Is the Von Neumann entropy related to heat transfer? The Von Neumann entropy of a QM system, as far as I understand it, is a measure of the information a particular observer has about that system. Is this definition of entropy directly related to heat transfer in a sense analogous to the classical viewpoint where $\Delta S\geq\frac{Q}{T}$ ? 
Note that I'm not asking about the equivalence of the mathematical form. My question is whether the amount of information one has affects the evolution of the system i.e., would the physics between two truly identical states (assuming that's even possible) be different if the level of knowledge of its starting conditions was different between the two?   
In the classical sense, information is an accounting tool and it doesn't affect how the state evolves over time. Also in the classical sense, entropy may be discussed among physicists and amateurs like me as akin to the amount of knowledge one has of the state but, as I just mentioned, physics doesn't care about how much information there is. Information is not a physical quantity, but entropy clearly is.
This Nature News article suggests there isn't an academic consensus answering the question I posed. I want to see what this community has to say.
http://www.nature.com/news/battle-between-quantum-and-thermodynamic-laws-heats-up-1.21720
 A: While, it's not your question, it's worth point out.  The Von-Neumann Entropy is the definition of thermo dynamic entropy for a canonical ensemble. Thus if you use the canonical-ensemble density-matrix in the Von-Neumann Entropy you will get the thermodynamic Entropy of the system:
$
-tr\left<\rho ln(\rho)\right>=tr\left<e^{-\beta H}/Z(\beta H-lnZ)\right>=S
$

My question is whether the amount of information one has affects the evolution of the system i.e., would the physics between two truly identical states (assuming that's even possible) be different if the level of knowledge of its starting conditions was different between the two? My question is whether the amount of information one has affects the evolution of the system i.e., would the physics between two truly identical states (assuming that's even possible) be different if the level of knowledge of its starting conditions was different between the two?

If the initial density matrices are different, then time time evolution will be different.  So if two density matrices have different von-neumann entropy, they must be different and have different evolutions.
Ignorance In Measurement
Now what your really interested in, does "knowing something about the system change it's evolution?"  To understand this, you have to think about how you'll know something about a system.  To lean something about the system you have to measure it by entangling it with another system.  This process of entangling is physical and thus has physical consequences.
Suppose the entanglement process entangles your system states $\left|n\right>_s$ with your measurement device states $\left|n\right>_m$.  Then the combined state will look something like
$
\sum_n c_n\left|n\right>_s \otimes \left|n\right>_m
$
with a pure state density matrix
$
\rho = \sum_{n,n'}c_n c^*_{n'} \left|n\right>_s \otimes \left|n\right>_m \left<n'\right|_s \otimes \left<n'\right|_m
$
Making a measurement in your measurement device is a projection onto the state you observe: $P_M = \left|M\right>\left<M\right|$.  Suppose $\left|M\right>=\sum_l M_l \left|l\right>_m$ This measurement will collapse your system into a mixed state:
$
\rho \rightarrow  \sum_{l,l'}c_{l}M_lc^*_{l'}M^*_{l'}\left|l\right>_s\left<l'\right|_s
$
Different measurements will collapse to different mixed states with different entropies, and will evolve differently.
Ignorance In Chaotic Motion
Essentially, by acquiring information about a system, we change it's state and subsequent evolution.  This sort of information is kinda different from what we consider in our loss of information in statistical mechanics, we usually attribute this to chaotic evolution and  your intuition from classical mechanics applies here.  If you start a quantum system in the exact same state, it will evolve to the exact same state.  But we can't predict what this state will be, here the entropy is describing how well we can predict the final state.  This sort of knowledge doesn't effect the evolution of the system just like in classical mechanics.
