Even more important can it be used to make certain encryption algorithms useless.

That is the thing I always care about.

And it is always the government agencies or government contractors that gets these kinds of systems first.

From an other article linked in the comments:

"There was a further limitation. Theoretically, the quantum computer should operate at a temperature of 0 kelvin, but such extreme cooling is impossible in practice, so D-Wave repeatedly ran the system at slightly above zero in the hope of reaching the lowest-energy state. Due to these higher temperatures the calculation got the right answer only 13 times after 10,000 attempts."

http://www.newscientist.com/blogs/shortsharpscience/2012/08/quantum...

So eventually you might end up with something like a couple of thousand guesses to decrypt certain data.

If that is true, that could be bad.

*Edited 2013-03-22 12:37 UTC*

Said improvements imperil current cryptography systems. However, it is not the end-all of cryptography -- the newest replacement in SSH and GPG security, for examples, include elliptic curves and another algorithm. These newer algorithms are not known to be attacked by quantum computation.

That's fine, no great, but the general public doesn't use it. They use HTTPS, maybe POP3S or IMAPS. Or even an IPSEC- or SSL VPN.

Over 25% of the top 200.000 'secure websites' still have SSL2 support enabled, even though it has been know to be insecure since 1995 or 1996:

https://www.trustworthyinternet.org/ssl-pulse/

I doubt any software comes installed with SSL2 enabled by default. So that means that 25% of the people who configure the top 200.000 'secure websites' don't know what they are doing. That is pretty insane.

I also doubt you can find anyone deploying IPSEC VPN in a corporate environment that even knows what Perfect forward secrecy or elliptic curve is and is using an IPSEC gateway that actually supports it.

EDIT/UPDATE correction: at least Windows 2008 servers have SSLv2 enabled by default. So maybe others too. Do we still need it for compatibility, I hope not.

*Edited 2013-03-23 10:24 UTC*

xiaokj,

"However, it is not the end-all of cryptography -- the newest replacement in SSH and GPG security, for examples, include elliptic curves and another algorithm. These newer algorithms are not known to be attacked by quantum computation."

Do you have a source for this? This very much interests me and I'd like to read more about it. It's a bit unintuitive as to why a quantum algorithm wouldn't work.

Wikipedia says this:

http://en.wikipedia.org/wiki/Elliptic_curve_cryptography#Quantum_co...

"Quantum computing attacks

Elliptic curve cryptography is vulnerable to a modified Shor's algorithm for solving the discrete logarithm problem on elliptic curves."

But it doesn't provide an online source.

"However, it is not the end-all of cryptography -- the newest replacement in SSH and GPG security, for examples, include elliptic curves and another algorithm. These newer algorithms are not known to be attacked by quantum computation."

Do you have a source for this? This very much interests me and I'd like to read more about it. It's a bit unintuitive as to why a quantum algorithm wouldn't work.

Wikipedia says this:

http://en.wikipedia.org/wiki/Elliptic_curve_cryptography#Quantum_co...

"Quantum computing attacks

Elliptic curve cryptography is vulnerable to a modified Shor's algorithm for solving the discrete logarithm problem on elliptic curves."

But it doesn't provide an online source.

Just found this paper while doing a quick web search with the "elliptic curve quantum" keywords, would it help ? Sounds like a more detailed explanation of what Wikipedia mentioned...

http://arxiv.org/abs/quant-ph/0301141

*Edited 2013-03-24 22:00 UTC*

This is from the article in the New York Times, so I can hardly blame Thom for repeating it verbatim. But this is so freaking wrong it makes my head explode.

A one can be one or a zero or both. Not everything in between. It can't be a floating point 0.48294302. Quantum means a small discrete value.

It's been artlessly stated, but there's more than a grain of accuracy in that line. A qubit can have a range of possible values; the basic values it can assume are zero or one. Physicists would say that the qubit (let's call it |q>) can be in either the state |0> or the state |1>.

However, it can also be in a linear superposition of these states. Given any two complex numbers c and d, the qubit can be scaled to become:

|q> = c|0> + d|1>.

There are normalization requirements to make the probabilities sum to unity but this is just really basic linear algebra on a complex vector space. It's in this sense,

*the sense of a continuous range of possibilities for the superposition over basis vectors*that I take the quote you mention to refer. In which case he's entirely accurate, if admittedly a little unclear.

