Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Fundamental limits to contrast reversal of self-fidelity correlations

Kyoungho Cho, Jeongho Bang

TL;DR

The paper addresses whether two quantum evolutions can be globally anti-correlated across a broad input ensemble by treating self-fidelity $X_j(|\psi\rangle)=|\langle\psi|\hat U_j|\psi\rangle|^2$ as a random variable on projective space and measuring opposition with the Pearson correlation $P$. It proves a fundamental unitary-geometric floor: perfect anti-correlation with $P=-1$ is unattainable for any nontrivial pair of unitaries, both in a single-qubit Bloch-sphere Ramsey model and in higher dimensions via Haar-moment invariants, implying a universal limit on anti-contrast in unitary sensing. In the qubit case, the result is $P=(3\cos^2\delta-1)/2$, with a strict minimum of $-1/2$ at orthogonal axes, independent of pulse amplitudes; in higher dimensions, $P$ depends on a small set of invariants and remains strictly greater than $-1$ due to unitary geometry. Short-time expansions relate the fluctuations of self-fidelity to state-dependent energy variances, reinforcing that the anti-contrast floor cannot be bypassed by parameter tuning alone. Collectively, these results provide a model-independent criterion for the limits of anti-contrast and a practical metric for calibrating and benchmarking unitary sensing protocols.

Abstract

In measurement design, it is common to engineer anti-contrast readouts -- two measurements that respond as differently as possible to the same inputs so that common-mode contributions are suppressed. To assess the fundamental scope of this strategy in unitary dynamics, we ask whether two evolutions can be made uniformly opposite over a broad input ensemble, or whether quantum mechanics imposes a structural limit on such opposition. We address this by treating self-fidelity (survival probability) as a random variable on projective state space and adopting the Pearson correlation coefficient as a device-agnostic measure of global opposition between two evolutions. Within this framework we establish the following theorem: For any nontrivial pair of unitaries, self-fidelity maps cannot be point-wise complementary correlation on the entire state space. Consequently, the mathematical lower edge of the correlation bound is not physically attainable, which we interpret as a unitary-geometric floor on anti-contrast, independent of hardware specifics and noise models. We make this floor explicit in realizable settings. In a single-qubit Bloch-sphere Ramsey model, a closed-form relation shows that a residual common-mode component persists even under nominally optimal tuning. In higher dimensions, Haar/design moment identities reduce ensemble means and covariances of self-fidelity to a small set of unitary invariants, yielding the same conclusion irrespective of implementation details. Taken together, these results provide a model-independent criterion for what anti-contrast can and cannot achieve in unitary sensing protocols.

Fundamental limits to contrast reversal of self-fidelity correlations

TL;DR

The paper addresses whether two quantum evolutions can be globally anti-correlated across a broad input ensemble by treating self-fidelity as a random variable on projective space and measuring opposition with the Pearson correlation . It proves a fundamental unitary-geometric floor: perfect anti-correlation with is unattainable for any nontrivial pair of unitaries, both in a single-qubit Bloch-sphere Ramsey model and in higher dimensions via Haar-moment invariants, implying a universal limit on anti-contrast in unitary sensing. In the qubit case, the result is , with a strict minimum of at orthogonal axes, independent of pulse amplitudes; in higher dimensions, depends on a small set of invariants and remains strictly greater than due to unitary geometry. Short-time expansions relate the fluctuations of self-fidelity to state-dependent energy variances, reinforcing that the anti-contrast floor cannot be bypassed by parameter tuning alone. Collectively, these results provide a model-independent criterion for the limits of anti-contrast and a practical metric for calibrating and benchmarking unitary sensing protocols.

