Quantum Gaussian memory channels

We study the information transmission through two different models of Gaussian memory channels: an additive Gaussian channel and a lossy bosonic memory channel. We then show that entangled inputs can enhance the transmission rate in such channels. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 152, No. 2, pp. 390–404, August, 2007.     

Quantum Bayesian computation

Quantum Bayesian computation is an emerging field that levers the computational gains available from quantum computers. They promise to provide an exponential speed-up in Bayesian computation. Our article adds to the literature in three ways. First, we describe how quantum von Neumann measurement provides quantum versions of popular machine learning algorithms such as Markov chain Monte Carlo and deep learning that are fundamental to Bayesian learning. Second, we describe quantum data encoding methods needed to implement quantum machine learning including the counterparts to traditional feature extraction and kernel embeddings methods. Third, we show how quantum algorithms naturally calculate Bayesian quantities of interest such as posterior distributions and marginal likelihoods. Our goal then is to show how quantum algorithms solve statistical machine learning problems. On the theoretical side, we provide quantum versions of high dimensional regression, Gaussian processes and stochastic gradient descent. On the empirical side, we apply a quantum FFT algorithm to Chicago house price data. Finally, we conclude with directions for future research.     

A Priori Estimates for Quantum Dynamical Semigroups

We obtain a priori bounds for the solutions of master Markov quantum equations in the Heisenberg and Schrödinger pictures.     

Quantum double constructions for compact quantum group

For a non-degenerate pair of compact quantum groups, we first construct the quantum double as an algebraic compact quantum group in an algebraic framework. Then by adopting some completion procedure, we give the universal and reduced quantum double constructions in the correspondence C＊-algebraic settings, which generalize Drinfeld＇s quantum double construction and yield new C＊-algebraic compact quantum groups.     

Maximal vectors in Hilbert space and quantum entanglement

Let V be a norm-closed subset of the unit sphere of a Hilbert space H that is stable under multiplication by scalars of absolute value 1. A maximal vector (for V) is a unit vector ξ∈H whose distance to V is maximum

     

基于异质预期的量子伯川德博弈复杂动力机制分析

针对伯川德双寡头垄断博弈经济系统中出现的混沌现象,利用量子博弈论,构建了基于有限理性与天真预期行为的量子伯川德动态博弈模型,分析了量子纠缠度对纳什均衡点稳定性及复杂动力行为的影响。结果表明:量子纠缠度能增强该系统的稳定性,企业价格调整速度达到某一程度时会导致该系统的复杂混沌特性,纠缠度可以有效控制混沌状态。最后利用数值模拟从分岔、最大李雅普诺夫指数、奇怪吸引子、初始条件敏感性及分数维数方面验证了理论准确性。     

Generalized Bell inequality and a method for its verification

We use the reduced density matrix of the two-particle spin state to construct a generalized Bell-Clauser-Horne-Shimony-Holt inequality. For each specific state and under a special choice of the vectors , this inequality becomes an exact equality. We show how such vectors can be found using the reduced density matrix. Both sides of this equality have a specific numerical value. We indicate the connection of this number with the measure of entanglement of the two-particle spin state. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 152, No. 3, pp. 488–501, September, 2007.     

On the maximal dimension of a completely entangled subspace for finite level quantum systems

LetH _ibe a finite dimensional complex Hilbert space of dimensiond _i associated with a finite level quantum system A_i for i = 1, 2, ...,k. A subspaceS ⊂ is said to becompletely entangled if it has no non-zero product vector of the formu ₁⊗u ₂ ⊗ ... ⊗u _k with u_i inH _i for each i. Using the methods of elementary linear algebra and the intersection theorem for projective varieties in basic algebraic geometry we prove that

Quantum approximation theorem

In this article a problem of relation between p-adic and ordinary quantum mechanics is discussed. The text was submitted by the author in English.     

Quantum duality in quantum deformations

In accordance with the quantum duality principle, the twisted algebra is equivalent to the quantum group and has two preferred bases: one inherited from the universal enveloping algebra and the other generated by coordinate functions of the dual Lie group . We show howthe transformation can be explicitly obtained for any simple Lie algebra and a factorable chain of extended Jordanian twists. In the algebra , we introduce a natural vector grading , compatible with the adjoint representation of the algebra. Passing to the dual-group coordinates allows essentially simplifying the costructure of the deformed Hopf algebra , considered as a quantum group . The transformation can be used to construct new solutions of the twist equations. We construct a parameterized family of extended Jordanian deformations and study it in terms of ; we find new realizations of the parabolic twist. Dedicated to the birthday of my teacher, Yurii Novozhilov __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 148, No. 1, pp. 112–125, July, 2006.     

