Mutual Information and Correlation Spreading

Mutual information measures every correlation retained by a chosen pair of regions, whereas a connected two-point function tests one declared pair of observables. This page compares the two after an exact local Gaussian phase kick in a regulated free-fermion chain. The calculation exposes a propagating kick–control contrast, but also a nonzero critical-state background and strong estimator dependence. Its contours are neither hard support edges nor signaling tests.

Required background. Entanglement growth supplies Gaussian reduced-state evolution and finite-size controls. Mutual information in QFT supplies the algebraic definition, ultraviolet conditions, and correlation inequality used here.

Helpful background. Front extraction supplies the frozen threshold, fit-window, held-out, and baseline protocol.

Time-dependent mutual information between separated intervals

For a type-I regulated pair of regions $A$ and $B$, with natural logarithms throughout,

$$I(A:B)=S(\rho_A)+S(\rho_B)-S(\rho_{AB}) =D(\rho_{AB}\Vert\rho_A\otimes\rho_B)\geq0.$$

The relative-entropy expression is the primary definition for continuum algebras Araki 1976, § 1, Eqs. (1.1)–(1.2), pp. 809–811. The three-entropy expression used below is legitimate because the lattice fixes a spatial tensor product and a finite ultraviolet regulator. A continuum limit would require separated algebras, compatible regulator removal, and a new convergence analysis; this finite-chain calculation does not take that limit.

For bounded $O_A$ and $O_B$, define

$$C_{AB}=\langle O_AO_B\rangle -\langle O_A\rangle\langle O_B\rangle.$$

Applying Hölder’s inequality and quantum Pinsker to $\delta=\rho_{AB}-\rho_A\otimes\rho_B$ gives

$$|C_{AB}|^2 \leq2\|O_A\|^2\|O_B\|^2 I(A:B).$$

Thus a nonzero bounded correlator certifies nonzero total correlation. The converse fails: a zero selected correlator can simply be blind. For example, the Bell state has $I(A:B)=2\log2$ but $C(X,Z)=0$. Conversely, a classically correlated bit has $I(A:B)=\log2$ and $C(Z,Z)=1$ while remaining separable. Mutual information measures total correlation, not entanglement alone. Wolf et al. 2008, Eq. (5) give the same norm-controlled connection.

An exact Gaussian local kick

Consider an open chain of $L=512$ fermionic sites,

$$H=-J\sum_{j=0}^{L-2} \left(c_{j+1}^{\dagger}c_j+c_j^{\dagger}c_{j+1}\right), \qquad a=J=\hbar=1,$$

initially in its exact half-filled ground state. At site $j_0=L/2$, apply the local number-phase unitary

$$U_\phi=e^{i\phi n_{j_0}}, \qquad \phi=\frac{\pi}{2}.$$

This unitary is generated by a quadratic operator, so it preserves Gaussianity. It is also a genuine local encoding, unlike a normalized annihilation and postselection protocol, whose conditioning can change a critical state nonlocally at the preparation time.

The source interval and right-moving target intervals are

$$A=\{j_0-6,\ldots,j_0+5\}, \qquad B_x=\{j_0+x,\ldots,j_0+x+11\}, \qquad x\geq12.$$

They are disjoint twelve-site fermionic regions; even the nearest pair has six intervening sites, or seven bonds between its facing edges. With $G_{ij}=\langle c_i^\dagger c_j\rangle$, the kick and time evolution are

$$G_\phi(0^+)=D_\phi^\dagger G_0D_\phi, \qquad G_\phi(t)=e^{iht}G_\phi(0^+)e^{-iht},$$

where $(D_\phi)_{jj}=e^{i\phi\delta_{j,j_0}}$ and $h$ is the one-particle hopping matrix. The calculation uses exact spectral evolution rather than time stepping. Eisler 2021, § 2, Eqs. (1)–(8), pp. 3–4 gives the corresponding free-chain covariance and entropy machinery.

This is explicitly a fermionic-site algebra calculation. A Jordan–Wigner string crosses the gap between two disjoint spin blocks, so their XX-spin reduced state is not silently identified with the two-interval fermionic reduced state Fagotti and Calabrese 2010, § 2, Eqs. (11)–(19), pp. 5–6.

