5.20.1 Tensor Product Linear Map Factor Selection

Understanding how tensor products enable the factorization of linear maps in algebraic structures.

Tensor Product Linear Map Factor Selection is the practice, within the tensor product of linear maps operation, of applying a nontrivial linear map to only one selected factor of a tensor product while leaving the other factor or factors fixed by the identity map, as in f ⊗ id_W : V ⊗ W → V′ ⊗ W, so that a transformation can act on a single chosen "slot" of a multi-factor tensor product without disturbing the others. This selective application is the standard way multi-index tensors are transformed one index at a time, and it underlies how operators acting locally on a subsystem are represented within a larger tensor product space.

Definition of a Factor-Selective Map

Acting on One Factor via the Identity on the Rest

Given a linear map f : V → V′ and the identity map id_W : W → W, the tensor product construction produces

f \otimes {id}_{W} : V \otimes W \to V^{'} \otimes W

acting on simple tensors by (f ⊗ id_W)(v ⊗ w) = f(v) ⊗ w, leaving the W factor of every simple tensor untouched while transforming only the V factor.

The Symmetric Case: Selecting the Other Factor

Analogously, id_V ⊗ g : V ⊗ W → V ⊗ W′ for a map g : W → W′ acts by (id_V ⊗ g)(v ⊗ w) = v ⊗ g(w), selecting the second factor for transformation instead of the first. Factor selection is thus a choice of which tensor slot receives the nontrivial map.

Recovering the General Tensor Product Map by Composition

Decomposing `f ⊗ g` into Two Factor-Selective Steps

Any general tensor product map f ⊗ g factors as the composition of two single-factor-selective maps:

f \otimes g = ({id}_{V^{'}} \otimes g) \circ (f \otimes {id}_{W})

so applying f to the first factor and g to the second, in either order, yields the same result as applying both simultaneously — the two factor-selective maps commute, since they act on independent tensor slots.

Why the Order of Selection Does Not Matter

Verifying on simple tensors, (id_{V′} ⊗ g)((f ⊗ id_W)(v ⊗ w)) = (id_{V′} ⊗ g)(f(v) ⊗ w) = f(v) ⊗ g(w), and the same value results from applying the maps in the opposite order, confirming commutativity and justifying the decomposition of any two-factor tensor map into independent, order-irrelevant single-factor selections.

Diagram of Factor-Selective Action

Extension to Multi-Factor Tensor Products

Selecting a Single Index Among Many

For an n-factor tensor product V₁ ⊗ V₂ ⊗ ... ⊗ Vₙ, factor selection generalizes to applying a map fₖ : Vₖ → Vₖ′ to the k-th slot while every other slot receives the identity:

{id}_{V_{1}} \otimes \dots \otimes f_{k} \otimes \dots \otimes {id}_{V_{n}}

with all n such single-slot maps pairwise commuting, so any combination of per-slot transformations can be built up by composing them in any order.

Relevance to Local Operators on Composite Systems

This form of factor selection is the standard way to represent an operator that acts "locally" on one subsystem of a composite system built as a tensor product, such as a quantum operator acting on a single particle within a multi-particle tensor product Hilbert space, without affecting the states of the other subsystems.

Matrix Perspective on Factor Selection

Kronecker Product with an Identity Block

In finite dimensions, if f is represented by a matrix A and id_W by the identity matrix Iₙ, the factor-selective map f ⊗ id_W is represented by the Kronecker product A ⊗ Iₙ, a block matrix consisting of scaled copies of Iₙ arranged according to the entries of A; symmetrically, id_V ⊗ g is represented by Iₘ ⊗ B. These two matrix families generate, under matrix multiplication, the full Kronecker product A ⊗ B, matching the composition decomposition described above.

Significance of Factor Selection

Isolating the Effect of a Single Transformation

Factor selection provides the precise algebraic language for stating that a transformation affects one component of a composite object and leaves the rest untouched, a distinction that is essential whenever a tensor product is used to model a system built from independent or weakly interacting parts.

Building Block for General Tensor Map Constructions

Because every tensor product map f ⊗ g (and its multi-factor generalizations) decomposes into commuting factor-selective maps, factor selection serves as the elementary building block from which all tensor product linear map operations are assembled, reducing the study of maps on tensor products to the simpler study of maps acting on one factor at a time.