In quantum mechanics, the density matrix is a mathematical tool used to describe the state of a quantum system. It provides a way to represent mixed states, which are states that have both classical and quantum uncertainties. The elements of the density matrix represent the probabilities of finding the system in different states.
In a pure state, the density matrix is a projection operator onto that state, and it has only one non-zero eigenvalue, while all other eigenvalues are zero. This means that the system is in a well-defined quantum state with no classical uncertainties.
On the other hand, in a mixed state, the density matrix has multiple non-zero eigenvalues. This indicates that the system is in a superposition of different states with classical uncertainties. The non-zero eigenvalues represent the probabilities of finding the system in each of those states.
Now, when you mention driving the entities down to zero in the density matrix, it seems like you are referring to the idea of completely removing the classical uncertainties and bringing the system into a pure state. However, in general, it is not possible to completely eliminate classical uncertainties in a quantum system.
According to the principles of quantum mechanics, there is always an inherent uncertainty associated with certain observables. This is captured by the Heisenberg uncertainty principle, which states that certain pairs of physical properties, such as position and momentum, cannot both be precisely determined simultaneously. This fundamental limit on precision arises due to the wave-like nature of quantum objects.
Even if you could somehow manipulate the system to minimize its classical uncertainties, the uncertainty principle still implies that there will always be some residual uncertainty present. This residual uncertainty prevents the entities from being driven completely down to zero in the density matrix, and the system will remain in a mixed state.
In practical terms, the inability to eliminate classical uncertainties entirely is a fundamental aspect of quantum mechanics. It is this inherent probabilistic nature that distinguishes quantum systems from classical systems and gives rise to the strange and fascinating phenomena observed in the quantum world.