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Genting Online Casino is run by one of the world's largest gaming groups (owners of the Genting and Circus casino brands). This casino is great for roulette with bags of choice (10 variants) including some of the best live dealer roulette games in the business.
They run a generous £/€ 150 first deposit match bonus with some very favourable wagering terms for roulette players. You'll also more than likely to find some Live Roulette specials going on- the last time we were here, they were running a live roulette Sunday Special where they were giving 10% cashback on all live roulette losses on Sundays.


What can I say about the supplier which has been in the W@W Top 10 for the longest time? Sympre they did well!
Although, I really cant say that I personally liked the food coz we really weren't able to eat during the reception. 😦 We only got to eat the food when we were in the hotel (good thing josiahs set aside some food for us).
Our AE – Ms Mherlyn is also very courteous and very easy to talk to.
The only thing that bothered me was naubusan daw ng food (like soup tapos yung sa side ni lui, naubos na daw yung ibang putahe) yung ibang guests… Which I think is impossible kasi marami di pumunta sa wedding. There were at least 4 tables vacant. Kaya medyo nagtaka lang ako.
There were only a few left-over that was handed over to my mom and to lui's mom… yung lengua, lahat nasira by the next day. Di ko lam kung mabilis tlaga masira or panget pagkakaluto.
My MOH also said that the water glasses were dirty that's why they didn't drink from it.
Pero overall, good feedback naman ang Josiahs.


Seventh Annual Irish Labor Awards, Friday September 8, 2017.
One of America's most respected labor figures, John Samuelsen. International President of transit workers union TWU, has been named as recipient of the Irish Echo's coveted Irish Labor Leader of the Year award.
This year we will be honoring 100 Irish-American men and women in today's labor movement, while at the same time reflecting on the work of those historical giants who paved the way to labor's central role in contemporary American life.


text:
Irish actress Jessie Buckley talks to IMAGE about 'Wild Rose'
by Meg Walker
Irish actress Jessie Buckley has been working solidly for the last 11 years, but her new film, Wild Rose, is sure to send her career into the stratosphere.
Born in Kerry, Jessie Buckley first burst onto our TV screens in the Andrew Lloyd-Webber West End talent contest I'd Do Anything when she was 18. Today, she's an acclaimed actress with a CV boasting award-winning films and television series, including Beast and BBC series War & Peace. This month sees her showcasing her vocal and acting talents in a starring role that is sure to not only move audiences, but have viewers sitting up and taking serious notice.
Meg Walker caught up with the Irish actress in Dublin to talk about Wild Rose, a film that sees Buckley play a single Glaswegian mum freshly released from prison who has a passion for country music and notable talent for singing, but two children who depend on her, and her dreams of going to Nashville to become a country singer are at war with her emotional ties to home.
Wild Rose, directed by Tom Harper and written by Nicole Taylor and starring Jessie Buckley, Julie Walters and Sophie Okonedo, hits Irish cinemas on April 12. Cert 15a.
The Lion King: First reactions to Disney's live action remake are here
Here are five great Netflix picks to binge-watch this month
WATCH: Renée Zellweger stars as Judy Garland in new biopic
12 brilliant, classic films you won't believe are THAT Old
The Little Mermaid: Fans predict who'll play the role of Prince Eric
Wait for it... We have 21 feel-good songs from musicals
Eight Netflix gems worth adding to your binge-watch lists
10 royal TV series and films worth binge-watching now
We love this first look of Saoirse Ronan in 'Little Women'

redpajama_set_name: RedPajamaCommonCrawl


text:
Used Cars, Trucks, & SUVs for Sale in Corbin KY, Manchester KY, Harrodsburg KY & London KY
make=Kia&model=Soul& 3 auto-used /used-inventory/index.htm
Model: Soul
Tim Short Price:$13,377
6-Speed Automatic with Sportmatic,
Front Bucket Seats (6-Way Adjustable Driver's Seat)
Tim Short Price:$8,900
Titanium Gray,
6-Speed Manual,
9511P1
Used Cars, Trucks, and SUVs for Sale in London, KY
We Have the High-Quality Cars, Trucks, and SUVs that Savvy Drivers from Corbin KY, Manchester KY, Williamsburg KY, Jellico TN, and Whitley City KY are Looking For
If you should choose to take a look in person at the complete inventory of pre-owned vehicles at the Tim Short Select Used Cars showroom, you should make sure to eat a solid lunch beforehand. You see, we offer a comprehensive selection of used cars, pickups, SUVs and vans, with literally hundreds of vehicles for you to consider, so you'll want to be sure you're up for the job!
Because we're the area's top source for Chevrolet, Chrysler, Dodger, Jeep and Ram vehicles, we've built quite a loyal following -- and our customers return to us time and again when they're looking to get behind the wheel of a new vehicle. Often, they'll use the equity they've built in their current car by trading it in as part of the down payment on their new vehicle. We, then, make that pre-owned vehicle available to other customers as a reliable, affordable transportation option.
We understand that not every driver in the Corbin KY, Manchester KY, or London KY area needs -- or even wants -- to purchase a brand-new car, pickup or SUV. And, honestly, there are times where it just makes good financial sense to purchase a late-model, gently-driven vehicle. Perhaps you're looking for a truck that can be relied on to survive weekend fun. Maybe you've got a newly-licensed teen driver who needs a trustworthy, affordable first car. You might even have a hankering to garage a sports car that you can take to the open roads when the weather's nice and you're feeling like cutting loose. Whatever your reason may be, you can count on the team at Tim Short Select Used Cars to have a vehicle that meets your needs.
From fuel-efficient coupes to family-friendly SUVs and hard-working pickups, we offer a tremendous assortment of used vehicles from such popular American automakers as Chevrolet, Lincoln, Cadillac, GMC, Buick, Ford, Chrysler, Dodge, Jeep, and Ram. We also feature an alluring selection of models from Honda, Toyota, Volkswagen, Nissan, Subaru, Hyundai, and Mazda, as well as luxury brands such as BMW, Lexus, Jaguar, Mercedes-Benz, Infiniti, Land Rover, Acura, Porsche, and Audi.
We think it's important that you know that you can trust in our inventory of used sedans, SUVs, vans, and pickups. We make sure to thoroughly examine every pre-owned vehicle we receive before we offer it for resale, verifying that it meets the highest performance, quality, and reliability standards to ensure that you'll enjoy years of driving pleasure. It's important to us that our customers who make the trek from Williamsburg KY, Jellico TN, or Whitley City KY are rewarded for their faith in us.
Go ahead and take a look at our extensive inventory of used vehicles, and make sure to use the various filtering options to narrow down the vast selection to the models that most appeal to you. You can either sign up to receive alerts if the price changes, or you can opt to save the car for future research. Then, when it's most convenient for you, just visit our London, KY used car dealership and meet with one of our sales associates to take a closer look and arrange for your test drive. We look forward to helping you!


Looking for Fairfield at Classic Interiors in Idaho Falls, ID?
Classic Interiors in Idaho Falls, ID is an authorized dealer of Fairfield Products. Fairfield Chair Company is a major U.S. manufacturer of fine upholstered seating for the home, office, and hospitality industries. Fairfield Chair Company's sofas, loveseats, sleepers, chairs, executive desk chairs, desks and tables are noted for their quality, style and value.
So if you are looking for Fairfield products in Idaho Falls, Jackson Hole, Rexburg, Pocatello, Blackfoot, Driggs, Sun Valley, Boise, Salt Lake City, Twin Falls and Logan, or if you have any questions about Fairfield products, please feel free to call us at (208) 524-8200 or simply stop by Classic Interiors at any time and we would be glad to help you.


Is this the beginning of the end of Apple?
Following yesterdays disappointing iPhone 4s announcement, it got us to thinking about Apple and it's long term prospects without Steve Jobs. With a boring keynote and a 4s product announcement that has been met with a nearly universal reaction of "that's all" basically Apple has become the follower in the phone market with Android leading the way, the 4S is basically just a catch-up to what is already out there in the Android and unbelievably Windows phone market.
So the company we have always thought of as the leader in the market is now playing catch up, imagine in another year how far behind Android the iPhone is going to be and it starts to look worse.
The conventional wisdom is that the applications will be what keeps the iPhone ahead, along with the iPad the ecosystem can't currently be beat. That is changing as well though as the Android developers are going to be going crazy soon with the release of the Kindle Fire which contrary to popular belief is not that much of a fork of Android and is running Android 2.3 which will allow developers to write one program for the Android phones as well as the Fire. Priced at $199.00 and with Amazon behind it the Fire is going to be a serious competitor to the iPad, and if you are used to all these Android apps, you are going to look to an Android phone as well especially if the features are better than an iPhone.
The sales numbers were leaked today and the Fire presales are higher than iPads. The Fire has over 250,000 presales already and is increasing at a rate of roughly 50,000 a day. That has to scare the heck out of Apple this years killer Christmas gift is not going to come from Apple, but from Amazon.
So today we have the market split between the Fire on the bottom and the iPad on top, that's where Apple has always done best, and if that were the only competitor Apple would probably come out just fine, but there is something else coming along that will put Apple in a bad spot.
That is the forthcoming Windows 8 and the day when you can have a complete operating system on your tablet. This is going to split the market three ways with the top end suddenly becoming Microsoft with a complete tablet computer at the top, iPad in the middle, and the Fire at the bottom. History tells us that in a three way market, the middle is in trouble and the iPad is going to fall squarely in the middle next year with very little other than screen size to differentiate it from the Fire and not enough power to compete with Windows 8.
That's when the iPad becomes just another shiny gadget that Apple fanbois will keep buying, but who else will?
That leaves just the computer line and Apple's current strategy seems to be to move more towards an iOS feel with the desktop, everyone keeps talking about a post pc world, but what we see coming is a pc+ world where your devices work together and Apple is way behind there. Having had our hands on the Windows 8 system and the way it will work between a desktop and a tablet is an amazing step forward and more in line with the way most people work.
So in the phone market we have Android ahead of Apple and Windows Mango behind, in tablets we will have Windows 8 ahead of Apple and the Kindle Fire behind and in the desktop we have Windows on top, Apple second and Linux behind. Leaving Apple in the middle in every market it competes in. You can say they are leaders in the music player market, but that is a dying market as phones are supplanting mp3 players at a rapid pace.
The middle is not where anyone wants to be. Apple was the cult of Steve. They were always one step ahead as Jobs had a knack for keeping Apple ahead of the pack, without Steve they are falling quickly behind already and in 2 more years they will become once again what they were before, an expensive alternative that delivers less than the competition. How long it will take for consumers to recognize this is the question no one can answer, but in the end, the consumer always gets it right.
It was a nice ride while it lasted Apple.
This entry was posted on Wednesday, October 5th, 2011 at 9:51 pm	and is filed under Amazon, Apple. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.


REGENSBURG, Germany, Feb. 22, 2013 — Osram Opto Semiconductors has announced that it is coordinating a three-year project to improve high-brilliance infrared laser systems performance and to reduce production costs.
The IMOTHEB (Integrated Micro-Optical and microTHhermal Elements for diode lasers of high Brilliance) project, part of the "Integrated Microphotonics" Initiative, runs from Oct. 1, 2012, to Sept. 30, 2015, and is supported by the German Federal Ministry for Education and Research. Project partners include Dilas Diode Laser and the Max Born Institute for Nonlinear Optics and Short Pulse Spectroscopy, a nonprofit research organization.
Prototype of a high-power laser bar, as developed by Osram Opto Semiconductors in the IMOTHEB project. Images courtesy of Osram.
Diode-pumped high-power laser systems for materials processing applications such as cutting and welding are playing an increasingly important role in industry. Their advantages over carbon dioxide lasers and lasers pumped with flashlamps include lower operating costs, greater efficiency and smaller size. Fiber lasers and fiber-coupled diode lasers have become more important for optical materials processing.
As these lasers become more widespread, the focus is shifting to the issue of cost. Developments that increase the performance of laser systems, and at the same time reduce production costs, are necessary to move this market forward. Infrared semiconductor laser diodes that can be used to pump fiber lasers are key components in this area. They offer enormous potential for automating production and miniaturizing the systems while reducing the number of semiconductor chips needed.
The project's objective is to investigate new approaches and technologies that could lead to significant reductions in the cost of pump modules, including the semiconductor lasers, and the cooling elements, optics and sensors. Plans to increase semiconductor laser output by 40 percent while retaining the same high beam quality are also in the works.
Osram Opto Semiconductors is coordinating the project and offering its know-how in semiconductors, laser diodes and laser bars; simulations are being subcontracted to the Fraunhofer Institute for Applied Optics and Precision Engineering.
Dilas is responsible for the assembly technology with improved thermal resistance and higher integration in laser modules, as well as for automation in module production.
The Max Born Institute is acting as a scientific partner, analyzing and characterizing the chips and modules.
Higher output from laser diodes and new automated assembly processes for laser systems are targeted at industrial applications such as laser welding and laser cutting, making them more efficient and more cost-effective.
Osram has set the goal of achieving a higher degree of integration at the semiconductor level, increasing the brilliance of infrared laser diodes by integrating micro-optical and microthermal elements directly on the chip. Over the course of the project, the target is to increase output by 40 percent, which will make laser chips suitable for fiber laser pump modules and for fiber-coupled diode lasers.