*Edited 2013-03-22 15:55 UTC*

The sentence could very well be correct, actually. Qubits, the basic unit of "information" in quantum computing, are quaternary in nature. As opposed to "traditional" digital bits, which are binary.

So a Qubit can be indeed, 0, or 1, or 0 and 1 simultaneously, or numerical coefficients representing the probability of each state.

Qubits are not quaternions (or indeed "quaternary"). There exists an interpretation of quantum information theory using a quaternion formalism that eventually leads to something called density operator theory, but this is obscure even for the field.

No. A qubit is a linear superposition of basis vectors in some two-dimensional complex vector space. Numerical factors representing probabilities occur only when one performs

*an operation on qubits*(specifically the inner product on the vector space).

I liked how you say "no" but your response implies "yes." ;-)

PS. I did not say qubits were quaternions, but that they were "quaternary in nature." If we're going to do the whole anal thing.

And yes, I should have perhaps said that "from a logic design standpoint Qubits could be viewed as being quaternary in nature." With that interpretation being mainly correlated with logic design expedience, since qubits could be interpreted to be ternary for example (as initial qubit interpretations tended to be...). But then "traditional" binary logic functions, which form the basis of the majority of our logic/computing designs, is not easily translated or represented using a ternary base. Etc, etc, etc.

Even if this is a quantum computer it can't do any of the more interesting things a real quantum computer can do in linear time. And people much smarter than me don't want to call it a quantum computer even if the manufacturers description of it is correct...

Quantum as in quantum computer and quantum physics doesn't mean a small range of values BTW.

My understanding was that quantum is just an adjective - allowing us to refer to the smallest possible discrete unit of something.

So the smallest possible unit of weight isn't 1 kilogram, because you can divide that into grams. Any number you define can be subdivided so you have to just bail on that scene, and just refer to quanta instead.

It's like the old child's game where you state the largest number, and finally someone says infinity, and then someone says infinity+1 - no dude, infinity is defined as the largest number, and quanta as the smallest discrete unit - as the infinitesimally small.

After this things get a bit murky, but apparently Einsteins theory of relativity and Quantum theory when taken together, can describe everything we know about matter and energy.

And you have other oddities like, Einsteins theory of relativity, Quantum Theory, and Selena Gomez, taken together somehow predicts the popularity of the 1990's hit Melrose place. A seemingly unrelated event.

Look, I don't want to be 'that guy' that guy that poo poo's everything he doesn't understand, and predicts everything to fail. because he doesn't get it. But I don't get it, and when I read all the people saying this is hocus pocus - what can I say, it seems like 'it kinda is' - they really need to make a bit more progress, then I'll take another looksy.

Well, Infinity + 1 is larger.

Infinity isn't "The largest number" because there is no "largest." (That implies that it stops and there is nothing larger).

Infinity + 1 is larger than Infinity in the same sense as "The set of real numbers" is larger than "the set of whole numbers." Both are infinite in size, but Set of Real Numbers contains elements that aren't in the Set of Whole Numbers.

Well, Infinity + 1 is larger.

Infinity isn't "The largest number" because there is no "largest." (That implies that it stops and there is nothing larger).

Infinity + 1 is larger than Infinity in the same sense as "The set of real numbers" is larger than "the set of whole numbers." Both are infinite in size, but Set of Real Numbers contains elements that aren't in the Set of Whole Numbers.

===>I can't help myself, while you are correct in that my statement was wrong - I admit defeat on that.

If I were to say to you, that in that I had this infinite set of whole numbers, plus I had the number 1.

I would still only have the infinite set of whole numbers, the number 1 is already included. I still fail to understand, how in the context of a child's game where you are attempting to say the highest number possibile and someone says infinity, how the number infinity plus 1 is larger.

Probably the correct answer is infinity is not a number, but a set of numbers. Therefore infinity plus 1 - if taken as infinity plus the number 1 - is just wrong, because infinity already included the number 1.

But if it means infinity but some number outside this set, then that is a larger set. A larger set - but not a larger number.

In reality, I'm going back and stating that in this game the word 'infinity' really did represent the largest number possible. It's a misuse of the word infinity - I get that now.

However it is absolutely unreasonable to me, to suggest the number 1 in the context of stating the largest number, is outside the set of numbers.

*Edited 2013-03-22 20:13 UTC*

I would still only have the infinite set of whole numbers, the number 1 is already included. I still fail to understand, how in the context of a child's game where you are attempting to say the highest number possibile and someone says infinity, how the number infinity plus 1 is larger.

