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Since the discovery of pulsars, the rotating-lighthouse model has been the choice of model to explain the radiation pulsation of pulsars. After discovering that some main sequence stars (e.g., CU Virginis) and ultracool dwarfs (e.g., TVLM 513-46546) also behave like pulsars, the lighthouse model was again adopted to explain their pulse signals. Our research found that if we use the magnetic-field oscillation (MO) model, we can explain the pulse radiation results better regardless of whether its source is a neutron star, a main-sequence star or an ultracool dwarf. We propose a verifiable prediction that can be used to evaluate the MO model. Our prediction is that there is a 90 degree phase lag between the magnetic field and radio signal of TVLM 513-46546, and the zero-crossing point of the magnetic field is the moment when the direction of the light's circular polarization is reversed. No new observations are needed to check this prediction, but certain existing data needs to be re-mined., Comment: 8 pages, 4 figures, accepted for publication in MNRAS

A new semi-supervised machine learning method for the discovery of structure-spectrum relationships is developed and demonstrated using the specific example of interpreting X-ray absorption near-edge structure (XANES) spectra. This method constructs a one-to-one mapping between individual structure descriptors and spectral trends. Specifically, an adversarial autoencoder is augmented with a novel rank constraint (RankAAE). The RankAAE methodology produces a continuous and interpretable latent space, where each dimension can track an individual structure descriptor. As a part of this process, the model provides a robust and quantitative measure of the structure-spectrum relationship by decoupling intertwined spectral contributions from multiple structural characteristics. This makes it ideal for spectral interpretation and the discovery of new descriptors. The capability of this procedure is showcased by considering five local structure descriptors and a database of over fifty thousand simulated XANES spectra across eight first-row transition metal oxide families. The resulting structure-spectrum relationships not only reproduce known trends in the literature, but also reveal unintuitive ones that are visually indiscernible in large data sets. The results suggest that the RankAAE methodology has great potential to assist researchers to interpret complex scientific data, test physical hypotheses, and reveal new patterns that extend scientific insight.

As machine learning (ML) methods continue to be applied to a broad scope of problems in the physical sciences, uncertainty quantification is becoming correspondingly more important for their robust application. Uncertainty aware machine learning methods have been used in select applications, but largely for scalar properties. In this work, we showcase an exemplary study in which neural network ensembles are used to predict the X-ray absorption spectra of small molecules, as well as their point-wise uncertainty, from local atomic environments. The performance of the resulting surrogate clearly demonstrates quantitative correlation between errors relative to ground truth and the predicted uncertainty estimates. Significantly, the model provides an upper bound on the expected error. Specifically, an important quality of this uncertainty-aware model is that it can indicate when the model is predicting on out-of-sample data. This allows for its integration with large scale sampling of structures together with active learning or other techniques for structure refinement. Additionally, our models can be generalized to larger molecules than those used for training, and also successfully track uncertainty due to random distortions in test molecules. While we demonstrate this workflow on a specific example, ensemble learning is completely general. We believe it could have significant impact on ML-enabled forward modeling of a broad array of molecular and materials properties., Comment: 24 pages, 16 figures

We investigate the diagonal entropy for ground states of the extended Kitaev chains with extensive pairing and hopping terms. The systems contain rich topological phases equivalently represented by topological invariant winding numbers and Majorana zero modes. Both the finite size scaling law and block scaling law of the diagonal entropy are studied, which indicates that the diagonal entropy demonstrates volume effect. The parameter of volume term is regarded as the diagonal entropy density, which can identify the critical points of symmetry-protected topological phase transitions efficiently in the studied models, even for those with higher winding numbers. The formulation of block scaling law and the capability of diagonal entropy density in detecting topological phase transitions are independent of the chosen bases. In order to manifest the advantage of diagonal entropy, we also calculate the global entanglement, which can not show clear signatures of the topological phase transitions. This work provides a new quantum-informatic approach to characterize the feature of the topologically ordered states and may motivate a deep understanding of the quantum coherence and diagonal entropy in various condensed matter systems.

