Local measurements and the entanglement transition in quantum spin chains

This paper demonstrates that performing local measurements of $G$-charge on infinite quantum spin chains initially in a non-trivial symmetry-protected topological phase drives a transition from short-range to long-range entangled states, resulting in post-measurement states that cannot be uniformly short-range entangled and exhibit maximally correlated almost local observables.

The Gist

The Big Picture: Turning a "Local" World into a "Connected" One

Imagine you have a very long line of people (a "quantum spin chain") holding hands. In this specific setup, everyone is only holding hands with their immediate neighbors. They are all in a special, organized state called a Symmetry-Protected Topological (SPT) phase.

Think of this state like a perfectly synchronized dance. Everyone knows exactly what to do based on their neighbor's move, but there are no "secret handshakes" connecting the person at the very beginning of the line to the person at the very end. In physics terms, this is Short-Range Entangled (SRE). The connections are local; the "magic" doesn't reach far.

The Big Question: What happens if we start peeking at these people? Specifically, what if we measure (look at) small groups of them one by one?

Usually, in quantum mechanics, looking at a system breaks its delicate connections. However, this paper discovers a surprising twist: Looking at the system in a specific way actually creates long-distance connections. It turns a local dance into a global telepathic network.

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The Analogy: The "Secret Code" Dance

Let's break down the key concepts using a story about a line of dancers.

1. The Setup: The Hidden Pattern

Imagine the dancers are following a complex rule. They are wearing red or blue hats. The rule is: "If your neighbor is red, you must be blue."

• The State: The whole line is perfectly coordinated.
• The Catch: The coordination is "hidden." If you look at just two people far apart, they seem random. But if you look at the whole path between them, you see a pattern. This is called String Order. It's like a secret code that only makes sense if you read the whole message, not just the first and last letters.

2. The Action: The "Local Measurement"

Now, imagine a judge walks down the line. Every time the judge stops at a group of dancers, they ask a specific question: "What is your 'charge'?" (In physics, this is measuring a property like spin).

• The judge forces the dancers to answer.
• Because of the secret code (the String Order), when the judge forces a group to reveal their answer, it creates a ripple effect.
• The Magic: The act of measuring the middle of the line forces the dancers at the far ends to become perfectly correlated. The "secret code" that was hidden in the middle gets pulled out and stretched across the whole line.

3. The Result: The Entanglement Transition

Before the judge arrived, the dancers at the ends didn't know anything about each other. After the judge measures the middle, the dancers at the ends are now Long-Range Entangled (LRE).

• They are now "telepathically" connected. If you change the state of the dancer at the start, the dancer at the end instantly reacts, even though they are miles apart.
• The paper proves that you cannot keep the system "local" (short-range) if you keep measuring larger and larger sections of it. The more you measure, the more the "local" connections break, and the more the "global" connections form.

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The Two Main Discoveries

The paper presents two main findings, which we can think of as two different ways to tell this story:

Discovery 1: The "Fuzzy" Measurement (The General Case)

Imagine the judge measures the dancers, but their eyes are a little blurry. They get the answer, but the "ripple" of information spreads out a bit.

• The Finding: Even with this blurriness, if you keep measuring longer and longer sections of the line, the "blur" of the connection gets worse and worse.
• The Metaphor: Imagine trying to keep a rubber band short while you keep stretching it. No matter how hard you try to keep it short, the more you pull (measure), the longer it must get. You can't keep the rubber band (entanglement) short forever. The paper proves mathematically that the "short-range" nature of the system must break down.

Discovery 2: The "Perfect" Measurement (The QCA Case)

Now, imagine the judge has laser-sharp vision and measures the dancers in perfect, non-overlapping blocks.

• The Finding: In this perfect scenario, the paper shows that you can create a state where the dancers at the very beginning and the very end are maximally connected.
• The Metaphor: It's like cutting a long rope into pieces and tying the ends of the pieces together in a specific way. If you do this perfectly, the two ends of the original rope become knotted together so tightly that they act as one single unit, regardless of how long the rope is.

