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Exact Localized Information Capacity of a Local Observable in a Hilbert-Space-Shattered System

This paper establishes a rigorous, closed-form expression for the exact localized information capacity of local observables in Hilbert-space-shattered systems, demonstrating that this capacity is determined by the projection onto the exact commutant algebra and yields specific, non-universal bit densities (such as $\sim 1.09$ bits/site for the dipole-conserving chain and $2/3$ bit/site for the $t$ - $J_z$ chain) that explain why complete memory in these systems arises from statistically localized data rather than strictly local integrals of motion.

Original authors: Hikaru Wakaura, Taiki Tanimae

Published 2026-07-16

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Hikaru Wakaura, Taiki Tanimae

Original paper licensed under CC BY 4.0 (https://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

The Quantum Memory Game: When the Universe Gets Stuck

Imagine you have a giant, chaotic ballroom filled with thousands of dancers. In a normal party, everyone eventually mixes up, forgets who they started with, and the room reaches a state of total, uniform chaos. This is how most quantum systems behave: they "thermalize," scrambling any initial information until it's gone forever. But sometimes, the rules of the dance floor are so strict that the dancers get stuck in tiny, isolated groups. They can't mix with the whole crowd; they can only shuffle with their immediate neighbors. This phenomenon is called Hilbert-space fragmentation. It's like the ballroom has been shattered into a million tiny, disconnected rooms, and once a dancer steps into one, they can never leave it.

To understand how much "memory" these stuck systems keep, scientists use a concept called information capacity. Think of this as asking: "If I look at just one dancer in the crowd after a billion years, how many bits of information can I still read about where they started?" In some systems, like those with "many-body localization," we know the answer is roughly estimated. In others, we have to run supercomputers to guess. But there's a special class of these "stuck" systems where the rules are so rigid that the math is exact. The big question has been: Can we count the exact number of bits of memory these systems hold, and can we do it without guessing?

The Exact Count of Frozen Memories

In this new research, the authors, Hikaru Wakaura and Taiki Tanimae, have finally cracked the code for a specific type of quantum system that is "shattered" by strict rules. They didn't just guess or run a simulation; they derived an exact, closed-form mathematical formula to count the memory.

Imagine a line of spin-1 particles (think of them as tiny magnets that can point up, down, or stay neutral). These particles are governed by a rule that conserves not just their total "charge" (how many are up or down) but also their "dipole moment" (a more complex balance of their positions). Because of this double rule, the system shatters into an enormous number of isolated "sectors." Once the system starts in one sector, it can never jump to another.

The authors defined a new measure called the Localized Information Capacity. In plain English, this is the maximum number of bits of information about the starting state that a single local observer can read out after infinite time. They proved that for these shattered systems, this number isn't a fuzzy estimate—it is an exact value determined solely by the structure of the disconnected rooms (the sectors), regardless of the specific energy rules (Hamiltonian) of the system.

The Silver Ratio Surprise
For their main model, the spin-1 dipole-conserving chain, the authors found a beautiful, surprising pattern. As the chain gets longer, the number of these isolated sectors grows in a very specific way related to the silver ratio (a number similar to the famous golden ratio, but involving $\sqrt{2}$ ).

They calculated that the number of sectors, $K_L$ , follows a formula involving Pell numbers.
This leads to a theoretical maximum density of information of about 1.27 bits per site (specifically $\log_2(1+\sqrt{2})$ ).
However, because the sectors aren't all the same size (some are huge, some are tiny), the actual average memory capacity is slightly lower. Through rigorous math, they proved the exact capacity converges to 1.09(1) bits per site.

To put this in perspective, they compared this to a simpler model called the t–Jz chain. In that system, the memory capacity is exactly 2/3 of a bit per site. This is a perfect, clean number: it's simply the density of particles times one bit of spin information that never gets scrambled.

The "Non-Universal" Excess
The paper also clarifies a common confusion. The total memory a system keeps (the autocorrelation) is often higher than this exact "capacity" number. The authors explain that the difference is "non-universal dynamical memory."

Think of the Exact Capacity as the floor of a building that is guaranteed by the laws of physics (the fragmentation rules). You can't go below this floor.
The Excess is the extra furniture you might find in the room. It depends on the specific details of the system (like disorder or random noise) and isn't guaranteed by the basic rules.
They showed that if you add random disorder to the system, this "extra" memory actually grows, proving it's a separate, dynamical effect, not part of the fundamental shattered structure.

Why Can't We Read It All Locally?
One of the most fascinating findings is why we can't read all this memory with a simple, strictly local rule. The authors used a clever analogy involving a "partially-commutative monoid" (a fancy math term for a set of moves where some swaps are allowed and others aren't).

Imagine trying to sort a deck of cards by only looking at two cards at a time. Sometimes, you can swap two cards if they are different; sometimes you can't.
The "complete label" of the system (which sector it is in) requires knowing the order of these swaps.
The authors proved that no strictly local "sweep" (looking at a small window and moving down the line) can perfectly identify the sector. You need statistically localized information. This means the memory is "frozen" in a way that is spread out and requires statistical knowledge of the whole chain to decode, rather than a sharp, local switch.

The Bottom Line
This paper turns the mysterious "memory" of shattered quantum systems into a precise, countable number. It shows that for these specific systems, the memory is an exact, analytic value (like 1.09 bits per site) that can be calculated from first principles. It separates the "guaranteed" memory from the "accidental" memory caused by disorder. Most importantly, it explains why the "keys" to these quantum rooms (the integrals of motion) are fuzzy and statistical rather than sharp and local, solving a long-standing puzzle about how these systems store information forever.

