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Information is Physical (?)

If you know anything about computers, then you probably have some idea of the way it stores and processes information. It is all about the 1’s and 0’s right? You write the 1’s and 0’s somewhere, the computer somehow reads it and BAM! It does what you want. Well, that is perhaps oversimplifying things just a little, but life is too short to care for the details. In any case, you have to store those 1’s and 0’s somewhere, perhaps on a microSD card or a hard disk drive or a compact disc or any one of the whole multitude of devices meant to store your digital information. In this sense, you probably already have an intuitive idea that information is physical because you always have to write them down on physical objects.

In 1961, Rolf Landauer took this (admittedly obvious) observation to another level by demonstrating a physical principle using it. It essentially states the following: the erasure of a single bit of information has to cost a minimum amount of energy called the Landauer limit (it has a numerical value of KTln2, see  Details aside, the essential argument revolves around a theoretical engine, named the Szilard Engine (Figure 1) after the scientist who studied it.  The engine appears to be able to do physical work without actually requiring any input of energy. This violates the 2nd law of thermodynamics, which says that this engine is giving us too good a deal and that a free lunch is impossible. The Szilard engine, however, does require you to make a measurement that gives you a single bit of information about the system in order to do work. Since the information has to be written down somewhere, and the actual writing on information appears to be doable without costing any energy, it should cost energy to erase this bit of information. In this way, there is no free lunch since you will eventually run out of space to write down the measurement and erasure of information has to be performed.

Figure 1: A rough Schematic of the Szilard Engine. You initially have a box containing a single particle in a heat bath. At first, you don’t know where the particle is inside the box. You then insert a movable partition inside the box and perform a measurement to find out if the particle is on the left or right of the partition. After this measurement, you know which direction the partition will move when the gas expands, therefore performing useful work.

These arguments are fairly convincing, and Landauer’s principle is accepted by many in the scientific community today. However, the issue of its validity is far from completely settled. One may for instance argue that the assumptions underlying the Szilard engine cannot be valid. Using the same set of assumptions, you can actually design a separate engine that does work without even performing a measurement (See Figure 2), also a clear violation of the 2nd Law of Thermodynamics. Is there a flaw in such a design

Figure 2: An engine design similar to the Szilard engine that performs work without measurement. For details see Mind, Matter and Methods by Paul K Feyerabend

So this casts some doubt as to whether Landauer’s principle is actually valid or not. Needless to say, this is an issue that is still being debated today, decades after the principle was proposed. Whatever your stand is on the issue, information remains physical, so keep that hard disk drive handy.

Conway’s game of life

One of the cooler things I discovered recently was that when you google search “Conway’s game of life” google automatically begins to play the game of life in the background of the search results page. It is rather mesmerizing,

The game of life is played on a grid of little squares; squares have two configurations “alive” and “dead” which are usually represented by using two different colours. A single game consists of a initial seed or starting configuration and a set of rules which govern how the grid updates:

(1) Any live cell will remain alive if it has 2 or 3 live neighbors; otherwise it will die.

(2) Any dead cell with 3 live neighbors will become alive.

The rules are roughly motivated by physical considerations; if the population density is too low or too high then individual members of a colony will die before reproducing. However three alive cells may reproduce to create a extra adjacent live cell.

The game is extremely popular because the seemingly overly simple rules can generate extremely complicated patterns. Recently it was demonstrated that certain initial conditions of the game of life can give rise to emergent behavior in the sense that we can aggregate large sections of the grid (consisting of multiple cells) and treat this section as an individual “macroscopic cell” which behaves non-trivially. There are no simple set of rules for how these macroscopic cells behave; all update

rules continue to apply to the small indivisible cells which constitute them. Nevertheless it is possible to  initialize the game of life so that the macroscopic squares effectively appear to evolve according to conditions (1) and (2). There is quite a spellbinding example of this available here. Check it out as it is unbelievable at first sight.

Emergent time coordinate

Recently I have been very interested in the concept of a thermal time. The basic premise is that unidirectional time coordinate we experience may not be a fundamental property of some GUT (grand unified theory) scale physics but is instead an emergent phenomenon. At first this sounds somewhat anarchical as physical laws mostly focus on predicting the state of a system at some future time and almost invariably involve some kind of description of the behaviors of a system under dynamical evolution.  The idea is at once fascinating but also unconventional enough to be something you need a really good reason for doing. If you are genuinely curious try reading Rovelli and Connes [1].