Well, if you find it difficult to understand how this could be so in the context of the natural numbers, try thinking about it using a different set.

Think of the largest possible set you can (the number of elements in your set is something that a mathematician would call the

*cardinality*of that set).

Each time you come up with a set containing a huge number of elements, I can counter you by constructing a set containing

*all of your elements, plus any other element that isn't already in your set*. Thus I can construct a set containing an arbitrarily large number of elements; this is, in essence, one type of infinity.

No. Infinity is not a number; it's a

*concept*.

*If I were to say to you, that in that I had this infinite set of whole numbers, plus I had the number 1.*

I would still only have the infinite set of whole numbers, the number 1 is already included. I still fail to understand, how in the context of a child's game where you are attempting to say the highest number possibile and someone says infinity, how the number infinity plus 1 is larger.

I would still only have the infinite set of whole numbers, the number 1 is already included. I still fail to understand, how in the context of a child's game where you are attempting to say the highest number possibile and someone says infinity, how the number infinity plus 1 is larger.

*"*

Well, if you find it difficult to understand how this could be so in the context of the natural numbers, try thinking about it using a different set.

Think of the largest possible set you can (the number of elements in your set is something that a mathematician would call the

Each time you come up with a set containing a huge number of elements, I can counter you by constructing a set containing

Probably the correct answer is infinity is not a number, but a set of numbers.

No. Infinity is not a number; it's a

Well, if you find it difficult to understand how this could be so in the context of the natural numbers, try thinking about it using a different set.

Think of the largest possible set you can (the number of elements in your set is something that a mathematician would call the

*cardinality*of that set).Each time you come up with a set containing a huge number of elements, I can counter you by constructing a set containing

*all of your elements, plus any other element that isn't already in your set*. Thus I can construct a set containing an arbitrarily large number of elements; this is, in essence, one type of infinity.No. Infinity is not a number; it's a

*concept*.I actually think you suffer from an understanding of language.

I can define a set as all possible numbers.

And then you cannot tell me there is an additional number outside the set.

And infinity is a concept - I said that. You said that. You can pretend we are in disagreement, but we are not.

I can define a set as all possible numbers.

And then you cannot tell me there is an additional number outside the set.

Perhaps (I'd actually dispute that by asking you precisely what type of numbers are in your set, but I digress).

Regardless, let me be kind and assume that you've got a set of numbers that is uncountable; perhaps you're thinking of, say, the set of all real numbers. You're now quite pleased with yourself because you've got a set with an infinite number of elements.

However, I come along and claim that I can define a set with even more elements in it than yours. I can even be kind to you and say that I'll restrict myself to working with a set containing real numbers. The question therefore is: do you believe that I can construct a set of real numbers that contains more elements than your - already infinitely large - set of real numbers?

Because I can do so quite simply by taking my set to consist of all possible subsets of the real numbers. Both of our sets have infinitely many elements, but mine has more than yours.

BeamishBoy is arguing from the standard construction of numbers and set in modern mathematics which, for somewhat subtle reasons, disallows defining the set of all possible numbers. Due to the simplifying assumption that everything is a set (otherwise you could run out of numbers for talking about the size of sets which is similar to the argument BeamishBoy makes in his sibling post), numbers are defined in such a way that the set of all possible numbers must contain itself and sets containing themselves is disallowed due to https://en.wikipedia.org/wiki/Russell%27s_paradox .

This means that there is obvious way to define the largest infinity and leads to two separate kinds of numbers which act rather differently for infinities:

https://en.wikipedia.org/wiki/Ordinal_numbers for counting things where n != n+1 even n is infinite and https://en.wikipedia.org/wiki/Cardinal_numbers for measuring the size of sets where n == n+1 if n is infinite.

Oh please do explain what those risks are - or would that be risky?

Seriously dude I was not representing myself as an expert - my words stating the opposite of that - did in fact represent what I mean, and weren't some clever ploy.

My explanatory style is meant to lay bare to the world what I might or may not understand on the subject, both to clarify my own thoughts and to invite commentary, and my experience has been that talking advances the subject more than being silent.

I appreciate the advice both the counterpoints on what quantum mechanics is about, and the little banter on the child's game of stating the largest number.

Peace, friend!

