Abstract— Some representations of states of quantum systems are discussed, and their equivalence is proved. In particular, an approach going back to L.D. Landau in which the density operator is constructed using a reduction of a pure state of a quantum system described by the tensor product of suitable Hilbert spaces is presented. Under these assumptions, changes in the states of subsystems of a quantum system caused by experiments are investigated.
Abstract— Some representations of states of quantum systems are discussed, and their equivalence is proved. In particular, an approach going back to L.D. Landau in which the density operator is constructed using a reduction of a pure state of a quantum system described by the tensor product of suitable Hilbert spaces is presented. Under these assumptions, changes in the states of subsystems of a quantum system caused by experiments are investigated.
Complex Hilbert spaces play an important role in the description of quantum systems. In fact, with every quantum system there is associated an infinite dimensional or a finite dimensional separable complex Hilbert space ℍ that consists of the states of the quantum system. In physics terminology, the Hilbert space ℍ is usually referred to as the space of (pure) states. Throughout this monograph, the mathematical description of a quantum system shall be based on a certain complex (separable) Hilbert space ℍ, and therefore the quantum system will simply be denoted by ℍ. The quantum system ℍ is said to be a finite-dimensional system if ℍ is a finite-dimensional complex Hilbert space. Otherwise, the quantum system ℍ is said to be an infinite-dimensional system. The Hilbert space ℍ representing a composite quantum system that consists of n subsystems is a tensor product of the Hilbert spaces of n component systems described by ℍ 1 , … ℍ n , i.e., ℍ = ℍ 1 ⊗ … ⊗ ℍ n . In the theory of Hilbert spaces, a tensor product of Hilbert spaces is a way to create a new Hilbert space out of 2 (or more) Hilbert spaces. In the following, we illustrate the concept and construction of tensor product ℍ ⊗ 핂 of only 2 Hilbert spaces ℍ and 핂. The concepts and constructions can be easily extended to more than 2 Hilbert spaces such as ℍ 1 ⊗ ℍ 2 ⊗ … ⊗ ℍ n . Recall that the tensor product of ℍ with 핂 denoted by ℍ ⊗ 핂 is a new Hilbert space that consists of elements ϕ ⊗ φ ( ϕ ∈ ℍ and φ ∈ 핂) and is equipped with the Hilbertian inner product 〈·, ·〉 ℍ⊗핂 :(ℍ ⊗ 핂) × (ℍ ⊗ 핂) → C defined by 〈 ϕ 1 ⊗ φ 1 , ϕ 2 ⊗ φ 2 〉 ℍ⊗핂 = 〈 ϕ 1 , ϕ 2 〉 ℍ 〈 φ 1 , φ 2 〉 핂 , ϕ i ∈ ℍ and φ i ∈ 핂 for i = 1, 2.
A state reduct,ion is the stat1e change caused by a lueasurement on a quan- tum system conditional upon the outcome.A rigorous theorv of the state reduction is developed with nlathematiCa1 formalism, I)hysiCa1 interpret.ation,and nιodels .A special em1JhasiS is on the pure state reduction which trans- forms a pure 1) riorStatCe to the pure posterior state for every outcome.Mathe- 1natiCa1 structure of general pure state reductions is discussed and it is 1) roVed that every pure state reduction is decomposed into just two types, called tlle von Neumann-Davies (ND)tyl)e and the Gordon-Louisell (GL) type; a state reduction $\psirightarrow\psi_{x}$ ' is of the ND type if the 1IlaP1in ) gψ ) ↦P(x∣ψ)1/2?l′x ' is linear, where P(x∣ψ′) is the probability density of the outcome x , and of the GL tyl ) e if ψx depends only on the outcome x (independent of the prior state )) .
Abstract In this chapter we explain how we can describe quantum theory within the general framework given in Chapter 1. We do not discuss various theories concerning the question of how quantum theory can (or cannot) be deduced from experimental facts or from a verifiable axiomatic system. Instead we start by introducing the assumptions of quantum theory. The main content is the algebraic viewpoint of quantum theory based on the theory of operator algebras, the GNS construction which supplies the fundamental relation between the description of the states by Hilbert space vectors and the description of the states in this general framework, the Wigner theorem which is basic for the mathematical description of quantum mechanical symmetries, and its generalizations. The basic definitions and theorems about Hilbert spaces and operator algebras are collected in the appendix.
Abstract Quantum nonlocality represents correlations between subsystems of a composite quantum system, usually including Bell nonlocality, steerability, and entanglement. According to the hypothesis of quantum mechanics, states of a quantum system Q described by a d -dimensional Hilbert space are denoted by density operators acting on . Under a basis e for the Hilbert space , every abstract density operator ρ of the system AB corresponds to a density matrix ρ e , which is a state of the complex Hilbert space . In this work, we discuss the consistency of quantum nonlocality of density operators ρ and their corresponding density matrices ρ e under the chosen basis e . It is proved that only when a basis e is a product one, a density operator ρ is entangled (respectively, Bell nonlocal, steerable) if and only if its density matrix ρ e is entangled (respectively, Bell nonlocal, steerable).
Abstract. In the present paper, we consider the marginal entropy of the a pure state in the tensor product of two Hilbert spaces. The expectation of the marginal entropy of a random pure state with respect to the Haar measure of unitary transformation on the tensor product Hilbert space depends on group integrals of unitary group. For group integrals of ordinary representation matrix elements of U(n), we provide a elementary method to reobtain Weingarton’s result of asymptotical behavior of group integrals. For group integrals of matrix elements of irreducible representations of U(n),we generalize the Weyl-Schur duality theorem and get an algorithm. 1. Mathematical Preliminaries We adopt the following standard conventions of quantum mechanics: The states of an n-level quantum system (and we include the case infinity) are represented by an n-dimensional complex Hilbert space H. In this familiar representation, the states of a composite of two systems A and B say, the first one A n-level, and the second B m-level, is then represented by the tensor product of the respective Hilbert spaces HA and HB. As a result, the composite system (AB) is an (nm)-level system. This also makes sense in the infinite case where we then use standard geometry for Hilbert space, and suitable choices of orthonormal bases (ONBs). For a fixed system with Hilbert space H, the corresponding pure quantum states are vectors in H of norm one, or rather equivalence classes of such vectors: Equivalent vectors v and v ′ in H yield the same rank-one projection operator P, i.e., the projection of H onto Key words and phrases. random pure state, marginal entropy,group integral, Weyl-Schur duality