Bra-ket notation and qubit issue.

[Relative0]

I am having trouble understanding the following:

U_f: |x>|y> → |x>|y $\oplus$f(x)>

$\oplus$ being a mod 2 operation (nand)? I suppose I don't understand how to read the "ket" states so well. As far as I understand we have that since x and y can be 0,1 only if |x=1>|y=1> then if f(x) = 1 then |y $\oplus$ f(x)> would come out to be → |1>|0>

but the main part is that if we are dealing with a quantum computer then we can chose the input state to the a superposition of |0> and |1>. That if the second qubit is initially prepared in the state 1/$\sqrt{2}$(|0> - |1> then... The issue is with this equation in that how does |0> - |1> mean superposition? This is the part of the bra-ket notation that I don't understand

 

[kith]

the main part is that if we are dealing with a quantum computer then we can chose the input state to the a superposition of |0> and |1>. That if the second qubit is initially prepared in the state 1/$\sqrt{2}$(|0> - |1> then... The issue is with this equation in that how does |0> - |1> mean superposition? This is the part of the bra-ket notation that I don't understand

What definition of superposition do you use then?

If you want to apply your gate U to a superposition |x₁ y₁>+|x₂ y₂>, you can simply apply it to each term because of the linearity of U. So U(|x₁ y₁>+|x₂ y₂>) = U|x₁ y₁> + U|x₂ y₂>

 

[Relative0]

Thanks kith. Well as far as superposition goes - I am not quite sure of the definition he (Preskill) is using in his Notes: www.theory.caltech.edu/people/preskill/ph229/notes/book.ps‎

I am guessing just some sort of overlap of states that is treated as a vector space. I say this as he talks about states being "ray's" in a Hilbert space. That these rays are vectors. He does say that every ray corresponds to a possible state so that given two spaces |θ>, |ψ> we can create a state a|θ> + b|ψ> by the superposition principle as he puts it (Pg. 38).

Is it just that the states |0> and |1> are perpendicular and thus 1/sqrt(2) is the normalization? I.e. in 2-D Euclidean space we would have 1x + 0y being denoted as |0> and 0x + 1y as |1> hence the distance between them is sqrt((1-0)^2 + (0-1)^2) = sqrt(2)?

He has something similar where |cat> = 1/sqrt(2) $\cdot$ |dead> + |alive> so since these are distinct (can't be both dead and alive) we have that they are perpendicular hence the normalization of 1/sqrt(2).

I suppose I am looking for a better understanding of |0> + |1>. are these indeed considered perpendicular? But furthermore I am somehow looking at a probability density in |0> and |1> respectively? So I would be looking at some sort of superposition of two perpendicular waves?

Thanks,

Brian

 

[kith]

I.e. in 2-D Euclidean space we would have 1x + 0y being denoted as |0> and 0x + 1y as |1> hence the distance between them is sqrt((1-0)^2 + (0-1)^2) = sqrt(2)?

Yes.

I suppose I am looking for a better understanding of |0> + |1>. are these indeed considered perpendicular?

Yes. |0> and |1> are eigenstates of a certain self-adjoint operator which corresponds to a physical quantity (usually spin). Such eigenstates have the property of being perpendicular.

But furthermore I am somehow looking at a probability density in |0> and |1> respectively?

Why a probability density? Do you know how probabilities are related to superpositions? Do you know the postulates of QM?

So I would be looking at some sort of superposition of two perpendicular waves?

Waves belong to infinite-dimensional spaces while our space is two-dimensional.