3 ms·

I like this description of it : https://www.smbc-comics.com/comic/the-talk-3 https://www.smbc-comics.com/comic/the-talk-3 key quote: 'Superposition doesn't m

by drdeca 8y ago

I like this description of it : https://www.smbc-comics.com/comic/the-talk-3 https://www.smbc-comics.com/comic/the-talk-3 key quote: 'Superposition doesn't mean "and", but it also doesn't mean "or".' 'It means a complex linear combination of a 0 state and a 1 state. You should think of it as a new ontological category: a way of combining things that doesn't really map onto any classical concept.'

SuoDuanDao 8y ago
I like that quote myself, but one thing I don't understand is - are the amplitudes always of a magnitude 1? Because if qbits are really always the same magnitude, why represent them as two-dimensional? Wouldn't the math be easier if we converted it to polar form and just had one measurement per superposition? Sorry if this is a dumb question, I'm not formally trained in any of this stuff...
- drdeca 8y ago
  There are 2 complex numbers, one of them the coefficient of |0> and |1>. The sum of the square of the magnitudes of these numbers is 1. Alternatively/equivalently , the magnitude of the vector taken as a whole, is 1. So if you have (a+bi)|0> + (c+di)|1>, the magnitude of this vector is a^2 + b^2 + c^2 + d^2 = 1 Edit: however, the vector only has to have magnitude 1 if it is describing the entire system, I think?
- millstone 8y ago
  It's a good question. The cartoon says that qubits are "unit vectors in 2D Hilbert space." However because the length is 1, you can think of it as a vector in 2D projective Hilbert space. Projective space is what you get when you throw away a vector's length and only talk about its angles. Some intuition: if you have a particle and you look everywhere for it, your chance of finding it must be 100%. It can't be 50% (where could it be?) and it can't be 200% (duh). QM in one dimension is formulated as "a 1d complex unit vector", i.e. a complex number of magnitude 1, which is routinely represented in polar form (a complex exponential). QM in two dimensions is a "2d complex unit vector" which is really 3d: you get three linearly independent components, and then the fourth one is decided for you. You can think of this as spherical coordinates, but complex numbers are much easier to manipulate, so that's what we do.
gpsx 8y ago
That's a funny cartoon, and _very_ fitting to the discussion. I'm a physicist and its true phsycist's don't always tell the whole truth when talking about quantum mechanics. I understand the different perspective between a person doing quantum computing and physicist thinking about the real world. In a quantum computer there are maybe 32 objects each with 2 states. In the real world something like an electric field has an infinite number of possibly values for each of an inifinte numbers of points in space. And then there are interactions between different objects which we can't hope to mathematcially solve but only approximate. And, on top of that, we have to relate the mathematcis to what people see around us. So generally we have to create a straw man for a specific scenario, hence the line in the comic where phsycists don't tell the whole truth. And that's why all these people can look at the straw man example I gave and then say "No, that's not right. It's not "OR". So, you can say a particle is in two places (AND) at the same time wihtin a given context. So that is fair. In the context of describing the connection of a field to its originating particle, that strawman is not useful. The elctric field is correclated with the position of the particle so the "OR" idea is helpful there, but if you try to take that example too far, sooner or later that strawman breaks down. So you could look at the statement in the comic instead about wave functions. Does that provide an clarity to how a field relates to the particle from which it originates? Probably not unless you already know the answer. Admitedly, I guess what I wrote didn't provide much clarity either. Any disucssion of quantum mechanics in a place like this usually devolves into everyone talking past each other.

by drdeca 8y ago

SuoDuanDao 8y ago

I like that quote myself, but one thing I don't understand is - are the amplitudes always of a magnitude 1? Because if qbits are really always the same magnitude, why represent them as two-dimensional? Wouldn't the math be easier if we converted it to polar form and just had one measurement per superposition? Sorry if this is a dumb question, I'm not formally trained in any of this stuff...

drdeca 8y ago
There are 2 complex numbers, one of them the coefficient of |0> and |1>. The sum of the square of the magnitudes of these numbers is 1. Alternatively/equivalently , the magnitude of the vector taken as a whole, is 1. So if you have (a+bi)|0> + (c+di)|1>, the magnitude of this vector is a^2 + b^2 + c^2 + d^2 = 1 Edit: however, the vector only has to have magnitude 1 if it is describing the entire system, I think?
millstone 8y ago
It's a good question. The cartoon says that qubits are "unit vectors in 2D Hilbert space." However because the length is 1, you can think of it as a vector in 2D projective Hilbert space. Projective space is what you get when you throw away a vector's length and only talk about its angles. Some intuition: if you have a particle and you look everywhere for it, your chance of finding it must be 100%. It can't be 50% (where could it be?) and it can't be 200% (duh). QM in one dimension is formulated as "a 1d complex unit vector", i.e. a complex number of magnitude 1, which is routinely represented in polar form (a complex exponential). QM in two dimensions is a "2d complex unit vector" which is really 3d: you get three linearly independent components, and then the fourth one is decided for you. You can think of this as spherical coordinates, but complex numbers are much easier to manipulate, so that's what we do.

gpsx 8y ago

That's a funny cartoon, and _very_ fitting to the discussion. I'm a physicist and its true phsycist's don't always tell the whole truth when talking about quantum mechanics. I understand the different perspective between a person doing quantum computing and physicist thinking about the real world. In a quantum computer there are maybe 32 objects each with 2 states. In the real world something like an electric field has an infinite number of possibly values for each of an inifinte numbers of points in space. And then there are interactions between different objects which we can't hope to mathematcially solve but only approximate. And, on top of that, we have to relate the mathematcis to what people see around us. So generally we have to create a straw man for a specific scenario, hence the line in the comic where phsycists don't tell the whole truth. And that's why all these people can look at the straw man example I gave and then say "No, that's not right. It's not "OR". So, you can say a particle is in two places (AND) at the same time wihtin a given context. So that is fair. In the context of describing the connection of a field to its originating particle, that strawman is not useful. The elctric field is correclated with the position of the particle so the "OR" idea is helpful there, but if you try to take that example too far, sooner or later that strawman breaks down. So you could look at the statement in the comic instead about wave functions. Does that provide an clarity to how a field relates to the particle from which it originates? Probably not unless you already know the answer. Admitedly, I guess what I wrote didn't provide much clarity either. Any disucssion of quantum mechanics in a place like this usually devolves into everyone talking past each other.