HAVRIX is indicated for active immunisation against hepatitis A virus (HAV) infection in susceptible subjects at risk of exposure to HAV. HAVRIX JUNIOR is indicated in subjects aged 2 to 15 years and HAVRIX 1440 is indicated in subjects aged 16 years and older. In areas of low prevalence of hepatitis A, immunisation with HAVRIX is particularly recommended in the following subjects: Travellers : Persons travelling to areas of intermediate or high endemicity for hepatitis A. These areas include Africa, Asia, India, the Pacific Islands, the Mediterranean basin, the Middle East, Central and South America. Armed Forces : Armed forces personnel who travel to higher endemicity areas or to areas where hygiene is poor, have an increased risk of HAV infection. Persons for whom hepatitis A is an occupational hazard or for whom there is an increased risk of transmission. These include: employees in day-care centres particularly in situations where children have not been toilet trained; teachers and other close contacts of the intellectually disabled; staff and residents of residential facilities for the intellectually disabled; health workers and teachers in remote Aboriginal and Torres Strait Islander communities; nursing staff and other healthcare workers in contact with patients in paediatric wards, infectious diseases wards, emergency rooms and intensive care units sewerage workers ; food handlers, since food hygiene procedures and food processing methods are not always adequate to protect from contamination from food handlers. Homosexual men : Increased incidence of hepatitis A infection among homosexual males suggests that the disease may be sexually transmitted in this group. Contacts of infected persons : Since virus shedding from infected persons may occur for a prolonged period, active immunisation of close contacts is recommended. The use of vaccine in outbreak control has been shown to be more effective than the use of immunoglobulin. Specific population groups known to have a higher incidence of hepatitis A: eg. Australian aboriginals, recognised community-wide HAV epidemics. Individuals with chronic liver disease and recipients of liver transplants, as hepatitis A infections is likely to be more severe in these groups. Many injecting drug users will have pre-existing liver disease from hepatitis B or hepatitis C infection. Recipients of blood products, such as Factor VIII eg. haemophiliacs. HAVRIX will not prevent hepatitis infection caused by other agents such as hepatitis B virus, hepatitis C virus, hepatitis D virus, hepatitis E or other pathogens known to infect the liver.
A turbid liquid with a slow settling white deposit.


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text:
Gavin and Stacey cast WILL reunite this Christmas but with a difference
The cast of Gavin and Stacey will reunite for a brand new Christmas special but it will be a lot different
Get ready to shout 'tidy' as loud as you can as the cast of Gavin and Stacey WILL reunite for a new Christmas special this year.
Gavin & Stacey's Christmas Special was the most watched TV show of last year, attracting some 18.5 million viewers to BBC1 on Christmas Day, well ahead of any other show on TV on December 25.
We were all left wondering if Smithy would accept Nessa's shock proposal and we still are.
Months after Gavin and Stacey won the Landmark accolade at this year's National Television Awards, it returned to primetime TV when the BBC aired repeats during the peak of lockdown.
Vicar of Dibley Christmas episodes confirmed with fans declaring show's return has 'made their year'
Quality Street truths exposed on how many of each chocolate you get leaving ITV viewers furious
Writers and Nessa's real life alias Ruth Jones devastated fans back in September when she shut down speculation of more episodes, mainly because James Corden is in America, but it has now been confirmed that Gavin and Stacey themselves, Matthew Horne and Joanna Page are bringing the cast back together over the festive season.
And, while a new episode won't air on TV, they will be joined by the likes of Larry Lamb, Alison Steadman, Melanie Walters, Steffan Rhodri and Robert Wilfort for a special on BBC Radio Wales.
James has also recorded a very special message from his home in Los Angeles.
Matt said: "Christmas Day is a special day and, particularly this year, it's been very challenging for a lot of people, so it's nice to feel like we might be bringing a little bit of joy," he added.
If you're not in Wales, you can listen to Radio Wales live online, or catch up with shows from the past 30 days, from the homepage - bbc.co.uk/radiowales.
Vera guest star Krissi Bohn thanks cast after 'taking on' Brenda Blethyn
VeraKrissi Bohn appeared in the latest episode of Vera on Sunday, with her character daring to take on DCI Stanhope
Ant McPartlin chokes back tears at live Britain's Got Talent show telling fans he missed them
Britain's Got TalentAnt and Dec returned to host the talent competition in front of a live audience for the first time in two years
Geordie Hospital star Kit, 4, thrives after heart transplant and should inspire organ donation discussion'
Freeman Hospital NewcastleGeordie Hospital's Kit Matthews features prominently in the show's opening episode as he waits for a heart transplant. His family - and the consultant who treated him - hope it can help people realise how vital transplants can be.

redpajama_set_name: RedPajamaCommonCrawl


text:
10 Top-Rated & Best Seller Ground Coffee
3 Top-Rated & Best Seller Nintendo Switch
10 Top‐Rated & Best Seller Waffle Makers
10 Top-Rated & Best Seller Coffee Makers
Cheapest keurig k575
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September 23, 2020 November 11, 2020 1 year ago Mark Prakanong	940 Views 0 Comments	Nintendo, Nintendo Switch, nintendo switch games, Nintendo Switch Lite, nintendo switch price
Nintendo, a Japanese multinational consumer electronics and video game company, started off as a card company in the early 1900s. From that onwards, the company slowly progresses by adapting and developing new products in accordance with the specific needs and wants of the customers that change over time. Nintendo was outstanding due to the astounding diversity, causing it to destroy numerous boundaries. One of them was breaking records of revenues, net sales, and profits in 2009 after the Nintendo DS and Wii release. However, in subsequent years, its revenue started to decline until Nintendo decided to unveil its latest product to bring back popularity and reputation.
In 2017, Nintendo Switch was released worldwide by Nintendo in the likes of bringing the company back to the limelight. Nintendo is unlike other consoles as it is a hybrid console that can be used as a home console and portable device. The specific Joy-con controller is wireless, with standard buttons and directional analog sticks for user input, motion sensing, tactile feedback, and can attach to both sides of the console to support handheld-style play. Not long after, Nintendo Switch was shipped nearly three million in the first month of its launch, and within a year, the Nintendo Switch achieved over 14 million units sold worldwide, outselling others by far. As of June 2020, the Nintendo Switch, together with other versions, sold more than 61 million units globally.
Nintendo has always believed in developing their products to suit the current times. Following that belief, Nintendo has released a number of different versions of the Nintendo Switch. Nevertheless, the top 3 rated editions are the Nintendo Switch with Neon Blue and Neon Red Joy-Con – HAC-001(-01), the Nintendo Switch Pro Controller – Xenoblade Chronicles 2 Edition, and the Nintendo Switch Lite.
1.Nintendo Switch with Neon Blue and Neon Red Joy-Con – HAC-001(-01)
2.Nintendo Switch Pro Controller – Xenoblade Chronicles 2 Edition
3.Nintendo Switch Lite
This Nintendo Switch comes with a color of neon blue and red, in which the model number is HAC-001(-01), corresponding to its name. It can be played in 3 different styles that are TV Mode, Tabletop Mode, Handheld mode. This version of the Nintendo Switch contains a long span of battery life at approximately 4.5 hours to over 9 hours, but it can vary depending on the software usage conditions. The multi-touch capacitive touch screen is 6.2 inches long that is an LCD screen with a resolution of 1280×720. The screen plus the Joy-Con will be around 4 inches high, 9.4 inches long, and 0.55 inches deep. It can be connected to the Wi-Fi for multiplayer gaming, and up to 8 consoles can be connected for local wireless multiplayer.
The outstanding robust point about this model is the extended battery life. As a handheld, it is quite powerful (the latest iteration nearly doubles the handheld's battery life, addressing the Switch's greatest weakness). This allows the player to play the same game on TV and keep playing as the Switch is taken out with the player quite so a unique ability. The uniqueness enables gaming on the go to be effortless, suiting larger varieties of people, mainly busy adults. By playing it, the busy adults can reminisce about the fun they had in the past, along with destressing themselves from work. Also, the Nintendo Switch is designed to be played by many players at once, and doing so will increase the family's enjoyment, creating long-life bonds between the members. Last but not least, the model comes with great graphics due to the fact that Nintendo has done high-quality ports from older consoles such as "Mario Kart 8" as well as several new titles such as "Super Smash Brothers Ultimate" or "Luigi's Mansion 3".
Neon Blue and Neon Red Joy-Con / Gray Joy-Con
Play Modes :
TV Mode, Tabletop, Handheld
Compatible Games :
All Nintendo Switch Games
Joy-Con Controllers Included :
Nintendo Switch Dock Included :
4" high 9.4" long .55" deep
Approximately .88 lbs
Screen :
6.2" Touch Screen
Battery :
4.5-9 hours
Check price & Spec & Ratings & Customer reviews
This Nintendo Switch Pro Controller has a present color of Xenoblade Chronicles 2 theme with a combination of black, purplish magenta, and little green color on it. It also includes a charging cable (USB-C to USB-A) by which amiibo is sold separately. Many available selections are a single controller, a controller with the 'Mario Kart 8' game, a controller with the 'Super Smash' game, and a controller with 'The Legend of Zelda' game. There are a few more advantages to this controller. Similarly, the Nintendo Switch Pro Controller – Xenoblade Chronicles 2 edition is well-known for its extended hour lifespan as on the box itself showed that it could go 40 hours without charging. However, the difference that this edition has compared to the others is effortless to charging. It charges over the included USB-C cable that can be connected to one of the two front-facing USB ports on the switch dock or any other USB that powers the device according to the specifications. This allows the controller to be charge relatively fast and can be charged anywhere. Another benefit to this controller is the unique grip compared to the Joy-Con, enabling the player to feel comfortable gripping for a few hours of gameplay. Besides comfortability, the controller is designed rather big that perfectly fits the player's hands, resulting in the player able to generate full capacity towards gaming, thereby increasing the player's performance. The Xenoblade Chronicles 2 edition is considered a "traditional" controller meaning that the great majority of people will instantly know how to use it and get used to it quite quickly, resolving in a much lesser time wasted to teach new players on how to use the controller.
Approximately 14.1 ounces
5.12 x 6.3 x 2.76 inches; 14.08 Ounces
The famous Nintendo Switch Lite comes in many different colors: coral, gray, turquoise, and yellow. Not only different colors, but also different styles, for instance just the lite console, the lite console with 128GB memory card, the lite console with 256GB memory card, and the lite console with 64GB memory card. The console is optimized for personal, handheld play. It is a small and light Nintendo Switch system, featuring a sleek, unibody design with fully integrated controls and a built-in control pad. It is compatible with all physical and digital Nintendo Switch games that support handheld mode. The handheld Nintendo Switch gaming comes true with a great price compared to the others, suiting it to be owned by every family member so there would not be any problems when using it. Even though the Nintendo Switch Lite is a portable version, it still has a sharp and clear display. The unit has a great feel in hand with a matte plastic finish, and it is terrific for not being a fingerprint magnet when touching the screen. This handheld device is light and not big, meaning that it is easy to move it around as it can be fitted in a pocket or a small bag. The lightweight allows the player to hold it for a more extended period without fatigue, making the player feel comfortable and genuinely healthy.
Furthermore, it comes with an exceptional D-pad, which makes it smooth and straightforward for the players. Funnily enough, the lite has a better internet connection and faster downloads than the original Switch though the specifications are similar. Lastly, the Nintendo Switch Lite is a handheld device, so it does not have to be connected to the TV; therefore, other people in the room can enjoy themselves while we enjoy ours.
Yellow / Turquoise / Gray
Handheld Mode
Handheld Supported Games
3.6" high 8.2" long .55" deep
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text:
English FA to review allowing betting firms to air cup games
LONDON (AP) — The English Football Association is reviewing its decision to allow broadcast rights for FA Cup games to be sold for streaming by betting companies. Online betting platforms are currently the only way to watch some FA Cup games in Britain despite the FA announcing two years ago that it would end partnerships with gambling firms.
Chelsea's Ross Barkley celebrates after scoring his side's second goal during an English FA Cup third round soccer match between Chelsea and Nottingham Forest at Stamford Bridge in London, Sunday, Jan. 5, 2020.
While Bet365 also aired games last season, criticism of the deal came about because the FA used third-round games last weekend to campaign around mental health. Claire Murdoch, a mental health director at the National Health Service, called the betting company's streaming of FA Cup games "another own-goal from the gambling industry."
The British government weighed into the discussion on Wednesday, with culture secretary Nicky Morgan writing on Twitter that "things have moved on since the contract was signed & I hope they will re-consider."
The FA is locked into a broadcasting contract that was signed before it decided to terminate a sponsorship with Ladbrokes amid concerns about the prevalence of gambling in soccer. The FA's contract is not directly with the Bet365 company. The governing body has a media rights agreement with IMG, which then sells the rights under terms agreed in 2017 that allow deals with betting firms.
"This deal was agreed before we made a clear decision on the FA's relationship with gambling companies in June 2017 when we ended our partnership with Ladbrokes," the FA said in a statement. "We will review this element of the media rights sales process ahead of tendering rights from the 2024-25 season."
IMG is in a joint venture with The Associated Press, operating the SNTV sports video news agency which carries footage from FA Cup games.
More AP soccer: https://apnews.com/Soccer and https://twitter.com/AP_Sports
Sports,
Soccer,
Government contracts,
Western Europe,
Playing catch up: Premier League round to air live in UK
AP Source: Champions League TV rights in Britain kept by BT
Chinese TV pulls Arsenal game coverage after Ozil criticism
Amazon secures rights to show Champions League in Germany
Ahead in the clouds: Alibaba plans to change the Olympics
McGregor Comeback
Personal Trainer App
Real Signs Wonderkid Reinier