Abstract

In measurement design, it is common to engineer anti-contrast readouts -- two measurements that respond as differently as possible to the same inputs so that common-mode contributions are suppressed. To assess the fundamental scope of this strategy in unitary dynamics, we ask whether two evolutions can be made uniformly opposite over a broad input ensemble, or whether quantum mechanics imposes a structural limit on such opposition. We address this by treating self-fidelity (survival probability) as a random variable on projective state space and adopting the Pearson correlation coefficient as a device-agnostic measure of global opposition between two evolutions. Within this framework we establish the following theorem: For any nontrivial pair of unitaries, self-fidelity maps cannot be point-wise complementary correlation on the entire state space. Consequently, the mathematical lower edge of the correlation bound is not physically attainable, which we interpret as a unitary-geometric floor on anti-contrast, independent of hardware specifics and noise models. We make this floor explicit in realizable settings. In a single-qubit Bloch-sphere Ramsey model, a closed-form relation shows that a residual common-mode component persists even under nominally optimal tuning. In higher dimensions, Haar/design moment identities reduce ensemble means and covariances of self-fidelity to a small set of unitary invariants, yielding the same conclusion irrespective of implementation details. Taken together, these results provide a model-independent criterion for what anti-contrast can and cannot achieve in unitary sensing protocols.

Paper Structure

This paper contains 9 sections, 7 theorems, 51 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Pearson Correlation Coefficient and Cauchy-Schwarz bound
  3. Self-Fidelity Correlation Model and Impossibility of Perfect Anti-Correlation
  4. Self-Fidelity in Practice: Bloch-sphere Ramsey Interferometry and Loschmidt Echo
  5. Single-qubit Bloch-sphere Ramsey interferometry
  6. Beyond qubits: Loschmidt echo
  7. Discussion
  8. Covariance---Haar measure approach
  9. $SU(2)$: Bloch sphere via Haar measure

Key Result

Theorem 1

Let $X, Y \in L^2(\mu)$ with $\Delta_X, \Delta_Y>0$. Then, with equality if and only if $Y=aX+b$ almost surely for some $a \neq 0$ and $b \in \mathbb{R}$. Here, $L^2(\mu)$ denotes the real Hilbert space of (equivalence classes of) square‑integrable random variables, i.e., $L^2(\mu):=\{Z:\Omega\to\mathbb{R}\ \mid\ \mathbb{E}[Z^2]<\infty\}$, equipped with the

Figures (2)

  • Figure 1: Self-fidelity fringe patterns on the single-qubit Bloch-sphere Ramsey interferometry model. For a single-qubit unitary $\hat{U}(\theta, \mathbf{n})=\exp\left(-\frac{i}{2} \theta \mathbf{n} \cdot \boldsymbol{\sigma} \right)$, the color encodes $X_{\hat{U}}$ in Eq. (\ref{['eq:qubit-fidelity-formula']}). Here, it is observed that the two bright caps are centered at $\pm\mathbf n$ and the dark belt follows the great circle $\mathbf r\!\cdot\!\mathbf n=0$. (a) $\theta=2.5$ (high contrast): the bright/dark separation is strong because the amplitude $A=\sin^2{(\tfrac{\theta}{2})}$ is large. (b) $\theta=1.5$ (lower contrast): the cap/belt fringe geometry is unchanged---only the contrast amplitude is reduced. Thus, $\mathbf{n}$ fixes the location of the bright/dark regions, while $\theta$ only rescales their contrast.
  • Figure 2: Qubit self-fidelity correlation $P$ versus $\delta$ between Ramsey axes. Perfect anti-correlation $P=-1$ is forbidden; the best anti-correlation is $P=-\tfrac{1}{2}$ at orthogonal axes.

Theorems & Definitions (10)

  • Theorem 1: Cauchy-Schwarz bound for PCC
  • proof
  • Theorem 2: No universal quantum inversion Buzek99Rungta2001Bang2012
  • proof
  • Corollary 1: No perfect complement
  • Theorem 3: Impossibility of $P=-1$ for self-fidelities
  • Proposition 1: Qubit geometric correlation
  • proof
  • Proposition 2
  • Proposition 3