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2 Qubit Bloch Chain

We normalize and explore the space spanned by 2 qubits, with basis vectors $\left|00\right>$, $\left|01\right>$, $\left|10\right>$, $\left|11\right>$ as follows:

$$ \left|\psi\right> = $$ $$ \cos{\left(\frac{\theta_0}{2}\right)}\left|00\right> + $$ $$ e^{i\phi_0}\sin{\left(\frac{\theta_0}{2}\right)}\cos{\left(\frac{\theta_1}{2}\right)}\left|01\right> + $$ $$ e^{i\phi_1}\sin{\left(\frac{\theta_0}{2}\right)}\sin{\left(\frac{\theta_1}{2}\right)}\cos{\left(\frac{\theta_2}{2}\right)}\left|10\right> + $$ $$ e^{i\phi_2}\sin{\left(\frac{\theta_0}{2}\right)}\sin{\left(\frac{\theta_1}{2}\right)}\sin{\left(\frac{\theta_2}{2}\right)}\left|11\right> $$

This is represented by a sequence of three($2^2 - 1$) pairs of angles $(\theta_0,\phi_0)$, $(\theta_1,\phi_1)$, and $(\theta_2,\phi_2)$ which rotate a sequence of vectors along which we display spheres. Then angles $\phi_i$ represent complex phases, and the angles $\theta_i$ are a factor of two from the generalized "lattitude" around the surface of a three sphere $\mathbb{S}^3$ embedded in a Euclidean four dimensional space $\mathbb{R}^4$. The factor of two in lattitude is so that a rotation of $\pi$ radians gets from one eigenstate to the next, and to make it consistent with the 1 qubit Bloch sphere.

$$ \alpha \equiv \left< *0\right> = \left< 00|00\right> + \left< 10|10\right> $$

$$ \beta \equiv \left< *1\right> = \left< 01|01\right> + \left< 11|11\right> $$

$$ \gamma \equiv \left< 0*\right> = \left< 00|00\right> + \left< 01|01\right> $$

$$ \delta \equiv \left< 1*\right> = \left< 10|10\right> + \left< 11|11\right> $$

These four probabilities are used to determine the opacity of four images, each representing either a one or a zero of either the first or second qubit.

$$ \alpha = \cos^2{\left(\frac{\theta_0}{2}\right)} + \sin^2{\left(\frac{\theta_0}{2}\right)}\sin^2{\left(\frac{\theta_1}{2}\right)}\cos^2{\left(\frac{\theta_2}{2}\right)} $$

$$ \beta = \sin^2{\left(\frac{\theta_0}{2}\right)}\cos^2{\left(\frac{\theta_1}{2}\right)} + \sin^2{\left(\frac{\theta_0}{2}\right)}\sin^2{\left(\frac{\theta_1}{2}\right)}\cos^2{\left(\frac{\theta_2}{2}\right)} $$

$$ \gamma = \cos^2{\left(\frac{\theta_0}{2}\right)} + \sin^2{\left(\frac{\theta_0}{2}\right)}\cos^2{\left(\frac{\theta_1}{2}\right)} $$

$$ \delta = \sin^2{\left(\frac{\theta_0}{2}\right)}\sin^2{\left(\frac{\theta_1}{2}\right)} $$

In a web browser each of these values determins an opacity of a Geometron symbol which represents that state. There are four of these, one for each state of each qubit.

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