Crossed product algebras and generalized entropy for subregions

We thank the referee again for their extremely careful reading of the manuscript. We believe there are two distinct issues being conflated here: the issue of regularization (e.g., by $1/N$) and the issue of renormalization (e.g., by vacuum subtraction).

The factor of $1/N$ arises in holography because the thermofield double state belongs to the canonical ensemble. There, thermal fluctuations scale like $N^2$, hence the need to work perturbatively with $1/N$ as a regularizer. In contrast, thermal fluctuations in Rindler space die out at infinity, so there is no need for this factor (see the penultimate paragraph on page 21 and surrounding discussion). Extensive discussion of this issue can be found in 2209.10454 (ref. [34] of this version), where the authors considered a microcanonical version of the TFD. There the fluctuations are $O(1)$, so the regularization by $1/N$ is similarly not required.

The vacuum subtraction (by which we meant the standard normal ordering prescription in our previous reply) is a separate issue, and is infinite even at finite $N$ (cf. eq. (2.2) of 2112.12828 (ref. [27] in this version), where the Hamiltonian has a factor of $N$ outside a divergent integral). In analogy with a more pedagogical example, the Hamiltonian VEV here suffers from a divergence with $N$ given that it scales as $N^2$, and the usual divergence arising from the integral of the Hamiltonian density that resembles the familiar divergence for a free scalar field. This makes the vacuum subtraction used in previous approaches purely formal, since -- as the referee correctly points out -- there is no known way to rigorously define this without resorting to some regularization scheme. (We emphasize again that here we refer to some hypothetical regularization scheme for the infinite vacuum divergence, not the thermal fluctuations regularized by $1/N$ above.)

In our approach, we do *not* use this subtraction scheme, for reasons discussed in the two paragraphs immediately below eq. (3.8). We also point out the possible shortcoming of this regularization in large-$N$ gauge theory under eq. (4.6). In contrast, our choice has the advantage of avoiding the question of regularization/renormalization raised by the referee, though it is still necessary to work formally. In either case, the (state-independent) shift in the vacuum energy gets absorbed into the constant $c$, cf. the penultimate paragraph on page 11.

To more accurately reflect the need to work formally, we propose to slightly soften the claim highlighted by the referee to "*formally* paving the way to the study of entanglement entropy..." We would also like to emphasize that our construction (as other works do) relies on the fact that the crossed product of a Type III$_1$ factor by its modular automorphism group generated by the modular Hamiltonian $H_\mathrm{mod}$, which is a well-defined operator in the Type III theory unlike its half-sided counterparts, is a semifinite factor. Generally, the crossed product is a way to incorporate, on the same Hilbert space, the representation of the original algebra and the unitary action of the modular group on this algebra, neither of which makes reference to the half-sided Hamiltonians. Our central claim is true regardless of the concrete way of physically implementing this crossed product as the proposal is to replace the infinite Type III factors with their associated semifinite Type II factors obtainable via the crossed product with the modular group.