Quantum Symmetries and Strong Haagerup Inequalities

In this paper, we consider families of operators \({\{x_r\}_{r \in \Lambda}}\) in a tracial C*-probability space \({({\mathcal{A}}, \varphi)}\) , whose joint *-distribution is invariant under free complexification and the action of the hyperoctahedral quantum groups \({\{H_n^+\}_{n \in \mathbb {N}}}\) . We prove a strong form of Haagerup’s inequality for the non-self-adjoint operator algebra \({{\mathcal{B}}}\) generated by \({\{x_r\}_{r \in \Lambda}}\) , which generalizes the strong Haagerup inequalities for *-free R-diagonal families obtained by Kemp–Speicher (J Funct Anal 251:141–173, 2007). As an application of our result, we show that \({{\mathcal{B}}}\) always has the metric approximation property (MAP). We also apply our techniques to study the reduced C*-algebra of the free unitary quantum group \({U_n^+}\) . We show that the non-self-adjoint subalgebra \({{\mathcal{B}}_n}\) generated by the matrix elements of the fundamental corepresentation of \({U_n^+}\) has the MAP. Additionally, we prove a strong Haagerup inequality for \({{\mathcal{B}}_n}\) , which improves on the estimates given by Vergnioux’s property RD (Vergnioux in J Oper Theory 57:303–324, 2007).     

Quantum Subgroups,Quantum Co-Set Spaces and Corresponding Differential Calculi

In this paper, quantum subgroups and quantum co-set spaces including corresponding dififerential calculi are discussed.     

Unitarity of Induced Representations from Coisotropic Quantum Subgroups

We study unitarity of the induced representations from coisotropic quantum subgroups which were introduced by Ciccoli in math.QA/9804138. We define a real structure on coisotropic subgroups which determines an involution on the homogeneous space. We give general invariance properties for functionals on the homogeneous space which are sufficient to build a unitary representation starting from the induced one. We present the case of the one-dimensional quantum Galilei group where we have to use in all generality our definition of quasi-invariant functional.     

Cyclotomic Integers,Fusion Categories,and Subfactors

Dimensions of objects in fusion categories are cyclotomic integers, hence number theoretic results have implications in the study of fusion categories and finite depth subfactors. We give two such applications. The first application is determining a complete list of numbers in the interval (2, 76/33) which can occur as the Frobenius-Perron dimension of an object in a fusion category. The smallest number on this list is realized in a new fusion category which is constructed in the Appendix written by V. Ostrik, while the others are all realized by known examples. The second application proves that in any family of graphs obtained by adding a 2-valent tree to a fixed graph, either only finitely many graphs are principal graphs of subfactors or the family consists of the A _n or D _n Dynkin diagrams. This result is effective, and we apply it to several families arising in the classification of subfactors of index less than 5.     

Writing Quantum Data into One-Way Quantum Computer by Fusion Gate

A scheme to write the quantum data into the one-way quantum computer by the type-I and type-II gates is given in this paper. It may provide a method to make full use of the resources including time and entanglement during quantum computation.     

对线性光学量子融合门的简化

作为一种特殊的多体纠缠态,簇态可以用来进行通用的量子计算.量子融合门可以有效地扩展簇态,将一维的簇态进行任意加长,或者把一维的簇态融合为多维的簇态从而获得具有我们所需要的构型的簇态.文章对两类线性光学量子融合门进行了讨论,并且在Ⅱ类融合门的基础上提出一种更加简洁的融合簇态的方法,精简了融合门中的线性光学元件,同时简化了计算过程.     

Braiding and Fusion Matrices for Quantum Octet Representation

The braiding and fusion matrices based on the octet representation of the quantum sl(3) enveloping algebra are computed. The multiplicity in the decomposition of the coproduct 8×8 manifests itself in those matrices.     

