Physicists experimentally realize a quantum Hilbert hotel

October 21, 2015 by Lisa Zyga feature
When the light “petals” (quantum states with an infinite number of values representing the infinite number of hotel rooms) in the top row are multiplied by 3, the number of petals in the bottom row is tripled—analogous to “tripling infinity.” Credit: Václav Potoček, et al. ©2015 American Physical Society

(Phys.org)—In 1924, the mathematician David Hilbert described a hotel with an infinite number of rooms that are all occupied. Demonstrating the counterintuitive nature of infinity, he showed that the hotel could still accommodate additional guests. Although clearly no such brick-and-mortar hotel exists, in a new paper published in Physical Review Letters, physicists Václav Potoček, et al., have physically realized a quantum Hilbert hotel by using a beam of light.

In Hilbert's thought experiment, he explained that additional rooms could be created in a hotel that already has an of rooms because the hotel manager could simply "shift" all of the current guests to a new room according to some rule, such as moving everyone up one room (to leave the first room empty) or moving everyone up to twice their current room number (to create an infinite number of empty rooms by leaving the odd-numbered rooms empty).

In their paper, the physicists proposed two ways to model this phenomena—one theoretical and one experimental—both of which use the infinite number of quantum states of a quantum system to represent the infinite number of hotel rooms in a hotel. The theoretical proposal uses the infinite number of energy levels of a particle in a potential well, and the experimental demonstration uses the infinite number of orbital angular momentum states of light.

The scientists showed that, even though there is initially an infinite number of these states (rooms), the states' amplitudes (room numbers) can be remapped to twice their original values, producing an infinite number of additional states. On one hand, the phenomena is counterintuitive: by doubling an infinite number of things, you get infinitely many more of them. And yet, as the physicists explain, it still makes sense because the total sum of the values of an infinite number of things can actually be finite.

"As far as there being an infinite amount of 'something,' it can make physical sense if the things we can measure are still finite," coauthor Filippo Miatto, at the University of Waterloo and the University of Ottawa, told Phys.org. "For example, a coherent state of a laser mode is made with an infinite set of number states, but as the number of photons in each of the number states increases, the amplitudes decrease so at the end of the day when you sum everything up the total energy is finite. The same can hold for all of the other quantum properties, so no, it is not surprising to the trained eye."

Steps to realizing the theoretical version of a quantum analogue of the Hilbert hotel, which is done by remapping the amplitudes shown in (a) (which initially have an infinite number of values) to twice their original values, as shown in (f), by expanding and shrinking the potential well. Credit: Václav Potoček, et al. ©2015 American Physical Society

The physicists also showed that the remapping can be done not only by doubling, but also by tripling, quadrupling, etc., the states' values. In the laser experiment, these procedures produce visible "petals" of light that correspond to the number that the states were multiplied by.

The ability to remap energy states in this way could also have applications in quantum and classical information processing, where, for example, it could be used to increase the number of states produced or to increase the information capacity of a channel.

Explore further: You don't exist in an infinite number of places, say scientists

More information: Václav Potoček, et al. "Quantum Hilbert Hotel." Physical Review Letters. DOI: 10.1103/PhysRevLett.115.160505

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30 comments

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TheWalrus
2 / 5 (2) Oct 21, 2015
"Phenomena" is a plural noun. The singular is "phenomenon."
SuperThunder
2.2 / 5 (5) Oct 21, 2015
This is incredible. I am excited about further applications of infinity mathematics in physics. Just, wow. I can't wait to see how quantum computer engineers exploit this.
anywallsocket
3.5 / 5 (2) Oct 21, 2015
"And yet, as the physicists explain, it still makes sense because the total sum of the values of an infinite number of things can actually be finite."

someone wanna help clarify this absurdity?
docile
Oct 21, 2015
This comment has been removed by a moderator.
Whydening Gyre
not rated yet Oct 21, 2015
"And yet, as the physicists explain, it still makes sense because the total sum of the values of an infinite number of things can actually be finite."

someone wanna help clarify this absurdity?

An infinite number of "things" with measurable properties ("values") can add up to a non infinite "value"...?
It is a little bit vague...
PhysicsMatter
5 / 5 (1) Oct 22, 2015
What a confusion in terms. The so-called number of states is a number of states that potentially a particle could be in and unless it is in one of such states, the state itself does not exists.

In classical physics an object could be in any state it wishes to be in, meaning infinite number of states are possible and by doubling the number of states you do not increase the number of possibilities since it is infinite.

The same is if you double the infinite series 1,2,3,4,5... (you see 5+ number) to 2,4,6,8,10... (you see 5+ numbers, the same as before) you do not decrease the total number of the existing numbers since they stay the same you just select different numbers to look at.

