Conservation Laws and Information Transport

A conservation law creates a slow hydrodynamic mode that retains information about a density long after other local observables have relaxed. It can control long-time tails, front broadening, charge fluctuations, symmetry-resolved entropies, and the operations available to a decoder. It does not force the von Neumann entanglement entropy to grow diffusively: ballistic entanglement can coexist with diffusive energy or charge transport.

Required background. Mutual-information and correlation spreading supplies the direct diagnostics.

Helpful background. Charge-resolved entanglement separates fluctuations among sectors from entanglement within sectors.

Continuity and hydrodynamic scale

For a locally conserved scalar density $q(x,t)$ and current $j(x,t)$ in one dimension,

$$\partial_t q+\partial_x j=0.$$

In a homogeneous rest frame with no coupled sound or advective mode, the leading diffusive constitutive relation is

$$j=-D\partial_x q+\cdots, \qquad \partial_t q=D\partial_x^2q+\cdots.$$

The diffusion constant has units $[D]=L^2/T$. A localized charge $Q$ evolves as

$$q(x,t)=\frac{Q}{\sqrt{4\pi Dt}} \exp\!\left(-\frac{x^2}{4Dt}\right),$$

with

$$\int_{-\infty}^{\infty}q(x,t)\,dx=Q, \qquad \frac1Q\int x^2q(x,t)\,dx=2Dt.$$

In more dimensions, use the appropriate diffusion tensor. If momentum is conserved, sound and heat modes can mix with the charge; if the system is integrable, ballistic Drude contributions can coexist with diffusion. Determine the transport law from densities, currents, and response functions before using it to interpret an information diagnostic.

Conservation also affects operator fronts and long-time tails. Controlled random-circuit and chaotic-chain studies show diffusive structures tied to a $U(1)$ mode Rakovszky, Pollmann, and von Keyserlingk 2018, §§ II–IV. This does not make every operator or correlation front diffusive: components orthogonal to the conserved density can spread ballistically while leaving a diffusive wake.

Exact charge-sector decomposition

Let the global state have fixed total charge and let $Q=Q_A+Q_{\bar A}$. Then the reduced state commutes with the regional charge,

$$[\rho_A,Q_A]=0.$$

With projectors $\Pi_q$ onto eigenspaces of $Q_A$, define

$$p_q=\operatorname{tr}(\Pi_q\rho_A), \qquad \rho_{A,q}=\frac{\Pi_q\rho_A\Pi_q}{p_q}$$

for $p_q>0$. The normalized block decomposition is

$$\rho_A=\bigoplus_q p_q\rho_{A,q}.$$

Its von Neumann entropy is exactly

$$S(\rho_A)=H(\{p_q\})+\sum_qp_qS(\rho_{A,q}), \qquad H(\{p_q\})=-\sum_qp_q\log p_q.$$

The first term is the number or charge-fluctuation entropy; the second averages entropy within fixed regional-charge sectors. Global purity is needed to call $S(\rho_A)$ bipartite entanglement entropy, but not for this block-entropy identity.

For the exact benchmark

$$(p_0,p_1)=\left(\frac14,\frac34\right), \qquad S(\rho_{A,0})=0, \qquad S(\rho_{A,1})=\log2,$$

one finds

$$H(p)=0.562335\ldots, \qquad \sum_qp_qS_q=0.519860\ldots, \qquad S(\rho_A)=1.082196\ldots.$$

Equivalently, a spectrum $(1/4,3/8,3/8)$ has that entropy. This independently checks block normalization and the classical-plus-conditional decomposition.

Goldstein and Sela 2018, Eqs. (1)–(2) develop symmetry-resolved entanglement as a diagnostic of many-body states. A superselection rule adds an operational question: if allowed operations must commute with charge, coherences or resources across sectors may be inaccessible. The block decomposition is a statement about the state; accessible entanglement is a statement about the allowed operation class Wiseman and Vaccaro 2003.

Entropy order matters

A diffusive conserved mode does not determine one universal entropy exponent. In $U(1)$-symmetric random circuits and a diffusive spin chain studied by Rakovszky, Pollmann, and von Keyserlingk 2019, Rényi entropies with $\alpha>1$ show a sub-ballistic $\sqrt t$ contribution governed by rare regions, while the von Neumann entropy $S_1$ remains predominantly linear.

This distinction is physically plausible because higher Rényi entropies weight the largest eigenvalues of $\rho_A$ more strongly. Rare low-entanglement configurations can control them without controlling the full spectral average in $S_1$. Therefore the statement

$$\text{diffusive charge}\quad\Longrightarrow\quad S_A(t)\propto\sqrt t$$

is false for von Neumann entropy in general. Always record the Rényi index.

The converse counterexample is equally important: Kim and Huse 2013 exhibit ballistic spreading of entanglement in a nonintegrable system with diffusive energy transport. The entropy front and the conserved-density mode answer different questions.

