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The sentence of Wikipedia :

"For example, there may be a 50% probability that the state vector is $| \psi_1 \rangle$ and a 50% chance that the state vector is $| \psi_2 \rangle$ . This system would be in a mixed state."

is false.

The difference between pure states and partially or completely mixed states, is only a difference of structure of the density matrix.

For a pure (supposed normed) state $\psi$, the density matrix is $\rho =|\psi\rangle \langle \psi|$, and this matrix has rank one, so in some basis, $\rho$ may be written $\rho = \text{Diag}(1,0,0.......0)$

Density matrix with rank different of one correspond to partially or completely mixed states.

Compare a pure and a mixed density matrix (in a basis $\psi_1 , \psi_2$):

$$\rho_{pure} =\frac{1}{2} \begin{pmatrix} 1&1\\1&1 \end{pmatrix}, \quad \quad \rho_{mixed } =\frac{1}{2} \begin{pmatrix} 1&0\\0&1 \end{pmatrix}$$ where the pure density matrix is build from a pure state $\psi = \frac{1}{\sqrt{2}}(\psi_1 + \psi_2)$, with $\langle \psi_1| \psi_2 \rangle = 0$, and where the mixed density matrix is a classical statistical matrix.

It is easy to see that the probability density to find the system in state $1$, is the same for the two density matrices :

$$p_1 = Tr(\rho P_1) = Tr (\rho |\psi_1\rangle \langle \psi_1|) = \rho_{11}=\frac{1}{2}$$

In the same way, one finds , for the two matrices, : $p_2 = \rho_{22}=\frac{1}{2}$

The sentence of Wikipedia :

"For example, there may be a 50% probability that the state vector is $| \psi_1 \rangle$ and a 50% chance that the state vector is $| \psi_2 \rangle$ . This system would be in a mixed state."

is false.

The difference between pure states and partially or completely mixed states, is only a difference of structure of the density matrix.

For a pure (supposed normed) state $\psi$, the density matrix is $\rho =|\psi\rangle \langle \psi|$, and this matrix has rank one, so in some basis, $\rho$ may be written $\rho = \text{Diag}(1,0,0.......0)$

Density matrix with rank different of one correspond to partially or completely mixed states.

Compare a pure and a mixed density matrix (in a basis $\psi_1 , \psi_2$):

$$\rho_\text{pure} =\frac{1}{2} \begin{pmatrix} 1&1\\1&1 \end{pmatrix}, \quad \quad \rho_\text{mixed } =\frac{1}{2} \begin{pmatrix} 1&0\\0&1 \end{pmatrix}$$ where the pure density matrix is build from a pure state $\psi = \frac{1}{\sqrt{2}}(\psi_1 + \psi_2)$, with $\langle \psi_1| \psi_2 \rangle = 0$, and where the mixed density matrix is a classical statistical matrix.

It is easy to see that the probability density to find the system in state $1$, is the same for the two density matrices :

$$p_1 = Tr(\rho P_1) = Tr (\rho |\psi_1\rangle \langle \psi_1|) = \rho_{11}=\frac{1}{2}$$

In the same way, one finds , for the two matrices, : $p_2 = \rho_{22}=\frac{1}{2}$

The sentence of Wikipedia :

"For example, there may be a 50% probability that the state vector is $| \psi_1 \rangle$ and a 50% chance that the state vector is $| \psi_2 > \rangle$ . This system would be in a mixed state."

is false.

The difference between pure states and partially or completely mixed states, is only a difference of structure of the density matrix.

For a pure (supposed normed) state $\psi$, the density matrix is $\rho =|\psi\rangle \langle \psi|$, and this matrix has rank one, so in some basis, $\rho$ may be written $\rho = Diag(1,0,0.......0)$

Density matrix with rank different of one correspond to partially or completely mixed states.

Compare a pure and a mixed density matrix (in a basis $\psi_1 , \psi_2$):

$$\rho_{pure} =\frac{1}{2} \begin{pmatrix} 1&1\\1&1 \end{pmatrix}, \quad \quad \rho_{mixed } =\frac{1}{2} \begin{pmatrix} 1&0\\0&1 \end{pmatrix}$$ where the pure density matrix is build from a pure state $\psi = \frac{1}{\sqrt{2}}(\psi_1 + \psi_2)$, with $\langle \psi_1| \psi_2 \rangle = 0$, and where the mixed density matrix is a classical statistical matrix.

It is easy to see that the probability density to find the system in state $1$, is the same for the two density matrices :

$$p_1 = Tr(\rho P_1) = Tr (\rho |\psi_1\rangle \langle \psi_1|) = \rho_{11}=\frac{1}{2}$$

In the same way, one finds , for the two matrices, : $p_2 = \rho_{22}=\frac{1}{2}$

The sentence of Wikipedia :

"For example, there may be a 50% probability that the state vector is $| \psi_1 \rangle$ and a 50% chance that the state vector is $| \psi_2 \rangle$ . This system would be in a mixed state."

is false.

