LOEM: Local Operation with Entangling Measurements

• LOEM is a quantum paradigm that combines local operations with entangling measurements to generate or reveal nontrivial entanglement across subsystems.
• It enables enhanced entanglement localization, resource conversion, and optimized protocols for quantum metrology and measurement-based engines.
• LOEM challenges conventional LOCC limits by leveraging joint measurement strategies, with successful implementations in photonic, optomechanical, and atomic systems.

Local Operation with Entangling Measurements (LOEM) is a quantum information paradigm wherein local operations—preparation, interactions, or manipulations initially restricted to subsystems or parties—are augmented or followed by measurements in entangled bases, potentially across multiple subsystems. This approach expands the operational toolkit beyond standard LOCC (local operations and classical communication) by exploiting the effects of entangling measurements, which enable novel forms of entanglement creation, localization, efficient measurement-based protocols, and resource conversion, without requiring entangled input states or nonlocal quantum channels during the operation.

1. Formal Definitions and Fundamental Principles

LOEM protocols are formally distinguished by the following structure:

• Local operations: Preparation of product states, local unitaries, or local measurements on individual subsystems. No initial entangled input is required.
• Entangling measurements: Quantum measurements performed in a basis that is entangled across multiple subsystems, typically implemented as joint rank-one projectors $\{|k\rangle\langle k|\}$ in a non-product basis. This distinguishes LOEM from conventional local or separable measurement strategies.

The operational power of LOEM arises from two interlocking phenomena:

• Entanglement localization: The measurement collapses the post-measurement state such that nontrivial entanglement is established or revealed between target subsystems, e.g., by measuring the environment or ancillary systems. The process can be optimized with respect to a chosen entanglement monotone.
• Entanglement by measurement (EbM): In systems of indistinguishable particles, even tensor-product measurements (formally "local" on the distinguishable-particle Hilbert space) can induce entanglement due to exchange symmetrization or antisymmetrization (Rendón et al., 2011).

Formal definitions for a multipartite pure state $\rho_{A_1\ldots A_n}$ partitioned into measured ($A$) and unmeasured ($B$) systems:

• Multipartite Entanglement of Assistance (MEA):

$$\mathrm{MEA}_E(\rho_{AB}) = \max_{\{M_k\}\in\mathcal{P}_\text{global}} \sum_k p_k\,E(\rho^k_B)$$

where $E$ is an entanglement monotone and the maximization is over global rank-one projectors on $A$.

• Localizable Multipartite Entanglement (LME):

$$\mathrm{LME}_E(\rho_{AB}) = \max_{\{M_k\}\in\mathcal{P}_\text{local}} \sum_k p_k\,E(\rho^k_B)$$

restricted to local (product) measurement operators (Vairogs et al., 6 Nov 2024).

2. LOEM in Entanglement Localization and Quantification

Key protocols for entanglement localization via LOEM have been studied in both few-body and many-body contexts.

2.1. Maximizing Localized Entanglement

Given a pure state $|\Psi\rangle$ on $U = S_1 + S_2 + E$, projective measurements in an orthonormal basis $\{|\xi_i\rangle^E\}$ of environment $E$ yield outcome probabilities $p_i$ and corresponding pure post-measurement states $|\xi_i\rangle^{S}$. The average localized entanglement is

$$\mathrm{LE}\{\xi^E\} = \sum_i p_i\,\mathcal{S}_i, \qquad \mathcal{S}_i = S\left(\mathrm{Tr}_{S_2}|\xi_i\rangle^{S}\langle\xi_i|\right)$$

where $S(\cdot)$ denotes von Neumann entropy. The choice of measurement basis is generally optimized to maximize LE, leading to a variational or "stationarity" condition that determines global maxima (Sahoo, 2012).

2.2. Canonical Measurements and Entanglement Length

A canonical approach is to choose the eigenbasis of the environment's reduced density matrix as the measurement basis ("canonical measurement"), often yielding closed-form and operationally significant results:

$$\mathrm{LECM} = \sum_i \lambda_i\,\underline{\mathcal{S}}_i$$

with $\lambda_i$ from the Schmidt decomposition. In translation-invariant systems (e.g., spin-1/2 chains), the spatial decay of $\mathrm{LECM}(R)$ with separation $R$ defines an operational entanglement length $\xi_E$ (Sahoo, 2012):

$$\xi_E^{-1} = -\lim_{R\to\infty}\frac{\ln[\Delta\,\mathrm{LECM}(R)]}{R}$$

3. LOEM in Multipartite Resource Control and Measurement

LOEM interpolates between the resources and limitations of strictly local measurements (LME) and fully global, coherent operations (MEA), especially in multipartite quantum networks.

• Seed entanglement measures: Typical choices include the even-qubit $n$-tangle $\tau_{N_B}$, the genuine multipartite concurrence $C_{GME}$, and concentratable entanglement $CE(s)$ for a subset $s$ (Vairogs et al., 6 Nov 2024).
• Operational access: Partial joint measurements on select subsystems drive protocols continuously from strictly local to maximally entangling measurements.
• Resource considerations: MEA protocols may require large ancilla spaces; LME schemes are readily parallelizable but are resource-limited. Interestingly, for certain graph state transformations (notably, protocols mapping line graph states to GHZ states), local measurement protocols (LME) nearly saturate the entanglement bound set by MEA, i.e., $\mathrm{LME}^{\tau} \approx \mathrm{MEA}^{\tau}$ (Vairogs et al., 6 Nov 2024).

