Open the box. What is the status of the 2 hundred dollar bills? — hypericin

The physical assumptions underlying this thought experiment could be questioned. For example, the no-cloning theorem says that it is impossible to make an exact copy of a physical system. And the no-hiding theorem says that information is never destroyed.

That aside, it's of course possible for a human being to lose track of which bill is the real one. That just means that the status of the bills is unknown, it doesn't change the fact that one is real and the other is counterfeit (per the original designation).

It doesn't have to be an identical copy. Merely close enough so that for any conceivable test of genuineness that the real dollar passes, the counterfeit passes as well.

I'm having a hard time wrapping my head around the no hiding theorem. Right now I'm thinking of a 10 digit number. I'm not repeating it, and there's no way I will remember it in 5 minutes. Could a sufficiently clever alien, arriving on a venus like earth 10 million years from now, retrieve it?

Would you answer the same way if the no hiding theorem turned out to be false? Or true, but information can still be irretrievably, in principle as well as practice, lost and inaccessible?

Right now I'm thinking of a 10 digit number. I'm not repeating it, and there's no way I will remember it in 5 minutes. Could a sufficiently clever alien, arriving on a venus like earth 10 million years from now, retrieve it? — hypericin

I think the no hiding theorem is more about quantum information than about memory.I think it is
totally possible to retrieve your memory if it is stored in the form of waves.

At some point, it becomes absolutely undecidable which is real, and which is the counterfeit. That knowledge is lost to the universe. — hypericin

Presumably the authentic bill is issued by a central bank that keeps records of the total legal tender in circulation. So that information exists. The knowledge isn't lost to the universe.

Then so far as the central bank is concerned, it just has to cut up one of the two bills.

Which one doesn't matter. But what does matter is that one is erased so that the records are kept right.

A better example would be a promissory note assigning a fixed amount ($100) worth from one person to another.

If it's molecularly similar there's no difference and only 'counterfeit' due to the fact it was not legally printed from a government-backed institution. The reason I say all this is because of the serial number. It's only valid for one note. Granted if you pass a bill with a duplicate or erroneous (random) serial number to some random clerk or individual it's not like they're going to or even can check to see if it's valid and alert that someone needs to be investigated for financial crimes.

If we're ever in a scenario where we can molecularly replicate something exactly, a paper note would be the last thing to do it with.

Suppose you have a $100 bill, and a molecule-for-molecule counterfeit of one. — hypericin

Contradiction! If the two $100 bills were indistinguishable right from the beginning then how would you know whether:

1. Both are real

2. One is real and the other counterfeit

3. Both are counterfeit

? :chin:

In fact, as far as you're concerned all three possibilities are true.

Suppose you have a $100 bill, and a molecule-for-molecule counterfeit of one — hypericin

If it is that exact of a copy then it shouldn't be considered a counterfeit and is there any other distinguishing feature you haven't mentioned.

I'm having a hard time wrapping my head around the no hiding theorem. Right now I'm thinking of a 10 digit number. I'm not repeating it, and there's no way I will remember it in 5 minutes. Could a sufficiently clever alien, arriving on a venus like earth 10 million years from now, retrieve it? — hypericin

In principle, yes. For example, suppose the Earth's system were isolated with respect to the alien. Since no information has been lost, the alien could perform a physical transformation on the Earth's system that, in effect, runs the laws of physics in reverse until it's back to you thinking of the number. That process would be akin to unscrambling a scrambled egg.

Would you answer the same way if the no hiding theorem turned out to be false? Or true, but information can still be irretrievably, in principle as well as practice, lost and inaccessible? — hypericin

I would answer the same way if the information were merely irretrievable, since the information does still exist. But if the information does not exist, then there would be no basis for a distinction between the real and counterfeit bills.

As it happens, exactly this issue arises in quantum mechanics - see the Hong–Ou–Mandel effect. Suppose the photon coming in from above represents the real bill and the photon coming in from below represents the counterfeit bill (see the four possibilities diagram). State 2 has both photons transmitting, while state 3 has both photons reflecting. In the experiment, states 2 and 3 destructively interfere (i.e., are not experimentally observed) which means that there is no "which-way" information distinguishing those two states. So for states 1 and 4 that are observed, there is no longer any basis for regarding one photon as representing the real bill and the other photon as representing the counterfeit bill - that information does not exist. There are simply two photons that have a shared history.