Multimodality Image Fusion Based on Quantum Wavelet Transform and Sum-Modified-Laplacian Rule

The parallelism and entanglement characteristics of quantum computation greatly improve the efficiency of image processing tasks. With the sharp increase of data size and requirement of real-time processing in image fusion application, rapid implementation using quantum computation will become the inexorable trend. A novel multimodality image fusion algorithm based on quantum wavelet transform (QWT) and proposed quantum version of sum-modified-laplacian (SML) rule is designed in this paper. The source digital images are firstly represented by flexible representation of quantum image (FRQI) model, and then the quantum form images are transformed with QWT to capture salient features of source images. The quantum version of SML rule is proposed to fuse wavelet coefficients, which has higher efficiency and runs faster than its classical counterpart. The final fused image is obtained by using inverse quantum wavelet transform. The simulations and theoretical analysis verify that the proposed algorithm is effective in the fusion of multimodality images.

     

量子非欧姆阻尼环境中的重核熔合概率及热核裂变速率

考虑处于量子非欧姆阻尼环境下的重核熔合及热核裂变系统的动力学,给出了数值模拟相应c数量子广义朗之万方程的方法。其中提出的产生任意关联量子色噪声的数值方法,适用于任意非马尔科夫过程噪声的产生。利用此方法计算了重核熔合概率,结果表明量子涨落对重核熔合具有“低抬高压”的效应:当粒子的初始动能小于（大于）临界初始动能时,量子涨落会增大（减小）粒子鞍点通过概率。非欧姆阻尼环境中粒子稳定通过概率随δ值的变化是非单调的,且当粒子初始动能小于（大于）临界初始动能,量子涨落会使稳定通过概率随δ值变化曲线的极大值位置向右（向左）漂移。此外,在热核裂变系统中,超欧姆阻尼环境会增大裂变速率,而量子涨落不仅显著增大裂变速率,还使裂变速率随δ值变化曲线的极大值位置发生漂移。Dynamics of heavy-ion fusion and nuclear fission system in a quantum non-Ohmic environment have been considered and a numerical simulation method to solve the corresponding c-number quantum generalized Langevin equation is proposed. The method of generating quantum colored noise with arbitrary correlation can be applied to generate noise of arbitrary non-Markov process. Calculating fusion probability of heavy nuclei with this method, the result has shown that the passing probability is enlarged (decreased) by the quantum fluctuation when the initial kinetic energy of the particle is less than (greater than) the critical initial kinetic energy. Steady passing probability of particle in non-Ohmic environment versus is nonmonotonic. Quantum fluctuation makes the maximum position of the curve drift towards right (left), when the initial kinetic energy of the particle is less than (greater than) the critical initial kinetic energy. Furthermore, nuclear fission rate is larger in super-Ohmic environment. Quantum fluctuation enlarges nuclear fission rate and makes the the maximum position of nuclear fission rate versus δ drift.     

近垒能量下重核熔合几率的量子路径积分计算

针对近垒能量下经典涨落耗散模型预期的重核熔合几率比实验结果偏小的问题, 发展了一种实时间路径积分方法并用于研究重核熔合激发函数, 给出了包含量子涨落效应的解析表达式。 计算了几个对称和近似对称反应系统的熔合几率, 结果表明理论结果与实验值符合较好。 还讨论了颈部增长对熔合障碍的影响。 A real time path integral approach is developed in order to work out a correct solution to a problem for the smaller result of the fusion probability of heavy nuclei based on the classical diffusion model at sub barrier energies. An analytical expression for the quantum fusion probability is proposed,which contains the quantum fluctuation effect. The fusion probabilities of several symmetrical and approximate symmetrical systems are calculated, the heoretical results are in agreement with the experimental data. The influence of the neck length and its fluctuation upon the fusion hindrance is discussed.     

Quantization of Co-Isotropic Subgroups

The purpose of this Letter is to show that the concept of co-isotropic subgroups of a Poisson–Lie group can be given a natural analogue in the context of quantum homogeneous spaces, explaining some of the special features of this theory. We will give examples related to some previously known and some new quantum homogeneous spaces of the two-dimensional Euclidean quantum groups.     

