Geometric measure for entanglement in N-qudit pure states
Ali Saif M. Hassan
*
and Pramod S. Joag
†
Department of Physics, University of Pune, Pune 411007, India
Received 13 June 2009; published 1 October 2009
We present a multipartite entanglement measure for N-qudit pure states, using the norm of the correlation
tensor which occurs in the Bloch representation of the state. We compute this measure for an important class
of N-qutrit pure states, namely, general GHZ states. We prove that this measure possesses all the essential and
many desirable properties expected of a good entanglement measure, including monotonicity. We also discuss
the feasibility of the experimental evaluation of this measure for an N-qutrit system.
DOI: 10.1103/PhysRevA.80.042302 PACS numbers: 03.67.Mn, 03.65.Ca, 03.65.Ud
I. INTRODUCTION
Entanglement has proved to be a vital physical resource
for various kinds of quantum information processing, includ-
ing quantum state teleportation 1,2, cryptographic key dis-
tribution 3, classical communication over quantum chan-
nels 4 –6, quantum error correction 7, quantum
computational speedups 8, and distributed computation
9,10. Further, entanglement is expected to play a crucial
role in the many particle phenomena such as quantum phase
transitions, transfer of information across a spin chain
11,12. Therefore, quantification of entanglement of multi-
partite quantum states is fundamental to the whole field of
quantum information and in general, to the physics of mul-
ticomponent quantum systems. Whereas the entanglement in
pure bipartite states is well understood, classification of mul-
tipartite pure states and mixed states, according to the degree
and character of their entanglement, is still a matter of in-
tense research 13–15. Principal achievements are in the set-
ting of bipartite systems. Among these, one highlights Woot-
ter’s formula for the entanglement of formation of two qubit
mixed states 16, which still awaits a viable generalization
to multiqubit case. Others include corresponding results for
highly symmetric states 17–19.
Interest in entangled states of higher than two level sys-
tems comes from the foundations of quantum mechanics as
well as the development of new protocols in quantum com-
munication. For example, it has been shown that maximally
entangled states of two quantum systems in a high dimen-
sional Hilbert space, qudits, violate local realism more
strongly than qubits, and entangled qudits are less affected
by noise than entangled qubits 20,21. In quantum cryptog-
raphy 22, the use of entangled qutrits 22,23 or qudits
24,25 instead of qubits is more secure against eavesdrop-
ping attacks. Furthermore, the protocols for quantum telepor-
tation or for quantum cryptography work best with maxi-
mally entangled states. These facts motivate the development
of techniques to generate entangled states among quantum
systems in a higher dimensional Hilbert space with good
entanglement characteristics. Technical developments in this
direction have been made. For example, four polarized en-
tangled photons have been used to form two entangled
qutrits 26. Entangled qutrits with two photons using an
unbalanced three-arm fiber optic interferometer or photonic
orbital angular momentum have been demonstrated 27,28.
Time-bin entangled qudits of up to 11 dimensions from
pump pulses generated by a mode-locked laser have also
been reported 29. In short, quantifying the entanglement of
N-qudit system is of physical interest. The issue of entangle-
ment in multipartite and higher dimensional states is far
more complex. Notable achievements in this area include
applications of the relative entropy 30, negativity 31,
Schmidt measure 32 and the global entanglement measure
proposed by Meyer and Wallach 33.
A measure of entanglement is a function on the space of
states of a multipartite system which is invariant on indi-
vidual parts. Thus a complete characterization of entangle-
ment is the characterization of all such functions. Under the
most general local operations assisted by classical communi-
cation LOCC, entanglement is expected to decrease. A
measure of entanglement which decreases under LOCC is
called an entanglement monotone. On bipartite pure states
the sums of the k smallest eigenvalues of the reduced density
matrix are entanglement monotones. However, the number of
independent invariants i.e., the entanglement measures in-
creases exponentially as the number of particles N increases
and complete characterization rapidly becomes impractical.
A pragmatic approach would be to seek a measure which is
defined for any number of particles scalable, which is eas-
ily calculated and which provides physically relevant infor-
mation or equivalently, which passes the tests expected of a
good entanglement measure 13,14.
In this paper, we present a global entanglement measure
for N-qudit pure states which is scalable, which passes most
of the tests expected of a good measure and whose value for
a given system can be determined experimentally for a small
number say 2 or 3 qutrits, without having a detailed prior
knowledge of the state of the system see Sec. IV D. The
measure is based on the Bloch representation of multipartite
quantum states 34. The corresponding measure for N-qubit
pure states has already been obtained 35.
The paper is organized as follows. In Sec. II, we give the
Bloch representation of a N-qudit quantum state and define
our measure E
T
. In Sec. III, we compute E
T
for an important
class of N-qutrit states, namely, GHZ states. In Sec. IV , we
prove various properties of E
T
, including its monotonicity,
expected of a good entanglement measure. In Sec. IV C, we
*
alisaif@physics.unipune.ernet.in
†
pramod@physics.unipune.ernet.in
PHYSICAL REVIEW A 80, 042302 2009
1050-2947/2009/804/0423029 ©2009 The American Physical Society 042302-1