Which one doesn't matter. But what does matter is that one is erased so that the records are kept right. — apokrisis

Yes. Which is to say, there's no difference between the bills that makes a difference. ;-)

I'm having a hard time wrapping my head around the no hiding theorem. — hypericin

Me too. That's some theorem. And it seems to me it must lead to some very strange consequences. For example, what are the bounds on the information available or storable? But I see this sentence from the wiki page:
"Thus, one may say that as energy keeps changing its form, the wave function keep moving from one Hilbert space to another Hilbert space. Since the wave function contains all the relevant information about a physical system, the conservation of wave function is tantamount to conservation of quantum information."

I suspect the "quantum" in "quantum information" makes a difference because it is not so-called classical information. This needs its own thread!

Exactly. Suppose there was a universe that consisted of a computer and a super battery. The computer continuously overwrites its hard drive with random 1's and 0's, and runs effectively forever. Where does all the information go? Quantum or no, at some point the universe will run out of room for it.

It doesn't matter if they are indistinguishable, so long as they can be individiually tracked.

I would answer the same way if the information were merely irretrievable, since the information does still exist. — Andrew M

But if it were irretrievable, from our perspective the situation is identical with that where it doesn't exist.

It doesn't matter if they are indistinguishable, so long as they can be individiually tracked. — hypericin

Can you tell them apart before you put them in the box?

No.

Can you tell them apart after you put them in the box?

No

In my humble opinion, there was no knowledge to begin with. So, no knowledge is lost.

All I have to do to tell them apart is to put the real one in my left pocket, and the counterfeit, fresh from the counterfeit machine, in my right.

All I have to do to tell them apart is to put the real one in my left pocket, and the counterfeit, fresh from the counterfeit machine, in my right — hypericin

:chin: However, note that after shaking the box, you're relying on information content of the bills themselves and nothing else to tell them apart but that information, as I said, never really existed.

I would answer the same way if the information were merely irretrievable, since the information does still exist.
— Andrew M

But if it were irretrievable, from our perspective the situation is identical with that where it doesn't exist. — hypericin

For all practical purposes, sure. But I think the conceptual distinction remains. Before the bills go in the box, you know which one is real and which one is the counterfeit. After the bills go in the box, you lose track of them, but they still have a definite history as their positions change over time.

So, in effect, all that has changed before and after is your knowledge of that definite history.

Do we know, of this "quantum information," if a) it can ever be what we call knowledge, i.e., known, and, b) can it always be known, in the sense of retrieved? Or not retrieved? Or not retrievable?

I imagine throwing a stone into the ocean thereby disturbing the water. And maybe that determines uniquely the future of the ocean. The ocean, then, stores a record of that disturbance. But how is that to be recognized as such, and how retrieved as to the particulars that make that what it is?

Might it be the case that the no-hide theorem merely means that change is in some sense permanent? Anyone?

Do we know, of this "quantum information," if a) it can ever be what we call knowledge, i.e., known, and, b) can it always be known, in the sense of retrieved? Or not retrieved? Or not retrievable? — tim wood

Whether something can be known (or retrievable) or not depends on what is being done. If a coin is flipped inside the event horizon of a black hole (and subsequently destroyed), then it is going to be essentially impossible for external observers to retrieve the result. On the other hand, if you send a photon into a suitably configured Mach-Zehnder interferometer, it can be predicted with certainty which detector the photon will arrive at (despite being impossible to predict using classical theory).

In both cases, quantum information is conserved. Here's a PBS Space Time video on this - Why Quantum Information is Never Destroyed. The idea of quantum computing, then, is to exploit the special characteristics of quantum information (i.e., qubits) to solve problems that are beyond the capabilities of classical computers.

I imagine throwing a stone into the ocean thereby disturbing the water. And maybe that determines uniquely the future of the ocean. The ocean, then, stores a record of that disturbance. But how is that to be recognized as such, and how retrieved as to the particulars that make that what it is? — tim wood

The way I would put it is that information about the stone hitting the water is exhibited in the ocean itself. We can see it in the ripple. Of course there is a lot that we don't see and measure, and the ripple soon disappears. But that information is nonetheless retained in the environment which is, in principle, measurable.

