Fidelity Strange Correlators for Average Symmetry-Protected Topological Phases

This paper introduces the "fidelity strange correlator" (FSC) as a boundary-free diagnostic tool to identify nontrivial Average Symmetry-Protected Topological (ASPT) phases in open quantum systems by revealing long-range correlations and establishing connections to statistical loop models for deriving exact scaling exponents.

The Gist

Imagine you are trying to identify a very special, secret club. In the world of quantum physics, this "club" is a Symmetry-Protected Topological (SPT) phase. These are special states of matter that look boring on the inside but have a hidden, magical structure that protects them from falling apart, as long as certain rules (symmetries) are followed.

Usually, to find out if a material is in this secret club, you have to look at its edges (boundaries). The edges glow with a special signal. But what if you can't see the edges? What if you only have a messy, noisy, middle chunk of the material? That's where this paper comes in.

Here is the story of their discovery, explained simply:

1. The Problem: The "Noisy Room"

In the real world, quantum computers and materials are never perfect. They are like a noisy room where people are shouting, doors are slamming, and the lights are flickering. This "noise" (decoherence or disorder) usually destroys the delicate quantum secrets.

However, the authors realized something amazing: Even if the noise breaks the rules locally (in one spot), the rules might still hold on average (if you look at the whole room over time). They call these new states Average Symmetry-Protected Topological (ASPT) phases. It's like a chaotic party where everyone is dancing randomly, but if you take a time-lapse video, you see a perfect pattern emerge.

2. The Old Tool: The "Strange Mirror"

Physicists already had a tool to find these secret clubs in perfect, quiet rooms. It's called a Strange Correlator.

• The Analogy: Imagine you have a mysterious, patterned rug (the quantum state). You want to know if it's special. You lay a plain, boring white rug next to it (the "trivial" reference). Then, you shine a flashlight through both.
• If the mysterious rug is just a normal rug, the light passes through and fades quickly.
• If the rug is a "Secret Club" rug, the light travels all the way across the room without fading. This "long-distance glow" tells you the rug is special.

3. The New Problem: The "Fuzzy Photo"

The old tool works on perfect, single images (pure quantum states). But our "noisy room" is a fuzzy photo (a mixed state). You can't just shine a flashlight through a blurry picture; the math gets messy. The old tool breaks because you can't define a single "image" when the system is noisy.

4. The New Solution: The "Fidelity Strange Correlator" (FSC)

The authors invented a new tool called the Fidelity Strange Correlator (FSC).

• The Analogy: Instead of comparing two single images, imagine you have a stack of photos (representing all the possible noisy versions of the system).
• The FSC asks: "If I take every photo in the stack of the mysterious rug and compare it to every photo in the stack of the boring white rug, how similar are they on average?"
• The Magic: If the mysterious rug is a true ASPT (a secret club member), this "average similarity" stays high even when you look at points far apart. If it's just a normal rug, the similarity drops to zero quickly.

This tool is special because it works directly on the "fuzzy photo" (the bulk density matrix) without needing to see the edges of the room.

5. The Secret Code: "Watermelons" and "Loops"

To prove their tool works, the authors looked at specific examples in 1D and 2D. They found a surprising connection to a game played with loops.

• Imagine drawing lines on a piece of paper. Sometimes the lines form loops.
• In these special quantum states, the "FSC" turns out to be exactly the same as calculating the probability of drawing a specific shape with these loops.
• They call this shape a "Watermelon Correlator" (imagine a watermelon sliced into a few segments; the lines connect the center to the outside).
• The Result: By using this loop game, they could predict exactly how the "glow" of the FSC should behave. They found that for these special states, the glow follows a specific mathematical power law (it fades slowly, like a long-distance radio signal), proving the state is indeed a secret club member.

6. How to Measure It in the Real World

You might ask, "Okay, but how do we actually measure this in a lab?"

