Note that in the fermionic case, forσa transposition interchanging two
particles,πacts on the factorH⊗Hby interchanging vectors, taking
w⊗w∈H⊗Hto itself for any vectorw∈H. Antisymmetry requires thatπtake this state to
its negative, so the state cannot be non-zero. As a result, one cannot have non-
zero states inH⊗ndescribing two identical particles in the same statew∈ H,
a fact that is known as the “Pauli principle”.
While the symmetry or antisymmetry of states of multiple identical particles
is a separate axiom when such particles are described in this way as tensor
products, we will see later on (chapter 36) that this phenomenon instead finds
a natural explanation when particles are described in terms of quantum fields.
9.3 Indecomposable vectors and entanglement
If one is given a functionfon a spaceXand a functiongon a spaceY, a
product functionfgon the product spaceX×Ycan be defined by taking (for
x∈X,y∈Y)
(fg)(x,y) =f(x)g(y)
However, most functions onX×Yare not decomposable in this manner. Sim-
ilarly, for a tensor product of vector spaces:
Definition(Decomposable and indecomposable vectors).A vector inV⊗W
is called decomposable if it is of the formv⊗wfor somev∈V,w∈W. If it
cannot be put in this form it is called indecomposable.
Note that our basis vectors ofV⊗W are all decomposable since they are
products of basis vectors ofVandW. Linear combinations of these basis vectors
however are in general indecomposable. If we think of an element ofV⊗W
as a dimV by dimW matrix, with entries the coordinates with respect to our
basis vectors forV⊗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 spaceHT=H 1 ⊗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 spaceHsystem, and