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text:
/*
************************************************************************
 ECE 362 - Mini-Project C Source File - Fall 2015
***********************************************************************

 Team ID: 21

 Project Name: Frame Perfect

 Team Members:

   - Team/Doc Leader: Kyle Diekhoff      Signature: Kyle Diekhoff

   - Software Leader: Jordan Huffaker     Signature: Jordan Huffaker

   - Interface Leader: Eric Colter     Signature: Eric Colter

   - Peripheral Leader: Diego De La Fuente    Signature: Diego De La Fuente

 Academic Honesty Statement:  In signing above, we hereby certify that we
 are the individuals who created this HC(S)12 source file and that we have
 not copied the work of any other student (past or present) while completing
 it. We understand that if we fail to honor this agreement, we will receive
 a grade of ZERO and be subject to possible disciplinary action.

***********************************************************************

 The objective of this Mini-Project is to intercept the communication
 between a GameCube and controller so that when a certain button combination
 is pressed, a frame perfect macro will be sent to the GameCube.

***********************************************************************

 List of project-specific success criteria (functionality that will be
 demonstrated):

 1. Ability to start macros with specific button combinations.

 2. Ability to switch the active macro button between X and Y.

 3. Ability to switch between schemes of macros with the D-Pad.

 4. Ability to run frame perfect macros read from flash memory.

 5. Ability to recover a connection when disconnected from the GameCube.

***********************************************************************

  Date code started: 11/21

  Update history (add an entry every time a significant change is made):

  Date: 11/25  Name: Jordan Huffaker   Update: Addition of microcontroller file.

  Date: 12/3  Name: Eric Colter   Update: Added GAL code and finished the working echo program.

  Date: 12/4  Name: Diego De La Fuente   Update: Added foundational microcontroller input and output drivers.

  Date: 12/5  Name: Jordan Huffaker   Update: Added main control logic and scheme functionality.

  Date: 12/5  Name: Kyle Diekhoff   Update: Added storage and recovery functionality.

  Date: 12/6  Name: Jordan Huffaker   Update: Added input masking and echoing functionality.

  Date: 12/7  Name: Diego De La Fuente   Update: Added some simple macros.

  Date: 12/7  Name: Kyle Diekhoff   Update: Now asserts macro frames 2 times instead of 3 times.

  Date: 12/8  Name: Jordan Huffaker   Update: Program now stores macros in flash memory.

  Date: 12/9  Name: Diego De La Fuente   Update: Added drill shine, fixed wave shine, and added moonwalk.

  Date: 12/10  Name: Jordan Huffaker   Update: Finalized schemes for Spark Challenge demonstration.

***********************************************************************
*/

#include "derivative.h"
#include "microcontroller.h"
#include "control.h"

#define CYCLESFOR40MICROWAIT 160

static void wait40MicroSeconds(void);

void main(void)
{
	initializations();

	for(;;)
	{
		// MicroController is working properly
		ClearWatchdogTimer();

		if (Transaction_Complete)
		{
			// Clear Transaction_Complete flag
			Transaction_Complete = 0;

			// Wait 40 Micro Seconds
			wait40MicroSeconds();

			// Get the next input from the control and send it out
			setOutputFrame(GetNextInput(getInputFrame()));
		}
	}
}

static void wait40MicroSeconds()
{
	int i;
	for (i = 0; i < CYCLESFOR40MICROWAIT; i++) {}
}

redpajama_set_name: RedPajamaGithub


text:
Related player
Arnautovic Marko
View Complete Profile →
El Shaarawy Stephan
Home » West Ham to swap Arnautovic for Maxi Gomez? What does this mean for your fantasy football draft?
West Ham to swap Arnautovic for Maxi Gomez? What does this mean for your fantasy football draft?
Blog, Top storiesxfcediJuly 8, 2019
Marko Arnautovic has finally secured his desired move to the Chinese Super League. This leaves West Ham in need of a new striker. The position is likely to be filled by Celta Vigo's Maxi Gomez. But, how will the deals affect your fantasy football draft?
1Arnautovic remains available in FootballCoin
2Maxi Gomez joining the Premier League?
Arnautovic remains available in FootballCoin
Marko Arnatutovic's departure was to be expected. In spite of the Austrian being West Ham's most prolific striker during the last couple of years, the player always seemed unsettled. Not only once did he hint at a desire to relocate to China. The deal will no doubt greatly increase Arnautovic's wages. But, it also leaves West Ham in a perilous situation. Some would argue that Arnautovic may have been the Hammer's only player showing real goal-scoring prowess for the last couple of years. Yes, we're counting Andy Carroll and Chicharito.
Marko Arnatutovic is represented as a three-star player in FootballCoin. Owners of the card will continue to be able to use the collectible card. The striker has moved to Shanghai SIPG. The team is part of the CSL, a league supported by the game. The CSL club is rumored to have paid £22.4m for the forward. It's one of many high-profile transfers secured by Chinese clubs in recent years.
Arnautovic is not the only international star moving to the CSL this summer. It has been revealed that AS Roma striker Stephan El Shaarawy is set to join Shanghai Shenhua for the sum of €18 million.
Maxi Gomez joining the Premier League?
Maxi Gomez is a free player in FootballCoin, currently representing Celta Vigo. Similarly to Arnautovic, Gomez seems unsettled at his current club. Celta narrowly avoided relegation, largely due to the attacking partnership between Iago Aspas and Gomez.
The Uruguayan scored 13 times in 35 appearances. This seems to have been enough to convince West Ham to splash on a transfer, something they're known to do. The Hammers may be spending-prone, to begin with. But, with Gomez they are looking to send a message by offering £45m for the transfer. If the deal goes through, it will likely be a club record.
Maxi Gomez would then be available for your fantasy EPL draft, most likely as a free player. Should he continue on the mark with his La Liga performances, Gomez is sure to be one of the best free striker options in fantasy Premier League.
Eduard Bănulescu
Celta VIgoCSLLa LigaPremier LeagueWest Ham
5 Best La Liga strikers in 2020
"Mbappe will remain at Monaco" says team's sporting director
Pique blames Barcelona's lack of transfers on Neymar
The Premier League teams with most conceded goals in a season
"Shape of things to come" says Tottenham's Pochettino after duel with Real Madrid
Four quality players in La Liga with free cards in FootballCoin


Meet Lynn Markwell. A classical musician, gardening enthusiast, and consummate mother, wife, and friend. When shortness of breath, a dry hacking cough, and exercise-induced asthma limited her daily five-mile walks and interfered with her career in music education, she knew something was wrong. After consulting with a physician and being prescribed medication to address the asthma, her symptoms didn't improve.
After consulting with numerous specialists, and undergoing a lung biopsy, Lynn was diagnosed with hypersensitivity pneumonitis (HP) – a type of interstitial lung disease that can permanently damage the lungs if not treated.
Her diagnosis and the day-to-day struggle with the illness significantly impacted her career and family life. Yet, amidst these challenges Lynn was committed to find educational and exercise resources to effectively combat her disease.
Shortly after the diagnosis, Lynn entered into a pulmonary rehabilitation (PR) program. In PR, Lynn found a community of fighters – focused on living their best lives and fighting various forms of chronic lung disease.
The pulmonary rehabilitation program of educational classes and supervised exercise sessions helped Lynn reclaim control of her life. Eight years removed from the diagnosis, you'll find Lynn doing power yoga, working out five days a week, and she still has the time and energy to garden on Saturdays.


Reuters reports that Attorney Michael Avenatti, was arrested on Monday. Avenatti was charged with extorting more than $20 million from Nike, federal prosecutors said. The U.S. Attorneys offices in New York and Los Angeles separately filed charges against Avenatti. The California case accuses him of embezzling a client's money to cover his own debts. Avenatti was also charged with using phony tax returns to obtain millions of dollars in loans from a bank.


text:
\section{Introduction}

The study of mutual entropy (information) and capacity in classical system
was extensively done after Shannon by several authors like Kolmogorov \cite
{Kol} and Gelfand \cite{GY}. In quantum systems, there have been several
definitions of the mutual entropy for classical input and quantum output
\cite{BS,Hol,Ing,Lev}. In 1983, the author defined \cite{O2} the fully
quantum mechanical mutual entropy by means of the relative entropy of Umegaki%
\cite{U}, and it has been used to compute the capacity of quantum channel
for quantum communication process; quantum input-quantum output \cite{OPW1}.

Recently, a correlated state in quantum syatems, so-called quantum entangled
state or quantum entanglement, are used to study quntum information, in
particular, quantum computation, quantum teleportation, quantum cryptography
\cite{BO,Ben,BBPSSW,Eke,IOS,JB,Sch1,Sch2}.

In this paper, we mainly discuss three things below:(1) We point out the
difference between the capacity of quantum channel and that of
classical-quantum-classical channel followed from \cite{O11}. (2) So far the
entangled state is merely defined as a non-separable state, we give a wider
definition of the entangled state and classify the entangled states into
three categories. (3)The quantum mutual entropy for an entangled state is
discussed. The above (2) and (3) are a joint work with Belavkin\cite{BO} .

\vspace{-4mm}

\section{Qunatum Mutual Entropy}

The quantum mutual entropy was introduced in \cite{O2} for a quantum input
and quantum output, namely, for a purely quantum channel, and it was
generalized for a general quantum system described by C*-algebraic
terminology\cite{O4}. We here review the quantum mutual entropy in usual
quantum system described by a Hilbert space.

Let $\mathcal{H}$ be a Hilbert space for an input space, $B%
\mathcal{(H)}$ be the set of all bounded linear operators on $\mathcal{H}$
and $\mathcal{\ S(H)}${\scriptsize \ }be the set of all density operators on
$\mathcal{H}.$ An output space is described by another Hilbert space $%
\stackrel{\sim }{\mathcal{H}}$ , but often $\mathcal{H=}\stackrel{\sim }{%
\mathcal{H}}$. A channel from the input system to the output system is a
mapping $\Lambda $* from $\mathcal{S(H)}$ to $\mathcal{S(\stackrel{\sim }{%
\mathcal{H}})}$ \cite{O1}. A channel $\Lambda $* is said to be completely
positive if the dual map $\Lambda $ satisfies the following condition: $%
\Sigma _{k,j=1}^{n}$ $A_{k}^{*}\Lambda (B_{k}^{*}B_{j})A_{j}\geq 0$ for any
$ n\in {\bf N}$ and any $A_{j}\in B(\mathcal{H}),B_{j}\in
B(\stackrel{\sim  }{\mathcal{H}})$.

An input state $\rho $ $\in \mathcal{S(H)}$ is sent to the output system
through a channel $\Lambda $*, so that the output state is written as $%
\stackrel{\sim }{\rho }\equiv \Lambda ^{*}\rho .$ Then it is important to
ask how much information of $\rho $ is correctly sent to the output state $%
\Lambda ^{*}\rho .$ This amount of information transmitted from input to
output is expressed by the mutual entropy in Shannon's theory.

