October 21, 2015 feature

# Physicists experimentally realize a quantum Hilbert hotel

(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 infinite number 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."

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.

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**More information:**Václav Potoček, et al. "Quantum Hilbert Hotel."

*Physical Review Letters*. DOI: 10.1103/PhysRevLett.115.160505

© 2015 Phys.org

**Citation**: Physicists experimentally realize a quantum Hilbert hotel (2015, October 21) retrieved 20 September 2019 from https://phys.org/news/2015-10-physicists-experimentally-quantum-hilbert-hotel.html

## User comments

TheWalrusSuperThunderanywallsocketsomeone wanna help clarify this absurdity?

docileWhydening GyreAn infinite number of "things" with measurable properties ("values") can add up to a non infinite "value"...?

It is a little bit vague...

PhysicsMatterIn 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_physorgTake 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).

Shabs42A simpler example that I believe is correct, adding together all integers: -1 + 1 + -2 + 2...= 0.

Jaymandocilegarciah926Order of operations is everything

The are most definitely CONVERGING infinite series

garciah926So 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

docilegarciah926For 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

garciah926Infinity 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

garciah926docilegarciah926The 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

garciah926I 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_physorgZeph 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)

docilegarciah926So 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.

garciah926Period. 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.

docilegarciah926docilegarciah926When 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.

docileenaskanenasI 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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