RESEARCH
Two and three particles entanglement; helicity and spin
with momentum
M. Aghaee
1
•
A. Rabeie
1
•
M. V. Takook
1
Received: 6 July 2017 / Accepted: 6 September 2017 / Published online: 21 September 2017
Ó The Author(s) 2017. This article is an open access publication
Abstract Construction of the reduced spin and helicity
density matrix for systems of two and three particles are
described by a wave packet with sharp and Gaussian
momentum distribution. The entropy for the spin and
helicity part of the systems is calculated from the viewpoint
of moving observers.
Keywords Entanglement Á Spin Á Helicity Á Entropy
Introduction
Special relativity effects on quantum entanglement and
quantum information are investigated by many authors.
Considerable efforts have also been expanded on the the-
oretical investigation of quantum information theory in
relativistic framework, which has gone beyond from pho-
tons to electrons, and from explicit examples calculated in
some specific cases to general framework formulated in
relativistic quantum mechanics and even relativistic
quantum field theory [1–31]. In practice, Lorentz trans-
formations in some special cases can change the entan-
glement of the spin and helicity of massive particles. Since
the helicity has an advantage in providing a smooth
transition to the massless case, it is the helicity rather than
spin that is more often under both theoretical consideration
and experimental detection in high energy physics. We
know the helicity states and the spin states can be the basis
of Hilbert space, they differ in the way of unitary trans-
formation under the action of Lorentz group. As a result,
the entanglement properties for helicity differs from spin
after we trace out the momentum degree of freedom. Thus,
for both theoretical completeness and possible implemen-
tation in high energy physics, it is intriguing and significant
to investigate quantum entanglement between helicity and
momentum in relativistic framework.
One of the early works in this area has considered a
massive spin 1/2 field and by calculating reduced spin and
helicity density matrix. It is shown that the resultant
helicity entropy is different from spin in general [32].
According to [32], the helicity and spin entropy for one
particle are S
ðhÞ
1
¼ 0:4917 and S
ðsÞ
1
¼ 1; respectively. This
paper is an effort for studying some special features of two
and three particles, spin and helicity entanglement in
inertial frames. We consider a quantum system of two and
three massive spin 1/2 particles that is observed by inertial
observers. The state of system for helicity state is supposed
to have an appropriate sharp momentum distribution and
for spin state the momentum distribution is supposed to be
Gaussian. We use the entropy as a measure for the helicity
entanglement as viewed in the boosted frame because the
entropy is one of the most important measure for entan-
glement. At the end, we have very interesting results.
Our paper is organized as follows: in Sects. 2 and 3 we
discuss entanglements for two and three particles’ spin and
helicity with momentum and calculating of spin and
helicity reduced density matrix and finally we obtained the
entropy of the states. In Sect. 4 we conclude our results.
& M. Aghaee
mohsenaghaee1388@gmail.com
A. Rabeie
rabeie@razi.ac.ir
M. V. Takook
takook@razi.ac.ir
1
Department of Physics, Razi University, Kermanshah, Iran
123
J Theor Appl Phys (2017) 11:263–268
https://doi.org/10.1007/s40094-017-0265-9
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