Observable-projected ensembles

TL;DR

This work introduces observable-projected ensembles, a framework for studying entanglement in a subsystem after measuring an extensive observable inside a region, yielding an ensemble of mixed states rather than pure states. The authors develop a field-theoretic approach, deriving exact results for the charge-projected ensemble in a free compact boson and showing that the measurement mostly preserves the total entanglement entropy of the targeted region with a calculable geometry-dependent correction; they also analyze projections of general Gaussian observables, discuss UV divergences and how to obtain universal quantities, and provide numerical checks on lattice models. A key practical contribution is a protocol using randomized measurements and classical shadows to experimentally probe the charge-projected ensemble, enabling scalable access to the ensemble properties on NISQ devices. The work uncovers a versatile path to connect entanglement structure, measurement-induced effects, and universal quantities in both Gaussian and interacting CFTs, with potential applications to deep thermalization and symmetry-resolved entanglement analyses. Overall, observable-projected ensembles offer a analytically tractable and experimentally viable lens on how partial observations shape quantum correlations in critical many-body systems.

Abstract

Measurements in many-body quantum systems can generate non-trivial phenomena, such as preparation of long-range entangled states, dynamical phase transitions, or measurement-altered criticality. Here, we introduce a new measurement scheme that produces an ensemble of mixed states in a subsystem, obtained by measuring a local Hermitian observable on part of its complement. We refer to this as the observable-projected ensemble. Unlike standard projected ensembles-where pure states are generated by projective measurements on the complement-our approach involves projective partial measurements of specific observables. This setup has two main advantages: theoretically, it is amenable to analytical computations, especially within conformal field theories. Experimentally, it requires only a linear number of measurements, rather than an exponential one, to probe the properties of the ensemble. As a first step in exploring the observable-projected ensemble, we investigate its entanglement properties in conformal field theory and perform a detailed analysis of the free compact boson.

Figures (7)

• Figure 1: The geometry we consider in this manuscript is the following: we measure an observable (such as the charge operator) in a region $B$ and we study the properties of the reduced density matrix $\rho_A$ after this operation.
• Figure 2: Mapping the replica geometry $\mathcal{R}_n$ to a single complex plane using Eq. (\ref{['eq:z_to_w']}).
• Figure 3: Log-log plot of the Holevo $\chi$-quantity as a function of $b-a=\ell_2$ for different values of subsystem size of $A$, $L$, and different $a=d+L$, i.e. different distances between $A$ and $B$. The $\chi$-quantity corresponds to the green line $n=1$ and it has been obtained by using a numerical approach. The black dashed line is the approximation in Eq. \ref{['eq:approx']}, valid for $\ell_2\gg \epsilon$. As a reference, we also plot different values of $n$ for $\frac{1}{2(1-n)}\log\frac{M_{11}^n}{\mathrm{det}M}$.
• Figure 4: Plot of the $C_n$ coefficient defined in Eq. \ref{['eq:gauss_gen2']}. In the left panel, we prove that its dependence on $n$ is nonlinear, as the disagreement between the exact numerical values of $C_n$ (symbols) and the best linear fit (dashed black line) shows. The inset also reveals a discrepancy $O(10^{-2})$ between the numerical data and the linear fit. In the right panel, we plot the non-trivial $L$ dependence of $C_2$ for the 2-replica computation. In both cases, we have used the operator with conformal weight $(1/8,1/8)$.
• Figure 5: A generic Feynman diagram contributing to $\langle Q_B^k \rangle$ . Black dots indicate the insertion of $Q_B$ and single black lines are $\mathcal{O}_s$ propagators. The red dot is the insertion of the interaction $S_I$. Multi-line emanating from it represent all possible fields leaving that vertex. The striped disk contain the rest of the diagram.