The Berry phase reveals geometric information of quantum wave functions via their phases acquired after an adiabatic cyclic process, and its concept has laid the foundation for understanding many topological properties of materials. The theory of Berry phase is built on pure quantum states. For example, the ground state fits the description as the limit of a statistical ensemble at zero temperature. At finite temperatures, the density matrix describes thermal properties of a quantum system by associating a thermal distribution to all the states of the system. Therefore, it is an important task to generalize the Berry phase to the realm of mixed quantum states. There have been several approaches to address this problem, among which the Uhlmann phase has attracted much attention recently since it has been shown to exhibit topological phase transitions at finite temperatures in several 1D, 2D, and spin-j systems. A key feature of those systems is the discontinuous jumps of the Uhlmann phase at the critical temperatures, signifying the changes of the underlying Uhlmann holonomy as the system traverses a loop in the parameter space. However, due to the complexity of the mathematical structure and physical interpretation, the knowledge of the Uhlmann phase is far less than that of the Berry phase in the literature. Moreover, only a handful of models allow analytical results of the Uhlmann phase to be obtained. The Berry phase is purely geometric in the sense that it does not depend on any dynamical effect during the time evolution of the quantum system of interest . Therefore, the theory of the Berry phase can be constructed in a purely mathematical manner. As a generalization, hydroponic bucket the Uhlmann phase of density matrices was built in an almost parallel way from a mathematical point of view and shares many geometric properties with the Berry phase.
We will first summarize both the Berry and Uhlmann phases using a fiber-bundle language to highlight their geometric properties. Next, we will present the analytic expressions of the Uhlmann phases of bosonic and fermionic coherent states and show that their values approach the corresponding Berry phases as temperature approaches zero. Both types of coherent states are useful in the construction of path integrals of quantum fields. While any number of bosons are allowed in a single state, the Pauli exclusion principle restricts the fermion number of a single state to be zero or one. Therefore, complex numbers are used in the bosonic coherent states while Grassmann numbers are used in the fermionic coherent states. Moreover, the Berry phases of coherent states can be found in the literature, and we summarize the results in Appendix A. Our exact results of the Uhlmann phases of bosonic and fermionic coherent states suggest that they indeed carry geometric information, as expected by the concept of holonomy and analogy to the Berry phase. We will show that the Uhlmann phases of both cases decrease smoothly with temperature without a finite-temperature transition, in contrast to some examples with finite-temperature transitions in previous studies. As temperature drops to zero, the Uhlmann phases of bosonic and fermionic coherent state approach the corresponding Berry phases. Our results of the coherent states, along with earlier observations, suggest the Uhlmann phase reduce to the corresponding Berry phase in the zero-temperature limit.
The correspondence is nontrivial because the Uhlmann phase requires full-rank density matrices, which cannot be satisfied only by the ground state at zero temperature. Moreover, the fiber bundle for density matrices in Uhlmann’s theory is a trivial one, but the fiber bundle for wavevfunctions in the theory of Berry phase needs not be trivial. A similar question on why the Uhlmann phase agrees with the Berry phase in certain systems as temperature approaches zero was asked in Ref. without an answer. In the last part of the paper, we present a detailed analysis of the Uhlmann phase at low temperatures to search for direct relevance with the Berry phase. With the clues from the previous examples, we present a conditional proof of the correspondence by focusing on systems allowing analytic treatments of the path-ordering operations. Before showing the results, we present a brief comparison between the Uhlmann phase and another frequently mentioned geometrical phase for mixed quantum states proposed in Refs. , which was originally introduced for unitary evolution but later extended to non-unitary evolution. This geometrical phase was inspired by a generalization of the Mach-Zehnder interferometry in optics and was named accordingly as the interferometric phase. It has a different formalism with a more intuitive physical picture and has been measured in experiments. In general situations, the interferometric phase can be expressed as the argument of a weighted sum of the Berry phase factors from each individual eigenstate. Thus, its relation to the Berry phase is obvious. However, the concise topological meaning of the interferometric phase is less transparent since it is not directly connected to the holonomy of the underlying bundle as the Uhlmann phase does. The reason has been discussed in a previous comparison between the two geometrical phases. The interferometric phase relies solely on the evolution of the system state while the Uhlmann phase is influenced by the changes of both the system and ancilla, which result in the Uhlmann holonomy. Although Uhlmann’s approach can be cast into a formalism parallel to that of the Berry phase as we will explain shortly, its exact connection to the Berry phase is still unclear. The Uhlmann-Berry correspondence discussed below will offer an insight into this challenging problem. The rest of the paper is organized as follows. In Sec. II, we first present concise frameworks based on geometry for the Berry and Uhlmann phases, using a fiber-bundle language. In Sec. III, we derive the analytic expressions of the Uhlmann phases of bosonic and fermionic coherent states and analyze their temperature dependence. Additionally, the Uhlmann phase of a three-level system is also presented. Importantly, the Uhlmann phases of both types of coherent states and the three-level system are shown to approach the respective Berry phases as temperature approaches zero. In Sec. IV, we propose the generality of the correspondence between the Uhlmann and Berry phases in the zerotemperature limit and give a conditional proof. In Sec. V, we discuss experimental implications and propose a protocol for simulating and measuring the Uhlmann phase of bosonic coherent states. Sec. VI concludes out work. The Berry phases of bosonic and fermionic coherent sates and the special cases with a 1D Hilbert space are summarized in the Appendix.The classical approach to the central limit theorem and the accuracy of approximations for independent random variables rely heavily on Fourier transform methods. However, the use of Fourier methods is highly limited without an independence structure, which makes it far less possible to capture the explicit bounds for the accuracy of approximations. In 1972, Charles Stein introduced a novel technique, now known as Stein’s method, for normal approximation. The method works for both independent and dependent random variables. The method also provides bounds of approximation accuracy. Extensive applications of Stein’s method to obtain uniform and non-uniform Berry–Esseen-type bounds for independent and dependent random variables can be found in, for example, Diaconis , Baldi et al. , Barbour , Dembo and Rinott , Goldstein and Reinert , Chen and Shao , Chatterjee , Nourdin and Peccati and Chen and Fang . In addition to the traditional study of Berry–Esseen bounds, new developments to Stein’s method have triggered a series of research on Cramér-type moderate deviations, stackable planters which address the relative error of two tail probabilities. See, for example, Raič , Chen et al. and Shao and Zhou , among others.
