Danny Boy
- 48
- 3
The following, regarding quantum measurement, is stated in the paper "Limitation on the amount of accessible information in a quantum channel" :
"Our discussion of measurement will be based on a specific physical model of measurement, to which we now turn. Suppose we have a quantum system ##Q## with an initial state ##\rho^{(Q)}##. The measurement process will involve two additional quantum systems an apparatus system ##A## and an environment system ##E##. The systems ##A## and ##E## are initially in a joint state ##\rho_{0}^{(AE)}##, so that the overall initial state of everything is ##\rho^{(AEQ)} = \rho_{0}^{(AE)} \otimes \rho^{(Q)} ##.
The measurement process proceeds in two successive stages :
1. A dynamical evolution including interactions among ##A, E## and ##Q## represented by a unitary operator ##U##:
$$\rho^{(AEQ) } \rightarrow \hat{\rho}^{(AEQ) } = U\rho^{(AEQ) } U^{\dagger} $$
2. Discarding of the environment, represented by a partial trace over the system ##E##:
$$\hat{\rho}^{(AEQ)} \rightarrow \hat{\rho}^{(AQ)} = \text{Tr} _{E}\hat{\rho}^{(AEQ)}$$
For the process to constitute a measurement, we require that after these two stages, the state ##\hat{\rho}^{(AQ)}## be of the following form:
$$\hat{\rho}^{(AQ) } = \sum_a P(a) |\phi_{a}^{(A) } \rangle \langle \phi_{a}^{(A) }| \otimes w_{a}^{(Q) }~~~~~~~~~~~~(*) $$
Where the states ##|\phi_{a}^{(A) }## are a fixed orthogonal set of apparatus states, independent of the input state ##\rho^{(Q) }##.
Questions:
1. Why does the system have to be in state ##(*)## for the process to constitute a measurement? I
2. It is further stated "Coherences between different measurement outcomes do not remain in the joint state of systems ##A## and ##Q##. Any such coherences have leaked away into the environment during the dynamical evolution." How would you define/interpret the word "coherence" in this context?
Thanks for any assistance.
"Our discussion of measurement will be based on a specific physical model of measurement, to which we now turn. Suppose we have a quantum system ##Q## with an initial state ##\rho^{(Q)}##. The measurement process will involve two additional quantum systems an apparatus system ##A## and an environment system ##E##. The systems ##A## and ##E## are initially in a joint state ##\rho_{0}^{(AE)}##, so that the overall initial state of everything is ##\rho^{(AEQ)} = \rho_{0}^{(AE)} \otimes \rho^{(Q)} ##.
The measurement process proceeds in two successive stages :
1. A dynamical evolution including interactions among ##A, E## and ##Q## represented by a unitary operator ##U##:
$$\rho^{(AEQ) } \rightarrow \hat{\rho}^{(AEQ) } = U\rho^{(AEQ) } U^{\dagger} $$
2. Discarding of the environment, represented by a partial trace over the system ##E##:
$$\hat{\rho}^{(AEQ)} \rightarrow \hat{\rho}^{(AQ)} = \text{Tr} _{E}\hat{\rho}^{(AEQ)}$$
For the process to constitute a measurement, we require that after these two stages, the state ##\hat{\rho}^{(AQ)}## be of the following form:
$$\hat{\rho}^{(AQ) } = \sum_a P(a) |\phi_{a}^{(A) } \rangle \langle \phi_{a}^{(A) }| \otimes w_{a}^{(Q) }~~~~~~~~~~~~(*) $$
Where the states ##|\phi_{a}^{(A) }## are a fixed orthogonal set of apparatus states, independent of the input state ##\rho^{(Q) }##.
Questions:
1. Why does the system have to be in state ##(*)## for the process to constitute a measurement? I
2. It is further stated "Coherences between different measurement outcomes do not remain in the joint state of systems ##A## and ##Q##. Any such coherences have leaked away into the environment during the dynamical evolution." How would you define/interpret the word "coherence" in this context?
Thanks for any assistance.