Weakly broken conservation

A small symmetry-breaking rate $\Gamma$ gives the reaction–diffusion equation

$$\partial_t q=D\partial_x^2q-\Gamma q.$$

For localized initial charge $Q\delta(x)$,

$$q(x,t)=\frac{Qe^{-\Gamma t}}{\sqrt{4\pi Dt}} \exp\!\left(-\frac{x^2}{4Dt}\right).$$

The total system charge decays as $Qe^{-\Gamma t}$, while the normalized conditional profile still has variance $2Dt$. The crossover occurs near $\Gamma t\sim1$. Data restricted to $t\ll\Gamma^{-1}$ can look exactly conserved even though no asymptotic conservation law survives.

Microscopically, the weak-breaking perturbation may also change the diffusion constant, couple sectors, or open inelastic channels. Fit $D$ and $\Gamma$ jointly and compare with an exactly symmetric control. Do not infer $\Gamma$ merely from a moving contour whose amplitude and width were not separated.

Matched symmetry comparison

Compare two local models at matched energy density, interaction scale, geometry, and preparation: one exactly $U(1)$ symmetric, one with a tunable breaking term. Measure

• the density profile, current, and structure factor;
• total and symmetry-resolved von Neumann and Rényi entropies;
• mutual information and bounded correlators for operators with and without charge overlap;
• operator-front position and diffusive wake;
• recovery performance with unrestricted and symmetry-respecting decoders.

First establish the hydrodynamic mode directly. Then identify which entropy correction, correlation tail, or decoder restriction changes with $\Gamma$. Preserve the same physical time window across models so that a preasymptotic symmetric regime is not compared with a post-crossover nonsymmetric one.

An open system needs an additional statement. The system Hamiltonian may conserve charge while a bath exchanges it. Specify whether the Lindblad operators commute with system charge, whether total system-plus-environment charge is conserved, and whether the record resolves transferred charge. Each choice produces a different sector structure.

The chapter orientation map separates the hydrodynamic effective mode from direct entropy and correlation diagnostics. Its failure controls require symmetry breaking and sector mixing to be varied, while the diagnostic comparison prevents a diffusion constant from being renamed an entanglement velocity.

Common pitfalls

Inferring entropy scaling from charge scaling. Measure each entropy order. Diffusive density and ballistic $S_1$ can coexist.

Leaving sector blocks unnormalized. $\rho_{A,q}$ has unit trace and $p_q$ appears outside it. Otherwise the entropy decomposition is wrong.

Calling a finite-time plateau exact conservation. Tune the breaking rate and collapse data against $\Gamma t$.

Exercises

Prove the entropy decomposition for $\rho_A=\bigoplus_qp_q\rho_{A,q}$.

Solution

If $\lambda_{qj}$ are eigenvalues of normalized $\rho_{A,q}$, then the eigenvalues of $\rho_A$ are $p_q\lambda_{qj}$. Therefore

$$\begin{aligned} S(\rho_A) &=-\sum_{qj}p_q\lambda_{qj} \log(p_q\lambda_{qj})\\ &=-\sum_qp_q\log p_q -\sum_qp_q\sum_j\lambda_{qj}\log\lambda_{qj}\\ &=H(p)+\sum_qp_qS(\rho_{A,q}). \end{aligned}$$

Verify the reaction–diffusion kernel’s total charge and conditional variance.

Solution

The Gaussian integrates to one, leaving total charge $Qe^{-\Gamma t}$. Dividing $q(x,t)$ by that total gives a centered normal distribution with variance $2Dt$. Breaking changes the total weight through $e^{-\Gamma t}$ but not the normalized width in this simple model.

Evaluate the sector benchmark with $p=(1/4,3/4)$ and within-sector entropies $(0,\log2)$.

Solution

The fluctuation term is

$$H(p)=-\frac14\log\frac14-\frac34\log\frac34 \approx0.562335.$$

The conditional term is $(3/4)\log2\approx0.519860$. Their sum is $1.082196$ nats.

References

• Goldstein, Moshe, and Eran Sela. “Symmetry-Resolved Entanglement in Many-Body Systems.” Physical Review Letters 120 (2018): 200602. DOI.
• Kim, Hyungwon, and David A. Huse. “Ballistic Spreading of Entanglement in a Diffusive Nonintegrable System.” Physical Review Letters 111 (2013): 127205. DOI.
• Rakovszky, Tibor, Frank Pollmann, and C. W. von Keyserlingk. “Diffusive Hydrodynamics of Out-of-Time-Ordered Correlators with Charge Conservation.” Physical Review X 8 (2018): 031058. DOI.
• Rakovszky, Tibor, Frank Pollmann, and C. W. von Keyserlingk. “Sub-Ballistic Growth of Rényi Entropies due to Diffusion.” Physical Review Letters 122 (2019): 250602. DOI.
• Wiseman, Howard M., and John A. Vaccaro. “Entanglement of Indistinguishable Particles Shared between Two Parties.” Physical Review Letters 91 (2003): 097902. DOI.