240
The
above observation indicates one way
to
progress
with
classification
of
mulipratitie
states:
different
entanglement
patterns
are
represented
by
equivalence classes
of
pure
states
related
by
local
unitary
actions. However,
already
in
the
case
of
a
tripartite
system,
even
in
its
simplest version con-
sisting
of
3
qubits,
the
classification
turns
out
to
be
much
more
complex
l5
,16
than
the
bipartite
one: local
unitary
equivalence yields
the
following
state
prototypes
with
Ai
~
0,
L:
A;
=
1,
a
:::;
e
:::;
1r.
A general scheme yielding similar
"normal
forms" for
arbitrary
multi-
partite
systems
is known
l9
, minimizing
the
number
of
summands
from a
product
orthonormal
basis,
yet
the
number
of
free
parameters
grows con-
siderably
with
the
number
of
parties.
This
substantial
increase
of
normal
form complexity suggests
the
re-
placement
of
local
unitary
equivalence
with
another,
more
crude
classifica-
tion
scheme.
It
can
be
realized
by
the
so-called
stochastic
LOCC
(SLOCC)
equivalence
20
where, roughly speaking, local
unitary
transformations
are re-
placed
with
local invertible ones.
It
is
the
SLOCC
classification
that
splits
the
set
of 3-qubit
pure
states
into
the
well-known six categories:
separable
states
represented
by
1000),3
types
of
biseparable
ones,
A-BC
represented
up
to
normalization
by
10)
Q9
(100)
+
111))
and
similarly for
AB-C
and
AC-B,
the
W-states
and
, finally,
the
GHZ
type
states
represented
by
(1).
However,
the
passage
to
a
4-qubit
system
reveals
again
the
whole con-
tinuum
of
SLOCC
equivalence classes,
just
like
in
the
LV classification
scheme: one
can
distinguish 9
structural
types
of
pure
state
entanglement
yet
the
complete
SLOCC
classification involves also 4
independent
complex
parameters21.
Natural
convex
structure
of
mixed
state
sets
yields still different
and
much
simpler classification scheme,
based
on
partitions
of
the
set
of
con-
stituents
of
the
system.
An
approach
of
such
type
has
been
developed in
detail
e.g. in
[22].
Suppose
that
the
parties
constituting
a
compound
quantum
system
are
labeled
by
AI"'"
An,
that
is H = HA,
...
A
n
'
Any
partition
of
this
set
of
labels, say
{A~,
.
..
, A;,,} U
...
U {Af, ... ,
A~k}'
corresponds
to
a specific
"coarse-grained" way
of
looking
at
the
entire
system,
i.e.
here
a
k-partite
one. A
pure
state
I'lj!)
E H is called k-separable iff
there