Quantum Entanglement Survives Measurement in up to Four Linked Particles

Researchers are increasingly focused on measurement-induced phase transitions (MIPTs) as a route to understanding emergent dynamical states of matter. Liuke Lyu, James Allen, and Yi Hong Teoh, from the Université de Montréal and University of Waterloo, alongside Roger G Melko and William Witczak-Krempa, demonstrate a large-scale numerical simulation of a trapped-ion native MIPT, establishing its connection to the Haar non-unitary conformal field theory. Their work is significant because it provides precise characterisation of the critical behaviour at these transitions, revealing robust algebraic decay of genuine multiparty entanglement and establishing lower bounds for both multiparty entanglement and mutual information, potentially offering new insights into the fundamental nature of quantum many-body systems.

Trapped-ion simulations reveal critical behaviour and entanglement structure in measurement-induced phase transitions

Researchers have demonstrated a significant advance in understanding measurement-induced phase transitions (MIPT), novel dynamical states of quantum matter arising from the interplay between quantum evolution and measurement. This work presents large-scale numerical simulations of a trapped-ion native MIPT, establishing its connection to the Haar non-unitary conformal field theory.
Through finite-size analysis, the critical measurement rate and correlation length exponent were determined, falling close to the value expected for percolation. These findings illuminate the behaviour of quantum systems under continuous monitoring, revealing how measurements can drive transitions between distinct phases of matter.

A key achievement of this study lies in the robust characterisation of genuine multiparty entanglement (GME) , a complex form of quantum correlation, as a function of spatial separation. By employing a computable monotone via semi-definite programming, researchers uncovered algebraic decay of GME for systems of 2, 3, and 4 parties.

The resulting critical exponents were found to be lower-bounded by those of the multiparty mutual information, which was determined for up to 4 parties and conjectured to follow a (k+2) relationship for k parties. This establishes a fundamental link between different measures of entanglement and provides constraints on their behaviour near the critical transition.

Furthermore, the research derives lower bounds for both GME and multiparty mutual information, solidifying the theoretical framework for understanding entanglement in MIPTs. Simulations were performed on chains containing up to 26 sites, allowing for robust finite-size scaling and extrapolation to the thermodynamic limit.

The observed exponents adhere to established inequalities, including monotonicity and subadditivity, validating the consistency of the results. This detailed analysis of multiparty entanglement offers new insights into the emergent properties of MIPTs and their potential applications in quantum technologies.

Circuit construction and simulation parameters for measurement-induced phase transition studies

A (1+1)-dimensional brickwork circuit model forms the basis of this work, designed to investigate measurement-induced phase transitions (MIPT) and balance generic unitary evolution with projective measurements. The circuit utilizes a small gate set native to trapped-ion quantum processors, consisting of layers of two-site entangling gates applied in a brickwork geometry, interspersed with layers of single-site projective measurements in the computational Z-basis with probability p.

Unitary gates are constructed from the Mølmer-Sørensen (MS) interaction, specifically gates of the form U j,j+1 = M j,j+1 R j ⊗R j+1 , where M is a fixed-angle MS gate defined as exp(−iπ/4X j X j+1 ) and R j ,R j+1 are single-qubit rotations drawn uniformly from a discrete set of 3π/2 rotations. To prepare the ensemble of states, simulations were performed on systems ranging in size from N = 18 to N = 26, each initialized in the product state ∣0⟩ ⊗N .

The circuit was applied for N full periods, where each period comprises an even unitary layer, a measurement layer, an odd unitary layer, and a second measurement layer, resulting in a total depth of 2N unitary layers and 2N measurement layers. One additional even unitary layer followed by 50% probability measurements was applied at the end of the evolution to mitigate any bias from the final layer, with confirmation that this depth is sufficient to reach a non-equilibrium steady state through comparison with simulations at 1.5N periods.

This specific gate set suppresses parity oscillations commonly found in standard Haar-random brickwork circuits, enabling reliable extraction of critical exponents for higher-order multiparty correlations. Genuine multiparty entanglement (GME) and multiparty mutual information are employed to characterize the complex structure of the critical state.