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Why Does This Matter?

This isn't just about abstract math; it has real-world implications for Quantum Computing.

1. One-Way Quantum Computers: There is a type of quantum computer that works by measuring a "cluster state" (our line of dancers). This paper explains why that works. It shows that the measurements aren't just destroying the quantum state; they are transforming a simple, local state into a powerful, long-range entangled resource that can do complex calculations.
2. New Materials: It helps physicists understand how to create new materials that have "long-range" properties (like superconductivity or topological protection) just by manipulating them locally.

The Takeaway

The paper tells us that measurement is not just a passive act of looking; it is an active act of creation.

In the quantum world, if you have a system that is locally connected but globally "boring" (short-range entangled), and you start measuring it locally, you don't just destroy the order. You transform it. You pull the hidden "strings" of the system out, turning a local neighborhood into a globally connected community.

In short: By looking at the middle, you force the ends to talk to each other. The "entanglement transition" is the moment the system goes from being a collection of isolated pairs to a single, giant, connected whole.

Technical Summary

1. Problem Statement

The paper addresses a fundamental question in the classification of quantum phases of matter: How do local measurements affect the entanglement structure of Symmetry-Protected Topological (SPT) states?

• Context: Traditionally, quantum phases are classified by their equivalence under Locally Generated Automorphisms (LGAs) or Quantum Cellular Automata (QCA). These dynamics preserve the distinction between Short-Range Entangled (SRE) and Long-Range Entangled (LRE) states; specifically, they cannot create long-range entanglement from an SRE state.
• The Paradox: It is known that local measurements (unlike unitary dynamics) can create long-range entanglement (e.g., in the one-way quantum computer). However, a rigorous mathematical characterization of this "entanglement transition" for infinite quantum spin chains, particularly within the framework of SPT phases, has been lacking.
• Specific Goal: The authors investigate the transition from an SRE state in a non-trivial SPT phase to a state with long-range correlations induced by measuring local charges of an Abelian subgroup $G$ within a larger symmetry group $G \times H$.

2. Methodology and Mathematical Framework

The authors employ the rigorous framework of $C^*$-algebras and quasi-local automorphisms to handle infinite quantum spin chains.

A. Mathematical Setup

• Algebra of Observables: The system is defined on an infinite lattice $\mathbb{Z}$ with local algebras $A_j \cong M_{d}(\mathbb{C})$. The global algebra $A$ is the $C^*$-closure of the union of finite tensor products.
• Split Property & SRE: A state $\omega$ is defined as Short-Range Entangled (SRE) if it can be obtained from a product state $\omega_0$ via a strongly continuous one-parameter group of *-automorphisms $\{\alpha_s\}$ satisfying the split property. This means $\alpha_s$ can be decomposed into left/right automorphisms and an almost-local unitary $U^j$ (Definition 1).
• Symmetry and SPT Phases: The system possesses a global symmetry group $G = \mathcal{G} \times \mathcal{H}$, where $\mathcal{G}$ is Abelian. SPT phases are classified by the second cohomology group $H^2(G, U(1))$. The authors focus on non-trivial mixed SPT phases, where the cohomology class is non-trivial specifically between elements of the subgroups $\mathcal{G}$ and $\mathcal{H}$ (i.e., $\sigma(g, h) \neq 1$).

B. The Measurement Protocol

• Local Measurements: The authors define measurements of the $\mathcal{G}$-charge on intervals. For an Abelian group $\mathcal{G}$, the measurement operators are spectral projectors $P_q$ of the unitary representation $U_g$.
• Post-Measurement States: The state after measuring a region $[-n, n]$ with outcomes $q$ is given by $\omega_{n,q}(A) = \frac{1}{N} \omega(P_{n,q} A P_{n,q})$.
• Key Assumption: The initial state is an SPT state where the $\mathcal{G}$-subgroup is in a trivial phase (to allow for specific commutation properties), but the full group $G$ is in a non-trivial mixed phase.

