} \cdots
\hat{J}_{L_1} \hat{S}_{L_1-0}
|\omega_1\rangle
\nonumber\\ & \propto &
\sqrt{P_e}\sqrt{1-P_e^L} | \alpha e^{-\frac{L}{2}} \rangle
- \sqrt{1-P_e}\sqrt{P_e^L} |\!-\!\alpha e^{-\frac{L}{2}} \rangle
\nonumber\\ & = &
e^{ -\frac{1}{2} |\alpha|^2 e^{-L} }
\sum^{\infty}_{m=0} \frac{1}{\sqrt{m!}} \left\{
\begin{array}{ll}
  \left( \sqrt{P_e}\sqrt{1-P_e^L} - \sqrt{1-P_e}\sqrt{P_e^L} \right)
  \left( \alpha e^{-\frac{L}{2}} \right)^m |m\rangle &
  \qquad \mbox{$m$ is zero or even} \\
  \left( \sqrt{P_e}\sqrt{1-P_e^L} + \sqrt{1-P_e}\sqrt{P_e^L} \right)
  \left( \alpha e^{-\frac{L}{2}} \right)^m |m\rangle &
  \qquad \mbox{$m$ is odd}
\end{array}
\right. \nonumber\\ & = &
\left(
- \frac{|\alpha|^2 e^{-2|\alpha|^2}}{\sqrt{1-e^{-4|\alpha|^2}}} e^{-L}
+ \cdots \right)
|0\rangle + \left( e^{-2|\alpha|^2} + \cdots \right)
\alpha e^{-\frac{L}{2}} |1\rangle + \cdots ,
\end{eqnarray}
\end{widetext}
where the last line also shows the terms up to the order of $e^{-L}$.
From Eqs.~(\ref{eq:omega_0_odd}) and (\ref{eq:omega_1_odd}),
we can check that these two possible outputs are still orthogonal
to each other and, even in the approximated form,
the orthogonality is satisfied in any order of $L$.
For example, the inner product between
$|\omega_0^L\rangle$ and $|\omega_1^L\rangle$ up to the order of $e^{-L}$
is computed as
\begin{eqnarray}
\label{eq:inner_product_approx}
\langle\omega_0^L|\omega_1^L\rangle & \approx &
- \sqrt{1-e^{-4|\alpha|^2}}
\frac{|\alpha|^2 e^{-2|\alpha|^2}}{ \sqrt{1-e^{-4|\alpha|^2}} } e^{-L}
\langle 0|0 \rangle
\nonumber\\ & &
+ |\alpha|^2 e^{-2|\alpha|^2} e^{-L} \langle 1|1 \rangle
\nonumber\\ & = & 0.
\end{eqnarray}

Then, upon taking the limit of $L \to \infty$ and normalizing
Eq.~(\ref{eq:omega_0_odd}), we know that $|\omega_0^L\rangle$
approaches the vacuum state. We also find from
Eq.~(\ref{eq:omega_1_odd}) that the state $|\omega_1^L\rangle$
converges to $|1\rangle$. However, since {\it all} the terms in
Eq.~(\ref{eq:omega_1_odd}) decrease exponentially for large $L$,
we know that $|\omega_1^L\rangle$ does not occur at all in the
limit of $L \to \infty$. Hence we have
\begin{eqnarray}
\label{eq:omega_0_odd_final}
|\omega_0^L\rangle
& \to & |0\rangle ,
\\
\label{eq:omega_1_odd_final}
|\omega_1^L\rangle
& \to & 0 ,
\end{eqnarray}
and thus, if the number of the totally detected photons $N_{\rm
tot}$ is zero or even, we can unambiguously identify the input
state $|\omega_0\rangle$.

On the other hand, when $N_{\rm tot}$ is odd, the final states are given by
\begin{eqnarray}
\label{eq:omega_0_even}
|\omega_0^L\rangle
& \to & 0 ,
\\
\label{eq:omega_1_even}
|\omega_1^L\rangle
& \to & |0\rangle ,
\end{eqnarray}
and hence, in this case, we can unambiguously infer the input
state $|\omega_1\rangle$. Let us finally emphasize that our model
may be also translated into the time domain by replacing the
spatial parameter $l$ by the time $t$. Therefore, our approach
provides an alternative derivation of the original Dolinar
receiver. This approach is relatively simple, because it only
relies upon the preservation of the orthogonality of the signals
during the entire measurement process.

