Localizable Entanglement as an Order Parameter for Measurement-Induced Phase Transitions

arXiv:2601.14185 · quant-ph, cond-mat.stat-mech · Submitted 2026-01-20 · Read on arXiv

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Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: I'm Kai, and with me are Mira and Lev, guest researcher.

Mira: Today's paper: "Localizable Entanglement as an Order Parameter for Measurement-Induced Phase Transitions".

Kai: The gist: Localizable entanglement (LE) is identified as an order parameter for measurement-induced phase transitions (MIPT),

Mira: First, who's behind it and why it matters.

Title and authors: Kai: Let’s start with the paper itself, "Localizable Entanglement as an Order Parameter for Measurement-Induced Phase Transitions." It’s written by Sourav Manna, Arul Lakshminarayan, and Vaibhav Madhok from IIT Madras.

Mira: The title tells you right away that they're proposing a specific quantity—localizable entanglement—as the order parameter to track these measurement-induced phase transitions.

Lev: So, what’s the big picture idea behind calling LE an order parameter in this context? Is it just a fancy way of saying it changes at the transition?

Kai: It’s more specific than that; they are identifying LE as a physically inspired correlation length within monitored quantum circuits, where it quantifies the maximal entanglement that can be concentrated between two sites through optimal local measurements on the rest of the system.

Mira: That definition is key because it frames LE not just as a static measure, but as a dynamic diagnostic of how far quantum information can propagate before being irreversibly lost to measurements.

Lev: So, instead of just looking at entanglement between two fixed points, they are measuring the *maximal* entanglement that can be accessed through local actions on the rest of the system. That’s a different kind of resource.

Kai: Right. And this concept informs how we characterize quantum phase transitions in models like Heisenberg or Ising chains because this length scale diverges as you approach the critical point.

Mira: The paper also gives a nice operational interpretation, connecting these measurement-induced transitions to classical percolation theory, which is usually about successful transport across a network.

Lev: So, when they compare it to percolation, what’s the actual distinction they are making between the two concepts?

Kai: They show that while classical percolation captures successful transport across a network as bond or site probability increases, MIPT characterized by LE quantifies the amount of quantum teleportation between two given nodes in a quantum circuit.

Mira: It suggests that instead of looking for macroscopic connectivity, we should look for the localization properties of entanglement itself.

Lev: That makes sense if we think about what it means for a system to be connected or not when you're applying these measurement operations.

The paper's summary: Kai: So, summarizing what the paper actually found, they use LE as an order parameter and show it has universal finite-size scaling with critical exponents that match previous MIPT results.

Mira: They also highlight a crucial finding about its spatial dependence: LE decays exponentially with distance in the area-law phase but stays nearly flat in the volume-law phase.

Lev: That difference between exponential decay and staying constant is what leads to the discovery of an intrinsic length scale, ξE, which diverges at the critical measurement probability pc.

Kai: So, this length scale ξE(p) is essentially a measure of how far entanglement can be accessed via local measurements before it gets lost in the area-law phase.

Mira: And they provide a clear mathematical description for this divergence: ξE(p) scales with p - pc to an exponent ν equal to one point three one, which is consistent with the MIPT results they’ve been seeing.

Lev: That number, one point three one, gives us a concrete prediction for how fast that entanglement correlation length grows as we approach the transition point pc = zero point one six on their representative data points in Figure two.

Kai: And this is all based on computing LE exactly across system sizes using random Clifford circuits and measurement rates ranging from zero point one four to zero point one eight, showing the transition clearly at p = zero point one six in Figure two(b) <ref:2601.14185#pg2>.

Mira: It’s a very detailed mapping of how the entanglement correlation function <CLE> behaves—distance-independent in the volume-law phase, intermediate at criticality, and exponentially decaying in the area-law phase.

Lev: So, if I were to ask a listener driving about this paper: "What is this actually telling me about quantum information propagation?" they would hear that entanglement localization is the key metric we use to understand how far quantum signals can travel before measurements cause them to vanish.

The paper's improvements: Kai: Now, looking at what the authors suggest as improvements, they propose a two-ancilla protocol for experimental access. This is a big step because it allows us to probe the transition without needing direct access to the system for detection.

Mira: They are using concurrence between two reference qubits alone—without ever touching the main system—to faithfully detect the MIPT based solely on these local measurements.

