Imaginarity measures induced by real part states and the complementarity relations

TL;DR

Addresses how to quantify imaginarity in quantum states by constructing measures from real part states. Introduces a fidelity-based imaginarity measure $M_Re( ho) = 1 - F( ho, Re( ho))$ and derives its basic bounds and properties, with an explicit analytic form for qubits. Establishes quantitative relationships between $M_Re$ and existing imaginarity measures such as trace-norm, relative-entropy, and geometric imaginarity, including a hierarchy $M_Re \,\le\, M_g$. Demonstrates a complementarity relation under a complete set of mutually unbiased bases in low dimensions and discusses extension to higher dimensions via real orthogonal mappings, highlighting the measurement-basis dependence of imaginarity.

Abstract

Complex numbers are indispensable in quantum mechanics and the resource theory of imaginarity has been developed recently. In this paper, we propose a method to construct imaginary measures by real part states. Specifically, we propose an imaginarity measure in terms of fidelity and explore its properties. The analytical expression of the imaginarity measure is presented in qubit systems. The relations between the proposed imaginarity measure and some other imaginarity measures (such as geometric imaginarity, Tsallis relative entropy imaginarity and trace norm imaginarity) are derived. The complementarity relations of the imaginarity measure under a complete set of mutually unbiased bases are provided in low-dimensional systems. This work not only highlights the prominent role of the real part state in the imaginarity resource theory, but also reveals the constraint of imaginarity on a complete set of mutually unbiased bases physically.

Figures (5)

• Figure 1: (Color online) The subfigure (a) is the visualization of the ternary function $M_{\mathrm{Re}}$ in Eq. (\ref{['eq mre 2']}) in qubit systems. The subfigure (b) is the contour plot of $M_{\mathrm{Re}}$ with $r_z=0$.
• Figure 2: (Color Online) The comparison of the imaginarity measures $M_{\mathrm{g}}\ ( M^\prime_{T,\frac{1}{2}})$ and $M_{\mathrm{Re}}\ (M_{\mathrm{g}}')$ for qubit states in Eq. (\ref{['eq ex rho-2-o']}), with $M_{\mathrm{g}}\ (M^\prime_{T,\frac{1}{2}})$ shown as the solid green curve and $M_{\mathrm{Re}}\ (M_{\mathrm{g}}')$ as the dashed blue curve.
• Figure 3: (Color Online) The visualization of the bounds for $M_{\mathrm{Re}}$ in Theorem \ref{['thm:trace norm']} for the qubit state in Eq. \ref{['eq ex rho-2-o']}. The red dashed curve above represents $f_1=1-(P(\rho)-M_{\mathrm{tr}}(\rho)^2)^{\frac{1}{2}}$ corresponding to the upper bound in Eq. \ref{['eq tradeoff']}. The yellow dashed curve below represents $f_2=1-\left[\frac{1}{d}+\frac{1}{d}\sqrt{(d-1)^2-(d-1)M_{\mathrm{tr}}(\rho)^2}\right]^{\frac{1}{2}}$ corresponding to the lower bound in Eq. \ref{['eq tradeoff']}. The blue solid curve represents $M_{\mathrm{Re}}$ and it coincides with the lower bound $f_2$.
• Figure 4: (Color Online) The green part above is the region available for the imaginarity measure $M_{\mathrm{Re}}$. Here the vertical axis $f= \left(1-M_{\mathrm{Re}}^{B_1}\right)^2+\left(1-M_{\mathrm{Re}}^{B_2}\right)^2+\left(1-M_{\mathrm{Re}}^{B_3}\right)^2$ corresponds to the left-hand side of Eq. (\ref{['eq trade-off for qubit']}), while the blue curve plots the function $\frac{\sqrt{(1-|\boldsymbol{r}|^2)(3-2|\boldsymbol{r}|^2)}+3+2|\boldsymbol{r}|^2}{2}$ with respect to $|r|^2$, corresponding to the right-hand side.
• Figure 5: (Color Online) Visualization of the complementarity relation in Eq. \ref{['eq compl qutrit pure state under complete mutually']} for 2000 randomly generated qutrits. The blue solid dots represent $f=\sum\limits_{k=1}^4 \left(1-M_{\mathrm{Re}}^{Z_k}\right)^2$ corresponding to the left-hand side of Eq. \ref{['eq compl qutrit pure state under complete mutually']} with respect to the purity $P$ for randomly generated qutrit states. The red line represents the function $\frac{31}{14}P$ corresponding to the right-hand side of Eq. \ref{['eq compl qutrit pure state under complete mutually']}.

Theorems & Definitions (17)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Corollary 1
• Example 1
• Theorem 5
• Theorem 6
• Example 2
• Theorem 7