Quantum state randomization constrained by non-Abelian symmetries

This paper demonstrates that for quantum systems with non-Abelian symmetries, the inability to achieve true Haar-like randomization at late times is primarily caused by experimental limitations on preparing low-entanglement initial states that fail to match the conserved operator moments of the Haar ensemble, rather than by the symmetry-constrained dynamics themselves.

The Gist

The Big Picture: Can a Quantum System Become Truly Random?

Imagine you have a giant box of marbles, each representing a tiny piece of a quantum system (like an atom). In the world of quantum physics, there is a special kind of "perfect chaos" called Haar randomness. If a system reaches this state, it is like a perfectly shuffled deck of cards: every possible arrangement is equally likely, and the system has "forgotten" everything about how it started. This is the gold standard for randomness.

For a long time, physicists thought that if you let a quantum system evolve naturally (like shaking that box of marbles), it would eventually become this perfectly random state, no matter how you started.

However, this paper says: "Not so fast."

The authors, Yuhan Wu and Joaquin Rodriguez-Nieva, discovered that if your system has certain complex rules (called non-Abelian symmetries, specifically SU(2) symmetry, which governs things like spin), it can never become truly perfectly random if you start with a simple, unentangled state. It will always retain a tiny "fingerprint" of how it began.

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The Analogy: The Dance Floor and the Rules

To understand this, let's use a dance floor analogy.

1. The Goal: The Perfect Shuffle (Haar Randomness)

Imagine a massive dance floor with $L$ dancers. The goal is to get them to dance so chaotically that if you look at any group of dancers, their positions look completely random. This is the Haar state. In this state, the dancers have no memory of where they started; they are a "perfect soup" of motion.

2. The Rules: The Symmetry Constraints

Now, imagine the dance floor has strict rules (Symmetries).

• Simple Rule (U(1) Symmetry): Imagine a rule that says, "The total number of people wearing red hats must stay the same." This is a simple constraint. The paper notes that even with this rule, the dancers can still shuffle around enough to look random to an observer, as long as they started with the right number of red hats.
• Complex Rule (SU(2) Symmetry): Now, imagine a much stricter, more complex rule. It's like saying, "The dancers must move in a way that preserves a specific 3D balance between their movements in the X, Y, and Z directions." This is the non-Abelian constraint. It's like trying to balance a spinning top while juggling; the rules are interconnected and rigid.

3. The Starting Point: The "Unentangled" State

In real experiments (like those with superconducting qubits or trapped ions), scientists usually start with a product state.

• Analogy: Imagine every dancer starts standing perfectly still in a neat row, each holding a specific pose. They are not holding hands or dancing together yet. They are "unentangled."
• The paper asks: If we let these neat rows start dancing under the complex SU(2) rules, will they eventually become the "perfect soup" of randomness?

The Discovery: The "Fingerprint" That Won't Wash Off

The authors found a surprising limitation:

If you start with a neat row of dancers (unentangled), you can never achieve the "perfect soup" of randomness, even after dancing for a very long time.

Here is why, using the Spin Variance concept:

• The Perfect Soup (Haar State): In a truly random state, the "wiggles" or fluctuations in the dancers' movements (variance) are huge. They are all over the place.
• The Neat Row (Product State): Because the dancers started in a neat row, their total "wiggle room" is limited by a mathematical law. The paper shows that for unentangled states, the sum of their wiggles in the X, Y, and Z directions is strictly limited.
  • The Math Metaphor: It's like saying, "You can wiggle your left hand a lot, but then your right hand must wiggle less." You can never wiggle everything at the maximum intensity simultaneously.
• The Consequence: Because the starting "wiggle room" is too small, the dancers can never spread out enough to fill the whole dance floor perfectly. They get stuck in a slightly smaller, slightly more ordered version of the dance floor.

The "Entanglement Entropy" Test

How do we know they aren't perfectly random? The paper uses a tool called Entanglement Entropy.

• Analogy: Imagine measuring how "mixed up" the dancers are. If they are perfectly random, the mixing score (Entropy) hits a maximum value known as the Page Entropy.
• The Result: The paper shows that for unentangled starting points, the mixing score always falls short of the maximum. It is lower by a small, fixed amount (an "O(1) correction") that doesn't disappear even if you add more and more dancers to the floor.
• The Takeaway: No matter how long you wait, the system remains distinguishable from a truly random system. It's like a song that is 99% shuffled, but you can still hear the original melody faintly in the background.

