D. M. Tong, E. Sjöqvist, L. C. Kwek, C. H. Oh
A kinematic approach to the geometric phase for mixed quantal states in nonunitary evolution is proposed. This phase is manifestly gauge invariant and can be experimentally tested in interferometry. It leads to well-known results when the evolution is unitary.
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The problem of geometric phase for an open quantum system is reinvestigated in a unifying approach. Two of existing methods to define geometric phase, one by Uhlmann's approach and the other by kinematic approach, which have been considered to be distinct, are shown to be related in this framework. The method is based upon purification of a density matrix by its uniform decomposition and a generalization of the parallel transport condition obtained from this decomposition. It is shown that the generalized parallel transport condition can be satisfied when Uhlmann's condition holds. However, it…
If Quantum Measurements Are Secretly Continuous Nonunitary Processes, Weak Measurements Can Detect It
本文探讨了量子测量的非幺正演化，通过弱后选择测量区分连续非幺正过程与瞬时塌缩，提出实验协议分析氢原子案例。
Geometric phases, arising from cyclic evolutions in a curved parameter space, appear in a wealth of physical settings. Recently, and largely motivated by the need of an experimentally realistic definition for quantum computing applications, the quantum geometric phase was generalized to open systems. The definition takes a kinematical approach, with an initial state that is evolved cyclically but coupled to an environment --- leading to a correction of the geometric phase with respect to the uncoupled case. We obtain this correction by measuring the nonunitary evolution of the reduced density …
In a differential approach elaborated, we study the evolution of the parameters of Gaussian, mixed, continuous variable density matrices, whose dynamics are given by Hermitian Hamiltonians expressed as quadratic forms of the position and momentum operators or quadrature components. Specifically, we obtain in generic form the differential equations for the covariance matrix, the mean values, and the density matrix parameters of a multipartite Gaussian state, unitarily evolving according to a Hamiltonian $\hat{H}$. We also present the corresponding differential equations which describe the nonun…
The metric underlying the mixed state geometric phase in unitary and nonunitary evolution [Phys. Rev. Lett. {\bf 85}, 2845 (2000); Phys. Rev. Lett. {\bf 93}, 080405 (2004)] is delineated. An explicit form for the line element is derived and shown to be related to an averaged energy dispersion in the case of unitary evolution. The line element is measurable in interferometry involving nearby internal states. Explicit geodesics are found in the single qubit case. It is shown how the Bures line element can be obtained by extending our approach to arbitrary decompositions of density operators. The…
If Mixed States Are Secretly Quickly Oscillating Pure States, Weak Measurements Can Detect It
本文探讨了混合态与快速振荡纯态的可区分性，证明仅靠强非后选择测量无法区分二者，但后选择系统中的弱测量可以产生不同结果，从而为探测黑洞辐射等高能物理现象提供了桌面实验方案。
We calculate the geometric phase for an open system (spin-boson model) which interacts with an environment (ohmic or nonohmic) at arbitrary temperature. However there have been many assumptions about the time scale at which the geometric phase can be measured, there has been no reported observation yet for mixed states under nonunitary evolution. We study not only how they are corrected by the presence of the different type of environments but also estimate the corresponding times at which decoherence becomes effective. These estimations should be taken into account when planning experimental …
Using a kinematic approach we show that the non-adiabatic, non-cyclic, geometric phase corresponding to the radiation emitted by a three level cascade system provides a sensitive diagnostic tool for determining the entanglement properties of the two modes of radiation. The nonunitary, noncyclic path in the state space may be realized through the same control parameters which control the purity/mixedness and entanglement. We show analytically that the geometric phase is related to concurrence in certain region of the parameter space. We further show that the rate of change of the geometric phas…
Off-diagonal geometric phases have been developed in order to provide information of the geometry of paths that connect noninterfering quantal states. We propose a kinematic approach to off-diagonal geometric phases for pure and mixed states. We further extend the mixed state concept proposed in [Phys. Rev. Lett. {\bf 90}, 050403 (2003)] to degenerate density operators. The first and second order off-diagonal geometric phases are analyzed for unitarily evolving pairs of pseudopure states.
Generalized Entropic Quantum Speed Limits
本文基于 α-z-Rényi 相对熵提出一类广义熵量子速度极限，适用于幺正与非幺正演化下的有限维量子系统，并给出其与能量涨落或 Kraus 算符变化率的关系及在量子计量等领域的应用前景。
Measuring Mixed-State Topological Invariant in Open Photonic Quantum Walk
该研究通过光子量子漫步实验，提出了一种测量混合态拓扑不变量的方法，利用非幺正动力学制备拓扑非平凡态，并通过机器学习提取几何相位。
In our previous work it has been shown the possibility to use the Aharonov-Anandan invariant as a tool in the analysis of disparate systems, including Hawking and Unruh effects, as well as graphene physics and thermal states. We show that the vacuum condensation, characterizing such systems, is also related with geometric phases and we analyze the properties of the geometric phase of systems represented by mixed state and undergoing a nonunitary evolution. In particular, we consider two level atoms accelerated by an external potential and interacting with a thermal state. We propose the realiz…
The geometric phase is of fundamental interest and plays an important role in quantum information processing. However, the definition and calculation of this phase for open systems remains a problem due to the lack of agreement on generalizations of the parallel transport condition to mixed state nonunity evolutions. Here we tackle this problem by associating the open system geometric phase with the parallel transport of the joint system-reservoir state. Our approach not only provides a way around the nonunitary evolution obstacle, but also sheds light on the relation between the geometric pha…
We calculate the geometric phase of a bipartite two-level system coupled to an external environment. We analyze the reduced density matrix for an arbitrary initial state of the composite system and compute the correction to the unitary geometric phase through a kinematic approach. In all cases considered, we observe a similar structure as a function of the degree of the entanglement of the initial state. Further, we compute the entanglement entropy and concurrence of the bipartite state and analyze if there is any relation among these quantities and the geometric phase acquired during the nonu…
In our investigation into the effects of the degree of polarization in modulation of partially polarized light we assume general settings of the interferometry of partially polarized lightwaves and perform theoretical analysis of the Uhlmann and the interferometric phases. We introduce the relative Uhlmann phase determined by the Uhlmann holonomies of interfering beams and show that the interferometric phase generalized to the case of nonunitary evolution can, similar to the Uhlmann phase, be cast into the holonomy defined form. By using the technique based on a two-arm Mach-Zehnder interferom…
Geometric Phase in Quantum Synchronization
本文研究自旋系统中量子极限环振荡器的几何相位，发现其随信号强度和失谐的结构与同步 Arnold 舌对应，并在同步阻塞区共同消失，同时给出慢旋转弱信号极限下的解析表达式。
Based on a kinematic approach in defining a geometric phase for a density matrix, we define the generalized Loschmidt overlap amplitude (GLOA) for an open system for arbitrary quantum evolution. The GLOA reduces to the Loschmidt overlap amplitude (LOA) with a modified dynamic phase for unitary evolution of a pure state, with the argument of the GLOA well-defined by the geometric phase, thus possessing similar physical interpretation to that of the LOA. The rate function for the GLOA exhibits non-analyticity at a critical time, which corresponds to the dynamical quantum phase transition. We obs…
A topological measure characterizing symmetry-protected topological phases in one-dimensional open fermionic systems is proposed. It is built upon the kinematic approach to the geometric phase of mixed states and facilitates the extension of the notion of topological phases from zero-temperature to nonzero-temperature cases. In contrast to a previous finding that topological properties may not survive above a certain critical temperature, we find that topological properties of open systems, in the sense of the measure suggested here, can persist at any finite temperature and disappear only in …
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