\subsection{Photonic-qubit signals}

In this subsection, we show that our measurement model is not only
applicable to coherent-state signals, but it may also be applied
to other types of signals, for example, photon-number qubits. As
an example, let us consider the projection measurement onto the
basis $|\omega_{0,1}\rangle = ( |0\rangle \pm |1\rangle )/\sqrt{2}$,
for which one can derive a no-go statement in terms of the
criteria of Ref.\cite{vanLoock03}.

As for the exact discrimination of $|\omega_{0,1}\rangle$, let us
first briefly discuss the no-go statement for any linear-optics
scheme based on photon counting, finite steps of conditional
dynamics, and arbitrary auxiliary states. Since the signal states
have an unfixed number of photons, using an auxiliary state
$|A\rangle$ with unfixed photon number or employing phase-space
displacements may help to exactly discriminate the signal states
\cite{vanLoock03}. For the signal states $|\omega_{0,1}\rangle$, the
necessary conditions in any conditional-dynamics scheme become
\cite{vanLoock03}
\begin{eqnarray}\label{criteria2}
\langle A|\langle\omega_0| (\hat c^\dagger)^n\hat c^n
|\omega_1\rangle|A\rangle = 0,\quad \forall n=0,1,2,...
\end{eqnarray}
Here, we use the same definitions as in the preceding section. By
inserting the decomposition of Eq.~(\ref{decompose}) into the
first-order $n=1$ condition, we obtain now
\begin{equation}\label{finalcondition2}
|\nu_1|^2 = 2i \,{\rm Im}\,\delta\,,
\end{equation}
where now ${\rm Im}\,\delta$ is the imaginary part of
\begin{equation}
\delta\equiv \nu_1(\gamma^* + \langle A|b_{\rm aux} \hat c_{\rm
aux}^\dagger|A\rangle)\,.
\end{equation}
Again, only the trivial solution $\nu_1=0$ exists. Thus, there is
no linear-optics scheme based on a finite first detection step
that achieves the exact discrimination of $|\omega_{0,1}\rangle$. Let
us now see how a continuous-measurement based scheme leads to the
perfect discrimination of the signal states.

The design for the measurement apparatus is the same as
that of the preceding subsection. So we send the states
$\{ |\omega_0\rangle,\,|\omega_1\rangle \}$ into the measurement
apparatus. Let us assume that the first photon is detected at the
$L_1$-th detector and no photon is counted during $[0,\,L_1)$. The
outcoming signals are then given by
\begin{eqnarray}
\label{eq:qubit_no-count_1-0}
\hat{S}_{L_1-0} |\omega_0\rangle & = &
\left( 1 + \int^{L_1}_0 {\rm d}l \, \beta^*(l) e^{-\frac{l}{2}} \right)
|0\rangle + e^{-\frac{L_1}{2}} |1\rangle ,
\nonumber\\
\\
\label{eq:qubit_no-count_1-1}
\hat{S}_{L_1-0} |\omega_1\rangle & = &
\left( 1 - \int^{L_1}_0 {\rm d}l \, \beta^*(l) e^{-\frac{l}{2}} \right)
|0\rangle - e^{-\frac{L_1}{2}} |1\rangle .
\nonumber\\
\end{eqnarray}
The orthogonality condition for Eqs.~(\ref{eq:qubit_no-count_1-0})
and (\ref{eq:qubit_no-count_1-1}) implies an appropriate local
oscillator function for $\{ |\omega_0\rangle,\,|\omega_1\rangle \}$,
namely
\begin{equation}
\label{eq:qubit_LO_L_1-0}
\beta_{L_1-0} (l) = \pm \frac{ e^{-\frac{l}{2}} }{
2 \sqrt{ 1-e^{-l} } } .
\end{equation}
When choosing the plus solution for $\beta_{L_1-0} (l)$,
Eqs.~(\ref{eq:qubit_no-count_1-0}) and
(\ref{eq:qubit_no-count_1-1}) can be rewritten as
\begin{eqnarray}
\label{eq:qubit_no-count_2-0} \hat{S}_{L_1-0} |\omega_0\rangle & =
& \left( 1 + \sqrt{1-e^{-L_1}} \right) |0\rangle +
e^{-\frac{L_1}{2}} |1\rangle ,
\nonumber\\
\end{eqnarray}
\begin{eqnarray}
\label{eq:qubit_no-count_2-1} \hat{S}_{L_1-0} |\omega_1\rangle & =
& \left( 1 - \sqrt{1-e^{-L_1}} \right) |0\rangle -
e^{-\frac{L_1}{2}} |1\rangle .\nonumber\\
\end{eqnarray}

