238
that
there
is no
straightforward
generalization
of
the
Schmidt
decomposi-
tion
for
the
3-partite
case,
Indeed,
any
such
I'lj!),
and
IGHZ)
in
particular,
becomes a separable mix-
ture
under
one-party
reduction
, while
the
W
state
does
not
- hence
it
cannot
be
cast
in such a "Schmidt form" by a choice
of
local bases.
This
picture
unfolds into
an
even more complex one
with
the
increase of
the
number
of
subsystems.
Distinguishing
entangled
and
separable
states
of
bipartite
systems re-
mains
to
be
one
of
the
most
challenging
open
problems
in
quantum
infor-
mation
theory.
The
discrimination
among
various genres of
multipartite
entanglement
appears
a still
harder
task.
In
experiments, Bell inequalities
are
among
commonly used tools
to
detect
entanglement
both
in bi-
and
multipartite
settings. Yet Bell inequalities prove
to
be
a faulty
detector
in
general: on
the
one
hand
, even
in
the
bipartite
case,
there
are
entan-
gled
states
not
violating
the
inequalities,
on
the
other
hand
the
degree
of
violation
of
Bell inequalities is
not
sufficiently "monotone" when going
from
"weaker" entangled biseparable
states
to
fully entangled
multipartite
ones
6
,
3.
Consequently,
it
is
not
an
adequate
tool for
the
classification of
various
multipartite
entanglement
patt
erns.
An
alternative
technique, recognized
to
be
quite
fruitful
both
theoretically as well as experimentally, is
the
use
of
entanglement
witnesses
1
,5,6,7,8,9,1O.
While originally7
,9,
entanglement
witnesses were de-
fined
to
distinguish
bipartite
entanglement
from
sep
arability,
they
proved
to
be
equally useful for
the
discrimination
among
various
types
of
multi-
partite
entanglement
ll
,5,6,
or
even as simple
entanglement
measures
a
.
Ac-
tually,
th
ere is a
number
of
interesting
relations between specific witnesses
and
standard
entanglement
measures,
the
former providing various useful
bounds
for
the
latt
er
12.
In
the
present
paper,
we
are
going
to
focus on a special
method
of
constructing
entanglement
witness using a physical observable whose spec-
trum
is
partially
known
10.
In
[1]
we have discussed
an
application of such
a
method
to
construct
entanglement
witness
out
of
the
Hamiltonian
of
a I-dimensional isotropic Heisenberg model. We have
demonstrated
the
aStrictly
speaking
every
witn
ess W gives rise
to
a
pseudo-m
eas
ure
of
entanglement
Ew
(p)
= max{O, -
Tr(W
p)}
rather
than
a
true
measure
, for in
general
it
yields zero
value for
some
e
ntangled
states.