A binary-matrix equation on the adjacency matrix of a graph state precisely characterizes when LOEM can extract post-measurement states of prescribed $n$-tangle. If the criterion admits no solution, no set of (including global) projective measurements can achieve nonzero $n$-tangle.

4. Enhancement of Measurement-Based Quantum Engines and Work Extraction

In measurement-based quantum engines, LOEM serves as an optimal measurement–feedback protocol, maximizing work extraction from quantum measurements:

• Cycle structure: Engine cycles consist of local state preparation, coherent interaction, an entangling measurement (often in the energy or Bell basis), feedback (conditional unitary), and erasure of measurement memory.
• Thermodynamic balance: The energy (quantum heat) injected by measurement, $Q$, and extracted work $W$ are related by

$W = Q - k_B T\,S_f$

where $S_f$ is the Shannon entropy of the outcome record (Mayo et al., 3 Jun 2024, Bresque et al., 2020). Collective entangling measurements reduce $S_f$ by generating correlations, improving efficiency over local measurement protocols by an amount proportional to the mutual information $\mathcal{I}$ induced by the measurement.

• Reversibility in the large $N$ limit: For $N$ identical two-level systems managed by LOEM, one obtains

$$\eta_{\sharp} \to 1 - \frac{k_B T}{N\Delta E_1} \frac{\ln N}{2}$$

with per-site work extraction remaining finite even as $\eta_\sharp \to 1$ for $N\to\infty$ (Mayo et al., 3 Jun 2024).

• Measurement as energetic fuel: The approach frames quantum measurements, especially entangling ones, as an effective fuel, with thermodynamic bounds set by Landauer erasure of measurement records.

5. Multiparameter Quantum Metrology via LOEM

LOEM allows for the saturation of the quantum Cramér–Rao bound (QCRB) in multiparameter quantum metrology without entangled probe states:

• Classical correlation in input, entangling measurement at output: Probe states are formed as a tensor product of mutually orthogonal pure states—no initial entanglement. After collective parameter encoding via parallel unitaries, a global measurement in an entangled basis is performed (Mi et al., 12 Sep 2025).
• Theoretical guarantees: For pure-state unitary encoding, the LOEM protocol ensures that the quantum Fisher information matrix is diagonal and that the weak commutativity condition holds, guaranteeing $F=Q$ (classical Fisher information equals quantum Fisher information), so the QCRB is saturated.
• Heisenberg scaling: Iterative protocols implementing $N$-fold interactions amplify parameters, yielding $1/N^2$ scaling in estimation error for each parameter. Experimental verification using entangled photonic measurements confirms both QCRB saturation and Heisenberg scaling.
• Advantages and limitations: The full efficiency gain is achieved without entangled inputs, only requiring collective entangling measurements. Challenges include implementing high-fidelity nonlocal measurement operations for large $d$.

6. Physical Realizations and Experimental Implementations

LOEM’s theoretical structure is reflected in diverse experimental modalities:

• Optomechanical systems: Entangling measurements on optical modes mediate entanglement swapping between distant mechanical resonators. Local operations, joint optical Bell measurements, and locally conditioned displacements realize LOEM, with entanglement certification via purity relations between subsystems (Abdi et al., 2013).
• Atomic physics: Local spin-exchange interactions (e.g., adiabatically merged optical tweezers with ground- and excited-state atoms) achieve entanglement, followed by entangling measurement protocols (e.g., parity oscillations under local magnetic gradients) for state verification (Kaufman et al., 2015).
• Measurement-based engines: Chains of qubits under sequential swap interactions, interrupted by local measurements and feedback, implement deterministic energy transport and conversion in agreement with LOEM cycle predictions (Bresque et al., 2020).
• Identical particle systems: Non-symmetrized local measurements generate entanglement in symmetric or antisymmetric many-body states, distinct from LOCC protocols (Rendón et al., 2011).

7. Theoretical and Operational Implications

LOEM reframes the operational boundary between "local" and "nonlocal" actions in quantum information theory:

• Beyond LOCC: LOEM highlights cases where the physical consequences of a formally local operation—due to indistinguishability, symmetrization, or post-measurement correlations—can exceed the entanglement capabilities of standard LOCC.
• Resource control and benchmarking: The figures of merit (MEA, LME) derived from LOEM protocols benchmark the capacities of multipartite measurement-based protocols and serve as sensitive probes of quantum phase transitions, e.g., in the transverse-field Ising model, where localizable entanglement sharply signals criticality (Vairogs et al., 6 Nov 2024).
• Limits and scalability: While entangling measurements offer formal and quantitative advantages, their implementation on large, distributed, or highly symmetric systems presents significant physical challenges, particularly as nonlocal measurement bases scale exponentially with subsystem number or dimension.

In summary, Local Operation with Entangling Measurements provides a unified, operationally motivated, and experimentally relevant framework for entanglement creation, localization, measurement-enabled thermodynamic cycles, and optimal quantum estimation, with rigorous theoretical backing and growing experimental validation across quantum platforms.