🔬 condensed matter

Measurement-induced phase transition in space

This paper introduces a spatial realization of measurement-induced phase transitions in a single monitored Clifford chain by imposing a deterministic measurement gradient, which creates a steady state with coexisting entanglement phases and enables the direct extraction of critical exponents through spatial scaling without Kibble-Zurek dynamics.

Original authors: Jia-Qiang Li, Liang-Jun Zhai, Shuo Liu, Shuai Yin

Published 2026-07-15

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CC BY 4.0

Original authors: Jia-Qiang Li, Liang-Jun Zhai, Shuo Liu, Shuai Yin

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a long line of tiny quantum coins, each one flipping and interacting with its neighbors in a chaotic dance. Usually, if you watch these coins too closely, you freeze their dance; if you ignore them, they get wildly tangled. This paper explores a special "phase transition" where the system suddenly switches from being highly tangled (a "volume-law" phase) to being mostly separate (an "area-law" phase), depending on how often you peek at the coins.

The Big Idea: A Gradient Instead of a Switch
In most experiments, scientists try to find this transition by setting up many different versions of the system: one where they peek 10% of the time, another where they peek 20%, and so on, until they find the exact tipping point. It's like trying to find the perfect temperature for a cake by baking a hundred separate cakes at different temperatures.

This paper suggests a smarter, more efficient way. Instead of baking a hundred cakes, imagine baking just one giant, long cake where the oven temperature changes smoothly from one end to the other. On the left side, the oven is cool (few measurements); on the right side, it's scorching hot (many measurements). In the middle, there's a specific spot where the temperature is just right to hit the critical tipping point.

By doing this, the researchers created a single "steady state" that contains all three zones at once:

The Volume-Law Zone: Where the coins are wildly entangled.
The Area-Law Zone: Where the coins are mostly separate.
The Critical Zone: The exact boundary in the middle where the magic happens.

What They Found (and What They Didn't)
The team ran detailed simulations using a specific type of quantum circuit called a "monitored Clifford chain." They didn't just guess; they measured the "entanglement entropy" (a fancy way of measuring how tangled the coins are) across this gradient.

Here is what their simulations revealed:

The Scaling Law: The transition follows a specific mathematical pattern. Just as a river flows differently depending on the slope, the entanglement changes in a predictable way based on the "steepness" of the measurement gradient.
The "Cut" is Key: The exact point where the measurement probability equals the critical value ( $p_c$ ) acts as a "spatial cut." If you look to the left of this cut, you see volume-law behavior. If you look to the right, you see area-law behavior.
The "Window" Effect: Because you can't measure a coin more than 100% of the time or less than 0% of the time, the gradient has to stop at the edges. This creates a "finite linear window." The researchers found that this physical limit actually helps them measure a crucial number called the correlation-length exponent ( $\nu$ ). In their simulations, this value was approximately 1.260(15).
No Time Travel: A common way to study these transitions is to sweep the measurement rate over time (like turning a dial faster and faster). This paper explicitly argues that their spatial method is different. There is no "Kibble-Zurek" time evolution here. The transition happens because of where you are in space, not how fast time is passing.

The "What If" They Ruled Out
The paper is careful to distinguish its findings from other methods. They argue against the idea that this spatial setup behaves exactly like a time-based sweep.

Not a Time Sweep: In time-based experiments, the system's behavior is limited by how fast it can relax (critical slowing down). In this spatial setup, the limit is purely geometric—the physical size of the chain and the fact that probabilities can't go above 1 or below 0.
Not a New Universe: The transition doesn't belong to a completely new "universality class" (a new category of physics). Instead, the spatial constraints just reorganize the math in a way that makes it easier to see the same underlying rules.

How Sure Are They?
The authors are very confident in their results, but it is important to note that these are numerical simulations, not a physical experiment in a real lab yet. They simulated a chain of 1,024 and 2,048 qubits.

They verified that the data "collapses" onto a single curve when plotted correctly, which is a strong sign the theory is right.
They tested their idea in two different ways: one where the critical point was in the middle of the chain, and another where it was anchored at the edge. Both gave the same results, suggesting the method is robust.
They measured the exponent $\nu$ to be 1.260(15) and the critical parameter $\alpha$ to be 1.57(1).

Why It Matters
If you want to study these quantum phases, you usually need to prepare a new state for every single measurement probability you want to test. That takes a lot of time and resources. This paper suggests that with a single, cleverly designed "gradient" setup, you can see the whole transition map in one go. It's like getting a topographical map of a mountain range by hiking just one trail that goes from the valley to the peak, rather than climbing a hundred different mountains.

While this is currently a simulation, the authors suggest that real quantum computers could eventually use this "spatial gradient" trick to study criticality more efficiently, saving time and reducing the need for constant recalibration. They even propose that the "area-law" side of the gradient could serve as a built-in reference to check for errors in the hardware.

In short, the paper shows that by arranging measurements in space rather than time, we can create a single, rich snapshot of a quantum phase transition, revealing the hidden rules of entanglement with surprising clarity.

