Disappearance of measurement-induced phase transition in a quantum spin system for large sizes

This paper demonstrates that while a measurement-induced phase transition appears at finite critical times for small quantum spin chains under global measurements, analytical scaling laws reveal that this transition vanishes ($\tau_c \to 0$) in the thermodynamic limit, with survival probability decay characteristics further distinguishing between paramagnetic and ferromagnetic ground states.

The Gist

The Big Idea: The "Magic Mirror" Experiment

Imagine you have a long line of 28 tiny magnets (spins) on a table. At the start, they are all pointing North.

Now, imagine a magical process that happens in a loop:

1. The Dance (Evolution): You let the magnets wiggle and interact with each other for a short time. This is like letting a crowd of people dance; they start mixing, holding hands, and getting entangled with their neighbors. In physics, this creates entanglement (a deep, spooky connection between particles).
2. The Check (Measurement): Every time the dance stops, you hold up a giant mirror and ask one specific question: "Are everyone still pointing North?"
   • If the answer is YES, the experiment ends for that group.
   • If the answer is NO, you throw away the groups that said "Yes" and keep the ones that said "No" to dance again.

The scientists wanted to see: How long can this group of magnets keep dancing without anyone noticing they've stopped pointing North?

The Discovery: A "Phase Transition"

In the world of quantum physics, there's a famous idea called a Measurement-Induced Phase Transition. It's like a traffic light for information.

• Green Light (Low Measurement): If you check the magnets rarely, the "dance" gets wild. The magnets get so tangled up that the whole system becomes a giant, complex web of connections. This is called a Volume Law (the more magnets you have, the more tangled it gets).
• Red Light (High Measurement): If you check the magnets constantly, you keep snapping the connections. The system stays simple and local. The magnets don't get to know their distant neighbors. This is called an Area Law (the complexity is limited to the surface).

Usually, scientists find a "tipping point" (a specific time $\tau$) where the system switches from Green to Red. It's like a switch flipping from a chaotic party to a quiet library.

The Twist: The "Small Room" vs. The "Mega-Stadium"

Here is where this paper gets interesting.

The researchers ran this experiment on a small chain of magnets (about 28). They found the tipping point! They saw the switch flip. It looked like a real, permanent phase transition. It was exciting because it happened even though they were checking the entire group at once (a "global" measurement), not just checking one magnet at a time.

But then, they did the math for a HUGE chain (up to 1,000 magnets).

And here is the shocker: The tipping point disappeared.

As the chain got bigger and bigger, the time it took to reach the "switch" got shorter and shorter.

• For 28 magnets, the switch happened at time $\tau = 0.2$.
• For 1,000 magnets, the switch happened at time $\tau = 0.006$.
• For an infinite chain (the real world), the switch happens at time zero.

The Analogy: The Whispering Game

Think of it like a game of "Whisper Down the Lane" (Telephone).

• The Small Group (L=28): If you have 28 people, and you whisper a message, it takes a little while for the message to get garbled. If you stop and check the message every few seconds, you can find a sweet spot where the message is still clear (Area Law) or completely scrambled (Volume Law). There is a distinct moment where it changes.
• The Huge Crowd (L=1000): Now imagine 1,000 people. The message gets scrambled almost instantly. If you try to find a "tipping point" where the message changes from clear to scrambled, you realize that it happens immediately. There is no "middle ground" anymore. The transition you saw in the small group was just an illusion caused by the group being too small to show the full picture.

Why Does This Matter?

1. The "Illusion" of Stability: Many previous studies on quantum computers and simulations used small systems (less than 40 particles). They saw these "phase transitions" and thought, "Aha! This is a fundamental law of nature!" This paper says, "Wait a minute. That might just be a side effect of the system being too small."
2. The Thermodynamic Limit: In physics, the "Thermodynamic Limit" means the system is infinitely large (like a real piece of metal, not a tiny chip). This paper suggests that for this specific type of measurement, the "Phase Transition" does not exist in the real, infinite world. It only exists in our small, finite simulations.
3. A New Warning: It prompts scientists to be very careful. Just because you see a cool transition in a small computer simulation doesn't mean it will survive when you scale it up to a real, massive quantum computer.

Summary in One Sentence

The paper shows that a "phase transition" (a sudden change in how quantum particles connect) that looks very real in small, toy-sized quantum systems actually vanishes and disappears when you look at the system on a truly massive scale, suggesting that what we thought was a fundamental rule might just be a trick of small numbers.

Technical Summary

1. Problem Statement

Measurement-Induced Phase Transitions (MIPT) are a phenomenon where the scaling behavior of entanglement entropy in a quantum many-body system changes abruptly (e.g., from a "volume law" to an "area law") based on the frequency or strength of measurements. While extensively studied in random quantum circuits with local, probabilistic measurements, the behavior of MIPT in deterministic, global measurements on interacting spin systems remains less understood.

A critical open question is whether these transitions persist in the thermodynamic limit (infinite system size, $L \to \infty$). Most existing studies focus on small system sizes ($L < 40$), where transitions are observed, but it is unclear if these are finite-size artifacts or robust features of the infinite system. This paper investigates an interacting spin-1/2 system under a specific protocol of global projective measurements to determine the fate of the transition as system size increases.

