Independent stabilizer Rényi entropy and entanglement fluctuations in random unitary circuits

Technical Summary: Independent Stabilizer Rényi Entropy and Entanglement Fluctuations in Random Unitary Circuits

Problem Statement
Quantum complexity is a central theme in quantum computing, distinguishing "simple" states (close to classical) from "complicated" states characterized by strong non-locality and quantum correlations. While entanglement entropy is a well-established measure of quantum resources and complexity, it does not capture the full picture. Specifically, highly entangled states can be non-complex if they are generated by classically simulable Clifford circuits. To address this, the concept of "magic" (or non-stabilizing power), quantified by the Stabilizer Rényi Entropy (SRE), has been introduced to measure the hardness of classically simulating a quantum state.

A critical open question remains regarding the relationship between these two distinct measures of complexity: magic and entanglement. While entanglement is instrumental in generating high-magic states, the existence of states with negligible magic but high entanglement (and vice versa) suggests a complex interplay. Previous studies have identified distinct regimes in the Hilbert space, but the statistical relationship between magic and entanglement in generic, Haar-random quantum states has not been fully characterized. This work investigates whether magic and entanglement are correlated or independent in the thermodynamic limit of random quantum circuits.

Methodology
The authors perform a numerical investigation of the joint distribution of magic ($M_2$) and entanglement entropy ($S$) in $N$-qubit Haar-random quantum states.

• System Generation: Random states are generated by applying random unitary operators $U \in U_N$ to a reference product state $|0\rangle = \bigotimes_{j=1}^N |0\rangle_j$. The study also considers brick-wall circuits of depth $D$, demonstrating that for $D \gtrsim 2N$, the distributions converge to those of generic Haar-random operators.
• Magic Quantification: The Stabilizer 2-Rényi Entropy (SRE), denoted as $M_2$, is calculated using the probability distribution of Pauli strings. For a state $|\psi\rangle$, the Pauli string probability is $\Xi_\psi(\sigma) = \frac{1}{d} |\langle \psi | \sigma | \psi \rangle|^2$, where $d=2^N$. The magic is defined as $M_2(|\psi\rangle) \equiv -\log_2 \sum_{\sigma \in P_N} \Xi_\psi^2(\sigma) - \log_2(d)$.
• Entanglement Quantification: The von Neumann entropy $S$ is computed by partitioning the system into two subsystems of equal size (or nearly equal for odd $N$).
• Statistical Analysis: The authors analyze the marginal distributions $P_N(M_2)$ and $P_N(S)$, as well as the joint distribution $P_N(M_2, S)$. They examine the scaling of the mean values, variances, and the covariance between the two quantities as the system size $N$ increases.

Key Contributions and Results

1. Exponential Localization of Marginal Distributions:

   • Magic: The distribution of magic $P_N(M_2)$ becomes exponentially sharp and localized around a typical value $\tilde{M}_2 \approx N - 2$ as $N \to \infty$. The variance scales as $\text{var}(M_2) \propto 4^{-N}$, which is exponentially faster than standard thermodynamic scaling ($\sim N$). This concentration is attributed to the sum of weakly correlated terms in the definition of SRE.
   • Entanglement: Similarly, the entanglement entropy distribution $P_N(S)$ concentrates exponentially around $\tilde{S} \approx N/2$ (consistent with the Page formula). The variance scales as $\text{var}(S) \propto 2^{-N}$.
   • Distribution Shape: For large $N$, both distributions converge toward a Gaussian with vanishing variance.

2. Asymptotic Independence of Fluctuations:

   • The core finding of the paper is the behavior of the joint distribution $P_N(M_2, S)$. While the typical values of magic and entanglement are strongly correlated (both scale linearly with $N$), their fluctuations become exponentially uncorrelated.
   • The covariance between magic and entanglement, $\text{cov}(M_2, S)$, decays exponentially with system size as $\propto 2^{-3N}$.
   • Crucially, this decay is faster than the geometric mean of the individual variances ($\sqrt{\text{var}(M_2)\text{var}(S)}$). This implies that the correlation coefficient vanishes in the thermodynamic limit, indicating that magic and entanglement fluctuations are asymptotically independent.

3. Rarity of Atypical States:

   • The study confirms that while states with vanishing magic but high entanglement (or high magic with low entanglement) exist, they represent an exponentially small fraction of the Hilbert space compared to typical states, which possess both large magic and large entanglement entropy.

Significance and Claims
The paper claims to resolve the ambiguity regarding the relationship between magic and entanglement in generic quantum states. By demonstrating that their fluctuations are exponentially uncorrelated in the thermodynamic limit, the authors establish a fundamental statistical independence between these two measures of quantum complexity.

• Theoretical Implication: The results extend previous analytical findings (which showed zero covariance for linear magic) to the non-linear Stabilizer 2-Rényi entropy and von Neumann entropy. This suggests that in large random systems, the "hardness" of simulation (magic) and the "amount" of non-locality (entanglement) are distinct resources that fluctuate independently.
• Resource Theory: The work highlights the predominance of highly complex quantum states in the Hilbert space, characterized by large values of both resources. It reinforces the utility of random unitary circuits as a framework for exploring quantum resource theories and understanding the statistical properties of key complexity measures.
• Modesty of Claims: The authors do not propose new experimental protocols or specific applications for quantum algorithms. Instead, they provide a rigorous statistical characterization of the landscape of quantum states, clarifying that while magic and entanglement are both necessary for high complexity, they are not statistically coupled in their deviations from typical values in random circuits.