A medium-scale quantum computer with full universal quantum computing capability is necessary for various practical aims and testing applications. Here we report a 34-qubit quantum virtual machine (QtVM) based on a medium server. Our QtVM can run quantum assembly language with graphic interfaces. The QtVM is implemented with single qubit rotation gate, single to multiple controlled NOT gates to realize the universal quantum computation. Remarkably, it can realize a series of basic functions, such as, the "if" conditional programming language based on single-shot projective measurement results, "for" iteration programming language, build in arithmetic calculation. The measurement can be single-shot and arbitrary number of multi-shot types. In addition, there is in principle no limitation on number of logic gates implemented for quantum computation. By using QtVM, we demonstrate the simulation of dynamical quantum phase transition of transverse field Ising model by quantum circuits, where 34 qubits with one million gates are realized. We also show the realization of programmable Shor algorithm for factoring 15 and 35., Comment: 11 pages, 9 figures. In V2, results of 34 qubits one million gates are presented. The speed of QtVM is corrected because 100 measurements are missing in V1

We propose a quantum tomography scheme for pure qudit systems which adopts random base measurements and generative learning methods, along with a built-in fidelity estimation approach to assess the reliability of the tomographic states. We prove the validity of the scheme theoretically, and we perform numerically simulated experiments on several target states including three typical quantum information states and randomly initiated states, demonstrating its efficiency and robustness. The number of replicas required by a certain convergence criterion grows in the manner of low-degree polynomial when the system scales, thus the scheme achieves high scalability that is crucial for practical quantum state tomography., Comment: 5 pages of main text including 5 figures, appended by 4 appendices including 1 figure. GitHub: https://github.com/congzlwag/BornMachineTomo

We investigate the optimal quantum state reconstruction from cloud to many spatially separated users by measure-broadcast-prepare scheme without the availability of quantum channel. The quantum state equally distributed from cloud to arbitrary number of users is generated at each port by ensemble of known quantum states with assistance of classical information of measurement outcomes by broadcasting. The obtained quantum state for each user is optimal in the sense that the fidelity universally achieves the upper bound. We present the universal quantum state distribution by providing physical realizable measurement bases in the cloud as well as the reconstruction method for each user. The quantum state reconstruction scheme works for arbitrary many identical pure input states in general dimensional system., Comment: 7 pages, 1 figure

We present a quantum algorithm solving the greatest common divisor (GCD) problem. This quantum algorithm possesses similar computational complexity with classical algorithms, such as the well-known Euclidean algorithm for GCD. This algorithm is an application of the quantum algorithms for the hidden subgroup problems, the same as Shor factoring algorithm. Explicit quantum circuits realized by quantum gates for this quantum algorithm are provided. We also give a computer simulation of this quantum algorithm and present the expected outcomes for the corresponding quantum circuit.

Quantum coherence and quantum correlations lie in the center of quantum information science, since they both are considered as fundamental reasons for significant features of quantum mechanics different from classical mechanics. We present a group of complementary relations for quantum coherence and quantum correlations; specifically, we focus on thermal discord and conditional information in scenarios of multiple measurements. We show that the summation of quantum coherence quantified in different bases has a lower bound, resulting from entropic uncertainty relations with multiple measurements. Similar results are also obtained for thermal discord and for post-measurement conditional information with multiple measurements in a multipartite system. These results indicate the general applications of the uncertainty principle to various concepts of quantum information., Comment: 7 pages

We study the entanglement in momentum space of the ground state of a disordered one-dimensional fermion lattice model with attractive interaction. We observe two components in the entanglement spectrum, one of which is related to paired-fermion entanglement and contributes to the long-range correlation in position space. The vanishing point of it indicates the localization phenomenon in the ground state of this model. Additionally, by method of entanglement spectrum, we provide a new evidence to show the transition of two phases induced by interaction, and fnd that this phase transition is not infuenced by the disorder. Our result show key characteristics in entanglement for different phases in the system, and provide a novel perspective to understand localization phenomena., Comment: 9 pages, 5 figures, published edition

We investigate the entanglement spectrum of the ground state of Su-Schrieffer-Heeger-Hubbard model. The topological phases of the model can be identified by degeneracy of the largest eigenvalues of entanglement spectrum. The study of the periodic boundary condition is enough to obtain the phase diagram of the model, without the consideration of the open boundary condition case. Physical interpretation about the bulk-edge correspondence in the entanglement spectrum is presented. The method of the entanglement spectrum can be applicable in studying other topological phases of matter.