After detecting a photon at $L_1$, the states become
\begin{eqnarray}
\label{eq:qubit_JS0}
\hat{J}_{L_1} \hat{S}_{L_1-0} |\omega_0\rangle & \propto &
\left( 1 - \sqrt{1-e^{-L_1}} \right) |0\rangle
+ e^{-\frac{L_1}{2}} |1\rangle ,
\nonumber\\
\\
\label{eq:qubit_JS1}
\hat{J}_{L_1} \hat{S}_{L_1-0} |\omega_1\rangle & \propto &
\left( 1 + \sqrt{1-e^{-L_1}} \right) |0\rangle
- e^{-\frac{L_1}{2}} |1\rangle ,
\nonumber\\
\end{eqnarray}
and these are still orthogonal to each other.

Let us consider the next interval $(L_1,\,L_2)$, where we assume
that the second photon is detected at $L_2$. Defining the local
oscillator function $\beta_{L_2-L_1} (l)$, we obtain
\begin{widetext}
\begin{eqnarray}
\label{eq:qubit_SJS0}
\hat{S}_{L_2-L_1} \hat{J}_{L_1} \hat{S}_{L_1-0} |\omega_0\rangle & \propto &
\left( 1 - \sqrt{1-e^{-L_1}} + e^{-\frac{L_1}{2}}
\int^{L_2}_{L_1} {\rm d}l \, \beta_{L_2-L_1}^* (l) e^{-\frac{l}{2}}
\right) |0\rangle + e^{-\frac{L_2}{2}} |1\rangle ,
\\
\label{eq:qubit_SJS1}
\hat{S}_{L_2-L_1} \hat{J}_{L_1} \hat{S}_{L_1-0} |\omega_1\rangle & \propto &
\left( 1 + \sqrt{1-e^{-L_1}} - e^{-\frac{L_1}{2}}
\int^{L_2}_{L_1} {\rm d}l \, \beta_{L_2-L_1}^* (l) e^{-\frac{l}{2}}
\right) |0\rangle - e^{-\frac{L_2}{2}} |1\rangle .
\end{eqnarray}
\end{widetext}
Now the orthogonality condition implies
\begin{equation}
\label{eq:qubit_LO2}
\beta_{L_2-L_1} (l) = - \frac{ e^{-\frac{l-L_1}{2}} }{
2 \sqrt{1-e^{-l}} } .
\end{equation}
More generally, the local oscillator function for the interval
$(L_j ,\, L_{j+1})$ is given by
\begin{eqnarray}
\label{eq:qubit_LOgeneral} \beta_{L_{j+1} - L_j} (l) = (-1)^j
\frac{ e^{-\frac{l-L_j}{2}} }{ 2 \sqrt{1-e^{-l}} } , \qquad
(L_0=0) .\nonumber\\
\end{eqnarray}