Lev: That’s very appealing from a hardware perspective because it reduces the complexity of what we need to measure down to just two qubits, which is much more manageable for near-term platforms.

Kai: They also point out that exploiting the structure of correlations and monogamy of entanglement can help sharpen this experimental detection further, adding another layer of sensitivity.

Mira: The paper also provides a way to extract this physical length scale ξE(p) by analyzing the entanglement correlation function in the area-law phase where it decays exponentially with distance.

Lev: So, so the improvement is twofold: first, a low-overhead experimental readout using two qubits, and second, a method to extract that specific physical length scale from those measurements.

Kai: It gives us a way to confirm if our experimental results align with the theoretical predictions for the scaling of ξE(p) as we approach pc.

Conclusion: Mira: To wrap up, the main implication of this paper is that they’ve successfully established localizable entanglement as an order parameter for MIPT with a clear operational interpretation in terms of quantum teleportation.

Kai: It sets a standard for characterizing these transitions by providing a measurable quantity that connects the dynamics of measurements to classical percolation in a way that's consistent with previous results.

Lev: For me, the most practical implication is this two-ancilla protocol; it means we can get an experimentally accessible readout of entanglement redistribution across the transition using only local measurements.

Mira: And I think that connection between MIPT and classical percolation is what makes this work so significant for connecting different fields in condensed matter physics.

Kai: So, to sum up the paper "Localizable Entanglement as an Order Parameter for Measurement-Induced Phase Transitions," we found LE behaves as an order parameter with a clear length scale ξE that diverges at pc.

Mira: And this divergence is captured by the exponent ν = one point three one, which gives us a concrete benchmark for verifying our understanding of these transitions.

Lev: For running this on real hardware, the main challenge remains isolating those two reference qubits reliably to see that signature clearly in the measurement statistics.

Kai: We’ll keep an eye on how this two-ancilla protocol works in practice and see if we can use it for diagnostics on more complex quantum circuits next.

Department of Physics and Center for Quantum Information, Communication and Computing, Indian Institute of Technology Madras

quant-ph, cond-mat.stat-mech

Submitted: 2026-01-20

Updated: 2026-02-10

Comments: 8 pages, 6 figures. Identifies localizable entanglement as an operational order parameter for measurement-induced phase transitions in monitored quantum circuits. Reveals an emergent entanglement correlation length whose divergence allows extraction of the critical exponent from single-system-size data.A two-ancilla protocol provides an experimentally accessible probe of the transition

Journal ref: Phys. Rev. A 114, 032222 (2026)

DOI: 10.1103/81zs-jht1

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Importance score: 92/100

The gist: The gist: Localizable entanglement (LE) is identified as an order parameter for measurement-induced phase transitions (MIPT), exhibiting universal finite-size scaling with critical exponents that

Key concepts

Localizable Entanglement (LE): LE measures the maximum entanglement that can be concentrated between two sites by performing optimal local measurements on the rest of the system. It acts as a physical correlation length indicating how far quantum information can travel before being lost to measurements.
Measurement-Induced Phase Transition (MIPT): This is a transition in quantum systems caused by continuous monitoring or measurement. The paper shows that LE changes its behavior dramatically at a critical measurement probability, signaling the shift between different types of entanglement phases.
Area-Law vs. Volume-Law Phase: These describe the spatial decay of entanglement. In the area-law phase, entanglement decays exponentially with distance, suggesting measurements destroy long-range connections. In contrast, the volume-law phase means entanglement persists across the entire system scale.
Critical Measurement Probability (pc): This is a specific measurement rate where the system undergoes a transition. The LE length scale diverges as this probability approaches pc, marking the point where long-range quantum connectivity emerges in this measurement-induced transition.

Terminology

Entanglement — 81× in this paper · explained in 3 episodes
Phase Transitions — 17× in this paper · explained in 2 episodes
Quantum Circuits — 9× in this paper · explained in 2 episodes
Stabilizer States — 5× in this paper · explained in 2 episodes
Fidelity — 3× in this paper · explained in 2 episodes
Measurement-induced phase transitions (MIPT) — 2× in this paper · explained in 2 episodes
Unitary Dynamics — 2× in this paper · explained in 2 episodes
Benchmark — 1× in this paper · explained in 2 episodes
Unified Framework — 1× in this paper · explained in 5 episodes

Summary

The gist: Localizable entanglement (LE) is identified as an order parameter for measurement-induced phase transitions (MIPT), exhibiting universal finite-size scaling with critical exponents that match previous MIPT results and giving a nice operational interpretation connecting MIPTs to classical percolation.