The story begins with the Tolman Ehrenfest effect. Static observers who measure Temperature in gravitational fields find that it depends on the gravitational potential at the point where the measurement is made so that


is a constant (interestingly this scaling is usually associated with the relation between the coordinate time and the proper time measured by a clock traveling along a world line) [2,3].  In other words a long column of fluid in thermal equilibrium will naturally have a temperature gradient with hotter liquid at the bottom.  The difference is tiny; nevertheless it implies general relativity is deeply intertwined with the temperature of your beer tower.

What if at some level statistical physics and thermodynamics are fundamental and what we perceive as time is an emergent property. We can postulate that the universe is described by a generally covariant theory which treats all coordinate directions  evenhandedly and time is an artifact of the thermal state that we are immersed in.

There is no place for a preferred time variable in general relativity; furthermore if we look at the physically observable quantity measured by a clock it is in fact the proper time along a world line.  The coordinate time which parameterizes the field variables,

and trajectories of relativistic particles is just a variable. Indeed there is one equivalent definition of t for each Lorentz transformation,
because the equations of motion are manifestly Lorentz covariant it really doesn’t matter which you choose. They are all just reparameterizations.

Intriguingly thermal states break Lorentz invariance. They pick out a preferred coordinate system; in fact a thermal bath preferences the Lorentz frame in which it is at rest. This thermal state may be used to define a preferred physical time. Yet the background theory remains generally covariant.

In thermal equilibrium our thermal states can be described by Gibbs states; these encode information about the Hamiltonian and the dynamical properties of the system are attributed to this thermal state rather than direct Hamiltonian evolution. To some extent it is valid to assume you are always working in a thermal regime as you stereotypically don’t have access to the full microscopic state of a system and hence you simply reconstruct the microscopic state from macroscopic observations

[1] Rovelli and Connes [arXiv: 9406019v1]

[2] Rovelli and Smerlak [arXiv:1005.2985v5]

[3]Tim-Torben Paetz [#://]




The Zeno Paradox

The Zeno Paradox

For fun this week we have a short story from Dag on The Zeno Paradox.

A side street in Tokyo. Neon lights in heavy rain. A shady bar with a barman who never speaks unless you don’t pay for your booze. A lonely guy sits in the darkest corner of the bar with a half empty bottle of Yamazaki. Cigarette smoke slithers around his unshaven face, eyes focus on some memories swirling in the dark behind the window. This is the place where men come to absolve their sins before disappearing into the night.

The bar door swings open. A man in a trench coat steps in, pauses to look around. His long shadow stretches towards the lonely guy as if trying to tighten its icy fingers around his throat. The barman gives the newcomer a quick glance only to get back to his world of endless nights when time stands still like the rows of bottles behind him.

“I hate rain”, he mutters to himself.

The newcomer sits in front of the lonely guy.

“William?”. The guy takes a long drag of his cigarette, savours the smoke for awhile, turns his head towards the newcomer and exhales straight into his face.

“Who’s asking?”, he says.

“Wilson. Do you have it?”

William doesn’t reply immediately. He pours himself a glass of whiskey, double shot, looks through it at Wilson, puts it on the table, adds more and then gulps it down like it is his last.

“Yeah,” says William, inhaling the cigarette.”I have it,” he adds, exhaling a thin streak of smoke.

“Give it to me.” Wilson’s voice sounds greedy. William looks straight into his eyes and says almost caringly,

“I’ll give it to you but you must listen to my story first.” “Keep it short, pal,” replies Wilson.

“I loved Gail more than anything, more than myself. I first saw her in a small dancing studio at night. It was raining like today.” William’s voice becomes shaky. He takes another shot of whiskey.

“She was practicing some moves in front of a big mirror. She looked so beautiful, like out of this world. Her body moved across the dance floor with a grace I’d never seen before. I was standing there, glued to that big window and I knew that Gail was the woman I wanted to be with.” He grabs Wilson by the arm and says feverishly “Can you understand that? Can you?!”. Wilson shakes off the hand.

“Take it easy, man” he says dryly.

“We were like Bonnie and Clyde. Lovers, friends. It was a blast but nothing good lasts for long in this twisted universe. Gail fell terminally ill.” William stops, lights another cigarette. Smoke seems to make it easier for him.

