We know that the angle between two orthogonal qubit states is 90 degrees. Why then, when we use the Bloch sphere, the angle becomes 180 degrees?


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The Bloch representation of states is what you get when you decompose density matrices in terms of a basis of orthonormal Hermitian operators. That is, you write $\rho=\sum_k \sigma_k \langle\sigma_k,\rho\rangle$ where $\langle\sigma_k,\rho\rangle\equiv\operatorname{Tr}(\sigma_k\rho)$ and $\{\sigma_k\}_k$ is some basis of Hermitian operators. For single qubits, the standard choice is to use Pauli matrices, and write the decomposition as $$\rho=\frac{I + c_x \sigma_x+ c_y \sigma_y + c_z \sigma_z}{2}.$$ Note that the coefficient of the identity matrix $I$ has been fixed to $\frac12$. This comes from the normalisation constraint on states, as $\langle I,\rho\rangle=\operatorname{Tr}(\rho)=1$.

Now, how does the inner product (and thus the geometry) between quantum states relate to the inner product between their Bloch representations? Given pure states $|\psi_1\rangle,|\psi_1\rangle$, and corresponding density matrices $\rho_1\equiv|\psi_1\rangle\!\langle\psi_1|$ and $\rho_2\equiv|\psi_2\rangle\!\langle\psi_2|$, with Bloch representations $$\rho_i = \frac{I + \vec r_i\cdot \vec\sigma}{2},$$ we can see that $$|\langle\psi_1|\psi_2\rangle|^2 = \langle\rho_1,\rho_2\rangle = \langle\vec v_1,\vec v_2\rangle,$$ where $\vec v_i=(\frac12,\frac12 \vec r_i)\in\mathbb{R}^4$, and thus $$\langle \vec v_1,\vec v_2\rangle = \frac12(1+\langle\vec r_1,\vec r_2\rangle).$$ From this we can see that orthogonality of quantum states corresponds to orthogonality of the vectors $\vec v_i$, which corresponds to the translated vectors $\vec r_i$ being collinear, that is, to $\langle\vec r_1,\vec r_2\rangle=-1$. The vectors $\vec r_i$ are what people typically mean when talking about the "Bloch sphere representation" of a state.

  • $\begingroup$ Why is the vector in the Bloch sphere, v_i, in R^4? Isn't the Bloch sphere in R^3? $\endgroup$
    – gary69
    Jan 24, 2023 at 17:34
  • 1
    $\begingroup$ @gary69 probably a typo, thanks. Fixed it $\endgroup$
    – glS
    Jan 24, 2023 at 18:14

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