Density Operators and Mixed States

A density operator is a positive, trace-one operator that represents a quantum state. It includes pure states described by rays, statistical mixtures of preparations, reduced states of entangled systems, thermal states, and states produced by noisy or open-system dynamics within one common formalism.

Density operators are not optional advanced notation. They are the natural state objects whenever preparation records are incomplete, subsystems are considered locally, outcomes are averaged, or evolution is described by quantum channels rather than one closed-system state vector.

Required background. State Vectors supplies the pure-state language, and Expectation Values supplies the probability calculations generalized here. Familiarity with finite-dimensional traces, eigenvalues, and positive semidefinite matrices is assumed.

What This Chapter Owns

This chapter is the canonical home for

• density operators as general quantum states;
• the pure-versus-mixed distinction and purity criteria;
• preparation ensembles and their nonunique decompositions;
• the trace rule for expectations and measurement probabilities;
• von Neumann entropy at an introductory formal level;
• the operational difference between coherent superpositions and incoherent mixtures.

Within this chapter, the Bloch-ball section gives the complete one-qubit geometry, while the reduced-state section introduces subsystem reduction and purification. Continue through Entangled States and Partial Trace for the composite-system treatment.

Continue to Quantum Operations for channels. Detailed entropy inequalities, master equations, many-body reduced-state hierarchies, and thermal ensembles require additional theory beyond this introductory density-operator treatment.

Definition and Spectral Form

A density operator $\rho$ on a Hilbert space satisfies

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

Positivity means

$$\langle\psi\rvert\rho\lvert\psi\rangle \ge0$$

for every state vector $\lvert\psi\rangle$. It implies that $\rho$ is self-adjoint. In infinite-dimensional Hilbert space, a density operator is also trace class. Its spectral decomposition has the form

$$\rho = \sum_k\lambda_k \lvert k\rangle\langle k\rvert, \qquad \lambda_k\ge0, \qquad \sum_k\lambda_k=1,$$

with the usual functional-analytic qualifications. The eigenvalues therefore form a probability distribution, but they should not automatically be identified with one unique laboratory preparation ensemble.

A density operator is basis independent. A density matrix is its array of components in a chosen basis. Under a unitary change of representation,

$$\rho' = U^\dagger\rho U.$$

Matrix entries change; eigenvalues, trace, positivity, purity, entropy, and all consistently transformed measurement probabilities do not.

Pure and Mixed States

A normalized vector defines the pure-state density operator

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

It is rank one and obeys

$$\rho_\psi^2 = \rho_\psi, \qquad \operatorname{Tr}(\rho_\psi^2)=1.$$

Conversely, a density operator is pure exactly when it is a rank-one projector. A mixed state is a density operator that is not pure. Its purity satisfies

$$\gamma(\rho) \equiv \operatorname{Tr}(\rho^2) \lt1.$$

In a $d$-dimensional space,

$$\frac1d \le \operatorname{Tr}(\rho^2) \le 1,$$

with the lower bound attained by the maximally mixed state $I_d/d$.

Pure states are the extreme points of the convex set of density operators. A mixed state can be written as a nontrivial convex combination of other states. “Mixed” does not mean that the matrix has off-diagonal entries; diagonality depends on basis, while purity does not.

Ensembles Are Not Unique

An ensemble preparation with probabilities $p_i$ and pure states $\lvert\psi_i\rangle$ produces

$$\rho = \sum_i p_i \lvert\psi_i\rangle\langle\psi_i\rvert, \qquad p_i\ge0, \qquad \sum_i p_i=1.$$

This decomposition is generally nonunique. For example, the maximally mixed qubit state can be prepared as an equal mixture of the computational-basis states or as an equal mixture of the two eigenstates of any spin direction. No measurement on that qubit alone distinguishes ensembles that yield the same $\rho$.

The density operator is therefore the operational state for all measurements on the system. It does not, by itself, identify which preparation labels existed. Extra classical records can distinguish preparation procedures even when the unconditional quantum state is the same.

Mixedness can arise because a classical preparation label was ignored or because a subsystem is entangled with another system. These origins matter when additional systems or records become accessible, but the same local density operator gives the same local statistics. The matrix alone does not encode a unique story of its origin.

Trace Rule

For an observable $A$, the expectation value is

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

For a projective outcome $P_a$ and a generalized effect $E_i$,

$$p(a) = \operatorname{Tr}(\rho P_a), \qquad p(i) = \operatorname{Tr}(\rho E_i).$$

When $\rho=\lvert\psi\rangle\langle\psi\rvert$, cyclicity of the trace recovers

$$\operatorname{Tr}(\rho A) = \langle\psi\rvert A\lvert\psi\rangle.$$

The trace rule is basis independent and linear in both the state and the measured effect. That linearity is what makes different ensemble decompositions of one density operator operationally equivalent.

Coherence Versus Mixture

Consider the coherent plus state

$$\lvert+\rangle = \frac{\lvert0\rangle+\lvert1\rangle}{\sqrt2}.$$

Its density matrix in the computational basis is

$$\rho_+ = \frac12 \begin{pmatrix} 1&1\\ 1&1 \end{pmatrix}.$$

The equal incoherent mixture is

$$\rho_{\mathrm{mix}} = \frac12 \lvert0\rangle\langle0\rvert + \frac12 \lvert1\rangle\langle1\rvert = \frac12 \begin{pmatrix} 1&0\\ 0&1 \end{pmatrix}.$$

Both give equal probabilities in the computational basis. They differ in another basis: an $X$-basis measurement gives the plus outcome with probability one for $\rho_+$ and probability one half for $\rho_{\mathrm{mix}}$. The off-diagonal entries in this basis encode coherence relative to the computational alternatives.