text:
\section*{Introduction}
One of the fundamental tasks of Physics is unification of apparently different
laws describing different phenomena into a single theoretical framework. There exist many examples of successful unification: Maxwell electrodynamics, theory of electroweak interactions, statistical approach to thermodynamics, quantum theory of chemical phenomena or general relativity. In the last few decades we became a witness of a new, very ambitious, in the philosophical perspective,  unification attempt of information theory and physics. The extreme formulation due to Wheeler reads:``.. every physical quantity, every it, derives
its ultimate significance from bits, binary yes-or-no indications, a conclusion which we epitomize in the phrase, it from bit" \cite{Wheeler}.
\par
This point of view has been accepted, for instance, by the majority of physicist working in the field of quantum information and resource theory in quantum thermodynamics, sometimes giving an impression that information acquired a status of ``substance" similarly to \emph{caloric} in XVIII - XIX century physics.
Perhaps, the strongest motivation for this philosophy is provided by the Szilard approach to \emph{Maxwell's demon} concept. \\
Originally, a Maxwell's demon \cite{Maxwell} uses manipulations at the molecular scale in order to reduce the thermodynamic entropy of a closed system, in violation of the Second Law of Thermodynamics (see \cite{Norton} for the review). More precisely, the demonic being by opening and closing a door in a  wall dividing a chamber with gas accumulates slow molecules on one side and fast molecules on the other.
\par
In the beginning of XX century the results of Einstein \cite{Einstein} and Smoluchowski \cite{Smoluchowski} finally convinced community of physicists that thermal fluctuations are observable. In principle, such fluctuations could lead to violation of the Second Law at microscopic and short-time scales. Hence, thermal fluctuations could provide a ``fuel" for a heat engine coupled to a single heat bath and employing Maxwell's demon operation principles. Indeed, a number of proposals, some of them intended for experimental implementations, has been discussed in the literature.
\par
The main message of the Smoluchowski's paper from 1912 \cite{Smoluchowski1912} was that all those  attempts to naturalize Maxwell's demon must fail.
Namely, the same thermal fluctuations which could reverse the direction of thermodynamical processes  allowing temporal violation of the Second Law act also on Maxwell's demon randomizing its operation and completely undoing the possible spontaneous decrease of entropy. Smoluchowski did not provide a general proof but studied several examples, among them the original Maxwell's proposal with the demon realized as a trapdoor controlled by a spring. Smoluchowski's arguments provide till now the most convincing exorcism of Maxwell's demon.
\par
In 1929, with the goal to clarify the idea of Maxwell's demon, Szilard proposed a model of an engine which consists of a box, containing only a single gas particle, in thermal contact with a heat bath, and a partition \cite{Szilard}.
The partition can be inserted into the box, dividing it into two equal volumes, and can slide without friction along the box.
To extract  $k_B T\ln 2$ of work in an isothermal process of gas expansion one connects up the partition to a pulley. Szilard assumes that in order  to realize work extraction it is necessary to know  ``which side the molecule is on'' what corresponds to one bit of information.
\par
In this picture energy used to insert/remove the partition is negligible  while the subjective lack of knowledge is treated as a real thermodynamical entropy reduced by the measurement. To avoid the conflict with the Second Law of Thermodynamics it is assumed that the reduction of entropy is compensated by its increase in the environment due to dissipation of at least $k_BT\ln 2$ work invested in the feed-back protocol of work extraction. This idea initiated a never ending discussion  about the place where the external work must be invested: in measurement \cite{Brillouin:1956}, in resetting memory \cite{Landauer:1961,Bennett:2003}, or both \cite{Sagawa:2012}. Most known version is the Landauer's principle :\\
\emph{Any logically irreversible manipulation of information, such as the erasure of a bit or the merging of two computation paths, must be accompanied by a corresponding entropy increase in non-information-bearing degrees of freedom of the information-processing apparatus or its environment.}\\
In particular it means that erasure of a bit in an environment at the temperature $T$ should cost at least $k_B T \ln 2$ of work.

\par
Although, the arguments in favor of this information-thermodynamics link  seem to be generally accepted, one can find in the literature several examples of doubts and criticism.\\
\begin{enumerate}
\item{
As already noticed  by Popper and Feyerabend \cite{Feyerabend}  and more recently by one of the authors \cite{Alicki:2014},\cite{Alicki:2014a}, one can design procedures of extracting work without knowing the position of the particle.  Various designs has been proposed to eliminate the presence of observer in work extraction.}
\item{
Jauch and Baron \cite{Jauch} were concerned with \emph{subjectivity} of the argument based on measurement process and stressed that the \emph{potential to do work} is present even before measurement is performed.}
\item{
A well-motivated criticism of the information-thermodynamics link has been presented by Norton (see e.g.\cite{Norton}). He argued (following the reasoning of Smoluchowski) that the Szilard engine cannot yield work  without the presence of external non-equilibrium reservoir. A moving piston, massive enough to suppress thermal fluctuations, can serve as such external source of work.}
\end{enumerate}

In the present paper we would like to provide a new evidence supporting the critique of the standard paradigm.
The paper both provides original results, as well as review some previous ones.