2. Methodology

System and Protocol:

• Model: A 1D chain of $L$ Ising spins evolving under the Transverse Field Ising Hamiltonian:
  $$H = -\sum_{j=1}^{L} \sigma_j^x \sigma_{j+1}^x - h \sum_{j=1}^{L} \sigma_j^z$$
• Initial State: All spins aligned in the $+Z$ direction ($|\psi_0\rangle = |00\dots0\rangle$).
• Dynamics: The system undergoes a discrete-time evolution consisting of:
  1. Unitary Evolution: Evolution under $H$ for a fixed time duration $\tau$.
  2. Global Projective Measurement: A deterministic measurement is performed with certainty ($P=1$) at every time step. The measurement asks: "Are all spins up?" (i.e., projecting onto the state $|\psi_0\rangle$).
  3. Post-Selection: If the answer is "Yes," the system is reset/removed from the ensemble. If "No," the system continues evolving. The study tracks the Survival Probability ($R_n$), defined as the probability that the system has not been detected (i.e., answered "No" to the measurement) after $n$ steps.

Quantities Analyzed:

1. Survival Probability ($R_n$): The probability of the system remaining in the subspace orthogonal to the initial state after $n$ steps.
2. Bipartite Entanglement Entropy ($S$): Calculated for a bipartition of the chain ($L/4$ vs. $3L/4$).
3. Generalized Geometric Measure (GGM): A measure of genuine multipartite entanglement.

Analytical Approach:

• The authors derived a recursion relation for the wavefunction coefficients to compute the survival probability for large system sizes ($L$ up to 1000), overcoming the exponential scaling of Hilbert space.
• They utilized the exact solution of the Transverse Ising model (via Jordan-Wigner transformation to free fermions) to derive closed-form expressions for the overlap functions required in the recursion.

3. Key Contributions

1. Analytical Derivation for Large Systems: Unlike previous works limited to small $L$ due to numerical constraints, the authors derived a recursion relation enabling the calculation of survival probabilities for $L \sim 1000$.
2. Finite-Size Scaling Analysis: They demonstrated that the critical point $\tau_c$ (where the phase transition occurs) is not constant but scales with system size as $\tau_c \sim 1/\sqrt{L}$.
3. Thermodynamic Limit Conclusion: They proved analytically and numerically that as $L \to \infty$, the critical point $\tau_c$ vanishes ($\tau_c \to 0$). This implies that the measurement-induced phase transition observed in finite systems is a finite-size effect and does not exist in the thermodynamic limit for this specific protocol.
4. Novelty in Measurement Mechanism: The study highlights a transition that occurs even though the global measurement does not always reduce entanglement (unlike in random circuit models where measurement suppresses entanglement). In some cases, the measurement actually increases entanglement, yet the transition signature remains.

4. Results

Numerical Findings (Small Systems, $L \le 28$):

• Survival Probability: The plot of $R_n$ vs. time step $n$ exhibits a "plateau" region. The height of this plateau ($H$) depends on $\tau$. The derivative $dH/d\tau$ shows a distinct peak at a critical value $\tau_c \approx 0.2$ (for $h=1/2$).
• Entanglement Transition: At this same $\tau_c$, the bipartite entanglement entropy transitions from an area law (flat with $L$) to a volume law (linear with $L$).
• Multipartite Entanglement: The GGM also shows a transition signature at the same $\tau_c$.
• Robustness: These transitions were observed for different transverse field strengths ($h=1/2, 1, 3/2$), indicating the phenomenon is independent of the ground state order of the Hamiltonian.

Large System Analysis ($L$ up to 1000):

• Scaling of Critical Point: As $L$ increases, the peak in $dH/d\tau$ shifts towards $\tau = 0$.
• Data Collapse: When plotting the derivative against the scaled variable $\sigma = \tau\sqrt{L}$, all curves for different $L$ collapse onto a single universal curve with a peak at $\sigma \approx 1$.
• Conclusion: This confirms the scaling law $\tau_c \propto 1/\sqrt{L}$. Consequently, in the thermodynamic limit ($L \to \infty$), $\tau_c = 0$. The system effectively remains in the "area law" phase (or the transition point moves to zero time), meaning the distinct phase transition observed at finite sizes disappears.

Supplementary Findings:

• For $h > 1$ (disordered ground state), the survival probability decays logarithmically at large times, a behavior not seen for $h < 1$.
• The transition point depends on the initial state; for initial states aligned with the X-axis, the survival probability does not capture the transition, though entanglement does.

5. Significance

• Challenging the Universality of MIPT: The paper provides strong evidence that measurement-induced phase transitions are not always robust in the thermodynamic limit. It suggests that the "volume-to-area" law transition observed in many numerical studies of small systems might be an artifact of finite size when global or specific deterministic measurements are used.
• Distinction from Random Circuits: It highlights a fundamental difference between random local measurements (where MIPT is believed to persist) and deterministic global measurements (where it may vanish).
• Experimental Relevance: The protocol involves global measurements, which are experimentally feasible in trapped-ion and superconducting qubit systems. The results suggest that experimentalists must carefully consider system size when interpreting MIPT signatures, as the transition may disappear in larger, more realistic setups.
• Theoretical Insight: The work underscores the importance of scaling analysis in quantum dynamics. It demonstrates that a transition appearing robust at $L \sim 20$ can completely vanish at $L \sim 1000$, urging a re-evaluation of the thermodynamic stability of MIPT in various models.

In summary, the authors identify a measurement-induced phase transition in a finite-size quantum spin chain but rigorously prove that this transition is a finite-size phenomenon that recedes to zero in the thermodynamic limit, challenging the assumption that such transitions are universally stable features of open quantum systems.