I read the documentation of qiskit and I can't understand the meaning of the city plot, like this: enter image description here

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?


1 Answer 1


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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.