⚛️ quantum physics

Information Critical Phases under Decoherence

This paper identifies and characterizes a novel "information critical phase" in decohered $\mathbb{Z}_{N}$ toric codes (for $N>4$ ), where diverging Markov length and enhanced symmetry lead to a gapless mixed-state phase that preserves a finite fraction of logical information, thereby extending the paradigm of quantum memory beyond conventional gapped topological phases.

Original authors: Akash Vijay, Jong Yeon Lee

Published 2026-08-11

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Akash Vijay, Jong Yeon Lee

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 you are trying to send a secret message across a noisy room. In the world of quantum computing, this message isn't just a string of zeros and ones; it's a delicate, shimmering state of matter that can exist in many places at once. The problem is that the "noise" of the real world—heat, stray magnetic fields, or just the universe being a bit chaotic—tends to scramble these messages instantly. Scientists have spent decades building "quantum error-correcting codes," which are like magical, self-healing envelopes designed to protect these fragile messages even when the room gets noisy.

To understand if these envelopes are working, scientists use two main rulers. The first is the correlation length, which measures how far a ripple of information can travel before it fades away. Think of it like shouting a secret to a friend; if the room is too loud, the sound dies out after a few feet. The second ruler is the Markov length, a more sophisticated tool that measures how much "context" you need to understand a message. If you know what your friend whispered to the person next to them, does that help you guess what they told you? In a normal, noisy room, both of these rulers are short; information gets lost quickly, and you can't recover the secret. But what if there was a weird, in-between state where the sound dies out immediately (short correlation length), yet you could still piece together the whole story if you had enough context (infinite Markov length)? That is the strange, new territory this paper explores.

The authors, Akash Vijay and Jong Yeon Lee, investigate a specific type of quantum code called the $Z_N$ toric code. You can imagine this code as a giant, flat donut (a torus) made of tiny quantum switches. These switches are arranged so that the information is stored not in any single switch, but in the way the whole donut is twisted and knotted. The researchers asked: What happens if we poke this donut with noise? They found that for codes with a certain level of complexity (specifically when the number of states $N$ is greater than 4), the system doesn't just break down or stay perfect. Instead, it enters a bizarre, middle-ground state they call an information critical phase.

In this new phase, the system behaves like a "fractional" memory. Imagine you have a safe that holds a million dollars. If the safe is broken, you lose everything. If it's perfect, you keep it all. But in this "information critical" phase, the safe is partially broken: you might lose the small bills, but the big bills stay safe. The researchers showed that even though the noise is strong enough to scramble the fine details of the message, a finite fraction of the logical information remains protected forever, no matter how big the system gets. It's as if the noise creates a fog that hides the small details but leaves the large shapes visible.

To prove this, the team used a clever trick. They translated the messy quantum problem into a game of statistical mechanics, similar to how magnets align in a material. They found that in this middle phase, the "magnetic spins" in their model don't just order or disorder; they form a "superfluid" state where the system and the environment dance together in a synchronized, gapless rhythm. This dance creates a new kind of order that is neither fully solid nor fully liquid. They also built a super-smart decoder (a computer program that tries to fix the scrambled message) and found that it hits a wall: it can perfectly fix the big, obvious errors, but it gets stuck guessing on the tiny ones. This confirms that the memory is indeed "fractional"—it works perfectly for some parts of the data and fails for others, but it never completely collapses.

The paper suggests that this isn't just a quirk of one specific code, but a fundamental new type of phase of matter for mixed states (states that are a mix of quantum and classical). It challenges the old idea that quantum memory is either "on" or "off." Instead, it reveals a landscape where memory can be "dimmed" but not extinguished, offering a new, robust way to store information even in a noisy world. While the results are based on sophisticated simulations and mathematical proofs rather than a physical experiment in a lab, the consistency of the findings across different methods suggests this "fractional" phase is a real and stable feature of quantum systems with enough complexity.

Technical Summary: Information Critical Phases under Decoherence

Problem Statement
Quantum information is inherently fragile to environmental noise, necessitating robust quantum error-correcting codes. While topological codes (like the toric code) offer a promising route to fault tolerance, the classification of their phases under decoherence remains challenging. In pure-state systems, phases are distinguished by correlation lengths and adiabatic connectivity. However, for mixed states (decohered systems), the absence of a parent Hamiltonian complicates this classification. Recent work suggests that the "Markov length" ( $\xi_M$ ), which governs the decay of Conditional Mutual Information (CMI), serves as the natural analog to the correlation length for mixed-state phases. A critical question arises: can mixed-state phases exist where the Markov length diverges (indicating long-range conditional correlations) while the ordinary correlation length remains finite? This paper investigates the existence and nature of such "information critical phases" in decohered $\mathbb{Z}_N$ toric codes.