In order to define the quantum mutual entropy, we first mention the entropy
of a quantum state introduced by von Neumann. For a state $\rho ,$ there
exists a unique spectral decomposition $\rho =\Sigma _{k}\lambda _{k}P_{k},$%
where $\lambda _{k}$ is an eigenvalue of $\rho $ and $P_{k}$ is the
associated projection for each $\lambda _{k}$. The projection $P_{k}$ is not
one-dimensional when $\lambda _{k}$ is degenerated, so that the spectral
decomposition can be further decomposed into one-dimensional projections.
Such a decomposition is called a Schatten decomposition, namely, $\rho
=\Sigma _{k}\lambda _{k}E_{k},$where $E_{k}$ is the one-dimensional
projection associated with $\lambda _{k}$ and the degenerated eigenvalue $%
\lambda _{k}$ repeats dim$P_{k}$ times; for instance, if the eigenvalue $%
\lambda _{1}$has the degeneracy 3, then $\lambda _{1}=\lambda _{2}=\lambda
_{3}<\lambda _{4}$. This Schatten decomposition is not unique unless every
eigenvalue is non-degenerated. Then the entropy (von Neumann entropy\cite{OP}%
) $S\left( \rho \right) $ of a state $\rho $ is defined by

\begin{equation}
\renewcommand{\theequation}{2.1}
S\left( \rho \right) =-tr\rho \log \rho .
\end{equation}

The quantum mutual entropy was introduced on the basis of the above von
Neumann entropy for purely quantum communication processes. The mutual
entropy depends on an input state $\rho $ and a channel $\Lambda ^{*}$, so
it is denoted by $I\left( \rho ;\Lambda ^{*}\right) $, which should satisfy
the following conditions:

(1) The quantum mutual entropy is well-matched to the von Neumann entropy.
Furthermore, if a channel is trivial, i.e., $\Lambda ^{*}=$ identity map,
then the mutual entropy equals to the von Neumann entropy: $I\left( \rho
;id\right) $ = $S\left( \rho \right) $.

(2) When the system is classical, the quantum mutual entropy reduces to
classical one.

(3) Shannon's fundamental inequality \\ 0 $\leq $ $I\left( \rho
;\Lambda ^{*}\right) \leq S\left( \rho \right) $ is held.

In order to define the quantum mutual entropy followed by the classical one
(see\cite{O11}for the details), we need the joint state (it is called
``compound state'' in the sequel) describing the correlation between an
input state $\rho $ and the output state $\Lambda ^{*}\rho $ and the quantum
relative entropy. A finite partition of the classical measurable space
corresponds to an orthogonal decomposition $\left\{ E_{k}\right\} $ of the
identity operator I of $\mathcal{H}$ in quantum case because the set of all
orthogonal projections is considered to make an event system for a quantum
system. It is known \cite{OP}that the following equality holds

\[
\sup \left\{ -\sum_{k}tr\rho E_{k}\log tr\rho E_{k};\left\{ E_{k}\right\}
\right\} =-tr\rho \log \rho ,
\]
and the supremum is attained when $\left\{ E_{k}\right\} $ is a Schatten
decomposition of $\rho .$ Therefore the Schatten decomposition is used to
define the compound state and the quantum mutual entropy.

The compound state $\theta _{E}$ (corresponding to joint state
(measure) in CS) of $\rho $ and $\Lambda ^{*}\rho $ was introduced
in
\cite{O2,O3}, which is given by

\begin{equation}
\renewcommand{\theequation}{2.2}
\theta _{E}=\sum_{k}\lambda _{k}E_{k}\otimes \Lambda ^{*}E_{k},
\end{equation}
where $E$ stands for a Schatten decomposition of $\rho ,$ so that the
compound state depends on how we decompose the state $\rho $ into basic
states (elementary events).

The relative entropy for two states $\rho $ and $\sigma $ is defined by
Umegaki and Lindblad, which is written as

\begin{eqnarray*}
&&S\left( \rho ,\sigma \right)\\
&&= \left\{
\begin{array}{ll}
tr\rho \left( \log \rho -\log \sigma \right) & \left( \mbox{when
}\overline{ ran\rho }\subset \overline{ran\sigma }\right) \\
\infty & \left( \mbox{otherwise}\right)
\end{array}
\right..\\
&&\mbox{\hspace{6cm}(2.3)}
\end{eqnarray*}

Then we can define the quantum mutual entropy by means of the compound state
and the relative entropy \cite{O2}, that is,

\begin{equation}
\renewcommand{\theequation}{2.4}
I\left( \rho ;\Lambda ^{*}\right) =\sup \left\{ S\left( \theta _{E},\rho
\otimes \Lambda ^{*}\rho \right) ;E=\left\{ E_{k}\right\} \right\} ,
\end{equation}
where the supremum is taken over all Schatten decompositions. Some
computations reduce it to the following form:

\begin{equation}
\renewcommand{\theequation}{2.5}
I\left( \rho ;\Lambda ^{*}\right) =\sup \left\{ \sum_{k}\lambda _{k}S\left(
\Lambda ^{*}E_{k},\Lambda ^{*}\rho \right) ;E=\left\{ E_{k}\right\} \right\}
\end{equation}
This mutual entropy satisfies all conditions (1)$\sim $(3) mentioned above%
\cite{O11}.

When the input system is classical, an input state $\rho $ is given by a
probability distribution or a probability measure, in either case, the
Schatten decomposition of $\rho $ is unique, namely, for the case of
probability distribution ; $\rho =\left\{ \lambda _{k}\right\} ,$

\begin{equation}
\renewcommand{\theequation}{2.6}
\rho =\sum_{k}\lambda _{k}\delta _{k},
\end{equation}
where $\delta _{k}$ is the delta measure, that is, $\delta _{k}\left(
j\right) =\delta _{k,j}=\{_{0(k\neq j)}^{1(k=j)},\forall j.$ Therefore for
any channel $\Lambda ^{*},$ the mutual entropy becomes

\begin{equation}
\renewcommand{\theequation}{2.7}
I\left( \rho ;\Lambda ^{*}\right) =\sum_{k}\lambda _{k}S\left( \Lambda
^{*}\delta _{k},\Lambda ^{*}\rho \right) ,
\end{equation}
which equals to the following usual expression of Shannon when it is
well-defined:

\begin{equation}
\renewcommand{\theequation}{2.8}
I\left( \rho ;\Lambda ^{*}\right) =S\left( \Lambda ^{*}\rho \right)
-\sum_{k}\lambda _{k}S\left( \Lambda ^{*}\delta _{k}\right) ,
\end{equation}
which has been taken as the definition of the mutual entropy for a
classical-quantum(-classical) channel \cite{BS,Hol,Ing,Lev}.

Note that the above definition of the mutual entropy (2.5) is
also written as

\begin{eqnarray*}
I\left( \rho ;\Lambda ^{*}\right)&=&  \sup \left\{ \sum_{k}\lambda
_{k}S\left( \Lambda ^{*}\rho _{k},\Lambda ^{*}\rho \right) \right. \\
&&\qquad\left. ;\rho =\sum_{k}\lambda _{k}\rho _{k}\in F_{o}\left(
\rho
\right) \right\} ,
\end{eqnarray*}
where $F_{o}\left( \rho \right) $ is the set of all orthogonal finite
decompositions of $\rho $ \cite{O11}$.$

More general mutual entoropy was defined in \cite{O4} based on
Araki's relative entoropy \cite{Ara}.

\section{Communication Processes}

The information communication process is mathematically set as follows: M
messages are sent to a receiver and the $k$th message $\omega ^{\left(
k\right) }$ occurs with the probability $\lambda _{k}$. Then the occurence
probability of each message in the sequence $\left( \omega ^{\left( 1\right)
},\omega ^{\left( 2\right) },\right.$ $\left.\cdot \cdot \cdot ,\omega ^{\left(
M\right) }\right) $of M messages is denoted by $\rho =\left\{ \lambda
_{k}\right\} ,$ which is a state in a classical system. If $\xi $ is a classical
coding, then $\xi \left( \omega \right) $ is a classical object such as an e
lectric
pulse. If $\xi $ is a quantum coding, then $\xi \left( \omega \right) $ is a
quantum object (state) such as a coherent state. Here we consider such a
quantum coding, that is, $\xi \left( \omega ^{\left( k\right) }\right) $ is
a quantum state, and we denote $\xi \left( \omega ^{\left( k\right) }\right)
$ by $\sigma _{k}.$ Thus the coded state for the sequence $\left( \omega
^{\left( 1\right) },\omega ^{\left( 2\right) },\cdot \cdot \cdot ,\omega
^{\left( M\right) }\right) $ is written as $\sigma =\sum_{k}\lambda
_{k}\sigma _{k}.$ This state is transmitted through a channel $\gamma $,
which is expressed by a completely positive mapping $\Gamma ^{*}$ from the
state space of $X$ to that of $\stackrel{\sim }{X}$ , hence the output coded
quantum state $\stackrel{\sim }{\sigma }$ is $\Gamma ^{*}\sigma .$ Since the
information transmission process can be understood as a process of state
(probability) change, when $\Omega $ and $\stackrel{\sim }{\Omega }$ are
classical and $X$ and $\stackrel{\sim }{X}$ are quantum, the process is
written as

\begin{equation}
\renewcommand{\theequation}{3.1}
P\left( \Omega \right) \stackrel{\Xi ^{*}}{\longrightarrow }\mathcal{S}
\left( \mathcal{H}\right) \stackrel{\Gamma ^{*}}{\longrightarrow }\mathcal{S(%
}\stackrel{\sim }{\mathcal{H}})\stackrel{\stackrel{\sim }{\Xi }^{*}}{%
\longrightarrow }P(\stackrel{\sim }{\Omega }),
\end{equation}
where $\Xi ^{*}$ $($resp.$\stackrel{\sim }{\Xi }^{*})$ is the channel
corresponding to the coding $\xi $ (resp. decoding $\stackrel{\sim }{\xi }$
).

We have to be care to study the objects in the above transmission process
(3.1). For instance, if we want to know the information capacity of a
quantum channel $\gamma (=\Gamma ^{*}),$ then we have to take $X$ so as to
describe a quantum system like a Hilbert space and we need to start the
study from a quantum state in quantum space $X\ $not from a classical state
associated to a message. If we like to know the capacity of the whole
process including a coding and a decoding, which means the capacity of a
channel $\stackrel{\sim }{\xi }\circ \gamma \circ \xi (=\stackrel{\sim }{\Xi
}^{*}\circ \ \Gamma ^{*}\circ \Xi ^{*})$, then we have to start from a
classical state$.$

\section{Channel Capacity}

We discuss two types of channel capacity in communication processes, namely,
the capacity of a quantum channel $\Gamma ^{*}$ and that of a classical
(classical-quantum-classical) channel $\stackrel{\sim }{\Xi }^{*}\circ \
\Gamma ^{*}\circ \Xi ^{*}.$

(1) {\it Capacity of quantum channel:} The capacity of a quantum
channel is the ability of information transmission of a quantum
channel itself, so that it does not depend on how to code a message
being treated as classical object and we have to start from an
arbitrary quantum state and find the supremum of the quantum
mutual entropy. One often makes a mistake in this point. For
example, one starts from the coding of a message and compute the
supremum of the mutual entropy and he says that the supremum is
the capacity of a quantum channel, which is not correct. Even when
his coding is a quantum coding and he sends the coded message to a
receiver through a quantum channel, if he starts from a classical
state, then his capacity is not the capacity of the quantum channel
itself. In his case, usual Shannon's theory is applied because he can
easily compute the conditional probability by a usual (classical) way.
His supremum is the capacity of a classical-quantum-classical
channel, and it is in the second category discussed below.

The capacity of a quantum channel $\Gamma ^{*}$ is defined as follows: Let $%
\mathcal{S}_{0}(\subset $ $\mathcal{S(H))}$ be the set of all states
prepared for expression of information. Then the capacity of the channel $%
\Gamma ^{*}$ with respect to $\mathcal{S}_{0}$ is defined by

\begin{equation}
\renewcommand{\theequation}{4.1}
C^{\mathcal{S}_{0}}\left( \Gamma ^{*}\right) =\sup \{I\left( \rho ;\Gamma
^{*}\right) ;\rho \in \mathcal{S}_{0}\}.
\end{equation}
Here $I\left( \rho ;\Gamma ^{*}\right) $ is the mutual entropy given in
(2.4) or (2.5) with $\Lambda ^{*}=\Gamma ^{*}.$ When $\mathcal{S}_{0}=%
\mathcal{S(H)}$ , $C^{\mathcal{S}(\mathcal{H)}}\left( \Gamma ^{*}\right) $
is denoted by $C\left( \Gamma ^{*}\right) $ for simplicity.

