Stability Rules for Complex Systems Revealed by New Information Theory Limits

Researchers investigating the behaviour of matter far from equilibrium have established new constraints on system stability and the pathways along which these systems evolve. Yu-Hsueh Chen and Tarun Grover, both from the Department of Physics at the University of California at San Diego, demonstrate these constraints using information inequalities and the concept of conditional mutual information. Their work reveals a non-perturbative stability criterion linked to this mutual information, suggesting that systems cannot spontaneously transition from more stable, low-correlation states to less stable, high-correlation states. Furthermore, they apply these findings to diverse scenarios, ranging from classical symmetry breaking to two-dimensional criticality, and offer insights into the behaviour of classical nonequilibrium steady states, representing a significant advance in understanding the fundamental principles governing systems driven out of equilibrium.

This work centres on conditional mutual information, or CMI, a measure quantifying correlations between spatially separated regions independent of their surroundings.

Assuming CMI remains finite at ultraviolet (UV) cutoffs, the study demonstrates that the scaling function associated with CMI decreases monotonically along the renormalization group flow. This finding establishes a non-perturbative stability criterion, asserting that a fixed point exhibiting lower CMI cannot be destabilised towards one with higher CMI.
Furthermore, the research bounds the CMI of a convex mixture of states by the CMI of its individual components. This inequality is then applied to demonstrate the perturbative stability of states undergoing spontaneous symmetry breaking against channels that explicitly disrupt this symmetry. These constraints are illustrated through examples encompassing decoherence-driven transitions in classical symmetry-broken states, strong-to-weak symmetry breaking criticality in two dimensions, and even transitions observed in pure quantum states.

The central focus of this study is conditional mutual information, defined as I(A: C|B) = S(AB) + S(BC) −S(B) −S(ABC), where S(X) represents the von Neumann entropy for the density matrix associated with region X. CMI, inherently positive due to strong subadditivity, captures correlations between regions A and C that are not mediated by an intermediary region B.

In many systems, CMI diminishes exponentially with the separation between A and C, exhibiting a characteristic decay length known as the Markov length, denoted as ξM. The work reveals that the scaling function associated with CMI is monotonic along the RG flow, implying a crucial stability criterion. A fixed point with smaller CMI cannot be destabilized toward one with larger CMI, offering a novel perspective on the behaviour of nonequilibrium systems. This constraint, derived from information inequalities, provides a powerful tool for understanding the landscape of phenomena in systems evolving away from equilibrium.

Entanglement quantification via conditional mutual information and spatial partitioning

Conditional mutual information (CMI), quantifying correlations between spatially separated regions, serves as the central tool in this work to derive constraints on renormalization group (RG) flows and stability in nonequilibrium systems. The research begins by partitioning a system in d spatial dimensions into regions A, B, and C, with B acting as a separating slab of size lB×ld−1 between A and C, both of equal size.

Open boundary conditions are imposed along one direction, with translational invariance assumed in the bulk, and boundary effects neglected in the limit of large system sizes. Von Neumann entropy, denoted as S(X) for density matrix ρX associated with region X, is used to define CMI as I(A: C|B) = S(AB) + S(BC) −S(B) −S(ABC).

Strong subadditivity (SSA) implies that CMI monotonically decreases with increasing separation lB, expressed as I(l|lB + 2∆, t) ≤I(l|lB, t) for all ∆ 0. This inequality stems from the fact that the difference in CMI values can itself be expressed as a non-negative CMI. Considering the limit where the ratio of system size l to separation lB approaches infinity, and l|t|ν approaches infinity, the CMI I(∞|lB, t) may depend on two dimensionless variables.

Assuming CMI is finite in the scaling limit, it can depend only on a single variable, x = lB t1/ν, leading to the key result: df(x) d|x| ≤0, where f(x) represents the CMI. This constraint implies a non-perturbative stability criterion; a fixed point with smaller CMI cannot be destabilized towards one with larger CMI.

