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