In \cite{OPW1,Mur-O1}, we also considered the pseudo-quant-um capacity
$C_{p}\left( \Gamma ^{*}\right)$ defined by (4.1) with the pseudo-mutual
entropy $I_{p}\left( \rho ;\Gamma ^{*}\right)$ where the supremum is taken
over all finite decompositions instead of all orthogonal pure
decompositions:
\begin{eqnarray*}
&&I_{p}\left( \rho ;\Gamma ^{*}\right)  \nonumber \\
&=&\sup \left\{ \sum_{k}\lambda _{k}S\left( \Gamma ^{*}\rho _{k},\Gamma
^{*}\rho \right) ;\rho =\sum_{k}\lambda _{k}\rho _{k},\right.  \nonumber \\
&&\qquad\left. \mbox{ finite decomposition}\right\} .
\quad\quad\quad\quad \mbox{(4.2)}
\end{eqnarray*}

However the pseudo-mutual entropy is not well-matched to the conditions
explained in Sec.2, and it is difficult to be computed numerically. The
relation between $C\left( \Gamma ^{*}\right) $ and $C_{p}\left( \Gamma
^{*}\right) $ was discussed in\cite{OPW1}. From the monotonicity of the
mutual entropy\cite{OP}, we have

\[
0\leq C^{\mathcal{S}_{0}}\left( \Gamma ^{*}\right) \leq C_{p}^{\mathcal{S}
_{0}}\left( \Gamma ^{*}\right) \leq \sup \left\{ S(\rho );\rho \in \mathcal{S%
}_{0}\right\} .
\]

(2) {\it Capacity of classical-quantum-classical channel:} The
capacity of C-Q-C channel $\stackrel{\sim }{\Xi }^{*}\circ \ \Gamma
^{*}\circ \Xi ^{*} $ is the capacity of the information transmission
process starting from the coding of messages, therefore it can be
considered as the capacity including a coding (and a decoding). As is
discussed in Sec.3, an input state $\rho $ is the probability
distribution $\left\{ \lambda _{k}\right\} $ of messages, and its
Schatten decomposition is unique, so the mutual entropy is written
by (2.7):

\begin{eqnarray*}
&&I\left( \rho ;\stackrel{\sim }{\Xi }^{*}\circ \ \Gamma ^{*}\circ \Xi
^{*}\right)   \\
&=&\sum_{k}\lambda _{k}S\left( \stackrel{\sim }{\Xi }^{*}\circ \
\Gamma ^{*}\circ \Xi ^{*}\delta _{k},\stackrel{\sim }{\Xi }^{*}\circ \
\Gamma ^{*}\circ \Xi ^{*}\rho \right)\\
&&
\mbox{\hspace{6cm}(4.3)}
\end{eqnarray*}
If the coding $\Xi ^{*}$ is a quantum coding, then $\Xi ^{*}\delta _{k}$ is
expressed by a quantum state. Let denote the coded quantum state by $\sigma
_{k}$ and put $\sigma =\Xi ^{*}\rho =\sum_{k}\lambda _{k}\sigma _{k}.$ Then
the above mutual entropy is written as

\begin{eqnarray*}
&&I\left( \rho ;\stackrel{\sim }{\Xi }^{*}\circ \ \Gamma ^{*}\circ \Xi
^{*}\right) \\
&=&\sum_{k}\lambda _{k}S\left( \stackrel{\sim }{\Xi
}^{*}\circ \
\Gamma ^{*}\sigma _{k},\stackrel{\sim }{\Xi }^{*}\circ \ \Gamma ^{*}\sigma
\right) . \quad\quad\mbox{(4.4)}
\end{eqnarray*}
This is the expression of the mutual entropy of the whole information
transmission process starting from a coding of classical messages. Hence the
capacity of C-Q-C channel is

\begin{eqnarray*}
&&C^{P_{0}}\left( \stackrel{\sim }{\Xi }^{*}\circ \ \Gamma ^{*}\circ \Xi
^{*}\right) \\
&=&\sup \{I\left( \rho ;\stackrel{\sim }{\Xi }^{*}\circ \ \Gamma
^{*}\circ \Xi ^{*}\right) ;\rho \in P_{0}\}, \quad\quad\mbox{(4.5)}
\end{eqnarray*}
where $P_{0}(\subset P(\Omega ))$ is the set of all probability
distributions prepared for input (a-priori) states (distributions or
probability measures). Moreover the capacity for coding free is found by
taking the supremum of the mutual entropy over all probability distributions
and all codings $\Xi ^{*}$:

\begin{eqnarray*}
&&C_{c}^{P_{0}}\left( \stackrel{\sim }{\Xi }^{*}\circ \ \Gamma ^{*}\right) \\
&=&\sup \{I\left( \rho ;\stackrel{\sim }{\Xi }^{*}\circ \ \Gamma ^{*}\circ \Xi
^{*}\right) ;\rho \in P_{0},\Xi ^{*}\}. \quad\,\mbox{(4.6)}
\end{eqnarray*}
The last capacity is for both coding and decoding free and it is given by

\begin{eqnarray*}
&&C_{cd}^{P_{0}}\left( \ \Gamma ^{*}\right) \\
&=&\sup \{I\left( \rho ;\stackrel{%
\sim }{\Xi }^{*}\circ \ \Gamma ^{*}\circ \Xi ^{*}\right) ;\rho \in P_{0},\Xi
^{*},\stackrel{\sim }{\Xi }^{*}\}. \mbox{(4.7)}
\end{eqnarray*}
These capacities $C_{c}^{P_{0}},$ $C_{cd}^{P_{0}}$ do not measure the
ability of the quantum channel $\Gamma ^{*}$ itself, but measure the ability
of $\Gamma ^{*}$ through the coding and decoding.

Remark that $\sum_{k}\lambda _{k}S(\Gamma ^{*}\sigma _{k})$ is finite, then
(4.4) becomes

\begin{eqnarray*}
&&I\left( \rho ;\stackrel{\sim }{\Xi }^{*}\circ \ \Gamma ^{*}\circ \Xi
^{*}\right) \\
&=&S(\stackrel{\sim }{\Xi }^{*}\circ \Gamma ^{*}\sigma
)-\sum_{k}\lambda _{k}S(\stackrel{\sim }{\Xi }^{*}\circ \Gamma ^{*}\sigma
_{k}). \quad\quad\mbox{(4.8)}
\end{eqnarray*}
Further, if $\rho $ is a probability measure having a density function $%
f(\lambda )$ and each $\lambda $ corresponds to a quantum coded state $%
\sigma (\lambda ),$ then $\sigma =\int f(\lambda )$ $\sigma (\lambda
)d\lambda $ and

\begin{eqnarray*}
\renewcommand{\theequation}{4.9}
&&I\left( \rho ;\stackrel{\sim }{\Xi }^{*}\circ \ \Gamma ^{*}\circ \Xi
^{*}\right)  \\
&=&S(\stackrel{\sim }{\Xi }^{*}\circ \Gamma ^{*}\sigma )-\int
f(\lambda )S(\stackrel{\sim }{\Xi }^{*}\circ \Gamma ^{*}\sigma
(\lambda ))d\lambda . \\
&&\mbox{\hspace{6cm}(4.9)}
\end{eqnarray*}
This is bounded by

\[
S(\Gamma ^{*}\sigma )-\int f(\lambda )S(\Gamma ^{*}\sigma (\lambda
))d\lambda ,
\]
which is called the Holevo bound and is computed in several ocassions\cite
{YO,OPW1}.

The above three capacities $C^{P_{0}},$ $C_{c}^{P_{0}},$ $C_{cd}^{P_{0}}$
satisfy the following inequalities
\begin{eqnarray*}
0 &\leq &C^{P_{0}}\left( \stackrel{\sim }{\Xi }^{*}\circ \ \Gamma ^{*}\circ
\Xi ^{*}\right) \leq C_{c}^{P_{0}}\left( \stackrel{\sim }{\Xi }^{*}\circ \
\Gamma ^{*}\right) \\
&\leq &C_{cd}^{P_{0}}\left( \ \Gamma ^{*}\right) \leq \sup \left\{ S(\rho
);\rho \in P_{o}\right\}
\end{eqnarray*}
where $S(\rho )$ is not the von Neumann entropy but the Shannon entropy: -$%
\sum \lambda _{k}\log \lambda _{k}.$

The capacities (4.1), (4.5),(4.6) and (4.7) are generally different. Some
misuses occur due to forgetting which channel is considered. That is, we
have to make clear what kind of the ability (capacity) is considered, the
capacity of a quantum channel itself or that of a
classical-quantum(-classical ) channel. The computation of the capacity of a
quantum channel was carried in several models in \cite{OPW1}

\section{Compound States and Entanglements}

Recently the quantum entangled state has been mathematically studied \cite
{BBPSSW,Maj,Sch1}, in which the entangled state is defined by a state not
written as a form $\sum_{k}\lambda _{k}\rho _{k}\otimes \sigma _{k}$ with
any states $\rho _{k}$ and $\sigma _{k}.$ A state written as above is called
a separable state, so that an entangled state is a state not belonged to the
set of all separable states. However it is obvious that there exist several
correlated states written as separable forms. Such correlated states have
been discussed in several contexts in quantum probability such as quantum
filtering \cite{B2}, quantum compound state \cite{O2}, quantum Markov state
\cite{A} and quantum lifting \cite{AO}. In \cite{BO}, we showed a
mathematical construction of quantum entangled states and gave a finer
classification of quantum sates.

For the (separable) Hilbert space $\mathcal{K}$ of a quantum system, let $%
\mathcal{A\equiv }$ $B\left( \mathcal{K}\right) $ be the set of all linear
bounded operators {on }$\mathcal{K}$. A normal state $\varphi $ on $\mathcal{%
\ \ A}$ can be expressed as $\varphi \left( A\right) =tr_{\mathcal{G}}\kappa
^{\dagger }A\kappa ,$ $A\in \mathcal{A}$, where $\mathcal{G}$ is another
separable Hilbert space, $\kappa $ is a linear Hilbert-Schmidt operator from
$\mathcal{G}$ to $\mathcal{K}$ and $\kappa ^{\dagger }$ is the adjoint
operator of $\kappa $ from $\mathcal{K}$ to $\mathcal{G}$ such that $\sigma
=\kappa \kappa ^{\dagger }$ is the (unique) density operator $\sigma \in
\mathcal{A}$ of the state $\varphi :\varphi \left( A\right) =trA\sigma $, $%
A\in \mathcal{A}$. This $\kappa $ is called the amplitude operator, and it
is called just the amplitude if $\mathcal{G}$ is one dimensional space
${\bf C}$, corresponding to the pure state $\varphi \left( A\right) =\kappa
^{\dagger }A\kappa $ for a $\kappa \in \mathcal{K}$ with $\kappa ^{\dagger
}\kappa =\Vert \kappa \Vert ^{2}=1$. In general, $\mathcal{G}$ is not one
dimensional, the dimensionality $\dim \mathcal{G}$ must be not less than $%
\mathrm{rank}\sigma $, the dimensionality of the range $\sigma \mathcal{K}$
of the density operator $\sigma .$

Since $\mathcal{G}$ is separable, $\mathcal{G}$ is realized as a subspace of
$l^{2}(\mathbf{N})$ of complex sequences $(i.e.,\zeta ^{\bullet }=\left(
\zeta ^{n}\right) ,$ $\zeta ^{n}\in {\bf C}$, $n\in \mathbf{N}$ with $\sum
\left| \zeta ^{n}\right| ^{2}<+\infty )$, so that any vector $\zeta
^{\bullet }=(\zeta ^{n}$) represents a vector $\zeta =\sum \zeta
^{n}|n\rangle $ in the standard basis $\left\{ |n\rangle \right\} \in
\mathcal{G}$ of $l^{2}(\mathbf{N})$ .