Mutual information from the restricted covariance

For any chosen site set $R$, let $G_R$ be the corresponding principal submatrix. Its eigenvalues $\nu_k\in[0,1]$ determine the entropy

$$S(R)=-\sum_k\left[ \nu_k\log\nu_k+(1-\nu_k)\log(1-\nu_k) \right].$$

Restricting separately to $A$, $B_x$, and the noncontiguous union $A\cup B_x$ therefore gives $I_\phi(A:B_x;t)$. This construction follows Peschel 2003, Eqs. (2), (5)–(12), pp. L205–L207.

The comparison observable is the normalized block charge

$$Z_R=\frac1{|R|}\sum_{j\in R}(2n_j-1), \qquad \|Z_R\|=1.$$

For disjoint $A$ and $B$ in a number-conserving Gaussian state, Wick’s theorem gives

$$C_Z(A,B)=\langle Z_AZ_B\rangle_c =-\frac{4}{|A||B|} \sum_{i\in A}\sum_{j\in B}|G_{ij}|^2.$$

The un-kicked ground state is stationary and supplies the matched control. Store the four quantities separately:

$$I_\phi,\quad I_0,\quad \Delta I=I_\phi-I_0, \qquad C_{Z,\phi},\quad C_{Z,0},\quad \Delta C_Z=C_{Z,\phi}-C_{Z,0}.$$

Only $I_\phi$ and $I_0$ are mutual informations. The difference $\Delta I$ is signed, can be negative, and obeys no general positivity or monotonicity theorem. Likewise, the norm inequality applies separately to each raw pair $(I_\phi,C_{Z,\phi})$ and $(I_0,C_{Z,0})$; subtracting those inequalities would be invalid.

At $t=0$, the kick is supported entirely within $A$. Local-unitary invariance therefore gives $I_\phi(A:B_x;0)=I_0(A:B_x)$ for every $x$. Because the phase kick also commutes with every occupation $n_j$, $\Delta C_Z(A,B_x;0)=0$. These are exact controls, not fitted observations.

Frozen mutual-information and density-correlation fronts

Profiles are retained at $tJ=12,16,\ldots,96$. For either signed excess $\Delta Q\in\{\Delta I,\Delta C_Z\}$, define a same-time relative contour by

$$|\Delta Q(x_q,t)| =q\max_x|\Delta Q(x,t)|, \qquad q\in\{0.02,0.05,0.10,0.20\}.$$

The algorithm takes the outermost right-moving crossing and interpolates linearly in $\log|\Delta Q|$ between adjacent sites. Mutual information and block charge are normalized to their own same-time maxima. Equal $q$ therefore compares contour geometry, not absolute sensitivity.

The primary $q=0.10$ fit is frozen on $tJ=16,20,\ldots,56$ and scored without refitting on $tJ=60,64,\ldots,96$. Three candidate trajectories are retained:

$$x(t)=c+vt, \qquad x(t)=c+vt+g t^{1/3}, \qquad x(t)=c+d\sqrt t.$$

The $t^{1/3}$ term is a finite-window lattice-edge candidate, not an assumed universal exponent. The figure asks two linked questions: in panel (a), compare the nonzero critical ground-state baseline with the localized kick–control contrast; in panel (b), check whether predictions frozen before $tJ=60$ reach the untouched late-time contours for both diagnostics.

Exact finite-chain local-kick benchmark. Panel (a) shows the stationary raw mutual-information background and the localized magnitude $|\Delta I(x,tJ=48)|$; the exact local-unitary control has $\max_x|\Delta I(x,0)|$ at numerical zero. Panel (b) shows the separately normalized $q=0.10$ contours. Filled markers were fitted over $16\leq tJ\leq56$ and open markers were untouched over $60\leq tJ\leq96$. The frozen $c+vt+gt^{1/3}$ fits give $v=1.99774$ with held-out RMSE $0.15582$ sites for $|\Delta I|$ and $v=2.00036$ with RMSE $0.07982$ sites for $|\Delta C_Z|$. Pure-$\sqrt t$ fits miss the held-out contours by $18.576$ and $18.526$ sites. The dashed $v_{g,\max}=2$ ray is only the infinite-chain maximum band-group-velocity comparator. The image is quantitative at the declared lattice regulator; it is not an exact support, signaling, or continuum cone.

