Fidelity Correlator in Quantum Dynamics

• Fidelity correlator is a quantitative metric defined via the Pearson correlation of self-fidelity maps from distinct unitary quantum evolutions.
• It establishes a universal anti-contrast floor, revealing fundamental limits imposed by quantum unitarity and noise constraints in measurement.
• The correlator is crucial for benchmarking quantum sensing protocols and guiding experimental design in various quantum platforms.

A fidelity correlator is a quantitative metric for assessing the global similarity or opposition between survival probability distributions—referred to as self-fidelity maps—induced by distinct unitary evolutions on quantum states. Operationally, it is defined as the Pearson correlation coefficient between the self-fidelity responses of two unitaries, parametrized over all pure states with respect to the unitarily invariant (Fubini–Study or Haar-induced) ensemble. The fidelity correlator establishes a model- and platform-independent floor to anti-contrast in quantum sensing and benchmarking, and reveals fundamental structural constraints imposed by quantum mechanical unitarity. Related notions, such as the fidelity out-of-time-order correlator (FOTOC), probe dynamical information scrambling in open quantum systems.

1. Mathematical Definition and Properties

Given two unitary operators $U, V \in U(d)$, their associated self-fidelity maps on pure states $|\psi\rangle$ are defined as $f_U(\psi) := |\langle\psi|U|\psi\rangle|^2$ and $f_V(\psi) := |\langle\psi|V|\psi\rangle|^2$. Treating $f_U$ and $f_V$ as real random variables on projective Hilbert space (endowed with the unitarily invariant measure $d\psi$), define their statistical moments: $$\mu_U = \mathbb{E}[f_U] = \int f_U(\psi)\, d\psi, \quad \sigma_U^2 = \mathrm{Var}[f_U] = \mathbb{E}[(f_U - \mu_U)^2]$$ (and similarly for $V$). The fidelity correlator is then

$$\rho(U, V) := \frac{\mathrm{Cov}[f_U, f_V]}{\sigma_U \sigma_V}, \quad \mathrm{Cov}[f_U, f_V] = \mathbb{E}[(f_U - \mu_U)(f_V - \mu_V)]$$

or, equivalently,

$|\psi\rangle$0

By construction, the Pearson coefficient is invariant under affine rescalings, so $|\psi\rangle$1 by the Cauchy–Schwarz inequality. This construction is explicitly device-agnostic, depending only on the unitary operators and the projective Hilbert space structure (Cho et al., 30 Sep 2025).

2. Structural Limit: The No-Minus-One Theorem

The mathematical upper and lower limits $|\psi\rangle$2 would require

$|\psi\rangle$3

for almost all $|\psi\rangle$4. After affine recentering, this reads $|\psi\rangle$5 for all $|\psi\rangle$6. However, such a relation would enforce $|\psi\rangle$7 universally, i.e., $|\psi\rangle$8 and $|\psi\rangle$9 are always orthogonal. This contradicts known results in quantum mechanics: by the “no universal inversion” theorem (Peres/Bužek–Hillery–Werner), there exist no nontrivial pairs of unitaries for which $f_U(\psi) := |\langle\psi|U|\psi\rangle|^2$0 for all $f_U(\psi) := |\langle\psi|U|\psi\rangle|^2$1. As a consequence (Cho et al., 30 Sep 2025): $f_U(\psi) := |\langle\psi|U|\psi\rangle|^2$2 This is the universal no-perfect anti-correlation theorem for self-fidelity maps, establishing a sharp lower bound strictly greater than $f_U(\psi) := |\langle\psi|U|\psi\rangle|^2$3 for all nontrivial unitary pairs.