Using Eq.~(\ref{eq:qubit_LOgeneral}), the signal states after the
whole detection process are described by $\hat{S}_{L-L_N}
\hat{J}_{L_{N_{\rm tot}}} \hat{S}_{L_{N_{\rm tot}}-L_{{N_{\rm tot}}-1}}
\cdots \hat{J}_{L_1}
\hat{S}_{L_1-0} |\omega_i\rangle$, $(i=0,1)$, where $N_{\rm tot}$ is the total
number of detected photons and $(L \to \infty,\, N_{\rm tot}=0,1 \cdots
,\infty)$. When $N_{\rm tot}$ is zero or even, the final states are
\begin{eqnarray}
\label{eq:qubit_odd}
|\omega_0^L\rangle & \propto &
\left( 1+\sqrt{1-e^{-L}} \right) |0\rangle
+ e^{-\frac{L}{2}} |1\rangle
\nonumber\\ & = &
\left( 2 - \frac{1}{2}e^{-L} - \cdots \right) |0\rangle
+ e^{-\frac{L}{2}} |1\rangle
\nonumber\\ & \to & |0\rangle ,
\\
|\omega_1^L\rangle & \propto &
\left( 1-\sqrt{1-e^{-L}} \right) |0\rangle
- e^{-\frac{L}{2}} |1\rangle
\nonumber\\ & = &
\left( \frac{1}{2} e^{-L} + \cdots \right) |0\rangle
- e^{-\frac{L}{2}} |1\rangle
\nonumber\\ & \to & 0 .
\end{eqnarray}
These are still orthogonal to each other in any order of $L$.
On the other hand, when $N_{\rm tot}$ is odd, we have
\begin{eqnarray}
\label{eq:qubit_odd}
|\omega_0^L\rangle & \to & 0 ,
\\
|\omega_1^L\rangle & \to & |0\rangle .
\end{eqnarray}
Eventually, one finds that it is possible to distinguish
$|\omega_0\rangle$ and $|\omega_1\rangle$ perfectly by checking the
parity of the totally detected photon number. Let us finally
mention that the projection onto arbitrary orthogonal
superpositions of the vacuum state $|0\rangle$ and the single
photon state $|1\rangle$,
\begin{eqnarray}
\label{eq:0pm1_basis}
|\omega_0\rangle & = &
f_0 |0\rangle + f_1 e^{i\varphi} |1\rangle,
\\
|\omega_1\rangle & = &
f_1 |0\rangle - f_0 e^{i\varphi} |1\rangle,
\end{eqnarray}
can be achieved in the same manner as described above.

\section{Conclusions}

In summary, we systematically derived a scheme which is equivalent
to the original Dolinar receiver for implementing the minimum
error discrimination of binary coherent-state signals. This scheme
corresponds to a spatial version of the Dolinar receiver, based
upon linear optics, photodetectors, continuous measurement and
feedforward. In our approach, as opposed to previous works, we
focus on the (asymptotically) perfect implementation of a given
projection measurement. In order to derive the Dolinar-type
scheme, we consider the projection measurement that corresponds to
the minimum error discrimination of binary coherent-state signals.
The derivation then relies on the constraint that, for
discriminating the orthogonal states of the measurement basis, the
conditional states in each detection and feedforward step must
remain orthogonal. This new derivation method does not require
complicated optimization procedures and is applicable to various
kinds of projection measurements. We showed that not only the
particular measurement associated with the coherent-state signals
treated in Dolinar's original proposal, but also other types of
projection measurements can be implemented via continuous
measurement and feedforward. Significantly, in our approach,
optimal measurement performance, i.e., the maximum measurement
efficiency allowed by quantum mechanics, is achieved without the
need of expensive entangled auxiliary resources. However, finite
feedforward plus arbitrarily many auxiliary photons must be
replaced by arbitrarily fast feedforward based on arbitrarily weak
measurements.

Although we considered only projection measurements of a
single-mode field in this paper, our methodology might be
applicable to more general scenarios as well, including
generalized measurements or joint projective measurements for more
modes. While the possibility of a linear-optics implementation of
such scenarios has been studied already
\cite{vanLoock03,Calsamiglia02}, it remains an open question
whether the approach based on continuous measurement and
feedforward allows for an optimal efficiency also in these more
general schemes.