Technical Summary: Measurement-Induced Phase Transition in Space

Problem and Motivation
Measurement-induced phase transitions (MIPTs) in monitored quantum circuits are typically characterized by preparing steady states at various uniform measurement probabilities ( $p$ ) and comparing ensemble-averaged observables to reconstruct the transition between volume-law and area-law entangled phases. While temporally driven MIPTs (where $p$ is swept in time) have been shown to obey finite-time scaling forms beyond the Kibble-Zurek mechanism, a direct spatial realization of this phenomenon remains unexplored. The authors ask whether a deterministic spatial gradient in measurement probability can encode the tuning parameter statically, thereby realizing critical scaling directly in space within a single steady state. This inquiry is motivated by the broader context of gradient-controlled quantum dynamics, such as Stark many-body localization, and the need for resource-efficient protocols to probe measurement-induced criticality.

Methodology
The authors investigate a (1+1)D monitored random Clifford circuit consisting of a chain of $L$ qubits with periodic boundary conditions on the unitary gates. The system evolves via brickwork layers of two-site random Clifford gates followed by local projective measurements in the $Z$ basis.

The core innovation is the imposition of a deterministic spatial gradient in the measurement probability $p(x)$ . Since $p(x)$ is a probability, it is physically bounded by $0 \le p(x) \le 1$ . The authors implement a truncated linear profile:
$p(x) = \max\{0, \min\{1, p_c + R(x - x_c)\}}$
where $p_c$ is the critical probability, $R$ is the spatial gradient, and $x_c$ is the position where $p(x_c) = p_c$ . This construction creates a single steady state containing three distinct regions: a volume-law region ( $p < p_c$ ), an area-law region ( $p > p_c$ ), and a critical region at $x_c$ , separated by saturated plateaus at $p=0$ and $p=1$ .

The study employs two partition protocols to probe the system:

Entanglement Distribution: The subsystem size $|A|$ is fixed while the base cut slides along the gradient to sample different local tuning parameters $g = p - p_c$ .
Critical Scaling: The base cut is anchored at $x_c$ , and the subsystem size is varied by extending into either the volume-law or area-law side.

Key observables include the subsystem entanglement entropy ( $S_A$ ), the ancilla entropy ( $S_Q$ ), and the tripartite mutual information ( $I_3$ ). The authors perform numerical simulations on chains of length $L=1024$ and $L=2048$ .

Key Contributions and Results

Spatial Scaling Form: The authors derive a spatial scaling ansatz for the subsystem entropy:
$S_A(R, g, |A|) = \alpha \ln |A| + F_s(R|A|^{r_s}, g|A|^{1/\nu})$
where the scaling dimension of the gradient is $r_s = 1 + 1/\nu$ . This form is structurally analogous to the finite-time scaling of temporally driven MIPTs but lacks Kibble-Zurek time evolution. The distinction arises because the spatial gradient is tied to the spatial variation of the tuning parameter rather than critical relaxation dynamics.
Cutoff-Controlled Asymptotics: A critical finding is that the physical bounds $0 \le p \le 1$ impose a finite linear window $A_{\text{max}} \sim R^{-1}$ . This geometric cutoff controls the large-scale asymptotics of the scaling function, differing from the dynamically controlled asymptotics of temporal sweeps.
- Volume-law side: The entropy scales as $S_A \sim A_{\text{max}}$ , leading to a power-law behavior in the scaling function with an exponent $\mu \approx \nu$ . The fitted exponent $\mu = 1.112(6)$ is consistent with the correlation-length exponent $\nu \approx 1.26$ .
- Area-law side: The entropy scales logarithmically with the gradient, $S_A \sim -\nu \ln R$ , yielding a slope $\delta \approx -\nu$ . The fitted slope $\delta = -1.24(1)$ confirms this relationship.
  These results demonstrate that the finite spatial window reorganizes the scaling function to directly expose the correlation-length exponent $\nu$ on both sides of the transition.
Robustness and Diagnostics: The scaling behavior is verified across multiple observables ( $S_A$ , $S_Q$ , $I_3$ ). Furthermore, the authors introduce an alternative boundary-anchored protocol where the base cut is fixed at the chain edge and the gradient profile is shifted. This confirms that the observed scaling and cutoff-controlled asymptotics are intrinsic to the bounded measurement profile and not artifacts of the specific choice of a centered critical point.

Significance
The paper establishes spatially inhomogeneous measurements as a controlled route to engineer and probe measurement-induced criticality within a single steady state. By encoding the tuning parameter in space, the protocol eliminates the need to prepare and compare multiple steady states at different uniform $p$ values, offering a resource-efficient method for experimental platforms.

The work highlights a fundamental distinction between temporal and spatial driving: while temporal sweeps are limited by critical slowing down (Kibble-Zurek), spatial protocols are limited by the physical bounds of the probability distribution. This geometric constraint converts the asymptotic behavior into a cutoff-controlled regime, allowing for the direct extraction of the correlation-length exponent $\nu$ from restricted spatial windows near the critical cut. The authors conclude that this approach deepens the understanding of criticality in spatially inhomogeneous monitored circuits and provides a practical framework for studying measurement-induced phenomena beyond Clifford circuits.

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