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# EXPLORING THE TWO-STATE SYSTEM

Now that we’ve established the definition of the Quantum state as well as the Axioms of Quantum Mechanics, we can begin meaningful discussion on the two-state system. Though it may not see like much of an upgrade, this system is highly complex and lots can be learned from this seemingly simple system!

In the world of Quantum Computing, I’m sure you can imagine a relevant system to investigate … the Quantum Bit (qubit). This is a system in which any general state can be written as,

We know that a complex number zz can be written equivalently in its polar form: z=reiϕ,(r,ϕ)∈Rz=reiϕ,(r,ϕ)∈R. Knowing this we can substitute that into our qubit definition,

|q⟩=r1eiϕ1|0⟩+r2eiϕ2|1⟩|q⟩=r1eiϕ1|0⟩+r2eiϕ2|1⟩

As discussed in the Axioms, we want our qubit to be normalized so,

⟨q|q⟩=⟨r1e−iϕ1⟨0|+re−Iϕ22⟨1|)(r1eIϕ1|0⟩+r2eiϕ2|1⟩)⟨q|q⟩=⟨r1e−iϕ1⟨0|+r2e−Iϕ2⟨1|)(r1eIϕ1|0⟩+r2eiϕ2|1⟩)

=r21⟨0|0⟩+r2r1ei(ϕ2−ϕ2)⟨1|0⟩+r1r2ei(ϕ1−ϕ2)⟨0|1⟩+r22⟨1|1⟩=r21⟨0|0⟩+r2r1ei(ϕ2−ϕ2)⟨1|0⟩+r1r2ei(ϕ1−ϕ2)⟨0|1⟩+r22⟨1|1⟩

=r21+0+0+r22=r21+0+0+r22

=r21+r22=r21+r22

Therefore, we have the constraint that r21+r22=1r21+r22=1. We can be clever and reduce this two parameter constraint into a one parameter constraint by letting r1=cos(θ2)r1=cos⁡(θ2) and r2=sin(θ2)r2=sin⁡(θ2). Another clever thing we can do is recognize that the local phase for each complex number does not affect measurement and instead we can define ϕ=ϕ2−ϕ1ϕ=ϕ2−ϕ1 resulting in the general form of the qubit as,

|q⟩=cos(θ2)|0⟩+sin(θ2)eiϕ|1⟩,(θ,ϕ)∈R|q⟩=cos⁡(θ2)|0⟩+sin⁡(θ2)eiϕ|1⟩,(θ,ϕ)∈R

Now, we can actually visualize how this state vector is changing in Hilbert Space. We can imagine this vector with unity length (normalized after all) sweeping the surface of a sphere. This shape has a name and it is known as the Bloch Sphere!