Localizable Entanglement as an Order Parameter

Localizable entanglement (LE) is identified as a physically inspired correlation length in monitored quantum circuits, where the LE between two sites quantifies the maximal entanglement that can be concentrated between them through optimal local measurements on the rest of the system This makes it a natural diagnostic of how far quantum information can propagate before being irreversibly lost to measurements LE informs us about the distance over which entanglement can be accessed via local measurements In models like the Heisenberg or Ising chains, LE helps characterize quantum phase transitions as this length diverges as one approaches the critical point Operationally, LE is a resource for quantum teleportation as it quantifies the entanglement that can be localized between distant parties via local measurements Using random Clifford circuits, we exploit the stabilizer–graph-state correspondence to compute LE exactly across system sizes and measurement rates Our results show that the LE falls off exponentially with distance in the area-law phase, but stays nearly flat in the volume-law phase thereby defining an intrinsic length scale ξE that diverges at the critical measurement probability pc

Operational Order Parameter and Critical Behavior

A natural order parameter for MIPT can be constructed from LE by letting R be the largest separation ratio i − j/(L − 1) for which the LE remains non-zero In the area-law phase we expect measurements to destroy long-range entanglement, implying finite value of maxi,j i − j; hence R → 0 as L → ∞ In contrast, in the volume-law phase, entanglement persists across the entire system, so LEij even at system-scale separations, therefore R ∼ 1 Figure 2 illustrates this order parameter The crossing of curves at pc ≈ 0.16 signals the transition from the entangling (volume-law) to the disentangling (area-law) phase This crossing demonstrates that captures the change in the spatial extent of LE across the transition As shown in Fig. 3, in the area-law phase p > pc, the averaged correlation function decays exponentially with distance, ∼ exp(-r/ξE(p)) (4) The data (points) follow a clear power-law divergence, p − pc ν, and the linear fit (solid line) yields an exponent ν = 1.31 consistent with the MIPT The divergence of this length scale upon approaching the transition is captured by a power-law form, ξE(p) ∼ p − pc −ν (5)

Experimental Probes of the Transition

The MIPT transition can be probed by employing a reference qubit initially maximally entangled with one site of the circuit as it undergoes hybrid measurement-unitary dynamics A finite LE between the reference and the system captures the volume-law phase and vanishes in the area-law phase, with the crossover sharpening with system size Further exploiting the structure of correlations and monogamy of entanglement, we introduce a second reference qubit The resulting concurrence between the two references alone—without access to the system—faithfully detects the MIPT This provides an experimentally feasible method for probing the transition based solely on two-qubit measurements

Conclusion and Significance

In summary, we have discovered an order parameter to characterize MIPT based on LE that has an operational interpretation in terms of quantum teleportation between distant sites in close analogy with how classical information propagates in classical networks Thus, unlike classical one-dimensional percolation where macroscopic connectivity exists only at zero bond breaking, the measurement-induced transition exhibits finite-rate emergence of long-range quantum connectivity—signaled by a nonzero —enabling macroscopic localizable entanglement and nonvanishing teleportation fidelity despite local measurements<ref:2601.

Improvements for AI systems

This paper enables an experimentally accessible readout of entanglement redistribution across the transition by proposing a two-ancilla protocol that uses concurrence between two reference qubits to detect the Measurement-Induced Phase Transition (MIPT) without needing access to the system for detection.
The improved AI system can perform near-term experimental diagnostics on quantum circuits by measuring only two reference qubits and treating the many-body system as a black box, which is especially suitable for near-term experimental platforms.
By identifying localizable entanglement (LE) as an operational order parameter, the system can quantify the maximal entanglement that can be concentrated between two sites through optimal local measurements on the rest of the system, thus providing a diagnostic of how far quantum information can propagate before being irreversibly lost to measurements.
The AI system gains a capability to distinguish phases by monitoring the behavior of correlation functions: in the volume-law phase, it will observe that LEij even at system-scale separations, therefore R ∼ 1, while in the area-law phase, it will see R → 0 as L → ∞.
The system can extract a physically motivated length scale by analyzing the entanglement correlation function: in the area-law phase, it will determine that ⟨CLE(r; p)⟩ ∼ exp(−r/ξE(p)), thereby defining an entanglement correlation length ξE(p).