The system in the picture has a box filled with $N$ classical particles which can either be on the left or the right side. Let this system have a Hamiltonian $$H= \frac{\mathbf p^2}{\sum 2m_i} + V(x, t) $$ where V becomes infinite at $\pm \frac{L}{2} $. Let it be represented by a point in $2N$-dimensional phase space $$ (x_1, p_1, ... , x_N, p_N) $$ If we say that a positive x coordinate corresponds to 1, and negative to 0, then this system will encode a string of bits $N$ digits long.
Suppose I want a quantum computer to track the information in the string of bits above. The computer will contain $M$ qubits where $M = \log N$. Assuming this is a perfect quantum computer with no errors, the general state of the computer will be a density matrix in $2^M$-dimensional Hilbert space. $$ \rho = \begin{bmatrix} c_{1,1} & c_{1,2} & \dots & c_{1,N} \\ c_{2,1} & c_{2,2} \\ \vdots & & \ddots \\ c_{N,1} & & & c_{N,N} \end{bmatrix}$$
Now we will quantify the amount of information in both systems.
In order to quantify how much information the box of particles contains, we can use the Shannon Entropy. For a random variable $P_i$ representing the probability of finding a one of a zero in the string at position $i$, the information entropy is$$ S = - \sum_i P_i \log P_i $$ If we assume the particles are sufficiently mixed up in the box, then every $P_i$ will have a probability $\frac{1}{2}$, and then total entropy will be $\frac{N}{2} \log 2$
Next we turn the quantum computer, where the measure of the amount of information is Von Neumann Entropy. Given a density matrix $\rho$ for the quantum computer, the amount of information is given by $$ S = - Tr(\rho \log \rho) $$
Question 1: Does there exist a density matrix that contains the same amount of information as the classical system?
Question 2: Does there exist a Hamiltonian to control the evolution of the quantum computer so that the information contained in both systems are always equal?