Mutual Information and Regulator-Independent Correlations

Mutual information measures the total correlation visible to two declared algebras. For separated QFT regions it can remain finite after the local entropy divergences cancel, and it bounds connected correlations. It is neither automatically an entanglement measure nor regulator independent when the regions touch.

Required background. Use Relative Entropy for QFT States. Helpful background. Cluster decomposition controls large separation, while regulated subregion entropy explains the cancellation formula.

Algebraic and regulated definitions

For commuting algebras $\mathfrak A$ and $\mathfrak B$ with a well-defined product state $\omega_A\otimes\omega_B$, define

$$I_\omega(A{:}B) =S\!\left(\omega_{AB}\Vert\omega_A\otimes\omega_B\right).$$

In a type-I regulator this equals

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

The algebraic definition is primary in the continuum. For regions at positive separation, split or regulator constructions often give the same finite limit because local boundary divergences cancel. When boundaries touch, new contact divergences can appear and the limit must be reconsidered.

The structural figure places mutual information on the correlation branch of relative entropy.

Mutual information compares the joint restriction with its product of marginals. Its finiteness and operational meaning depend on the two algebras and their separation; it does not by itself quantify distillable entanglement. Schematic.

Correlation bounds and separation

For bounded $X\in\mathfrak A$ and $Y\in\mathfrak B$, Pinsker’s inequality implies a bound of the form

$$\frac{\lvert\omega(XY)-\omega(X)\omega(Y)\rvert^2} {2\lVert X\rVert^2\lVert Y\rVert^2} \leq I_\omega(A{:}B),$$

up to the chosen logarithm convention. Thus small mutual information forces every bounded connected correlator to be small. The converse is not automatic because a set of low-order correlators need not determine the full joint state.

For two separated intervals in a free field, compute the three regulated entropies with one lattice, boundary prescription, and center choice, then extrapolate their combination. At large separation, cluster decomposition and the lightest exchanged operators control the decay. Wolf et al. 2008, pp. 070502-1–070502-4 made the general correlation-bound relation explicit for quantum many-body systems.

Contact and algebra dependence

The failure map records why mutual information is not a property of two geometric sets alone.

Separated regions with matched local subtractions can yield a regulator-independent limit. Touching boundaries, changing a gauge-theory center, or using different regulators in the three entropy terms changes the quantity and can spoil cancellation. Schematic.

Because separable states can have nonzero classical correlation, $I(A{:}B)>0$ does not certify quantum entanglement. Conversely, statements about distillation require an allowed local-operation class and resources beyond the mutual-information value.

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. Open preprint.

Further reading

• Araki, Huzihiro. “Relative Entropy of States of von Neumann Algebras.” Publications of the Research Institute for Mathematical Sciences 11 (1976): 809–833. DOI.