• The Challenge: Measuring quantum states usually requires taking a million photos of every single particle, which takes forever (exponentially hard).
• The Trick: The authors suggest using a technique called Classical Shadow Tomography.
• The Analogy: Instead of taking a high-definition photo of the whole room, you take a few quick, random snapshots with a blurry camera. You then use a computer algorithm to "reconstruct" the most important features of the room from these few snapshots.
• This allows scientists to estimate the FSC quickly and efficiently, making it possible to test these theories on real, noisy quantum computers.

Summary

This paper is like finding a new pair of glasses that allows us to see hidden quantum patterns in a messy, noisy world.

1. The World: Noisy, imperfect quantum systems.
2. The Mystery: Are there hidden topological phases that survive the noise?
3. The Tool: The Fidelity Strange Correlator, which compares "stacks of fuzzy photos" to find long-range connections.
4. The Proof: It connects to a mathematical game of loops, proving these states exist and have unique properties.
5. The Future: We can now measure this on real quantum devices using "quick snapshots" (Classical Shadows).

This work bridges the gap between abstract theory and the messy reality of experimental physics, giving us a way to find "magic" in the noise.

Technical Summary

1. Problem Statement

Symmetry-Protected Topological (SPT) phases are characterized by short-range entanglement protected by symmetries. Traditionally, these phases are identified via boundary signatures (gapless edge modes) or bulk correlation functions that decay exponentially. However, identifying SPT phases becomes challenging in open quantum systems or disordered systems where:

• The system is described by a mixed state (density matrix $\rho$) rather than a pure wavefunction, due to decoherence or ensemble averaging over disorder.
• Symmetries may be broken locally but preserved only on average (Average Symmetry-Protected Topological or ASPT phases).
• Access is limited to a single bulk density matrix without physical boundaries.

Existing tools like the Strange Correlator (defined for pure states) do not directly apply to mixed states. Furthermore, standard correlation functions of local observables in mixed states often decay exponentially even in non-trivial topological phases, failing to distinguish ASPTs from trivial states. The paper addresses the need for a robust, bulk-based diagnostic tool to detect non-trivial ASPT phases in noisy or disordered environments.

2. Methodology

The authors introduce a novel diagnostic tool called the Fidelity Strange Correlator (FSC).

• Definition: The FSC extends the concept of the strange correlator to mixed states using quantum fidelity. For a mixed state $\rho$ and a reference trivial state $\rho_0$ (both preserving exact and average symmetries), the FSC is defined as:
  $$C(r, r') = \frac{F(\rho, \phi(r)\phi(r')\rho_0\phi(r)\phi(r'))}{F(\rho, \rho_0)}$$
  where $F(\rho, \sigma) = \text{Tr}\sqrt{\sqrt{\rho}\sigma\sqrt{\rho}}$ is the fidelity, and $\phi$ are local operators.
• Theoretical Framework:
  • The authors utilize the decorated domain wall picture. An ASPT state is viewed as an ensemble of symmetry defect configurations (domain walls) of the average symmetry, where each defect is "decorated" with a lower-dimensional SPT state protected by the exact symmetry.
  • By expanding the density matrix in a basis of symmetry-breaking configurations, the FSC is shown to be a statistical average of pure-state strange correlators, weighted by quantum corrections arising from the wavefunction overlap between the decorated SPT and the trivial state.
• Mapping to Statistical Models:
  • The authors map the FSC calculation to loop models (specifically $O(n)$ loop models) with quantum corrections.
  • The "quantum correction" manifests as a modification to the loop fugacity ($n$) and loop tension ($x$) in the statistical model.
  • Long-range or power-law behavior in the FSC corresponds to the system being in a dense loop phase or at a critical point of the mapped statistical model.
• Experimental Proposal:
  • To measure the FSC experimentally, the authors propose using Classical Shadow Tomography.
  • Since fidelity is non-linear and hard to measure directly, they utilize a series expansion of fidelity involving polynomials of density matrices (e.g., $\text{Tr}(\rho\sigma)$, $\text{Tr}((\rho\sigma)^2)$).
  • Classical shadow tomography allows for the efficient estimation of these polynomial terms using randomized measurements, providing a scalable pathway for experimental verification.