Finally, an important question is whether our scheme can be used
in a real, necessarily finite, physical implementation. In order
to address this question, one would have to consider a discrete
analogue of our scheme, including finite feedforward and weak, but
finite measurements. One may then determine bounds in terms of
suitable figures of merit, e.g. the minimum average error or the
maximum success probability, for given feedforward resources.

\begin{acknowledgements}
The authors thank M.~Ban for helpful discussions.
This work was supported by the DFG under the Emmy-Noether
program, the EU FET network RAMBOQ and the network of competence QIP of
the state of Bavaria.
\end{acknowledgements}

redpajama_set_name: RedPajamaArXiv


What an awesome spacecraft it has been — launching the Hubble Space Telescope, Ulysses and John Glenn! It returned NASA to manned spaceflight following both the Challenger and Columbia accidents. And now this week, we say goodbye to Space Shuttle Discovery. On Sunday night, Feb. 27, I took a look at it (perhaps my last), and the ISS, as they flew over together in an ideal pass across our sky. Even in binoculars you could see the shape of the ISS — or at least that it had a shape and was not just a star, as all other satellites appear. A great sight was watching the craft fly into Earth's shadow and go deep red then fade out. That's why the bright streak the ISS/Shuttle left on the frame during the time exposure fades out at left, as they fly to the east and experience sunset.
To the west (right) note the Zodiacal Light glow reaching up from the western horizon. Orion and Canis Major are at bottom in the south. To the northwest is my house. I purposely left the porch light on around the other side of the house to light up the yard and the photogenic old house on the property. It features in many of my backyard shots.
This shot is a two-image stack of 4-minute exposures, with the Canon 5D MkII at ISO 800 tracking the sky on an equatorial mount. The lens is one of my favourites, the Sigma 8mm full-circle fish-eye lens. Some haze in the sky knocked back the contrast, so the Milky Way doesn't pop out as well as it should. Still, this was the first clear night in a while when it wasn't -25°C!


