Volume-Law Entangled Phase

• Volume-law entangled phase is a quantum state where entanglement entropy grows linearly with subsystem size, characterized by a nonzero entanglement density.
• It arises in systems such as monitored quantum circuits and frustration-free Hamiltonians, exhibiting sharp transitions to area-law or logarithmic scaling.
• These phases underpin robust quantum thermalization and error correction, serving as key models for information scrambling and many-body coherence.

A volume-law entangled phase characterizes quantum many-body states where the bipartite entanglement entropy of subsystems exhibits extensive scaling with subsystem size. For a region $A$ composed of $|A|$ contiguous sites or degrees of freedom in a large quantum system, the volume-law criterion is $\lim_{|A|\to\infty} \frac{S(A)}{|A|} = s > 0$ for an appropriate entanglement functional $S(A)$ (e.g., von Neumann or Rényi entropy), and a nonzero entanglement density $s$. These phases arise in various contexts: monitored quantum circuits, measurement-only dynamics, highly excited eigenstates, non-Hermitian evolution, and certain ground or excited states in frustration-free Hamiltonians. The volume-law regime is sharply distinguished from area-law ($S_A=O(1)$) and logarithmic-law ($S_A \sim \log |A|$) scaling, with implications for thermalization, quantum information capacity, and many-body quantum criticality.

1. Formal Definition and Canonical Scaling Laws

For a contiguous region $A$ in a quantum system, the bipartite von Neumann entropy is $$S(A) = -\mathrm{Tr} \left[ \rho_A \log \rho_A \right]$$ where $\rho_A$ is the reduced density matrix of $|A|$0. In a volume-law phase, $|A|$1 with $|A|$2 a non-universal parameter dependent on circuit, measurement, or Hamiltonian details (Bao et al., 2021, Sang et al., 2020, Han et al., 2022). This extensive scaling persists for various moments of the entanglement spectrum, including Rényi entropies, in the thermodynamic and long-time limit.

Volume-law behavior is not unique to random/unitary circuits: it also appears in certain frustration-free models in higher spatial dimensions, measurement-only protocols, and specific families of excited eigenstates (Mohapatra et al., 2024, Mukherjee et al., 23 Jan 2025, Zhang et al., 2022, Zhang, 5 May 2025). General forms include

• $|A|$3 ($|A|$4 for KPZ universality corrections in 1D circuits)
• $|A|$5 or $|A|$6 for structured 2D models (Zhu et al., 2023, Zhang et al., 2022)
• $|A|$7 for non-thermal volume-law states with error-correction structure (Ippoliti et al., 2021)

2. Physical Realizations and Dynamical Mechanisms

Monitored Quantum Circuits and Measurement-Induced Transitions

Monitored quantum circuits, built from alternating layers of random local unitaries and projective measurements, generate a rich steady-state entanglement phase diagram as measurement rate $|A|$8 is varied. For $|A|$9, the system approaches a unique steady state with extensive volume-law entanglement, $\lim_{|A|\to\infty} \frac{S(A)}{|A|} = s > 0$0. At higher $\lim_{|A|\to\infty} \frac{S(A)}{|A|} = s > 0$1, area-law behavior ($\lim_{|A|\to\infty} \frac{S(A)}{|A|} = s > 0$2) dominates (Bao et al., 2021, Han et al., 2022).

Hybrid circuits endowed with global symmetries can exhibit multiple distinct volume-law phases, differentiated via order parameters such as discrete symmetry breaking, Edwards-Anderson correlators, or parity variances (Bao et al., 2021, Sang et al., 2020). In 1D systems, these phases can further bifurcate depending on measurement patterns and symmetry structure.

The phase transitions between these regimes are accompanied by universal critical scaling and critical exponents, often matching those of directed percolation or spin-glass transitions (e.g., $\lim_{|A|\to\infty} \frac{S(A)}{|A|} = s > 0$3 for entanglement transition, $\lim_{|A|\to\infty} \frac{S(A)}{|A|} = s > 0$4 for spin-glass universality) (Sang et al., 2020, Côté et al., 2021).

