Purified States of Random Matrices

• Purified states of random matrices are rigorously defined by isolating specific spectral and entanglement features to clearly distinguish localized versus extended eigenstates.
• Techniques such as replica formalism, spectral decomposition, and symmetry-based projections enable precise extraction of quantum number subspaces from complex ensembles.
• Static and dynamic purification approaches, including population dynamics and stochastic evolution, enhance the analysis of disordered systems and the quantification of entanglement.

A purified state of a random matrix refers to a rigorous extraction or separation of specific spectral, structural, or entanglement-related features from ensembles of random matrices, often in the context of quantum statistical physics, disordered systems, or quantum information theory. Approaches to purification include the identification of localized spectral modes, the construction of dimension-reduced subspaces with good quantum numbers, or the realization of quantum states with maximally mixed (locally "uninformative") reduced density matrices. Across these domains, purified states play a fundamental role in clarifying the relationship between randomness, symmetry, and structural order in high-dimensional systems.

1. Purification via Spectral Decomposition and Localization

For ensembles of sparse random matrices—such as those defined on graphs with arbitrary degree distributions—purification is naturally linked to the separation of the spectrum into continuous (delocalized) and singular (localized) components. The spectral density

$$\rho_N(\lambda) = \frac{1}{\pi N} \Im\, \mathrm{Tr} \langle (\lambda - i\epsilon - M)^{-1} \rangle$$

can be computed using a Gaussian path integral and the replica formalism. Replica-symmetric solutions reduce the order parameter to a distribution over effective local inverse variances, captured as a superposition of Gaussians determined via population dynamics or fixed-point self-consistency equations.

A crucial insight is the decomposition of the joint distribution $P(a, b)$—where $a$ and $b$ are the real and imaginary parts of certain local field combinations—into

$P(a, b) = P_0(b)\, \delta(a) + \tilde{P}(a, b)$

The delta-function at $a=0$ signals the existence of pure-point spectral weight associated with localized eigenstates, while the regular part describes the continuous (absolutely continuous) band. This separation constitutes a purification of the spectral data, delineating those eigenstates that remain confined (localized) versus those that are extended. The procedure is applicable to matrices on Poissonian, regular, and scale-free graphs (case-dependent via the degree distribution $p_c(k)$), and is further refined by analyzing the inverse participation ratio (IPR) of eigenvectors, which is $O(1)$ for localized states and $O(1/N)$ for extended ones.

In matrices with row constraints (e.g., graph Laplacians), the purification procedure adapts by modifying the couplings and the conjugation relation for auxiliary variables in the self-consistency equations. The interplay of disorder and constraint tunes the emergence and weight of localized, purified states—e.g., affecting Lifshitz tails in the spectral density.

2. Projection and Symmetry: Purification onto Quantum Number Subspaces

In random matrix models with imposed symmetries, purification is achieved by projecting the ensemble onto subspaces characterized by "good" quantum numbers. For example, a Hermitian matrix $H(\phi, \phi')$ invariant under U(1) symmetry is parameterized by

$$H(\phi, \phi') = F(\phi - \phi') = \frac{1}{2\pi}\sum_{m} h_m \cos(m(\phi - \phi'))$$

where the coefficients $h_m$ serve as effective Hamiltonian blocks within angular momentum $m$ subspaces. The variance for $m = 0$ is twice that of $m \neq 0$, favoring low-$m$ ground states upon sampling.

Similarly, under SU(2) symmetry, projection onto total angular momentum $J$ subspaces uses Legendre polynomial expansions; the variance of each block $h_J$ is computed by integrating $F(\omega)$ against $(P_J(\cos \omega))^2$. The larger variance in the $J=0$ sector leads to an enhanced likelihood of finding the ground state there. This symmetry-driven purification yields ensemble states (or spectra) that are "pure" with respect to the good quantum number, and underpins observed phenomena like ground state dominance of $J=0$ in even-even nuclear systems.

3. Entanglement, Reduced States, and Statistical Mechanics of Purification

For random quantum states in large Hilbert spaces, purification is paramount to quantifying entanglement properties. In bipartite systems with a random pure state $|\psi\rangle\in \mathcal{H}_A\otimes\mathcal{H}_B$, the distribution of Schmidt coefficients (the eigenvalues of the reduced density matrix) is fully specified by fixed-trace random matrix models. The average Rényi entropy

$$\langle \mathcal{S}_q\rangle = \frac{1}{1-q}\log\left( N\int_0^1 dx \, x^q \rho_{N, M}(x) \right)$$

(where $N, M$ are subsystem dimensions and $\rho_{N, M}(x)$ is the exact eigenvalue density for the chosen symmetry class) quantifies the degree of purification (entanglement) of the reduced state. Explicit Laplace transform and inverse combinatorial constructions yield closed formulas, with differences between the orthogonal (real, $\beta=1$) and unitary (complex, $\beta=2$) classes manifest in suppressed average entropies for $\beta=1$.