An interesting take on often misleading interpretation of the concepts of quantum physics I found at:

https://questforn...-quanta/
antialias_physorg
3.4 / 5 (5) Oct 22, 2015
"And yet, as the physicists explain, it still makes sense because the total sum of the values of an infinite number of things can actually be finite."


someone wanna help clarify this absurdity?


Take an infinite series (an infinite number of things) e.g. the series from n=1 to n=infinity, where the nth member in the series is 1/(2^(n))
If you add up all the items in this infinite series you get a finite value (1).
Shabs42
2.5 / 5 (2) Oct 23, 2015
"And yet, as the physicists explain, it still makes sense because the total sum of the values of an infinite number of things can actually be finite."


someone wanna help clarify this absurdity?


Take an infinite series (an infinite number of things) e.g. the series from n=1 to n=infinity, where the nth member in the series is 1/(2^(n))
If you add up all the items in this infinite series you get a finite value (1).


A simpler example that I believe is correct, adding together all integers: -1 + 1 + -2 + 2...= 0.
Jayman
not rated yet Oct 23, 2015
Or is all this just mathematical jugglery? Curve fitting maybe? Surely, such a disconnect between quantum objects and visible objects just cannot exist !
docile
Oct 23, 2015
This comment has been removed by a moderator.
garciah926
1 / 5 (2) Oct 23, 2015
This is how we know your are a total crackpot docile.

Order of operations is everything

The are most definitely CONVERGING infinite series
garciah926
3 / 5 (2) Oct 23, 2015
A better argument would be to argue what the definition of convergence of an infinite sequence is: An infinite series converges if there exists a finite limit to the sequence of partial sums as the terms go to infinity.

So whats the definition of a limit? In the simplest terms, it means that the limit of a sequence is the value such that, no matter how close the sequence comes to that value, I can always take the sequence further and find the values of the sequence tending even closer to the limit.

So in a sense there's an argument to be made as to whether a convergent infinite series actually EQUALS the value it converges to. This is more of a matter of the definition of what it means for a convergent infinite series to "equal" a value.

However, the series posted by shabs not only converges, but its absolutely equal to zero. It is a series of zeros.

You gave an example of when a series doesnt converge. So what? We know some series do not converge
docile
Oct 23, 2015
This comment has been removed by a moderator.
garciah926
3 / 5 (2) Oct 23, 2015
*explained the limit of the sequence a bit off above. Its the value that, no matter how close I want my sequence to approach that value, I can always take my sequence far enough to be within my desired "closeness"

For example:
lim 1/n as n goes to infinity is 0. Why? Suppose want my sequence to be .000001 within the range of 0. I can let "n" go to 1,000,001 and the value of the 1,000,001th term of my sequence is 0.000000999 which is closer to zero than .000001
garciah926
3 / 5 (2) Oct 23, 2015
yes infinity + (-infinity) is undefined because infinity is not a number. So not only is that undefined, SO IS THE IDEA OF HAVING INFINITY OR -INFINITY AS TAIL ENDS OF SEQUENCES OF NUMBERS!!!

Infinity and -infinity are not the tail ends of the sequence that shabs showed. There is no "tail end" to that sequence, what you do have is a sequence of zeros. We can write his sequence as:

Sum_(n=1)^(infinity) (n+(-n)),

write out the terms first, they will be the terms Shabs wrote. Now, before writing the terms, simplify the expression in the series, you get zero. so

Sum_(n=1)^(infinity) (n+(-n)) = Sum_(n=1)^(infinity) 0

Now write out the terms of that series, its a series of zeros
garciah926
3 / 5 (2) Oct 23, 2015
In fact, even if we question whether a convergent infinite series ALWAYS equals the value it converges to (depends on your definition of equality), we certainly know that the series will never add up to be GREATER than the value it converges to, so there is a finite limit to the size of a convergent infinite series.
docile
Oct 23, 2015
This comment has been removed by a moderator.
garciah926
3.7 / 5 (3) Oct 23, 2015
Infinity and -infinity are not the tail ends of the sequence that shabs showed
Which one? "Shabs42" did propose the adding all integers together : 0= -1 + 1 + -2 + 2... +.... +(-��ž) +��ž. This sequence indeed ends with sums of infinity + (-infinity), the result of which is undefined.