The study further bounds the CMI of a convex mixture of states by the CMI of its individual components, enabling the inference of perturbative stability of spontaneous symmetry breaking states against symmetry-breaking perturbations. In one spatial dimension, application of SSA demonstrates that the CMI can only increase as the system size l increases at a fixed separation lB, implying a non-negative anomalous dimension if the CMI takes a specific form. The research anticipates that CMI remains UV finite in systems described by local space-time actions, mirroring evidence from ground states of quantum field theories and extending to nonequilibrium systems governed by local actions like the Martin-Siggia-Rose-Janssen-De Dominicis functional.

Conditional mutual information bounds stability in nonequilibrium systems

Researchers established constraints on renormalization group flows and stability in nonequilibrium systems utilising information inequalities. Conditional mutual information, or CMI, was central to this work, quantifying correlations between spatially separated regions independent of their surroundings.

Assuming CMI is ultraviolet finite, the scaling function associated with CMI demonstrates monotonic behaviour along the renormalization group flow. This implies a non-perturbative stability criterion whereby a fixed point exhibiting smaller CMI cannot be destabilised towards one with larger CMI. Furthermore, the study bounded the CMI of a convex mixture of states in terms of the CMI of its individual components.

This inequality was then applied to infer perturbative stability of spontaneous symmetry breaking states against symmetry-breaking channels. Investigations encompassed decoherence-driven symmetry breaking in classical states, strong-to-weak symmetry breaking criticality in two dimensions, and even pure states, alongside implications for classical nonequilibrium steady states.

In many systems, CMI decays exponentially with the separation between regions A and C, following the relationship I ∼e−lB/ξM, where ξM represents the Markov length. Recent discussion highlights the essential role of Markov length in defining mixed state phases of matter. Notably, Gibbs states of local Hamiltonians at non-zero temperature exhibit finite Markov length, and exactly zero for commuting, local Hamiltonians.

The research presents two key results: an inequality for CMI along a renormalization group flow and a bound on the CMI of convex mixtures. Specifically, the study demonstrates that CMI is monotonically decreasing with increasing separation length lB: I(l|lB + 2∆, t) ≤I(l|lB, t) for all ∆ 0. Considering the limit where l/lB approaches infinity and l|t|ν approaches infinity, the CMI I(∞|lB, t) depends on two dimensionless variables, but assuming UV finiteness, it depends only on x, where x = tl1/ν B.

This leads to the finding that the derivative of CMI with respect to the absolute value of x is less than or equal to zero. In one spatial dimension, the research shows that the CMI can only increase as length l increases at a fixed lB, implying a non-negative anomalous dimension η. For a Gibbs state perturbed by a channel of strength p, the CMI decays as I ∼I0e−lB/ξ, where ξ is the Markov length and I0 is a finite constant. Consequently, perturbations of this fixed point cannot destabilise the system towards a state with nonzero CMI.

Conditional mutual information governs renormalization group stability and symmetry breaking

Researchers have established constraints on renormalization group flows and stability in systems far from equilibrium, utilising information inequalities to quantify correlations between spatially separated regions. These constraints centre on conditional mutual information, a measure of correlation not mediated by the surrounding environment, and demonstrate that its scaling function diminishes monotonically along the renormalization group flow.

This monotonicity implies a crucial stability criterion: a fixed point exhibiting lower conditional mutual information cannot be destabilized towards a state with higher values. Furthermore, the work bounds the conditional mutual information arising from mixtures of states based on the conditional mutual information of the individual components.

This inequality was then applied to demonstrate perturbative stability of states undergoing spontaneous symmetry breaking when subjected to symmetry-breaking influences. Examples illustrating these constraints include classical symmetry-broken states subject to decoherence, strong-to-weak symmetry breaking criticality in two dimensions, and even pure states, alongside implications for classical nonequilibrium steady states.

The validity of these findings relies on the assumption that conditional mutual information remains finite at ultraviolet cutoffs, a condition generally met by systems describable with a local spacetime action. The authors acknowledge that systems lacking a conventional thermodynamic limit, such as those exhibiting fractonic or subsystem symmetry, fall outside the scope of this analysis. Future research could explore the application of these constraints to more complex nonequilibrium systems and investigate the behaviour of conditional mutual information in systems where the ultraviolet cutoff is not finite, potentially extending the understanding of stability and RG flows in a broader range of physical scenarios.