Might it be the case that the no-hide theorem merely means that change is in some sense permanent? Anyone? — tim wood

In terms of the no-hiding theorem, if a system is randomized, information about the randomization operation will be stored in the environment external to the system. That information can then be used to perfectly reconstruct the original system. Here's an experiment demonstrating this:

In order to make the first qubit “lose” its information, the scientists had to make the system undergo a bleaching process. In their experiment, they bleached the system through quantum state randomization, in which the qubit transforms from a pure state to a mixed state. Although the randomization operation causes the qubit to appear to lose the information contained in the pure state, the scientists showed that the information could be found in one of the two ancilla qubits. They also demonstrated how to use the ancilla qubits to reconstruct the original state, showing that no information was hiding in the correlations between the original qubit and the ancilla qubits, which is the essence of the no-hiding theorem. — Quantum no-hiding theorem experimentally confirmed for first time

:up:
I had watched the video. To my classical mind it all does not quite make sense. However, it apparently makes perfect sense, and necessarily!

Whatever quantum information (QI) is, it seems it must exist in some sense. That is, QI is always somewhere, somehow, someway - and apparently never isn't! Two very simple statements of this:
1) QI must be, to be.
2) QI always is.
And not just any QI, but each and every QI.

And again, whatever QI is, to be information in any sense must (yes?) mean that it "contains" something that it itself is not, that can be extracted from it, apparently non-destructively, which implies repeatedly. And that something in every case is part of a recoverable path to the unbounded future and the unbounded past, somehow, someway.

And it seems quickly clear that this kind of language and thinking is not adequate for this task. If the theorem is true - one supposes it is - then the language has to be very tightly defined and constrained. I suspect past the breaking point. Information must finally reduce to mere being, and being as information leads to some ferocious paradoxes.

And again, whatever QI is, to be information in any sense must (yes?) mean that it "contains" something that it itself is not, that can be extracted from it, apparently non-destructively, which implies repeatedly. And that something in every case is part of a recoverable path to the unbounded future and the unbounded past, somehow, someway. — tim wood

Conservation of information also applies in classical physics. Here's a couple of snippets from Leonard Susskind's lectures on statistical mechanics and classical mechanics (videos available on Youtube):

Proper laws of physics are reversible and therefore preserve the distinctions between states - i.e. information. In this sense, the conservation of information is more fundamental than other physical quantities such as temperature or energy. — Statistical Mechanics - Entropy and conservation of information - Susskind

Liouville's theorem can be thought of as information conservation. The laws of mechanics are equivalent to the rules governing state transition. — Classical Mechanics - Liouville’s theorem - Susskind

To give an example, consider a simple system with six states, like the sides of a die. The dynamical laws in this system for each time step are:

1 -> 2 -> 3 -> 4 -> 5 -> 6 -> 1

Information is conserved. For any given state, the next state and the previous state can be predicted. Whereas in the following system, information is not conserved since the state prior to state 3 can't be predicted.

1 -> 3 <- 2

And it seems quickly clear that this kind of language and thinking is not adequate for this task. If the theorem is true - one supposes it is - then the language has to be very tightly defined and constrained. I suspect past the breaking point. Information must finally reduce to mere being, and being as information leads to some ferocious paradoxes. — tim wood

Information (or state) is an abstraction of physical systems. Quantum information is just a different abstraction than we're used to with classical information. A classical bit is just a 0 or a 1. Whereas a qubit is a linear combination of 0 and 1 that, when measured, collapses to a 0 or a 1. That seems like information has been erased. But the lost information has simply moved elsewhere (e.g., to an ancilla qubit or heat in the environment).

A good line that surfaces from time-to-time is, "We can do or we can teach, what's you pleasure?" In our present context I read that as, "We can learn or we can debate." And as it happens I think that's a false alternative because both are necessary, the trial by debate, intra-personally as learning, and inter-personally as debate, each a name of a different part of the digestive process that yields knowledge. But we needn't be and I hope are not on opposite sides, but rather put such understandings that we generate and evolve through "to the question" as simply a part of a shared goal of getting a handle on an idea that at first look seems to me fatally problematic.

