Comparing quantum and classical finite state generators

The gist The temporal correlations of classical and quantum finite state generators are qualitatively different, leading to findings that standard correlation measures like Bell-CHSH inequalities are inappropriate for benchmarking temporal processes.

Comparison of Correlation Benchmarking

Bell-CHSH-like inequalities have been successful in benchmarking spatial quantum correlations, but they are generally not sufficient for temporal quantum correlations because classical machines can exceed the Tsirelson bound of 2√2 due to their fundamental structure. However, when considering a time delay between consecutive measurements, quantum machines outperform their classical counterparts by maintaining correlations longer under generally scrambling operations. The authors conclude that CHSH-like inequalities may not be the correct candidate for studying quantum temporal correlations because the limits obtained for classical and quantum regimes are different from those set by spatial correlations.

Modeling Classical Temporal Processes

Classical stochastic processes can be modeled using two types of finite state generators: Markov models and Hidden Markov Models. Markov models are characterized by transition matrices T of the form T = p(−1 − 1) p(−1 + 1) p(+1 − 1) p(+1 + 1). Hidden Markov Models are a generalization where the output symbol is no longer an indication of the internal state, characterized by eight parameters pi(yx). The transition matrices for Hidden Markov Models are parameterized by six independent parameters a, b, c, d, e and f

Modeling Quantum Temporal Processes

Quantum temporal stochastic processes are modeled using Hidden Quantum Markov Models (HQMMs), which are the quantum analogues of Hidden Markov Models. These models are characterized by eight independent real parameters derived from the Kraus operators K(i). The Kraus operators K(i) describe a generalized measurement which is performed on a single qubit and causes the internal state of the machine to evolve. The completeness relation for Kraus operators is X i=±1 K(i)†K(i) = I

Performance Comparison and Limitations

When comparing the CHSH scores, it is observed that the general temporal quantum evolutions rarely manages to surpass S = 2, but can achieve this in principle. For projective cases of quantum machines, saturating the CHSH inequality with S = 2√2 is more likely than in general temporal cases For classical processes, a simple example shows that they can even surpass the limit of S = 2√2, achieving S = 3 This demonstrates that the classical and quantum bounds of S in case of temporal CHSH scenarios can get violated by classical stochastic processes that are describable by macrorealistic theories and noncontextual models. The paper suggests that CHSH scores lack the adequacy in characterizing temporal processes because classical machines do not require the orthonormal basis generation constraint that quantum machines do, which allows them to possess stronger correlations.

Longevity of Correlations

A key advantage of quantum temporal processes over classical ones is their ability to maintain long-term correlations While classical Hidden Markov Models can only assume one of two possible hidden states, the analogous Hidden Quantum Markov Models can be prepared in a superposition of two orthogonal quantum states with complex coefficients depending in a much more complex way on the history of the machine. This feature is illustrated by showing that the CHSH score is preserved somewhat in the quantum case even under multiple loss operations, while this is not the case for classical machines. This behavior can be understood by the presence of long range correlations in quantum systems. However, when considering only a single time step without scrambling, classical processes can be more powerful than the quantum ones because they are able to obtain a higher CHSH score.

Conclusion and Future Directions

The paper concludes that non-classical temporal quantum correlations which are easier to realise than quantum entanglement can lead to a quantum enhancement but only if the right scenarios are considered. To fully characterize temporal processes and distinguish classical from quantum, researchers need to consider other measures such as the out of time ordered correlations (OTOC) formalism or probabilities for the occurrence of certain words. Understanding different types of stochastic processes is key for identifying useful quantum resources and utilizing them in quantum technology applications. The results emphasize that CHSH scores lack the adequacy in characterizing temporal processes.

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1. The improved system can distinguish between classical and quantum temporal processes by calculating a quantity S that can be used to distinguish quantum from classical temporal correlations of Alice and Bob, as mentioned in the introduction. This allows for identifying novel resources for quantum technology applications based on whether the observed correlations exceed the limits set by CHSH-like inequalities.

2. The AI can utilize out-of-time ordered correlators (OTOC) formalism [41–44] to characterize temporal processes, allowing it to identify correlations that are more complex than the possible correlations of the Markov Models, which is key for distinguishing quantum from classical behavior in long-term dynamics.

3. The system can perform a comparison between quantum and classical machines by observing how they behave under noise, as shown in Figure 3 where the quantum versions better maintain their correlations than in the classical version when subjected to a tertiary evolution channel corresponding to Charlie.

4. The improved system can exploit the difference between spatial and temporal limits by recognizing that for CHSH scenarios, the limits for classical and quantum regimes yield by the spatial and temporal inequalities are the same, which is 2√2, concluding that CHSH-like inequalities may not be the correct candidate for studying quantum temporal correlations.

5. The system can be designed to leverage the structural difference between classical and quantum processes by recognizing that classical machines do not have the requirement that they generate an orthonormal basis, enabling them to possess stronger correlations because of this, potentially leading to higher calculated CHSH scores, such as S=3, beyond the quantum limit.

Bell-CHSH-like inequalities have been very successful in benchmarking spatial quantum correlations. However, as this paper illustrates, they are in general not sufficient for benchmarking temporal quantum correlations. To show this, we parametrise classical and quantum stochastic finite state generators based on a single bit and a single qubit, respectively, and compare the temporal correlations of their output sequences using a Bell-CHSH-like inequality. We find that for sequential measurements by two observers, Alice and Bob, classical machines can exceed the Tsirelson bound of 2 sqrt 2, due to their fundamental structure. However, when we consider a time delay between consecutive measurements, we find examples where the quantum machines outperform their classical counterparts by maintaining correlations longer under generally scrambling operations. Our result can be used to distinguish quantum from classical processes and to identify novel resources for quantum technology applications.