Density Matrix Conventions

The default symbol for a density operator is $\rho$. The phrase “density matrix” refers to a matrix representation of the density operator in a chosen basis. When basis dependence matters, use “density operator” for the abstract object and “density matrix” for its representation.

The default validity conditions are

$$\rho^\dagger=\rho, \qquad \rho\ge0, \qquad \operatorname{Tr}\rho=1.$$

Positivity means

$$\langle\psi\vert\rho\vert\psi\rangle\ge0$$

for every vector $\lvert\psi\rangle$ in the relevant Hilbert space.

Pure and Mixed States

A normalized pure state $\lvert\psi\rangle$ corresponds to

$$\rho_\psi=\lvert\psi\rangle\langle\psi\rvert.$$

An ensemble written as states $\lvert\psi_k\rangle$ prepared with probabilities $p_k$ is represented by

$$\rho=\sum_k p_k \lvert\psi_k\rangle\langle\psi_k\rvert, \qquad p_k\ge0, \qquad \sum_k p_k=1.$$

Different ensembles may produce the same $\rho$. The density operator, not the chosen ensemble story, determines all measurement statistics.

Matrix Elements

In an orthonormal basis $\{\lvert n\rangle\}$, use

$$\rho_{mn}=\langle m\vert\rho\vert n\rangle.$$

Diagonal entries in a physically meaningful basis are often called populations. Off-diagonal entries are often called coherences in that basis. Both labels are basis-dependent.

Trace Rules

Expectation values use

$$\langle A\rangle_\rho=\operatorname{Tr}(\rho A).$$

For a projective outcome with projector $P_a$,

$$P(a)=\operatorname{Tr}(\rho P_a).$$

For a general POVM effect $E_a$,

$$P(a)=\operatorname{Tr}(\rho E_a).$$

Use cyclicity of the trace only when all products are mathematically well-defined. In finite-dimensional pages this is automatic; in infinite-dimensional pages, trace-class conditions may matter.

Purity and Bloch Vectors

The purity is written

$$\operatorname{Tr}(\rho^2).$$

For a finite-dimensional pure state, $\operatorname{Tr}(\rho^2)=1$. For a mixed state, $\operatorname{Tr}(\rho^2)<1$.

For one qubit, the default Bloch form is

$$\rho=\frac12(I+\mathbf r\cdot\boldsymbol\sigma),$$

with $\lVert\mathbf r\rVert\le1$.

Common Mistakes

• Treating the diagonal entries as probabilities without specifying the basis.
• Confusing one ensemble decomposition with the density operator itself.
• Forgetting positivity and checking only trace one.
• Calling every mixed state ignorance about an underlying pure state.
• Using pure-state expectation formulas when the state is a subsystem or thermal state.

References

• J. von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton University Press, 1955.
• L. E. Ballentine, Quantum Mechanics: A Modern Development, 2nd ed., World Scientific, 2014.
• M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, 2010.

Exercises

1. Check whether

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

has trace one and compute its purity.

Solution

The trace is $1/2+1/2=1$. Also,

$$\rho^2= \begin{pmatrix} 1/4&0\\ 0&1/4 \end{pmatrix},$$

so

$$\operatorname{Tr}(\rho^2)=\frac12.$$

2. Why are the words “population” and “coherence” basis-dependent?

Solution

They refer to diagonal and off-diagonal matrix entries. A change of basis changes which entries are diagonal, so a term that appears as a coherence in one basis may contribute to populations in another.