Information (or state) is an abstraction of physical systems (PS). — Andrew M

Would you agree, on reflection, that in this context (hereinafter to be understood if not stated), that either this is exactly wrong, or needs qualification to be right? If information corresponds to state, then information just is itself and cannot be other than itself - without being other. If information is descriptive of any PS, then it is not that PS, but its own distinct PS, and if that PS is the idea of the thing, then that itself becomes difficult.

Am I right or on the wrong track in this simplification: if information is conserved, then it is possible as a matter of fact (not to say possible for us as a matter of fact) that from all the highways of the world - the impressions on the pavement, gravel, or dirt - it is possible to recover what traveled on them, when, and what was carried, even to the brand names on the labels on the cans, where the labels were printed and with what inks. Or in short, given any event at any level of granularity, the exact details of that event are recoverable from whatever is part of the event. (Which anticipates problems with propagation and the speed of propagation.)

Or another, that from the metamorphic rock from under an ancient and long gone streambed it is possible to recover what stone what sauron kicked into it on a day 100,000,000 years ago, and the configuration of the splash. Of course some of that evidence went up as water vapor, so part of the recovery must involve the entire atmosphere of the earth - or not? Is the information complete in parts or does it require the whole?

And these as nothing to the "information" of the cosmos.

I mean in these to evoke a sense of the aporia I think intrinsic to the problem. If information is just state, then it is at the moment and not otherwise. If information is knowable, then the state-as-information must also create some kind of meta-information/state (or something without yet a name) that travels through time, or endures through time, that preserves the original state, somehow. And that asks as to the question of meta-meta-...-meta information/states.

For the theorem to be meaningful it must cut through all of this, yes?

But we needn't be and I hope are not on opposite sides, but rather put such understandings that we generate and evolve through "to the question" as simply a part of a shared goal of getting a handle on an idea that at first look seems to me fatally problematic. — tim wood

:up:

Information (or state) is an abstraction of physical systems (PS).
— Andrew M

Would you agree, on reflection, that in this context (hereinafter to be understood if not stated), that either this is exactly wrong, or needs qualification to be right? If information corresponds to state, then information just is itself and cannot be other than itself - without being other. If information is descriptive of any PS, then it is not that PS, but its own distinct PS, and if that PS is the idea of the thing, then that itself becomes difficult. — tim wood

Be careful not to reify information. Consider a coin that has landed tails-up on a table. The state of the coin (tails) is not itself a system, it's the form of the system in a specific context (where form is the linguistic root of information).

Or another, that from the metamorphic rock from under an ancient and long gone streambed it is possible to recover what stone what sauron kicked into it on a day 100,000,000 years ago, and the configuration of the splash. Of course some of that evidence went up as water vapor, so part of the recovery must involve the entire atmosphere of the earth - or not? Is the information complete in parts or does it require the whole? — tim wood

It depends. For the stone-kicking question, perhaps the whole atmosphere (or even light cone) is required. Whereas the question of whether it rained earlier today could be answered from more easily accessible evidence.

Information is conserved in both cases and accessible in principle, if not in practice.

Consider again the no-hiding experiment referenced earlier where the system was randomized via a bleaching process. The original state of the system was recoverable from the ancilla qubits. But if that information had instead leaked into the external environment, then it would not be recoverable by present technology, but that information would still be somewhere.

I mean in these to evoke a sense of the aporia I think intrinsic to the problem. If information is just state, then it is at the moment and not otherwise. If information is knowable, then the state-as-information must also create some kind of meta-information/state (or something without yet a name) that travels through time, or endures through time, that preserves the original state, somehow. And that asks as to the question of meta-meta-...-meta information/states.

For the theorem to be meaningful it must cut through all of this, yes? — tim wood

I'm not sure I understand your objection. But let's apply your comments to the six-state die system that I presented in the previous post (which cycles from 1 to 6). Given the rules of that system and a current state (e.g., 3), then the previous state is necessarily implied (i.e., 2). I don't see how a meta-state arises here.

Counterfeit/fake refers to it having been made illegally moreso than its molecular arrangement, yes?

Been watching Good Girls, ? :)