Various extensions of Stein’s idea have been applied to many other probability approximations, most notably to Poisson, Poisson process, compound Poisson, binomial approximations and more recently to multivariate, combinatorial and discretized normal approximations. Stein’s method has also found diverse applications in a wide range of fields, see for example,Arratia et al. , Barbour et al. and Chen . Expositions of Stein’s method and its applications in normal and other distributional approximations can be found in Diaconis and Holmes , Barbour and Chen . We also refer to Chen et al. a thorough coverage of the method’s fundamentals and recent developments in both theory and applications. The paper is organized as follows. In the next section, we give a brief review on recent developments on Stein’s method. In Section 3, we present the main results in this paper, the Berry–Esseen bounds and Cramér type moderate deviations for Studentized nonlinear statistics. Applications to Studentized U-statistics and L-statistics are discussed in Section 4. The proofs of the main results are in Section 5, while other technical proofs are postponed to Appendix.A sea change in Colorado politics has vaulted the Democratic Party to unprecedented majorities in the state legislature and a stranglehold on statewide elected office. Democratic dominance to this degree appeared unlikely at the turn of the century when Republicans held majorities in both chambers of the General Assembly, a 4-2 advantage in the state’s U.S. House delegation, both U.S. Senate seats, and the governorship. In the razor-thin 2000 presidential election, Colorado cast its eight electoral votes for Texas Governor George W. Bush who comfortably carried the state by eight percentage points. Republican preeminence in state and federal electoral politics disappeared in less than a generation as Colorado became a solidly blue state with an adrift Republican party unable to wage competitive statewide campaigns. Near supermajority status in the General Assembly and firm control over all statewide executive offices has positioned Colorado Democrats with exceptional political power. While the negative economic effects of the COVID-19 pandemic continue to linger, the Colorado economy has generally rebounded from the great upheaval more rapidly than most other states . Economic growth has propelled increases in revenue; however, the Taxpayer’s Bill of Rights imposes substantial constraints on the total amount of funds available for policymakers to distribute. Spending commitments, such as mandatory increases in K-12 education funding as required by Amendment 23, further cut into the total amount available to appropriate, which creates difficult choices for members of the Joint Budget Committee. Ratified into Article X, Section 20 of the state constitution by voters in 1992, TABOR imposes restrictions on both revenue and spending. Because TABOR limits revenue collections to the prior year’s amount plus population growth and inflation, Colorado taxpayers have received $8.2 billion in TABOR refunds since its enactment including $525.5 million in 2021 and a record $3.7 billion in tax refunds in 2022 . Although it is difficult to amend the state constitution, Colorado voters have considered ballot measures proposing TABOR reform in nearly every election cycle since its adoption. Few have succeeded. The approval of just 11 of the 36 ballot measures to amend TABOR corresponds to a failure rate of nearly 70% . As a result of this unsuccessful track record to modify or repeal TABOR, its shadow continues to loom large over budgetary politics in the Centennial state. An exception to the general inability of reformers to modify TABOR occurred in 2005 when voters narrowly approved referendum C with 52% voting yes. This notable exemption to TABOR permitted the state to spend all revenue collected across the next five fiscal years, which resulted in nearly $3.6 billion in spending that would have otherwise returned to taxpayers during this time frame . Beginning in fiscal year 2010, referendum C permits the General Assembly to retain and spend all funds collected up to the “Referendum C cap.” The passage of referendum C provided greater opportunities for financial investment in areas such as health care, education, and transportation, as well as greater support for police, fire fighters, and other first responders. In retrospect, the successful passage of this reform in an off-year election was anomalistic as voters have since rejected several ballot measures to modify TABOR spending limits . Voters have also opposed an array of tax increases on a dozen occasions including proposals to fund public schools and transportation . Sensing an opportunity to capitalize on the public’s desire for property tax relief, Democrats unsuccessfully sought to connect a reduction in the property tax rate with further erosion of TABOR in 2023. Despite slowing population growth, property values across Colorado continue to soar.