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arXiv:quant-ph/0003079 · Journal of Physics A Mathematical and General

Subnormal operators regarded as generalized observables and compound-system-type normal extension related to 𝔰𝔲(1,1)

Masahito Hayashi, Fuminori Sakaguchi

Published Oct 19, 2000• 12 citations
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Abstract

In this paper, subnormal operators, not necessarily bounded, are discussed as generalized observables. In order to describe not only the information about the probability distribution of the output data of their measurement but also the framework of their implementations, we introduce a new concept of a compound-system-type normal extension , and we derive the compound-system-type normal extension of a subnormal operator, which is defined from an irreducible unitary representation of the algebra (1,1). The squeezed states are characterized as the eigenvectors of an operator from this viewpoint, and the squeezed states in multi-particle systems are shown to be the eigenvectors of the adjoints of these subnormal operators under a representation. The affine coherent states are discussed in the same context, as well.

DOI
10.1088/0305-4470/33/43/309

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Subnormal operators regarded as generalized observables and compound-system-type normal extension related to su(1,1)\mathfrak{su}(1,1)su(1,1) (New Development of Infinite-Dimensional Analysis and Quantum Probability)
In this paper, subnormal operators, not necessary bounded, are discussed as generalized observables.In order to describe not only the information about the probability distribution of the output data of their measurement but also a framework of their imple- mentations, we introduce a new concept 'compound-system-type normal extension', and we derive the 'compound-system-type normal extension' of a subnormal operator defined by an irreducible unitary representation of the algebra ϵu(1,1)\epsilon \mathrm{u}(1,1)ϵu(1,1) .The squeezed states are characterized as the eigenvectors of an operator from this viewpoint, and the squeezed states in multi-particle systems are shown to be the eigenvectors of the adjoints of these subnormal operators under a representation.The affine coherent states are discussed in the same context as well.
Three basic properties (eigenstate, orbit and intelligence) of the canonical squeezed states (SS) are extended to the case of n arbitrary observables. The SS for n observables X i can be constructed as eigenstates of their linear complex combinations or as states which minimize the Robertson uncertainty relation. When X i close a Lie algebra L the generalized SS could also be introduced as orbit of Aut ( L C ). It is shown that for the nilpotent algebra h N the three generalizations are equivalent. For the simple su (1, 1) the family of eigenstates of uK - + vK + ( K ± being lowering and raising operators) is a family of ideal K 1 – K 2 SS, but it cannot be represented as an Aut ( su C (1, 1) orbit although the SU (1, 1) group related coherent states (CS) with symmetry are contained in it. Eigenstates | z, u, v, w ; k ⟩ of general combination of uK - + vK + + wK 3 the three generators K j of SU (1, 1) in the representations with Bargman index k = 1/2, 1, ... , and k = 1/4, 3/4 are constructed and discussed in greater detail. These are ideal SS for K 1,2,3 . In the case of the one mode realization of su (1, 1) the nonclassical properties (sub-Poissonian statistics, quadrature squeezing) of the generalized even CS | z, u, v ; + ⟩ are demonstrated. The states | z, u, v, w ; k = 1/4, 3/4 ⟩ can exhibit strong both linear and quadratic squeezing.
A new formalism of the squeeze operator has been found by integration within normal ordered products. Based on this formalism, the essence of the formation of the squeezed state can be revealed directly and clearly. Various representations and properties of the squeezed state can be derived easily. It has been proved that the squeezed state is a coherent one in the quasiparticle space. The transformation relationship between squeezed states and eigenstates of both coordinate and momentum operators are presented. Also presented is the Wigner distribution function of the squeezed state.
The quantum entropy was introduced by von Neumann around 1932, which describes the amont of information of the quantum state itself. It was extended by Ohya for C*‐systems. The quantum relative entropy was first defined by Umegaki for σ ‐finite von Neumann algebras, which denotes a certain difference between two states. It was extended by Araki and Uhlmann for general von Neumann algebras and *‐algebras, respectively. The development of communication theory is closely connected with study of entropy theory. In order to discuss the commuinication processes, we need at least two systems, such as input and output systems. The signal of the input system is carried through a physical device, which is called a channel. The mathematical representation of the channel is a mapping from the input state space to the output state space. The mutual entropy (information) defined by using the relative entropy between the product state consist of the input state and the output state and the compound state given by the communication process, expresses the amount of information correctly transmitted through the channel from the input system to the output system. In classical communication theory, the mutual entropy was formulated by using the joint probability distribution between the input system and the output system. The (semi‐classical) mutual entropies for classical input and quantum output were defined by several researchers. In fully quantum system, there does not exist the joint probability distribution in general. Instead of the joint probability distribution, Ohya invented the quantum (Ohya) compound state, and he introduced the fully quantum mutual entropy (information), which is called Ohya mutual entropy, for quantum input and output systems, describes the amount of information correctly sent from the quantum input system to the quantum output system through the quantum channel. It was extended by Ohya for general quantum systems. Recently Shor and Bennet et al took the coherent entropy and defined the mutual type entropy to discuss a sort of coding theorem for communication processes.In this paper, we concretely calculate these mutual entropy‐type measures for the attenuation channel in order to assay these abilities.