Fractional Conformal Map, Qubit Dynamics and the Leggett-Garg Inequality

1. Problem Statement

The paper addresses the need for a unified mathematical framework to describe diverse quantum dynamics of a qubit, ranging from standard unitary evolution to non-unitary and non-linear processes (often associated with open systems or measurement-induced dynamics). Specifically, the authors aim to:

• Characterize the temporal correlations of these dynamics using the Leggett-Garg Inequality (LGI).
• Determine under what conditions these dynamics respect or violate the Lüders bound (the upper bound of $K_3 = 3/2$ for the LG parameter).
• Investigate the relationship between the linearity/non-linearity of the evolution operator on the Hilbert space and the violation of macroscopic realism (via No-Signaling in Time and Arrow of Time conditions).

2. Methodology

The authors employ a geometric approach mapping quantum states to the complex plane:

• Geometric Representation: They utilize stereographic projection to map the Bloch sphere (representing pure qubit states) to the extended complex plane (Riemann sphere, $\tilde{\mathbb{C}}$).
• Fractional Linear Conformal (FLC) Maps: The time evolution of the qubit is modeled by successive applications of Möbius transformations (fractional linear maps):
  $$f(z) = \frac{az + b}{cz + d}, \quad ad - bc \neq 0$$
  where $z$ is the complex coordinate of the state.
• Classification of Dynamics: The parameter space of these maps is analyzed to categorize the induced dynamics into three classes based on their action on the Hilbert space:
  1. Unitary: Linear and norm-preserving ($|a|^2 + |b|^2 = 1$).
  2. Non-Unitary but Linear: Linear action but with scaling ($|a|^2 + |b|^2 = r \neq 1$).
  3. Non-Unitary and Non-Linear: The map acts non-linearly on the state space (often arising from post-selection or non-Hermitian evolution).
• Leggett-Garg Analysis: The authors calculate the LG parameter $K_3 = C_{12} + C_{23} - C_{13}$ for a sequence of three projective measurements ($t_1, t_2, t_3$) on the dichotomic observable $\hat{Q} = \sigma_z$. They derive analytical expressions for the two-time correlation functions $C_{ij}$ based on the FLC map parameters.
• Consistency Checks: They evaluate the No-Signaling in Time (NSIT) and Arrow of Time (AoT) conditions to test for Macroscopic Realism (MR).

3. Key Contributions

• Unified Framework: The paper establishes that FLC maps serve as a comprehensive framework encompassing unitary, non-unitary linear, and non-unitary non-linear quantum dynamics.
• Analytical Derivation of Correlations: The authors provide explicit analytical formulas for the correlation functions $C_{ij}$ in terms of the map coefficients ($a, b, c, d$) and the initial state parameters.
• Lüders Bound Characterization: They identify specific ratio constraints on the map elements that guarantee the LG parameter $K_3$ remains within the Lüders bound ($K_3 \leq 3/2$), regardless of whether the dynamics are linear or non-linear.
  • The condition is: $|a_{ij}/c_{ij}| = |d_{ij}/b_{ij}|$, which implies $y_{ij} + z_{ij} = 1$ in their derived correlation formula.
• Violation Mechanisms: The study demonstrates that violating the Lüders bound (reaching $K_3 > 3/2$) is possible specifically in non-unitary non-linear dynamics when the aforementioned ratio constraints are not satisfied. This aligns with recent findings in non-Hermitian systems.
• NSIT Violation: The authors show that for most parameter spaces of FLC maps, the NSIT conditions are violated, indicating a breakdown of Macroscopic Realism, even when the Lüders bound is respected.

4. Key Results

• Three Distinct Classes of Dynamics:
  1. Linear Action, Respects Lüders Bound: Includes standard unitary evolution and certain non-unitary linear maps.
  2. Non-Linear Action, Respects Lüders Bound: Occurs when specific ratio constraints among map coefficients are met.
  3. Non-Linear Action, Violates Lüders Bound: Occurs when ratio constraints are broken, allowing $K_3$ to exceed $1.5$.
• Independence from Initial State: The maximization of the Lüders bound for $K_3$ is shown to be independent of the initial qubit state on the Bloch sphere.
• NSIT and AoT:
  • All AoT conditions are satisfied for the proposed dynamics.
  • NSIT conditions are generally violated, confirming the non-classical nature of the dynamics.
  • The paper provides a specific example where NSIT is violated throughout the parameter space (except for a one-parameter family), confirming that $K_3 > 1$ (non-classical behavior) is the norm when NSIT is violated.
• Table of Valid Maps: The paper provides a table (Table I) listing specific forms of FLC maps (e.g., $f(z) = \frac{az \pm b}{bz \pm a}$) that satisfy the ratio constraints and thus respect the Lüders bound.

5. Significance

• Theoretical Unification: By linking conformal geometry with quantum dynamics, the paper offers a powerful tool to visualize and classify complex quantum evolutions that go beyond standard Schrödinger dynamics.
• Clarification of Quantum vs. Classical Bounds: It clarifies the conditions under which the Lüders bound (often considered a limit for classical realism in LG tests) can be violated. The results suggest that non-linearity combined with specific structural constraints in the evolution operator is the key driver for super-Lüders violations.
• Experimental Relevance: The findings are relevant for experimental tests involving non-Hermitian systems, open quantum systems, and measurement-induced dynamics, where observing violations of the Lüders bound serves as a signature of non-classicality beyond standard unitary quantum mechanics.
• Foundational Insight: The work reinforces the connection between the violation of temporal inequalities (LGI) and the failure of Macroscopic Realism, specifically highlighting how non-linear effective dynamics (common in post-selected experiments) can lead to stronger violations than linear dynamics.