Our new results are about the work cost of inserting partition in Szilard engine. We discuss the case, where the partition is modeled by potential in Schrodinger equation \cite{Zurek-well}.
Then it is known that the work cost is negligible (in particular it does not grow with $T$). Next we consider a partition modeled by physical field, and obtain that if we want to stabilize the position of the particle to be either on the left or on the right, we need to spend nonnegligible work, which must be even greater than $k_BT$.
Moreover, we show that after insertion of the partition, the position of the particle (i.e. whether it is on the left or on the right) is correlated with two macroscopic states of the barrier. Thus, to draw work, demon is not needed anymore. Overall, in  the full cycle, the work is put and drawn in a different place than it is commonly thought.

Based on this example, we advocate the more general view, according to which the {\it stable} information
is not equivalent to work, i.e. that information-thermodynamics link ceases to hold for information that can be stably encoded, stored and transmitted.  Moreover we review the previous results studied in a series of mostly unpublished papers
\cite{Alicki:2014a},\cite{Alicki:2013},\cite{Alicki:2014b}, which support the thesis that stabilizing
information needs cost that exceeds $k_BT$, and grows along with the required stability.  We also revisit the interpretation
of experiments that  have aimed to confirm Landauer principle, and show that the results of these indeed ingenious measurements are
compatible with the view that we advocate in this paper.

We further reconcile the latter view with mathematical form of Landauer principle, which is correct, near-tautological statement, by pointing out that assumptions behind the statement are not satisfied in the case of stable information - namely, the stably encoded bit, is usually not in a product state with environment (as we obtain  in our analysis of Szilard box).

We also discuss the implication of the discrepancy between the two models of partition in Szilard box for
foundations of quantum mechanics.

\par

\section{The physics of quantum Szilard engine}
\label{sec:engine}
Most of the criticism of information-thermodynamics link motivated by the analysis of Szilard engine is related to the energy balance in the process of inserting/removing partition. The problem is quite tricky because it involves the interaction of a microscopic system - a single molecule - with a system which must be large enough to provide a stable barrier suppressing molecule's transition from one half of a box to another one.
Moreover, partition should be also a part of environment at thermal equilibrium with the rest of the single heat bath. We discuss two types of quantum models for molecule-partition dynamics. Quantum models are simpler from the mathematical point of view and lead to some interesting questions concerning the very foundation of quantum mechanics.

\subsection{Models with potential barrier}

The process of inserting partition has been discussed in several papers in terms of a single particle in one-dimensional potential describing a fixed box with a thin potential barrier in the middle with a time-dependent magnitude.  Two examples produce the exact solutions: harmonic well \cite{Viana} or infinite square potential \cite{Joglekara} well with Dirac-delta potential at the origin parametrized as $g \delta(x)$. For the coupling $g=0$ both systems posses discrete non-degenerated energy spectrum : $E_0 < E_1 < E_2 <...$ with the corresponding even eigenfunctions  $\psi_{2k}(x)$ and odd ones  $\psi_{2k+1}(x)$, $k= 0,1,2,...$. Increasing the magnitude of the potential barrier we see that
the odd states $\psi_{2k+1}$ and their energies $E_{2k+1}$ do not change, while each even state  is deformed into ${\psi^g}_{2k}$ and its energy $E_{2k}(g)$ approaches the energy of the next odd state, i.e. $\lim_{g\to\infty}{E}_{2k}(g) = E_{2k+1}$.  The deformation is quite interesting, a sinus in the middle is formed, the state remains even but the left half of the asymptotic state $\psi^{\infty}_{2k}$ coincides with the left half of
$\psi_{2k+1}$ and its right half with the right half of $-\psi_{2k+1}$. Their linear combinations
\begin{equation}
\phi^L_k = \frac{1}{\sqrt{2}}\bigl[\psi^{\infty}_{2k} + \psi_{2k+1}\bigr],\quad \phi^R_k =
\frac{1}{\sqrt{2}}\bigl[\psi^{\infty}_{2k} - \psi_{2k+1}\bigr]
\label{LRstates}
\end{equation}
are completely localized in the corresponding parts of the box.
\par
According to the standard interpretation of  quantum mechanics both orthonormal basis: $\{\psi_{2k+1} ,\psi^{\infty}_{2k}; k =0,1,2,...\}$ and $\{\phi^L_k ,\phi^R_k ; k =0,1,2,...\}$ are equivalent and can be used to represent e.g. the Gibbs state in two equivalent ways.
Moreover, the insertion of the barrier shifts the energies of even states by the factor which goes down with increasing size of the box.
Therefore, one can argue that, thermodynamically, the insertion/removal of the barrier is pretty harmless and should not essentially modify free energy of the system.
\par
On the other hand a strong enough potential barrier changes dramatically the physical properties of the system. A particle is localized always in a fixed part of the box and hence oscillates with a doubled frequency. The pressure executed by the particle on the walls is doubled.  We expect also that superpositions of left and right states, $\phi^L_k, \phi^R_{k'}$ are never observed.
This is related to the notorious problems of \emph{Schr\"odinger cat}, \emph{molecular structure}, \emph{decoherence induced superselection rules}, etc., (see e.g. \cite{Joos}). For example, as shown in \cite{Bender} the existence of such superpositions for the discussed model leads to rather non-acceptable instantaneous energy transfer on arbitrary distance. Moreover, according to the Boltzmann's definition of entropy as a logarithm of the number of \emph{accessible} states, here after insertion of partition the density of accessible states is reduced by a factor of 2 and hence a free energy of the Gibbs state is increased by $k_B T \ln 2$ providing a ``potential to work".
\par

  Thus we have a discrepancy between the model, that does not predict any substantial work cost of  inserting the barrier, and the physical intuition about such system.
In the next Section we discuss a model of partition being a large system with many degrees of freedom, which illustrates the possible solution of the paradox of above.

\section{Spin-boson model}

The potential barrier model of partition from the previous Section is rather unphysical, in particular in the context of thermodynamical analysis. The partition must be treated as a large enough stable quantum system driven by the external forces which can execute work. It must be also at thermal equilibrium at the same temperature as the rest of environment, otherwise the argument based on the Second Law cannot be used.
\par
The proposed model consists of a particle which can occupy only two states (spin-1/2) - the left $|+\rangle$ and the right $|-\rangle$ - which are eigenstates of the discrete ``position" operator $\sigma_3$. The partition is a solid body described by a family of quantum oscillators corresponding to its different  deformation modes. The total time-dependent Hamiltonian is given by ($\hbar \equiv 1$)
\begin{equation}
H(t) =  \sum_k \omega_k a_k^{\dagger} a_k +  \lambda(t) \sigma_3 \otimes \sum_k \bigl( \bar{f}_k a_k + f_k a_k^{\dagger} \bigr) ,
\label{Ham}
\end{equation}
where $\{\omega_k\}$ are frequencies of deformation modes, $\lambda(t)$ is an overall magnitude of particle-partition coupling and $f_k$ describe the shape of partition's deformation by the presence of particle. The time-dependence of $\lambda$ describes the process of inserting or removing
of partition driven by an external agent which provides a necessary amount of work. In the Hamiltonian \eqref{Ham} we neglect all physical interaction leading to transitions between the particle states which can be described by the term
\begin{equation}
H_1 =  \sigma_1 \otimes R ,
\label{Ham1}
\end{equation}
where $R$ is generally an environment observable. These hopping processes will be discussed later on. One should also notice that the model of partition can contain enough degrees of freedom to represent a whole (thermal) reservoir.
\par
For a fixed value of $\lambda$ the Hamiltonian \eqref{Ham} possesses two degenerated ground states of the form
\begin{equation}
|g\pm\rangle =  |\pm\rangle \otimes |[\pm \lambda f/\omega]\rangle,
\label{coherent1}
\end{equation}
where the global coherent state $|[\pm \lambda f/\omega]\rangle $ is a product of single-mode coherent states $|[\pm \lambda f_k/\omega_k]\rangle$ satisfying
\begin{equation}
a_k |[\pm \lambda f_k/\omega_k]\rangle = \pm \lambda\frac{f_k}{\omega_k}|[\pm \lambda f_k/\omega_k]\rangle .
\label{coherent2}
\end{equation}
The ground state energy reads
\begin{equation}
E_g(\lambda) = - \lambda^2 \sum_k \frac{|f_k|^2}{\omega_k} .
\label{groundenergy}
\end{equation}
The important parameter is the overlap of the coherent states appearing in \eqref{coherent}
\begin{equation}
\langle[ \lambda f/\omega]|[-\lambda f/\omega]\rangle =  e^{-2\lambda^2\|f/\omega\|^2} , \quad  \|f/\omega\|^2 = \sum_k \frac{|f_k|^2}{\omega_k^2}.
\label{overlap}
\end{equation}
When partition is treated as a large system described in the thermodynamic limit the sums in \eqref{groundenergy} and \eqref{overlap} are replaced by integrals and could be infinite, typically due to the increasing contributions from the low frequency modes (infrared catastrophe).
There exists always an ultraviolet natural cut-off - Debye frequency, hence the possible  divergences are always due to ``soft phonons" . In the continuous limit one can use  ``form-factor" depending on  $\omega$ only with the power-like scaling for small $\omega$'s
\begin{equation}
|f(\omega)| \sim \omega^{\kappa}
\label{form}
\end{equation}
For $ 0 < \kappa \leq 1$ , where $\kappa = 1$ corresponds to the Ohmic case, the ground state energy $ E_g$ is finite but the integral $\int_0^{\epsilon} |f(\omega)|^2/ \omega^2 \, d\omega$ diverges. Hence, the ground states given formally by coherent states do not overlap and therefore must be disjoint, i.e. cannot belong to a common Fock space of the bosonic system (so-called van Hove phenomenon). The most interesting regime relevant to stability of ground states is a strongly subohmic  one $  \kappa < 0$ ,
for which the ground states remain disjoint and $E_g$ diverges with infrared cut-off frequency $\omega_{min}$
\begin{equation}
-E_g \sim \omega_{min}^{\kappa} .
\label{energy_div}
\end{equation}
As the minimal frequency $\omega_{min}$ corresponds to the longest acoustic waves which can be supported by the system it scales like $L^{-d}$
where $L$ is a linear dimension of the system and $ d > 0 $, Therefore, the energy barrier $E_g$ separating both ground states grows with the size of the system. Such relation is always observed for systems displaying transition from quantum to classical world. For example stability of different conformations of molecules, like e.g. optical isomers, grows with their size. Moreover, ubiquitous presence of ``$1/f$- noise" phenomena in macroscopic systems suggests existence of strongly subohmic tails in spectral densities for low frequency excitations in the macroscopic world.\\
Notice that  exactly the same mechanisms (disjointness of different equilibrium phases in the thermodynamic limit and energy barrier between phases increasing with the system's size) explain the origin of phase transitions in quantum statistical mechanics. All that supports the claim that the localized states $\{\phi^L_k ,\phi^R_k ; k =0,1,2,...\}$ from the previous Section which belongs to physically separated ergodic components of the total system should be treated as disjoint and their superpositions as meaningless.
\par
Another approach involves dynamical origin of superselection rules for finite systems weakly interacting with large environment. A model essentially equivalent to our spin-boson one has been presented in an unpublished preprint \cite{Alicki:2013} and briefly described in one of the next Sections. This models shows the dynamical origin of metastable, macroscopically distinguishable pointer states
in the presence of heat bath. The energy barrier between such states introduces the Boltzmann factor which suppress both the error in distinguishability of those states and the tunneling rate between them. Moreover ``Schr\"odinger cat states" formed from pointer states decohere quickly to their mixtures with the rate proportional to  the square of the ``distance" between them.
\par

