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Results in Physics

On complementarity and distribution of imaginarity in finite dimensions

Bin Chen, Xiaofen Huang, Shao-Ming Fei

Published Apr 12, 2024• 6 citations
Publisher

Abstract

We investigate the complementarity relation among the $l_{1}$ norm of imaginarity, linear entropy and the Brukner-Zeilinger invariant information in the presence of conjugate symmetry. By using an equality derived from the complementarity relation, we define a quantity named index of imaginarity distribution with respect to the imaginarity, linear entropy and invariant information in finite dimensional quantum systems. We show that the index of imaginarity distribution is nonnegative and upper bounded by the square of the $l_{1}$ norm of imaginarity, and serves as an invaluable tool in characterizing effectively the imaginarity distribution. Detailed examples are presented to illustrate our theoretical analysis.

DOI
10.1016/j.rinp.2024.107671

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We investigate the complementarity relation among the l1l_{1}l1​ norm of imaginarity, linear entropy and the Brukner-Zeilinger invariant information in the presence of conjugate symmetry. By using an equality derived from the complementarity relation, we define a quantity named index of imaginarity distribution with respect to the imaginarity, linear entropy and invariant information in finite dimensional quantum systems. We show that the index of imaginarity distribution is nonnegative and upper bounded by the square of the l1l_{1}l1​ norm of imaginarity, and serves as an invaluable tool in characterizing effectively the imaginarity distribution. Detailed examples are presented to illustrate our theoretical analysis.
To quantify the intrinsic information content in a quantum state, Brukner and Zeilinger introduced the concept of operationally invariant information in terms of the outcome probabilities of measuring a complete set of mutually complementary observables [\ifmmode \check{C}\else \v{C}\fi{}. Brukner and A. Zeilinger, Phys. Rev. Lett. 83, 3354 (1999)]. This information quantity has basic significance and implications, and the present work is devoted to some further studies of it. We first introduce the Brukner-Zeilinger invariant information in the presence of conjugate symmetry or antisymmetry, which are motivated by considerations of fundamental issues concerning conjugate symmetry in quantum mechanics. Then we prove that both the Brukner-Zeilinger invariant information with conjugate symmetry and that with conjugate antisymmetry are convex in the quantum state, and we show that they constitute a natural decomposition of the Brukner-Zeilinger invariant information. We further relate them to the imaginarity (i.e., the usage of a complex number field) of quantum mechanics and evaluate their extreme values.
We investigate the trade-off relations between imaginarity and mixedness in arbitrary ddd-dimensional quantum systems. For given mixedness, a quantum state with maximum imaginarity is defined to be a "maximally imaginary mixed state" (MIMS). By using the l1l_{1}l1​ norm of imaginarity and the normalized linear entropy, we conclusively identify the MIMSs for both qubit and qutrit systems. For high-dimensional quantum systems, we present a comprehensive class of MIMSs, which also gives rise to complementarity relations between the 111-norm of imaginarity and the 111-norm of mixedness, as well as between the relative entropy of imaginarity and the von Neumann entropy. Furthermore, we examine the evolution of the trade-off relation for single-qubit states under four specific Markovian channels: bit flip channel, phase damping channel, depolarizing channel and amplitude damping channel.
Abstract Complex numbers provide a remarkably powerful formalism in quantum mechanics. In this paper, we propose a method to construct imaginarity 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 (MUBs) 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 MUBs physically.
In the last few years, the l1{l}_{1}l1​ norm has been used as a quantum coherence measure by many authors. In this work, we introduce a variable that mediates the complementarity relation between the l1{l}_{1}l1​ norm of coherence, linear entropy, and the predictability of a quantum state of dimension greater than two. We show that this variable indicates the coherence distribution among the pairs of base states expanding the Hilbert space. We also show that a uniform quantum coherence distribution creates a direct complementarity between the l1{l}_{1}l1​ norm, predictability, and linear entropy.
Quantum imaginarity is an essential physical resource that underpins key functionalities of modern quantum technologies. We improve imaginarity witnessing via the prior knowledge of imaginarity-witness operators. To this end, we derive a rigorous upper bound on the maximum expectation value of an imaginarity witness operator over the set of all free (real) quantum states, which is given by the spectral norm of the real component of the corresponding witness operator. We demonstrate via detailed examples that this bound substantially improves imaginarity detection. We further classify all imaginarity-witness operators into four distinct families based on this bound. For these four witness classes, we perform a comprehensive analysis of their completeness and finite completeness, the joint detection of shared imaginary quantum states by different witnesses, and the conditions for distinct witnesses to identify identical imaginary states. Our results advance the fundamental understanding of imaginarity detection and offer useful insights for both theoretical studies and experimental implementations of quantum imaginarity.