3. Explicit Forms and Dimension-Dependence

Single-Qubit (Bloch-Sphere Ramsey) Model

For $f_U(\psi) := |\langle\psi|U|\psi\rangle|^2$4, $f_U(\psi) := |\langle\psi|U|\psi\rangle|^2$5, with $f_U(\psi) := |\langle\psi|U|\psi\rangle|^2$6 the Bloch axes and $f_U(\psi) := |\langle\psi|U|\psi\rangle|^2$7, the self-fidelity map for pure input state Bloch vector $f_U(\psi) := |\langle\psi|U|\psi\rangle|^2$8 reads

$f_U(\psi) := |\langle\psi|U|\psi\rangle|^2$9

Ensemble averages over the sphere are

$f_V(\psi) := |\langle\psi|V|\psi\rangle|^2$0

With $f_V(\psi) := |\langle\psi|V|\psi\rangle|^2$1,

$f_V(\psi) := |\langle\psi|V|\psi\rangle|^2$2

The Pearson correlator reduces to

$f_V(\psi) := |\langle\psi|V|\psi\rangle|^2$3

with $f_V(\psi) := |\langle\psi|V|\psi\rangle|^2$4 at $f_V(\psi) := |\langle\psi|V|\psi\rangle|^2$5 (“orthogonal Ramsey axes”). Thus, for qubits, the geometric anti-correlation floor is $f_V(\psi) := |\langle\psi|V|\psi\rangle|^2$6.

Higher Dimensions: Haar-Moment Analysis

For all finite $f_V(\psi) := |\langle\psi|V|\psi\rangle|^2$7, projective 2- and 4-design (Haar-moment) techniques express cross-moments in terms of unitary invariants: $f_V(\psi) := |\langle\psi|V|\psi\rangle|^2$8

$f_V(\psi) := |\langle\psi|V|\psi\rangle|^2$9

with explicit $f_U$0. Thus, variances and covariances, and hence $f_U$1, are fully determined by these invariants. Direct analysis shows that the strict lower bound $f_U$2 persists for any nontrivial $f_U$3 in all finite dimensions (Cho et al., 30 Sep 2025).

In the short-time expansion for $f_U$4,

$f_U$5

Perfect anti-correlation ($f_U$6) would require $f_U$7 a.s. with $f_U$8, which is only possible if $f_U$9 with $f_V$0, which cannot hold for Hermitian $f_V$1.

4. Implications for Quantum Measurement and Sensing

• Model-Independent Anti-Contrast Floor: The lower limit on $f_V$2 (“unitary-geometric floor”) is set by the structure of quantum theory and cannot be overcome by hardware, encoding, or noise engineering. In any purely unitary axis tuning, perfect differential readout—complete removal of common-mode contributions—is impossible.
• Experimental Design Guidance: Optimization of control parameters (Bloch axes separation, pulse area, interrogation time) should be used to push $f_V$3 as low as possible (e.g., for $f_V$4 in qubits), with the recognition that $f_V$5.
• Benchmarking and Cross-Platform Comparison: As a device-agnostic, rescaling-invariant metric, the fidelity correlator enables direct assessment of the theoretical anti-contrast limit for gate protocols, echo sequences, and related operations, providing a universal standard for benchmarking (Cho et al., 30 Sep 2025).
• Readout Noise Constraint: In differential measurement schemes of the form $f_V$6, the minimal achievable variance is $f_V$7 unless $f_V$8, imposing a fundamental irreducible noise floor.

5. The Fidelity Out-of-Time-Order Correlator (FOTOC)

A related but distinct quantity is the fidelity out-of-time-order correlator (FOTOC), which probes information scrambling and dynamical sensitivity in open quantum systems. For pure state $f_V$9, perturbation operator $d\psi$0, and $d\psi$1, the FOTOC is

$d\psi$2

In the linear regime,

$d\psi$3

In the unbiased spin-boson model at zero temperature, FOTOC displays an initial exponential growth indicating “pure scrambling,” after which system dynamics induce adulteration, compressing observable timescales. The FOTOC then advances ahead of conventional observables (such as polarization), acting as a short-time predictor (Chen, 2023). Necessary physical conditions include zero temperature, unbiased Hamiltonian, and continuous (sub-)ohmic spectral density.

6. Summary Table

Quantity	Definition	Structural Bound
Fidelity correlator	$d\psi$4	$d\psi$5
Qubit minimum	$d\psi$6	$d\psi$7
FOTOC	$d\psi$8	Exponential growth, then decay

The fidelity correlator provides a rigorous, model-independent quantification of global (anti-)contrast between quantum evolutions, with universal lower bounds derived from unitary geometry and direct implications for the design, benchmarking, and evaluation of quantum sensing protocols (Cho et al., 30 Sep 2025, Chen, 2023).