Comparing quantum and classical finite state generators

1. Problem Statement

The paper addresses a fundamental gap in quantum information theory: the benchmarking of temporal quantum correlations. While spatial quantum correlations (entanglement between distant particles) are well-characterized using Bell-CHSH inequalities, the same tools have been applied to temporal correlations (sequential measurements on a single system) with limited success.

The authors argue that:

• Standard Bell-CHSH inequalities, which set a classical limit of $S \le 2$ and a quantum Tsirelson bound of $S \le 2\sqrt{2}$ for spatial scenarios, may not be sufficient or appropriate for distinguishing classical from quantum temporal processes.
• Classical stochastic processes (specifically Hidden Markov Models) possess structural properties that allow them to mimic or even exceed quantum temporal correlations in specific short-term scenarios, potentially leading to false positives when using CHSH scores as a witness for quantumness.
• There is a need to identify scenarios where quantum processes genuinely outperform classical ones, particularly regarding the longevity of correlations under noisy or scrambling conditions.

2. Methodology

The authors construct a comparative framework using stochastic finite state generators:

• Classical Models:
  • Markov Models (MM): Machines with no internal memory where the output directly reveals the internal state.
  • Hidden Markov Models (HMM): Machines where the output symbol does not reveal the internal state. The internal state evolves based on transition probabilities $p(y|x)$ and output probabilities. The authors parametrize these using transition matrices $T^{(i)}$ for outputs $i = \pm 1$.
• Quantum Models:
  • Hidden Quantum Markov Models (HQMM): The quantum analogues of HMMs, utilizing a single qubit.
  • Generalized Measurements: Instead of projective measurements, the authors use Kraus operators ($K^{(i)}$) to model general measurements. This allows for a broader class of quantum dynamics where the internal state can remain in a superposition after measurement, preserving history.
  • Parametrization: The authors derive an over-complete parametrization of Kraus operators using 12 real parameters (via unitary evolution on a system-ancilla pair), ensuring all possible generalized measurements are covered.
• The CHSH Test:
  • Two observers, Alice and Bob, perform sequential measurements on the same system (qubit or bit).
  • Alice chooses between observables $A_1, A_2$; Bob chooses $B_1, B_2$.
  • The CHSH score is calculated as $S = |C_{11} + C_{12} + C_{21} - C_{22}|$, where correlators $C_{nm}$ are derived from joint probabilities of sequential outcomes.
• Simulation Strategy:
  • The authors numerically sample $10^8$ random instances of classical (HMM) and quantum (HQMM) machines.
  • They calculate the distribution of $S$ scores for both.
  • Scrambling Test: To test robustness, they introduce a third party, "Charlie," who applies a random scrambling channel (Markovian evolution or Kraus operators) between Alice's and Bob's measurements for a duration $t$.

3. Key Contributions

1. Re-evaluation of CHSH for Temporal Correlations: The paper demonstrates that the CHSH inequality is not a definitive witness for temporal quantumness because classical HMMs can violate the standard classical bound ($S > 2$) and, in specific constructed cases, even exceed the Tsirelson bound ($S > 2\sqrt{2}$).
2. Structural Difference Identification: The authors identify that classical machines can achieve higher correlations in short-term scenarios because they are not constrained by the requirement to generate orthogonal basis states (unlike quantum projective measurements). Classical transition matrices can be constructed to create "stronger" correlations that are mathematically impossible for standard quantum projective measurements.
3. Quantum Advantage in Longevity: The paper establishes that while classical machines may win in short-term, noise-free scenarios, quantum machines are superior at maintaining correlations over time when subjected to scrambling operations. The quantum superposition allows the system to retain memory of the initial state longer than classical probabilistic mixing.

4. Key Results

• Violation of Classical Limits: In the temporal CHSH scenario, classical Hidden Markov Models can produce scores $S > 2$. The authors provide a specific counter-example where a classical machine achieves $S = 3$, surpassing the quantum Tsirelson bound of $2\sqrt{2} \approx 2.82$.
• Distribution of Scores:
  • General Quantum Processes: Most randomly sampled HQMMs rarely surpass $S=2$.
  • Projective Quantum Processes: When restricted to projective-like measurements, quantum machines can reach the Tsirelson bound ($2\sqrt{2}$), but this is less common in the general parameter space than the classical violations.
  • Classical Processes: A significant portion of the sampled classical HMMs exceed $S=2$, and a small fraction exceed $2\sqrt{2}$.
• Robustness to Scrambling (The "Charlie" Effect):
  • When a scrambling channel is introduced between measurements, the CHSH score for classical machines drops rapidly toward the classical limit.
  • Quantum machines, particularly those using projective-like operators, maintain a higher CHSH score for longer durations ($t$).
  • Conclusion: Quantum systems exhibit long-range temporal correlations that are more resilient to information scrambling than their classical counterparts.

5. Significance and Implications

• Limitations of CHSH: The paper concludes that the CHSH score is an inappropriate quantitative measure for distinguishing quantum from classical temporal processes in general. The fact that classical models can exceed quantum bounds in specific setups renders the inequality insufficient as a "witness" for quantumness in time.
• Need for New Metrics: To properly characterize temporal quantum correlations, the authors suggest moving beyond CHSH. They propose using:
  • Out-of-Time-Ordered Correlators (OTOC): To study information scrambling and chaos.
  • Word Probabilities: Analyzing the probability of specific sequences (words) of outcomes, as suggested in previous work [34].
• Quantum Resources: The study highlights that temporal quantum correlations (which do not require entanglement) are a distinct and valuable resource. While they may not always beat classical machines in short-term correlation strength, their ability to preserve information against noise makes them superior for applications like quantum feedback control, metrology, and cryptography where long-term coherence is essential.
• Foundational Physics: The results challenge the assumption that quantum mechanics is always "stronger" than classical mechanics in correlation tasks, showing that the definition of "strength" depends heavily on the temporal structure and the presence of noise/scrambling.

In summary, the paper argues that while classical stochastic generators can mimic or exceed quantum temporal correlations in idealized, short-term settings, quantum systems possess a unique advantage in information retention over time, a feature that standard Bell-CHSH inequalities fail to capture.