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arXiv:2110.00793 · International Journal of Geometric Methods in Modern Physics

On states of quantum theory

Amir R. Arab

Published Jul 29, 2022• 1 citation
arXivHTMLPDF

Abstract

In this paper, the generalized quantum states, i.e. positive and normalized linear functionals on [Formula: see text]-algebras, are studied. First, we study normal states, i.e. states which are represented by density operators, and singular states, i.e. states cannot be represented by density operators. It is given an approach to the resolution of bounded linear functionals into quantum states by applying the Gelfand–Neumark–Segal (GNS) construction, i.e. the fundamental result of Gelfand, Neumark and Segal on the representation theory of [Formula: see text]-algebras, and theory of projections. Second, it is given an application in quantum information theory. We study covariant cloners, i.e. quantum channels in the Heisenberg and the Schrödinger pictures which are covariant by shifting, and it is shown that the optimal cloners cannot have a singular component. Finally, we discuss on the representation of pure states in the sense of the Gelfand–Pettis integral. We also give physical interpretations and examples in different sections of this work.

DOI
10.1142/s0219887822502218

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In this paper, the generalized quantum states, i.e. positive and normalized linear functionals on [Formula: see text]-algebras, are studied. First, we study normal states, i.e. states which are represented by density operators, and singular states, i.e. states cannot be represented by density operators. It is given an approach to the resolution of bounded linear functionals into quantum states by applying the Gelfand–Neumark–Segal (GNS) construction, i.e. the fundamental result of Gelfand, Neumark and Segal on the representation theory of [Formula: see text]-algebras, and theory of projections. Second, it is given an application in quantum information theory. We study covariant cloners, i.e. quantum channels in the Heisenberg and the Schrödinger pictures which are covariant by shifting, and it is shown that the optimal cloners cannot have a singular component. Finally, we discuss on the representation of pure states in the sense of the Gelfand–Pettis integral. We also give physical interpretations and examples in different sections of this work.
The purpose of this paper is to build a new bridge between category theory and a generalized probability theory known as noncommutative probability or quantum probability, which was originated as a mathematical framework for quantum theory, in terms of states as linear functional defined on category algebras. We clarify that category algebras can be considered to be generalized matrix algebras and that the notions of state on category as linear functional defined on category algebra turns out to be a conceptual generalization of probability measures on sets as discrete categories. Moreover, by establishing a generalization of famous GNS (Gelfand–Naimark–Segal) construction, we obtain a representation of category algebras of †-categories on certain generalized Hilbert spaces which we call semi-Hilbert modules over rigs. The concepts and results in the present paper will be useful for the studies of symmetry/asymmetry since categories are generalized groupoids, which themselves are generalized groups.
G.A. RAGGIO Abstract. We give a brief and incomplete survey of the problem of entan- glement of of composite quantum systems. 1960's - 1980's there were serious attempts (all of them anticipated by J. von Neu- mann) to do away with the underlying Hilbert space and work directly with the abstract algebraic structure modeled by C � -algebras or W � -algebras (abstract von Neumann algebras)(1, 2, 3). This was particularly fruitful when dealing with sys- tems of infinitely many degrees of freedom (quantum fields, thermodynamic limits, etc.). The problem of entanglement which we want to address here can be formu- lated quite straightforwardly in this algebraic framework of quantum theory. But we will stick to the quantum mechanics of the 1930's and keep the Hilbert space. Mainly because the results which are available concern the finite dimensional case. The states of the quantum system specified by H are associated with the lin- ear functionals f : B(H) → C which are positive (i.e., f(a) ≥ 0 for all positive a ∈ B(H)), normalized (i.e., f(1) = 1) and normal (i.e., f(supn{an}) = supnf(an) for any increasing family of selfadjoint operators {an : n = 1,2, � � � } which is uni- formly bounded). It is this normality condition which guarantees that the are in one-to-one correspondence with positive trace class operators d of unit trace (density operators) via the formula B(H) ∋ a 7→ tr(da). This gives an extremely convenient representation of and one often confuses the state f as a linear functional with the associated density operator d for which f(a) = tr(da). States are automatically continuous, and satisfy f(a � ) = f(a). The complex number f(a) is interpreted probabilistically as the expected value of the observable associated with the operator a when the system is in the state f. The normalization condi- tion is thus necessary for the consistency of this interpretation and the normality condition is seen as a non-commutative version of the �-additivity of probability measures. Clearly states, which we will denote by �(H), form a convex set which is closed