The rigid ballistic fits have held-out RMSEs $0.24352$ sites for mutual information and $0.44394$ sites for block charge. The subleading $t^{1/3}$ term improves prediction on this window, but it does not identify an asymptotic broadening law. More importantly, both selected contours translate ballistically and decisively reject a purely diffusive $\sqrt t$ trajectory on untouched times.

Pre-existing correlations as the adversary

The half-filled ground state is critical, so its raw mutual information does not begin from zero. In the $L=512$ control, $I_0(A:B_x)$ is approximately $0.03541$, $0.01050$, $0.003357$, and $0.002047$ nats at $x=32,64,128,192$. A raw threshold would therefore report correlation at every one of those separations at $t=0$.

This behavior is not a finite-chain accident. For two intervals of lengths $\ell_A$ and $\ell_B$ separated by a gap $r$ in the massless Dirac vacuum, the exact continuum result implies

$$I_0(A:B)=\frac13\log\!\left[ \frac{(\ell_A+r)(\ell_B+r)} {r(\ell_A+\ell_B+r)} \right] \sim\frac{\ell_A\ell_B}{3r^2}.$$

It is positive at every finite $r$ Casini and Huerta 2009, § 3.1.5, Eqs. (176)–(179), pp. 23–24. The formula is quoted only for that continuum vacuum; it is not used as a finite-chain fit.

Matched subtraction removes this stationary background because the kick and control share the Hamiltonian, initial ground state, lattice, regions, and ultraviolet regulator. It does not turn $\Delta I$ into a new information measure. Contours use $|\Delta I|$ only as a declared estimator and retain its sign in the complete record. General correlation-growth bounds likewise contain a term inherited from the initial state; an exponentially clean correlation cone requires an appropriate clustering hypothesis Kastner 2015, § 5, Eqs. (16)–(24), pp. 7–8; § 6.1, Eqs. (28)–(29), pp. 9–10.

Observable, threshold, and geometry controls

The two diagnostics do not have comparable amplitudes. From $tJ=12$ to $96$, the sampled peak $|\Delta I|$ falls from about $0.3821$ to $0.03828$ nats, whereas the peak $|\Delta C_Z|$ falls from $1.231\times10^{-3}$ to $4.859\times10^{-5}$. A chosen density correlator can therefore be much less sensitive than total correlation even when its normalized contour is sharp.

Changing the analysis also moves the inferred slope. Across advancing training windows, the broadened-fit velocity ranges from $1.98661$ to $2.01389$ for mutual information and from $1.99769$ to $2.04164$ for block charge. The $q=0.02$ contour is visibly more vulnerable to lattice oscillations and low-amplitude tail sensitivity than the primary $q=0.10$ contour. These are deterministic estimator-sensitivity ranges, not confidence intervals.

The $L=384$ control is compared with $L=512$ only through $tJ=64$, before the leading packet can return from an open boundary. At $q=0.10$, the cross-size contour RMSEs are $0.03966$ sites for mutual information and $0.05928$ sites for block charge; the maximum shifts are $0.06250$ and $0.08089$ sites. This supports the reported finite-window contours while leaving the thermodynamic and continuum limits open.

Correlation is not signaling

Neither raw mutual information nor a connected correlator asks whether a receiver can detect a sender’s choice. A signaling test compares two encodings $a$ and $a'$ with one receiver measurement $M_b$:

$$\Delta p(b|a,a')= \operatorname{tr}\!\left[M_b\rho_B^{(a)}(t)\right] -\operatorname{tr}\!\left[M_b\rho_B^{(a')}(t)\right].$$

This is a contrast of receiver states, not a correlation between $A$ and $B$. In a relativistic QFT, microcausality constrains it outside the causal future. In this finite hopping chain, Lieb–Robinson reasoning supplies exponentially small tails rather than an exact relativistic support edge. Bravyi, Hastings, and Verstraete 2006, Eq. (1) and surrounding derivation, p. 050401-2 make the operational distinction explicit.

