Pure versus Mixed States and the Purity Criterion

Purity is the headline number on every quantum processor's characterization report. State fidelity and purity are what a tomography run quotes after preparing a target state: a purity below $1$ flags decoherence, and its rate of decay is the $T_2$ time that gates a device's usable circuit depth. Purity-based entanglement witnesses turn (1) into a test — for a pure global state, the purity of a subsystem directly measures entanglement (developed in Entanglement Entropy and Measures of Entanglement by name), and randomized-measurement protocols now estimate subsystem purity on hardware without full tomography, a standard tool in cold-atom and superconducting-qubit labs. In many-body physics, the second Rényi entropy $-\ln\mathrm{Tr}\rho^2 = -\ln P$ is measured directly by interfering two copies of a state, making purity an experimentally accessible probe of entanglement growth after a quench. The undergraduate purity dial has become a laboratory instrument.

A pure state is a maximal description: knowing $|\psi\rangle$, quantum mechanics tells you everything it permits. A mixed state carries, on top of that, classical ignorance — and purity measures how much. The extremes are sharp. A pure state has purity $1$: all the "weight" of $\rho$ sits on a single eigenvector, and squaring an idempotent operator changes nothing. The maximally mixed state $\mathbb 1/d$ has purity $1/d$: the weight is spread as thinly as it can be, one $\tfrac1d$ on each of $d$ equal eigenvalues, and squaring each thins it further. Everything else lies between, and purity is the dial reading where. The reason purity is so useful is that it sees only the spectrum. Rotate your basis, relabel your states, apply any unitary — the eigenvalues do not move, so $P$ does not change. This is exactly right physically: a unitary is a reversible change of description, and it cannot make a pure state mixed or vice versa. What can lower purity is an irreversible process that redistributes eigenvalues toward equality — decoherence. So the purity of a laboratory state is a live readout of how much coherence it has lost, and its decay in time is the quantity a $T_2$ measurement reports.

The purity is a scalar built from $\rho$ in two matrix multiplications and a trace, and it depends only on the spectrum: $P = \mathrm{Tr}\,\rho^2 = \sum_k\lambda_k^2$, where the $\lambda_k$ are the eigenvalues (a legal $\rho$ has $\lambda_k\ge0$, $\sum_k\lambda_k = 1$, recalled from the credentials owned by The Density Operator and Mixed States). The whole content of the purity criterion is the range of this functional over the set of legal spectra.

:::derivation[The purity functional theorem — why P lives in [1/d, 1]] We prove that as $\rho$ ranges over all $d$-dimensional density operators, $P = \sum_k\lambda_k^2$ takes every value in $[1/d, 1]$, hitting the ceiling exactly at pure states and the floor exactly at $\mathbb 1/d$. The tool is convexity on the probability simplex $\Delta = \{\lambda_k\ge0,\ \sum_k\lambda_k = 1\}$; because $P$ depends only on the eigenvalues, the problem is entirely about points of $\Delta$.

Step 1. Fix the domain. The eigenvalues of a density operator form a point of $\Delta$, and conversely any point of $\Delta$ is realized by some $\rho$ (diagonal, say). So extremizing $P$ over states is extremizing $\sum_k\lambda_k^2$ over $\Delta$ — a purely classical problem about probability vectors.

Step 2. The maximum is $1$, at the vertices. Since $\lambda_k\le\sum_j\lambda_j = 1$, each $\lambda_k\le1$, hence $\lambda_k^2\le\lambda_k$, so $P = \sum_k\lambda_k^2 \le \sum_k\lambda_k = 1$, with equality iff every $\lambda_k^2 = \lambda_k$, i.e. every $\lambda_k\in\{0,1\}$. On the simplex that forces exactly one eigenvalue equal to $1$ — a vertex, which is $\rho = |\psi\rangle\langle\psi|$, pure. Why this is the idempotence criterion: $\lambda_k^2 = \lambda_k$ for all $k$ is exactly $\rho^2 = \rho$; the two statements $P = 1$ and $\rho^2 = \rho$ are the same fact read on eigenvalues versus operators.