"Monetization & Trade Program Available"
(8) Other Assets may be considered on case-by-case basis.
The Assets can be "Assigned", "Leased", or "OWNED", But we Require to have the Complete Contact Info of the Actual Title/Holder Owner of the Asset for the Transaction to happen. NO EXCEPTIONS.
Profit/Returns are Good to Excellent and "Guaranteed by Contract". Funds can be used for any legal purpose and do NOT have to be repaid, since they are Non-Recourse Funding.
Serious Inquiries ONLY Accepted from High Net Worth Clients and/or their DIRECT Legal Rep with your complete contact info.
"Multiple Currency Exchange Possibilitites Available"
We have Providers who'll exchange their currencies, such as EURO, HKD, JPY, RMB, and USD for your (OWNER) Currencies of USD, EURO, RMB, JPY, HKD, EURO.
Unlimited Quantities are Available and Accepted, and MOST preferred. Bonuses are paid.
Excellent Financial Opportunities for all parties involved.
Serious Inquiries ONLY from Owners of Currencies and/or their Direct Legal Rep with your complete contact info.
We offer a mix and match service under which we are able to team up investors with entrepreneurs seeking funds for profitable ventures. We are in a position to help you finance your project. Please contact us with a feasible business plan and we shall arrange the desired finance either through high street bank or through carefully selected private investors, who are willing to invest in high yielding projects located anywhere in the world.
Entrepreneurs looking for investors may contact us for completing their dream projects. Funds can also be arranged for working capital or to take an existing business to the next level.
By optionally appointing you as the Co-Director of our special purpose company as the investment vehicle. We will via 2 transactions securely enhance your funds well beyond $247m. Thereby giving you the option to enter a managed buy-sell contract at the $100m+ minimum investment amount. IF and only if, you so choose. As alternatively you can simply walk away with your proceeds.
If you prefer to access this 2 step 130 day transaction via our US-based LLC instead of being made a UK company Director to contract directly with us. You can alternatively subscribe via our LLC from (10,000 Euros) to $27,778 per slot and have the LLC sign the relevant contracts with us on your behalf since they are already a Co-Director of our special purpose company as the investment vehicle. If you want details of this LLC option then please also state your investment amount.
This is the program for properties above 150 crores only minimum in India . There is no investment from the property owner's side at all. There are 2 options available.
*Bullet Payment Method: This is a one-time settlement option of 25%. The one time settlement will have 2 tranches of payment. First in approximately 10 to 15 days from contract signing and the final tranche after 45 days.
* Monthly Payment Method: The second option is monthly payments of 15% of the valuation amount for 12 times.
Blocking. For other banks we can offer 150%-160% bullet-payment for 45 days Blocking.
If client is interested in 40 week program they get 15%-20% weekly for 40 weeks. Or trader purchases FDs that are 1 year old (minimum).
Similar to SKR, 40 weeks program which will yield 50%-55% per week for 40 week split 50-50% basis between trader and investor.
Instruments must be confirmed via swift MT799 and MT760. If client will have problem their bank advance our platform will assist in finding a solution with their bankers.
Principals only can receive further details per email.
I have spent most of the first quarter raising private capital for our other business. We have a solid, signed, funding commitment for $250,000.00. We will only get the funds in the middle of May. The funds can only be used for our other business. Therefore, I need to personally raise about 45K for 90 days. I am willing to pay speed, candor and privacy: 10K for the money. I need a private lender who will do a signature loan. Fico Score 633. No real estate or other collateral. SOURCE OF REPAYMENT:PRACTICE OF LAW: (Areas: Federal Securities law; Offshore Corps.; Probate Administration) (No cash flow lenders, no Fintechs, no upfront fees, no junk fees.) Prefer East Coast lender(s). Brokers, facilitators, finders are welcome.
Heritage funds program. No LTV, whole amount can be entered into trades. 100% return per month.
Tearsheet program. 85% return per month. 100m minimum to enter.
Found a direct banker that can do all tough and non rated banks, Asian banks, Russian banks, Indian banks, African banks, mostly all non rated banks globally are all acceptable. Minimum entry of 100m cash, BGs, SBLC and Bank Drafts etc are all acceptable.
From Trader in London: Quote "For Applicants with funds at some of the more challenging banks, like Sberbank, Mahkota, Bank of West Indies, Bank of China, other Chinese Banks or Commercial Development Bank, as long as they have at least USD 100M in their account, we can take it. Corporate Pension funds may also be acceptable.
mt799 or 760. That's all.
LTV will range from 10%-90% then they can do 100-320% per month.
72 hrs front-to-back contract. 2-3 weeks later advance. monthly trade.
70k euro program. 1.5m net to client per year. We can also scale and do ...x per year for life.
$50,000 pays out $100,000 in 90 days.
$100,000 pays out $300,000 in 90 days.
$250,000 pays out $1,000,000 in 90 days.
$500,000 pays out $2,500,000 in 90 days.
$750,000 pays out $4,500,000 in 90 days.
$500,000 pays out $2,000,000 in 60 days.