“I couldn’t watch her body wasting away”, he pauses, eyes fixated on the swirling cigarette smoke.

“Have you heard about the Zeno paradox?” “No” replies Wilson.

“Zeno claimed that nothing moves because to get from A to B you need to cover half the distance, then the half of the half and so on. Every half requires a finite time to travel but there are infinitely many of them so you won’t cover the distance in a finite time.”

“Nonsense” says Wilson.

“Yeah…. Infinitely many pieces can give you a finite thing”, William pauses, “Not in the quantum world.”

“What do you mean?” William gets Wilson’s attention.

“In the quantum world there is no reality. Observation creates it and this means you can manipulate reality by simply looking at physical systems” William puts out the cigarette. “If you observe them frequently enough you can freeze them forever.”

“That’s how the machine works?” interrupts Wilson.

“Yeah, something like that.”

“Where are the blueprints?” Wilson’s eyes flicker with greed.

“I haven’t finished yet.” William lights another cigarette. “I thought I could keep Gail in a state of suspension until they found a cure.”


“I asked her to dance for me one more time and…” he swallows tears.

“What?” asks Wilson impatiently, pouring William another drink. William ignores it.

“Then I set this… machine… in motion.” William’s voice quivers again. He gulps down the glass of whiskey and goes motionless like a mechanical toy with a discharged battery.

“And?” Wilson prompts him.

“At first it worked beautifully. Gail’s body froze in time… She looked so beautiful.”

“And?!” asks Wilson’s impatiently.

“A few days later I noticed some small changes in her face. Blemishes.” He pauses. “The blemishes started to become fuzzy and larger, slowly transforming Gail’s body into… into…” William swallows hard, his Adam’s apple forcing its way up and down like a piston of a worn out engine, “into something undefined, smeared in space.” William’s hand wipes some invisible grease off his face.

“Couldn’t you stop the machine?” interrupts Wilson.

“It was too late. I would have had to reverse the whole time evolution but I didn’t have enough computational power.” William takes out a notebook. “Here’s the blueprint for the machine.” He throws it on the table. “Can I go now?”

“Where is she now? I mean Gail” asks Wilson ignoring William’s question.

“I’d like to believe that she’s become entangled with the rest of the universe” he pauses, looks into the night behind the window. “And that one day I’ll be able to bring her back, see her dancing again…”

Wilson picks up the blueprint and puts it into an internal pocket of his trench coat.”You know I can’t let you go. We need your expertise. Without you it would take us too long to build the machine.” Wilson wraps his fingers around William’s arm. “Just don’t do anything stupid.”

William looks at Wilson and smiles, his eyes hidden in the shadow.

A side street in Tokyo. Neon lights in heavy rain. A shady bar with a barman who never speaks unless you don’t pay for your booze. A lonely guy sits in the darkest corner of the bar with a half empty bottle of Yamazaki. Cigarette smoke slithers around his unshaven face. A fuzzy, slowly expanding blemish appears at the corner of his eye.

Check out more at Quantum Shorts 2013: Zeno Paradox 

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Can we really change the past?

As the previous post did a great job of explaining Wheeler’s delayed choice experiment, I thought a natural follow-up would be to talk about the delayed choice quantum eraser experiment. But while writing this post I became disturbed by the implications of the original experiment, so I ended up addressing that as well.

Going back to the ubiquitous double-slit experiment, we recall that each photon seems to go through a single slit (particle behaviour) only when we the observer are actually looking at what is going on each slit. Otherwise it goes through both slits simultaneously, and we observe a wave interference pattern at the screen. As the previous post explained, this is true even when the choice to look at the slits or not is made after the photon is past the slits, which seems to imply the mind-blowing conclusion that our choice to look or not look has somehow altered the photon’s history. More will be said on this at the end..

But first, the quantum eraser. (Or you could skip to the end right now – perhaps your choice will affect what was written in between :)) This experiment was proposed by Scully and Drühl in 1982 and first performed by Kim et. al. in 1999. It seeks to answer the question: If we do choose to look at the slits individually but then somehow erase this information, will we recover the interference pattern?

