What may I read from DensityMatrix charts when running simple Qiskit code?

Question

Let's say I am running this code and I obtain picture below using Qiskit DensityMatrix class.

from qiskit.quantum_info import DensityMatrix
from qiskit.visualization import plot_state_city
qc2 = QuantumCircuit(2,2)
qc2.h(0)
qc2.cx(0,1)
qc2.draw('mpl')
state2 = DensityMatrix.from_instruction(qc2)
plot_state_city(state2.data)

How may I interpret this 'DensityMartix' chart? For what cases is it useful?

I have searched in Qiskit Documentation but did not find something beginner could digest. Will be grateful for any hints and directions.

Answer 1

The bars represent the amplitude of the elements in the desnity matrix, $\rho$.

So for instance, this particular plot

tell us that the element $\rho_{0,0} = \rho_{0,3} = \rho_{3,0} = \rho_{3,3} = 1/2$ and the rest are zero. That is, it meant to represent this matrix:

$$ \rho = \begin{pmatrix} \rho_{0,0} & \rho_{0,1} & \rho_{0,2} & \rho_{0,3}\\ \rho_{1,0} & \rho_{1,1} & \rho_{1,2} & \rho_{1,3}\\ \rho_{2,0} & \rho_{2,1} &\rho_{2,2} & \rho_{2,3}\\ \rho_{3,0} & \rho_{3,1} & \rho_{3,2} & \rho_{3,3}\end{pmatrix}= \begin{pmatrix} \rho_{0} & \rho_{1} & \rho_{2} & \rho_{3}\\ \rho_{4} & \rho_{5} & \rho_{6} & \rho_{7}\\ \rho_{8} & \rho_{9} &\rho_{10} & \rho_{11}\\ \rho_{12} & \rho_{13} & \rho_{14} & \rho_{15}\end{pmatrix}= \begin{pmatrix} 1/2 & 0 & 0 & 1/2\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 1/2 & 0 & 0 & 1/2 \end{pmatrix} $$

Note that on the graph, the state $0000$ corresponds to the element $\rho_{0,0}$ which is the $0th$ element of $\rho$.

The state $0011$ corresponds to the $3rd$ element of the density matrix which is the element $\rho_{0,3}$. You can see the binary value $0011$ converts to $3$ in decimal.

Similarly, $1100$ converts to $12$ which corresponds to the $12th$ element of $\rho$ which is the element $\rho_{3,0}$.

And lastly, $1111$ converts to $15$ which is the last element of $\rho$, which is also $\rho_{3,3}$.

So in conclusion, the bars in the graph just to represent the amplitude of the density matrix $\rho$. And the size of the grid indicate how big your system is. The examples you gave all have matrix of size $4 \times 4$, hence it is a two qubit system. But if you have a 3 qubit system then your grid would be $8 \times 8$.

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Another point to note is that the matrix

$$ \rho = \begin{pmatrix} 1/2 & 0 & 0 & 1/2\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 1/2 & 0 & 0 & 1/2 \end{pmatrix} $$

can be constructed from taking the outer product of the state $|\psi \rangle = \dfrac{|00\rangle + |11 \rangle}{\sqrt{2}} = \begin{pmatrix} 1/\sqrt{2} \\ 0 \\ 0 \\ 1/\sqrt{2} \end{pmatrix}$.

That is,

$$\rho = |\psi \rangle \langle \psi | = \begin{pmatrix} 1/\sqrt{2} \\ 0 \\ 0 \\ 1/\sqrt{2} \end{pmatrix} \begin{pmatrix} 1/\sqrt{2} & 0 & 0 & 1/\sqrt{2} \end{pmatrix} = \begin{pmatrix} 1/2 & 0 & 0 & 1/2\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 1/2 & 0 & 0 & 1/2 \end{pmatrix}$$

notice that the state $|\psi \rangle$ is precisely the state you created with your circuit.

qc2 = QuantumCircuit(2,2)
qc2.h(0)
qc2.cx(0,1)