The maximum single-particle band velocity follows from $\epsilon(k)=-2J\cos k$:

$$v_{g,\max}=\max_k|\partial_k\epsilon(k)|=2J.$$

Agreement of a fitted contour slope with $2J$ is therefore a useful model-specific comparison. It is not a measurement of the Lieb–Robinson velocity, an entanglement velocity, a butterfly velocity, or a communication capacity.

Reproducibility and stop rule

Download the quantitative SVG, complete signed profiles, fits, and controls, compact CSV view used by the figure, and JSON calculation, schema, diagnostics, and claim boundary. The compact view is derived deterministically from the complete record and introduces no fitted value.

What survives every declared control is deliberately narrow: in this exact finite open free-fermion chain, a local unitary changes interval mutual information and a bounded block-charge correlator along estimator-dependent ballistic contours whose fitted slopes lie near the model’s maximum band group velocity. The calculation stops short of exact support, a common diagnostic speed, an asymptotic exponent, a continuum limit, interacting universality, or signaling.

The chapter orientation map places both quantities among direct diagnostics. Its failure controls require baseline, observable, threshold, and window changes, while the diagnostic comparison keeps total correlation distinct from entanglement and communication.

Common pitfalls

Treating $\Delta I$ as mutual information. Each raw $I$ is nonnegative; their difference need not be. Keep the sign and call it an excess or contrast.

Applying Pinsker after subtraction. Pinsker bounds a raw relative entropy and raw connected correlator. It does not survive term-by-term subtraction into a bound between $\Delta I$ and $\Delta C_Z$.

Using one weak correlator as a no-correlation theorem. A nonzero bounded correlator lower-bounds $I$, but a small or vanishing selected correlator does not upper-bound it.

Calling a relative threshold a detection probability. The contour is normalized to the same-time sampled peak of one diagnostic. It is an estimator, not an experimental noise model or confidence level.

Renaming the $v_{g,\max}$ comparator. A near-$2J$ contour in this free lattice is not an exact support, signaling, entanglement, butterfly, hydrodynamic, or universal velocity.

Exercises

Derive the raw connected-correlator bound from quantum Pinsker and Hölder’s inequality. Why can it not be applied to $\Delta I$ and $\Delta C_Z$?

Solution

Set $\delta=\rho_{AB}-\rho_A\otimes\rho_B$. Then

$$C_{AB}=\operatorname{tr}[\delta(O_A\otimes O_B)].$$

Hölder gives $|C_{AB}|\leq\|\delta\|_1\|O_A\|\|O_B\|$, while Pinsker gives $I(A:B)\geq\|\delta\|_1^2/2$. Combining them yields

$$|C_{AB}|^2\leq 2\|O_A\|^2\|O_B\|^2I(A:B).$$

The proof uses one density operator $\rho_{AB}$ and its own product of marginals. A kick-minus-control difference is not a relative entropy between those objects, and subtracting two valid inequalities does not preserve their order.

Compare a classically correlated bit with a Bell pair. For $\rho_{\rm cl}=(|00\rangle\langle00|+|11\rangle\langle11|)/2$, compute $I$ and $C(Z,Z)$. For $|\Phi^+\rangle$, compute $I$ and $C(X,Z)$. What do the two examples rule out?

Solution

For $\rho_{\rm cl}$, both marginals and the joint state have entropy $\log2$, so $I=\log2$. The one-point functions vanish and $\langle Z\otimes Z\rangle=1$, hence $C(Z,Z)=1$. The state is a convex mixture of product states, so neither value certifies entanglement.

For $|\Phi^+\rangle$, the joint state is pure and both marginals are maximally mixed, giving $I=2\log2$. Direct evaluation gives $C(X,Z)=0$. A chosen correlator can therefore miss total correlation completely.