\section{Thermodynamic cost of operating partition}
\label{sec:cost-partition}
When the partition is treated as a large system with many degrees of freedom one can expect that the process of operating partition involves work provided by the external driving which changes the free energy of the particle-partition system and partially is dissipated
in a form of heat. For simplicity we consider first the zero temperature case.
\par
Inserting partition for the model of the previous section is realized by increasing the coupling $\lambda(t)$ from the initial value $\lambda(0)=0$ to the final one $\lambda(t_0) = \lambda$. The initial state of the particle - partition system is assumed to be $|\pm\rangle \otimes |[0]\rangle $. Because the Hamiltonian \eqref{Ham} is quadratic the exact solution of the Schr\"odinger equation can be easily found. It is equal to  $|\Psi_{\pm}(t)\rangle =  |\pm\rangle \otimes |[\pm\chi(t)]\rangle $ where
\begin{equation}
\chi_k(t) =  \int_0^t e^{-i\omega_k (t-s)} \lambda(s) f_k \, ds = \lambda\frac{f_k}{\omega_k} - e^{-i\omega_k t} \Bigl(\int_0^{t_0} \dot{\lambda}(s) e^{i\omega_k s} ds\Bigr) \frac{f_k}{\omega_k} .
\label{evolution}
\end{equation}
Notice, that the first term on the RHS of \eqref{evolution} describes a static deformation of inserted partition while the second, time-dependent term corresponds to traveling ``acoustic wave" excited by the process of partition's insertion. Choosing for simplicity a linear slope of $\lambda (t)$ we obtain, for $t > t_0$
\begin{equation}
\chi_k(t) =  \lambda\frac{f_k}{\omega_k} - \lambda e^{-i\omega_k t} \xi_k \frac{f_k}{\omega_k} ,\quad \xi_k = \frac{1}{i\omega_k t_0}\bigl(e^{i\omega_k t_0} - 1\Bigr) , \quad |\xi_k| < 1.
\label{evolution1}
\end{equation}
The energy of the  state after insertion has form
\begin{equation}
E(t> t_0) = E_g  +  \lambda^2\sum_k |\xi_k|^2 \frac{|f_k|^2}{\omega_k} < 0
\label{energy}
\end{equation}
Because the dynamics of the total system is a Hamiltonian one it means that $E(t> t_0)$ is equal to work performed by the external driving.
As $|\xi_k|^2 < 1$ this work is negative what can be quite easily understood. Namely, to stabilize the partition inside the box we need a kind of a ``lock" with a low enough energy of  the final ``locked state". Hence, $-E_g$ must be much larger than $k_BT$ in the case of thermal noise. During this process a part of energy  equal to $\lambda^2\sum_k |\xi_k|^2 \frac{|f_k|^2}{\omega_k} $ is carried away by ``acoustic waves" (``click of the lock") and dissipated as a heat.

Consider now, as above, the strongly subohmic case with the energy barrier scaling \eqref{energy_div}. Then in the continuous
approximation the dissipated energy is given  by $\lambda^2 \int_{\omega_{min}}^{\omega_{max}} |\xi(\omega)|^2 \frac{|f(\omega)|^2}{\omega} $
and the main contribution to the integral comes from low frequencies $\omega \simeq \omega_{min}$. Therefore,  under the condition $\omega_{min} t_0 \leq 1$, $|\xi(\omega)|^2 \simeq 1$  for the relevant small frequencies and the dissipated energy is comparable to $-E_g$ and hence much higher than $k_B T$. On the other hand, the condition $\omega_{min} t_0 \leq 1$ must be satisfied because the time dependence of $\lambda(t)$ follows from the dynamics of the total macroscopic system including the partition and the control machinery. In fact  $\omega_{min}$ is the minimal frequency of the total system what implies that such system  cannot generate processes varying on the time scale longer than $\omega_{min}^{-1}$.

Finally, to ``unlock" the partition we start from the new ground state $|\pm\rangle\otimes|[\pm \lambda f/\omega]\rangle$ and we must invest at least $-E_g >> k_B T$  of work to reach the initial state $|\pm\rangle\otimes|[0]\rangle$. Again a substantial portion of work is dissipated by ``acoustic waves".
\par
Actually, the operation scheme of the engine proposed by Szilard is even more complicated. To transform a partition into a ``piston" a motion along the box must be ``unlocked" by the external agent with a little help of small contribution from the single-particle gas expansion. Finally, to return to the initial state several steps of ``locking" and ``unlocking" of various degrees of freedom must be used.
One can also imagine different designs where insertion of partition cost work which is then partially recovered by its removal or the cases where both operations need work provided by external machinery.
\par
It is not difficult to generalize the model to finite temperatures. Instead of evolving pure states $|\Psi_{\pm}(t)\rangle =  |\pm\rangle \otimes |[\pm\chi(t)]\rangle $ we have two density matrices $ |\pm\rangle\langle \pm| \otimes \rho_{T}[\pm\chi(t)] $  with deformed oscillator Gibbs states at the temperature $T$, which are stable in the absence of the hopping Hamiltonian \eqref{Ham1}. The energy  balance is similar to the zero-temperature case - again an amount of work much higher than $k_BT$ is involved in the process of inserting/removing of partition. \\
The influence of hopping Hamiltonian does not change the conclusions as well. In the beginning of the insertion  process, when
 $\lambda(t) \ll \lambda$, particle is hopping between the left and right positions making unpredictable in which state is finally trapped. With increasing $\lambda(t)$  hopping probability becomes exponentially suppressed by the Boltzmann factor. At the end of removing process
hopping is switched on again what leads to complete thermalization of the system. Similarly to the pure states, the density matrices $\rho_{T}[\pm\lambda f/\omega] $  for high enough $\lambda$ must be treated as disjoint, i.e. respecting superselection rule.
\par

Summarizing, inserting the barrier involves two effects:
\begin{itemize}
\item[(i)] final state is a correlated state of the particle and of the macroscopic state barrier
\item[(ii)] work much larger than $k_BT$ is dissipated during the process.
\end{itemize}
In section \ref{sec:recon} we shall discuss in more detail how these results (dis)agree with Landauer principle.
Here let us mention, that once we are given the box with the barrier already inserted, drawing
work does not contradict the Second Law. The barrier already measured where is the particle, and it is the barrier
which place the role of demon. Thus one can expand the single-particle gas, conditioned on the state of the barrier, and perform $k_BT \ln 2$
work.

And of course, if we want to talk about complete cycle, then no positive work can be obtained (in agreement with the argument
of \cite{Pusz} based on passivity),  as one then has to reset the barrier state. We will discuss this is more in detail in
section \ref{sec:recon}. Here we want just to emphasize, that according to a common view, the external agent, who holds a box  with a
particle, but does not know where the particle is, cannot use the box to draw work, without gathering the knowledge.
In contrast, here we obtain, that subjective knowledge possessed by the agent is not important, as the position of the particle has been imprinted in the state of the barrier.

