量子理论中的虚部在描述量子相干性中发挥着关键作用，被广泛应用于态分辨、伪随机数生成和量子计量等量子信息任务中。Fernandes等人近期的一篇论文[C. Fernandes, R. Wagner, L. Novo, and E. F. Galvão, Phys. Rev. Lett. 133, 190201 (2024)]展示了如何利用Bargmann不变量来见证一组量子态的虚性。在本工作中，我们深入研究了Bargmann不变量的结构及其在量子比特系统中的量子实现。首先，我们针对一般的n个量子态的集合，给出了一类特殊Bargmann不变量集合的刻画（Fernandes等人曾针对四个态的集合研究过这类不变量）。随后，我们研究了相关Bargmann不变量集合$\boldsymbol{\text{数学符号：B}}_n$的性质及其在量子比特系统中的量子实现。我们的结果为Bargmann不变量的结构提供了新见解，有助于推动量子信息技术的发展，尤其是在量子比特系统领域。
英文摘要：
The imaginary in quantum theory plays a crucial role in describing quantum coherence and is widely applied in quantum information tasks such as state discrimination, pseudorandomness generation, and quantum metrology. A recent paper by Fernandes et al. [C. Fernandes, R. Wagner, L. Novo, and E. F. Galvão, Phys. Rev. Lett. 133, 190201 (2024) ] showed how to use the Bargmann invariant to witness the imaginarity of a set of quantum states. In this work, we delve into the structure of Bargmann invariants and their quantum realization in qubit systems. First, we present a characterization of special sets of Bargmann invariants (also studied by Fernandes et al. for a set of four states) for a general set of $n$ quantum states. Then, we study the properties of the relevant Bargmann invariant set $\mathcal{B}_n$ and its quantum realization in qubit systems. Our results provide new insights into the structure of Bargmann invariants, contributing to the advancement of quantum information techniques, particularly within qubit systems.
The geometrical arrangement of a set of quantum states can be completely characterized using relational information only. This information is encoded in the pairwise state overlaps, as well as in Bargmann invariants of higher degree written as traces of products of density matrices. We describe how to measure Bargmann invariants using suitable generalizations of the SWAP test. This allows for a complete and robust characterization of the projective-unitary invariant properties of any set of pure or mixed states. As applications, we describe basis-independent tests for linear independence, cohe…
We develop the widest possible generalisation of the well-known connection between quantum mechanical Bargmann invariants and geometric phases. The key notion is that of null phase curves in quantum mechanical ray and Hilbert spaces. Examples of such curves are developed. Our generalisation is shown to be essential to properly understand geometric phase results in the cases of coherent states and of Gaussian states. Differential geometric aspects of null phase curves are also briefly explored.
We investigate the geometric phases and the Bargmann invariants associated with a multi-level quantum systems. In particular, we show that a full set of `gauge-invariant' objects for an $n$-level system consists of $n$ geometric phases and ${1/2}(n-1)(n-2)$ algebraically independent 4-vertex Bargmann invariants. In the process of establishing this result we develop a canonical form for U(n) matrices which is useful in its own right. We show that the recently discovered `off-diagonal' geometric phases [N. Manini and F. Pistolesi, Phys. Rev. Lett. 8, 3067 (2000)] can be completely analysed in te…
Characterizing local unitary equivalence of multipartite states by local unitary Bargmann invariants
School of Mathematics and Data Science, Jiangnan University（江南大学数学与数据科学学院）；School of Mathematical Sciences, Beihang University（北京航空航天大学数学科学学院）；School of Mathematical Sciences, Capital Normal University（首都师范大学数学科学学院）
We present a study of the properties of Bargmann Invariants (BI) and Null Phase Curves (NPC) in the theory of the geometric phase for finite dimensional systems. A recent suggestion to exploit the Majorana theorem on symmetric SU(2) multispinors is combined with the Schwinger oscillator operator construction to develop efficient operator based methods to handle these problems. The BI is described using intrinsic unitary invariant angle parameters, whose algebraic properties as functions of Hilbert space dimension are analysed using elegant group theoretic methods. The BI-geometric phase connec…
We propose Lagrangian formulation for the particle with value of spin fixed within the classical theory. The Lagrangian turns out to be invariant under non-abelian group of local symmetries. As the gauge-invariant variables for description of spin we can take either the Frenkel tensor or the BMT vector. Fixation of spin within the classical theory implies $O(\hbar)$-corrections to the corresponding equations of motion.
We reinvestigate Bargmann's superselection rule for the overall mass of $n$ particles in ordinary quantum mechanics with Galilei invariant interaction potential. We point out that in order for mass to define a superselection rule it should be considered as a dynamical variable. We present a minimal extension of the original dynamics in which mass it treated as dynamical variable. Here the classical symmetry group turns out to be given by an $\reals$-extension of the Galilei group which formerly appeared only at the quantum level. There is now no obstruction to implement an action of the classi…
Pairwise detection, chiral conversion: Resource theory of quantum multistates
College of Science, Zhejiang University of Science and Technology（浙江科技学院理学院）；School of Mathematical Sciences, Hangzhou Dianzi University（杭州电子科技大学数学科学学院）
The Schroedinger equation is up-to-a-phase invariant under the Galilei group. This phase leads to the Bargmann's superselection rule, which forbids the existence of the superposition of states with different masses and implies that unstable particles cannot be described consistently in non-relativistic quantum mechanics. In this paper we claim that Bargmann's rule neglects physical effects and that a proper description of non-relativistic quantum mechanics requires to take into account this phase through the Extended Galilei group and the definition of its action on spacetime coordinates.
Many-body chirality of topological stabilizer states
Purdue University（普渡大学）；Perimeter Institute for Theoretical Physics（理论物理前沿研究所）；University of Waterloo（滑铁卢大学）；IBM Quantum, T.J. Watson Research Center（IBM量子，T.J.沃森研究中心）；University of Toronto（多伦多大学）
