Worked Problems in Physics

Nielsen and Chuang Exercise 2.58

Posted in Nielsen/Chuang by rpmuller on March 16, 2010

Exercise 2.58 Supposed we prepare a quantum system in an eigenstate {\left|\psi\right>} of some observable {M}, with corresponding eigenvalue {m}. What is the average observed value of {M}, and the standard deviation.

As long as the state does not decohere, the average value will always be exactly {\left<\psi|M|\psi\right>=m}, and the standard deviation will be 0. This does not violate the uncertainty principle, which is about pairs of noncommuting observables.

Nielsen and Chuang Exercise 2.57

Posted in Nielsen/Chuang by rpmuller on March 16, 2010

Exercise 2.57 Cascaded measurements are single measurements. Suppose {\{L_l\}} and {\{M_m\}} are two sets of measurement operators. Show that a measurement defined by the measurement operators {\{L_l\}} followed by a measurement defined by the measurement operators {\{M_m\}} is physically equivalent to a single measurement defined by the measurement operators {\{N_{lm}\}} with the representation {N_{lm}=M_mL_m}.

Suppose we consider the matrix element {\left<\mu|M_mL_l|\nu\right>} of the specified cascaded measurement. We can insert a complete set of states between the two operators via {\sum_\lambda\left<\mu|M_m|\lambda\right>\left<\lambda|L_l|\nu\right>}, making this equivalent to the matrix multiplication of elements of {N_{lm}=M_mL_l}.