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arXiv:1202.3969 · Journal of Physics A Mathematical and Theoretical

Reduction of Lie–Jordan Banach algebras and quantum states

Fernando Falceto, Leonardo Ferro, Alberto Ibort, Giuseppe Marmo

Published Dec 3, 2012• 27 citations
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Abstract

In this paper, it is shown that the concept of dynamical correspondence for Jordan Banach algebras is equivalent to a Lie structure compatible with the Jordan one. Then a theory of reduction of Lie–Jordan Banach algebras in the presence of quantum constraints is presented and compared to the standard reduction of C *-algebras of observables of a quantum system. The space of states of the reduced Lie–Jordan Banach algebra is characterized in terms of Dirac states on the physical algebra of observables and its GNS representations described in terms of states on the unreduced algebra.

DOI
10.1088/1751-8113/46/1/015201

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In this paper, it is shown that the concept of dynamical correspondence for Jordan Banach algebras is equivalent to a Lie structure compatible with the Jordan one. Then a theory of reduction of Lie–Jordan Banach algebras in the presence of quantum constraints is presented and compared to the standard reduction of C *-algebras of observables of a quantum system. The space of states of the reduced Lie–Jordan Banach algebra is characterized in terms of Dirac states on the physical algebra of observables and its GNS representations described in terms of states on the unreduced algebra.
The main objective of this dissertation is the study of the theory of Lie-Jordan Banach algebras, their role in the framework of classical and quantum mechanics, and their applications to different aspects of quantum systems. One of the main results of this thesis is the novel proof of a theorem which characterizes the Jordan (-Banach) algebras that are in a unique correspondence with C*-algebras. The novelty lies in the introduction of a Lie structure on the algebra, which is required to be compatible with the Jordan one. The physical interpretation of this connection with the Lie structure reflects the dual role played by the observables: they are measurable quantities but also the generators of motions of the state space. Whenever constraints are imposed on a quantum system or symmetries are present, both dynamical or gauge, some reduction on the state space must be considered either because not all states are physical and/or because families of states are equivalent. We address this problem in algebraic terms and obtain a reduction procedure which is free from the problems of gauge anomalies which is often encountered in a heuristic approach to the quantization of constraints. Then we study the composition of physical systems and we prove, from algebraic considerations only, the uniqueness of the Planck's constant. This is a strong result, since he existence of multiple Planck's constants would lead to violations of basic space-time conservation laws, which are not observed experimentally. Finally we address the problem of characterizing classical and quantum states, which has become particularly relevant for the field of quantum information and quantum computation, since a number of tasks in computation and communication can be performed only if quantum resources are available. By using the Jordan product we are able to provide a general definition of classical states and the associated quantumness witness which is suitable to infinite-dimensional systems. Then we propose a measure of nonclassicality based on the incompatibility of states relative to each other, rather than on correlations. Surprisingly, this new measure turns out to be experimentally suitable to a direct estimation by using a quantun circuit based on an interference experiment. We finally describe this experiment which is realizable with current technology.
The idea that symmetries simplify or reduce the complexity of a system has been remarkably fruitful in physics, and especially in quantum mechanics. On a mathematical level, symmetry groups single out a certain structure in the Hilbert space that leads to a reduction. This structure is given by the irreducible representations of the group, and in general it can be identified with an operator algebra (a.k.a. C*-algebra or von Neumann algebra). The primary focus of this thesis is the extension of the framework of reductions from symmetries to operator algebras, and its applications in finite-dimensional quantum mechanics. Finding the irreducible representations structure is the principal problem when working with operator algebras. We will therefore review the representation theory of finite-dimensional operator algebras and elucidate this problem with the help of two novel concepts: minimal isometries and bipartition tables. One of the main technical results that we present is the Scattering Algorithm for analytical derivations of the irreducible representations structure of operator algebras. For applications, we will introduce a symmetry-agnostic approach to the reduction of dynamics where we circumvent the non-trivial task of identifying symmetries, and directly reduce the dynamics generated by a Hamiltonian. We will also consider quantum state reductions that arise from operational constraints, such as the partial trace or the twirl map, and study how operational constraints lead to decoherence. Apart from that, we will extend the idea of reduction beyond operator algebras to operator systems, and formulate a quantum notion of coarse-graining that so far only existed in classical probability theory. We will also characterize how the uncertainty principle transitions to the classical regime under coarse-grained measurements and discuss the implications in a finite-dimensional setting.
Reduced density matrices are a powerful tool in the analysis of entanglement structure, approximate or coarse-grained dynamics, decoherence, and the emergence of classicality. It is straightforward to produce a reduced density matrix with the partial-trace map by tracing out part of the quantum state, but in many natural situations this reduction may not be achievable. We investigate the general problem of identifying how the quantum state is reduced given a restriction on the observables. For example, in an experimental setting, the set of observables that can actually be measured is usually modest (compared to the set of all possible observables) and their resolution is limited. In such situations, the appropriate state-reduction map can be defined via a generalized bipartition, which is associated with the structure of irreducible representations of the algebra generated by the restricted set of observables. One of our main technical results is a general, not inherently numeric, algorithm for finding irreducible representations of matrix algebras. We demonstrate the viability of this approach with two examples of limited-resolution observables. The definition of quantum state reductions can also be extended beyond algebras of observables. To accomplish this task we introduce a more flexible notion of bipartition, the partial bipartition, which describes coarse grainings preserving information about a limited set (not necessarily algebra) of observables. We describe a variational method to choose the coarse grainings most compatible with a specified Hamiltonian, which exhibit emergent classicality in the reduced state space. We apply this construction to the concrete example of the one-dimensional Ising model. Our results have relevance for quantum information, bulk reconstruction in holography, and quantum gravity.
In this chapter we will discuss the notion of a “dynamical correspondence”, which is applicable in both JB and JBW contexts. This generalizes the correspondence of observables and generators of one-parameter groups of automorphisms in quantum mechanics. It is closely related to Connes’ concept of orientation [36, Definition 4.11] that he used as a key property in his characterization of the natural self-dual cones associated with von Neumann algebras (i.e., the cones P ξ ♮ of Tomita—Takesaki theory). Both can be thought of as ways to specify possible Lie structures compatible with a given Jordan structure.