Given the amplitude operator $\kappa $, one can define not only the states $%
\sigma \equiv \kappa \kappa ^{\dagger }$ and $\rho \equiv $ $\kappa
^{\dagger }\kappa $ on the algebras $\mathcal{A}\left( =B\left( \mathcal{K}%
\right) \right) $ and $\mathcal{B}\left( =B\left( \mathcal{G}\right) \right)
$ but also an entanglement state $\Theta $ on the algebra $\mathcal{B}%
\otimes \mathcal{A}$ of all bounded operators on the tensor product Hilbert
space $\mathcal{G}\otimes \mathcal{K}$ by

\[
\Theta \left( B\otimes A\right) =tr_{\mathcal{G}}B\kappa ^{\dagger }A\kappa
=tr_{\mathcal{K}}A\kappa B\kappa ^{\dagger },
\]
for any $B\in \mathcal{B}$. This state is pure as it is the case of $%
\mathcal{F}={\bf C}$ in the theorem below, and it satisfies the marginal
conditions: For any $B\in \mathcal{B},A\in \mathcal{A}$,

\[
\Theta \left( B\otimes I\right) =tr_{\mathcal{G}}B\rho ,\quad \Theta \left(
I\otimes A\right) =tr_{\mathcal{K}}A\sigma .\quad
\]

\bigskip

\noindent
{\bf Theorem 5.1.} {\sl
Let $\Theta :\mathcal{B}\otimes \mathcal{A}\rightarrow {\bf C}$ be a state
\[
\Theta \left( B\otimes A\right) =tr_{\mathcal{E}}\psi ^{\dagger }\left(
B\otimes A\right) \psi ,
\]
defined by an amplitude operator $\psi $ on a separable Hilbert space $%
\mathcal{E}$ into the tensor product Hilbert space $\mathcal{G}\otimes
\mathcal{K}$ ; $\psi :\mathcal{E}\rightarrow \mathcal{G}\otimes \mathcal{K}$
with $tr_{\mathcal{F}}\psi ^{\dagger }\psi =1$. Then there exist a Hilbert
space $\mathcal{F}$ and an amplitude operator $\kappa :\mathcal{G}%
\rightarrow \mathcal{F}\otimes \mathcal{K}$ with
\begin{equation}
\renewcommand{\theequation}{5.1}
\kappa ^{\dagger }\left( I\otimes \mathcal{A}\right) \kappa \subset \mathcal{%
\ B},\;tr_{\mathcal{F}}\kappa \mathcal{B}\kappa ^{\dagger }\subset \mathcal{A%
}
\end{equation}
such that the state $\Theta $ can be achieved by an entanglement
\begin{eqnarray*}
\Theta \left( B\otimes A\right) &=&tr_{\mathcal{G}}B\kappa ^{\dagger }\left(
I\otimes A\right) \kappa \\
& =&tr_{\mathcal{F}\otimes \mathcal{K}}\left( I\otimes
A\right) \kappa B\kappa ^{\dagger } \mbox{\hspace{1.3cm}{\rm (5.2)}}
\end{eqnarray*}
The entangling operator $\kappa $ is uniquely defined up to a unitary
transformation of the minimal space $\mathcal{F}$.}

\bigskip

Note that the entangled state (5.2) is written as
\begin{equation}
\renewcommand{\theequation}{5.3}
\Theta \left( B\otimes A\right) =tr_{\mathcal{G}}B\phi \left( A\right) =tr_{%
\mathcal{K}}A\phi _{*}\left( B\right) ,
\end{equation}
where $\phi \left( A\right) =\kappa ^{\dagger }\left( I\otimes A\right)
\kappa $ is in the predual space $\mathcal{B}_{*}\subset \mathcal{B}$ of all
trace-class operators in $\mathcal{G}$, and $\phi _{*}\left( B\right) =tr_{%
\mathcal{F}}\kappa B\kappa ^{\dagger }$ is in $\mathcal{A}_{*}\subset
\mathcal{A}$. The map $\phi $ is the Steinspring form of the general
completely positive map $\mathcal{A}\rightarrow \mathcal{B}_{*}$, written in
the eigen-basis $\left\{ \left| n\right\rangle \right\} $ of $\mathcal{G}
\subseteq l^{2}\left( {\bf N}\right) $ of the density operator $\rho =\phi
\left( I\right) $ as
\begin{equation}
\renewcommand{\theequation}{5.4}
\phi \left( A\right) =\sum_{m,n}|m\rangle \kappa _{m}^{\dagger }\left(
I\otimes A\right) \kappa _{n}\langle n|,\quad A\in \mathcal{A}
\end{equation}
where $\kappa _{n}$ is the vector in $\mathcal{F}\otimes \mathcal{K}$ such
that $\kappa =\sum_{n}\kappa _{n}\langle n|$. The dual operation $\phi _{*}$
is the Kraus form of the general completely positive map $\mathcal{B}%
\rightarrow \mathcal{A}_{*}$, given in this basis as
\begin{equation}
\renewcommand{\theequation}{5.5}
\phi _{*}\left( B\right) =\sum_{n,m}\left\langle n\right| B\left|
m\right\rangle tr_{\mathcal{F}}\kappa _{n}\kappa _{m}^{\dagger },\quad B\in
\mathcal{B}.
\end{equation}
It corresponds to the general form of the density operator
\begin{equation}
\renewcommand{\theequation}{5.6}
\theta _{\phi }=\sum_{m,n}|n\rangle \langle m|\otimes tr_{\mathcal{F}}\kappa
_{n}\kappa _{m}^{\dagger }
\end{equation}
for the entangled state $\Theta $, characterized by the weak orthogonality
property
\begin{equation}
\renewcommand{\theequation}{5.7}
tr_{\mathcal{K}}\kappa _{n}\kappa _{m}^{\dagger }=p_{n}\delta
_{n}^{m}=\kappa _{m}^{\dagger }\kappa _{n}.
\end{equation}

\bigskip

\noindent
{\bf Definition 5.2.} {\sl
The dual map $\phi _{*}:\mathcal{B}\rightarrow \mathcal{A}_{*}$ to a
completely positive map $\phi :\mathcal{A}\rightarrow \mathcal{B}_{*}$,
normalized as $tr_{\mathcal{G}}\phi \left( I\right) =1$, is called the
quantum entanglement of the state $\rho =\phi \left( I\right) $ on $\mathcal{%
B}$ to the state $\sigma =\phi _{*}\left( I\right) $ on $\mathcal{A}$. The
entanglement by $\phi \left( A\right) =\sigma ^{1/2}A\sigma ^{1/2}$ of the
state $\rho =\sigma $ on the algebra $\mathcal{B}=\mathcal{A}$ given by the
standard entangling operator $\kappa =\sigma ^{1/2}$ is called standard.}

\bigskip

\section{d-Entanglements and Correspondences}

A compound state, playing the similar role as the joint input-output
probability measures in classical systems, was intorduced in \cite{O2} as
explained in Sec.2. It corresponds to a particular diagonal type
\[
\theta _{\phi }=\sum_{n}|n\rangle \langle n|\otimes \kappa _{n}\kappa
_{n}^{\dagger }
\]
of the entangling map (5.6) in the eigen-basis of the density operator
$%
\rho =\sum p_{n}|n\rangle \langle n|$, and is discussed in this section.
Therefore the entangled states, generalizing the compound state, also play
the role of the joint probability measures.

The diagonal entanglements are quantum correspondences of classical symbols
to quantum, in general not orthogonal and pure, states. The general
entangled states $\Theta $ are described by the density operators $\theta
_{\phi }$ of the form (5.6) which is not necessarily diagonal in the
eigen-representation of the density operator $\rho =\sum_{n}p_{n}|n\rangle
\langle n|$. Such nondiagonal entangled states were called in \cite{O4} the
quasicompound (q-compound) states, so we can call also the nondiagonal
entanglement the quantum quasi-correspondence (q-correspondece) in contrast
to the d-correspondences, described by the diagonal entanglements, giving
rise to the d-compound states.

Let us consider a finite or infinite input system indexed by the natural
numbers $n\in \mathbf{N}$. The associated space $\mathcal{G}\subseteq
l^{2}\left( \mathbf{N}\right) $ is the Hilbert space of the input system
described by a quantum projection-valued measure $n\mapsto |n\rangle \langle
n|$ on $\mathbf{N}$ giving an orthogonal partition of unity $I=\sum
|n\rangle \langle n|$ $\in \mathcal{B}$ of the finite or infinite
dimensional input Hilbert space $\mathcal{G}$. Each input pure state,
identified with the one-dimensional density operator $|n\rangle \langle
n|\in \mathcal{B}$ corresponding to the elementary symbol $n\in \mathbf{N}$,
defines the elementary output state $\omega _{n}$ on $\mathcal{A}$. If the
elementary states $\omega _{n}$ are pure, they are described by pure output
amplitudes $\upsilon _{n}\in \mathcal{K}$ satisfying $\upsilon _{n}^{\dagger
}\upsilon _{n}=1=tr\omega _{n}$, where $\omega _{n}=$ $\upsilon _{n}\upsilon
_{n}^{\dagger }$ are the corresponding output one-dimensional density
operators. If these amplitudes are non-orthogonal $\upsilon _{m}^{\dagger
}\upsilon _{n}\neq \delta _{n}^{m}$, they cannot be identified with the
input amplitudes $|n\rangle $.

The elementally joint input-output states are given by the density operators
$|n\rangle \langle n|\otimes \omega _{n}$ in $\mathcal{G}\otimes \mathcal{K}$%
, and their mixtures
\begin{equation}
\renewcommand{\theequation}{6.1}
\theta =\sum_{n}|n\rangle \langle n|\otimes \sigma _{n},\quad \sigma
_{n}=p_{n}\upsilon _{n}\upsilon _{n}^{\dagger }
\end{equation}
define the compound states on $\mathcal{B}\otimes \mathcal{A}$, giving the
quantum correspondences $n\mapsto |n\rangle \langle n|$ with the
probabilities $p_{n}$. Here we note that the quantum correspondence is
described by a classical-quantum channel, and the general d-compound state
for a quantum-quantum channel in quantum communication can be obtained in
this way due to the orthogonality of the decomposition (6.1),
corresponding to the orthogonality of the Schatten decomposition $\rho
=\sum_{n}p_{n}|n\rangle \langle n|$ of $\rho =tr_{\mathcal{K}}\theta $.

The comparison of the general compound state (5.6) with (6.1)
suggests that the quantum correspondences are described as the
diagonal entanglements
\begin{equation}
\renewcommand{\theequation}{6.2}
\phi _{*}\left( B\right) =\sum_{n}p_{n}\langle n|B|n\rangle \upsilon
_{n}\upsilon _{n}^{\dagger }
\end{equation}
which are dual to the orthogonal decompositions
\begin{equation}
\renewcommand{\theequation}{6.3}
\phi \left( A\right) =\sum_{n}p_{n}|n\rangle \upsilon _{n}^{\dagger
}A\upsilon _{n}\langle n|.
\end{equation}
\newline
These are the entanglements with the stronger orthogonality
\begin{equation}
\renewcommand{\theequation}{6.4}
tr_{\mathcal{F}}\kappa _{n}\kappa _{m}^{\dagger }=p_{n}\omega _{n}\delta
_{n}^{m},
\end{equation}
for the amplitudes $\kappa _{n}\in \mathcal{F}\otimes \mathcal{K}$ of the
decomposition $\kappa =\sum_{n}\kappa _{n}\langle n|$ in comparison with the
weak orthogonality of $\kappa _{n}$ in (5.7). The orthogonality (6.4)
can be achieved in the following manner: Take $\kappa _{n}=|n\rangle
\otimes \psi _{n}$ with $\psi _{n}=p_{n}^{1/2}\upsilon _{n}$ so that
\[
\kappa _{m}^{\dagger }\left( I\otimes A\right) \kappa _{n}=\left\langle
m\mid n\right\rangle \psi _{m}^{\dagger }A\psi _{n}=p_{n}\upsilon
_{n}^{\dagger }A\upsilon _{n}\delta _{n}^{m}
\]
for any $A\in \mathcal{A}$. Then, we have the following theorem.

\bigskip

\noindent
{\bf Theorem 6.1.} {\sl
Let $\mathcal{F=\oplus }_{n}\mathcal{F}_{n}$ and let $\psi _{n}$ be the
operators, defining a compound state of the diagonal form
\begin{equation}
\renewcommand{\theequation}{6.5}
\Theta \left( B\otimes A\right) =\sum_{n}\langle n|B|n\rangle tr_{\mathcal{F}%
_{n}}\psi _{n}^{\dagger }A\psi _{n}
\end{equation}
Then it corresponds to the entanglement by the orthogonal decomposition
\begin{equation}
\renewcommand{\theequation}{6.6}
\phi \left( A\right) =\sum_{n}|n\rangle \kappa _{n}^{\dagger }\left(
I\otimes A\right) \kappa _{n}\langle n|,
\end{equation}
mapping from the algebra $\mathcal{A}$ into a diagonal subalgebra of $%
\mathcal{B}$.}

\bigskip

Thus the entanglement (5.5) corresponding to (6.5) is given by
the dual to (6.6) diagonal map
\begin{equation}
\renewcommand{\theequation}{6.7}
\phi _{*}\left( B\right) =\sum_{n}\langle n|B|n\rangle \psi _{n}\psi
_{n}^{\dagger }
\end{equation}
with the density operators $\sigma _{n}=\psi _{n}\psi _{n}^{\dagger }$
normalized to the probabilities $p_{n}=tr_{\mathcal{K}}\psi _{n}\psi
_{n}^{\dagger }$.

