A review is made of the multitude of different mathematical formalizations of the physical concept ‘two observables (or two systems) are independent’ that have been proposed in quantum theories, particularly relativistic quantum field theory. The most basic mathematical formulation of independence in any quantum theory is what one may call kinematical independence: the two observables, resp. the observables of the two quantum systems, which are represented by elements of a C*-algebra, resp. two subalgebras of a C*-algebra, are required to commute. This is related to a mathematical formulation of the notion of the coexistence (or compatibility) of two observables. Another basic notion of independence, generally called statistical independence in the literature, is, roughly speaking, two quantum systems are said to be statistically independent if each can be prepared in any state, how ever the other system has been prepared. There are numerous mathematical formulations of this notion and their interrelationships are explained. Statistical independence and kinematical independence are shown to be logically independent. Additional notions such as strict locality and their relation to statistical independence are discussed. The mathematics of a more quantitative measure of statistical independence, Bell’s inequalities, is reviewed and its relations with previously introduced notions are indicated. All of these notions are then viewed in application to relativistic quantum field theory.
A review is made of the multitude of different mathematical formalizations of the physical concept ‘two observables (or two systems) are independent’ that have been proposed in quantum theories, particularly relativistic quantum field theory. The most basic mathematical formulation of independence in any quantum theory is what one may call kinematical independence: the two observables, resp. the observables of the two quantum systems, which are represented by elements of a C*-algebra, resp. two subalgebras of a C*-algebra, are required to commute. This is related to a mathematical formulation of the notion of the coexistence (or compatibility) of two observables. Another basic notion of independence, generally called statistical independence in the literature, is, roughly speaking, two quantum systems are said to be statistically independent if each can be prepared in any state, how ever the other system has been prepared. There are numerous mathematical formulations of this notion and their interrelationships are explained. Statistical independence and kinematical independence are shown to be logically independent. Additional notions such as strict locality and their relation to statistical independence are discussed. The mathematics of a more quantitative measure of statistical independence, Bell’s inequalities, is reviewed and its relations with previously introduced notions are indicated. All of these notions are then viewed in application to relativistic quantum field theory.
In the case of systems composed of identical particles, a typical instance in quantum statistical mechanics, the standard approach to separability and entanglement ought to be reformulated and rephrased in terms of correlations between operators from subalgebras localized in spatially disjoint regions. While this algebraic approach is straightforward for bosons, in the case of fermions it is subtler since one has to distinguish between micro-causality, that is the anti-commutativity of the basic creation and annihilation operators, and algebraic independence that is the commutativity of local observables. We argue that a consistent algebraic formulation of separability and entanglement should be compatible with micro-causality rather than with algebraic independence.
From the outset statistical mechanics will be framed in the language of quantum theory. The typical macroscopic system is composed of multiple constituents, and hence described in some many-particle Hilbert space. In general, not much is known about such a system, certainly not the precise preparation of all its microscopic details. Thus, its description requires a more general notion of a quantum state, a so-called mixed state. This chapter begins with a brief review of the basic axioms of quantum theory regarding observables, pure states, measurements, and time evolution. Particular attention is paid to the use of projection operators and to the most elementary quantum system, a two-level system. The chapter then motivates the introduction of mixed states and examines in detail their mathematical representation and properties. It also dwells on the description of composite systems, introducing, in particular, the notions of statistical independence and correlations.
Bell correlations are the hallmark of quantum non-locality, and a rich context for analysing them is provided by the algebraic approach to quantum field theory (AQFT): the basic idea is to associate with each bounded region O of Minkowski spacetime an algebra A(O) of operators, of which a self-adjoint element P ∈ A(O) represents a physical quantity pertaining to that part of the field system lying in O, that is measurable by a procedure confined to O. The violation of Bell inequalities in AQFT is known to be "generic", as regards the choices of regions O, and of quantities P, and of states. Furthermore, they are typically "maximal" and "indestructible" in a sense that can be made mathematically precise. The prospects for “peaceful coexistence” between quantum non-locality and relativity theory’s requirement of no action-at-a-distance are also explored. The purpose of this Essay is to review the developments in these areas.