C. Analytical Tools

• String Order Parameters: The authors utilize the existence of "string order" in SPT states. They construct almost-local operators $W^j_g$ that act as edge operators on half-chains.
• Projective Representations: They analyze the action of symmetry on half-infinite chains, showing that the symmetry action induces a projective representation characterized by a 2-cocycle $\sigma$.
• Decay Estimates: They use Lieb-Robinson-type bounds and decay functions $f \in \mathcal{F}$ (vanishing faster than any power) to quantify "almost-locality."

3. Key Contributions and Results

Main Result 1: Deterioration of Locality Bounds (Theorem 1)

The paper proves that while every finite post-measurement state $\omega_n$ remains mathematically SRE (it can be mapped to a product state by some automorphism), the locality bounds of the required automorphisms deteriorate as the measurement region grows.

• Statement: If $\omega$ is in a non-trivial mixed SPT phase, and $\omega_n$ are the post-measurement states, then the sequence of decay functions $f_{\alpha_n}$ associated with the automorphisms $\alpha_n$ (where $\omega_n = \omega_0 \circ \alpha_n$) cannot be uniformly bounded by a single function in $\mathcal{F}$.
• Implication: The post-measurement states cannot be uniformly short-range entangled. As $n \to \infty$, the "cost" to disentangle the state becomes infinite, signaling a transition to long-range order.

Main Result 2: Emergence of Long-Range Entanglement (Theorem 2)

Under the stronger assumption that the initial entangler is a Quantum Cellular Automaton (QCA) (strictly local propagation), the authors construct a limiting state that is genuinely Long-Range Entangled (LRE).

• Method: They perform measurements on blocked sites (blocks of size $2N+1$ matching the QCA range).
• Result: In the limit of infinite volume, the resulting state $\nu$ exhibits maximal correlations between local observables at arbitrary distances. Specifically, for operators $W^i$ and $W^j$ separated by distance $|i-j|$:
  $$|\nu(W^i W^j) - \nu(W^i)\nu(W^j)| = 1$$
• Mechanism: The measurement "unlocks" the hidden string order of the SPT state. The string order parameter, which was non-local in the original state, becomes a local observable in the post-measurement state, effectively converting the topological order into classical long-range order.

Technical Lemma: Preservation of String Order

A crucial intermediate result (Lemma 8 and Lemma 12) shows that the expectation value of the string order parameter remains unity after measurement, while the expectation value of the individual edge operators vanishes. This forces the correlation between distant edge operators to be maximal.

4. Significance and Impact

1. Rigorous Entanglement Transition: The paper provides the first rigorous mathematical proof (using $C^*$-algebraic methods) that local measurements can induce a transition from SRE to LRE in infinite systems, resolving a question raised by previous conjectures.
2. Beyond Matrix Product States (MPS): Unlike previous works that relied heavily on the Matrix Product State formalism (which is specific to 1D and often requires finite bond dimension assumptions), this work uses general cohomological indices and split automorphisms. This suggests the results may generalize to higher dimensions.
3. Connection to Kennedy-Tasaki: The authors draw a parallel between their measurement-induced transition and the Kennedy-Tasaki transformation (a non-local unitary that maps SPT to symmetry-broken phases). They show that local measurements perform a similar "unhiding" of order but via a distinct, probabilistic mechanism.
4. Implications for Quantum Computing: The results reinforce the power of measurement-based quantum computation (MBQC). They demonstrate that measurements are not just a tool for reading out information but can fundamentally alter the topological class of a state, converting hidden topological order into accessible long-range correlations.

Summary Conclusion

Bachmann, Rahnama, and Tournaire demonstrate that local measurements of symmetry charges in a non-trivial SPT phase act as a catalyst for an entanglement transition. While finite measurement regions yield states that are technically SRE, the locality required to disentangle them diverges with system size. In the thermodynamic limit (specifically for QCA-generated states), this results in a state with genuine long-range entanglement, characterized by maximal correlations between local observables. This work bridges the gap between the abstract classification of topological phases and the physical reality of measurement-induced dynamics.