9.3 Indecomposable vectors and entanglement

9.3 Indecomposable vectors and entanglement

Concept links · terms present in this machine draft; source roles are unverified: vector space

If one is given a function $f$ on a space $X$ and a function $g$ on a space $Y,\mathrm{a}$ product function $fg$ on the product space $X\times Y$ can be defined by taking (for $x\in X,y\in Y)$

$$(fg)(x,y)=f(x)g(y)$$

However, most functions on $X\times Y$ are not decomposable in this manner. Similarly, for a tensor product of vector spaces:

Definition (Decomposable and indecomposable vectors)

$A$ vector in $V\otimes W$ is called decomposable if it is of the form $v\otimes w$ for some $v\in V,w\in W$ . If it cannot be put in this form it is called indecomposable.

Note that our basis vectors of $V\otimes W$ are all decomposable since they are products of basis vectors of $V$ and W. Linear combinations of these basis vectors however are in general indecomposable. If we think of an element of $V\otimes W$ as a dim $V$ by dim W matrix, with entries the coordinates with respect to our basis vectors for $V\otimes W$ , then for decomposable vectors we get a special class of matrices, those of rank one.

In the physics context, the language used is:

Definition (Entanglement)

An indecomposable state in the tensor product state space ${\mathcal{H}}_T={\mathcal{H}}_1\otimes {\mathcal{H}}_2$ is called an entangled state.

The phenomenon of entanglement is responsible for some of the most surprising and subtle aspects of quantum mechanical systems. The Einstein-Podolsky-Rosen paradox concerns the behavior of an entangled state of two quantum systems, when one moves them far apart. Then performing a measurement on one system can give one information about what will happen if one performs a measurement on the far removed system, introducing a sort of unexpected apparent non-locality.

Measurement theory itself involves crucially an entanglement between the state of a system being measured, thought of as in a state space ${\mathcal{H}}_{system}.$ , and the state of the measurement apparatus, thought of as lying in a state space ${\mathcal{H}}_{apparatus}$ The laws of quantum mechanics presumably apply to the total system ${\mathcal{H}}_{system}\otimes {\mathcal{H}}_{apparatus},$ , with the counter-intuitive nature of measurements appearing due to this decomposition of the world into two entangled parts: the one under study, and a much larger for which only an approximate description in classical terms is possible. For much more about this, a recommended reading is chapter 2 of [75].

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原书 PDF · 印刷页 109、110、111、112、113、114、115、116、117、118、119、120

来源版本：2025-10-20

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