Density Operators and Mixed States

A density operator describes pure states, statistical mixtures, and the local states of entangled systems in one language. It is especially useful when a preparation label is unavailable or when only part of a larger system can be measured. This chapter explains what the matrix means before introducing its geometric and information-theoretic uses.

Required background. State Vectors supplies pure states, and Expectation Values supplies the probability calculations generalized here. You should also be familiar with traces, eigenvalues, and positive matrices.

A superposition is not a random choice

In the $0,1$ basis, compare

$$\rho_+=\lvert+\rangle\langle+\rvert =\frac12\begin{pmatrix}1&1\\1&1\end{pmatrix}, \qquad \rho_{\mathrm{mix}}=\frac12\begin{pmatrix}1&0\\0&1\end{pmatrix}.$$

Here $\lvert+\rangle=(\lvert0\rangle+\lvert1\rangle)/\sqrt2$. The first matrix describes a coherent pure state. The second describes, for example, equal random preparations of $\lvert0\rangle$ and $\lvert1\rangle$ with the label ignored.

Both give $0$ and $1$ with equal probabilities. A measurement in the $\lvert\pm\rangle$ basis distinguishes their predictions: $p(+)=1$ for $\rho_+$ and $p(+)=1/2$ for $\rho_{\mathrm{mix}}$. A single choice of measurement therefore cannot always tell two states apart.

A valid density operator obeys $\rho\ge0$ and $\operatorname{Tr}\rho=1$. In finite dimensions its purity is $\operatorname{Tr}(\rho^2)$, equal to one precisely for pure states. Diagonal entries alone do not determine purity, and off-diagonal entries do not by themselves imply it.

Probabilities follow the trace rule $p(a)=\operatorname{Tr}(\rho E_a)$ for a measurement effect $E_a$. Different preparation ensembles can give the same $\rho$ and therefore exactly the same statistics for every measurement on the system alone.

Reading sequence

The first four articles establish the state and how to use it. The remaining articles give geometry, subsystem applications, and ways to distinguish different kinds of uncertainty.

Question	Article	What you will learn
Why describe a state by an operator?	Density Operators	construct a state and test its validity
When does one vector suffice?	Pure vs Mixed States	use rank, purity, and their hypotheses
What does a preparation record add?	Ensembles and Preparation Procedures	separate an ensemble from its average state
How do I predict a detector reading?	Trace Rule for Expectation Values	calculate probabilities, means, and variances
How do qubit matrix entries become geometry?	Bloch Sphere	read mixedness and measurement directions
What can one subsystem predict?	Reduced Density Matrices	contract the unobserved index and check local statistics
How can mixedness arise from a pure whole?	Purification Overview	construct an auxiliary system
What does entropy measure?	Entropy Overview	distinguish spectral, measurement, and preparation entropy
How can equal populations hide different states?	Classical Mixtures vs Quantum Superpositions	reveal coherence through interference

For the full subsystem theory, continue to Entangled States and Partial Trace. Quantum Operations explains how physical processes transform these states.

Try it. Calculate both purities. Does switching basis change either result?

Solution

$\rho_+^2=\rho_+$ gives purity one. Since $\rho_{\mathrm{mix}}=I_2/2$, its purity is $\operatorname{Tr}(I_2/4)=1/2$. Unitary changes of basis preserve these traces, although individual matrix entries change.

References

• I. Bengtsson and K. Życzkowski, Geometry of Quantum States, 2nd ed., Cambridge University Press, 2017.
• M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, 10th anniversary ed., Cambridge University Press, 2010.
• A. Peres, Quantum Theory: Concepts and Methods, Kluwer, 1995.
• J. von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton University Press, 1955.
• J. Watrous, The Theory of Quantum Information, Cambridge University Press, 2018.
• M. M. Wilde, Quantum Information Theory, 2nd ed., Cambridge University Press, 2017.