Scheme to Detect the Strong-to-weak Symmetry Breaking via Randomized Measurements

The Gist

Imagine you have a complex, noisy machine (a quantum system) that is supposed to follow strict rules of symmetry, like a perfectly balanced mobile. Sometimes, this machine gets "noisy" or "decohered" by its environment. In the past, scientists knew about two ways a machine could break its symmetry: either it stays perfectly balanced (symmetric), or it falls over completely (broken symmetry).

However, this paper introduces a fascinating new middle ground called "Strong-to-Weak Symmetry Breaking" (SW-SSB). Think of it like this:

• Strong Symmetry: The machine is perfectly balanced, and even if you look at it from every angle, it looks the same.
• Weak Symmetry: The machine looks balanced from the outside, but if you peek inside, the internal gears are actually spinning out of sync.
• The Break: The machine starts in a "Strong" state but, due to noise, slips into a "Weak" state where the internal order is lost, even though the outside still looks okay.

The problem is that detecting this specific "slip" is incredibly hard. It's like trying to hear a whisper in a hurricane. Traditional methods require taking a "snapshot" of the machine twice and comparing them perfectly, which is nearly impossible to do in a real lab.

The New "Randomized Guessing" Trick

The authors propose a clever, practical way to detect this slip using a method they call Randomized Measurements. Here is the analogy:

Imagine you have two identical decks of cards (representing the quantum state).

1. The Original Deck: You shuffle the deck randomly and look at the cards.
2. The "Twisted" Deck: You take a second deck, but before shuffling, you secretly swap a few specific cards (this represents applying a "charged operator" or a Z-gate). Then you shuffle and look at these cards too.

Instead of trying to compare the decks card-by-card perfectly (which is hard), the authors suggest a game of "Hamming Distance" (counting differences):

• You perform this random shuffle-and-look game millions of times.
• Each time, you count how many cards are different between the two decks you looked at.
• If the system is in the Symmetric Phase (no breaking), the "Twisted" deck will look very different from the original deck most of the time. The "difference count" will be high and distinct.
• If the system is in the SW-SSB Phase (the breaking has happened), the "Twisted" deck will surprisingly look very similar to the original deck, even after the swap. The "difference count" will drop and look just like the original deck's pattern.

By running this game many times and looking at the statistics of the differences, they can tell if the symmetry has broken, without needing perfect, impossible measurements.

The "Small Sample" Shortcut

The paper also notes a practical hurdle: To get a perfect answer, you might need millions of samples, which takes a long time. However, the authors found a clever shortcut.

They realized that even with a small number of samples (like a quick glance rather than a long stare), you can still tell if the system is breaking symmetry. They use a mathematical tool called KL Divergence (think of it as a "Similarity Score").

• If the "Similarity Score" between the two decks is high, the system is in the new "Strong-to-Weak" phase.
• If the score is low, it's still in the normal symmetric phase.

They tested this on a simulated model (a chain of quantum bits, like a row of spinning tops) and found that even with a small system and a small number of guesses, their method could accurately draw the map of where the symmetry breaking happens.

Why This Matters (According to the Paper)

The authors claim this is a practical protocol that can be run on current, state-of-the-art quantum devices (like those using atoms or ions). It opens the door for experimentalists to actually see and study this new type of quantum phase (SW-SSB) in a real lab, rather than just talking about it in theory. They specifically mention that their method works well for systems with "all-to-all" connections, which are common in modern quantum computers.

In short: They invented a "statistical guessing game" that lets scientists detect a subtle, hidden change in quantum systems using random measurements, even when they don't have enough time or data to do a perfect measurement.

Technical Summary

Technical Summary: Scheme to Detect Strong-to-Weak Symmetry Breaking via Randomized Measurements

Problem Statement
Recent theoretical developments have identified a novel phase of matter in mixed quantum states known as Strong-to-Weak Spontaneous Symmetry Breaking (SW-SSB). In this scenario, the "strong" symmetry of a density matrix (where the symmetry operator commutes with the state up to a phase, $U\rho = e^{i\theta}\rho$) spontaneously breaks down to a "weak" symmetry (where $U\rho U^\dagger = \rho$). The standard theoretical diagnostic for SW-SSB is the Rényi-2 correlator, defined as:
$$C^{(2)}(j, k) = \frac{\text{tr}[O_j O_k^\dagger \rho O_k O_j^\dagger \rho]}{\text{tr}[\rho^2]}$$
where $O$ is a charged operator. While the denominator (purity) can be measured via randomized measurements, the numerator involves a complex overlap between the original state $\rho$ and a state evolved by charged operators ($\tilde{\rho} = O_j O_k^\dagger \rho O_k O_j^\dagger$). Currently, there is a lack of practical experimental protocols to measure this numerator for generic density matrices, hindering the experimental study of SW-SSB.

