- #1
Danny Boy
- 49
- 3
The Fundamental Theorem of Quantum Measurement is stated as follows:
Every set of operators ##\{ A_n \}## ##n =1,...,N## that satisfies ##\sum_n A_n^{\dagger}A_n = I## describes a possible measurement on a quantum system, where the measurement has ##n## possible outcomes labeled by ##n##. If ##\rho## is the state of the system before the measurement and ##\tilde{\rho}_n## is the state of the system upon obtaining measurement result ##n##, and ##p_n## is the probability of obtaining result ##n##, then $$\tilde{\rho}_n = \frac{A_n \rho A_n^{\dagger}}{p_n}~~\text{and}~~p_n = \text{Tr}[A_n^{\dagger}A_n \rho]$$
Question: Since ##p_n = \text{Tr}[A_n^{\dagger}A_n \rho]## represents the probability of obtaining measurement result ##n##, I assume that this is a real number (in the interval ##[0,1]##) rather than complex, but I fail to see how it is guaranteed that this will be a real number. Am I missing something?
Every set of operators ##\{ A_n \}## ##n =1,...,N## that satisfies ##\sum_n A_n^{\dagger}A_n = I## describes a possible measurement on a quantum system, where the measurement has ##n## possible outcomes labeled by ##n##. If ##\rho## is the state of the system before the measurement and ##\tilde{\rho}_n## is the state of the system upon obtaining measurement result ##n##, and ##p_n## is the probability of obtaining result ##n##, then $$\tilde{\rho}_n = \frac{A_n \rho A_n^{\dagger}}{p_n}~~\text{and}~~p_n = \text{Tr}[A_n^{\dagger}A_n \rho]$$
Question: Since ##p_n = \text{Tr}[A_n^{\dagger}A_n \rho]## represents the probability of obtaining measurement result ##n##, I assume that this is a real number (in the interval ##[0,1]##) rather than complex, but I fail to see how it is guaranteed that this will be a real number. Am I missing something?