A How Do Quantum Measurements Impact System Coherence?

[Danny Boy]

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.

 

[kith]

The following, regarding quantum measurement, is stated in the paper "Limitation on the amount of accessible information in a quantum channel"

This link doesn't work for me.

1. Why does the system have to be in state $(*)$ for the process to constitute a measurement?

I have only skimmed the passage but what about turning this around: modify the state in different ways and think about the consequences. For example, introduce more indices.

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?

In open quantum dynamics, the term "coherences" often refers to the off-diagonal elements of the density matrix. (The on-diagonal elements are called "populations".)

In your example, it refers to the fact that only one index appears in each term of the sum.