Methodology
The authors employ a combination of analytic diagonalization, statistical-mechanical mapping, and advanced numerical simulations:

Analytic Diagonalization: The decohered density matrix of the $\mathbb{Z}_N$ toric code is diagonalized in the anyon basis. The authors demonstrate that the spectrum of the decohered state can be expressed in terms of partition functions of two-dimensional disordered $\mathbb{Z}_N$ clock models (specifically, random bond clock models) evaluated along the Nishimori line.
Information-Theoretic Diagnostics:
- Coherent Information ( $I^{(c)}$ ): Used to quantify the robustness of encoded logical information. It is expressed as a disorder average of Shannon entropies of conditional winding number distributions.
- Conditional Mutual Information (CMI): Used to define the Markov length ( $\xi_M$ ). The authors analyze the spatial decay of CMI to distinguish between phases.
Statistical-Mechanical Mapping: The problem of characterizing mixed-state phases is mapped to the thermodynamic phases of disordered clock models. For $N > 4$ , these models are known to exhibit a Quasi-Long-Range Ordered (QLRO) phase.
Numerical Simulations:
- Worm Algorithm: Used to sample flow configurations in the current-loop expansion of the disordered clock model to compute conditional winding number probabilities.
- Parallel Tempering: Implemented to overcome critical slowing down in the QLRO phase, allowing for efficient sampling across topological sectors.
- Maximum Likelihood Decoding: A decoder is constructed using the worm algorithm to determine the most probable logical sector given a syndrome, providing an operational test of decodability.
Dual Models: The authors utilize Kramers-Wannier (KW) duality to map the decohered toric code to an "ungauged" $\mathbb{Z}_N$ spin model, interpreting the phase diagram in terms of Strong-to-Weak Spontaneous Symmetry Breaking (SWSSB).

Key Contributions and Results

Discovery of the Information Critical Phase:
- For $N \le 4$ (e.g., $\mathbb{Z}_2$ , $\mathbb{Z}_4$ ), the decohered toric code exhibits a single transition between a decodable (ordered) and non-decodable (disordered) phase.
- For $N > 4$ (e.g., $\mathbb{Z}_6$ , $\mathbb{Z}_8$ ), the system hosts a distinct Information Critical Phase intervening between the decodable and non-decodable phases.
- In this phase, the Markov length diverges ( $\xi_M = \infty$ ), indicating long-range conditional correlations, while the ordinary correlation length remains finite ( $\xi < \infty$ ). This separation of scales is an intrinsically mixed-state phenomenon.
Fractional Quantum Memory:
- In the information critical phase, the coherent information saturates to a non-zero fractional value in the thermodynamic limit.
- This indicates that a finite fraction of the logical information remains protected, while the rest is lost.
- The protection is "charge-dependent": logical information stored in superpositions of states with large charge separation remains protected, while information in superpositions with small charge separation is destroyed.
- The logical error rate, when decoded via maximum likelihood, saturates to a fractional value rather than zero or the random-guessing limit.
Structure of the Mixed State:
- Using the Minimally Entangled Typical Thermal State (METTS) ansatz, the authors show that the decohered density matrix in the information critical phase decomposes into a convex sum of "Coulombic" pure states.
- These states exhibit power-law correlations of string operators and support gapless gauge modes (emergent photons), effectively behaving as an ensemble of quantum critical pure states.
Dual Interpretation (SWSSB):
- In the KW dual model, the information critical phase corresponds to a regime where the spontaneously broken off-diagonal $\mathbb{Z}_N$ symmetry is effectively enhanced to a continuous $U(1)$ symmetry.
- This results in a novel superfluid phase with quasi-long-range order (QLRO), where gapless modes involve collective excitations of both the system and the environment.
- The phase diagram is interpreted as a sequence: Strong $\mathbb{Z}_N$ Symmetry $\to$ $U(1)$ QLRO (Information Critical) $\to$ Weak $\mathbb{Z}_N$ Symmetry (SWSSB).
Self-Duality and Critical Points:
- The two critical points bounding the information critical phase satisfy a self-duality relation derived from the coherent information of raw qudits: $\log N = H(T_c^{(1)}) + H(T_c^{(2)})$ .
- The second critical point (transition to the non-decodable phase) is found to be approximately independent of $N$ .

Significance
The paper claims to extend the conventional paradigm of quantum memory phases by identifying a gapless analog for mixed-state phases. Unlike traditional gapped topological memories where logical information is either fully preserved or fully lost, the information critical phase acts as a fractional topological quantum memory. This work broadens the landscape of mixed-state phases, demonstrating that divergent Markov lengths can exist without divergent correlation lengths, and provides a concrete physical realization of this phenomenon in decohered topological codes. The findings suggest that topological protection can persist in a "fractional" form even in the presence of significant noise, provided the system possesses sufficient internal symmetry ( $N > 4$ ).

Technical Summary: Information Critical Phases under Decoherence

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