Measurement-Only and Dissipative Constructions

Absence of unitaries does not preclude volume-law scaling: repeated, non-commuting local measurements alone, or suitably engineered dissipative channels with chiral symmetry, can stabilize volume-law entangled pure or mixed states (Bera et al., 17 Sep 2025, Pocklington et al., 2021). In these settings, local one-body measurements in overlapping blocks suffice, while imposing additional kinetic constraints (e.g., three-body checks) can suppress or destroy volume-law behavior.

Dissipative preparation with localized pairing operators and symmetry-constrained nearest-neighbor hoppings can generate “rainbow” volume-law entangled states robust to Hamiltonian integrability breaking (Pocklington et al., 2021).

Random Tensor Networks

Random stabilizer tensor network models in two dimensions, when boundary bond dimension $\lim_{|A|\to\infty} \frac{S(A)}{|A|} = s > 0$5, generate boundary states with $\lim_{|A|\to\infty} \frac{S(A)}{|A|} = s > 0$6 and KPZ correction exponents $\lim_{|A|\to\infty} \frac{S(A)}{|A|} = s > 0$7, $\lim_{|A|\to\infty} \frac{S(A)}{|A|} = s > 0$8. Breaking bulk bonds at random implements entanglement transitions between volume-law and area-law regimes, with criticality governed by percolation universality (Yang et al., 2021).

Non-Hermitian and Non-Unitary Quasicrystals

Certain non-Hermitian extensions of the Aubry-André-Harper model reveal volume-law scaling in steady-state entanglement, strictly when the single-particle spectrum remains delocalized and PT-symmetric. At criticality, the scaling law collapses discontinuously to area-law (Zhou, 2023).

3. Statistical Mechanics and Universality of Fluctuations

Universal subleading corrections in volume-law entangled phases are controlled by effective statistical mechanics mappings:

• Directed Polymer in a Random Environment (DPRE): The minimal cut in a monitored circuit maps to a fluctuating domain wall, whose free energy encodes both mean entanglement and sample-to-sample variance. The fluctuation exponent ($\lim_{|A|\to\infty} \frac{S(A)}{|A|} = s > 0$9) and “wandering” exponent ($S(A)$0) reflect the KPZ universality class (Li et al., 2021, Han et al., 2022, Yang et al., 2021).
• Rare Event and Error Correction Exponents: Rare configurations dominate subleading terms; e.g., the mutual information between a single site and complement decays as $S(A)$1 (point-to-point) in the volume-law phase, indicating robust error-correcting code properties (Li et al., 2021).

This universality extends to both Clifford and Haar-random circuits, and to random unitary circuits subject to depolarizing noise and pinning transitions (Li et al., 2021).

4. Internal Structure, Code Properties, and Quantum Information

Volume-law phases are not merely extensive in entanglement—they also encode robust quantum information properties:

• Error-Correcting Capabilities: The steady-state in the volume-law regime can serve as a dynamical code, with contiguous code distances exhibiting sublinear power-law scaling with system size ($S(A)$2) (Han et al., 2022). The transition to non-error-correcting behavior can be driven by depolarizing noise ("pinning") (Li et al., 2021).
• Decodability Transitions: Within the volume-law regime, there exists a sharp crossover between “decodable” and “scrambled” subphases. The boundary of decodability does not coincide with the measurement-induced entanglement transition, and exhibits mean-field-type universality ($S(A)$3) (Paszko et al., 18 Aug 2025).
• Hierarchical Sector Structure: Certain quenched or time-evolved states display a hierarchy of entanglement transitions as function of Rényi index $S(A)$4: $S(A)$5 (area-law), $S(A)$6 (volume-law), with recursively hidden O(1)-dimensional sectors at lower cuts showing similar transitions at $S(A)$7 (Grover, 6 May 2026).

Table: Error-correction and information-theoretic features in volume-law phases

Architecture	Property	Scaling (1D)
Random Clifford circuit	Decodable volume-law phase	$S(A)$8
Hybrid QA/KPZ circuit	Subleading entropy fluctuations	$S(A)$9
Measurement-only, 1-body	Emergent steady-state coding	$s$0
Dissipative rainbow protocol	Maximally extensive codewords	$s$1

Quantum information in these states is highly scrambled, but not maximally: the entanglement density $s$2 is generically less than the corresponding infinite-temperature “Page” value, except for specially constructed or dimerized cases (Vidmar et al., 2018, Mohapatra et al., 2024).