However, more importantly, according to (ii) much higher than $k_B T\ln 2$ amount of work must be involved in a stable and deterministic operation of the macroscopic parts of Szilard's engine. Hence the delicate balance of quantities like $k_BT \ln2$ is completely
overshadowed by the cost of creating system with two stable states.

  Thus, replacing the potential from sec. \ref{sec:engine} with the physical field, we obtain higher thermodynamical cost
of processing. This suggests, that all the results, where ultimate bounds on work cost of operations were analysed
in the model where Schrodinger equation with the potential was used, should be reexamined by treating the potential physically,
i.e. as some field.

\section{Consequences for thermodynamics and foundations of quantum mechanics}

  In previous section, we have considered an example showing how the interaction with environment can create effective
superselection rules for quantum systems.
For all practical purposes it is convenient to formulate a consistent description in terms of the system alone, with the influence of environment described by  mixed states obeying superselection rules and suitable effective irreversible dynamics. For a system with superselection rules the Hilbert space is decomposed into a direct sum of subspaces, $\mathcal{H} = \oplus_j \mathcal{H}_j$ corresponding to ergodic components.
\par
In our context the mathematically precise definition of ergodic component always involves some limiting procedures similarly to the definition of phases in the theory of phase transitions. Typically, a thermodynamic limit must be performed or/and cut-offs must be removed leading, for example, to infrared catastrophe. Practically, by ergodic component we mean a system with a smallest Hilbert subspace of states which are metastable with respect to a class of admissible evolutions. The states are metastable if the probability of escape per unite time from the given Hilbert subspace is negligible. The class of admissible evolutions must be defined by suitable physical requirements. For example, we begin with a fixed Hamiltonian of the system and perturb it by ``small" or ``local" perturbations.  Another method is to allow weak coupling of the system to a heat bath with restricted range of temperatures.
\par
Remembering the above physical criteria, which must be satisfied by a non-ergodic system, we can define its state by a density matrix possessing a block diagonal structure
\begin{equation}
\rho = \oplus_j p_j \rho_j , \quad \mathrm{Tr} \rho_j = 1 ,\quad  \sum_j p_j = 1
\label{dmatrix}
\end{equation}
where the support of each density matrix $\rho_j$ is in $\mathcal{H}_j$. Here, the probabilities $p_j$ are uniquely determined and describe our \emph{subjective lack of knowledge} about system's localization.
  As the example from previous section suggests, the capability of drawing work does not depend on this subjective knowledge, as the environment  "knows" itself which ergodic component the system is in.   Therefore, the objective, physical entropy $S_{\mathrm{ph}}$ which enters all thermodynamical expressions must be defined as an averaged entropy over all ergodic components (see \cite{Ishioka} for the very similar arguments)
\begin{equation}
S_{\mathrm{ph}}[\rho ]= k_B \sum_j p_j  S_{\mathrm{vn}}(\rho_j) = k_B S_{\mathrm{vn}}(\rho)  - k_B I[\mathbf{p}],
\label{physentropy}
\end{equation}
where $S_{\mathrm{vn}}(\sigma) = - \mathrm{Tr}(\sigma \ln\sigma)$ is the von Neumann entropy of a density matrix $\sigma$ and $I[\mathbf{p}] = -\sum_j p_j \ln p_j $ can be interpreted as \emph{information-theoretical entropy}. Indeed, any non-ergodic system which is decomposed into $n$ergodic components, can serve as a carrier of $\log_2 n$ bits of information. However,   as argued,
the Shannon entropy $I[\mathbf{p}]$ does not contribute to the physical (thermodynamical) one: indeed, for the Szilard engine, the state of the particle after inserting partition has lower thermodynamical entropy and hence possesses a ``potential to work".
\par

  Let us emphasize, that this does not nullify the widespraed connections between information and thermodynmics
\cite{Goold-thermo-inf-review} including e.g. the fluctuation relations with feedback of Ref. \cite{Sagawa-Ueda}, with the important caveat,
that the type of information which can be thermodynamically relevant is not stable. To consider most trivial example:
the entropy of momenta of gas molecules in a container does not  represent a useful information. In contrast the position of a single molecule in Szilard box  is a meaningful information - such a box is a register that can store one bit of information. Yet, according to our view, this information is not relevant thermodynamically.

Let us consider two particular examples.

{\it Example 1.}
In the paper \cite{MandalJarzynski-rotor}  the demon produces work at the expense of polluting the tape, originally filled with zeros.
The role of demon is here played by the Markov process between the three state system and the tape (memory of the demon).
The model is a Markov process, that is not related to physical realization,  apart from the fact, that energy change is attributed to state changes, and the detailed balance is used, to relate probability transitions with Gibbs factor.

If we are interested in physical implementation of the model, we should ask in particular about the physics of the tape – how it is possible to have tape filled with zeros.  There are three possibilities:
(i)	the tape is decoupled with heat bath (or the coupling is so small, that we for long time, we may consider it as effectively uncoupled);
(ii)	the tape is in equilibrium with the heat bath of low temperature;
(iii)	the bits of tape are stable, because they are ergodic components.

The case (i) is the situation "coupling on demand" which is widely used in literature, but physically is of limited use (see also discussion in section \ref{sec:recon}). The case (ii) describes heat engine operating between two temperatures, hence it fills into standard paradigm of thermodynamics, where Landauer erasure need not be invoked to save the Second Law. Finally the situation (iii) is related to our model of Szilard engine: the zero of each bit in the tape is imprinted in the nearby environment. Our results suggest that it is then not relevant, in what states are the bits of tape on one hand, and changing bits costs more than $kT_B ln2$.

{\it Example 2.}
In Ref. \cite{Quan-supercond} drawing work by Maxwell demon is demonstrated in terms of two superconducting qubits. One of them ($S$) interacts with hot heat bath, and the other ($D$) places a role of Maxwell demon. By means of a short circuit, the information
from the $S$ is mapped into $D$. The information is reset in result of the contact of the system $D$ with cold bath. The situation corresponds to case (ii) described above, i.e. the qubits $S$ and $D$ are always coupled to their heat baths. So there is no stable information processed here. The information processed by means of circuit is {\it temporal} random data  from the heat bath transferred  to the the qubit $S$, and the system $D$ does not write the information in some stable memory, but it releases them to the low temperature heat bath. Thus the circuit operates on the information that is never stably stored. Also, as mentioned in Example 1,
since we deal here with engine operating between two heat baths, and offering efficiency of Otto cycle, Landauer principle need not be invoked to save the Second Law, as simply its standard formulation is not violated to begin with.
One can of course interprete the situation in Landauer spirit as is done in Ref. \cite{Quan-supercond},
which is an interesting way of thinking about the thermodynamics as information flow. Yet as said, it is not a flow of stable information,
but of some temporal, dynamically instable data, that immediately after emerging are sunk in the cold bath.

\subsection{  Ergodicity versus superpositions. }
The discussion on mixture of ergodic component suggests that a new postulate should be added to the standard formulation of
quantum mechanics:\\
\emph{Superpositions of (pure) quantum  states belonging to different ergodic components do not exist, and the corresponding occupation probabilities reflect the subjective lack of knowledge, i.e. the system occupies always a given ergodic component of the Hilbert space.}
\par
Fortunately, this new postulate allows to eliminate the standard one - \emph{von Neumann projection postulate}:\\
\emph{Immediately after a measurement the state of the system is the corresponding eigenstate associated with that eigenvalue.}
\par
Indeed, the spin-boson model used in the previous Sections can be also treated as a model of a measurement process of the observable
$\sigma^3$. The Hamiltonian \eqref{Ham} describes the coupling of the measured spin-1/2 with the essentially macroscopic apparatus with many
degrees of freedom represented by harmonic oscillators. The possible final states $|\pm\rangle\otimes|[\pm \lambda f/\omega]\rangle$ contain
the stable and essentially disjoint \emph{pointer states} \cite{Zurek:2003}  $|[\pm \lambda f/\omega]\rangle$ of the apparatus strictly
correlated with the spin eigenstates of $\sigma^3$. Then according to the new postulate the total system occupies for sure one of the states
$|\pm\rangle\otimes|[\pm \lambda f/\omega]\rangle$ and hence spin-1/2 can be certainly found in the corresponding eigenstate $|\pm\rangle$.
\par
Let us emphasize, that the proposed postulate is not fundamental, in the sense that it does not apply to quantum field theory.
Rather it is designed to Schr\"odinger quantum mechanics which involves phenomenological potential. Infinite potential barrier
has to be modeled on the level of quantum field theory, and the postulate emerges within  the effective picture.
\par

Let us finally remark, that there is a question, whether in general stabilizing information must involve superselection rule, i.e.
whenever we have stable bit, environment must be correlated with the values of the state of the bit.
If self-correcting quantum memory can be built, this would be a counterexample: Indeed, the stable qubit means that superpositions between
states are preserved, which means, that the information "which state" cannot be imprinted in environment.
At present it is an open question, whether such self-correcting quantum memory exists in three spatial
dimensions \cite{rev-top-mem}.  In spatial dimension four, there exist stable topological quantum memories based on
Kitaev model \cite{Dennis-top-mem,AlickiHHH-4d}, however
these architectures are still not physical at the moment: the Hamiltonian  is ultralocal - admitting, in particular, no transport.