The difference between pure states and partially or completely mixed states, is only a difference of structure of the density matrix.

For a pure (supposed normed) state $\psi$, the density matrix is $\rho =|\psi\rangle \langle \psi|$, and this matrix has rank one, so in some basis, $\rho$ may be written $\rho = \text{Diag}(1,0,0.......0)$

Density matrix with rank different of one correspond to partially or completely mixed states.

Compare a pure and a mixed density matrix (in a basis $\psi_1 , \psi_2$):

$$\rho_{pure} =\frac{1}{2} \begin{pmatrix} 1&1\\1&1 \end{pmatrix}, \quad \quad \rho_{mixed } =\frac{1}{2} \begin{pmatrix} 1&0\\0&1 \end{pmatrix}$$ where the pure density matrix is build from a pure state $\psi = \frac{1}{\sqrt{2}}(\psi_1 + \psi_2)$, with $\langle \psi_1| \psi_2 \rangle = 0$, and where the mixed density matrix is a classical statistical matrix.

It is easy to see that the probability density to find the system in state $1$, is the same for the two density matrices :

$$p_1 = Tr(\rho P_1) = Tr (\rho |\psi_1\rangle \langle \psi_1|) = \rho_{11}=\frac{1}{2}$$

In the same way, one finds , for the two matrices, : $p_2 = \rho_{22}=\frac{1}{2}$

The sentence of Wikipedia :

"For example, there may be a 50% probability that the state vector is $| \psi_1 \rangle$ and a 50% chance that the state vector is $| \psi_2 > \rangle$ . This system would be in a mixed state."

is false.

The difference between pure states and partially or completely mixed states, is only a difference of structure of the density matrix.

For a pure (supposed normed) state $\psi$, the density matrix is $\rho =|\psi\rangle \langle \psi|$, and this matrix has rank one, so in some basis, $\rho$ may be written $\rho = Diag(1,0,0.......0)$

Density matrix with rank different of one correspond to partially or completely mixed states.

Compare a pure and a mixed density matrix :

$$\rho_{pure} =\frac{1}{2} \begin{pmatrix} 1&1\\1&1 \end{pmatrix}, \quad \quad \rho_{mixed } =\frac{1}{2} \begin{pmatrix} 1&0\\0&1 \end{pmatrix}$$ where the pure density matrix is build from a pure state $\psi = \frac{1}{\sqrt{2}}(\psi_1 + \psi_2)$, with $\langle \psi_1| \psi_2 \rangle = 0$, and where the mixed density matrix is a classical statistical matrix.

It is easy to see that the probability density to find the system in state $1$, is the same for the two density matrices :

$$p_1 = Tr(\rho P_1) = Tr (\rho |\psi_1\rangle \langle \psi_1|) = \rho_{11}=\frac{1}{2}$$

In the same way, one finds , for the two matrices, : $p_2 = \rho_{22}=\frac{1}{2}$

The sentence of Wikipedia :

"For example, there may be a 50% probability that the state vector is $| \psi_1 \rangle$ and a 50% chance that the state vector is $| \psi_2 > \rangle$ . This system would be in a mixed state."

is false.

The difference between pure states and partially or completely mixed states, is only a difference of structure of the density matrix.

For a pure (supposed normed) state $\psi$, the density matrix is $\rho =|\psi\rangle \langle \psi|$, and this matrix has rank one, so in some basis, $\rho$ may be written $\rho = Diag(1,0,0.......0)$

Density matrix with rank different of one correspond to partially or completely mixed states.

Compare a pure and a mixed density matrix (in a basis $\psi_1 , \psi_2$):

$$\rho_{pure} =\frac{1}{2} \begin{pmatrix} 1&1\\1&1 \end{pmatrix}, \quad \quad \rho_{mixed } =\frac{1}{2} \begin{pmatrix} 1&0\\0&1 \end{pmatrix}$$ where the pure density matrix is build from a pure state $\psi = \frac{1}{\sqrt{2}}(\psi_1 + \psi_2)$, with $\langle \psi_1| \psi_2 \rangle = 0$, and where the mixed density matrix is a classical statistical matrix.

It is easy to see that the probability density to find the system in state $1$, is the same for the two density matrices :

$$p_1 = Tr(\rho P_1) = Tr (\rho |\psi_1\rangle \langle \psi_1|) = \rho_{11}=\frac{1}{2}$$

In the same way, one finds , for the two matrices, : $p_2 = \rho_{22}=\frac{1}{2}$