Bloch sphere is a very handy representation used in the realm of
quantum information and quantum computation. In quantum information and quantum computation, we mostly deal with a two-level system, i.e, a
qubit which is a quantum analog of the classical bits 0 and 1. A qubit is differentiated from a classical bit in a sense that it can take any value between the values between 0 and 1, which quantum mechanically are states $\left. |0\right >$ and $\left. |1\right >$. A general state of the system can thus be written as a superposition of the qubits as:
where by probability conservation we have:
To note, the coefficients C1 and C2 are complex. So for convinience, we can make the coefficients real and positive by introducing a phase factor as:
where $\Delta $ is the phase. It is all good to write the superposition state like this, however, Bloch sphere gives a visualisation. It just makes things easier, just as the reciprocal space in solid state physics.
Now one can start thinking of constructing a sphere to represent a state like in (3) by looking at equation (2), which tells that whatever value of C1 and C2 (real and positive) are chosen, it must lie on a circle of unit radius.
Next, from (3) we get that when C1 = 1 and C2 = 0, we get the basis state |0i and so we can label that point as shown in figure.
Let any state $\left. |\psi \right >$ be denoted by a phasor which makes angle θ with the vertica axis. Now thinking in terms of spherical coordinates let $\phi $, representing the phase in (3), be the azhimuthal angle.
Now, since C1 and C2 are real and posiive (negative values will just give a negative phase, but in quantum mechanics we are only interested in state. Try putting C1 = -1 and C2 = 0), it can be easily seen that we just need to consider the upper half of the sphere. Thus we can deal in terms of $\frac{\theta }{2}$ instead, such that
which will comply with our general considerations in the spherical coordinate system where θ ranges from [0,π]. And now our general state in (3) can be written as
So, now given any state in the form of (1) we can convert it into a form like (4) and locate the state on the Bloch’s sphere. It is further seen that with θ = π/2, φ = 0 and φ = π/2 gives states,
respectively. So now we can construct the final representation of the Bloch sphere with the help of six bases $\left\lbrace \left. |\pm \right >, \left. |\pm i\right >, \left. |0\right > \& \left. |1\right >\right\rbrace$.
For the sake of completion, I would also add that the similar reperesentation can be made in the Cartesian coordinate system, using the transformations,
with the above relations and using (4), it is not difficult to show that
The take away point is this: given any state in the form of (1), convert it into one like (4) and then plot it on a sphere in the spherical or cartesian coordinate system. The representation of a quantum state on the Bloch sphere turns very useful when dealing with operations on a state in a quantum algorithm. In this representation, a rotation of a state simply becomes of a vector on a sphere.
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