Methodology
The authors propose a concrete protocol to estimate the numerator of the Rényi-2 correlator using the randomized measurement toolbox. The scheme operates on a system of $N$ qubits and proceeds as follows:

1. Random Basis Selection: For each qubit $i$, a random Pauli direction $\hat{n}_i \in \{\hat{x}, \hat{y}, \hat{z}\}$ is chosen independently.
2. Measurement on Original State: The system is prepared in state $\rho$, and measurements are performed along $\hat{n}_i$, yielding a bit string outcome $s$.
3. Measurement on Evolved State: The system is prepared in the state $\tilde{\rho}$ (obtained by applying $Z$ gates to qubits $j$ and $k$ on $\rho$). Measurements are performed along the same random directions $\hat{n}_i$, yielding a bit string outcome $s'$.
4. Statistical Estimation: By repeating this process $N_a$ times, the numerator $P_{ZZ} = \text{tr}[\tilde{\rho}\rho]$ is estimated using the statistical correlation between $s$ and $s'$:
   $$P_{ZZ} \approx \frac{1}{N_a} \sum_{a=1}^{N_a} 2^N (-2)^{-D(s_a, s'_a)}$$
   where $D(s, s')$ is the Hamming distance between the bit strings. The authors prove that this estimator is unbiased in the limit of infinite samples.

To demonstrate the protocol, the authors utilize a solvable model: a transverse-field Ising chain subject to all-to-all decoherence. The decoherence channel is defined such that it preserves strong symmetry at low strength but drives a transition to SW-SSB at high strength. The critical point $\mu_c$ is derived analytically using the Choi-Jamiolkowski isomorphism and saddle-point approximation.

Key Results

• Unbiased Estimation: Numerical simulations using exact diagonalization for small system sizes ($N \lesssim 7$) confirm that the proposed protocol yields an unbiased estimate of $P_{ZZ}$ with a variance comparable to that of the purity measurement.
• Sample Complexity Limitations: The authors find that for larger system sizes ($N \gtrsim 7$), the standard deviation of the estimator becomes comparable to the expectation value when using a moderate number of samples ($N_a \sim 10^6$). This makes the direct calculation of the Rényi-2 correlator impractical for larger systems with current sample constraints.
• Alternative Detection via KL Divergence: To bypass the high sample complexity of direct correlator calculation, the authors propose an alternative criterion based on the distributions of Hamming distances, $p_I(D)$ (for purity) and $p_{ZZ}(D)$ (for the numerator).
  • In the symmetric phase, $P_{ZZ} \ll P_I$, leading to distinguishable distributions and a significant Kullback-Leibler (KL) divergence.
  • In the SW-SSB phase, $P_{ZZ}$ and $P_I$ are of the same order, causing the distributions to converge and the KL divergence to vanish.
• Phase Boundary Estimation: Using the KL divergence as a proxy, the authors successfully reconstructed the phase diagram of the decohered Ising model. Remarkably, this alternative approach provided a reasonable estimate of the phase boundary using a small system size ($N=7$) and a relatively small number of samples ($N_a = 10^4$), matching the theoretical prediction derived from Matrix Product State (MPS) simulations.

Significance and Claims
The paper claims to provide the first practical protocol for detecting SW-SSB in experiments using randomized measurements, a method accessible to current state-of-the-art quantum devices such as Rydberg atom arrays and trapped ion systems. The authors emphasize that their scheme does not require non-local operations or the preparation of multiple replicas simultaneously, making it feasible for near-term experiments.

The work opens a pathway for the experimental study of novel quantum phases in mixed states. By demonstrating that the phase boundary can be estimated with high accuracy even with limited sample sizes via the KL divergence criterion, the authors suggest that experimental verification of SW-SSB is within reach. The paper concludes by noting that while the focus is on the Rényi-2 correlator, similar schemes could potentially be extended to other SW-SSB definitions (e.g., fidelity or Wightman correlators), though this remains a subject for future work.