In the case of systems composed of identical particles, a typical instance in quantum statistical mechanics, the standard approach to separability and entanglement ought to be reformulated and rephrased in terms of correlations between operators from subalgebras localized in spatially disjoint regions. While this algebraic approach is straightforward for bosons, in the case of fermions it is subtler since one has to distinguish between micro-causality, that is the anti-commutativity of the basic creation and annihilation operators, and algebraic independence that is the commutativity of local observables. We argue that a consistent algebraic formulation of separability and entanglement should be compatible with micro-causality rather than with algebraic independence.
Introduction Quantum probability is a generalization of both classical probability theory and quantum mechanics that allows to describe the probabilistic aspects of quantum mechanics. This generalization is formulated in two steps. First, the theory is reformulated in terms of algebras of functions on probability spaces. Therefore, the notion of a probability space (Ω, ℱ , P ) is replaced by the pair ( L ∞ (Ω), E (.) = ∫ Ω . d P ) consisting of the commutative von Neumann algebra of bounded random variables and the expectation functional. Then, the commutativity condition is dropped. In this way we arrive at the notion of a (von Neumann) algebraic probability space ( N ,Φ) consisting of a von Neumann algebra N and a normal (faithful tracial) state Φ. As we have seen this includes classical probability spaces in the form ( L ∞ (Ω), E ), it also includes quantum mechanical systems modeled by a Hilbert space H and a pure state ψ ∊ H (or a mixed state ρ ∊ S ( H )), if we take N = B ( H ) and Ф the state defined by Ф( X ) = 〈 ψ, X ψ 〉 (or Ф ( X ) = tr(ρ X )) for X ∊ B ( H ). Note that in this course we shall relax the conditions on N and Φ and work with involutive algebras and positive normalized functionals, that is, *-algebraic probability spaces. A striking feature of quantum probability (also called noncommutative probability) is the existence of several notions of independence. This is the starting point of this course, which intends to give an introduction to the theory of quantum stochastic processes with independent increments. However, before we come to these processes, we will give a general introduction to quantum probability. In Section 1.2, we recall the basic definitions of quantum probability and discuss some fundamental differences between classical probability and quantum probability. In Section 1.3 we address the question ‘Why do we need Quantum Probability?’ We discuss the EPR experiment and the Kochen–Specker Theorem, which show that we cannot model quantum physics with classical probability spaces because the values of observable quantities do not exist unless we specify which measurement we will carry out and which quantities we will determine. In this sense quantum physics requires a more radical description of chance.
INTRODUCTION The main purpose of this chapter is to prove a lemma about random variables, and then to apply this lemma to the characterization of local theories of hidden variables by Bell (1964, 1966) and Wigner (1970), which are focused around Bell's inequality. We use the results of the lemma in two different ways. The first is to show that the assumptions of Bell and Wigner can be weakened to conditional statistical independence rather than conditional determinism because determinism follows from conditional independence and the other assumptions that are made about systems of two spin-½ particles. The second direction is to question the attempt of Clauser and Home (1974) to derive a Bell-type inequality for local stochastic theories of hidden variables which use an assumption of conditional statistical independence for observables. The main thrust of our analysis obviously arises from the probabilistic lemma we prove. Roughly speaking, this lemma asserts that if two random variables have strict correlation, that is, the absolute value of the correlation is one, and it is in addition assumed that their expectations are conditionally independent when a third random variable λ is given, then the conditional variance of X and Y given λ is zero. In other words, given the hidden variable λ the observables X and Y are strictly determined. The lemma itself, of course, depends on no assumptions about quantum mechanics. It may be regarded as a limitation on any theories that assume both strict correlation between observables and their conditional independence on the basis of some prior or hidden variable.