Uncertainty Relations for Coherence

Quantum mechanical uncertainty relations are fundamental consequences of the incompatible nature of noncommuting observables. In terms of the coherence measure based on the Wigner-Yanase skew information, we establish several uncertainty relations for coherence with respect to von Neumann measurements, mutually unbiased bases(MUBs), and general symmetric informationally complete positive operator valued measurements(SIC-POVMs),respectively. Since coherence is intimately connected with quantum uncertainties, the obtained uncertainty relations are of intrinsically quantum nature, in contrast to the conventional uncertainty relations expressed in terms of variance,which are of hybrid nature(mixing both classical and quantum uncertainties). From a dual viewpoint, we also derive some uncertainty relations for coherence of quantum states with respect to a fixed measurement. In particular, it is shown that if the density operators representing the quantum states do not commute, then there is no measurement(reference basis) such that the coherence of these states can be simultaneously small.     

New Application of Noncanonical Maps in Quantum Mechanics

One invertible and one unitary operator can be used to reproduce the effect of a q-deformed commutator of annihilation and creation operators. The original annihilation and creation operators are mapped into new operators, not conjugate to each other, whose standard commutator equals the identity plus a correction proportional to the original number operator. The consistency condition for the existence of this new set of operators is derived, by exploiting the Stone theorem on 1-parameter unitary groups. The above scheme leads to modified equations of motion which do not preserve the properties of the original first-order set for annihilation and creation operators. Their relation with commutation relations is also studied.     

Theory of nonlinear optical response of gauge invariant currents to linear and nonlinear couplings

When a gauge field interacts with a quantum condensed matter system, at first order of the gauge field it couples to the current operator of the electrons. Higher orders of the gauge field couple to electrons through other operators such as the stress tensor, etc. On the other hand, when one performs a measurement on a quantum system, not only the current operator, but also stress tensor operator of the electrons, etc. are hidden in the measurement, as they contribute to the gauge invariant current. We formulate a general problem of nonlinear optical response of the gauge invariant currents in presence of nonlinear couplings. We show that the new couplings along with new responses arising from field current have a very simple structure which can be formulated as time ordered multi-particle correlation functions. We also obtain their Lehman representation and thereby show that one need not use non-equilibrium formulations to deal with them. These new correlation functions suggest that in nonlinear optical response many new processes are possible. The experimental detection of the new terms in the current operator, and application corresponding multi-photon processes needs further theoretical and experimental investigations.     

Efficient method for computing the maximum-likelihood quantum state from measurements with additive Gaussian noise

We provide an efficient method for computing the maximum-likelihood mixed quantum state (with density matrix ρ) given a set of measurement outcomes in a complete orthonormal operator basis subject to Gaussian noise. Our method works by first changing basis yielding a candidate density matrix μ which may have nonphysical (negative) eigenvalues, and then finding the nearest physical state under the 2-norm. Our algorithm takes at worst O(d(4)) for the basis change plus O(d(3)) for finding ρ where d is the dimension of the quantum state. In the special case where the measurement basis is strings of Pauli operators, the basis change takes only O(d(3)) as well. The workhorse of the algorithm is a new linear-time method for finding the closest probability distribution (in Euclidean distance) to a set of real numbers summing to one.     

Why error-prone quantum measurements have outcomes

To solve the quantum measurement problem it is necessary to construct quantum mechanical models of measurement interactions to show why properly conducted measurements always yield definite outcomes. The main barrier to a solution has been the interpretive principle that a quantum system has a definite value for an observable only if it may be described by a quantum eigenstate of the corresponding operator. I have recently proposed a solution to the measurement problem based on alternative interpretive principles. The present paper defends this proposal against recent criticisms which seek to show that it fails to solve the problem unless quantum measurements meet highly idealized conditions which no actual measurement could hope to meet. Several models of error-prone measurements are shown to lead to definite outcomes, and a general defense of the appropriateness of these models is sketched.     

关于量子测量问题的一点思考

量子测量原理是量子力学的重要组成部分。具体的测量实验是否构成量子测量,是有商榷的余地的。并不是所有可观测量的本征值都具有实在的意义。量子测量原理中论及的经典—量子世界分界处之扰动的作用,可改述为量子测量需要加入统计原理的考量,这其实正印证了“统计原则高于量子原则”的现实。类似双缝干涉和Stern—Gerlach实验这样的宏观实验同量子测量原理是相融洽的,可能反映的恰是量子测量原理建立的历史背景和心理基础。本文的目的在于引起对量子测量问题的关注,并深信对该问题严肃、深入的讨论是有意义的。     

基于量子相干性的四体贝尔不等式构建

贝尔不等式在定域性和实在性的双重假设下,对于被分隔的粒子同时被测量时其结果的可能关联程度建立了一个严格的限制,违反贝尔不等式确保量子态存在纠缠.本文利用量子相干性的l1和相对熵测度构建了四体量子贝尔不等式,发现一般实系数Greenberger-Horne-Zeilinger纯态和簇纯态总是违反四体相对熵相干性测度贝尔不等式,因此违反四体相对熵相干性测度贝尔不等式的这些态是纠缠态.     