We study the entanglement R\'{e}nyi $\alpha$-entropy (ER$\alpha $E) as the measure of entanglement. Instead of a single quantity in standard entanglement quantification for a quantum state by using the von Neumann entropy for the well-accepted entanglement of formation (EoF), the ER$\alpha $E gives a continuous spectrum parametrized by variable $\alpha $ as the entanglement measure, and it reduces to the standard EoF in the special case $\alpha \rightarrow 1$. The ER$\alpha $E provides more information in entanglement quantification, and can be used such as in determining the convertibility of entangled states by local operations and classical communication. A series of new results are obtained: (i) we can show that ER$\alpha $E of two states, which can be mixed or pure, may be incomparable, in contrast to the fact that there always exists an order for EoF of two states; (ii) similar as the case of EoF, we study in a fully analytical way the ER$\alpha $E for arbitrary two-qubit states, the Werner states and isotropic states in general d-dimension; (iii) we provide a proof of the previous conjecture for the analytical functional form of EoF of isotropic states in arbitrary d-dimension., Comment: 11 pages, 4 figures

We present the entropic uncertainty relations for multiple measurement settings in quantum mechanics. Those uncertainty relations are obtained for both cases with and without the presence of quantum memory. They take concise forms which can be proven in a unified method and easy to calculate. Our results recover the well known entropic uncertainty relations for two observables, which show the uncertainties about the outcomes of two incompatible measurements. Those uncertainty relations are applicable in both foundations of quantum theory and the security of many quantum cryptographic protocols., Comment: 11 pages (including the appendix), 2 figures

Quantum state can be teleported to a remote site by only local measurement and classical communication if the prior Einstein-Podolsky-Rosen quantum channel is available between the sender and the receiver. Those quantum channels shared by multiple nodes can constitute a quantum network. Yet, studies on the efficiency of quantum communication between nodes of quantum networks remain limited, which differs from classical case in that the quantum channel will be consumed if teleportation is performed. Here, we introduce the exclusive quantum channels (EQC) as the measure of efficiency of quantum information transmission. It quantifies the amount of quantum information which can be teleported between nodes in a quantum network. We show that different types of EQC are local quantities with effective circles. Significantly, capacity of quantum communication of quantum networks quantified by EQC is independent of distance for the communicating nodes. Thus, the quantum network can be dealt as the isotropic medium where quantum communication is no-decaying. EQC are studied by both analytical and numerical methods. The EQC can be enhanced by transformations of lattices of quantum network via entanglement swapping. Our result opens the avenue in studying the quantum communication of the quantum networks., Comment: 8 pages, 7 figures

Each of the two pulsars in the double pulsar PSR J0737-3039A/B system exhibits not only the pulses emanating from itself, but also displays modulations near the pulse period of the other. Freire et al. (2009, MNRAS, 396, 1764) have put forward a technique using the modulation of B by A to determine the sense of rotation of pulsar A relative to its orbital motion, among other quantities. In this paper, we present another technique with the same purpose. While the Freire et al. approach analyzes pulse arrival times, ours instead uses periods or frequencies (their inverses), which can be experimentally determined via power spectral analysis similar to that used in pulsar searches. Our technique is based on the apparent change in spin period of a body when it is measured from an orbiting platform (the other pulsar), and is shown to be entirely analogous to the difference between the sidereal and solar spin period of the Earth (i.e., the sidereal and solar day). Two benefits of this approach are its conceptual and computational simplicity. The direct detection of spin with this technique will observationally validate the rotating lighthouse model of pulsar emission, while the detection of the relative directions of spin and orbital angular momenta has important evolutionary implications. Our technique can be used on other binary systems exhibiting mutually induced phenomena., Comment: 7 pages, 2 figures, 2 tables, accepted by MNRAS, this article supersedes arXiv:1210.6561

We propose a quantum fitting scheme to estimate the magnetic field gradient with $N$-atom spins preparing in W state, which attains the Heisenberg-scaling accuracy. Our scheme combines the quantum multi-parameter estimation and the least square linear fitting method to achieve the quantum Cram\'{e}r-Rao bound (QCRB). We show that the estimated quantity achieves the Heisenberg-scaling accuracy. In single parameter estimation with assumption that the magnetic field is strictly linear, two optimal measurements can achieve the identical Heisenberg-scaling accuracy. Proper interpretation of the super-Heisenberg-scaling accuracy is presented. The scheme of quantum metrology combined with data fitting provides a new method in fast high precision measurements., Comment: 7 pages, 2 figures