\bigskip

\noindent
{\bf Definition 6.2.} {\sl
The positive diagonal map
\begin{equation}
\renewcommand{\theequation}{6.8}
\phi _{*}\left( B\right) =\sum_{n}\langle n|B|n\rangle \sigma _{n}
\end{equation}
into the subspace of trace-class operation $\mathcal{K}$ with $tr_{\mathcal{G%
}}\phi _{*}\left( I\right) =1$, is called quantum d-entanglement with the
input probabilities $p_{n}=tr_{\mathcal{K}}\sigma _{n}$ and the output
states $\omega _{n}=p_{n}^{-1}\sigma _{n}$, and the corresponding compound
state (2.2) is called d-compound state. The d-entanglement is called
c-entanglement and compound state is called c-compound if all density
operators $\sigma _{n}$ commute: $\sigma _{m}\sigma _{n}=\sigma _{n}\sigma
_{m}$ for all $m$ and $n$. }

\bigskip

Note that due to the commutativity of the operators $B\otimes I$ with $%
I\otimes A$ on $\mathcal{G}\otimes \mathcal{K}$, one can treat the
correspondences as the nondemolition measurements in $\mathcal{B}$ with
respect to $\mathcal{A}$. So, the compound state is the state prepared for
such measurements on the input $\mathcal{G}$. It coincides with the mixture
of the states, corresponding to those after the measurement without reading
the sent message. The set of all d-entanglements corresponding to a given
Schatten decomposition of the input state $\rho $ on ${\rm A}$ is
obviously convex with the extreme points given by the pure elementary output
states $\omega _{n}$ on $\mathcal{A}$, corresponding to a not necessarily
orthogonal decompositions $\sigma =\sum_{n}\sigma _{n}$ into one-dimensional
density operators $\sigma _{n}=p_{n}\omega _{n}.$

The orthogonal Schatten decompositions $\sigma =\sum_{n}p_{n}\omega _{n}$
correspond to the extreme points of c-entanglements which also form a convex
set with mixed commuting $\omega _{n}$ for a given Schatten decomposition of
$\sigma $. The orthogonal c-entanglements were used in \cite{AO} to
construct a particular type of Accardi's transition expectations \cite{A}
and to define the entropy in a quantum dynamical system via such transition
expectations\cite{BO}.

Thus we classified the entangled states into three categories, namely,
q-entangled state, d-entangled state and c-entangled state, and their
rigorous expressions were given.

\section{Quantum Mutual Entropy via Entanglements}

Let us consider the entangled mutual entropy by means of the above three
types compound states. We denote the quantum mutual entropy of the compound
state $\Theta $ achieved by an entanglement $\phi _{*}:$ $\mathcal{B}%
\rightarrow \mathcal{A}_{*}$ with the marginals
\begin{equation}
\renewcommand{\theequation}{7.1}
\Theta \left( B\otimes I\right) =tr_{\mathcal{G}}B\rho ,\;\Theta \left(
I\otimes A\right) =tr_{\mathcal{K}}A\sigma
\end{equation}
by $I_{\phi }\left( \rho ,\sigma \right) $ or $I_{\phi }\left( \mathcal{A},%
\mathcal{B}\right) $ and it is given as
\begin{equation}
\renewcommand{\theequation}{7.2}
I_{\phi }\left( \mathcal{\rho },\mathcal{\sigma }\right) =tr\theta _{\phi
}\left( \log \theta _{\phi }-\log \left( \rho \otimes \sigma \right) \right)
.
\end{equation}
Besides this quantity describes an information gain in a quantum system $%
\left( \mathcal{A},\sigma \right) $ via an entanglement $\phi _{*}$ with
another system ($\mathcal{B},\rho ),$ it is naturally treated as a measure
of the strength of an entanglement, having zero the value only for
completely disentangled states (7.1), corresponding to $\theta _{\phi
}=\rho \otimes \sigma $.

\bigskip

\noindent
{\bf Definition 7.1.} {\sl
The maximal quantum mutual entropy for a fixed state $\sigma $
\begin{equation}
\renewcommand{\theequation}{7.3}
H_{\sigma }\left( \mathcal{A}\right) =\sup \{I_{\phi }\left( \mathcal{A},%
\mathcal{B}\right) ;\phi _{*}\left( I\right) =\sigma \}
\end{equation}
is called q-entropy of the state $\sigma $. The differences
\begin{eqnarray*}
H_{\phi }\left( \mathcal{B}|\mathcal{A}\right)  &=&H_{\sigma }\left(
\mathcal{A}\right) -I_{\phi }\left( \mathcal{A},\mathcal{B}\right) , \\
D_{\phi }\left( \mathcal{B}|\mathcal{A}\right)  &=&S\left( \mathcal{\sigma }%
\right) -I_{\phi }\left( \mathcal{A},\mathcal{B}\right)
\end{eqnarray*}
are respectively called the q-conditional entropy on $\mathcal{B}$ with
respect to $\mathcal{A}$ and the degree of disentanglement for the compound
state $\phi $.}

\bigskip

$H_{\phi }\left( \mathcal{B}|\mathcal{A}\right) $ is obviously positive,
however\\ $D_{\phi }\left( \mathcal{B}|\mathcal{A}\right) $ has the positive
maximal value $S\left( \mathcal{\sigma }\right) =$ $\sup \left\{ D_{\phi
}\left( \mathcal{B}|\mathcal{A}\right) ; \phi _{*}\left( I\right)
=\sigma
\right\} $ and can achieve also a negative value
\begin{equation}
\renewcommand{\theequation}{7.4}
\inf \left\{ D_{\phi }\left( \mathcal{B}|\mathcal{A}\right) ;\phi _{*}\left(
I\right) =\sigma \right\} =S\left( \mathcal{\sigma }\right) -H_{\sigma
}\left( \mathcal{A}\right)
\end{equation}
for the entangled states \cite{BO}.

\bigskip

\noindent
{\bf Theorem 7.2. }{\sl
Let $\mathcal{A}$ be a discrete decomposable algebra $\oplus B\left(
\mathcal{K}_{i}\right) $ with a normal state $\sigma =\oplus \sigma _{i}$ ,
and $\mathcal{C}\subseteq \mathcal{A}$ be its center with probability
distribution $\mu =\oplus \mu _{i}$ induced by $\sigma .$ Then the q-entropy
is given by
\begin{equation}
\renewcommand{\theequation}{7.5}
H_{\sigma }\left( \mathcal{A}\right) =\sum_{i}\left( \mu _{i}\ln \mu
_{i}-2tr_{\mathcal{K}_{i}}\sigma _{i}\ln \sigma _{i}\right) ,
\end{equation}
It is positive, $H_{\sigma }\left( \mathcal{A}\right) \in [0,\infty ]$, and
if $\mathcal{A}$ is finite dimensional, it is bounded, $H_{\sigma }\left(
\mathcal{A}\right) \leq \dim \mathcal{A}$.}

\bigskip

Let us consider $\mathcal{G}$ as a Hilbert space describing a quantum input
system and $\mathcal{K}$ as its output Hilbert space. A quantum channel $%
\Lambda ^{*}$ sending each input state defined on $\mathcal{G}$ to an output
state defined on $\mathcal{K}.$ A deterministic quantum channel is given by
a linear isometry $\Upsilon $ $\mathrm{:}\mathcal{G}\rightarrow
$$\mathcal{K}$ with $\Upsilon ^{\dagger }\Upsilon =I_{0}$ ($I_{0}$
is the identify operator in $\mathcal{G}$) such that each input state
vector $\eta
\in \mathcal{G}$, $\left\| \eta \right\| =1$ is transmitted into an output
state vector $\Upsilon \eta \in \mathcal{K}$, $\left\| \Upsilon \eta
\right\| =1$. The mixtures $\rho =\sum_{n}p_{n}\omega _{n}$ of the pure
input states $\omega _{n}=\eta _{n}\eta _{n}^{\dagger }$ are sent into the
mixtures $\sigma =\sum_{n}p_{n}\sigma _{n}$ with pure states $\sigma
_{n}=\Upsilon \omega _{n}\Upsilon ^{\dagger }$. A noisy quantum channel
sends pure input states $\omega $ into mixed ones $\sigma =\Lambda
^{*}\omega $ given by the dual of the following completely positive map $%
\Lambda $%
\begin{equation}
\renewcommand{\theequation}{7.6}
\Lambda \left( A\right) =\Upsilon ^{\dagger }\left( I_{1}\otimes A\right)
\Upsilon ,\mathrm{\quad }A\in \mathrm{\mathcal{A}}
\end{equation}
where $\Upsilon $ is a linear isometry from $\mathcal{G}$ to $\mathcal{F}
_{1}\otimes \mathcal{K}$, $\Upsilon ^{\dagger }\left(
I_{1}\otimes I\right) \Upsilon =I_{0}$, and $I_{1}$ is the identity operator
in a separable Hilbert space $\mathcal{F}_{1}$ representing the quantum
noise. Each input mixed state $\rho $ $\in B\left( \mathcal{G}\right) $ is
transmitted into the output state $\sigma =\Lambda ^{*}\rho $ on $\mathcal{A}%
\subseteq B\left( \mathcal{K}\right) $, which is given by the density
operator
\begin{equation}
\renewcommand{\theequation}{7.7}
\sigma =tr_{\mathcal{F}_{1}}\Upsilon \rho \Upsilon ^{\dagger }\equiv \Lambda
^{*}\rho \in \mathcal{A}_{*}.
\end{equation}

We apply the proceeding discussion of the entanglement to the above
situation containing a channel $\Lambda ^{*}.$ For a given Schatten
decomposition $\rho =\sum_{n}p_{n}|n\rangle \langle n|\ $and the state $%
\sigma \equiv \Lambda ^{*}\rho ,$we can construct three entangled states of
the preceeding section:

(1) q-entanglement $\phi _{*}^{q}$ and q-compound state $\theta _{\phi }^{q}$
are given as

\begin{eqnarray*}
\phi _{*}^{q}(B) &=&\sum_{n,m}\left\langle n\mid B\mid m\right\rangle tr_{%
\mathcal{F}}\kappa _{n}\kappa _{m}^{\dagger } \\
\theta _{\phi }^{q} &=&\sum_{m,n}|n\rangle \langle m|\otimes tr_{\mathcal{F}%
}\kappa _{n}\kappa _{m}^{\dagger }
\end{eqnarray*}
with the marginals $\rho =\sum_{n}p_{n}|n\rangle \langle n|,$ $\sigma \equiv
\Lambda ^{*}\rho =tr_{\mathcal{G}}\theta _{\phi }^{q}$ and $tr_{\mathcal{K}%
}\kappa _{n}\kappa _{m}^{\dagger }=p_{n}\omega _{n}\delta _{n}^{m}=\kappa
_{m}^{\dagger }\kappa _{n}$ for $\omega _{n}=\Lambda ^{*}|n\rangle \langle
n|.$ Let $\mathcal{E}_{q}$ be the convex set of all completely positive maps
$\phi ^{q}$ .

(2) d-entanglement $\phi _{*}^{d}$ and d-compound state $\theta _{\phi }^{d}$
are given as

\begin{eqnarray*}
\phi _{*}^{d}(B) &=&\sum_{n}\left\langle n\mid B\mid n\right\rangle tr_{%
\mathcal{F}}\kappa _{n}\kappa _{n}^{\dagger } \\
\theta _{\phi }^{d} &=&\sum_{n}|n\rangle \langle n|\otimes tr_{\mathcal{F}%
}\kappa _{n}\kappa _{n}^{\dagger }
\end{eqnarray*}

\noindent with the same marginal conditions as (1). Let $\mathcal{E}_{d}$ be
the convex set of all completely positive maps $\phi ^{d}.$

(3) c-entanglement $\phi _{*}^{c}$ and c-compound state $\theta _{\phi }^{c}$
are same as those of (2) with commuting $\left\{ \omega _{n}\right\} .$ Let $%
\mathcal{E}_{c}$ be the convex set of all completely positive maps $\phi
^{c} $ .

Now, let us consider the entangled mutual entropy and the capasity of
quantum channel by means of the above three types of compound states.