The experiment proceeds as follows. After passing through yet another double slit apparatus, a photon is split into two entangled photons by a beta barium borate crystal. One photon (called the signal photon) proceeds to detector D0, while the other (the idler photon) passes through a combination of beam splitters and ends up at one of four possible detectors. Essentially detector D3 can only be reached if the original photon went through slit B, D4 slit A, and detectors D1 and D2 can be reached with equal probability form either slit. One thus says the which-path information has been erased if the photon arrives at D1 or D2.

As you might expect by now, after conducting the experiment D1 and D2 recorded an interference pattern, and D3 and D4 did not. Evidently erasing the which-path information had indeed seemed to retroactively make the photon “choose” to go through both slits. Even more confusingly, when Kim et. al. used a coincidence counter to match each idler photon with its signal counterpart reading at D0, they found that each entangled pair displayed the same behaviour (i.e. if the idler photon showed interference at D1, so did its partner at D0.) The mystery was that the signal photon arrived at D0 before its partner even entered the multi-beam splitter contraption to have its path history erased or not erased. We seem to be left with a choice of two disturbing conclusions: either by manipulating the idler photon we affected the original photon’s history, or the original photon somehow “knew” what was going to happen in the future and behaved accordingly.

The explanation I favour that avoids all this evident causality violation is as thus: the photon does not “choose” to go through slit 1 or slit 2 (or both) at all! Returning to the original delayed choice experiment as promised, after passing through the slit apparatus, the photon is in a superposition of “went through slit 1” and “went through slit 2”. Removing the screen “at the last second” to reveal his two telescopes (or detectors) does not retroactively collapse the photon’s wavefunction when it was passing through the slits earlier. In other words, detector 1 registering a reading does not mean “the photon went through slit 1 earlier”. Choosing whether to use the screen or the two detectors is a choice of measurement basis. The photon then collapses into an eigenstate compatible with the measurement, and we have the observed wave or particle behaviour. We can apply a similar reasoning to the quantum eraser experiment, with more complexity involved. This is still decidedly strange non-classical behaviour, but at least it helps one sleep a little easier at night.

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Second law of thermodynamics and heat engines

The second law of thermodynamics is believed to be one of the most fundamental postulates of modern physics. Its most famous formulation is as the statement: the entropy of an isolated system never decreases; and it can be conveniently used as an excuse to never clean your room. This argument is ingeniously simple. Entropy of a macroscopic state is a measure of the number of possible corresponding microscopic configurations. In this case the macrostate is ‘a messy room’ and each microscopic configuration labels the positions of every object in that room such as the 10 books and a computer currently cohabiting your bed. Note that cleaning up a room involves ordering the objects in it. Since there are many possible configurations of these objects for which we say the room is messy but there may be only a small number of orderings for which we consider it clean; your efforts locally decrease the entropy of the room. However we know by the second law of thermodynamics that while we are doing the work to reordering the room we dissipate heat to the environment causing the universe to become more entropic globally; or in other words cleaning up your room contributes to the heat death of the universe and should only be undertaken at your own peril. Check out xkcd’s ideal room configuration.

This illustrates a perhaps more physical reinterpretation of the second law of thermodynamics; it is often restated as the impossibility of convert heat directly into work which ties into the following construction of a Heat engine. In this graphical representation we can see that not all the heat Q H transferred from the hot reservoir can be converted directly into useful work, only a fraction Q H – QC  can be accessed; the remainder must be dissipated to the cold reservoir thereby increasing the entropy of the cold reservoir. If we could completely convert Q H  directly into useful work then we could use this work to reduce the entropy (for example we could erase some information), at the same time we would have QC equal to zero so there would be no counterbalancing increase in the entropy of the cold reservoir. This would lead to a global entropy decrease thereby violating the second law. In our example of a messy room we realize that you are functioning as the hot reservoir and the universe (empty space has a temperature of about 3 Kelvin) is functioning as the cold reservoir. We can clearly see that it is sensible to define the amount of useful work which can be extracted from a system a concept called the Helmholtz free energy F. Making it possible to succinctly summarize everything we have said above by  the statement that  the change in Helmholtz free energy of the system can be at most equal to the average work done on a system: <W>  ≥  ΔF. The averaging over the work distribution in this statement arises because in finite systems we  can have quite large statistical fluctuations. This is an extremely powerful statement and a key tool in many active fields of research particularly in its relation to Szilard engines and generalized forms of the second law like Jarzynski equality which extends the second law of thermodynamics (relationship between free energy and work) to higher moments of the work distribution (the average work is the first moment).


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