We formulate a series of non-trivial equalities which are satisfied by all no-signaling correlations, meaning that no faster-than-light communication is allowed with the resource of these correlations. All quantum and classical correlations satisfy these equalities since they are no-signaling. By applying these equalities, we provide a general framework for solving the multipartite "guess your neighbor's input" (GYNI) game, which is naturally no-signaling but shows conversely that general no-signaling correlations are actually more non-local than those allowed by quantum mechanics. We confirm the validity of our method for number of players from 3 up to 19, thus providing convincing evidence that it works for the general case. In addition, we solve analytically the tripartite GYNI and obtain a computable measure of supra-quantum correlations. This result simplifies the defined optimization procedure to an analytic formula, thus characterizing explicitly the boundary between quantum and supra-quantum correlations. In addition, we show that the gap between quantum and no-signaling boundaries containing supra-quantum correlations can be closed by local orthogonality conditions in the tripartite case. Our results provide a computable classification of no-signaling correlations., Comment: 7 pages of article (including appendix), 4 figures; major revision; accepted by PRA

Measurement based quantum computation (MBQC) is an effective paradigm for universal quantum computation. In this scheme, the universal set of quantum gates are realized by only local measurements on the prior prepared cluster states. The inevitable decoherence is harmful to the realization of those quantum gates. Here, we investigate the performance of the quantum gates exposed to different type of noises. We find that some errors may not influence the success of the quantum gates, in contrast, some others may destroy their realization. We show that there is a controlling pattern that can protect quantum gates from certain types of noises and thus can improve the success probability of the gates implementation., Comment: 6 pages, 6 figures

A model was constructed to explain the orbit modulation of PSR J0737-3039B (referred to as B), which included the following suppositions: (1) B has arrived at the aged period and generally cannot emit observable light; (2) The PSR J0737-3039A (referred to as A) is dynamic and is encircled by dense charged particles. When A revolves along its orbit, some of the particles are scattered and form a particle cloud band along its orbit; (3) When passing the two intersection points of the orbits, B goes through the cloud band and drives the cloud particles to emit light; (4) Generally, B passes twice through the cloud band every orbit cycle; hence, there are two bright phases along the orbit. (5) It is the complexities of the distribution of the cloud particles and B's emission region that cause the bright phases to drift away from the orbit intersections. We also provide two predictions for the coming performance of B to test our model., Comment: 6 pages, 3 figures

We propose a quantum method to judge whether two spatially separated clocks have been synchronized within a specific accuracy $\sigma$. If the measurement result of the experiment is obviously a nonzero value, the time difference between two clocks is smaller than $\sigma$; otherwise the difference is beyond $\sigma$. On sharing the 2$N$-qubit bipartite maximally entangled state in this scheme, the accuracy of judgement can be enhanced to $\sigma\sim{\pi}/{(\omega(N+1))}$. This criterion is consistent with Heisenberg scaling that can be considered as beating standard quantum limit, moreover, the unbiased estimation condition is not necessary., Comment: 5 pages, 1 figure

No-cloning theorem is fundamental for quantum mechanics and for quantum information science that states an unknown quantum state cannot be cloned perfectly. However, we can try to clone a quantum state approximately with the optimal fidelity, or instead, we can try to clone it perfectly with the largest probability. Thus various quantum cloning machines have been designed for different quantum information protocols. Specifically, quantum cloning machines can be designed to analyze the security of quantum key distribution protocols such as BB84 protocol, six-state protocol, B92 protocol and their generalizations. Some well-known quantum cloning machines include universal quantum cloning machine, phase-covariant cloning machine, the asymmetric quantum cloning machine and the probabilistic quantum cloning machine etc. In the past years, much progress has been made in studying quantum cloning machines and their applications and implementations, both theoretically and experimentally. In this review, we will give a complete description of those important developments about quantum cloning and some related topics. On the other hand, this review is self-consistent, and in particular, we try to present some detailed formulations so that further study can be taken based on those results., Comment: 98 pages, 12 figures, 400+ references. Physics Reports (published online)

We investigate the schemes of quantum network teleportation for quantum information distribution and concentration which are essential in quantum cloud computation and quantum internet. In those schemes, the cloud can send simultaneously identical unknown quantum states to clients located in different places by a network like teleportation with a prior shared multipartite entangled state resource. The cloud first perform the quantum operation, each client can recover their quantum state locally by using the classical information announced by the cloud about the measurement result. The number of clients can be beyond the number of identical quantum states intentionally being sent, this quantum network teleportation can make sure that the retrieved quantum state is optimal. Furthermore, we present a scheme to realize its reverse process, which concentrates the states from the clients to reconstruct the original state of the cloud. These schemes facilitate the quantum information distribution and concentration in quantum networks in the framework of quantum cloud computation. Potential applications in time synchronization are discussed., Comment: 7 pages, 1 figure

