Interpreting von Neumann Entropy in Schrodinger's Cat-type Experiment

Main Question or Discussion Point

***EDIT: This is nonsensical. I'm new to density matrices, and I had a sign error that led to some confusion. In addition... this just doesn't make sense, reading it over. I apologize for the post. I'd appreciate it if a moderator removed this, please.

I'm reading about the von Neumann entropy, and I'm having trouble understanding the relationships between the joint quantum, conditional quantum, and classical thermodynamic entropy. If the von Neumann entropy of a system is an extension of the classical entropy, and the joint quantum entropy of a set of subsystems is equal the von Neumann entropy of the system containing these subsystems, then I'm not sure how to interpret the von Neumann in certain situations. Let me try to apply the von Neumann entropy to a Schrodinger's cat-type experiment.

The Schrodinger's cat experiment describes the evolution of an isolated macroscopic interacting with a (similarly isolated) pure state. We put these elements in a "box" which isolate them from the lab. For simplicity, let's say that the "cat-killing" subsystem which is in a pure state is described by a spin superposition before measurment:

$|\psi>=\frac{1}{\sqrt{2}}(|+>+|->)$

The density matrix of this state is:

$\widehat{\sigma}=|\psi><\psi|$

Let's say that the rest of the system inside the box is described by the density matrix $\widehat{\rho}$, and so the entire system in isolation is described by:

$\widehat{\rho}_{box}=\widehat{\sigma}\widehat{\rho}$

The expectation values for the entropy operator of the subsystem containing everything but the "cat-killing" pure state is generally some positive value:

$<\widehat{S}_{\rho}>\geq0$

The expectation value for the entropy operator of the "cat-killing" bit is zero:

$<\widehat{S}_{\sigma}>=0$

Because of inclusion of the pure state, the joint entropy of the system is zero:

$<\widehat{S}_{\sigma \rho}>=0$

Let's say that the "cat-killing" bit started as just a thermalized particle inside the box, so the joint entropy was non-zero. Then we seal the box. Then, the apparatus inside prepares a single particle in a pure state, immediately reducing the joint entropy of the system to zero.

If the joint entropy of all subsystems is equal to the total von Neumann entropy of the complete system, and the von Neumann entropy of the whole system is a generalization of the classical entropy of the whole system, how do we interpret the sudden decrease in total entropy of the system? If the box was at the same temperature as the lab before we sealed it, do we expect the box to suddenly give off heat? I wouldn't expect so, but I'm not sure how to interpret the joint entropy.

The entropy of the subsystems I considered still adds to what I would expect the total value to be for the classical entropy. However, they don't add up to the joint entropy. I understand that quantum information and entropy can sometimes be negative; the conditional quantum entropy appears to be negative in this case. I understand that the conditional quantum entropy has something to do with state merging, but again I'm not sure how to interpret this.

My biggest question is, what value(s) stand in for the thermodynamic entropy in this case if not the joint entropy of the system? Do we really expect the thermodynamic entropy of the entire isolated system to drop to zero in this case?

(Also, I'd love it if anyone could suggest a good book on quantum information theory while they're at it. Thank you!)

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