Step 3. The minimum is $1/d$, at the barycenter. $P$ is strictly convex ($\sum\lambda_k^2$ has positive-definite Hessian $2\mathbb 1$), so on the convex set $\Delta$ its minimum is at the unique stationary point of the constrained problem. A Lagrange multiplier for $\sum_k\lambda_k = 1$ gives $2\lambda_k = \mu$ for all $k$, i.e. all eigenvalues equal, $\lambda_k = 1/d$ — the barycenter $\rho = \mathbb 1/d$, maximally mixed, with $P = d\cdot(1/d)^2 = 1/d$. Why equality is forced: any spread away from uniform increases $\sum\lambda_k^2$ (variance of the $\lambda$'s adds to their mean-square), so the flattest distribution is the least pure.

Step 4. Fill the interval by continuity. $P$ is continuous on the connected simplex, so it attains every value between its min $1/d$ and max $1$ — the range is exactly $[1/d, 1]$. The purity dial is complete: $1$ at one end (pure), $1/d$ at the other (maximal ignorance), and a bijective-per-spectrum reading of mixedness in between. Verified numerically: for $d = 2, 3, 4$, pure spectra give $P = 1$, the uniform spectrum gives $P = 1/d$, and 2000 random legal spectra per dimension all yield $P\in[1/d, 1]$.

For a qubit the theorem draws itself, and the drawing is a two-line derivation.

Purity is spectrum-only, and the states with the same purity form a shell, so purity does not uniquely identify $\rho$ — it grades it. Two facts sharpen its role. First, a common confusion: does a dephased state have lower purity than the coherent state with the same populations? The question is malformed, because purity depends on eigenvalues, not diagonal entries in a chosen basis. Dephasing lowers purity precisely because it changes the spectrum — it slides eigenvalues toward each other — not because it "kills off-diagonals" in some fixed basis; the two descriptions are the same event, but the eigenvalue one is the honest cause. Second, purity is the leading approximation to entropy. The linear entropy

is the "impurity," ranging from $0$ (pure) to $1 - 1/d$ (maximally mixed), and it is the first term in the expansion of the von Neumann entropy $S = -\sum_k\lambda_k\ln\lambda_k = -\mathrm{Tr}(\rho\ln\rho)$ about the maximally mixed point. The two agree at the endpoints (both zero for pure, both maximal for $\mathbb 1/d$) and differ in between; the full entropy, with its operational meaning, is owned by Entanglement Entropy and Measures of Entanglement by name. A related graduate application — telling two mixed states apart by measurement — is bounded by the Helstrom limit, a distance on the space of $\rho$'s seeded by the inner-product geometry of Inner Product and Norm of States by name.

Example 1 — three states with identical $\langle\sigma_z\rangle$, sorted by purity (confidence builder). All three have $\langle\sigma_z\rangle = 0$ (so $r_z = 0$), yet they are wildly different states. Rank them.

Step 1. The pure state $|+x\rangle$: $\mathbf r = \hat x$, $|\mathbf r| = 1$, so $P = \tfrac12(1+1) = \boxed{1}$, $S_L = 0$ — pure, on the equator of the Bloch ball. It gives $\langle\sigma_x\rangle = +1$ with certainty.

Step 2. The equal mixture $\tfrac12|+z\rangle\langle+z| + \tfrac12|-z\rangle\langle-z| = \mathbb 1/2$: $\mathbf r = 0$, $P = \tfrac12(1+0) = \boxed{\tfrac12}$, $S_L = \tfrac12$ — maximally mixed, at the center. It gives coin flips on every axis.

Step 3. A tilted mixture with $\mathbf r = (0.5, 0, 0)$: partial coherence along $x$, $|\mathbf r| = 0.5$, so $P = \tfrac12(1 + 0.25) = \boxed{0.625}$, $S_L = 0.375$, eigenvalues $(1\pm0.5)/2 = 0.75, 0.25$ — a real mixture, closer to pure than $\mathbb 1/2$. All three share $\langle\sigma_z\rangle = 0$, but purity orders them $1 > 0.625 > 0.5$: a single $z$-measurement cannot tell them apart, purity instantly can (all machine-verified).