In the study of d-dimensional quantum channels (d⩾2), an assumption which includes many interesting examples, and which has a natural physical interpretation, is that the corresponding Kraus operators form a representation of a Lie algebra. Physically, this is a symmetry algebra for the interaction Hamiltonian. This paper begins a systematic study of channels defined by representations; the famous Werner-Holevo channel is one element of this infinite class. We show that the channel derived from the defining representation of su(n) is a depolarizing channel for all n, but for most other representations this is not the case. Since the standard Bloch sphere only exists for the qubit representation of su(2), we develop a consistent generalization of Bloch’s technique. By representing the density matrix as a polynomial in Lie algebra generators, we determine a class of positive semidefinite matrices which represent quantum states for various channels defined by finite-dimensional representations of semisimple Lie algebras. We also give a general method for finding positive semidefinite matrices using Lie algebraic trace identities. This includes an analysis of channels based on the exceptional Lie algebra g2 and the Clifford algebra.
In the second part of our work on observables we have shown that quantum observables in the sense of von Neumann, i.e.bounded selfadjoint operators in some von Neumann subalgebra RRR of L(H)L(H)L(H), can be represented as bounded continuous functions on the Stone spectrum Q(R)Q(R)Q(R) of RRR. Moreover, we have shown that this representation is linear if and only if RRR is abelian, and that in this case it coincides with the Gelfand transformation of RRR. In this part we discuss classical observables, i.e. measurable and continuous functions, under the same point of view. We obtain results that are quite similar to the quantum case, thus showing up the common structural features of quantum and classical observables.
Introduction.The theory of information, created by Shannon [23], is developed by Feinstein, Kullback, MacMillan, Wiener and other American statisticians (e. g., cf.[10]), and also advanced into the ergodic theory by Gelfand, Khinchin, Kolmogorov, Yaglom and other Russian probabilists (e. g., cf.[8]).Through recent years, the theory is regarded as a new chapter in the theory of probability.Recently, Segal [22] gave a mathematical formulation of the entropy of state of a von Neumann algebra, which contains both the cases for the theory of information and the theory of quantum statistics.Segal's theorem was reformulated in operator algebraic form by Nakamura and Umegaki [16] and independently by Davis [3].Since the summer in 1954, Nakamura and Umegaki have investigated the concept of the conditional expectation in von Neumann algebra as a noncommutative extension of probability theory (cf.for example [13~18] and [25 ~28]), and in the most recent paper [18] it was applied to the theory of measurements of quantum statistics which is regarded as a non-commutative case of the theory of entropy and information.Furthermore, it may be very intersesting to develope the theory of information under functional-analysistic and operator-theoretic methods.From these points of views, we shall discuss the measure of information of integrable operators or of normal states of a von Neumann algebra.Davis [3] has independently studied on the almost same theme with Nakamura-Umegaki [16] and [18], in which he developed the theory of entropy and he simplified the proof of the theorem relative to the operatorentropy.Now, we shall give the basic notations and describe the fundamental concepts in a von Neumann algebra which will be used throughout the present paper.Let A be a von Neumann algebra, that is, A is a weakly closed self-adjoint algebra of bounded operators acting over a complex Hubert space H, which contains the identity operator 7. A linear functional p of A is said to be positive if p(aa*) ^ 0 for every a ε A. Such p is said to be state if p(I) = 1, to be normal in the terminology of Dixmier [4] if p(a a )tp(a) for a a ^a, and to be trace if p(ab) = p(ba) for every pair α, b ε A. The normality of state is equivalent to the complete additivity: Σ p(p a ) = p(Σ Pa) for any disjoint family of projections {p a }aA (cf.Dixmier [4]).