Stationary Nonequilibrium Entangled States

• Stationary nonequilibrium entangled states are robust quantum states maintained out of Gibbs equilibrium by continuous dissipative influences, exhibiting persistent correlations.
• They underpin advances in open-system dynamics and quantum information processing by leveraging non-Abelian symmetry constraints and engineered environmental interactions.
• Quantitative analyses show that non-Abelian symmetries yield logarithmic entanglement scaling while quantum fragmentation can drive a volume-law behavior in Renyi negativities.

A stationary nonequilibrium entangled state is a long-time, nonthermal, entangled quantum state reached under the continuous influence of dissipative mechanisms or external drivings that maintain the system out of Gibbs equilibrium. Such states arise in open or driven quantum systems or chains that interact with structured environments or are subjected to nonequilibrium boundaries or engineered symmetry constraints. They exhibit persistent correlations, including genuine quantum entanglement between subsystems, even though energy or other quantities may continuously flow through the system.

1. Symmetry Structure and Commutant Algebras

The presence and nature of conservation laws strongly determine the possible structure and entanglement properties of stationary nonequilibrium states. In open quantum evolutions described by unital channels, if the system possesses strong non-Abelian symmetries (e.g., SU(N)), the space of stationary states is governed by the commutant algebra—the set of all operators commuting with the action of the symmetry group. This commutant may be, for example, the universal enveloping algebra (UEA) of a Lie algebra or, in special cases with quantum fragmentation, the so-called Read-Saleur commutant, whose dimension can grow exponentially with system size (Li et al., 12 Jun 2024).

A fundamental distinction emerges:

• Abelian symmetry (e.g., U(1)): The commutant is diagonalizable with a product basis, and all stationary states within each symmetry sector are separable.
• Non-Abelian symmetry (e.g., SU(N)): The commutant has nontrivial, higher-dimensional irreducible representations (irreps). The stationary states in each symmetric subspace (e.g., the singlet subspace) are inherently entangled, as the noncommuting conservation laws forbid any product basis across subsystems.

Quantum fragmentation, as exemplified by the Read-Saleur commutant, further enhances this effect by producing an exponentially large number of dynamically disconnected symmetry sectors, each supporting stationary states with rich entanglement structure.

2. Explicit Entanglement Quantification: Negativities and Scaling Laws

For stationary states restricted to a symmetric subspace (notably the trivial or singlet irrep), exact closed-form expressions for mixed-state entanglement measures are derivable:

• Orthonormal Basis Structure: States take the form

$$|\lambda_{\mathrm{tot}} = 0; \lambda; a, b\rangle = \frac{1}{\sqrt{d_{\lambda}}} \sum_m \eta_{(\lambda, m)} |\lambda, m; a\rangle \otimes |\bar{\lambda}, \bar{m}; b\rangle$$

where $d_{\lambda}$ is the dimension of the irrep $\lambda$, and $a, b$ run over the "bond algebra" multiplicities.

• Logarithmic Negativity:

$$\mathcal{E}_n = \log\left[\frac{1}{D_0^{(L)}}\sum_\lambda d_\lambda D_\lambda^{(L_A)} D_{\bar{\lambda}}^{(L_B)}\right]$$

• Renyi Negativities (for integer $n$):

$$\mathcal{R}_n = -\log\left[\frac{1}{D_0^{(L)}}\sum_\lambda \frac{D_\lambda^{(L_A)} D_{\bar{\lambda}}^{(L_B)}}{d_\lambda^{n-1}}\right]$$

• Generalized Renyi Negativity:

$$\tilde{\mathcal{R}}_n = \frac{1}{2-n}\log\left[\frac{1}{D_0^{(L)}}\sum_\lambda \frac{D_\lambda^{(L_A)} D_{\bar{\lambda}}^{(L_B)}}{d_\lambda^{n-2}}\right]$$

For Abelian symmetries ($d_\lambda=1$), all negativities vanish, indicating separability. For non-Abelian symmetries, these measures are determined entirely by the representation theory of the commutant and the chosen bipartition (Li et al., 12 Jun 2024).

A general upper bound exists for all bipartite entanglement measures:

$$\mathcal{E}_n,\, \mathcal{R}_n,\, S_{\mathrm{op}} \leq \log[\dim(\mathcal{C}_{\min})]$$

where $\mathcal{C}_{\min}$ is the commutant algebra on the smaller subsystem of the bipartition.

