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Revisited Quantification of the Resource Theory of Imaginarity

Yue Han, Naihong Hu

Abstract

In this paper, we investigate the decay behaviors of three imaginarity-related metrics, specifically the $l_1$-norm-based imaginarity measure, imaginarity robustness, and imaginarity relative entropy, for arbitrary single-qubit pure initial states under three typical quantum channels: dephasing, generalized amplitude damping, and phase-amplitude damping. Furthermore, we extend our analysis to higher-dimensional systems by examining the decay trends of the aforementioned imaginarity metrics for several key two-qubit states under two-qubit channels. We also generalize the concept of the maximal imaginary state (originally defined for single qubits in the resource theory of imaginarity) to separable two-qubit states. In addition, we extend the definitions of imaginary power and de-imaginary power for single-qubit channels to two-qubit channels acting on separable two-qubit states. Finally, we compute the imaginary and de-imaginary powers for several common two-qubit channels.

Revisited Quantification of the Resource Theory of Imaginarity

Abstract

In this paper, we investigate the decay behaviors of three imaginarity-related metrics, specifically the -norm-based imaginarity measure, imaginarity robustness, and imaginarity relative entropy, for arbitrary single-qubit pure initial states under three typical quantum channels: dephasing, generalized amplitude damping, and phase-amplitude damping. Furthermore, we extend our analysis to higher-dimensional systems by examining the decay trends of the aforementioned imaginarity metrics for several key two-qubit states under two-qubit channels. We also generalize the concept of the maximal imaginary state (originally defined for single qubits in the resource theory of imaginarity) to separable two-qubit states. In addition, we extend the definitions of imaginary power and de-imaginary power for single-qubit channels to two-qubit channels acting on separable two-qubit states. Finally, we compute the imaginary and de-imaginary powers for several common two-qubit channels.
Paper Structure (10 sections, 2 theorems, 51 equations, 10 figures)

This paper contains 10 sections, 2 theorems, 51 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Framework of the Resource Theory of Imaginarity
  4. Decay of Imaginarity of Quantum States After Passing Through Different Quantum Channels
  5. Single-Qubit Case
  6. Two-Qubit Case
  7. Decay of Imaginarity of the Pure State $\ket{\gamma} = \alpha\ket{00} + \beta\ket{11}$ under a Two-Qubit Bit-Flip Channel
  8. Decay of Imaginarity of a Dual-Rail Two-Qubit State under an Amplitude Damping Channel
  9. Imaginary Power and De-imaginary Power of Quantum Channels
  10. Conclusion

Key Result

Theorem 2.1

(Existence of Operator-Sum Representation) All quantum operations $\varepsilon$ on a Hilbert space of dimension $d$ can be generated by an operator-sum representation with $d^{2}$ elements, where $1 \le M \le d^{2}$.

Figures (10)

  • Figure 1: Decay of $\mathscr{F}_{l_{1}}$, $\mathscr{F}_{R}$, and $\mathscr{F}_{r}$ under a Dephasing Channel
  • Figure 2: Decay of $\mathscr{F}_{l_{1}}$, $\mathscr{F}_{R}$, and $\mathscr{F}_{r}$ under a Generalized Amplitude Damping Channel
  • Figure 3: Decay of $\mathscr{F}_{l_{1}}$, $\mathscr{F}_{R}$, and $\mathscr{F}_{r}$ under a Phase-Amplitude Damping Channel
  • Figure 4: Decay of $\mathscr{F}_{l_{1}}$, $\mathscr{F}_{R}$, and $\mathscr{F}_{r}$ for the pure state $\ket{\gamma} = \alpha\ket{00} + \beta\ket{11}$ under a two-qubit bit-flip channel
  • Figure 5: Decay of $\mathscr{F}_{l_{1}}$, $\mathscr{F}_{R}$, and $\mathscr{F}_{r}$ for a dual-rail two-qubit state under an amplitude damping channel
  • ...and 5 more figures

Theorems & Definitions (5)

  • Theorem 2.1
  • Definition 4.1
  • Theorem 4.1
  • Definition 4.2
  • Definition 4.3