Density-Matrix Examples

Density matrices describe pure states, mixed states, reduced states, and open-system states in one notation. Examples are most useful when they keep ensemble mixtures, coherent superpositions, and subsystem reductions separate.

Example Routes

Need	Learn	Main Check
Definition and tests	Density Operators	Hermitian, positive, trace one
Pure versus mixed	Pure vs Mixed States	$\rho^2=\rho$ only for pure states
Trace-rule average	Trace Rule Expectation Values	Operator and state on same Hilbert space
Reduced state	Partial Trace	Tensor-factor ordering
Practice reductions	Partial Trace Exercises	Basis labels and index sums
Entropy	Entropy Overview	Logarithm base and eigenvalues
Channel model	Depolarizing Channel	Kraus normalization
Formula card	Density Matrix Expectation	Trace convention

Minimal Worked Check

For a classical mixture of $\lvert0\rangle$ with probability $p$ and $\lvert1\rangle$ with probability $1-p$,

$$\rho =p\lvert0\rangle\langle0\rvert +(1-p)\lvert1\rangle\langle1\rvert = \begin{pmatrix} p & 0\\ 0 & 1-p \end{pmatrix}.$$

For $Z=\lvert0\rangle\langle0\rvert-\lvert1\rangle\langle1\rvert$,

$$\operatorname{Tr}(\rho Z)=p-(1-p)=2p-1.$$

This is not the same state as the coherent superposition $\sqrt p\lvert0\rangle+\sqrt{1-p}\lvert1\rangle$, whose density matrix has off-diagonal terms.

Common Mistakes

• Confusing a mixture with a superposition that has the same diagonal probabilities.
• Taking a partial trace without fixing tensor-factor order.
• Checking trace one but forgetting positivity.
• Using a reduced density matrix as if it described a pure state of the subsystem.
• Treating entropy as a property of a state vector without specifying the subsystem or ensemble.
• Writing a channel map without checking trace preservation.

Reference Companions

• Density Matrix Expectation
• Partial Trace
• Kraus Map
• Von Neumann Entropy
• Bloch Vector

References

• M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, 2010.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.

Exercises

1. For the mixture above, what is $\operatorname{Tr}\rho^2$?

Solution

Since $\rho$ is diagonal, $\operatorname{Tr}\rho^2=p^2+(1-p)^2$. This equals one only when $p=0$ or $p=1$, so intermediate probabilities give mixed states.