Freire et al. (2009, MNRAS, 396, 1764) have put forward a technique for timing the double pulsar system PSR J0737-3039A/B (hereafter A and B, respectively). Their technique can be used to determine the sense of rotation of A relative to its orbital plane. In this paper, we present another technique with the same purpose. Two well-known periods, the sidereal day and solar day, are often used to define the spin period of the earth. Their difference is caused by a kinematic effect which correlated with earth's rotation and revolution. We think that this kinematic effect should exist in the double pulsar system and can be used to determine the sense of rotation of the pulsars. Commonly, B's modulation frequency is considered to be equal to A's rotation frequency, because B's signal is modulated by A's energy flux. When the kinematic effect is considered, B's modulation frequency will be, similar to the solar day and sidereal day, a little higher or lower than A's rotation frequency. If this frequency offset can be found, A's rotation sense will be determined and the position of the lighthouse model will be consolidated., Comment: This paper has been withdrawn by the author because the content, abstract, title, and the authors need to change. A revolutionary new vision has been submitted

From the frozen-in field lines concept, a highly conducting fluid can move freely along, but not traverse to, magnetic field lines. We discuss this topic and find that in the study of the frozen-in field lines concept, the effects of inductive and capacitive reactance have been omitted. When admitted, the relationships among the motional electromotive field, the induced electric field, the eddy electric current, and the magnetic field becomes clearer and the frozen-in field line concept can be reconsidered. We emphasize the importance of isomagnetic surfaces and polarization charges, and show analytically that whether a conducting fluid can freely traverse magnetic field lines or not depends solely on the magnetic gradient in the direction of fluid motion. If a fluid does not change its density distribution and shape (can be regarded as a quasi-rigid body), and as long as it is moving along an isomagnetic surface, it can freely traverse magnetic field lines without any magnetic resistance no matter how strong the magnetic field is. When our analysis is applied, the origin of the magnetic field of sunspots can be interpreted easily. In addition, we also present experimental results to support our analysis., Comment: 12 pages, 12 figures, 4 tables

We study the minimal input sets which can determine completely the universal and the phase-covariant quantum cloning machines. We find that the universal quantum cloning machine, which can copy arbitrary input qubit equally well, however can be determined completely by only four input states located at the four vertices of a tetrahedron. The phase-covariant quantum cloning machine, which can copy all qubits located on the equator of the Bloch sphere, can be determined by three equatorial qubits with equal angular distance. These results sharpen further the well-known results that BB84 states and six-states used in quantum cryptography can determine completely the phase-covariant and universal quantum cloning machines. This concludes the study of the power of universal and phase-covariant quantum cloning, i.e., from minimal input sets necessarily to full input sets by definition. This can simplify dramatically the testing of whether the quantum clone machines are successful or not, we only need to check that the minimal input sets can be cloned optimally., Comment: 7 pages, 4 figures

This paper is withdrawn. We study the quantum key distribution (QKD) protocol based on a quantum retrodiction protocol, namely the so-called mean king problem. The security is analyzed by considering the eavesdropping on both the preparation of the entangled pair and the transmission of the quantum state. This QKD protocol can generate efficiently a bit of raw key in every single run. We find that, for qubit system, it is more secure than the QKD Bennett-Brassard 1984 protocol and the six-state protocol. This QKD protocol works also for higher dimensional system., Comment: This paper is withdrawn by the authors and it will be incorporated into a new paper

We study a general quantum key distribution protocol in higher dimension. In this protocol, quantum states in arbitrary $g+1$ ($1\le g\le d$) out of all $d+1$ mutually unbiased bases in a d-dimensional system can be used for the key encoding. This provides a natural generalization of the quantum key distribution in higher dimension and recovers the previously known results for $g=1$ and $d$. In our investigation, we study Eve's attack by two slightly different approaches. One is considering the optimal cloner for Eve, and the other, defined as the optimal attack, is maximizing Eve's information. We derive results for both approaches and show the deviation of the optimal cloner from the optimal attack. With our systematic investigation of the quantum key distribution protocols in higher dimension, one may balance the security gain and the implementation cost by changing the number of bases in the key encoding. As a side product, we also prove the equivalency between the optimal phase covariant quantum cloning machine and the optimal cloner for the $g=d-1$ quantum key distribution.