The "Best" You Can Do

The paper also explores: Is there a way to start that gets us closer to randomness?

• The "Ising" Start: If you start with dancers perfectly aligned in one direction (like a straight line), the system behaves as if it only has one simple rule. It gets stuck with a large "fingerprint" of order.
• The "IsoVar" Start: The authors found a "sweet spot." If you start the dancers with their poses distributed evenly across all three directions (X, Y, and Z) in a specific balanced way (called IsoVar), the system gets closer to randomness than any other starting point.
  • However, even this "best" start still falls short of the perfect random state. The "fingerprint" is smaller, but it is still there.

Why Does This Matter?

1. For Quantum Computers: We are building quantum computers to do things classical computers can't. These machines often start with simple, unentangled states. This paper tells us that if we use systems with these complex symmetries, we might not be able to generate the "perfect randomness" needed for certain advanced algorithms or benchmarks.
2. For Understanding Nature: It changes how we view thermalization (how things heat up and settle). We used to think, "Give it enough time, and it becomes random." Now we know, "Give it enough time, and it becomes random only if you started with the right kind of 'wiggles'." If you start too simple, the system remembers its past forever.

Summary in One Sentence

Even if you let a quantum system dance chaotically for a billion years, if you started with a simple, unentangled state and the system has complex symmetry rules, it will never become truly random; it will always carry a permanent, measurable scar of how it began.

Technical Summary

1. Problem Statement

The emergence of randomness from unitary quantum dynamics is fundamental to statistical mechanics and quantum information. While generic quantum-chaotic systems are expected to evolve into "featureless" states indistinguishable from Haar-random states (pure random states), physical systems are often constrained by symmetries.

• The Gap: Previous work established that systems with Abelian symmetries (e.g., $U(1)$ charge conservation) can still reproduce the statistical moments of Haar-random states, provided the initial state matches the conserved charge's moments. However, the behavior of systems with non-Abelian symmetries (specifically $SU(2)$, where charges do not commute) remains less understood.
• The Core Question: Can time-evolved states under $SU(2)$-symmetric dynamics reproduce Haar-like randomness at the level of finite statistical moments (accessible via experiments with finite copies), or do the non-commuting nature of the symmetries and experimental constraints on state preparation (specifically unentangled product states) fundamentally limit this randomization?

2. Methodology

The authors employ a combination of analytical ensemble theory and extensive numerical simulations to address the problem.

• Theoretical Framework:

  • Constrained Ensembles: They construct a theoretical ensemble $\Phi$ of states that preserve $SU(2)$ symmetry (conserving total spin $S$ and magnetization $M$) but are otherwise randomized. They compare the statistical moments of this ensemble to the Haar ensemble using trace distance.
  • Moment Matching: They analyze the conditions under which an initial state $|\psi_0\rangle$ must match the moments of conserved operators ($S_\alpha$) in the Haar ensemble to achieve effective randomization.
  • Entanglement Entropy (EE) as a Probe: They use the mean entanglement entropy $S_A$ and its fluctuations as a sensitive diagnostic for randomness, specifically looking for deviations from the Page curve (the EE of Haar-random states).

• Numerical Models:

  • Random Quantum Circuits (RQCs): They simulate $SU(2)$-symmetric brickwork circuits using random two-qubit gates of the form $e^{i\theta \vec{S}_i \cdot \vec{S}_{i+1}}$.
  • Hamiltonian Dynamics: They simulate a non-integrable spin chain with nearest- and next-nearest-neighbor interactions, including a spin-chirality term to ensure quantum chaos while preserving $SU(2)$ symmetry.
  • Initial Conditions: They generate unentangled product states with zero total magnetization but tunable spin variances ($\sigma_x^2, \sigma_y^2, \sigma_z^2$) subject to the constraint $\sigma_x^2 + \sigma_y^2 + \sigma_z^2 = L/2$.