Coherence is basis dependent, but the distinction between these two states is not. They have different spectra, purity, and statistics for suitable measurements. A change of coordinates transforms the matrix; physical dephasing changes the state.

Bloch-Ball Representation

Every qubit density operator can be written

$$\rho = \frac12 \left( I+\mathbf r\cdot\boldsymbol\sigma \right), \qquad \lVert\mathbf r\rVert\le1.$$

Pure states lie on the surface $\lVert\mathbf r\rVert=1$, mixed states lie inside, and the maximally mixed state is at $\mathbf r=0$. The purity is

$$\operatorname{Tr}(\rho^2) = \frac12 \left( 1+\lVert\mathbf r\rVert^2 \right).$$

The Bloch ball is complete only for one qubit. Higher-dimensional state spaces are not ordinary balls, and not every vector of generalized expansion coefficients represents a positive density operator.

Reduced States and Purification

For a composite state $\rho_{AB}$, the reduced state

$$\rho_A = \operatorname{Tr}_B(\rho_{AB})$$

contains exactly the information needed for all $A$-local expectation values. A pure entangled $\rho_{AB}$ can yield a mixed $\rho_A$. This does not make the global state mixed.

Conversely, every density operator can be represented as the reduction of a pure state on a larger Hilbert space. If

$$\rho_A = \sum_k\lambda_k \lvert k\rangle\langle k\rvert,$$

one purification is

$$\lvert\Psi\rangle_{AR} = \sum_k\sqrt{\lambda_k} \lvert k\rangle_A \lvert k\rangle_R.$$

Purifications are not unique; sufficiently large purifying systems are related by isometries on the reference system. The Reduced States continuation explains purification scope and subsystem applications; this gateway does not supply a full proof of the purification theorem.

Entropy and Evolution

The von Neumann entropy is

$$S(\rho) = -\operatorname{Tr}(\rho\log\rho) = -\sum_k\lambda_k\log\lambda_k.$$

Pure states have zero entropy. The maximally mixed state in dimension $d$ has entropy $\log d$. The logarithm base fixes the units: base two gives bits, while the natural logarithm gives nats unless a factor such as Boltzmann’s constant is included.

Closed-system unitary evolution acts as

$$\rho(t) = U(t,t_0)\rho(t_0)U(t,t_0)^\dagger.$$

It preserves the spectrum of $\rho$, and therefore preserves purity and von Neumann entropy. Reduced dynamics, measurements with unread outcomes, and general quantum channels can change these quantities. Entropy increase is not automatic for every channel or every subsystem evolution; the physical assumptions must be stated.

Page Map

Question	Core page	Canonical scope
What is a general quantum state?	Density Operators	canonical definition and basic rules
How is the formalism used for composite systems?	Entangled States	pure and mixed separability across a subsystem split
How is a subsystem state obtained?	Partial Trace	canonical reduction map and worked calculations
How do physical processes act on density operators?	Quantum Operations	completely positive maps and channel structure

Suggested Routes

First systematic pass

Begin with State Vectors, then Projectors, Probability Amplitudes, the Born Rule, and Expectation Values. Then read Density Operators for the full state and trace-rule formalism.

Composite-systems route

Continue through Entangled States and Partial Trace. This sequence separates joint-state structure from the operation that produces local states.

Open-systems route

After partial trace, use Quantum Operations for dynamics that need not be unitary on the retained system. No-Broadcasting Theorem is an advanced application of the same density-operator and channel language.

Density-Operator Checks

• Check Hermiticity, positivity, and unit trace.
• Verify eigenvalues are nonnegative and sum to one.
• Confirm that purity lies in its allowed dimension-dependent range.
• Transform density matrices by unitary conjugation under basis changes.
• Test ensemble calculations against the directly constructed density operator.
• Check partial-trace dimensions, trace, and local expectation values.
• Distinguish off-diagonal coherence from basis-independent mixedness.
• In numerics, monitor small negative eigenvalues and trace drift rather than relying only on entrywise inspection.

Common Mistakes

• Treating a density matrix as basis independent entry by entry. The operator is invariant; its matrix representation is not.
• Calling every diagonal matrix mixed. A pure projector is diagonal in its eigenbasis.
• Calling every matrix with off-diagonal entries pure. Coherence and purity are different properties.
• Assuming one ensemble decomposition is physically unique. Many preparations can yield the same density operator.
• Equating mixedness with classical ignorance alone. Entanglement with another system can produce the same local state.
• Inferring entanglement from one mixed reduced state when the global state is mixed. The pure-global-state assumption is essential.
• Forgetting positivity. Hermiticity and trace one do not by themselves define a physical state.
• Assuming all evolution preserves purity or entropy. Only unitary conjugation preserves the density-operator spectrum generally.
• Treating entropy as a complete description of a state. Different spectra can share one entropy value, and eigenvectors also matter for observables.

Cross-Links

• States and Representations
• Probability and the Born Rule
• Composite Systems and Entanglement
• Measurement and Open Quantum Systems
• No-Broadcasting Theorem

References

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