Hadamard gate

H=1/Sqrt[2]{{1,1},{1,-1}};
MatrixForm[H];
MatrixForm[Inverse[H]];

on={0,1};
off={1,0};
H.on;

Therefore the Hadamard gate acting on  |on⟩ = |1 ⟩ ={0,1} =

(1/(Sqrt[2])) |1 ⟩ - (1/(Sqrt[2])) |0 ⟩ 

in terms of our physical picture of cbits what kind of state is this
it appears to be a superposition

((-1)/(Sqrt[2]))  |  ⟩ +  (1/(Sqrt[2]))  |⟩

what does this mean ??

In the quantum theory this type of object does have an interpretation, we can also
find classical analogies to such a superpostion.

Electromagnetic wave

Linear polarized light

electricfieldA[t_]:={Cos[θ],Sin[θ]}Exp[2I Pi t];
Ex[t_]=Re[electricfieldA[t][[1]]];
Ey[t_]=Re[electricfieldA[t][[2]]];

graph1[t_]:=Show[Graphics[{Blue,Thick,Arrow[{{0,0},{Ex[t],Ey[t]}}]}],PlotRange->{{-1.1,1.1},{-1.1,1.1}}];

Manipulate[graph1[t] /. θ->Pi/2,{t,0,1}]
(* Linear vertical θ=π/2  and horizontal θ=0 polarized light *)

electricfieldA[t_]:={Cos[θ],I Sin[θ]}Exp[I 2Pi t];
Ex[t_]=Re[electricfieldA[t][[1]]];
Ey[t_]=Re[electricfieldA[t][[2]]];

graph2[t_]:=Show[Graphics[{Blue,Thick,Arrow[{{0,0},{Ex[t],Ey[t]}}]}],PlotRange->{{-1.1,1.1},{-1.1,1.1}}];

Manipulate[graph2[t] /. θ->Pi/4,{t,0,1}]

A fictional world scenario


Suppose we all had, without our knowledge, polarized filters in our eyes i.e. we could only see either linear horizontaly polarized light, or  linear vertical polarized light. We call them |H⟩ and |V⟩ light. All experimental studies are consistent with these two variables, until  the discovery of a new type of filter that has the  following remarkable property.


statistical mixture of |H⟩  and  |V⟩ light ⟶ V filter ⟶ V light ⟶ Magic filter ⟶ both V and H
light are measuremed (sometimes V and sometimes H).


What is this "magic" filter doing? A graduate student, Alberta Zweistein (she goes on to win the Nobel prize for this insight) points out that electromagnetic radiation can be polarized in any direction that is mathematically described as a linear combination of |H⟩ and |V⟩. The student points out that we only 
see  |H⟩ or  |V⟩ light,  but nature makes full use of this superposition, hence the magic filter.
The student calls this the "hidden variable theory" of light.

Another important Pauli gate is

PauliY={{0,-I},{I,0}};
MatrixForm[PauliY];

What is the imaginary number doing there? Does this make sense. Consider the following
gate

BZ=1/Sqrt[2]{{1,-I},{I,1}};
MatrixForm[BZ];

It sort of looks like the Hadamard gate but contains imaginary entries so that

BZ.on;

Or 
BZ  |1 ⟩ =  (1/(Sqrt[2])) |1 ⟩ + ((-i)/(Sqrt[2])) |0 ⟩ 

is there an EM analogy ?

Hyperlink["http://www.youtube.com/watch?v=RFkyvkBV5dM"]