130 7 Entangled multipartite systems
Let us, therefore, turn to the problem of classifying multipartite states
by finding equivalence classes under state transformations rather than via
their utility as resources for various tasks. One can find equivalence classes
under local unitary transformations (LUTs) of the statistical operator and
equivalently under (local) rotations of the Stokes tensor, as compound states
are equivalent in their nonlocal properties if they can be transformed into each
other by such operations. Each group, G, of transformations acts transitively
on an orbit O = G/S,whereS is the stabilizer subgroup of the orbit and is a
subgroup of G.
13
This requirement is equivalent to invariance under the choice
of local Hilbert space basis. Lower bounds on the number of parameters needed
to describe equivalence classes have been provided that show the insufficiency
of the total set of state descriptions of local systems for specifying the state
description of the compound system they comprise.
14
In particular, because states of N qubits are equivalent in entanglement
when they lie on the same orbit under LUTs of the statistical operator, each
such orbit corresponds to a single entanglement class with characteristic in-
variant quantities. The orbits have specific dimensionalities, dimO,givenby
the dimension of the stabilizer subgroups, dimS, of states on the orbit and
the dimension dimG of the group in question:
dimO =dimG−dimS , (7.29)
where for LUTs, G, being local, has elements of the form U
1
⊗U
2
⊗···⊗U
N
so
that each unitary transformation U
i
acts on a Hilbert space corresponding to
a component of the total system in the possession of a single party in its local
laboratory. The dimension of the orbit is just the number of real parameters
required to specific the location of a state in the orbit. The Hilbert space of
pure states of N parties, each in possession of a single qubit is, as we have
seen previously,
H
(N)
= C
2
⊗ C
2
⊗···⊗C
2
. (7.30)
Any pure state of the compound system is therefore described by 2(2
N
− 1)
real parameters, because there are 2
N
complex parameters and so 2
N+1
real
parameters describing any state on this space, of which normalization reduces
the number of real parameters by one, as does the freedom of global phase.
The number of parameters describing a state thus grows exponentially with
the number of components, N. Quantities invariant on an orbit thus specify
nonlocal equivalence classes of states, as discussed in the next section.
In contrast to situations described by LUTs, in LOCC each agent can per-
form generalized measurements on its local subsystem and classically com-
municate measurement outcomes to other agents. The other agents can then
choose their local transformations in way conditioned by these outcomes.
15
13
Consider S be the vector subspace kept fixed by a subgroup of elements of G
n
,
which is defined by Eq. 10.17, below; this subgroup is the stabilizer, S,ofS.
14
The extra parameters are known as “hidden nonlocalities;” see, for example [247].
15
Such a method is used, for example, in entanglement distillation; see Fig. 9.1.