1.2.33 Pure Tensor Definition

A pure tensor is a basic element in tensor algebra, representing multilinear relationships in mathematical structures.

Pure Tensor Definition is the characterization of a tensor that factors completely into a single product of one vector from each of the spaces contributing to a tensor product, with no residual sum required to express it. The term pure tensor is most common in contexts where tensor product spaces describe composite systems — particularly in quantum mechanics and in the representation theory of algebras — and it emphasizes the idea that such a tensor carries no internal correlation or entanglement between its factors, in direct parallel to the notion of a pure state.

Formal Definition

Let $V_{1}, \dots, V_{k}$ be vector spaces over a field $F$ , and let

W = V_{1} \otimes \dots \otimes V_{k}

be their tensor product space. An element $T \in W$ is called a pure tensor if it can be written as

T = v_{1} \otimes v_{2} \otimes \dots \otimes v_{k}

for some vectors $v_{i} \in V_{i}$ . This is the same class of elements referred to elsewhere as elementary or simple tensors; the choice of terminology reflects the surrounding context rather than any difference in mathematical content, with "pure" being the standard term whenever the tensor product describes a composite of independently prepared or independently generated components.

Motivation for the Terminology

Analogy with Pure States

In quantum mechanics, the state space of a composite system built from two subsystems with individual state spaces $V$ and $W$ is modeled by the tensor product $V \otimes W$ . A pure tensor $v \otimes w$ in this space describes a composite state in which the first subsystem is definitely in state $v$ and the second is definitely in state $w$ , independent of one another. This is the sense in which the tensor is "pure": it carries no mixing or correlation between the two subsystems, unlike a general element of $V \otimes W$ , which may describe a state in which the subsystems are entangled and cannot be assigned independent definite states of their own.

Distinguishing Purity from Simplicity in Broader Use

While "pure tensor" and "simple tensor" denote the identical mathematical set within a given tensor product space, the word "pure" is preferred in settings emphasizing the physical or representation-theoretic interpretation of factorizability, whereas "simple" and "elementary" are more common in purely algebraic treatments of the tensor product construction itself. All three terms nonetheless refer to exactly the same rank-one elements of the tensor product space.

Characterization and Basic Properties

A pure tensor satisfies the scalar redistribution identity

(a v_{1}) \otimes v_{2} \otimes \dots \otimes v_{k} = v_{1} \otimes (a v_{2}) \otimes \dots \otimes v_{k}

for any nonzero scalar $a$ , so its factorization into individual vectors is not unique; only the equivalence class of factorizations related by such scalar redistribution, together with the one-dimensional spans generated by each factor, is intrinsic to the pure tensor itself. The set of all pure tensors in $W$ , together with the zero element, forms a cone rather than a subspace, since the sum of two pure tensors is in general not pure.

Detecting Purity: The Rank-One Criterion

A pure tensor is, equivalently, a tensor of rank exactly one or zero, where tensor rank denotes the minimum number of terms needed to express the tensor as a sum of factorizable products. For a two-factor tensor product with components $T_{i j}$ relative to bases of $V$ and $W$ , purity is equivalent to the coordinate matrix having rank at most one, which can be tested directly by checking that all its $2 \times 2$ minors vanish. This gives a concrete, finite computational test for purity in the two-factor case that generalizes, with greater difficulty, to tensor products of more than two factors.

Non-Pure Tensors

An element of a tensor product space that cannot be written as a single product is called non-pure, mixed, or entangled, depending on context. Such an element is always expressible as a finite sum of pure tensors, since the pure tensors span the entire tensor product space, but no single term in that sum captures the element on its own. In the quantum-mechanical setting, a non-pure composite state exhibits genuine quantum correlation between its subsystems that cannot be reduced to an independent description of each part, which is the physical phenomenon of entanglement.

Role Within Tensor Algebra

Pure tensors occupy the same foundational role as simple and elementary tensors within the algebraic structure of a tensor product space: they generate the space by finite linear combinations, they are the natural elements on which the universal property of the tensor product is first verified, and the distinction between pure and non-pure tensors — measured by tensor rank — organizes the internal structure of the space into strata of increasing algebraic and, in applied contexts, physical complexity.