I read the documentation of qiskit and I can't understand the meaning of the city plot, like this:
Why do we need a 3D plot? Why can't we just use a 2D plot, where $ | 00\rangle$, $ | 01 \rangle$, $ |10 \rangle$, $ |11 \rangle$ would lie on X-axis, and their amplitudes (or magnitudes) would lie along Y-axis?
What does the intersection of $ | 00 \rangle$ and $ | 11 \rangle$ show on such city plot?
A density matrix $\rho$ on two qubits has 16 complex amplitudes (although not all are free variables due to constraints from normalization and Hermeticity), so the City plot is showing those amplitudes as well. The $|00\rangle\langle 11|$ and $|11\rangle\langle 00|$ amplitudes shown are not going to directly impact your measurement if you were to measure in the Z basis, but would come into effect due to basis transformations.
As a simple example of these off diagonal elements, Think about a single qubit density matrix representing a $|+\rangle$ state:
$$\rho = |+\rangle\langle +| = \frac{1}{2}(|0\rangle + |1\rangle )(\langle 0| + \langle 1|) = \frac{1}{2}(|0\rangle\langle 0| + |1\rangle\langle 0| + |0\rangle\langle 1| + |1\rangle\langle 1|)$$
When measured in the Z basis this state gives a 50% chance of 0 and a 50% chance of 1, which is identical to the mixed state:
$$\rho ' = \frac{1}{2}(|0\rangle\langle 0| + |1\rangle\langle 1|)$$
However in the X basis our $|+\rangle$ state will always give an eigenvalue of 1, while the mixed state will give other results.
As an aside, these off diagonal elements are important when checking the purity of a quantum state:
$$\rho^2 = \rho$$
If this equation is satisfied then your state is pure, which is only possible due to the off diagonal elements in the density matrix.