3. Key Contributions and Results

A. Symmetry Constraints vs. Haar Randomness

The authors prove that symmetry alone does not prevent Haar-like randomization at the level of finite moments.

• If an initial state matches the full distribution of moments for all conserved operators ($S_x, S_y, S_z$) as found in the Haar ensemble (specifically $\langle S_\alpha \rangle = 0$ and $\langle S_\alpha^2 \rangle = L/4$), the time-evolved state becomes indistinguishable from a Haar-random state for any finite-order probe (polynomial in system size).
• The trace distance between the symmetry-constrained ensemble and the Haar ensemble is exponentially small in the number of degrees of freedom.

B. The Limitation of Unentangled Initial States

The central finding is that experimentally relevant unentangled (product) initial states cannot satisfy the conditions for Haar randomization in the presence of non-Abelian symmetries.

• The Variance Constraint: For a product state of $L$ spin-1/2 particles with zero total magnetization, the spin variances must satisfy:
  $$\sigma_x^2 + \sigma_y^2 + \sigma_z^2 = \frac{L}{2}$$
• The Haar Requirement: A Haar-random state requires:
  $$\sigma_x^2 + \sigma_y^2 + \sigma_z^2 = \frac{3L}{4}$$
• Conclusion: Since $L/2 < 3L/4$, it is impossible for any unentangled product state to match the second moments of the Haar ensemble. Consequently, the system can never fully randomize to the Haar limit, regardless of how long it evolves.

C. Quantifying the Entanglement Deficit

The paper quantifies the maximum achievable entanglement entropy (EE) for these constrained systems.

• Scaling Law: The late-time EE for a subsystem of fraction $f$ is given by:
  $$\langle S_A \rangle = \langle S_A \rangle_{\text{Haar}} + \sum_{\alpha=x,y,z} \frac{f + \log(1-f)}{2} g\left(\frac{\sigma_\alpha}{\sigma_{\text{Haar}}}\right)$$
  where $g(x)$ is a universal scaling function with $g(0)=1$ (maximal deviation) and $g(1)=0$ (Haar limit).
• Optimal vs. Worst Initial States:
  • Ising-like (Point a): If the initial state has a resolved charge in one direction (e.g., $\sigma_z^2=0, \sigma_x^2=\sigma_y^2=L/2$), the system behaves as if it only conserves a single $U(1)$ charge. The EE is reduced by a finite $O(1)$ offset relative to the Page value.
  • IsoVar (Point c): If the variance is distributed isotropically ($\sigma_x^2 = \sigma_y^2 = \sigma_z^2 = L/6$), the deviation from the Page value is minimized but remains finite ($O(1)$) in the thermodynamic limit.
• Finite Thermodynamic Offset: Unlike Abelian cases where certain initial conditions can reach the Page limit, for $SU(2)$ with product states, the difference between the late-time EE and the Page entropy remains finite even as $L \to \infty$.

4. Significance and Implications

• Fundamental Limits on Randomization: The work establishes that non-Abelian symmetries, when combined with the practical constraint of preparing unentangled initial states, impose a fundamental barrier to achieving full quantum chaos (Haar randomness). The system retains a "memory" of the initial state's spin variances in its late-time entanglement structure.
• Experimental Relevance: This is critical for modern programmable quantum simulators (superconducting qubits, Rydberg atoms, trapped ions) which typically initialize systems in product states. The results suggest that even in strongly chaotic regimes, these systems will exhibit distinct, non-thermal statistical features in entanglement entropy and higher-order moments that differ from standard random matrix theory predictions.
• Beyond Thermalization: The paper refines the understanding of thermalization. While coarse-grained observables (local operators) may still appear thermal, fine-grained probes (entanglement entropy, full counting statistics) reveal that the system has not fully explored the Hilbert space due to the interplay of non-commuting symmetries and initialization constraints.
• Protocol Design: The authors suggest that to achieve Haar-like randomization in $SU(2)$ systems, one would need to prepare initial states that are already entangled (to satisfy the higher variance requirements), as simple product states are insufficient.

In summary, the paper demonstrates that non-Abelian symmetry constraints prevent unentangled initial states from ever reaching the Haar-random limit, resulting in a persistent, system-size-independent deficit in entanglement entropy compared to pure random states.