量子群的基变换与范畴同构

令Ｍ是Ｚ［ｖ］的由ｖ－１和奇素数ｐ生成的理想，Ｕ是Ａ＝Ｚ［ｖ］Ｍ上相伴于对称Ｃａｒｔａｎ矩阵的量子群， Ａ－Γ是环同态， Ｕг＝ＵＡΓ［Ｕг］是Ｕг的量子坐标代数，本文建立了量子坐标代数的基变换：即在相关约束条件下有Г－Ｈｏｐｆ同构 Ａ［Ｕ］ＡГ≌Г［Ｕг］．我们证明了有限秩 Ａ自由 １型可积 Ｕ模范畴和有限秩 Ａ自由 Ａ［Ｕ］余模范畴是同构的．特别，当 Г是域时，局部有限 １型 Ｕг模范畴和Г［Ｕг］余模范畴是同构的．最后，我们还证明了在［１］中定义的诱导函子和Ｂ．Ｐａｒｓｈａｌｌ与王建磐博士在［２］中研究的诱导函子的一致性．     

Stability of Quantum Electrodynamics and Quantum Chromodynamics

We consider the vacuum energy in QED viewed as in a system of charged fermions and bosons and in QCD viewed as in a system of quarks (fermions) and gluons (bosons) in a self-dual field with a constant strength. We show that the cause of instability is the instability of bosons in the self-dual vacuum field. For the global stability of a system consisting of fermions and bosons, the number of fermions should be sufficiently large. The nonzero self-dual field leading to the confinement of fermions realizes the minimum of the vacuum energy in the case where the boson has the smallest mass in the system. Confinement therefore does not arise in QED, where the fermion (electron) has the smallest mass, and does arise in QCD, where the boson (gluon) has the smallest mass.     

Quantum three-body problems

A scheme for dealing with the quantum three-body problem is presented to separate the rotational degrees of freedom completely from the internal ones. In this method, the three-body Schrodinger equation is reduced to a system of coupled partial differential equations, depending only upon three internal variables. For arbitrary total orbital angular momentum / and the parity (− 1) ^l+λ (λ = 0 or 1), the number of the equations in this system isl = 1 −λ. By expanding the wavefunction with respect to a complete set of orthonormal basis functions, the system of equations is further reduced to a system of linear algebraic equations.     

Quantum Differentials and the q-Monopole Revisited

The q-monopole bundle introduced previously is extended to ageneral construction for quantum group bundles with nonuniversaldifferential calculi. We show that the theory applies to several otherclasses of bundles as well, including bicrossproduct quantum groups, thequantum double and combinatorial bundles associated with covers of compactmanifolds.     

Quantum Measurements and Kolmogorov's Theory of Probability

A connection between the probability space measurability requirement and the complementarity principle in quantum mechanics is established. It is shown that measurability of the probability space implies that the results of the quantum measurement depend not only on properties of the quantum object under consideration but also on classical characteristics of the measuring instruments used. It is also shown that if the measurability requirement is taken into account, then the hypothesis that the objective reality exists does not lead to the Bell inequality.     

Quantum spaces

In this paper, a more general concept of quantum space is given by modifying the original concept defined by Borceux and Bossche. We show that a quantum space is a topological analogue of a quantale defined by Mulvey, and also a non-commutative generalization of the Zariski spectrum of a commutative ring. But quantum spaces are not good enough to have much of the properties of topological spaces, such as product spaces and quotient spaces.     

A Method to Construct Oriented Quantum Algebras

In this note, we mainly give a method to construct the oriented quantum algebra structure on the tensor product of two different oriented quantum algebras, generalizing Radford ([8 Radford , D. E. ( 2007 ). On the tensor product of an oriented quantum algebra with itself . J. Knot Theory Ramifications 16 : 929 – 957 . [Google Scholar]], Theorem 4.1).     

表示张量积与量子对称

利用量子对称群的概念，我们给出Ｕ＿ｑ（ｓｌ＿ｎ）－多项式、Ｕ＿ｑ（ｓｌ＿ｎ）－外积的定义，说明Ｕ＿ｑ（ｓｌ＿ｎ）的ｑ－振子、ｑ－旋量表示不过是Ｕ＿ｑ（ｓｌ＿ｎ）标准表示的量子对称、量子反称张量积，这和经典情形是一致的。