I have had a look at a book on 'Quantun Measurement' by Braginsky and Khalili$^1$. In it appears an equation that I would like confirmation of. The equation seems odd, in that it sets a probability for the result of a measurement to the trace of some matrix.The equation in question is (2.7) ,see the book, Section 2.5 'von Neumann's postulate of reduction'. I quote

$$w_n=Tr(|q_n\rangle\langle q_n|\hat{\rho}_{init}) \tag{2.7}$$

My question is. Is (2.7) correct?


1) Vladimir B Braginsky and Farid Ya Khalili, Quantum Measurement, Ed Kip S Thorne, Cambridge University Press, First paperback edition (with corrections) 1995.

Other Info

In the book the second part of the general form of the postulate is (2.8), this applies to the density operator asociated with the state of the system after measurement, it is

$$\hat{\rho}_n = \frac{1}{w_n} |q_n\rangle\langle q_n| \hat{\rho}_{init} |q_n\rangle\langle q_n| \tag{2.8}$$

(2.7) is supposed to apply to a quantum system which is initially in a state with associated density operator $\hat{\rho}_{init}$. The system may consist of two particles (or I assume an arbitrary number). In it, $w_n$ is the probability of obtaining the result of measurement $q_n$, for an observable $q$ with an associated operator $\hat{q}$, this operator has a discrete set of eigenvalues.We have $$ \hat{q} |q_n\rangle= q_n |q_n\rangle $$ The book also has the one degree of freedom version of von Neumann's postulate, in this the probability of a measurement is given in (2.6) as

$$w_n=\langle q_n|\hat{\rho}_{init}|q_n\rangle \tag{2.6}$$


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