Entanglement Efficiencies in PT -Symmetric Quantum Mechanics
Christian Zielinski
1, ∗
and Qing-hai Wang
2, †
1
Institut für Theoretische Physik, Universität Heidelberg,
Philosophenweg 16, 69120 Heidelberg, Germany
2
Department of Physics, National University of Singapore,
117542, Singapore
(Dated: November 12, 2018)
The degree of entanglement is determined for an arbitrary state of a broad class of PT -symmetric
bipartite composite systems. Subsequently we quantify the rate with which entangled states are
generated and show that this rate can be characterized by a small set of parameters. These relations
allow one in principle to improve the ability of these systems to entangle states. It is also noticed
that many relations resemble corresponding ones in conventional quantum mechanics.
PACS numbers: 03.67.Bg, 03.65.Ud, 03.65.Ta
Keywords: entanglement generation, entanglement efficiency, entanglement entropy, PT -symmetric quantum
mechanics
I. INTRODUCTION
In conventional quantum mechanics, one demands that
the Hamiltonian H generating the time evolution has a
real spectrum and that the corresponding time evolution
operator U is unitary. These conditions are fulfilled if the
Hamiltonian is Hermitian, i.e. H = H
†
, which is usually
considered an axiom of quantum mechanics. However,
the condition of Hermiticity can be weakened. In the
class of so-called PT -symmetric Hamiltonians, one can
ensure real eigenvalues and a unitary time evolution even
for explicitly non-Hermitian Hamiltonians [2, 3]. A thor-
ough review of the foundations of PT -symmetric quan-
tum mechanics can be found in [1].
In the following we will investigate entanglement phe-
nomena in bipartite systems within this framework. An
entangled state is a quantum state where two or more
degrees of freedom are intertwined, so that they are not
independent anymore. In this context a couple of histori-
cal discussions took place and gave deep insights into the
nature of quantum mechanics, like the Einstein-Podolsky-
Rosen paradox [8]. These states have a wide range of
applications, for example in quantum information theory
and quantum computing. The question of entanglement
generation and entanglement efficiencies for conventional
quantum mechanics was addressed earlier in [7].
Using PT -symmetric quantum mechanics to describe
entanglement phenomena was also done in [13]. Espe-
cially for the case of bipartite systems, relations for the
degree of entanglement of given states and the entangle-
ment capability of certain systems could be found. Al-
though we confirmed many of these results, we have some
different and new findings. For a particular initial state,
as well as for general states, we give relations between
the efficiency of the system to generate entangled states
∗
Electronic address: zielinski@thphys.uni-heidelberg.de
†
Electronic address: phywq@nus.edu.sg
and the parameters of the Hamiltonian describing the
dynamics of the system.
We will first introduce a measure for entanglement and
a certain class of PT -symmetric Hamiltonians. We then
quantify the degree of entanglement of an arbitrary and
of a generalized Einstein-Podolsky-Rosen state. Subse-
quently, we are dealing with the question of entangle-
ment generation of a particular PT -symmetric state and
generalize for arbitrary states. The question we try to
answer is how to characterize the rate of entanglement
generation for this class of systems.
II. BIPARTITE SYSTEMS
A. A measure for entanglement
Let H
i=1,...,N
denote a set of Hilbert spaces. We call a
state of a composite system H = H
1
⊗···⊗H
N
entangled,
if there is no decomposition of the form |Ψi = |χ
1
i⊗···⊗
|χ
N
i with suitable |χ
i
i ∈ H
i
. In the following we restrict
ourselves to bipartite systems, i.e. N = 2.
A measure of entanglement are the entropies
E(Ψ) = −tr
1
(ρ
1
log
2
ρ
1
) = −tr
2
(ρ
2
log
2
ρ
2
) , (1)
where ρ
1
= tr
2
ρ and ρ
2
= tr
1
ρ are the reduced den-
sity matrices, ρ = |ΨihΨ| is the density matrix itself and
tr
i
denotes the partial trace over the i
th
subsystem [12].
Here E(Ψ) ∈ [0, 1] and E(Ψ) = 0 if and only if |Ψi ∈ H
is not entangled.
B. General entanglement content
Consider the Hamiltonian
H =
re
iΘ
s
s re
−iΘ
, r, s, Θ ∈ R, (2)
with s
2
> r
2
sin
2
Θ. Observe that H is in general not
represented by a Hermitian matrix, H
†
6= H. But it
arXiv:1106.3856v3 [hep-th] 15 May 2012