  \section{ Proposed revision of information-thermodynamics link }
\label{sec:correct}
  Our model for Szilard engine, analysed in section \ref{sec:engine} suggests that {\it stable} information
is not a thermodynamic resource.
However, there still exist interesting physical questions of the fundamental nature related to information processing. Any physical system which, by means of external controlled constraints, can be decomposed into $n$ ergodic components can serve as an universal model of information carrier with the capacity of $\log_2 n $ bits. However, the constrains are never perfect because they are always realized by finite energy barriers. In the presence of thermal noise any finite barrier is penetrable and its shape determines the life-time of encoded information at the given temperature. This life-time can be estimated using Kramers theory of reaction kinetics. For the same reasons the system localization in a given ergodic component is also never perfect and  probability tails penetrating the other components yield the error of information encoding.

\subsection{Operation cost of a switch}

The interesting problem is  the thermodynamic cost of information processing. As an elementary ingredient of information processing hardware one can take a \emph{switch} - the device which can move the  system from one ergodic component to another and back. Two models of a switch have been studied independently: a classical electronic switch by Kish \cite{Kish} and a quantum switch, described by the spin-oscillator model, in \cite{Alicki:2013},\cite{Alicki:2014b}. The latter is very similar to the spin-boson model used in this paper  and we briefly discuss its basic features.\\
The system consists of a spin-$1/2$ described by the standard Pauli matrices $\hat{\sigma}^k , k=1,2,3,\pm$ interacting with the harmonic oscillator with the following Hamiltonian
\begin{equation}
{H_S} = \omega_0 {a}^{\dagger}{a} - \omega_0 g({a}^{\dagger}+{a}){\sigma}^3  ,\quad  \omega_0 , g > 0 .
\label{ham_TLSQB}
\end{equation}
The following notation for spin states, oscillator coherent states and joint spin-oscillator states is used
\begin{equation}
{\sigma}^3 |\pm\rangle = \pm |\pm\rangle, \quad {a}|\alpha\rangle = \alpha |\alpha\rangle, \quad \alpha\in\mathbf{Z},\quad |\mu;\alpha\rangle \equiv |\mu\rangle|\alpha\rangle, \quad \mu= \pm.
\label{coherent}
\end{equation}
A unitary \emph{polaron transformation} $U = e^{g( {a}- {a}^{\dagger}){\sigma}^3}$ leads to new parametrization ${a} \mapsto {b} = a-  g\sigma^3,\quad {\sigma}^k \mapsto {\tau}^{k} \equiv \sigma^k $ and transforms the Hamiltonian \eqref{ham_TLSQB} into
${H}_S=\omega_0 {b}^{\dagger}{b} - \omega_0 g^2 $ possessing degenerated ground states  $|\pm;\pm g\rangle$ with the energy $E_g = - \omega_0g^2$.\\
The spin-oscillator system (SOS) weakly interacts with the heat bath at the temperature $T$ by means of coupling linear in $\sigma^k$ and $a , a^{\dagger}$ (we disregard first environmental tunneling $\sim \sigma^1$). Using standard derivation one obtains the following quantum Markovian master equation for the density matrix of SOS
\begin{eqnarray}
\frac{d{\rho}}{dt} = -i[{H} , {\rho}]  + \frac{1}{2}\gamma\bigl([{b}, {\rho}{b}^{\dagger}] + [{b} {\rho}, {b}^{\dagger}]\bigr)+ \frac{1}{2}\gamma e^{-\omega_0/k_B T}\bigl([{b}^{\dagger}, {\rho}{b}] + [{b}^{\dagger} {\rho}, {b}]\bigr) -\frac{1}{2}\Gamma [{\tau}^3,[{\tau}^3, {\rho}]]
\label{MME_T}
\end{eqnarray}
with the dissipation rate $\gamma $, and the pure decoherence rate $\Gamma $. Any initial SOS state $\hat{\rho}$ tends asymptotically to the mixture $p_{+}\hat\rho^{(+)} + p_{-}\hat\rho^{(-)}$ of stationary \emph{biased Gibbs states}
\begin{equation}
\rho^{(\pm)} =  \bigl(1-e^{-\omega_0/k_B T}\bigr)|\pm\rangle\langle\pm|\,e^{-\frac{\omega_0}{k_B T}{b}^{\dagger}{b}} = \bigl(1-e^{-\omega_0/k_B T}\bigr)|\pm\rangle\langle\pm|\,e^{-\frac{\omega_0}{k_B T}({a}^{\dagger}\mp g)({a}\mp g)}
\label{gibbs_biased}
\end{equation}
For $g >> 1$, the harmonic oscillator components  $\rho^{(\pm)}_{\mathcal{P}} =  \bigl(1-e^{-\omega_0/k_T}\bigr)e^{-\frac{\omega_0}{k_B T}({a}^{\dagger}\mp g)({a}\mp g)}$ of SOS state \eqref{gibbs_biased} represented the semi-classical \emph{pointer states} which encode a bit of information with the error given by the overlap defined as the transition probability (see \cite{Scutaru:1998} for the derivation)
\begin{equation}
\epsilon = \mathrm{Tr}\Bigl(\sqrt{\sqrt{\hat\rho_{\mathcal{P}}^{(+)}}\hat\rho_{\mathcal{P}}^{(-)}\sqrt{\hat\rho_{\mathcal{P}}^{(+)}}}\Bigr) =  \exp\Bigl\{-4 D^2\tanh\bigl(\frac{\omega_0}{2k_B T}\bigr)\Bigr\} .
\label{trans_prob}
\end{equation}
This  probability of error in the process of pointer states discrimination (measurement error)  can be rewritten as a ``Boltzmann factor"
\begin{equation}
\epsilon =  \exp\Bigl\{-\frac{\bar{2|E_g|}}{k_B\Theta}\Bigr\} , \quad k_B\Theta= \frac{\omega_0}{e^{\omega_0/k_B T} -1} + \frac{\omega_0}{2}.
\label{highT}
\end{equation}
Here $2|E_g|$ is the energy gap between pointer states and their excitations by spin reversal, and  $\Theta \equiv \Theta[T,\omega_0]$ is the effective temperature which in the semiclassical regime, i.e. for $\frac{\omega_0}{k_B T} << 1$, is equal to the temperature $T$, and in the low temperature regime includes quantum fluctuations. \\
Description of tunneling process between the SOS states generated by the coupling to the heat bath through $\sigma^1$ is quite complicated and involves the derivation of the additional term to the Master equation \eqref{MME_T}, followed by a sequence of approximation. The leading order expression for the tunneling rate $\Gamma_{tun}$ contains again the Boltzmann factor and the pure decoherence rate $\Gamma_1$ for a $\sigma^1$ basis, computed for the spin alone coupled to the same bath
\begin{equation}
\Gamma_{tun} \simeq \Gamma_1  e^{-\frac{2|E_g|}{k_B \Theta}}.
\label{tunrate_A3}
\end{equation}
Taking into account that to overcome the energy barrier $2|E_g|$ one needs a comparable amount of work one can propose the following link between information and thermodynamics:\\
\emph{The minimal work $W_{min}$ needed to encode or change a bit of information with an error probability $\epsilon$ under the influence of combined thermal and quantum noise at the effective noise temperature $\Theta$ is given by
\begin{equation}
W_{min} \simeq  k_B\Theta \ln{\frac{1}{\epsilon}}.
\label{minwork}
\end{equation}
}
It it interesting that the formula \eqref{minwork} has been obtained by Brillouin a long time ago \cite{Brillouin:1956}, for a particular example of measurement involving photons and with $\Theta = T$. In that case $W_{min}$ was a minimal work necessary to perform this measurement with the accuracy $\epsilon$.

\subsection{The minimal cost of long computation}
\label{sec:minwork}

The formula \eqref{minwork} is in agreement with our everyday experience, the more accurate and stable is information carrier the more work has to be used for information processing - ``engraving on a stone tablet" costs much more than making `` a mark on paper".
It allows also to answer a practical question: \emph{What is the minimal work needed to perform an algorithm which consists of $N$ elementary logical steps?}
\par
The details of estimation based on the quantum switch model can be found in \cite {Alicki:2013}. The final formula valid for $N\to\infty$ and high temperatures ($\Theta \simeq T$ ) reads
\begin{equation}
\bar{W}_N \simeq k_B T  N  \bigr(  \ln N +  \ln \frac{1}{\delta}+  \ln \frac{1}{\kappa}\bigr) ,
\label{comp_error1}
\end{equation}
where  $\delta$ is an overall probability of failure and  $\kappa < 1$ is a ratio of decoherence time to dissipation time computed for the unprotected information carrier using the above model of a switch.  One can notice the differences between the formula \eqref{comp_error1} and the prediction based on the Landauer's principle $\bar{W}_N^{(L)} \simeq k_B T  N \ln 2$. The additional term $\ln N$ is necessary to keep the overall error probability fixed. For example, a modern supercomputer performing $10^{16}$ logical gates per second and working for a day executes an algorithm with $N \simeq 10^{21}$. Such large $N$ allows to neglect the terms dependent on $\delta$  and $\kappa$. Then, at the room temperature $300 K$ the total minimal work  $\bar{W}_N \simeq 10^2 J$ what is still much lower than the actual energy consumption, of the order of $10^{10} J$, but essentially higher than $\sim 1 J$ predicted by the Landauer's formula.