3. Key Contributions and Results

A. Theoretical Development of FSC

• Generalization: Successfully generalized the strange correlator from pure states to mixed states, creating a basis-independent diagnostic for ASPTs.
• Long-Range Detection: Proved that for non-trivial ASPTs, the FSC exhibits long-range order (constant) or power-law decay at large distances, whereas trivial states exhibit exponential decay.

B. Specific Examples and Scaling Exponents

The paper derives exact scaling exponents for the FSC in several 1D and 2D examples by mapping them to loop models:

1. 1D Average Cluster State ($Z_2 \times Z_2^A$):

   • The FSC of $Z$ operators on even sites is found to be long-range ordered ($C=1$), confirming the non-trivial nature of the decohered cluster state.

2. 2D Bosonic ASPT ($Z_2 \times Z_2 \times Z_2^A$) with 1D Decoration:

   • The decorated domain walls carry 1D cluster states.
   • The wavefunction overlap introduces a quantum correction, renormalizing the loop fugacity to $n=2$.
   • The FSC maps to the 2-leg watermelon correlator in an $O(2)$ loop model.
   • Result: In the dense loop phase, the FSC decays as a power law: $C(r, r') \sim |r-r'|^{-2\Delta_2}$ with exponent $\Delta_2 = 1/2$.
   • In the dilute phase, the standard FSC decays exponentially, but an FSC defined with average-symmetry charged operators ($\sigma$) remains long-range ordered.

3. 2D Fermionic ASPT ($Z_2^f \times Z_2^A$) with 1D Decoration:

   • Domain walls carry 1D Kitaev chains (Majorana modes).
   • The overlap yields a loop fugacity of $n=\sqrt{2}$.
   • The FSC maps to the 2-leg watermelon correlator in an $O(\sqrt{2})$ model.
   • Result: Power-law decay with exponent $\Delta_2 = 1/3$ (dense phase) or $3/5$ (dilute critical point).

4. 2D Bosonic ASPT ($Z_3 \times Z_3^A$) with 0D Decoration:

   • Vortices of the average symmetry are decorated with $Z_3$ charges.
   • The FSC maps to the 3-leg watermelon correlator in an $O(2)$ model.
   • Result: Power-law decay with exponent $\Delta_3 = 9/8$.

C. Experimental Feasibility

• The paper provides a concrete protocol for measuring the FSC using Classical Shadow Tomography.
• It demonstrates that the leading-order approximation of the FSC (Type-II strange correlator) and higher-order corrections can be estimated efficiently, bypassing the exponential cost of full Quantum State Tomography (QST).

4. Significance

• Bridging Theory and Experiment: The work provides a practical, experimentally accessible method (via classical shadows) to detect topological order in noisy, open quantum systems, which are the reality of current NISQ (Noisy Intermediate-Scale Quantum) devices.
• New Phase of Matter: It solidifies the theoretical framework for ASPT phases, distinguishing them from trivial mixed states and showing that topological properties can persist (or even emerge) in the presence of decoherence and disorder.
• Universality: The connection between FSCs and statistical loop models with quantum corrections offers a powerful analytical tool to derive exact critical exponents for topological phases in mixed states, extending the reach of conformal field theory and statistical mechanics to open quantum systems.
• Diagnostic Power: The FSC serves as a robust bulk diagnostic that does not require physical boundaries, making it applicable to systems where edge states are obscured or inaccessible.

In summary, this paper establishes the Fidelity Strange Correlator as a definitive tool for identifying and characterizing topological phases in open and disordered quantum systems, linking abstract quantum information concepts with concrete statistical mechanics models and experimental measurement protocols.