*Edited 2013-03-22 20:56 UTC*

I didn't imply you were an expert. I was just letting you know that if you do not understand a concept, or field of study, and all you have to go with is some odd (and perhaps equally uninformed hearsay) assumptions... It is then perhaps kind of silly to put a set of requirements about what those who are actually working on the concept, or field, should or should not do before they can hope to achieve your magnificent seal of approval ;-)

Oh in that case - just get bent.

Let me try to help you a bit understand what QM is and how it works.

One of the core tasks of physics is to describe what makes things move and how, whether the things in questions are tiny electrons or huge nebulas. Newton's laws of motion are a simple description of this which works pretty well at our scale, whereas Einstein's special and general theories of relativity are more accurate at high speed and large scales, and quantum mechanics (or QM) is the best at small scales.

QM and relativity can both be seen as extensions of Newton's laws to extreme scales, since if we try to apply them to "regular" objects, we'll get similar results as with Newton's laws. However, merging them into one single unified theory has proved to be extremely difficult for theoretical physicists. At this point, finding a satisfactory quantum description of gravitation, as described by the theory of general relativity, remains an open problem.

Now, how does the quantum description of the world differs from the one offered by classical mechanics? In a nutshell, it allows for some physical situations which are perfectly impossible in a Newtonian world, and simultaneously forbids some things which Newton's laws are perfectly fine with. Here are a few examples:

-Quantum objects' properties are not assumed to be perfectly known. What QM describes instead is the probabilities of finding something in a given state: at a given position, with a given speed...

-All these possible states of a system are not just statistical odds. Unless a measurement procedure leads the system to collapse in a single state, multiple "possible" states of a system exist simultaneously, and may thus interact with each other.

-QM's laws of motion are based on this latter effect, describing the probabilities of finding a system in a given state as changing in space and time in a manner similar to that a sound or like wave.

-All properties of a system are not independent, and it's impossible to simultaneously know all of them at once. This stem from the mathematics used to describe quantum states, which happen to be matching some real-world behaviors.

The reason why we're dealing with a theory that violates common sense in such a brutal way is that we have failed to find a saner description of the world which matches experiment so far. As an example, if the microscopic world followed Newton's laws of motion or their relativistic equivalents, electrons would crash into atom nuclei, many magnetic materials would have totally different properties, and we would still have to argue upon whether light is a wave or a stream of particles instead of having a maths that unify both descriptions.

*Edited 2013-03-22 21:07 UTC*

@Neolander, thanks for the explanation, although if I could request of you a more specific explanation of quantum computing, that would be great.

From my understanding regular computing would have to be based on the same laws of mechanics and properties of our world as everything else, its not as if we can step outside of it, simply because we are ignorant of it.

So, there must be something special about quantum computing that separates it from merely a theoretical description of how things work - to a practical engineering difference.

I'm not going to pretend that I didn't already go to wikipedia and try to make some sense of entanglement and superposition. The problem is - as much as they might try to be simple and easy to understand, they aren't quite using the right words for me.

Regular computers use transistors and attempt to represent information in binary, 1' and 0's. My understand is quantum computers attempt to use quantum properties to represent and manipulate data.

Ok, but then that's not enough to get it for me, and frankly I was joking earlier, but since you all are trying to help - what's the missing piece here?

It's not enough to just state the states can be 1, 0, or inbetween, what are the states, how do you manipulate them?

I can program in assembler just a little bit - so at least in my head, I get how regular computers work - shouldn't I be able to understand quantum computers. Lets say I want to program a quantum computer - what are my steps to do so?

From my understanding regular computing would have to be based on the same laws of mechanics and properties of our world as everything else, its not as if we can step outside of it, simply because we are ignorant of it.

So, there must be something special about quantum computing that separates it from merely a theoretical description of how things work - to a practical engineering difference.

As I mentioned above, it takes rather small systems for quantum phenomena to become significant. More precisely, what you actually need is something that has as little physical interactions as possible with its surroundings.

That is because when quantum systems are coupled to a macroscopic environment, it causes many quantum effects to vanish over time, in a process known as decoherence. As an example, if you put a system in a "half-zero, half-one" state, it will decay either to the zero or one state. The stronger the coupling, the faster the decay. One way to describe it from the point of view of physics is as information or energy being diffused into the environment and never coming back.

Thus, people who try to build quantum computers have the difficult task of building quantum chips whose state may easily be read on demand, but which are still too weakly linked to the outside world for the aforementioned decoherence effects to become problematic.