\subsection{The meaning of experimental results}
\label{sec:exp}

One can find few papers
\cite{Toyabe,Lutz,Jun,Hong} the authors of which claim to test experimentally Landauer's principle. However, a closer look at those,
in fact, ingenious experiments shows that at least in the first three papers the authors confirmed only that increasing or reducing a volume occupied by a Brownian particle reduces or increases  its free energy by $k_B T \ln 2$. The change of the volume is performed by controlled deformation of the external potential either in the form of double well or staircase. The potentials are generated by the external macroscopic devices which by each operation step consume energy of many orders of magnitude higher than $k_BT$. However this cost is not included in the energy balance. As explicitly, but incorrectly, stated by one of the authors \cite{Toyabe}: ``.. in the ideal case, energy to place the block can be negligible ". In reality, ``placing a block'' (i.e. modifying external potential) costs at least several $k_BT$ of work. Similarly, in the paper \cite{Lutz} only the energy dissipated by the Brownian particle is taken into account, while the work invested into control of the varying double-well potential, corresponding to the partition is disregarded (see also \cite{Jun} for more higher precision experiment).
\par
The precise experiment of \cite{Hong} deals with a single-domain magnetic dot (nanomagnet) as an information carrier. Magnetization along the ``easy axis" is used to encode a bit of information and the ``hard axis" is related to degrees of freedom providing the potential barrier of ``partition". In fact, using the Holdstein-Primakoff theory of ferromagnets one could  obtain a description of this system similar to our spin-boson model. In the theoretical analysis of the experimental data the authors compute work performed by time-dependent external magnetic fields from the areas of two hysteresis loops.  They treat a \emph{difference} of two contributions, indeed close
to $k_BT \ln 2$, as the cost
of the logical operation. However, any process with hysteresis is irreversible and accompanied by \emph{internal friction}. As illustrated above by the spin-boson model even if an operation step should in principle extract work from the partition, most of this work is dissipated by emission of energy quanta (magnons in the case of ferromagnet) Therefore, the work dissipated in one hysteresis loop cannot feed the other irreversible process associated with the second hysteresis loop. Actually, the total cost should be rather estimated by the \emph{sum} of areas of the loops which is essentially higher than $k_BT$.

\section{Reconciliation with Laundauer principle}
\label{sec:recon}

Having said all that we would like to make make sure that our claims are compatible  with the quantum information approach, where the Landauer principle is accepted as a simple to prove tautology, in seemingly completely general setup. Let us present a variant of
Reeb and Wolf \cite{Reeb} formulation and proof of Landauer principle.
For arbitrary (not necessarily equilibrium) state $\rho$ on the system with Hamiltonian $H$, we define free energy as
\begin{equation}
F(\rho)= Tr (H \rho) - k_B T S_{\mathrm{vn}}(\rho).
\end{equation}

\begin{prop}
\label{prop:Landauer}
Consider a system $S$ and a reservoir $R$, with the initial state $\rho^{in}_{RS}=\rho^\beta_R \otimes \rho_S$   where $\rho_R^\beta$ is Gibbs state ($\beta = 1/k_B T$),
and the final state denoted by $\rho^{out}_{RS}=U \rho^{in}_{RS} U^\dagger$, with $U$ being an arbitrary unitary. Define heat consumed by a reservoir by change of its energy
\begin{equation}
\Delta Q=Tr(H_R (\rho_R^{out} - \rho_R^{in})).
\end{equation}
Then we have
\begin{equation}
\Delta Q \geq \Delta F_S
\end{equation}
where
\begin{equation}
\Delta F_S= F(\rho_S^{out})-F(\rho_S^{in}).
\end{equation}
\end{prop}
{\bf Proof.}
For the total system $RS$ the free energy is equal to average energy, so that
$\Delta W_{RS} =  \Delta F_{RS}$. We now use formula $ F(\rho)=F(\rho_\beta)  + k_B T S(\rho|\rho_\beta)$, where $S(\rho | \sigma ) \equiv Tr (\rho \ln \rho  - \rho \ln \sigma)$,
which implies that
\begin{equation}
\beta\Delta F_{RS} = S(\rho^{out}_{RS} | \rho^\beta_{RS}) - S(\rho^{in}_{RS} | \rho^\beta_{RS}).
\end{equation}
Since $\rho^{in}_{RS}=\rho_R^\beta \otimes \rho^{in}_S$, and $\rho^\beta_{RS}=\rho_R^\beta \otimes \rho^\beta_S$ by monotonicity of
relative entropy we obtain
\begin{equation}
\beta\Delta F_{RS} \geq S(\rho_S^{out}|\rho^\beta_S) -  S(\rho_S^{in}|\rho^\beta_S) = \beta \Delta F_S.
\end{equation}
This ends the proof.

The above result seems to be completely general - we do not assume any model, any particular coupling. Moreover, it seems to
put the thermodynamical entropy (related to ergodicity) and information entropy - related to lack of knowledge - on the same footing.
Indeed, the amount of work does not depend on the origin of the entropy. There is one assumption though that restricts
the scope of applications of so formulated Landauer principle. Namely it is assumed that the initial state is product with the bath.
This assumption is used in the vastly developing domain of application resource theoretic approach to thermodynamics
\cite{Janzing,HO}

However, it clearly does not apply to   our model of the   Szilard engine for a simple reason: the states "left" and "right" are strongly, almost perfectly  correlated with states of  partition. We can imagine, that the partition bows on the left or right depending where is the particle, due to a pressure of the particle, see Fig. \ref{fig:bow}.

\begin{figure}[ht]
	\centering

	\includegraphics[width=8cm]{fig-bow.png}
	\caption{Partition as a Maxwell Demon: the shape of the partition depends on the location of the particle }
	\label{fig:bow}
\end{figure}

Thus the partition is the Maxwell demon itself, and therefore the potential of drawing work from such a system does not change
when some additional observer will get to know where the particle is.
This is related to emergence of objectivity studied in many models \cite{Zurek, object}.

If we now allow to apply an arbitrary unitary to the Szilard engine, as is allowed by the assumptions of the above theorem, we can clearly reset the value of the bit, {\it without measuring  where the particle is}.
Thus, while the commonly accepted  application of Landauer principle to Szilard model is incorrect  -- as it says that to draw work
the observation must be done to know  where is the particle -- still,
in essence Landauer principle applies, only that the partition has become a demon
(in Fig. \ref{fig:szilard} we compare those two situations).

\begin{figure}[ht]
	\centering
	\includegraphics[width=0.7\textwidth]{fig-Szilard-standard.pdf}
	\includegraphics[width=0.7\textwidth]{fig-Szilard-physical-v2.pdf}
	\caption{Comparison of standard description of the operating cycle of Szilard engine with the proposed physical description. Note, that in
	the idealized setup -- where arbitrary unitary is allowed -- the cost of erasing shape and position of the barrier is just $k_BT\ln 2$, because the shape is correlated with the position.}
	\label{fig:szilard}
\end{figure}

However, application of such an operation seems unphysical. Physically, the measurement performed by the partition
costs more than $k T_B\ln 2$, as evaluated in \eqref{minwork}.
Also, the step of erasure, we believe, should require the amount of work  to be dissipated of the same order as
the cost of gate operation, as to reset bit, we need to destabilize it and stabilize again, which
will incur the cost of \eqref{minwork}, as indicated on the bottom of Fig. \ref{fig:szilard}.

There is a question, what is the scope of application of the Landauer principle as formulated in Proposition \ref{prop:Landauer}.
Clearly, it is applicable in a  situation, where a system can be coupled to and decoupled from environment on demand.
Indeed, then the state of the system is stable just because it is not always coupled to reservoir, and there is no additional cost
of stabilizing it. Moreover, in such situation the total state can be product with environment: indeed, the correlations described in Szilard model arise, because we want stable state despite it being coupled to the reservoir. However the situation of coupling on demand is not likely to be found in practice, as it requires some resources to keep system uncoupled to heat bath - usually it is done by cooling that requires massive energy expenditure - incomparably big with $kT \ln 2$, as we have mentioned while discussing experiments claiming to achieve Landauer's limit.   It is not excluded though, that for concrete physical setups and in some time scales, we may treat some systems as effectively not coupled to the heat bath, which makes room for applications of the standard Landauer paradigm.

To summarize, we believe that there is a universal cost of erasure, however, it is not given by $kT \ln 2$, but it is generically much greater, depending  on stability we want to have. Moreover, it does not come from the basic relations like unitarity of total dynamics
and monotonicity of relative entropy, but comes from the physics of interactions. Thus the real {\it devil is in details}: if we want to estimate in more generality the cost of erasure, we need to study models, which describe situation on a more fundamental level than Schrodinger equation with some a priori potential.