Technical Summary: Exact Localized Information Capacity in Hilbert-Space-Shattered Systems

Problem Statement
The paper addresses a gap in quantifying the memory of closed quantum systems that fail to thermalize due to strong Hilbert-space fragmentation (shattering). While Many-Body Localization (MBL) memory is quantified via the dynamical Holevo quantity (relying on approximate integrals of motion) and generic systems via numerical subsystem information capacity, fragmented systems lack a rigorous, analytical measure of "localized information capacity" in bits. Existing characterizations for fragmented systems—such as infinite-time autocorrelation Mazur bounds, statistically localized integrals of motion (SLIOMs), or counting symmetry-protected qubits—do not provide a closed-form, bits-valued measure of the information a local observable retains about the initial state. Furthermore, the relationship between the exact commutant algebra of fragmented systems and this information capacity has not been established.

Methodology
The authors define the localized information capacity of a local observable $A$ as the Holevo/mutual information carried by the infinite-time-retained state about the initial sector label.

Theoretical Framework: They utilize the fact that strong fragmentation splits the Hilbert space into dynamically disconnected Krylov sectors $\{P_\alpha\}$ , which form a complete orthogonal basis for the commutant algebra $\mathcal{C}$ .
Universal Capacity Formula: They derive a closed-form expression for the squared Hilbert-Schmidt norm of the projection of $A$ onto the commutant:
$C_{\text{univ}}(A) = \frac{1}{D} \sum_\alpha \frac{[\text{Tr}(AP_\alpha)]^2}{\text{Tr}P_\alpha}$
This quantity is shown to be Hamiltonian-independent, relying solely on the sector decomposition.
Information-Theoretic Mapping: For infinite-temperature states, the retained part of the system is diagonal. The authors map the sector entropy $H(\{p_\alpha\})$ (where $p_\alpha$ is the weight of sector $\alpha$ ) to the total information capacity $\chi$ , and define the region-resolved capacity $\chi_R$ as the mutual information between the sector label and local configurations.
Model Analysis: The framework is applied to two specific models:
- Spin-1 Dipole-Conserving Chain: A system with local states $S^z_i \in \{-1, 0, +1\}$ conserving charge $N$ and dipole moment $P$ . The dynamics are mapped to a height field $h_j$ where moves correspond to adjacent transpositions of heights differing by one.
- $t-J_z$ Chain: A model of hard-core spinful particles hopping without spin exchange.

Key Contributions and Results

Exact Closed-Form Capacity: The paper establishes that in fragmented systems, the localized information capacity is not a numerical estimate but an exact, closed-form value derived from the sector structure. It serves as a rigorous lower bound on the true infinite-time autocorrelation, with any excess attributed to non-universal dynamical memory.
Spin-1 Dipole Chain Analysis:
- Sector Counting: The authors derive the exact number of Krylov sectors $K_L$ using a transfer matrix formalism on sector boundary types. The count follows a recurrence relation yielding $K_L = 2P_{L+1} - 1$ (where $P_n$ are Pell numbers), leading to a silver-ratio growth rate: $\lim_{L\to\infty} \frac{\log_2 K_L}{L} = \log_2(1+\sqrt{2}) \approx 1.2716$ .
- Capacity Density: Due to exponentially non-uniform sector sizes, the exact capacity density converges to $\chi_\infty = 1.09(1)$ bits per site, strictly below the equal-size bound.
- Autocorrelation: The fragmentation-specific structure nearly doubles the autocorrelation floor guaranteed by polynomial multipole charges alone.
$t-J_z$ Chain Analysis:
- The sector label corresponds to the spin word. The authors show the capacity closes fully to exactly $2/3$ bit per site, calculated as the particle density times one retained spin bit.
Nature of Integrals of Motion:
- The authors identify that the complete labeling of retained bits resists strict locality. While the dynamics can be described as a partially-commutative (trace) monoid, the "free-trace" invariants (computable by a strictly local sweep) are provably incomplete.
- The missing labels correspond to "frozen orderings" of commuting pairs, which constitute the statistically localized data. This provides a first-principles explanation for why the complete integrals of motion in shattered Hilbert spaces are statistically localized rather than sharply local.
Hierarchy of Bounds: The study demonstrates a clear hierarchy in lower bounds for autocorrelation: Global symmetries < Linear span of SLIOMs < Full algebra of sector projectors ( $C_{\text{univ}}$ ) < Total infinite-time autocorrelation ( $C_\infty$ ). The gap between $C_{\text{univ}}$ and $C_\infty$ represents non-universal dynamical memory that grows with disorder.

Significance
The paper claims to resolve the structural question of how to quantify memory in fragmented systems by turning the "memory of a shattered Hilbert space" into an exact bit count. Its significance lies in:

Analytical Precision: Providing the first exact, Hamiltonian-independent, closed-form expression for information capacity in fragmented systems, distinguishing them from MBL (approximate) and generic systems (numerical).
Separation of Memory: Clearly separating the universal, symmetry-guaranteed memory floor from non-universal dynamical memory.
Fundamental Insight: Explaining from first principles why complete integrals of motion in these systems must be statistically localized, as strictly local rules cannot capture the frozen orderings required for complete sector labeling.
Measurability: Asserting that this capacity is directly measurable in site-resolved platforms (e.g., tilted Fermi-Hubbard chains, superconducting processors) via mutual information between prepared sector labels and snapshot statistics.

The Quantum Memory Game: When the Universe Gets Stuck

The Exact Count of Frozen Memories

Technical Summary: Exact Localized Information Capacity in Hilbert-Space-Shattered Systems

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