Locality in the Everett Interpretation of Heisenberg-Picture Quantum Mechanics

Bell's theorem depends crucially on counterfactual reasoning, and is mistakenly interpreted as ruling out a local explanation for the correlations which can be observed between the results of measurements performed on spatially-separated quantum systems. But in fact the Everett interpretation of quantum mechanics, in the Heisenberg picture, provides an alternative local explanation for such correlations. Measurement-type interactions lead, not to many worlds but, rather, to many local copies of experimental systems and the observers who measure their properties. Transformations of the Heisenberg-picture operators corresponding to the properties of these systems and observers, induced by measurement interactions, label each copy and provide the mechanism which, e. g., ensures that each copy of one of the observers in an EPRB or GHZM experiment will only interact with the correct copy of the other observer(s). The conceptual problem of nonlocality is thus replaced with a conceptual problem of proliferating labels, as correlated systems and observers undergo measurement-type interactions with newly-encountered objects and instruments; it is suggested that this problem may be resolved by considering quantum field theory rather than the quantum mechanics of particles.     

COHERENT INFORMATION ON THERMAL RADIATION NOISE CHANNEL: AN APPROACH OF INTEGRAL WITHIN ORDERED PRODUCT OF OPERATORS

An analytical expression is given to the coherent information of the thermal radiation signal transmitted over the thermal radiation noise channel, one of the most essential quantum Gaussian channels. Focusing on the single normal mode of the thermal radiation signal and noise, we resolve the entangled state density operator, which characterizes quantum information transmission, into a direct product of two parts, with each part being a thermal radiation density operator. The calculation is aided by the technique known as "integral within ordered product of operators".     

A Non-Associative Quantum Mechanics

A non-associative quantum mechanics is proposed in which the product of three and more operators can be non-associative one. The multiplication rules of the octonions define the multiplication rules of the corresponding operators with quantum corrections. The self-consistency of the operator algebra is proved for the product of three operators. Some properties of the non-associative quantum mechanics are considered. It is proposed that some generalization of the non-associative algebra of quantum operators can be helpful for understanding of the algebra of field operators with a strong interaction.     

两个力学量算符具有共同本征态系的条件

量子力学是人们了解微观世界的一门重要的课程.在量子力学中,由于任何实物体都具有波粒二象性,必须用算符来表示力学量,因此算符理论显得尤为重要.在研究多个算符时,有一个非常重要的定理:两个力学量算符能够具有共同本征函数系的充分必要条件是这两个个力学量算符能够互相对易.本文分析了在使用该定理时可能存在的一些问题,并对这些问题进行了澄清.     

Topological Boundary Invariants for Floquet Systems and Quantum Walks

A Floquet systems is a periodically driven quantum system. It can be described by a Floquet operator. If this unitary operator has a gap in the spectrum, then one can define associated topological bulk invariants which can either only depend on the bands of the Floquet operator or also on the time as a variable. It is shown how a K-theoretic result combined with the bulk-boundary correspondence leads to edge invariants for the half-space Floquet operators. These results also apply to topological quantum walks.     

Algebras of Measurements: The Logical Structure of Quantum Mechanics

In quantum physics, a measurement is represented by a projection on some closed subspace of a Hilbert space. We study algebras of operators that abstract from the algebra of projections on closed subspaces of a Hilbert space. The properties of such operators are justified on epistemological grounds. Commutation of measurements is a central topic of interest. Classical logical systems may be viewed as measurement algebras in which all measurements commute. PACS: 02.10.-V.     

也谈量子力学的基础

本文通过在不同边界条件下动量算符的厄密性和自伴性的区分,给出了对量子力学的一维无限深方势阱中基态粒子的动量分布问题的严格分析,论证了量子力学基本假设的严谨性和自洽性.     

On Realization of Partially Ordered Abelian Groups

The paper is devoted to algebraic structures connected with the logic of quantum mechanics. Since every (generalized) effect algebra with an order determining set of (generalized) states can be represented by means of an abelian partially ordered group and events in quantum mechanics can be described by positive operators in a suitable Hilbert space, we are focused in a representation of partially ordered abelian groups by means of sets of suitable linear operators. We show that there is a set of points separating ?-maps on a given partially ordered abelian group G if and only if there is an injective non-trivial homomorphism of G to the symmetric operators on a dense set in a complex Hilbert space $\mathcal{H}$ which is equivalent to an existence of an injective non-trivial homomorphism of G into a certain power of ?. A similar characterization is derived for an order determining set of ?-maps and symmetric operators on a dense set in a complex Hilbert space $\mathcal{H}$ . We also characterize effect algebras with an order determining set of states as interval operator effect algebras in groups of self-adjoint bounded linear operators.     

Relativity of topology and dynamics

Recent developments in quantum set theory are used to formulate a program for quantum topological physics. The world is represented in a Hilbert space whose psi vectors represent abstract complexes generated from the null set by one bracket operator and the usual Grassmann (or Clifford) product. Such a theory may be more basic than field theory, in that it may generate its own natural topology, time, kinematics and dynamics, without benefit of an absolute timespace dimension, topology, or Hamiltonian. For example there is a natural expression for the quantum gravitational field in terms of quantum topological operators. In such a theory the usual spectrum of possible dimensions describes only one of an indefinite hierarchy of levels, each with a similar spectrum, describing nonspatial infrastructure. While c simplices have no continuous symmetry, the q simplex has an orthogonal group 0(m, n). Because quantum theory cannot take the universe as physical system, we propose a third relativity:The division between observer and observed is arbitrary. Then it is wrong to ask for the topology and dynamics of a system, in the same sense that it is wrong to ask for the the psi vectors of a system; topology and dynamics, like psi vectors, are not absolute but relative to the observer.