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Detection of quantum imaginarity using moments and its interferometric realization

Sudip Chakrabarty, Saheli Mukherjee, Ananda G. Maity, Bivas Mallick

Abstract

Complex numbers, intrinsic to the formulation of quantum theory, play a pivotal role in enabling advantages across a broad range of quantum information-processing tasks. Despite their fundamental importance, practical and scalable criteria for detecting quantum imaginarity remain relatively underexplored, particularly methods that enable its identification with reduced experimental overhead. In this work, we propose a realistic and experimentally feasible method to detect quantum imaginarity using moment-based approach. Our framework relies on experimentally accessible moments of the Kirkwood-Dirac quasiprobability distribution, enabling scalable detection in many-body and high-dimensional systems without requiring full state tomography. We then present an illustrative example to support our detection scheme. Finally, we present an interferometric scheme for measuring these moments, paving the way for experimental implementation of our detection protocol.

Detection of quantum imaginarity using moments and its interferometric realization

Abstract

Complex numbers, intrinsic to the formulation of quantum theory, play a pivotal role in enabling advantages across a broad range of quantum information-processing tasks. Despite their fundamental importance, practical and scalable criteria for detecting quantum imaginarity remain relatively underexplored, particularly methods that enable its identification with reduced experimental overhead. In this work, we propose a realistic and experimentally feasible method to detect quantum imaginarity using moment-based approach. Our framework relies on experimentally accessible moments of the Kirkwood-Dirac quasiprobability distribution, enabling scalable detection in many-body and high-dimensional systems without requiring full state tomography. We then present an illustrative example to support our detection scheme. Finally, we present an interferometric scheme for measuring these moments, paving the way for experimental implementation of our detection protocol.

Paper Structure

This paper contains 19 sections, 5 theorems, 128 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Quantum imaginarity
  4. Kirkwood-Dirac quasiprobability distribution
  5. Partial transposition moments
  6. Overview of a Mach–Zehnder interferometer
  7. Detection of imaginarity using moments
  8. Removal of real coherences by the $Y$-twirl map
  9. Equivalence between quantum coherence and nonpositivity of the extended KD distribution
  10. KD moment-based detection of quantum imaginarity
  11. Illustrative qubit example
  12. Interferometric Scheme for the realization of moments and imaginarity detection
  13. Realization of moments in an interferometric setup
  14. Direct detection of imaginarity from interferometric visibility
  15. Conclusion
  16. ...and 4 more sections

Key Result

Lemma 1

Let $\mathcal{H}$ be a $d$-dimensional Hilbert space with orthonormal basis $\{\ket{n}\}_{n=0}^{d-1}$. For each pair of distinct indices $p<q$, define the antisymmetric operator which acts non-trivially only on the two-dimensional subspace spanned by $\{\ket{p},\ket{q}\}$. Then, define the linear Y-twirl map $\mathcal{E}:\mathcal{B}(\mathcal{H}) \to \mathcal{B}(\mathcal{H})$ by For any density o

Figures (3)

  • Figure 1: Schematic representation of the Mach–Zehnder interferometric setup used to estimate the visibility $V$. The path degree of freedom of the interferometer acts as the control qubit, while the internal system carrying the state $\rho$ serves as the target. The interferometer consists of two $50$-$50$ beam splitters (BS I and BS II), two mirrors, a phase shifter (PS) introducing a phase $\theta$ in the arm along $\ket{0}$, and a controlled unitary (U). A measurement (M) in the computational basis at the output port along the path $\ket{0}$ gives the visibility $V$ along path $\ket{0}$.
  • Figure 2: Amount of imaginarity for the state $\rho$ with $\theta= \pi/2$, $(M_{\ell_1 }{(\rho)}=|\sin\alpha|)$ (black) and the minimal-order Hankel determinants detecting the entire region with non-zero imaginarity is plotted as a function of $\alpha$. For $\beta=\pi/2$, detection occurs at first order ($H_1$). For $\beta=0$, detection requires $H_2$. For $\beta=\pi/4$, first and second orders fail, and detection occurs at $H_3$. All curves vanish only at $\alpha=0,\pi,2\pi$.
  • Figure 3: Generalized Mach–Zehnder interferometric scheme for measuring the $n$th-order extended KD moment. The input state is $\rho'_1 \otimes\rho'_2 \otimes \dots \otimes \rho'_n$, which represents $n$ copies of the Y-twirled state $\rho'$. The controlled operation implements the unitary $S_n$ acting on the internal degrees of freedom. The visibility of the resulting interference pattern yields $V^{(n)}$, which corresponds to the $n$th-order moment. The remaining terminologies follow the same convention as in Figure \ref{['fig:inteferometer']}.

Theorems & Definitions (11)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Definition 1
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • ...and 1 more