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Holographic Relative Entropy in Infinite-Dimensional Hilbert Spaces

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We reformulate entanglement wedge reconstruction in the language of operator-algebra quantum error correction with infinite-dimensional physical and code Hilbert spaces. Von Neumann algebras are used to characterize observables in a boundary subregion and its entanglement wedge. Assuming that the infinite-dimensional von Neumann algebras associated with an entanglement wedge and its complement may both be reconstructed in their corresponding boundary subregions, we prove that the relative entropies measured with respect to the bulk and boundary observables are equal. We also prove the converse: when the relative entropies measured in an entanglement wedge and its complement equal the relative entropies measured in their respective boundary subregions, entanglement wedge reconstruction is possible. Along the way, we show that the bulk and boundary modular operators act on the code subspace in the same way. For holographic theories with a well-defined entanglement wedge, this result provides a well-defined notion of holographic relative entropy.

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  1. The causal complement of a region R, denoted by \(R^\prime \), is defined to be all of the points in spacetime which are spacelike separated from every point in R. A region R is causally complete if \(R^{\prime \prime } = R\). Note that any von Neumann algebra M satisfies \(M = M^{\prime \prime }\), where the \(\prime \) denotes the commutant.

  2. This means that u is a norm-preserving map. The map u need not be a bijection. In general, \(u^\dagger u\) is the identity on \(\mathcal {H}_{code}\) and \(uu^\dagger \) is a projection on \(\mathcal {H}_{phys}\).

  3. This is because we may act with an operator in \(M_{code}\) to send \({|{\Psi }\rangle }\) to a vector arbitrarily close to \({|{\Omega }\rangle }\), and we may act with an operator in \(M_{phys}\) to send \(u{|{\Omega }\rangle }\) arbitrarily close to any vector in \(\mathcal {H}_{phys}\).

  4. With the relation for the Tomita operators we derived above, we obtain a relation for the relative modular operators \(\Delta ^c_{\Psi | \Phi }\) and \(\Delta ^p_{u \Psi | u \Phi }\) to be \(u\Delta ^c_{\Psi | \Phi } = \Delta ^p_{u \Psi | u \Phi } u\).

  5. We apply the spectral theorem separately for the restriction of \(\Delta ^p_{u \Psi | u \Phi }\) to \(\text {Im } u\) and \((\text {Im } u)^\perp \).

  6. We use the relation \(\Delta ^c_{\Psi | \Phi } = u^\dagger \Delta ^p_{u \Psi | u \Phi } u\). For the projections, \(\Omega \) denotes a measurable subset of \({\mathbb {R}}\)

  7. Note that \(\mathcal {O}\) need not be self-adjoint.

  8. To be explicit, we have that

    $$\begin{aligned} {\langle {(\mathcal {O}- \lambda I)\chi |(\mathcal {O}- \lambda I)\chi }\rangle } = {\langle {(\mathcal {O}- \lambda _1 I)\chi |(\mathcal {O}- \lambda _1 I)\chi }\rangle } + {\langle {\lambda _2\chi | \lambda _2 \chi }\rangle } + i \lambda _2 {\langle {\chi |(\mathcal {O}- \lambda _1 I )\chi }\rangle } - i \lambda _2 {\langle { (\mathcal {O}- \lambda _1I)\chi |\chi }\rangle }. \end{aligned}$$

    The last two terms cancel because \(\mathcal {O}\) is self-adjoint and \(\lambda _1\) is real.

  9. For any \({|{\psi }\rangle } \in D(\mathcal {O})\), the integral \(\int _{\mathbb {R}} \lambda d(P^\mathcal {O}_\lambda {|{\psi }\rangle })\) with vector-valued measure \(P^\mathcal {O}_\Omega {|{\psi }\rangle }\) converges in the Hilbert space norm to \(\mathcal {O}{|{\psi }\rangle }\). The integral does not converge for \({|{\psi }\rangle } \notin D(\mathcal {O})\).

  10. In other words, \(e_\Psi \) is an injective map.

  11. \(S_{\Psi | \Phi }\) is well-defined if and only if \(\lim _{n \rightarrow \infty } \mathcal {O}_n {|{\Psi }\rangle } = 0 \implies \lim _{n \rightarrow \infty } \mathcal {O}_n^\dagger {|{\Psi }\rangle } = 0\). See footnote 14 of [14] for a proof of why this is true. \(S_{\Psi | \Phi }\) is densely defined because \({|{\Psi }\rangle }\) is cyclic with respect to M.

  12. \((\Delta ^p_{u \Psi | u\Phi })|_{\text {Im } u}\) denotes the restriction of \(\Delta ^p_{u \Psi | u\Phi }\) to the closed subspace \(\text {Im } u\).

  13. Assuming that the time-slice axiom [15] holds, \(\mathcal {A}(\mathcal {U})\) should really be associated with the domain of dependence of \(\mathcal {U}\), as operators in the domain of dependence are related to operators in \(\mathcal {U}\) via an equation of motion. Note that the time-slice axiom does not hold for generalized free fields [28], which we consider in Sect. 6.2.

  14. If a subalgebra \(\mathcal {S}\) of bounded operators contains the identity and is closed under hermitian conjugation, then its double commutant, \(S^{\prime \prime }\), is the von Neumann algebra generated by \(\mathcal {S}\). Von Neumann algebras are naturally associated with causally complete subregions [14, 23].

  15. Associating a set of operators with a subregion in the bulk is highly nontrivial due to nonlocal effects in the bulk [29]. This is addressed in [30], which studies information measures for sets of operators that are not closed under multiplication. We do not consider this subtlety in our analysis.

  16. It will be interesting to generalize equation (5.4) in [31] to an expression that uses infinite-dimensional von Neumann algebras.

  17. In certain contexts, the entanglement wedge reconstruction proposal must be nonperturbatively approximate (see [32, 33]). The approximate reconstruction with nonperturbative gravity corrections is rigorously formulated and analyzed in [36], that further expands the scope of our algebraic framework.

  18. The fact that all correlation functions may be expressed in terms of two-point functions arises from large-N factorization in the boundary CFT.

  19. This statement is also true for conformal transformations of such regions. For these boundary regions, the causal wedge is the same as the entanglement wedge [37].

  20. Technically, a spatial slice is not an open subregion of spacetime.

  21. If \(\mathcal {H}_0\) is a proper subset of \(\mathcal {H}_{code}\), we should redefine \(\mathcal {H}_{code}\) to be \(\mathcal {H}_0\) for Theorem 1.1 to apply.

  22. The definition of a type \(\hbox {III}_1\) factor is given in [41].


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The authors are grateful to Daniel Harlow, Temple He, Sungkyung Kang, and Kai Xu for discussions. M.J.K. and D.K. would like to acknowledge support from NSF grant PHY-1352084.

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Correspondence to Monica Jinwoo Kang.

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Kang, M.J., Kolchmeyer, D.K. Holographic Relative Entropy in Infinite-Dimensional Hilbert Spaces. Commun. Math. Phys. 400, 1665–1695 (2023).

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