5.14.4 Tensor Canonical Map Elementary Output

Tensor Canonical Map Elementary Output is the simple tensor v ⊗ w produced by applying the canonical map ⊗ to a single pair (v, w) ∈ V × W, considered specifically as an output value rather than as a formula or a property of the map. Studying the elementary output focuses attention on what these individual results look like, how they relate to one another, and why they form only a special subset of the full tensor product space rather than exhausting it.

Characterizing Elementary Outputs

Definition of a Simple Tensor

An elementary output, more commonly called a simple or elementary tensor, is any element of V ⊗ W of the form:

for some specific v ∈ V and w ∈ W. The set of all such outputs is precisely the image of the canonical map.

Elementary Outputs Are Not Closed Under Addition

A defining and often surprising feature of elementary outputs is that their sum is generally not itself an elementary output. For instance, given linearly independent vectors e₁, e₂ ∈ V and f₁, f₂ ∈ W, the sum:

cannot be rewritten as a single simple tensor v ⊗ w, which is verified using the rank criterion described below.

Detecting Whether an Element Is an Elementary Output

The Rank Criterion

An element t ∈ V ⊗ W, expressed in coordinates relative to bases {eᵢ} and {fⱼ} as a matrix of coefficients t = Σ tᵢⱼ (eᵢ ⊗ fⱼ), is an elementary output if and only if the coefficient matrix (tᵢⱼ) has rank at most one.

Rank as the Minimal Number of Terms

More generally, the minimal number of simple tensors needed to express any given element t as a sum equals the rank of its coefficient matrix, so an elementary output is exactly the case where this minimal number equals one (or zero for the zero element).

Uniqueness and Non-Uniqueness of the Representation

Elementary Outputs Are Not Uniquely Represented

A single elementary output v ⊗ w can arise from infinitely many different pairs (v', w'), since for any nonzero scalar c, (cv) ⊗ (c⁻¹w) = v ⊗ w. The elementary output as a value in V ⊗ W does not remember which specific pair produced it.

Fibers of the Canonical Map over an Elementary Output

The set of all pairs (v', w') mapping to a fixed nonzero elementary output v ⊗ w forms a single orbit under the scaling relation (v', w') ~ (cv, c⁻¹w), which can be visualized as a hyperbola-like curve of equivalent factorizations rather than a single point.

Visualization

Role Within the Tensor Product

Spanning Set, Not Ambient Space

Elementary outputs are important precisely because they span V ⊗ W even though they do not constitute all of it. This spanning property is what allows the universal property to be verified by checking behavior only on elementary outputs, while the existence of non-elementary elements is what gives the tensor product its genuinely richer structure compared to the domain product V × W.

Relation to Decomposability in Applications

In applications such as quantum mechanics, an elementary output corresponds to what is called a separable or product state, while a non-elementary element corresponds to an entangled state, illustrating that the algebraic distinction between elementary and non-elementary outputs has direct physical significance in fields that make heavy use of tensor product spaces.

Consequences for Computation

Efficient Representation of Elementary Outputs

Because an elementary output is fully specified by a single pair (v, w), it can be stored and manipulated using far less data than a general element of V ⊗ W, which in the worst case requires specifying dim(V) · dim(W) independent coefficients.

Elementary Outputs in Higher-Order Tensor Products

The same notion of elementary output extends to iterated tensor products, where a simple tensor v₁ ⊗ v₂ ⊗ ... ⊗ vₙ is again characterized by a generalized rank-one condition, and the gap between elementary and general elements grows increasingly significant as the number of factors n increases.