Mutual Information and Correlation Spreading

Mutual information measures all correlations accessible to two chosen subregion algebras, while a connected correlator tests one pair of observables. Tracking both after a localized excitation distinguishes a blind operator from a genuinely absent correlation signal. Neither quantity alone is a universal measure of causal influence.

Required background. Entanglement growth supplies the evolving reduced states, and mutual information in QFT supplies the finite relative-entropy definition.

Helpful background. Front extraction supplies threshold and window controls.

Two correlation diagnostics

For disjoint regions $A$ and $B$,

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

Leading local boundary divergences cancel at nonzero separation. For bounded operators $O_A,O_B$, Pinsker-type reasoning bounds the connected correlator by mutual information:

$$\left|\langle O_AO_B\rangle -\langle O_A\rangle\langle O_B\rangle\right|^2 \le 2\|O_A\|^2\|O_B\|^2 I(A{:}B).$$

Thus a nonzero connected correlator certifies nonzero mutual information, but a zero correlator for one operator pair does not imply $I=0$. For unbounded field operators, smear them and establish an energy or norm cutoff before using this bound.

Wolf et al. 2008, Theorem 1 and Eqs. (4)–(6) use mutual information to connect total correlations and area-law structure.

Localized Gaussian excitation

Prepare a finite-energy wavepacket near $A$ in a Gaussian field and evolve it toward a separated region $B$. At each time compute:

• $I(A{:}B;t)$ from covariance-matrix symplectic spectra;
• connected field and momentum correlators with declared smearing;
• the change relative to the unexcited state;
• an intervention contrast if signaling is the question.

Extract arrival contours using the same threshold protocol. If the field correlator is blind because the excitation is momentum-like, the momentum correlator or mutual information may still respond. Conversely, pre-existing vacuum mutual information may be nonzero before the excitation; use $\Delta I$ or a matched control rather than calling the baseline a transmitted signal.

Mutual information and connected correlators occupy the direct-diagnostic layer. Their comparison helps identify operator blindness and pre-existing correlations without assigning either a universal transport velocity. The map is schematic.

Long-range initial correlations

Critical and long-range-entangled initial states can have algebraically decaying $I(A{:}B;0)$. A quench changes that correlation pattern, but nonzero spacelike $I$ is compatible with microcausality. To claim transported information, specify a local encoding choice and show that receiver statistics depend on it only inside the causal future.

An apparent early front can be a nonzero initial correlation, a threshold artifact, or a response of only one chosen operator. Matched initial-state and observable controls distinguish these cases. The map is schematic.

Common pitfalls

Equating correlation with signaling. Mutual information can be nonzero in the vacuum. Signaling requires dependence on a localized intervention choice.

Using unsmeared fields in a norm bound. Local fields are unbounded distributions. Smear and regulate them before applying finite-operator inequalities.

References

• 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.