For two single fermionic sites with restricted covariance

$$G_{AB}=\begin{pmatrix} 1/2&r\\ r&1/2 \end{pmatrix}, \qquad |r|\leq\frac12,$$

derive the mutual information and evaluate it at $r=1/4$. Check the raw block-charge correlator bound.

Solution

Each one-site covariance has eigenvalue $1/2$, so $S(A)=S(B)=\log2$. The joint eigenvalues are $1/2\pm r$. With $h(p)=-p\log p-(1-p)\log(1-p)$,

$$I=2\log2-h(1/2+r)-h(1/2-r).$$

At $r=1/4$, $I\approx0.261624$ nats. For one site per region, $C_Z=-4|r|^2=-1/4$ and $\|Z_A\|=\|Z_B\|=1$. Thus $|C_Z|^2=0.0625\leq2I\approx0.523248$, as required.

Show that the phase kick leaves $I(A:B_x)$ and $C_Z(A,B_x)$ unchanged at $t=0$. Then explain why a nonzero raw ground-state $I_0(A:B_x)$ still makes baseline subtraction necessary for a front plot.

Solution

Because $j_0\in A$, the kick factorizes as $U_A\otimes\mathbf1_B$ on $A\cup B$. It conjugates $\rho_A$ and $\rho_{AB}$ without changing their spectra, and it leaves $\rho_B$ unchanged. Therefore $I_\phi(A:B;0)=I_0(A:B)$. The kick is generated by $n_{j_0}$ and commutes with all occupations, so every equal-time density moment, including $C_Z$, is also unchanged.

Nevertheless, $I_0(A:B_x)$ is positive at finite separation in the critical ground state. A raw contour would then be present before the intervention. The matched difference removes that stationary background while remaining a signed contrast rather than a new mutual information.

The primary mutual-information contour has a frozen broadened-ballistic held-out RMSE of $0.15582$ sites and a pure-$\sqrt t$ RMSE of $18.576$ sites. The fitted slope is $1.99774$, while $v_{g,\max}=2$. Which conclusions follow, and which do not?

Solution

On the declared finite chain, threshold, and time window, the untouched late-time contour is accurately predicted by a ballistic trajectory with a subleading correction and is incompatible with the fitted purely diffusive trajectory. Its slope is close to the maximum band group velocity.

The comparison does not establish a hard support edge, a Lieb–Robinson, signaling, entanglement, or butterfly velocity, equality with the block-charge contour, a universal $t^{1/3}$ correction, an interacting result, or a continuum limit. Those claims require different observables, limits, or theorems.

References

• Araki, Huzihiro. “Relative Entropy of States of von Neumann Algebras.” Publications of the Research Institute for Mathematical Sciences 11 (1976): 809–833. DOI.
• Bravyi, Sergey, Matthew B. Hastings, and Frank Verstraete. “Lieb–Robinson Bounds and the Generation of Correlations and Topological Quantum Order.” Physical Review Letters 97 (2006): 050401. DOI.
• Casini, Horacio, and Marina Huerta. “Entanglement Entropy in Free Quantum Field Theory.” Journal of Physics A: Mathematical and Theoretical 42 (2009): 504007. DOI.
• Eisler, Viktor. “Entanglement Spreading after Local and Extended Excitations in a Free-Fermion Chain.” Journal of Physics A: Mathematical and Theoretical 54 (2021): 424002. DOI.
• Fagotti, Maurizio, and Pasquale Calabrese. “Entanglement Entropy of Two Disjoint Blocks in XY Chains.” Journal of Statistical Mechanics: Theory and Experiment 2010 (2010): P04016. DOI.
• Kastner, Michael. “Entanglement-enhanced spreading of correlations.” New Journal of Physics 17 (2015): 123024. DOI.
• Peschel, Ingo. “Calculation of Reduced Density Matrices from Correlation Functions.” Journal of Physics A: Mathematical and General 36 (2003): L205–L208. DOI.
• Wolf, Michael M., Frank Verstraete, Matthew B. Hastings, and J. Ignacio Cirac. “Area Laws in Quantum Systems: Mutual Information and Correlations.” Physical Review Letters 100 (2008): 070502. DOI.