The tail end of the series is not "infinity," THERE IS NO TAIL END OF THE SEQUENCE, docile. The sequence isnt:

1+(1+-1)+(2+-2)+...+(infinity+-infinity)

its

1+(1+-1)+(2+-2)+...

where "..." is more and more pairs of integers and thier additive inverses, as many as you want. You want a million of them? list a million! You want a billion of them? List a billion of them. You want a googleplex of them? List a googleplex of them! Keep going! Then when you get there, keep going some more! Go as far as you'd like...but you wont ever get to a place where theres a stop sign that says "Stop adding here, you've reached infiniti+-infinity
garciah926
3 / 5 (2) Oct 23, 2015
No matter how many terms you list, each pair is equivalently zero, so each term can be listed as 0. No matter how many zeros you add together, its still zero.

I believe you have a fundamental misunderstanding of what "infinity" is in math.

So when I say lim_(n -> infinity) or Sum_(n=1)^(infinity)

I truly doesnt not mean "as n tends to infinity" as if it were a place to go to. It means "no matter how large n get" or "let n be as arbitrarily large as you want it to be" or "It doesnt matter how large n gets"
antialias_physorg
3 / 5 (2) Oct 23, 2015
1+(1+-1)+(2+-2)+...+(infinity+-infinity)

Zeph is even more wrong than usual. Infinity isn't a number - it's a limit. So you can never have the "+(infinity+-infinity)" case in this sequence because you're adding/subtracting numbers - not limits. (because no matter the n you choose in the sequence the sequence member is always finite. The case n= infinite isn't a valid number)
docile
Oct 23, 2015
This comment has been removed by a moderator.
garciah926
3.7 / 5 (3) Oct 23, 2015
A physical theory must make testable predictions. You must say that under these exact conditions, this will be the exact result. Conditions and results must be measured, and quantified, with numbers.

So a physical theory must be, mathematical in nature.

A physical theory can never be proven to be true, just simply more accurate than then one before it. Again, accuracy must be measured and quantified, by numbers.

So perhaps at best a physical theory can be regarded as a mathematical model. The mathematical model has a structure of its own, based on what has been discovered/deduced in the field of mathematics.

Infinities may be ugly and not sexy in physics, but nature need not obey our sensibilities, or our likes and dislikes.

Time can be handled perfectly well by mathematical models via variables/parameters.
garciah926
3.7 / 5 (3) Oct 23, 2015
You have a fundamental misunderstanding of math: abstract math makes no claims that the abstract structures we study exist is "real life." The only claim in math is: if a structure satisfies definition X, then properties Y,Z,..., must be true about the structure.

Period. End Of Story.

So for example: If a set of things satisfies the definition of what it means to be a Hilbert Space, then there is a whole set of properties that those things must also obey. This is pure abstract math.

It so happens that in QM, the set of all possible states of a quantum system satisfies the requirements of a Hilbert Space, therefore that set of states MUST obey all the results that Hilbert Spaces do.

QM COULD BE WRONG, and maybe the set of all possible states of a quantum system (under a new theory) don't form a HIlbert Space. But if QM is correct, then the all possible states of QM system is a Hilbert Space, end of story.
docile
Oct 23, 2015
This comment has been removed by a moderator.
garciah926
1 / 5 (1) Oct 23, 2015
Not sure what you are talking about now. But without measurement and experimentation, you aren't doing science, you are simply speculating.
docile
Oct 23, 2015
This comment has been removed by a moderator.
garciah926
3.7 / 5 (3) Oct 23, 2015
YOU ARE OBSERVING, by noting your observations, drawing pictures of phases, you are measuring (the curve of the shadow, how much of Venus is seen), even if its rudimentary and not exact, you have a margin of error!

When you are observe the phases of Venus you make a distinction between how much of the planet can be seen in one month an how much of the planet you can see in another. You note they are not the same. How much "not the same" may be rudimentary, but the measurement is made by simply noticing the DIFFERENCE.

Please docile/zephyr stop trolling for the sake of being trolling.
docile
Oct 23, 2015
This comment has been removed by a moderator.
EnsignFlandry
not rated yet Oct 26, 2015
"And yet, as the physicists explain, it still makes sense because the total sum of the values of an infinite number of things can actually be finite."

someone wanna help clarify this absurdity?


Absurdities are non-clarifiable. This is a theorem I proved about five minutes ago.
enaskanenas
not rated yet Nov 09, 2015
"And yet, as the physicists explain, it still makes sense because the total sum of the values of an infinite number of things can actually be finite."

someone wanna help clarify this absurdity?


I believe that this line refers to a relevant wikipedia article that explains how divergent series can be assingned with a certain value. It has to do with the so called "Ramanujan summation" and an example of this fact is that you can assign the value -1/12 to the divergent series 1+2+3+4+...

Crazy stuff but utterly awesome !!!

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