Regular computers use transistors and attempt to represent information in binary, 1' and 0's. My understand is quantum computers attempt to use quantum properties to represent and manipulate data.

Ok, but then that's not enough to get it for me, and frankly I was joking earlier, but since you all are trying to help - what's the missing piece here?

It's not enough to just state the states can be 1, 0, or inbetween, what are the states, how do you manipulate them?

A state, in a quantum mechanical sense, is a set of physical parameters that fully describe a system. As an example, electrons in solids can be described by their momentum, energy, and magnetic "spin", while a photon of light can be described by its momentum and polarization.

Often, these parameters are only allowed to take a discrete range of value, which as you can probably guess is particularly useful for information processing purpose : just make somehow sure that only two parameter values are accessible, and you got yourself your "zero" and "one" physical states. And since these are quantum states which we are talking about, nothing prevents a system from being in both states at the same time, with a certain probability of finding it in each during a measurement.

Effectively, such two-level systems have been built in a wide range of physical systems, ranging from atoms in optically reflective cavities to bunch of ions trapped using electrostatic fields, with superconducting micro-circuits somewhere inbetween. The two states in question are often chosen of different energy, so that the system cannot trivially switch between both without exchanging energy with its environment.

But at this point, I have to point out that such a quantum memory or "qubit" is useless by itself. Sure, your qubit may be simultaneously in its "zero" and "one" states, but since you need to perform lots of measurements to know for sure, nothing has really been gained. Besides, as mentioned above, quantum information storage tends to be short-lived due to decoherence effects, with data remaining reliably stored for at most a few miliseconds in the best experiments if memory serves me well.

So the next difficulty in building a quantum computer is to make quantum memories interact with each other, so as to perform computations. And just like with classical computing, the simplest thing which one can think of building this way is logic gates.

Where things become interesting here is that a quantum logic gate is able to act on all possible states of its input at the same time, and return a statistical superposition of all the results at the same time. So if we take, as an example, an unsorted database search algorithm of the form "examine every database entry, return a pointer to the entry that possesses a given key and NULL otherwise", then the only nonzero result is the pointer to our database entry, and we may naively imagine a quantum circuit which finds the right database entry in constant time.

Things sadly aren't quite as simple in practice because all logic gates which destroy information (such as, say, AND and OR) are forbidden in the quantum world. One thus has to deal with other logic gates like the CNOT and Hadamard gates, which arguably reorder information more than they do computation, and thus make programmers' lives more difficult. Due to this, an actual quantum database search algorithm, as derived by Grover et al., ends up with an algorithmic complexity of O(sqrt(N)), which is still a quadratic speedup over classical ones.

Other kinds of search-oriented tasks can benefit from a quantum device's ability to act on multiple inputs at the same time. Perhaps the most well-known one is Shor's algorithm, which factorizes numbers in O((log N)^3) time and is the worst nightmare of everyone who ever implemented RSA-based cryptographic systems. That being said, not all of cryptography is based on the difficulty of factorizing prime numbers, and so far some algorithms have proven to be pretty robust against the minds of quantum computing theoreticians.

Did that answer some of your questions ?

At this point, I don't think anyone has actually built a truly programmable quantum computer, so it's hard to answer this question. For now, most works I've heard of are based on special-purpose circuitry which aims at implementing a specific algorithm, so programming a quantum computer would be kind of like "writing" code using a bunch of logic gates wired as you wish and synced against a common clock signal.

I think that implementing a programmable quantum ALU is a dream of many quantum computing experimentalists, but is still out of reach for now. Theoreticians tend to be ahead in this domain, so I'm sure multiple designs have already been published in scientific literature.

Lots of "I think" in this last part, I know, but I haven't followed this realm closely enough to be perfectly sure about the state of current research. My current field of research (superconductivity) sure involves QM, but it's not quantum computing itself

*Edited 2013-03-24 21:49 UTC*

For those of you really interested in learning this subject (or teaching it) I downloaded the Quantum Fog emulator for Mac.

It's possible there is a better emulator out there - but until someone with more knowledge weighs in on the subject, I'm just on my own...but I thought I'd share, this quantum fog program, while it is clearly an older program - it's nice.

I don't know how well it would track with what else is available, however the guy that wrote it clearly went to considerable lengths to document how the simulator is supposed to work.

It has tons of information, and some example programs.

Anyway - I've put out the request for additional info from anyone who wants to share.

Folks its entirely reasonable for anyone to wait before jumping into QC programming.