We raise a general question of quantum information theory whether the quantum phase information can be compressed and retrieved. A general qubit contains both amplitude and phase information, while an equatorial qubit contains only a phase information. We study whether it is possible to compress the phase information of n equatorial qubits into m general qubits with m being less than n, and still those information can be retrieved perfectly. We prove that this process is not allowed by quantum mechanics., Comment: 4 pages, 1 figure

We present a unified universal quantum cloning machine, which combines several different existing universal cloning machines together including the asymmetric case. In this unified framework, the identical pure states are projected equally into each copy initially constituted by input and one half of the maximally entangled states. We show explicitly that the output states of those universal cloning machines are the same. One importance of this unified cloning machine is that the cloning procession is always the symmetric projection which reduces dramatically the difficulties for implementation. Also it is found that this unified cloning machine can be directly modified to the general asymmetric case. Besides the global fidelity and the single-copy fidelity, we also present all possible arbitrary-copy fidelities., Comment: 4 pages, 2 figures

In the paper Pulsar Electrodynamics, published in 1969, Goldreich and Julian propose some basic properties of pulsars, such as the oft-cited Goldreich-Julian density, light cylinder, open and closed magnetic field lines, corotation and so on. However, inspection of their mathematics reveals three mistakes: first, the relative velocity is irrelevantly replaced by the corotation velocity; second, a hypothesis in their theory is contradictory to Maxwell's equations; and third, their theory neglected a special solution of the frozen-in field equation which is of particular importance for pulsar research. We additionally describe the results of a series of magnetohydrodynamic experiments that may be beneficial to the understanding of pulsar electrodynamics., Comment: 7 pages, 7 figures

Periodic changes in a stellar magnetic field can be explained in two ways: the oblique rotator model or the solar cycle model. Although many papers favor the oblique rotator model, there has not been enough evidence to rule out the solar cycle model definitively. This paper presents five new observations (together with a old one) that can be carried out to distinguish magnetic field rotation and magnetic field reversal.

We constructed the magnetic field oscillation model (hereafter the MO model) by analogizing the periodically reversing phenomenon of the solar magnetic field to pulsars. Almost all kinds of pulsar radiation phenomena are best explained using the MO model, especially polarization characteristics, glitch, generation rate, the geodetic precession of pulsars and the configuration of pulsar-wind nebula of the Crab. The MO model also provides satisfactory explanation for other characteristics of pulsars, e.g., interpulse, spin-down, pulse nulling, beat and pulse drift, the loss rate of the rotating energy, and the accuracy of frequency. We present eight verification methods for the MO model. In addition to pulsars, our MO model can also be used to explain the pulse emission from non-compact stars such as the ultracool dwarf TVLM 513-46546 and the magnetic chemically peculiar star CU Virginis., Comment: 21 pages, 4 figures. One reference error was corrected

The shape phase structure and its transition of the nucleus in the transitional region between the U(5) and SU(3) symmetries is restudied within the framework of coherent-state theory with angular momentum projection in IBM-1. The certain angular momentum (or rotation-driven) effect on the nuclear shape is discussed. A coexistence of prolate and oblate shapes is found for the ground states of the transitional nuclei. A phase diagram in terms of the deformation parameter and angular momentum is given., Comment: 14 pages, 6 figures

By analyzing the potential energy surface, the shape phase diagram and the energy spectrum of the nucleus in U(5) symmetry in the IBM, we propose that the U(5) symmetry with parameters $(A+B) < 0$ may be a model to describe the rotation driven vibrational to axially rotational phase transition along the yrast line. With such a model, we have described successfully the observed rotation driven shape phase transition along the yrast line of individual nucleus and proposed some other empirical evidences., Comment: 17pages, 5figures

With the energy surface of the nucleus in U(5) symmetry being analyzed in the framework of thermodynamics, the vibration and rotation phase diagram in terms of the angular momentum and deformation parameter is given. Together with examining the energy spectrum, we propose a theoretical approach to describe the vibrational to axially rotational phase transition along the yrast line. By analyzing the available experimental data we show that the vibrational to rotational shape phase transition along the yrast line takes place in many nuclei., Comment: 13 pages, 4 figures