Example 2 — a qubit in a dephasing channel (exam level). Start from the pure coherent state $|+\rangle$ and apply the dephasing channel $\mathcal E_p(\rho) = (1-p)\rho + p\,\sigma_z\rho\,\sigma_z$, which mixes in a phase-flipped copy with weight $p$. Track purity as $p$ runs from $0$ to $1$.

Step 1. Effect on the coherence. $\sigma_z$ flips the sign of the off-diagonal, so $\mathcal E_p$ scales the coherence of $\rho = \tfrac12\begin{pmatrix}1&1\\1&1\end{pmatrix}$ by $(1 - 2p)$, leaving populations fixed: $\rho_p = \tfrac12\begin{pmatrix}1&1-2p\\1-2p&1\end{pmatrix}$.

Step 2. Purity as a function of $p$. The Bloch radius is $|r_x| = |1 - 2p|$, so

At $p = 0$: $P = 1$ (untouched, pure). At $p = \tfrac14$: $P = \tfrac12(1 + \tfrac14) = \boxed{0.625}$ — the half-way point in mixing weight, purity three-quarters of the way down. At $p = \tfrac12$: $P = \tfrac12$ — fully dephased, coherence exactly zero, the maximally mixed $\mathbb 1/2$ (all machine-verified).

Step 3. The subtlety worth banking: at $p = 1$ the channel returns $\sigma_z|+\rangle\langle+|\sigma_z = |-\rangle\langle-|$, purity $1$ again — a pure state, because a deterministic $\sigma_z$ is a reversible unitary, not decoherence. Maximal dephasing sits at $p = \tfrac12$, where the two branches are mixed with equal weight and the coherence cancels; the purity minimum is the honest signature of lost coherence, and it does not live at the endpoint of the mixing parameter but at its symmetric midpoint.

References. Nielsen & Chuang, Quantum Computation and Quantum Information, §2.4, §8.3 (purity, distance measures) · Cohen-Tannoudji, Diu & Laloë, Complement $E_{III}$ · Sakurai & Napolitano, §3.4 · Bengtsson & Życzkowski, Geometry of Quantum States, Ch. 2, 8 · Preskill, Lecture Notes, Ch. 2 · Helstrom, Quantum Detection and Estimation Theory (1976).

Provenance. All results are standard. The purity criterion, Bloch ball, and the three credentials of $\rho$ are owned by The Density Operator and Mixed States (formalism register twin) and recalled by name. The purity functional theorem — $P\in[1/d,1]$, ceiling at pure, floor at $\mathbb 1/d$ — is proved here by convexity and machine-verified for $d = 2,3,4$ (pure $\to1$, uniform $\to1/d$, 2000 random spectra per dimension in-range). The Bloch-radius bridge $P = \tfrac12(1+|\mathbf r|^2)$ is derived from the eigenvalues $(1\pm|\mathbf r|)/2$ and checked on 500 random Bloch vectors. The linear entropy $S_L = 1 - P$ and its relation to the von Neumann entropy are stated; the entropy proper and its operational meaning are owned by Entanglement Entropy and Measures of Entanglement by name (checked: $S_L$ and $S_{vN}$ agree at pure/maximally-mixed, differ at $\lambda = (0.8, 0.2)$: $S_L = 0.32$ vs $S_{vN} = 0.5004$). Worked numbers recomputed: Example 1's three states at $\langle\sigma_z\rangle = 0$ have $P = 1, 0.5, 0.625$; Example 2's dephasing channel gives $P(p) = \tfrac12(1 + (1-2p)^2)$ with the minimum $\tfrac12$ at $p = \tfrac12$ and the $p = 1$ subtlety ($\sigma_z$ is a unitary, purity $1$). Time evolution of purity under decoherence is owned by Properties and Time Evolution of the Density Matrix and Quantum Decoherence and the Density Matrix; the entanglement-as-source-of-local-mixedness story by The Reduced Density Matrix and Partial Trace and Quantum Entanglement and Non-Separability — all by name.

Builds on: The Density Operator and Mixed States · Spin-1/2 and the Pauli Matrices · Inner Product and Norm of States · The Reduced Density Matrix and Partial Trace

Leads to: Entanglement Entropy and Measures of Entanglement · The Schmidt Decomposition · Quantum Entanglement and Non-Separability · The Bell States and Maximally Entangled Pairs