Michael J. W. Hall
Generalised observables (POM observables) are necessary for representing all possible measurements on a quantum system. Useful algebraic operations such as addition and multiplication are defined for these observables, recovering many advantages of the more restrictive Hermitian operator formalism. Examples include new uncertainty relations and metrics, and optical phase applications.
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Advances in quantum technology require scalable techniques to efficiently extract information from a quantum system, such as expectation values of observables or its entropy. Traditional tomography is limited to a handful of qubits and shadow tomography has been suggested as a scalable replacement for larger systems. Shadow tomography is conventionally analysed based on outcomes of ideal projective measurements on the system upon application of randomised unitaries. Here, we suggest that shadow tomography can be much more straightforwardly formulated for generalised measurements, or positive o…
The Onsager algebra is one of the cornerstones of exactly solvable models in statistical mechanics. Starting from the generalised Clifford algebra, we demonstrate its relations to the graph Temperley-Lieb algebra, and a generalisation of the Onsager algebra. We present a series of quantum lattice models as representations of the generalised Clifford algebra, possessing the structure of a special type of the generalised Onsager algebra. The integrability of those models is presented, analogous to the free fermionic eight-vertex model. We also mention further extensions of the models and physica…
Generalised Geometric Phase: Mathematical Aspects
该研究为基于物理提出的广义几何相位提供数学基础，揭示量子系统中不同的射线空间与纤维丛结构，并将其理解为联络的（非）和乐，底层空间为伪凯勒流形。
Generalised Geometric Phase
本文提出纯态的广义几何相位，通过可观测量矩阵元辐角定义广义相对相位并推广自然联络，揭示其在时间演化、能谱移动和前向散射中的物理表现。
Generalised dichotomic quantum measurements are fully characterised by two real parameters, dubbed as sharpness parameter and biasedness parameter. The trade-off between the degree of joint measurability, sharpness and biasedness of generalised measurements was known in the case of pairs of qubit observables. In the present work we generalise the notion of sharpness and biasedness measure of multi-outcome generalised measurements pertaining to multilevel systems. A trade-off between the amount of quantum mechanical (QM) violation of macrorealism (MR), sharpness and biasedness is established. S…
It is shown that mean value of any observable with bounded spectrum can be uniquely determined from binary statistics of the measurement performed on {\it single} qubit ancilla coupled to a given system. The observable structure is fully encoded in the corresponding POVM. The method is generalised to the case of distant labs paradigm and discussed in the context of entanglement detection with few local measurements. The results are also discussed in the context of quantum programming.
A proof that minimum uncertainty states of the simplest periodic quantum system exist in a state space that is represented by a Colombeau algebra of generalised functions but not in Hilbert space or in the space of Schwartz distributions is given. There are two significant generalisations of Hilbert space that lead to the special Colombeau algebra. The first step is to a vector space of Schwartz distributions that contains the eigenstates of operators with continuous spectra, the second is to a Colombeau algebra that provides a solution for a differential equation with non-constant coefficient…
Given a topological group $G$ and a unitary representation $U$ of $G$, we consider the problem of classifying the positive operator measures which are based on a $G$-homogeneous space $X$ and covariant with respect to the representation $U$. We completely solve the problem in two cases: 1) when $G$ is abelian; 2) when $G$ is not abelian, $X$ has compact stbiliser in $G$ and the representation $U$ is irreducible. We then give an application of the above results to the characterisation of covariant position and momentum observables in quantum mechanics, and to the determination of those covarian…
To each unital C*-algebra A we associate a presheaf Σ^A, called the spectral presheaf of A, which can be regarded as a generalised Gelfand spectrum. We develop a categorical notion of local duality and show that there is a contravariant functor from the category of unital C*-algebras to a suitable category of presheaves containing the spectral presheaves. We clarify how much algebraic information about a C*-algebra is contained in its spectral presheaf. A nonabelian unital C*-algebra A that is neither isomorphic to C^2 nor to B(C^2) is determined by its spectral presheaf up to quasi-Jordan iso…
In arXiv:1212.2613, we associated a presheaf Σ^A with each unital C*-algebra A. The spectral presheaf Σ^A generalises the Gelfand spectrum of an abelian unital C*-algebra. In the present article, we consider one-parameter groups of automorphisms of the spectral presheaf, in particular those arising from one-parameter groups of inner automorphisms of the algebra. We interpret the spectral presheaf as a (generalised) state space for a quantum system and show how one can use flows on the spectral presheaf and on associated structures to describe the time evolution of non-relativistic quantum syst…
From reasonable postulates to generalised Hamiltonian systems
本文从量子与经典力学共同满足的正则对称性和内积不变性等公设出发，在广义概率论框架下推导出统一的哈密顿系统模型，并讨论多种替代理论，深化了对哈密顿量的理解。
Foundations of the formal series $*$ -- calculus in deformation quantisation are discussed. Several classes of continuous linear functionals over algebras applied in classical and quantum physics are introduced. The notion of nonnegativity in formal series calculus is proposed. Problems with defining quantum states over the set of formal series are analysed.
Motivated by several approaches to quantum gravity, there is a considerable literature on generalised uncertainty principles particularly through modification of the canonical position-momentum commutation relations. Some of these modified relations are also consistent with general principles that may be supposed of any physical theory. Such modified commutators have significant observable consequences. Here we study the noisy behaviour of an optomechanical system assuming a certain commonly studied modified commutator. From recent observations of radiation pressure noise in tabletop optomecha…
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