Modular Group Representations and Fusion in Logarithmic Conformal Field Theories and in the Quantum Group Center

The SL(2, ℤ)-representation π on the center of the restricted quantum group at the primitive 2p^th root of unity is shown to be equivalent to the SL(2, ℤ)-representation on the extended characters of the logarithmic (1, p) conformal field theory model. The multiplicative Jordan decomposition of the ribbon element determines the decomposition of π into a ``pointwise' product of two commuting SL(2, ℤ)-representations, one of which restricts to the Grothendieck ring; this restriction is equivalent to the SL(2, ℤ)-representation on the (1, p)-characters, related to the fusion algebra via a nonsemisimple Verlinde formula. The Grothendieck ring of at the primitive 2p^th root of unity is shown to coincide with the fusion algebra of the (1, p) logarithmic conformal field theory model. As a by-product, we derive q-binomial identities implied by the fusion algebra realized in the center of .     

Categories of coherence pathways for the CPMG sequence

The CPMG sequence has been extremely useful for efficient measurements of NMR signal, spin-spin relaxation, and diffusion, particularly in inhomogeneous magnetic fields, such as when samples are outside the magnet and RF coil. Due to the inaccuracy of the pulses and the off-resonance effects, the CPMG echoes have contributions from the Hahn echo as well as signals that are similar to stimulated echoes. The systematic understanding of the CPMG pulse sequence requires decomposing the magnetization dynamics into different coherence pathways. In this paper, we describe a method to classify the CPMG coherence pathways and illustrate the nature of these types of pathways. This classification shows that direct echo and stimulated echoes are the major contribution to the CPMG signal. It also provides a clear understanding of the effect of restricted diffusion in porous media.     

Quantum Bit Commitment and the Reality of the Quantum State

Quantum bit commitment is insecure in the standard non-relativistic quantum cryptographic framework, essentially because Alice can exploit quantum steering to defer making her commitment. Two assumptions in this framework are that: (a) Alice knows the ensembles of evidence E corresponding to either commitment; and (b) system E is quantum rather than classical. Here, we show how relaxing assumption (a) or (b) can render her malicious steering operation indeterminable or inexistent, respectively. Finally, we present a secure protocol that relaxes both assumptions in a quantum teleportation setting. Without appeal to an ontological framework, we argue that the protocol’s security entails the reality of the quantum state, provided retrocausality is excluded.     

Subgroups of index 4 of the 2-dimensional space groups

There are 281 subgroups of index 4 for the 17 types of 2-dimension space groups: p1:7, p2:31, pm:19, pg:7, cm:11, p2mm:67, p2mg:23, p2gg:11, c2mm:31, p4:11, p4mm:31, p4gm:11, p3:4, p3m1:4, p31m:4, p6:4, p6mm:5. Every subgroup has been identified by its conventional cell disposition with respect to the conventional cell of the starting group. These subgroups may be arranged in 96 automorphism classes and 184 conjugation classes according to their starting groups.     

On the Existence of Quantum Signature for Quantum Messages

Quantum signature (QS) is used to authenticate the identity of the originator, ensure data integrity and provide non-repudiation service with unconditional security. Depending on whether a trusted third party named arbitrator is involved or not, QS is classified as arbitrated QS and true QS. This paper studies existence problem about the two kinds of QS and contributes to two points: (1) a basic framework is provided to analyze the possibility of arbitrated QS on signing quantum messages; (2) disagreement between the impossibility of true QS and an existing true QS scheme is solved.     

On Gauging Symmetry of Modular Categories

We develop a non–relativistic twistor theory, in which Newton–Cartan structures of Newtonian gravity correspond to complex three–manifolds with a four–parameter family of rational curves with normal bundle \({\mathcal {O} \oplus \mathcal {O}(2)}\). We show that the Newton–Cartan space-times are unstable under the general Kodaira deformation of the twistor complex structure. The Newton–Cartan connections can nevertheless be reconstructed from Merkulov’s generalisation of the Kodaira map augmented by a choice of a holomorphic line bundle over the twistor space trivial on twistor lines. The Coriolis force may be incorporated by holomorphic vector bundles, which in general are non–trivial on twistor lines. The resulting geometries agree with non–relativistic limits of anti-self-dual gravitational instantons.