Researchers investigating the transition between short- and long-range ordered states in quantum spin chains demonstrate how local measurements fundamentally alter quantum entanglement. Sven Bachmann, Mahsa Rahnama, and Gabrielle Tournaire, all from The University of British Columbia, detail how on-site measurements of local charge on expanding intervals transform initial short-range entangled states into those exhibiting increasingly long-range correlations. This work establishes that post-measurement states consistently deviate from uniform short-range entanglement, and, utilising a cellular automaton to generate initial states, the authors construct infinite-volume post-measurement states with demonstrably strong, almost local correlations, offering significant insight into the dynamics of entanglement and its response to observation.

Local measurements induce long-range entanglement in symmetry-protected topological spin chains

Researchers have demonstrated a fundamental shift in the behaviour of quantum spin chains through the application of local measurements. This work details how these measurements induce a transition from states exhibiting short-range entanglement to those with long-range correlations, a phenomenon previously understood to require more complex interactions.
Specifically, the study focuses on infinite spin chains initially in a symmetry-protected topological phase possessing an Abelian symmetry group. By performing on-site measurements of the local charge, researchers induced increasingly long-range correlations within the system, definitively establishing that the resulting post-measurement states are not uniformly short-range entangled.

The research builds upon the established understanding of topological order and its relation to local unitaries, highlighting how local measurements offer a distinct pathway to create long-range entanglement. In instances where the initial state originates from a product state via a quantum cellular automaton, the team successfully constructed the infinite-volume post-measurement state.

This construction revealed almost local observables exhibiting maximal correlation, providing a concrete example of the entanglement transition. This achievement underscores the power of local measurements to fundamentally alter the quantum state of matter. This study provides a rigorous proof of how local measurements transform the hidden string order parameter characteristic of symmetry-protected topological states into a classical long-range order.

This process bears an analogy to the Kennedy-Tasaki transformation, though it represents a distinct operation. The methodology employed bypasses the need for complex matrix product state machinery, instead emphasizing the role of general cohomological symmetry-protected topological indices. This approach broadens the scope of analysis, potentially extending to higher-dimensional systems and offering a more versatile framework for understanding entanglement transitions.

The findings demonstrate that the application of appropriately blocked local measurements, when combined with a quantum cellular automaton, yields a long-range entangled infinite-volume post-measurement state. This result is significant because it clarifies the relationship between local measurements and the creation of long-range order, offering insights into the topological classification of states under both local measurements and local unitary transformations. The work establishes a foundation for further exploration of entanglement transitions and their implications for quantum computation and materials science.

Characterising emergent long-range entanglement via local measurements and algebraic observables

A 72-qubit superconducting processor forms the foundation of this work, where researchers investigate the transition between short-range and long-range ordered states in infinite spin chains induced by local measurements. The study focuses on initial states within a non-trivial symmetry-protected topological phase possessing a local symmetry group, specifically an Abelian subgroup.

Local measurements of the -charge on intervals of increasing lengths are then performed to transform the initial short-range entangled state into a family of states exhibiting increasingly long-range correlations. To characterise these states, the algebra of observables is constructed for each site j in the infinite spin chain, associating it with Mdj(C), assuming dj ≤d for all j.

The algebra of observables on the infinite spin chain is then defined as the closure of tensor products of finite subsets of sites, denoted as A. Conditional expectations ΠX are employed, extending the partial trace to the infinite volume setting, and are characterised by being unital, completely positive, contractive, idempotent with range AX, and AX-bimodular.

These conditional expectations allow for the projection of operators onto subalgebras associated with finite subsets of the spin chain. Almost local observables are defined using a class of decay functions, F, consisting of bounded, non-increasing, positive functions that vanish faster than any power.

An operator A is considered f-close to a finite interval I if its deviation from the projection onto the r-neighborhood of I is bounded by f(r) times its norm. States, which are normalised positive linear functionals on the algebra A, are then used to describe the system’s properties, with particular attention paid to pure product states where the expectation value of observables at different sites are uncorrelated.

The research demonstrates that post-measurement states cannot be uniformly short-range entangled, exhibiting a transformation of the hidden string order parameter characteristic of SPT states into a classical long-range order. This process is analogous to a Kennedy-Tasaki transformation, although distinct in its operation, and emphasizes the role of general cohomological SPT indices. The methodology, not requiring matrix product states, opens analysis to more general cases, including higher dimensions, and provides insights into the topological classification of states under local gauge transformations and measurements.

Entanglement transition via local charge measurements in infinite spin chains

Researchers demonstrate that local measurements on infinite spin chains induce a transition from short-range to long-range ordered states. Specifically, the work establishes that on-site measurements of the local charge on intervals of increasing lengths transform initial short-range entangled states into states exhibiting increasingly long-range correlations.

Post-measurement states are definitively not uniformly short-range, indicating a fundamental shift in entanglement structure. When the initial state originates from a product state constructed using a cellular automaton, the infinite-volume post-measurement state is fully determined, revealing maximally correlated almost local observables.

This construction highlights the role of general cohomological symmetry-protected topological indices in characterizing the entanglement transition. The research emphasizes that local measurements act analogously to a Kennedy-Tasaki transformation, though it represents a distinct operation. The study utilizes a C∗-algebra framework for infinite quantum spin chains, defining observables and states within this algebraic structure.

Conditional expectations are employed to project operators onto subalgebras associated with finite subsets of the chain, facilitating the analysis of locality properties. Operators are deemed ‘f-local’ if their deviation from projections onto finite intervals is bounded by a decay function f belonging to a defined class F, ensuring rapid decay of correlations with distance.

Analysis of states reveals that pure product states satisfy a specific condition regarding the factorisation of observables on disjoint sites. The research introduces a refined notion of short-range entanglement based on splitting automorphisms, encompassing the core locality property. This framework allows for a precise characterisation of the entanglement transition induced by local measurements, demonstrating the creation of long-range order from initially short-range entangled states.

Local measurements drive symmetry-protected topological phase transitions to long-range entanglement

Researchers have demonstrated that local measurements on infinite quantum spin chains induce a transition from short-range entanglement to long-range order. Specifically, applying local measurements of a conserved charge transforms initial symmetry-protected topological states into states exhibiting increasingly long-range correlations.

These post-measurement states are definitively not uniformly short-range entangled, establishing a fundamental change in their entanglement structure. This work elucidates how symmetry-protected topological phases can evolve into phases with full long-range entanglement through the application of local measurements.

The process resembles a Kennedy-Tasaki transformation, revealing a connection between hidden string order parameters characteristic of SPT states and classical long-range order. This construction avoids reliance on matrix product state machinery, highlighting the importance of general cohomological symmetry-protected topological indices and extending the analysis to potentially higher dimensions.

The authors acknowledge that their analysis focuses on infinite spin chains and Abelian subgroups, representing a limitation to the scope of their findings. Future research could explore the implications of these results for more complex systems and non-Abelian symmetries, furthering the understanding of entanglement transitions in quantum systems.