3. Scaling Behavior: Logarithmic and Volume Laws

The structure of the commutant determines the scaling of entanglement with system size:

• Abelian Symmetry (U(1), classical fragmentation): Logarithmic negativity and all Renyi negativities vanish—stationary states are separable regardless of subsystem size.
• Non-Abelian Lie Algebras (e.g., SU(2), SU(N)): The irreps have $d_\lambda > 1$. In half-chain bipartitions of length $L$, the logarithmic negativity typically scales as $\mathcal{E}_n \sim c \log L$; for SU(2), $c=1/2$, with higher $c$ for larger $N$.
• Quantum Fragmentation (Read-Saleur Communtants): The commutant has exponentially large dimension in $L$. Here, the logarithmic negativity and generalized Renyi negativities with $n<2$ display volume law scaling (i.e., $\propto L$), while for $n>2$ all Renyi negativities remain logarithmic in $L$. This reveals a novel crossover in the scaling behavior as one varies the Renyi index, reflecting an unusual mixed-state entanglement structure.

The table below summarizes these representative scaling behaviors.

Symmetry/Fragmentation	Scaling of Logarithmic Negativity	Scaling of $\mathcal{R}_n$ ($n>2$)	Scaling of $\tilde{\mathcal{R}}_n$ ($n<2$)
Abelian (U(1)/classical frag)	zero	zero	zero
Non-Abelian (SU(N))	$\propto \log L$	$\propto \log L$	$\propto \log L$
Read-Saleur (quantum frag)	$\propto L$ (volume law)	$\propto \log L$	$\propto L$ (volume law)

4. Quantum Fragmentation and Exponential Commutants

In quantum fragmented systems—where the commutant possesses a Hopf algebra structure with exponential dimension (e.g., Read-Saleur commutants), the stationary state, although mixed and obtained by purely dissipative dynamics, can have entanglement measures (e.g., logarithmic negativity and generalized Renyi negativities for $n<2$) scaling with the total volume. This distinguishes quantum fragmentation sharply from both non-Abelian Lie algebra symmetries (which only allow logarithmic scaling) and classical fragmentation. The scaling crossover at $n=2$ for the Renyi index is a particularly notable feature found only in the quantum fragmented setting.

5. Consequences for Open System Dynamics and Realizations

The presence of strong symmetries, and in particular non-Abelian conservation laws, ensures that even in the presence of unital (highly mixing) channels, open quantum systems can stabilize highly entangled stationary states. The algebraic structure of the commutant directly gives rise to basis states for symmetric subspaces that are maximally entangled across any nontrivial bipartition. These results are robust:

• They apply to any Lindbladian dynamics (or more general open quantum evolutions) that strongly enforces noncommuting symmetries.
• The formulas for entanglement measures hold not only for compact Lie algebras but also for finite groups and quantum groups, provided the infinite-system commutant possesses a Hopf algebra structure (Li et al., 12 Jun 2024).
• In systems with Abelian symmetries or classical fragmentation, no stationary entanglement is possible in any symmetric subspace.
• Volume-law stationary entanglement, otherwise rare in open-system settings, is possible in fragmented systems due to the exponential proliferation of symmetry sectors.

These findings emphasize that the algebraic nature of the symmetry constraints—specifically, the commutant’s structure—fully determines and enables the remarkable entanglement properties of stationary nonequilibrium states in open system dynamics.

6. Broader Impact, Contingencies, and Future Directions

The demonstration that non-Abelian symmetry constraints (or exponential commutants in fragmented systems) can stabilize volume-law or large logarithmic entanglement in stationary states holds significant implications for quantum information processing, dissipative state engineering, and the classification of mixed-state quantum phases. It suggests a path for harnessing symmetry and fragmentation in scalable entanglement generation, independently of fine-tuned coherent controls.

A plausible implication is that future protocols for dissipative entanglement generation in large quantum devices will explicitly exploit non-Abelian symmetry, possibly tuned to exploit quantum fragmentation for maximal entanglement. Open questions include precise operational thresholds for scaling crossovers in Renyi negativities as fragmentation patterns become more complex, and the universality of these scaling laws under less idealized or non-unital system-bath couplings.