B Probability of spin in the +x direction

  • Thread starter srallaba
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Consider a qubit.
Given a state equation, how to calculate the probability of finding spin in +x direction?
 

vanhees71

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If your system is prepared in the state ##\hat{\rho}## then the probability for the ##x##-component of the spin being ##\sigma_x## is
$$P(\sigma_x)=\langle \sigma_x|\hat{\rho}|\sigma_x \rangle.$$
 

olgerm

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How to do it if I know spinor instead of ##\hat{\rho}##?
 

vanhees71

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If your spin is prepared in a pure state, represented by the ket ##|\psi \rangle##, then the statistical operator is the projection operator ##\hat{\rho}=|\psi \rangle \langle \psi|##, and the general formula gives you the Born rule for pure states, as it should be:

$$P(\sigma_x)=\langle \sigma_x \psi \rangle \langle \psi|\sigma_x \rangle=\psi(\sigma_x) \psi^*(\sigma_x)=|\psi(\sigma_x)|^2.$$
 

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