\bigskip

\noindent
{\bf Definition 7.3.} {\sl
The mutual entoropy $I_{q}\left( \rho ,\Lambda ^{*}\right) $ and
q-capacity $C_{q}\left( \Lambda ^{*}\right) $ for a quantum
channel $\Lambda ^{*}$are defined by the supremums
\begin{eqnarray*}
I_{q}\left( \rho ,\Lambda ^{*}\right)  &=&\sup \left\{ S(\theta _{\phi
}^{q},\rho \otimes \Lambda ^{*}\rho );\phi ^{q}\in \mathcal{E}_{q}\right\} ,
\\
\;C_{q}\left( \Lambda ^{*}\right)  &=&\sup \left\{ I_{q}\left( \rho ,\Lambda
^{*}\right) ;\rho \right\} .
\end{eqnarray*}
The d-mutual entropy, d-capacity and c-mutual entropy, c-capacity are
defined as above using $\theta _{\phi }^{d}$ and $\theta _{\phi }^{c}$,
respectively.}

\bigskip

Note that due to $\mathcal{E}_{c}\subseteq \mathcal{E}_{d}\subseteq
\mathcal{E}_{q}$, we have the inequalities
\begin{eqnarray*}
I_{q}\left( \rho ,\Lambda ^{*}\right)  &\geq &I_{d}\left( \rho ,\Lambda
^{*}\right) \geq I_{c}\left( \rho ,\Lambda ^{*}\right) ,\; \\
C_{q}\left( \Lambda ^{*}\right)  &\geq &C_{d}\left( \Lambda ^{*}\right) \geq %
C_{c}\left( \Lambda ^{*}\right)
\end{eqnarray*}
for a deterministic channel ($\Lambda ^{*}=id$), the two lower mutual
entropies coincide with the von Neumann entropy:
\[
I_{d}\left( \rho ,id\right) =-tr\rho \log \rho =I_{c}\left( \rho ,id\right) .
\]
The capacity for such a channel is finite if $\mathcal{A}$ has a finite
rank, $C_{d}\left( \Lambda ^{*}\right) \leq \dim \mathcal{K}$. On the other
hand, the q-mutual entropy can achieve the q-entropy
\[
I_{q}\left( \rho ,id\right) =-2tr\rho \log \rho
\]
and its capacity is bounded by the dimension of the algebra $\mathcal{A}$,
$C_{q}\left( \Lambda ^{*}\right) \leq \dim \mathcal{A}$ which doubles the
d-capacity dim$\mathcal{K}$ when $\mathcal{A}=B\left( \mathcal{K}\right)$.

\footnotesize

redpajama_set_name: RedPajamaArXiv


text:
namespace Salient.ReliableHttpClient
{
    public enum RequestMethod
    {
        /// <summary>
        /// safe, idempotent: can cache, can retry
        /// </summary>
        GET,

        /// <summary>
        /// safe, idempotent: can cache, can retry
        /// </summary>
        HEAD,

        /// <summary>
        /// not safe, idempotent: do not cache, can retry
        /// </summary>
        PUT,

        /// <summary>
        /// not safe, idempotent: do not cache, can retry
        /// </summary>
        DELETE,

        /// <summary>
        /// not safe, not idempotent: do not cache, do not retry
        /// </summary>
        POST
    }
}


text:
Q: Is there a way to embed a logstash in java There is a test in an application I'm on that embeds an Elasticsearch instance (based on a .zip file) and uses it to tst if a class writes correctly to that Elasticsearch like this :
 embeddedElastic = EmbeddedElastic.builder()
            .withElasticVersion("5.5.2")
            .withInResourceLocation("./libs/elasticsearch-5.5.2.zip")
            .withStartTimeout(300L, TimeUnit.SECONDS)
            .withSetting(PopularProperties.HTTP_PORT, elasticPort)
            .build()

It's using this library : https://github.com/allegro/embedded-elasticsearch
Is there an equivalent of this library for logstash ?
Thanks

A: You could execute logstash as command line from a Java program

redpajama_set_name: RedPajamaStackExchange


Post here and a DM will help you out as soon as they possibly can.
Merchant store 11 - am now getting messages about paying the rent ???
Thought this was being handled by the DM's ?
Or is it a case that the server has not been reset in almost 2 weeks ?
Yea, apologies about that, been afk and missed paying everyones rent Will reimburse you the 50k as soon as I can.
Also when we get our new server we will be getting rid of the whole rent business.
So I turned in 10 gems of enhancement on my character Damien Bloodthorn, as well as 1 uncommon token, and 2 common tokens, all at once, and when I took the first level I gained from that (7 to 8 ) It took all the excess xp I earned from all those gems and tokens and erased it, I lost 23724 xp, and it did not reapply it, could I please be given that xp back? Pretty sure I just lost 2-3 levels right there that I had gained.
NecroDude wrote: So I turned in 10 gems of enhancement on my character Damien Bloodthorn, as well as 1 uncommon token, and 2 common tokens, all at once, and when I took the first level I gained from that (7 to 8 ) It took all the excess xp I earned from all those gems and tokens and erased it, I lost 23724 xp, and it did not reapply it, could I please be given that xp back? Pretty sure I just lost 2-3 levels right there that I had gained.
EDIT2: It also forgot all my progress, luckily enough I wasn't doing much, but a crafter would be really pissed.
EDIT3: Yeah, it's really laggy, I'm just gonna log off for now.
just had a look and seems fine at the moment. Also paid all stores rent.
Have sent another email off to see how close we are to getting the new server, just waiting for a reply.
If I dont hear back early this week I will get back on it and explain the urgency as the server is getting more borked each week.
Got them paid! There were a few stores that appear to be empty.
Okay... Umm how flexible is the DM group on requesting the list items and stuff that I can recall?
Sorry Lor we no longer have access like we did for the time being and if you can just send a list to me that will be fine.
Sooo... I logged on with my master crafter (Leopoldo da Quinci), but before I reached the swords, the game dropped out, and when I re-connected to the server the crafter was not listed, but a 1st level barbarian with no profile pic had appeared.
What are my chances of recovering this character?
Follow up edit: While I'm disappointed at having lost hundreds of hours of effort earning crafting levels on Leo, I'm looking at this as a chance to start afresh. I challenged myself to level up a character solely from crafting - never lifting a finger against a living thing.
I'd also be very happy if I could secure one of the unoccupied merchant stores to start flogging the stuff I make - I'll start with Alchemy so I can supply reasonably priced Heal pots to fund my workshop expenses.
hey did you ever get a store?
do we still have the character rebuild tower I remember from way back? I want to wind back my current char and redo her feats. I dont want to start a new character as it is levelling up with a current group playing now. I just ballsed up the feat selection and need a redo....its been a while.
Radhaz wrote: do we still have the character rebuild tower I remember from way back? I want to wind back my current char and redo her feats. I dont want to start a new character as it is levelling up with a current group playing now. I just ballsed up the feat selection and need a redo....its been a while.


text: typedef struct sqlite3_backup sqlite3_backup;


The 4th of July is a day to celebrate. Parades, barbecues and get-togethers of all types! And it's all about the kids, too! Red, white and blue streamers, hair ribbons, T-shirts, hand-held flags and adorned flip-flops proclaim our Independence Day!
In between the parades and fireworks, here's some fun for the kids!
Just sign-up below to get your 10-page guide for FREE!


blinking hard, as if to wake up and try it all again.
I guess he saw them first.
clipped my nails, hid my fur.
hunted me when I hurt them.
every third Tuesday when I would not sleep.
I ate as he ate, when one fell behind.
I grew teeth as he'd bite.
merging with the instincts of their parents.
of a living room lamp.


text:
CMX Winter Soirée: The Event About Community-Led Events
Wed, Dec 8, 2021, 10:00 AM (PST)
How do you combine events, engagement, education, fun, and the holidays? CMX Winter Soirée: The Event About Community-Led Events. Mix and mingle with the CMX community on Wednesday, December 8, starting at 10:00am PT! Like the title says, we will discuss all things community-led events, so no matter what holiday you celebrate, or if you celebrate a holiday at all, we just want to celebrate learning with YOU!
We've come to the end of another year and what have we learned about events? That community-led events are the future! We've teamed up with a few experts to share their learnings on driving engagement through specific event programming, balancing mental health and community-led event management, growing event programming year over year, and keeping diversity top of mind in the events and community industry.
Oh and let's not forget the fun:
Exclusive sneak peek (for what you ask? Come and find out!)
Festive photo booth
Cheerful networking
Heartfelt community stories
A 'walk' through a Winter Wonderland
See you at the event showcasing the hottest trends this winter!
Keneisha Williams
Black in Events Network
Jenny Sauer-Klein
Scaling Intimacy
Founder + CEO
Karlena Waugh
The Biz Mixer
Founder, Event Producer
Marsha Druker
Fuckup Nights Toronto
Corina Gheonea
Director, UiPath Community
Cristina Vidu
Regional Community Manager EMEA
Alexa Meyer
Co Founder & CEO
Alex Ranahan
Alex Ranahan Virtual Events Entertainment
Company Director / Virtual Event Host
Beth McIntyre
Head of Community
Keneisha was born and raised in the city of Toronto. She is an accomplished international Event professional with nearly 10 years of experience. Helping private and public organizations execute small to large-scale events within the social, not-for-profit and corporate spaces. She is also a consultant who is passionate about helping professionals and businesses become leaders through the power of their brand and create social impact. She attended Ryerson University studying Communications and Marketing, and would later study Events Management at the Humber College graduates program. She founded and managed SHESOFAB between 2010-2015 - a lifestyle company dedicated to empowering women through online content, campaigns and social events. In 2020 she founded The Black in Events Network – a globally inclusive initiative to amplify and create economic, educational social, opportunity for Black professionals working in the events industry. She has led keynotes, spoken on both live and virtual stages. She has also been seen in both IGNITE and Forbes magazine. Keneisha has volunteered for a number of places including the Canadian Cancer Society, Big Brother and Big Sisters, Ryerson Tri-Mentoring to name a few. She gives back her time as a mentor at both the Scarborough Women's Center and the Content Marketing Institute (CMI).
Jenny Sauer-Klein is passionate about designing transformational experiences that prioritize connection over content, and regularly consults for organizations like Google, Airbnb, Dropbox and the Haas School of Business. She is the founder and CEO of the Scaling Intimacy School of Experience Design, which trains event leaders to strategically weave connection into the fabric of their events. She created Play On Purpose, a video tutorial library with over 80 of her best virtual and in-person team building games, to support facilitators to confidently engage and connect groups and teams. In her previous career as the co-founder of AcroYoga, she successfully scaled an international brand, community and organization with millions of practitioners. She is a frequent presenter at international conferences and has been featured in The New York Times, Forbes, Fast Company, and Inc., as well as Tim Ferriss' book Tools of Titans.
Karlena is a millennial entrepreneur who is passionate about community building and collaboration in business. With a diverse background in residence life, student affairs and community building, Karlena utilizes these skills to effortlessly build rapport with clients, create a sense of community, and nurture strong business relationships.As an event producer, Karlena creates spaces for entrepreneurs, small business owners and women to showcase their businesses, learn from fellow entrepreneurs, and build relationships across industries through The Biz Mixer initiative.
Marsha Druker is the Founder of Fuckup Nights Toronto, a community that de-stigmatizes failure. Previously, she held various Marketing roles in companies ranging from consumer packaged goods to tech startups. She is also the host of the Create Community podcast where she chats with fellow community builders to define what community truly means.
Corina is the Director of the UiPath Community, passionate about building meaningful systems, playgrounds and connections for people and communities to thrive. For more than three years in charge of the automation community in the tech space for UiPath and for more than 10 years into community building, developing communities for tech, new educational systems and entrepreneurs. Before UiPath, she co-founded and managed communities for The Entrepreneurship Academy and the Alternative University while offering consultancy to organisations interested in better internal communities and learning programs.
Cristina thinks of herself as a consulting wizard and marketing professional. She loves brands, videos, events, but, above all, the wonder that lies people's powers and abilities. She joined Community at UiPath after an extended experience in events, business communities and project facilitation. Being a PolSci and Philosophy major, for her it's always about angles, the common good, and win-win situations. She loves the ecosystem view that only community can provide, and all the paths and threads it ties together. She's also passionate about design and user experience, having co-founded the first Design Thinking Conference in Romania. Together we can build any reality, therefore I build Community at UiPath.
Alexa is the Co-Founder & CEO of Coa, the gym for mental health. Coa offers therapist-led emotional fitness classes & 1:1 therapy, all grounded in community. After trying to take care of her own mental wellbeing, she was frustrated with the lack of options and asked: "Why isn't there a gym for mental health?" This sent her on a mission to make emotional fitness as inviting and community-based as physical fitness. She started the first "Mental Health Pop-Up" and helped thousands of people across the country try therapy. She continues this mission at Coa where she uses her background in start-ups, product, and branding to change how the world thinks about emotional wellbeing. Prior to Coa, Alexa was a VP at a fast-paced tech startup, worked at the world's largest brand strategy firm, Interbrand, and studied psychology and consumer behavior in Canada (her home country!)
Winner of the Leading Provider of Virtual Entertainment Events UK and Virtual Events Entertainment Provider of the Year awards this year, Alex's mission is to bond and bring people together across the world with ecstatic energy, joy, and magical moments through participating in his virtual game shows and workshops; so you can unite your team and communications with passion, excitement and a shared connection from a positive, immersive experience.
Beth is the Head of Community at Bevy and CMX. She oversees the engagement, strategy, and structure of all CMX Community spaces, as well as the CMX Connect program – an in-person distributed events program turned virtual in 2020. Beth lives in the incomparable Rocky Mountains in Jasper, Canada and spends most of her free time in total awe of their majesty.
inSided
Create one central hub for all customer needs with the industry's favorite community platform.
Build relationships, not spreadsheets.
Circosphere is a women-owned SF bay area based entertainment company offering unique, interactive, and wow-factor experiences for live and virtual events. Circosphere offers a wide variety of services for digital events including virtual cirque shows, classes and workshops, interactive activities for teams, family entertainment and much more. Make your next event event shine with our extensive preplanning, technical support staff, and astounding talent.
Tribute - The World's Most Meaningful Gift
Tribute would love to support you in bringing the most meaningful gift (acknowledgement, gratitude, and meaningful sentiments) to your community, company, or organization. Ali, our Director of People & Partnerships, can support you in learning more about how Tribute loves working with partners - ali@tribute.co.
Discourse is a powerful, customizable, open-source platform designed to facilitate civilized discussion for your community. We're re-imagining what a modern discussion platform should be. With an automatic trust system, powerful community moderation tools, and a focus on conversation at its core, Discourse gives you the tools to create and scale a healthy, vibrant, and fun place to connect with your customers, superfans, or teammates online.
Grow and measure your community across any platform with Orbit, mission control for your community. Orbit gives you a single view of your community by integrating data from everywhere your community interacts—including social, forums, chat, and other sources. We track members and interactions, providing a 360° view of who's doing what and where. These trigger alerts that automate common tasks and our reporting unlocks new insights.
Kissflow Community Platform. A fully managed community platform.
Superlinked
Superlinked connects your community members and encourages collaboration. From onboarding to match-making, we offer the smoothest member experience by integrating with your existing software and data. And best of all, we highlight your most helpful members to drive the flywheel of awesomeness. Let's build the future together!
Scale your community while keeping it personal.
Every Organization Needs a Community.
Khoros
Khoros, built from Spredfast + Lithium, is the leading customer engagement platform built to turn siloed knowledge into enterprise value, and customers into contributors. By connecting consumer insights across all departments, Khoros gives companies the ability to run their business with their customers, anticipating their needs and accelerating sales, loyalty, and innovation.
We're obsessed with engagement. Create Your Thriving Online Community.
Play on Purpose
Strong communities start with meaningful connection.