5. Distinctions, Order Parameters, and Multiple Volume-Law Phases

Volume-law scaling does not uniquely imply a single phase: distinct phases may exhibit the same leading entropy scaling, yet differ in their internal structure, order, and operational properties.

• Symmetry Enriched and Broken Symmetry Phases: Global symmetries (e.g., $s$3, Ising) enable distinct volume-law regimes, classified by long-range order diagnostics (Edwards-Anderson order, parity variance, topological invariants), with transitions demarcated by sharp changes in two-point or multipartite correlators (Bao et al., 2021, Sang et al., 2020).
• Spin Glass vs Paramagnetic Volume-Law Regimes: Clifford circuit ensembles can exhibit phase transitions between paramagnetic and spin-glass volume-law subphases, separated by sharp order parameter jumps and nonanalytic changes in entanglement density slopes (Côté et al., 2021).
• Separable vs Fully Entangled Volume-Law: Non-local monitored circuits support a transition between a “fully entangled” (no pure subsystems) and a “separable” volume-law regime, the latter being a tensor product over O(N) finite clusters. The order parameter distinguishing them is the entangling power: the capacity for external measurements to boost mutual information between finite subsystems (Vijay, 2020).
• Structured Volume-Law in 2D and Higher: Certain measurement-only or frustration-free models exhibit volume-law entanglement with subleading $s$4 or more exotic scaling (e.g., structured KPZ or quantum Lifshitz corrections), indicating nontrivial internal liquid or CFT-like order within the volume-law phase (Zhu et al., 2023, Zhang et al., 2022, Zhang, 5 May 2025).

6. Extensions Beyond One Dimension and Generalizations

Volume-law scaling is not confined to 1D circuits. Recent advances include:

• 2D frustration-free Hamiltonians supporting a tunable transition between area-law, subvolume-law, and full $s$5-volume-law phases via stochastic surface-growth correspondences (Zhang, 5 May 2025).
• Measurement-induced volume-law scaling in 2D monitored stabilizer circuits, where phase transitions and universality match or extend percolation-CFT predictions (Yang et al., 2021).
• Structured volume-law phases in interacting Majorana spin liquids and hybrid models, where the leading entropy is supplemented by $s$6 corrections, with precise spherical phase boundaries in parameter space (Zhu et al., 2023).
• Arbitrary-graph and higher-dimensional scarred eigenstates with extensive entanglement constructed via spectral reflection symmetry and dimerization (Mohapatra et al., 2024, Mukherjee et al., 23 Jan 2025).

7. Implications, Applications, and Outlook

Volume-law entangled phases serve as paradigms for quantum thermalization, information scrambling, and the stability of many-body quantum coherence. Their operational implications are significant:

• Robustness to measurements and emergent error correction are intrinsic to monitored circuit volume-law regimes (Li et al., 2021, Paszko et al., 18 Aug 2025).
• Quantum storage and cryptography: Scrambled volume-law states function as high-rate stabilizer codes, with applications in quantum error correction and message hiding (Paszko et al., 18 Aug 2025).
• Resource states for quantum computation and simulation: Explicit 2D volume-law states constructed from classical stochastic processes serve as benchmarks for tensor-network limits and measurement-based computation (Zhang, 5 May 2025).
• Criticality and universality: Volume-law phases encode nontrivial critical exponents and multifractality at phase boundaries, enabling the identification of quantum critical points and spin-glass universality classes from highly excited state entanglement (Côté et al., 2021, Grover, 6 May 2026).

Volume-law scaling is thus a unifying structural feature across non-equilibrium quantum phases, entanglement transitions, and applications in quantum information science. Ongoing research seeks universal classification schemes for these phases, elucidates the interplay with subsystem symmetries and measurement protocols, and develops explicit experimental protocols for their preparation and characterization.