$1,000,000 pays out $7,000,000 in 60 days.
YOU CAN DO ONE, GET PAID AND COME BACK IN TO REACH YOUR FUNDING GOALS AS LONG AS THE PROGRAM IS OPEN.
We have already paid 17 clients in this way.
We are Al-Talal Petroleum Services Co. Ltd.
Based on a 70% LTV monetization of a 100MM Bank draft.
Which is 70MM, you would earn 25% per week gross minus fees.
Client would get hard real estate assets.
ROI is 15%-18% per month (80 banking days) + Return of principal.
ROI first year is overall potentially up to 5x capital (Compounding allowed).
Funds move to trade groups account.
Client has option to do TTM in Toronto.
Investment opportunity is here for you to invest as low as $1,000 USD and get a profit return of $... USD in seven business working days. Contact us now for more information it's all about investing into crude oil and gas.
why wasting your precious time online looking for a loan, when there is an opportunity for you to invest with 500usd and get a profit returns of 5,000usd after ... business working days, contact us today for more information if interested in learning on how you can earn big with just little amount it is all about investing into crude oil and gas.
Trade Finance & Invoice Finance solutions for businesses to trade, import or export goods, both domestically and globally.
we are also able to provide financial banking instruments as BGs, SBLCs and comfort letter of credits for any projects requiring such supports.
WE have various provider who offer excellent prices for currencies, in their currency exchange programs. We have USD, EURO, HKD, JPY, RMB, and more and we can exchange vice versa. Unlimited quantities available, Large Contracts are Preferred. Serious Inquiries ONLY from Direct Currency Exchangers and/or their Direct Legal Rep with your complete contact info.
We are issuing corporate notes for 375K. We need a guarantee for the notes. We will pay 100K for the guarantee. 50K to start and 50K at finish. We are raising capital meet our obligation under 25 million dollar gold contract. We need a clean and clear source of economic support from a guarantor. We want a funder/ guarantor from the states. This is a small transaction (375K) so we do not want funders from Europe or Middle East. The due diligence is too difficult for alleged funders from Middle East and United Kingdom. The guarantee will cover the face value and interest on corporate notes. Total exposure is 375K. Corporate and personal guarantees are available from borrower. (No upfront fee or junk fees, No BGs, No MTNs or SBLCs). U.S. Treasuries, AAA Bonds, T Strips or cash.
April 2019 - we have new Deals!
We are offering fixed deposit Accounts with the best interest on the market.
Only limited! Without fees! Per month!
Buyer is Licensed Fiduciary and Forex Exchange Company in Boston MA USA. is Final Buyer of VENEZUELAN CURRENCY Bolivars, Not a intermediary or Broker. Willing to move first securing Seller's Bolivars Bank to Bank prior to paying the Bolivars by The Seller to my BOD or Banesco Account in Caracas Venezuela based on my Buying procedure as explained step by step on the Copy of my purchasing Contract. I can pay USD to the Sellers nominated Bank Account anywhere in the World except War Thorn Countries either from my Wells Fargo NA or BOA USA Bank.I only buy within Venezuela for now not outside. My buying rate today is 250+ 5 BSS to 1 USD with 5 BSS per USD as commission to any intermediary or Broker that bring the right seller to me in Venezuela making my Buying price to be 250 BSS to 1 USD Nett to me the final Buyer. Am buying this very cheap rate because am using the purchased Bolivars in Venezuela to move my Euro out of Venezuela to my Company Account in Spain via my direct contact at the Central Bank of Venezuela.
$50,000K USD Required To Pay UP FRONT ENTRY FEE To Facilitate Obtaining A Non-Recourse Loan Transaction In The Amount Of $100,000,000 Million Dollars USD. Yes, This Is Real And Legitimate And You Could And Would Benefit Greatly By Co-Operating With Me On My Entry Into This Program. Are You Intelligent Enough To Even Contact Me For Details? One Would Think So. There Is Nothing Complicated Nor Difficult About The Logistics Of Entering This Program And Then Receiving The Rewards! It ONLY Requires Having The Necessary Up Front Fee For Entry Into The Program.
100% per month. No LTV, full or whole amount of funds to enter trade.
Client received 100% per day for 5 days = 500% gross returns, then those returns entered a 40 week program.
These trades organized by Europe Central Bank through their Members banks (Deutsch, HSBC or Barclays, etc). Your money stays in your account therefore they are not at risk.
Client nets 4m in 30 to 35 banking days.
Clients will not have to move their funds if in a top tier world bank.
Clients will block their funds with the trader.
150% to 300% Returns per month paid monthly.
30 days to 40 weeks duration available.
There is a Bank Draft Program with a Monetizer/Trader as per the following:This opportunity is for clients with funds in Smaller banks that are less able or willing to issue MT799s.But, this program can also be utilized by clients with funds in the Major Banks.It is possible the LTV for the Bank Draft could be 70% to 80%.The client can request up to Ten percent (10%) advance against LTV – usually 3 days after closing.
Possible Profits may be as much as: 25% per week.The Bank Draft/Cheque method is simple, safe and inexpensive and quick. The Bank Draft/Cheque is delivered bank to bank by Bonded courier – not by MT799/760.