How to Distinguish Pure, Mixed, Entangled, and Separable States

Purity and entanglement are different questions. Purity asks whether a state is represented by one state vector or by a statistical mixture. Entanglement asks whether a state of several subsystems can be prepared as a product state, or as a mixture of product states. A state can be pure and entangled, mixed and separable, or mixed and entangled.

For an unknown system, these properties are generally inferred from measurements on many identically prepared copies. A single measurement on one copy usually cannot settle either question.

Test whether a state is pure or mixed

Represent the state by its density operator $\rho$. It is positive, has trace one, and has eigenvalues $p_i$ satisfying $p_i\geq 0$ and $\sum_i p_i=1$. Its purity is

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

The state is pure exactly when one eigenvalue is 1 and the others are 0, so $\operatorname{Tr}(\rho^2)=1$. It is mixed when $\operatorname{Tr}(\rho^2)<1$. Equivalently, a pure density operator has rank one and can be written $\rho=|\psi\rangle\langle\psi|$.

In an experiment, you usually do not know $\rho$ in advance. You can perform quantum state tomography: measure a suitable set of observables on many copies, estimate the density operator, and calculate its purity. The estimate has statistical uncertainty, especially with limited data. Purity can also be estimated by suitable measurements on two copies, including a measurement related to the SWAP operator.

Test whether a bipartite state is entangled

First specify the division into parties, such as two qubits with state space $\mathcal{H}_A\otimes\mathcal{H}_B$. A pure bipartite state is separable if it factors as $|a\rangle\otimes|b\rangle$. For a mixed bipartite state, separability means that it can be written as

$$\rho_{AB}=\sum_k q_k\,\rho_A^{(k)}\otimes\rho_B^{(k)}, \qquad q_k\geq 0,\quad \sum_k q_k=1.$$

If no such decomposition exists, the state is entangled. A mixed separable state need not be one product state; it may be a classical mixture of several product states.

For a pure bipartite state, a convenient test is to trace out one party. The state is entangled exactly when either reduced state, such as $\rho_A=\operatorname{Tr}_B(\rho_{AB})$, is mixed. In other words, $\operatorname{Tr}(\rho_A^2)<1$ signals entanglement for a pure joint state. This shortcut does not apply to mixed joint states: a mixed reduced state alone does not prove that the joint state is entangled.

For mixed states, useful tests include:

• Partial transpose (PPT) test: Take the transpose on one party, giving $\rho_{AB}^{T_B}$. If this operator has a negative eigenvalue, the state is entangled. The test is both necessary and sufficient for two qubits and for systems of dimensions $2\times 3$. In larger dimensions, a positive partial transpose does not rule out entanglement.
• Entanglement witnesses: An observable $W$ is chosen so that all separable states satisfy a bound, such as $\operatorname{Tr}(W\rho)\geq 0$. A measured value violating that bound proves entanglement. A witness detects states it was designed for, not necessarily every entangled state.
• Specialized measures and criteria: For two qubits, concurrence can quantify entanglement. Bell-inequality violations can demonstrate nonclassical correlations, but not every entangled state violates a given Bell inequality.

Tomography followed by a suitable criterion is a general practical route, though full tomography can be costly for large systems. If you only need to establish entanglement, a well-chosen witness can require fewer measurement settings.

Worked example: two states with the same local states

Compare the Bell state $|\Phi^+\rangle=(|00\rangle+|11\rangle)/\sqrt{2}$ with the mixed state

$$\rho_{\mathrm{mix}}=\frac12|00\rangle\langle00|+\frac12|11\rangle\langle11|.$$

For the Bell state, $\rho_{\mathrm{Bell}}=|\Phi^+\rangle\langle\Phi^+|$, so $\operatorname{Tr}(\rho_{\mathrm{Bell}}^2)=1$: it is pure. Tracing out either qubit gives $I/2$, whose purity is $1/2$. Since the overall state is pure but its reduced state is mixed, the Bell state is entangled.

For $\rho_{\mathrm{mix}}$, the two nonzero eigenvalues are $1/2$ and $1/2$, so its purity is $1/2$. It is mixed, but it is separable by its displayed decomposition into product states. Both examples have the same reduced state $I/2$ for each qubit. Thus, measurements on either qubit alone cannot distinguish these joint states; correlations between the parties matter.

The partial transpose also distinguishes them: the Bell state’s partial transpose has eigenvalues $1/2,1/2,1/2,-1/2$, so it is entangled. The partial transpose of $\rho_{\mathrm{mix}}$ is positive, consistent with separability.

Common mistakes

• Treating mixed as synonymous with separable. Mixed states can be entangled, and pure states can be entangled too, as the Bell-state example shows.
• Using reduced-state mixedness as a general entanglement test. It is decisive for a pure bipartite joint state, but not for an arbitrary mixed joint state.
• Assuming a positive partial transpose proves separability in every dimension. That conclusion is valid for $2\times2$ and $2\times3$, not in general.
• Expecting one measurement to classify an unknown state. Measurements have probabilistic outcomes. Reliable state estimates or witness tests require repeated preparations and careful treatment of statistical error.