Quantum information measures for restricted sets of observables

TL;DR

This work develops information-theoretic tools for quantum systems where the set of accessible observables ${\cal A}$ is not closed under multiplication, a situation that arises in gravitational contexts where locality is approximate. The authors construct a little Hilbert space ${\cal H}_{\psi}$, define the modular operator ${\Delta_{\psi}}$ and the relative modular operator ${\Delta(\psi|\phi)}$, and extract spectra-based distance measures that quantify how distinguishable two states are using only two-point data of ${\cal A}$. They show that these measures—such as the Araki-like relative entropy, the normed entropy, and the ${\chi}^{\|}$ distance—are basis-independent, monotone under contractions of ${\cal A}$, and insular, enabling a notion of entanglement even when a density matrix is not available; LOCC properties and additivity considerations are also discussed. The framework is then applied to coarse- and fine-grained subregion dualities in AdS/CFT, yielding insight into how boundary data constrains bulk regions via information measures and clarifying when entanglement wedges can be reconstructed from restricted observables. Overall, the paper provides a principled, spectrum-based approach to quantum information in non-algebraic observable settings with concrete implications for holography and bulk reconstruction.

Abstract

We study measures of quantum information when the space spanned by the set of accessible observables is not closed under products, i.e., we consider systems where an observer may be able to measure the expectation values of two operators, $\langle O_1 \rangle$ and $\langle O_2 \rangle$, but may not have access to $\langle O_1 O_2 \rangle$. This problem is relevant for the study of localized quantum information in gravity since the set of approximately-local operators in a region may not be closed under arbitrary products. While we cannot naturally associate a density matrix with a state in this setting, it is still possible to define a modular operator for a state, and distinguish between two states using a relative modular operator. These operators are defined on a little Hilbert space, which parameterizes small deformations of the system away from its original state, and they do not depend on the structure of the full Hilbert space of the theory. We extract a class of relative-entropy-like quantities from the spectrum of these operators that measure the distance between states, are monotonic under contractions of the set of available observables, and vanish only when the states are equal. Consequently, these distance-measures can be used to define measures of bipartite and multipartite entanglement. We describe applications of our measures to coarse-grained and fine-grained subregion dualities in AdS/CFT and provide a few sample calculations to illustrate our formalism.

Figures (4)

• Figure 1: The coarse-grained dual of a time-band obtained by following the procedure summarized in \ref{['coarsedual']}.
• Figure 2: A cross section of global AdS at constant time with the annular region $r > r_0$ marked in red.
• Figure 3: A plot of $S^{\|}_{{\cal A}}{(\psi|\phi)}$ (left) and $\chi^{\|}_{{\cal A}}{(\psi|\phi)}$ for a system with $N$ spins probed with polynomials of order at most $K$ in the individual spin operators.
• Figure 4: A plot of $S^{\|}_{{\cal B}}{(\psi|\phi)}$ (left) and $\chi^{\|}_{{\cal B}}{(\psi|\phi)}$ for a system that originally has ${\cal D}$ operators probed with a ${\cal D}'$-dimensional subset of these operators. (Note that successive values of ${\cal D}'$ are joined leading to the impression of a continuous curve.)