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Cambridge University Press eBooks

Composite systems and tensor products

Robert B. Griffiths

Published Nov 29, 2001• 1 citation
Publisher

Abstract

Introduction A composite system is one involving more than one particle, or a particle with internal degrees of freedom in addition to its center of mass. In classical mechanics the phase space of a composite system is a Cartesian product of the phase spaces of its constituents. The Cartesian product of two sets A and B is the set of (ordered) pairs {( a, b )}, where a is any element of A and b is any element of B . For three sets A, B , and C the Cartesian product consists of triples {( a, b, c )}, and so forth. Consider two classical particles in one dimension, with phase spaces x 1 , p 1 and x 2 , p 2 . The phase space for the composite system consists of pairs of points from the two phase spaces, that is, it is a collection of quadruples of the form x 1 , p 1 , x 2 , p 2 , which can equally well be written in the order x 1 , x 2 , p 1 , p 2 . This is formally the same as the phase space of a single particle in two dimensions, a collection of quadruples x, y, p x , p y . Similarly, the six-dimensional phase space of a particle in three dimensions is formally the same as that of three one-dimensional particles. In quantum theory the analog of a Cartesian product of classical phase spaces is a tensor product of Hilbert spaces. A particle in three dimensions has a Hilbert space which is the tensor product of three spaces, each corresponding to motion in one dimension. The Hilbert space for two particles, as long as they are not identical, is the tensor product of the two Hilbert spaces for the separate particles.

DOI
10.1017/cbo9780511606052.007

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Introduction A composite system is one involving more than one particle, or a particle with internal degrees of freedom in addition to its center of mass. In classical mechanics the phase space of a composite system is a Cartesian product of the phase spaces of its constituents. The Cartesian product of two sets A and B is the set of (ordered) pairs {( a, b )}, where a is any element of A and b is any element of B . For three sets A, B , and C the Cartesian product consists of triples {( a, b, c )}, and so forth. Consider two classical particles in one dimension, with phase spaces x 1 , p 1 and x 2 , p 2 . The phase space for the composite system consists of pairs of points from the two phase spaces, that is, it is a collection of quadruples of the form x 1 , p 1 , x 2 , p 2 , which can equally well be written in the order x 1 , x 2 , p 1 , p 2 . This is formally the same as the phase space of a single particle in two dimensions, a collection of quadruples x, y, p x , p y . Similarly, the six-dimensional phase space of a particle in three dimensions is formally the same as that of three one-dimensional particles. In quantum theory the analog of a Cartesian product of classical phase spaces is a tensor product of Hilbert spaces. A particle in three dimensions has a Hilbert space which is the tensor product of three spaces, each corresponding to motion in one dimension. The Hilbert space for two particles, as long as they are not identical, is the tensor product of the two Hilbert spaces for the separate particles.
The tensor product postulate of quantum mechanics states that the Hilbert space of a composite system is the tensor product of the components' Hilbert spaces. All current formalizations of quantum mechanics that do not contain this postulate contain some equivalent postulate or assumption (sometimes hidden). Here we give a natural definition of a composite system as a set containing the component systems and show how one can logically derive the tensor product rule from the state postulate and from the measurement postulate. In other words, our Letter reduces by one the number of postulates necessary to quantum mechanics.
One of the crucial differences between mathematical models of classical and quantum mechanics (QM) is the use of the tensor product of the state spaces of subsystems as the state space of the corresponding composite system. (To describe an ensemble of classical composite systems, one uses random variables taking values in the Cartesian product of the state spaces of subsystems.) We show that, nevertheless, it is possible to establish a natural correspondence between the classical and the quantum probabilistic descriptions of composite systems. Quantum averages for composite systems (including entangled) can be represented as averages with respect to classical random fields. It is essentially what Albert Einstein dreamed of. QM is represented as classical statistical mechanics with infinite-dimensional phase space. While the mathematical construction is completely rigorous, its physical interpretation is a complicated problem. We present the basic physical interpretation of prequantum classical statistical field theory in Sec. II. However, this is only the first step toward real physical theory.
One of the crucial differences between mathematical models of classical and quantum mechanics is the use of the tensor product of the state spaces of subsystems as the state space of the corresponding composite system. (To describe an ensemble of classical composite systems one uses random variables taking values in the Cartesian product of the state spaces of subsystems.) We show that, nevertheless, it is possible to establish a natural correspondence between the classical and quantum probabilistic descriptions of composite systems. Quantum averages for composite systems (including entangled) can be represented as averages with respect to classical random fields. It is essentially what Albert Einstein was dreamed of. Quantum mechanics is represented as classical statistical mechanics with infinite-dimensional phase space. While the mathematical construction is completely rigorous, its physical interpretation is a complicated problem (which will not be discussed in this paper).
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.
An axiomatic approach to the mathematical formalism of quantum mechanics, based upon a certain concept of conditional probability, has been proposed in two recent papers by the author. It leads to Jordan operator algebras and thus comes rather close to the standard Hilbert space model of quantum mechanics, but still includes the so-called exceptional Jordan algebras, for which a Hilbert space representation does not exist. This approach is now extended by defining a mathematical model of composite systems. Such a model is required for the study of the joint distribution of two quantum observables. A very general type of observables (not only the real-valued observables corresponding to the self-adjoint operators) is considered. The joint distribution is defined, using the concept of conditional probability, and exhibits a certain dependence on the succession of the observations which is different from the classical case and unknown so far in quantum mechanics. Finally, it turns out that, at least in the finite-dimensional case, a really satisfying model of the composite system exists only if each single system is modeled by a complex Jordan matrix algebra (or a direct sum), and the model then becomes the tensor product. This provides some reasoning why the exceptional Jordan algebras can be ruled out, why quantum mechanics needs the complex numbers and the complex Hilbert space, and why the tensor product is the right choice for the model of a composite system.
Quantum mechanics of composite systems, gives rise to certain special states called entangled states. A physical system, that is in an entangled state displays an intricate correlation between its subsystems. There are also some composite quantum states (classically correlated states or separable states) that are not entangled. It is generally claimed, often without a rigorous proof to support, that these intricate correlations of an entangled state cannot occur in a classical system. This expository article, provides an elementary proof that entangled states cannot arise in the setting of classical mechanics. In addition, a detailed description of the origin of entanglement in quantum systems is included. The mathematical concepts that are necessary for this purpose are presented. The absence of entanglement in the classical setting is due to the fact that every pure classical state of a composite system is a product state, that is, a tensor product of two pure states of the subsystems. In contrast, there are pure composite quantum states that cannot be expressed in the form of a product state or even by a convex sum of product states. Roughly speaking, this is because classical states are positive valued functions on the phase-space while quantum states are positive linear operators. The structure of the tensor product between two commutative spaces of scalar valued functions is drastically different from that of the tensor product between two non-commutative spaces of linear operators. In other words, entanglement is a non-commutative phenomenon.