\section{Concluding remarks}

  The  discussion of above suggests the following picture. If by information we mean the stable information i.e. one that can be reliably stored and transmitted, rather than information carried by fast changing random configurations of physical systems, we expect that the following holds:

I) \emph{Stable  information is an abstract entity}\\
 Stable Information $I(\mathbf{p})$, measured for simplicity in \emph{nats} ($\log_2 e \simeq1.443$ bits), is a
\emph{disembodied abstract entity independent of its physical carrier}. In particular, multiplying $I$ by the Boltzmann constant $k_B$ or thermal energy  $k_B T$ does not yield thermodynamical entropy or meaningful energy which should be taken into account in the energy-entropy balance related to the First and Second Law of Thermodynamics. The quantities like  $k_B\times I$ or $k_B T\times I$ possess no physical meaning. In particular, stable information is not convertible into energy. Similar statement illustrated by a number of classical measurement schemes can be found in \cite{Kish:2018}.

II) \emph{Only information carriers are physical}\\
It is true that, as Landauer wrote :\emph{ "[Information] is always tied to a physical representation. It is represented
by engraving on a stone tablet, a spin, a charge, a hole in a punched card, a mark on paper, or some
other equivalent. This ties the handling of information to all the possibilities and restrictions of our real
physical word, its laws of physics and its storehouse"}.\\
However, the legitimate questions concern the physical properties of \emph{information carriers} like  "stone tablet, a spin, a charge, a hole in a punched card, a mark on paper", but not the information itself.

III) \emph{How to encode information?}\\
To encode information one needs physical systems possessing a number of distinguishable and {\it stable} with respect to thermal and quantum noise states. Therefore, for example, gas of atoms may possess a well-defined entropy but does not encode any information. The distinguishability of states can be quantified  in terms of their overlap, well-defined for classical and quantum systems. Only well-distinguishable states can be cloned with a high accuracy and  protected against noise for long times \cite{Alicki:2006}. Because our present day computers are based on macroscopic elements, which are highly stable on human time-scale and are applied to moderately large inputs and outputs appropriate again to the human scale, we can still disregard, to a large extend, physical limitations and apply abstract complexity theory to efficiency problems. \\

  We were not of course able to give a general no-go result, in favor of the above picture. However we believe that our paper will be stimulating to consider more detailed models in order to verify whether the presented view  indeed is the correct description of the
relations between thermodynamics and stable information.

{\bf Acknowledgement.} MH is supported by National Science Centre, Poland, grant OPUS 9. 2015/17/B/ST2/01945.

redpajama_set_name: RedPajamaArXiv


The money will be used for renovating a mosque in the capital Phnom Penh and building a 2-hectare (5-acre) Islamic center near the mosque, in addition to several schools and mosques in a northeastern Cambodian province, Ahmad Yahya said.
Kuwait has pledged US$5 million to help support religious activity and education for Cambodia's Muslims, a government official said Friday (8 Aug).
Prime Minister Sheik Nasser Al-Mohammed Al-Ahmed Al-Jaber Al-Sabah made the pledge during his visit here earlier this week, said Ahmad Yahya, a government adviser whoCambodian Muslim Community Developmentthe Cambodian Muslim Community Development.
He said the fund was in response to a request Cambodian Muslim leaders brought to Kuwaiti officials during the visit.
Kuwait and other Gulf states, especially Saudi Arabia, have long funded schools and mosques in Cambodia and neighboring Thailand, both of which have small and generally poor Muslim populations.
Cambodia has an estimated 700,000 Muslims, representing about 5% of the country's 14 million people, according the U.S. State Department's annual report on religious freedom.
During his visit, the Kuwaiti leader also expressed his country's interest in investing in Cambodian farmland to produce food for the oil-rich Gulf state.
The two countries also signed agreements to foster economic cooperation, protect investments, and establish direct flights between them.
Cambodia also asked Kuwait for assistance in training its workers in offshore oil exploration and production.


am soooo jealous you get to do a thailand trip.
am also jealous you have managed to find a clothing store with good changing room mirrors.
i cannot remember the last time I tried something on in a store. i usually buy, take home to see if it fits and then if it doesn't return it. i have learnt to try on clothes without removing tags and keeping it delicate. it's a gift!
it was my 1st experience with such mirrors 🙂 Maybe also had something to do with lots of flu medication!


Isla Mujeres, Q.R. — According to Riviera Maya News, the Secretariat of the Navy of Mexico says four people were rescued south of Isla Mujeres after the regional captain received a call.
The rescue was made after the Regional Captaincy of Puerto Juárez informed the navy that a call had been received requesting help for a small vessel named SOL. The vessel belonged to a local diving company who needed help returning two tourists and two members of their crew back to land.
They report that the boat was diving near Punta Cancun when it lost contact with three of its people. That, coupled with low nighttime visibility made it difficult to locate them, so they called for assistance.
A vessel from the Naval Search, Rescue and Maritime Surveillance Station of Isla Mujeres along with Federal Maritime Reserves patrol of the Regional Captaincy of Puerto Juarez were ordered to locate the divers who were still in the water.
PHOTO BY RIVIERA MAYA NEWS ON FEBRUARY 8.
Navy personnel were able to quickly find the divers. All were found in good health approximately 7 nautical miles (13 kilometers) south of Isla Mujeres. The stranded divers were a woman and two men of Mexican nationality of 53, 25 and 44 years of age, respectively, as well as a 44-year-old woman of Spanish nationality.
They were transferred to the pier of Playa Tortugas in Cancun safe and sound.


Wanting to be a positive role model in people's lives………………………………………..?
We are committed to safeguarding and promoting the welfare of the people we support. All successful candidates will be subject to an Enhanced DBS**/PVG check and Regulated Activity which will include check against the Barred adult list.


text:
Q: Log handled MassTransit Saga exception I'm trying to figure out how to log handled exceptions. Currently the exception thrown from an Activity will be swallowed (for example if there are DI errors when trying to create the Acitivity).
What can I do to log the caught exceptions?
Initially(
    When(UpdateRequested)
        .Activity(x => x.OfType<HandleUpdateRequestActivity>())
        .TransitionTo(Updating)
        .Catch<Exception>(then => then.Finalize())
);

(Sorry for spamming the MassTransit tag lately :) )

A: First, simply to correct the terminology, it isn't swallowed, it's handled.
The code above is catching the exception. If you want to log it, well, log it.
.Catch<Exception>(then =>
    then.Then(context => Log(context.Exception))
        .Finalize())

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The official sourcebook of church government and bylaws, history, doctrine, ritual, and policy of the Church of the Nazarene. Contains all new legislation enacted by the delegates to the 2017 General Assembly.
Would you recommend "Church of the Nazarene Manual 2017-2021"?


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The Endoscopy Department normally operates Monday through Friday 7:30 am to 4:00 pm. After-hours appointments are also available.


text:
Shipping's response to Paris climate deal left in disarray
Governments last month failed even to agree on developing a work plan to determine shipping's 'fair share' contribution to meeting the goals of the Paris deal. Despite there being a clear majority in support of the move, a minority led by China, Brazil, Russia, South Africa and the Cook Islands blocked a consensus to move forward. The issue was put back on the agenda of the International Maritime Organisation's (IMO) environment committee for when it next meets in October.
Bill Hemmings
Senior Adviser, Aviation and Shipping
Pacific island nations, developed countries and much of the shipping industry had supported a proposal to develop a post-Paris work plan on what emissions cuts would be needed. This call was accompanied by a proposal from industry for shipping to make a Paris-style contribution pledge on CO2 emissions reduction. The so-called BRICS countries opposed the move claiming that the IMO's decision, also taken at the meeting, to agree mandatory reporting requirements for ship fuel burn and CO2 emissions was more than enough progress.
During the debate the new IMO secretary-general Ki Tack Lim felt it necessary to appeal to governments to continue the climate discussions, while France warned that failure to agree would expose the UN shipping body 'to ridicule on the very day the Paris agreement was being signed in New York'. The European Parliament had also sent a delegation to the IMO meeting and MEP Jytte Guteland later pointed out that both international and EU measures are necessary to abate shipping emissions.
Bill Hemmings, T&E shipping director, said: 'How extraordinary it is that the IMO can't agree that the Paris climate deal will require the shipping industry to even assess what it needs to do in response. Key developing countries seem to be in denial.'
The meeting did agree to the mandatory reporting of ship fuel burn and CO2 emissions – a result that would have been unthinkable a couple of years ago. China, the Cook Islands and others argued that the IMO should wait and collect this data, then analyse it and only at that 'step three' commence a discussion as to what further measures to take. T&E said the measure lacked transparency and was inadequate because the measure of cargo carried – how much weight a ship can carry – was based on deadweight tonnage as a proxy rather than real operational efficiency data such as cargo carried and distance travelled.
Sotiris Raptis, T&E shipping policy officer, said: 'Transparency provided for in the regional EU system of design efficiency scores, or efficiency metrics, is missing at IMO. There should also be a requirement to collect real data, not proxies for cargo carried. Τhe reported CO2 data must be transparent so that the market can differentiate between efficient and less efficient ships. Only transparency of data will break down the industry barriers to greater efficiency.'
The EU's monitoring, reporting and verification regulation came into force in July 2015. It will lead to the creation of a searchable database of ships, and their emissions and energy performance, allowing ship users to take into account fuel consumption before making chartering decisions. IMO data on the other hand, will be aggregated, anonymised and accessible only by the IMO secretariat, not by flag states, the public or stakeholders.
The IMO's much-needed review of its design efficiency standard for new ships, the EEDI, will continue after the failure of efforts to shut it down without a strengthening of the efficiency requirements. The review will be discussed again in October.
Ship air pollution