Bures Metric - Two-Level System

Two-Level System

The state of a two level system can be parametrized with three variables as

\rho = \frac{1}{2}( \sigma_0 + x^1 \sigma_1 + x^2 \sigma_2 + x^3 \sigma_3 )

with . The components of the Bures metric in this parametrization can be calculated as

g_{jk} = \frac{1}{4(1 - (x^1)^2 - (x^2)^2 - (x^3)^2)} \begin{pmatrix} 1 - (x^2)^2 - (x^3)^2 & x^1 x^2 & x^1 x^3 \\ x^1 x^2 & 1 - (x^1)^2 - (x^3)^2 & x^2 x^3 \\ x^1 x^3 & x^2 x^3 & 1 - (x^1)^2 - (x^2)^2 \end{pmatrix} .

The Bures measure can be calculated by taking the square root of the determinant to find

dV_B = \frac{dx^1 dx^2 dx^3}{8\sqrt{ 1 - (x^1)^2 - (x^2)^2 - (x^3)^2 }},

which can be used to calculate the Bures volume as

V_B = \int_{-1}^{1}dx^1 \int_{-\sqrt{1-(x^1)^2}}^{ \sqrt{1-(x^1)^2} }dx^2 \int_{-\sqrt{1-(x^1)^2-(x^2)^2}}^{\sqrt{1-(x^1)^2-(x^2)^2}}dx^3 \frac{1}{8\sqrt{ 1 - (x^1)^2 - (x^2)^2 - (x^3)^2 }} = \frac{\pi^2}{8}

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