Tinder continues to implement important changes in its platform. Half a year ago came the web version , for those who want to keep looking for a partner while they are on their computer.
Months later announced Tinder Gold , a feature that lets you know which users have slid their finger to the right in the application. Now, this feature reaches the whole world , ensuring that it is a "first class experience".
When activating Tinder Gold, we will have unlimited access to 'Like', five Super-Likes (which automatically notifies the person sending it to you) and a monthly Boost (it makes your profile appear 60% more. more interesting is to be able to see which users have liked your photography.
We know you You live in a world where every moment counts, in which speed means success. You always walk from one place to another, and you can not allow anything to slow you down. Especially your Tinder feed.
It must be made clear that this option is not retroactive , so you can only see which people have liked your profile from the moment you activate it.
Currently, this functionality is available to anyone who uses Tinder on iOS , and they hope to activate it soon on Android. You will have to pay five dollars a month to be able to activate it, which is added to the five dollars that are paid by Tinder Plus .


I am working as a Senior Consultant for Deloitte and am supporting the claims application for a major healthcare company of the US.
I am Gwenaelle BAIVIER, French, 28. I speak professionally English and Japanese. I have work experience in Japan and would like to work in Japan.
I am a native German speaker with a passion for written language. I am fluent in spoken and written English, as well as speaking Japanese on a daily basis. Currently living in Osaka, Japan.
I worked in japan 4 years .tokyo dome hotel .
Kitchen helpers . House keeping .agriculture's. Food paking .fish paking.i worked in Hitachi.


We are very pleased to announce the release of our new look website for your perusal.
As always our aim is to keep the information informative and not too technical.
We believe it is our shop window, where visitors can find out what ITC Concepts is really about and how we have developed as a business this last 23 years. It details the nature of projects we undertake, the different sectors and clients that we serve, and the amazing talented staff that we employ.
Our ITC Newspage will keep visitors up to date with current projects and corporate news.
Additionally, we hope you find our leaning towards more information about our teams and individuals within the business, who you may deal with but don't actually know, an informative insight!
We hope you enjoy the change and we welcome any feedback you may have.


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Q: Ubuntu loads and then monitor loses signal? Alright so about a couple of years ago I built a PC and loaded a version of Ubuntu onto it in order to learn more about it. Got frustrated and kind of forgot about it. Fast forward to a couple of years later and now I'm back trying to get it to work again.
Here are the specs

*

*CPU: intel core i5

*Motherboard: Gigabyte GA-Z97X-UD3H-BK

*GPU: Radeon XFX R9 290

*RAM: 8GB Gskill Ripjaws DDR3 (1)

*Ubuntu Version: unknown

Alright so the first problem that I am currently tackling is getting the thing to even boot up.  The splash screen for Ubuntu comes up with the dots, the dots load up and then the monitor loses signal.  Here is a link to a video that made which shows exactly what is going on.  Any help is greatly appreciated!
https://www.youtube.com/watch?v=P1OJfr_yD6A

A: Most likely it's either a wrong display mode which the monitor cannot handle, or Ubuntu, for some reason, switches from the graphics adapter you use to another one.
Since you have an i5 CPU with integrated graphics and a motherboard with DVI/HDMI, etc connectors, the easiest thing to try first is swapping your display cable from your graphics card and plug it into the motherboard connector (I would try this before boot). If you're already using the motherboard connector, try this reversed - naturally.
Now if this doesn't help, you'll have to get to a terminal prompt and reconfigure the GUI display settings.
Most likely you can get to a terminal by pressing CTRL-ALT-F1. Then login and follow one of the several answers here about re-configuring or resetting the GUI, for example:
How to reset monitor settings to default through terminal?
Now... if this really is a old "testing machine" with a old Ubuntu installation - the easiest way to solve all your problems is probably going to be a full installation of a newer version of Ubuntu. Just a thought.
At least do a full upgrade before attempting a fix; there might be never drivers in the new kernel, more up to date settings in X configs, etc.

A: Too less information.
Blank screen means that there is problem with video output, but not to system itself.
Possible causes:

*

*monitor video mode out of range (resolution/frequency)

*after splash screen switched to onboard intel/radeon video card

Try:

*

*after boot (on black screen), try to press Ctrl+Alt+F1 (F1 to F8 consoles, F8 - default graphical console, others ar terminal consoles), if this works, then You will be able to login in terminal and read log files

*after boot (on blank screen) try to connect video cable to onboard intel/radeon

*if nothing works, try to remove radeon card and then start ubuntu (should work with intel integrated video card)
try to get ubuntu version, Xorg log for errors.
You can boot your computer from cdrom/usb flash with ubuntu install media to access version (/etc/lsb_release  or /etc/os-release), get logs:
/var/log/Xorg.0.log /var/log/messages /var/log/syslog  and dmesg, check for errors related to video.

A: What fixed it for me:

*

*Find out what driver you have:
lspci -nnk |grep -iA3 vga
In my case it was Kernel modules: gma500_gfx

*Blacklist that driver in /etc/modprobe.d/blacklist.conf by adding
blacklist gma500_gfx

*Run this command to take it into effect:
sudo update-initramfs -u


text:
Johannes Lauridsen (25 December 1930, Frederikshavn – 25 October 2006) was a Danish long distance runner. Lauridsen took part in the 1960 Summer Olympics in Rome, placing 41st in the marathon with a time of 2:32:32. He won the 8 km race at the 1956 .

References

1930 births
2006 deaths
Danish male marathon runners
Athletes (track and field) at the 1960 Summer Olympics
Olympic athletes of Denmark
People from Frederikshavn
Sportspeople from the North Jutland Region

redpajama_set_name: RedPajamaWikipedia


One of three players to average double figures as he finished with 10.3 points per game ... Shot 44.3 percent from the floor, 38.8 percent beyond the arc and 72 percent at the charity stripe ... Dished out 81 assists against 60 turnovers while leading the team in minutes played (1085) ... Reached double figures in 16 total games in which A-State was 13-3 ... Tied for the team high with six rebounds in season opener at North Dakota State (Nov. 11) ... Had game high 18 points on 5-of-7 shooting in home opener against Central Baptist (Nov. 14) ... Logged 12 points, three rebounds and three assists in road win at Georgetown (Nov. 17) ... Netted nine points in win at Army (Nov. 19) ... Posted seven points in win over Chattanooga (Nov. 20) and hit a trey for the fifth-straight game ... One of three Red Wolves in double figures as he had 11 points along with five rebounds and four assists in win against SIUE (Nov. 22) ... Logged 11 points, five assists and four rebounds against Lehigh (Nov. 27) ... Managed eight points, two rebounds and three assists in win over Cleveland State (Nov. 30) ... Netted nine points, pulled down three rebounds and dished out season best six assists in road win at UT Martin (Dec. 14) ... Posted eight points, four rebounds and two assists in road win at Central Arkansas (Dec. 16) ... Posted 11 points on 5-of-8 shooting at Minnesota (Dec. 23) ... Poured in career high 27 points in league opener against Louisiana (Dec. 31) ... Had career high eight rebounds in addition to nine points and three assists against ULM (Jan. 2) ... Reached double figures with 10 points and added four rebounds and two assists against Troy (Jan. 21) ... Had 10 points and two steals in win against South Alabama (Jan. 23) ... Led A-State with 15 points and added two rebounds and two assists in road win at ULM (Jan. 28) ... Posted 17 points, four rebounds and three assists in win at Louisiana (Jan. 30) ... Played all 40 minutes and had 11 points, one rebound and three assists against Appalachian State (Feb. 4) ... Had game high 19 points along with three rebounds and four assists at UT Arlington (Feb. 11) ... Logged 10 points and four assists in win over Little Rok (Feb. 17) ... Had 12 points on 4-of-9 shooting against Georgia State (Feb. 27).
Garnered First Team All-Region 16 recognition last season at MACC (Moberly, Mo.), leading the Greyhounds to the Sweet 16 of the NJCAA Division I Tournament and a 25-11 record ... Averaged 12.6 points, 3.9 assists and 3.3 rebounds over 36 games while shooting 45.5 percent (165-363) from the field and 38.1 percent (56-147) beyond the arc ... Shot 85.7 percent (66-77) from the charity stripe to rank 21st nationally ... Helping the Greyhounds to their first appearance in the NJCAA Tournament since 2005, he scored double figures in 22 games, including each of the last eight contests ... Sscored 26 points in Moberly's victory over State Fair in the Region 16 championship game and added 19 points in the District IV win over Sauk Valley (Ill.) to clinch the Greyhounds spot in the national tournament ... In the NJCAA Tournament opener, he scored 19 points and added six boards as Moberly topped Indian Hills ... The Greyhounds advanced to play tournament host Hutchinson and fell to the eventual national runners-up 84-74 ... Had 18 points and five assists while shooting 50 percent (7-14) from the floor.
Averaged 11.9 points, 4.5 assists and 3.0 rebounds in 31 games ... Helped the Greyhounds to a 22-10 record and received First Team All-Region 16 accolades ... Advanced to the Region 16 championship game, falling to Mineral Area College 69-53 ... Had 16 points and seven rebounds in the championship game and finished his freshman season with 21 games in double figures.
Prepped at McCluer North High School (Florissant, Mo.) and averaged 21.5 points a game leading the Stars to a 19-8 record in his senior season ... He also averaged 3.6 rebounds and 2.7 assists while shooting 47 percent from the field and 77 percent from the charity stripe ... As a junior, he averaged 21.1 points in leading the Stars to the quarterfinals of the Class 5 state tournament ... He was a four-year varsity starter and attended and played basketball at Mary Institute Country Day School as a freshman and sophomore before moving to the McCluer North school district.


Upper Room Hotel serves tasty and healthy all-you-can-eat breakfast buffet. The breakfast is included in the accommodation rate; it will give you energy and set you in a good mood! The buffet presents cornflakes, various sorts of cheese, sausage and ham, smoked salmon, fresh fruit and fruit salad, jams and marmalades. The beverage menu consists of juice, milk, coffee and different sorts of tea.
Breakfast is served from Monday till Friday 6:30 am until 10 am; Saturday and Sunday 6:30 – 10:30.