You are not going to be writing your qc program, and people running that on their quantum computing based cellphone - with its near absolute zero cooling, any time soon.

however, as I'm pecking away at writing my first quantum computing program for the simulator I have to admit, I've discovered something - this is freakin' awesome.

I have zero clue why some of you - well that one dude, seems to dislike me. But look, I'm not a great programmer - I have no illusions about it.

Most of you will be better quantum computing programmers than I will ever dream to be. But, I am writing my first quantum computing program as we speak.

So - I will be the first in my family to do quantum computing and that's something right.

hahaha, lol.I'm joking I'd be the first in my family to do that, even if I waited 30 years to do it.

OK, anyway - it is freakin awesome. I had no idea how freakin awesome this was just this morning. Now, yeah - it's cool.

Now I admit, I'm having to dust off some old textbooks, because my math is a little rusty, but I'd say dont be put off by it. It's just a bit of shall we say - overhead.

Now anyway, I am out...because of the danger that I will like to argue, and to what extent I like to argue or contributed to the argument, I will now remove myself from the entire discussion.

I'm out, so until the next time - good day.

Scenario:

Two 10 year old boys are playing 'name the largest number'

In the game of name the largest number, the child that names the largest number wins.

The largest 'set' has absolutely no relevance in this game and never has.

Child 1: says 100

Child 2: says 1,000

Child 3: says 1 million

Child 1: says 1 billion

Child 2: says infinity

Child 3: says infinity+1

In this game, child 1 has won. They said 1 billion, the largest number stated in the game.

infinity is accepted as an answer in the real world, but is not in actuality a number, but a concept.

As stated earlier infinity+1 has no importance, because the game was never about having the largest set. However, infinity+1 is not, necessarily a larger set.

An example of a set:

The unique numbers in (1,2,3,4,5)

Now lets add the "number" 1 to this set:

The unique numbers in: (1,2,3,4,5,1)

The set size has not changed.

Lets do a more creative set:

I define this set as: all numbers in the world, real or imagined, that can exist, or cannot exist. I specifically define as already being included in this set, any numbers you attempt to add to it later, and it contains all the numbers in any set you compare to it.

That's my definition, and just add salt to the wound, in this intellectual exercise, this set is labeled "infinity" doesn't matter if you like it or not.

Now what??? Bring it on bitch.

Look, I don't get to decide the parameters - the children's game of name the highest number - infinity is accepted as an answer - it's accepted as being 'whatever the highest number is in theory' -

So infinity +1 has no additional meaning.

The last paragraph is controversial, you may argue against it. Please do.

One thing we can all agree on, the very fact that I wrote this nonsense at 3:34am in the morning, is a sign of mental - well lets be polite, of being mental.

So, yeah, I lose. Later guys!

“So infinity +1 has no additional meaning.”

If the infinities in both cases are equal, then yes it does. Different size infinites are one of the fundamental principals of calculus. It's also one of the most difficult to understand.

Don't think in terms of sets where each number can only be used once. That's a purely artificial restriction that has absolutely no meaning whatsoever in this context. If you have a set of all reals, for example, you have a set with infinite members, none of which has a value of infinity. We are actually talking about infinite series. Not sets.

Lets say we have the following summation,

S1 = 1+2+3+4 + · · ·

We can all agree that the above goes to infinity. Now lets compare it with the following series,

S2 = 2+4+6+8 + · · ·

so that each number in the series is 2*n. Both are infinite. However, S2/S1 is exactly 2.

Because each term in S2 is divisible by 2, it is absolutely correct to write,

S2 = 2*(1+2+3+4 + · · ·)

In S2/S1 the series cancel out, their value (infinite or otherwise) is completely irrelevant.

S2, despite being infinite, is *exactly* twice S1, which is also infinite.

Whether a number has already been used is irrelevant. It's a sum. I can add the same value any number of times.

1+1+2+3+4 + · · · > 1+2+3+4 + · · ·

I think that's tripping some people up is treating "infinity" as though it were a discrete number that can be compared. In discrete calculus, we were always careful to say a sequence could "approach" infinity faster than another sequence, which is both valid and fairly easy to understand.

The moment you treat "infinity" like a discrete number and manipulate it with discrete operators like comparison, you break the concept of infinity. Nothing is bigger than infinity. Infinity plus one isn't a discrete number, neither is infinity minus one. Sequences do not "equal" infinity because the transient property of equality would imply that all sequences approaching infinity are equal, which they're not.