In statistical mechanical treatments, the local purity $\pi = \mathrm{Tr}(\rho_A^2)$ is treated as an energy parameter in a partition function. Both global pure and global mixed state ensembles can be sampled, the latter via "balanced purification"—embedding $\rho$ into a larger system and imposing $\mathrm{Tr}\, \rho^2 = x$. The average local purity is given by

$$\mathcal{M}_1^A(x, 0) = \frac{N_A(N_B^2-1)}{N_A^2 N_B^2 - 1} + \frac{N_B(N_A^2-1)}{N_A^2N_B^2 - 1} x$$

illustrating its dependence on the global purity $x$.

4. Purified Ensembles: Static and Dynamic Approaches

Purified ensembles in the context of random density matrices can be constructed by microcanonical slicing ("static") or by stochastic dynamics ("dynamic"). In the static approach, a convex function (e.g., purity) is fixed: $$P(\vec{\lambda}) \propto \delta\left( \sum_i \lambda_i-1\right)\, \delta\left(\sum_i \lambda_i^2 - P\right)\, \prod_i \lambda_i^{|m-n|}\, \prod_{i<j} (\lambda_i - \lambda_j)^2$$ constructing a measure on the eigenvalues of density matrices with specified purity. This selection purifies the ensemble with respect to the chosen convex statistic.

The dynamic approach evolves an initial product state via a (random) Hamiltonian until a target purity is reached in the reduced system. Agreement between the static and dynamic ensembles is robust in the regime of strong or moderate coupling, revealing the typical spectral features of purified states, notably "cleaved peaks" around degenerate eigenvalues (Werner-like states).

These constructions generalize to other convex criteria such as the von Neumann entropy, with the resulting purified ensembles displaying sharply peaked features in high dimensions.

5. Purification in Random Band Matrices and Localized Regimes

Purification mechanisms are sharply visible in disordered systems, notably 1D or 2D random band matrices (RBMs). In the localized regime (fixed, finite bandwidth), purification manifests through the law of local eigenvalue statistics: the rescaled local spectrum converges to a Poisson point process, with intensity set by the infinite-volume density of states $n_\infty(E_0)$. The limiting DOS itself converges uniformly and smoothly to the semicircle law up to bandwidth-dependent corrections, as rigorously proved using spectral averaging, Wegner and Minami estimates, and localization bounds: $n_n(E) \xrightarrow[N \to \infty]{} n_\infty(E)$ Thus, after purification (removal of finite-volume or disorder-specific noise), the intrinsic (asymptotic) spectral statistics emerge, making the purified picture foundational for understanding Anderson localization and universality in disordered systems.

Advanced techniques—such as supersymmetric functional integrals and transfer operator constructions—allow for refined control over fluctuation corrections and smoothness bounds on the DOS, even in low-dimensional or marginal cases, enabling identification of "almost deterministic" purified spectral states.

6. Multisite and Multipartite Purification: k-Uniform States and Orthogonal Arrays

Multipartite generalizations rely on k-uniform mixed states, where every reduction to k parties is maximally mixed. Purified states in this sense achieve maximal "lack of information" (maximal local entropy) while constraining the global state to have the highest possible purity. Techniques based on orthogonal arrays (OAs) and their orthogonal partitions yield explicit constructions:

• In qubit systems, generators in GF(4) allow combinatorial construction with maximal purity for the given k-uniformity.
• For qudits, the OA formalism ensures that each k-party marginal is locally maximally mixed, and the symmetric mixture over array-partitioned pure states attains the optimal purity.

The construction of such states has direct implications: they are highly entangled, robust to local noise, and are foundational in quantum error correction and quantum cryptographic applications.

7. Outlook and Implications

Purification in random matrix theory encompasses spectral, structural, and operational aspects. It enables the precise separation of localized from extended states, the isolation of symmetry-respecting subspaces, and the systematic construction of maximally uninformative reduced states with maximal global purity. These mechanistic insights facilitate applications from quantum statistical mechanics and quantum information science to the rigorous analysis of localization phenomena and the engineering of exotic multipartite quantum states.

Advances in algorithmic generation (e.g., by population dynamics, ensemble sampling, or combinatorial design) and analytical tools (supersymmetry, fixed-trace random matrix theory, replica and path-integral methods) continue to expand the reach and power of purified state analysis in random matrices and complex quantum systems.