The study leverages a monotone computable via semi-definite programming to uncover robust algebraic decay of GME versus separation for 2, 3, and 4 parties. Lower bounds for both GME and multiparty mutual information were derived, with the scaling exponents α k for genuine multiparty negativity and multiparty mutual information extracted at p = 0.17 for various system sizes N, showing approximate N →∞ limits where convergence is observed, such as α GMN 2 = 6.79 for N = 18, increasing to 9 + 1.0 for N = 26. The space-time evolution of multiparty entanglement was also studied by numerically evaluating entanglement-weighted graphs, allowing construction of entanglement clusters and establishing the importance of measurement halos in creating long-range entanglement.

Genuine multiparty entanglement and mutual information scaling in a brickwork circuit model

Scaling exponents αk for genuine multiparty negativity and multiparty mutual information were extracted at p = 0.17 for various system sizes N, ranging from 18 to 26, with approximate N →∞ limits observed. Table I details these exponents, showing values of 5.32, 5.47, 5.71, 5.94, and 5.96 for αMI, alongside values of 6.0 with a margin of error of plus or minus 0.4 and minus 0.1.

The research employed a (1+1)-dimensional brickwork circuit model constructed from a small gate set native to trapped-ion quantum processors to investigate universal properties of measurement-induced phase transitions. This circuit consists of layers of two-site entangling gates applied to nearest neighbors, interspersed with layers of single-site projective measurements with probability p.

The study leveraged a monotone computable via semi-definite programming to uncover robust algebraic decay of genuine multiparty entanglement (GME) versus separation for 2, 3, and 4 parties. This revealed lower bounds for both GME and multiparty mutual information, providing insights into the structure of correlations in the critical state.

Specifically, the tripartite mutual information was utilized to determine the critical point of the ensemble, and a distance scale was specified for conformal finite-size analysis. Researchers simulated system sizes ranging from N = 18 to N = 26, initializing each realization in the product state ∣0⟩⊗N and evolving the circuit for a total depth of 2N unitary layers and 2N measurement layers.

To mitigate bias, an additional even unitary layer was applied, followed by measurements with a probability of 50% at the end of the evolution. This depth was confirmed to be sufficient to reach the non-equilibrium steady state, demonstrating excellent agreement with simulations performed at a depth of 1.5N periods. The work employed the k-party mutual information, Ik, to quantify total correlations among subsystems, and genuine multiparty entanglement to isolate strictly non-classical contributions.

Measurement-induced criticality and algebraic decay of multipartite entanglement

Researchers have demonstrated the existence of a measurement-induced phase transition in a trapped-ion system, revealing novel dynamical states arising from the interplay between quantum evolution and measurement processes. Large-scale numerical simulations were performed to characterise this transition, identifying behaviour consistent with a specific type of non-unitary conformal field theory known as the Haar theory.

Finite-size analysis established the critical measurement rate and a correlation length exponent approximating the value expected for percolation. Furthermore, the study uncovered robust algebraic decay of genuine multiparty entanglement with increasing separation between parties, for groups of two, three, and four.

Critical exponents governing this decay were determined and found to satisfy inequalities related to multiparty mutual information, with a conjecture proposed for the general case of k parties: αMI k = k + 2. Lower bounds were also established for both genuine multiparty entanglement and multiparty mutual information, providing quantitative constraints on entanglement structure.

The authors acknowledge limitations in the convergence of four-party genuine multiparty entanglement exponents and the focus on equal separations between parties. Future research could improve the precision of these exponents by exploring less symmetric configurations and investigating the entanglement structure at different measurement-induced phase transitions.

A key open question concerns the possibility of predicting genuine multiparty entanglement and multiparty mutual information exponents directly from the underlying conformal field theory, potentially offering a deeper theoretical understanding of these phenomena. These findings contribute to the growing field of measurement-induced phase transitions and provide insights into the behaviour of entanglement in complex quantum systems.