One might be tempted to say infinity minus infinity is zero, but that's not semantically valid because infinity isn't a discrete number.

S1=1+2+3+4...

S2=2*S1

Both sequences are infinite, but neither are equal, nor do they "equal infinity". S2-S1 doesn't equal zero, it equals S1.

**This is completely incorrect.**

Infinity is a well-defined concept. It's so well defined that one can talk about (i) different kinds of infinity, and (ii) how one version of infinity is "larger" than another.

I addressed this in another post by pointing out that the number of elements contained in the set of real numbers is one kind of infinity, and how this is smaller than the number of elements contained in the set of all subsets of real numbers.

I know this stuff is confusing but it's been understood by mathematicians for over a century.

BeamishBoy,

"Infinity is a well-defined concept. It's so well defined that one can talk about (i) different kinds of infinity, and (ii) how one version of infinity is 'larger' than another."

I didn't say it wasn't a well defined concept. It's just that the concept is being abused when we treat infinity as a discrete number as though it could be compared.

"I know this stuff is confusing but it's been understood by mathematicians for over a century."

It's really not that confusing, if you attempt to solve a discrete number for "infinity", anyone else can conceive of a number a number which is factually higher, therefor the concept of infinity rules out the possibility of any discrete number equaling infinity.

What we can do is compare sums of series which approach infinity at different rates, like you did earlier. Discrete calculus (which as you noted is well understood) allows us to solve the rate for any given iteration of the sequence and confirm via inductive proofs that it continues infinitely.

There are an infinite number of unique sequences who's sums add up towards infinity given an infinite number of iterations. Because of the transitive properties of mathematical equality, one cannot claim any of these infinite sequence sums "equal infinity", for the simple reason that they are not equal to each other.

We might be careless and say a given sum "equals infinity" and still understand one another, but we don't actually mean it in the true mathematical sense.

This is what my earlier example was trying to illustrate. If infinity could be treated as a discrete number, then mathematically this would be sound:

A=infinity

B=infinity

B-A = 0

But in fact the earlier example showed that B-A=A.

To be sure, I am arguing semantics in the conversation as a whole and not just you. I don't want us to talk over each other, and I don't think we have all that much to disagree on, despite your bolded statement that "This is completely incorrect."

*Edited 2013-03-24 20:57 UTC*

Well, first of all, there are two different concepts of infinity that had been discussed in some of this threads. One is the idea of infinity as a number that is larger than all the real numbers. The other is the idea of a set having an infinite number of elements. They are *not* the same concept. Let me give a brief description of both to see the difference:

1.- For certain applications, it is convenient to extend the real numbers to allow the values infinity and -infinity. (For example, this is a very common thing to do in measure theory and integration.) In this case infinity is (in a sense) a quantity that is bigger than all others quantities. The drawback of this approach is that infinity doesn't behave well under the usual arithmetic operations. For example infinity+1=infinity, while infinity-infinity is always undefined. In this setting you can't have anything bigger than infinity. Trying to manipulate divergent sequences as if they were the quantity infinity and doing algebraic manipulations with them is incorrect, and can quickly get you to absurd statements. For example, Riemann showed that if you have a conditionally convergent sequence, then you can rearrange its terms to make it have any limit you want.

2.- In set theory there is the notion of the "cardinality" of a set, which is a measure of how big it is. Two sets are said to be of the same cardinality if you can give a bijection between them. Of course, given any set you can always add one element to it to make it bigger. (This is based on the fact that there is no such a thing as "the set of all sets".) But having a set with more elements doesn't automatically means that it has a higher cardinality. For example the set of positive integers has the same cardinality as the set of even positive integers. (You just enumerate your positive even integers to get the bijection.) However, Cantor showed that the set of real numbers *does* have a bigger cardinality than the set of integers. This is very shocking. It means that it doesn't matter how you try to enumerate the real numbers, at the end there is always an infinite number of real numbers that you left out. Given any set, the set of all its subsets always has a higher cardinality, so there is no biggest cardinal number. (Again this is related to the fact that there is no "set of all sets".)

Here are some notes of my thesis advisor on quantum computing. They focus more on the algorithms than in the quantum mechanics, so if you are willing to take some things on faith you can find some cool material. They even have a description of the prime factorization algorithm that would make classical cryptography obsolete with a quantum computer.

http://www.math.ucsd.edu/~nwallach/Venice4.pdf