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OPTIMAL FIDELITY OF TELEPORTATION USING TWO-MODE GAUSSIAN STATES IN A THERMAL BATH AS A RESOURCE

ALEXEI ZUBAREV1,3,a, MARINA CUZMINSCHI2,3,b, AURELIAN ISAR2,3,4,c 1National Institute for Laser, Plasma and Radiation Physics, Magurele, Romania 2"Horia Hulubei" National Institute for Physics and Nuclear Engineering, Magurele, Romania

3Faculty of Physics, University of Bucharest, Romania 4Academy of Romanian Scientists, 54 Splaiul Independentei, Bucharest, Romania

E-mail a: E-mail b:

E-mail c:

Received May 9, 2019

Abstract. We describe the time evolution of the fidelity of teleportation and logarithmic negativity in a system composed of two coupled bosonic modes in contact with a thermal bath, in the framework of the theory of open systems based on completely positive quantum dynamical semigroups. We take two-mode Gaussian entangled states as initial resource states for the quantum teleportation of a coherent state. The evolution of the system is described in terms of covariance matrices. We consider both resonant and non-resonant cases and show that the optimal fidelity of teleportation decreases with the increase of the average number of thermal photons, therefore the squeezed vacuum state is the best resource state for quantum teleportation. We estimate also the optimal values of the squeezing and coupling parameters to obtain the maximal values of the fidelity of teleportation.

Key words: Fidelity of teleportation, entanglement, Gaussian states, open quantum systems.

1. INTRODUCTION

Improvement of quantum teleportation efficiency represents an important goal of quantum information processing. Quantum teleportation is at the basis of quantum secure communication [1], development of the quantum internet [2], and distributed quantum computing [3]. Quantum teleportation implies the transfer of the quantum state of a particle to another particle situated in another place. For a successful teleportation the information about the quantum state is transmitted via dual classical and quantum EPR communication [4]. For the teleportation to occur, the two parties between which the state is teleported should share an entangled state.

The first quantum teleportation protocol was proposed by Bennet [4] for a discrete variable system. After that followed a lot of theoretical and experimental investigations for discrete variable quantum teleportation [5, 6]. Later, the teleportation was developed also for continuous variable cases by Vaidman [7], and Braunstein

Romanian Journal of Physics 64, 108 (2019)

Article no. 108

Alexei Zubarev, Marina Cuzminschi, Aurelian Isar

2

and Kimble [8]. Experimental manifestation of quantum teleportation of coherent states using a system of continuous variables was accomplished by Furusawa [9], and teleportation of a squeezed thermal state was realised by Takei [10]. Quantum teleportation using, as a resource, entangled Gaussian states was obtained using diverse setups of atomic ensembles [11] and quantum optics devices [12?14]. Private communication over Gaussian channels was investigated by Laurenza [15].

The success of a quantum teleportation protocol can be estimated by evaluating the fidelity of teleportation, which is defined as the degree of similarity between the initial and teleported states [16, 17]. It can be much influenced by the dissipative effects that occur due to the interaction with the environment [18]. To reduce dissipation influence on the quantum teleportation protocols, special techniques are implemented. As example graphene coating of ion traps significantly reduces dissipation role [19], due to the excellent electron mobility of this material [20, 21]. The effects of noise in Gaussian channels on the fidelity of teleportation was investigated by Olivares [22], where there were found the squeezed states for which the teleportation is optimal in such conditions. Special interest present cases when the fidelity is larger than 0.5 (classical limit) [23] and 2/3 (no- cloning limit) [24]. Fidelity larger than 2/3 is one of the necessary conditions for secure quantum teleportation [25], and an important research goal is to find the optimal fidelity of teleportation for different setups [26, 27]. The second necessary condition is the two-way steering [25, 28]. Experimentally it was achieved a fidelity of teleportation up to 0.83 [11].

In a previous paper we investigated, in the framework of the theory of open systems based on completely positive quantum dynamical semigroups, the Markovian dynamics of the Uhlmann fidelity for a system consisting of two non-coupled bosonic modes embedded in a thermal bath [29]. In Ref. [30] it was studied the broadcasting of entanglement using telecloning of entanglement. Employing the same formalism of open quantum systems we described for a similar system the dynamics of quantum steering and quantum interferometric power [31?33]. In this paper we evaluate the fidelity of teleportation using as a resource the Gaussian state of two coupled bosonic modes placed in a thermal bath, and determine the system properties that ensure the best fidelity values. The paper is organised as follows. In Sec. 2 it is described the general formalism of quantum open systems based on the Gorini-KossakowskiLindblad-Sudarshan master equation. In Sec. 3 we provide the expressions of the fidelity of teleportation using a two-mode Gaussian state as a resource, and of the logarithmic negativity, which quantifies the degree of entanglement. In Sec. 4 we describe the evolution in time of the fidelity of teleportation, as a function of environment temperature, squeezing and coupling between the two modes. Conclusions are presented in Sec. 5.

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Quantum fidelity of two-mode Gaussian states in a thermal reservoir

Article no. 108

2. MASTER EQUATION FOR TWO BOSONIC MODES IN A THERMAL BATH

We investigate the evolution of a system composed of two coupled bosonic modes interacting with a thermal bath, in the framework of the theory of open systems based on completely positive quantum dynamical semigroups. We use the Gorini-Kossakowski-Lindblad-Sudarshan quantum Markovian master equation, which for a density operator (t) has the form:

d(t) i

1

= - [H, (t)] +

dt

2

(2Vj(t)Vj - {(t), VjVj}+).

(1)

j

H denotes the Hamiltonian of the two coupled bosonic modes of equal mass m with

frequencies 1 and 2 and coupling q:

H

=

1 2m

(p2x

+

p2y )

+

m 2

(12x2

+ 22y2)

+

qxy.

(2)

x, y and px, py denote the coordinates, respectively the momenta of the two modes, and Lindblad operators Vj describe their interaction with the environment.

The temporal evolution of the covariance matrix of the considered system is

given by the following equation [34, 35]:

d(t) = Z(t) + (t)ZT + 2D,

(3)

dt

where ( denotes the dissipation coefficient)

- 1/m 0

0

Z

=

-m12 0

- 0

-q 0

-

1/m

(4)

-q

0 -m22 -

and the matrix of diffusion coefficients is (we set = 1):

2m1

0

0

0

D=

0 0

m1 2

0

0

2m2

0 0

.

(5)

0

0

0

m2 2

The asymptotic covariance matrix is obtained from the equation

Z() + ()ZT = -2D,

(6)

and the solution of Eq. (3) is given by [34, 35]:

(t) = S(t)[(0) - ()]ST(t) + (),

(7)

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Article no. 108

Alexei Zubarev, Marina Cuzminschi, Aurelian Isar

4

3. DYNAMICS OF FIDELITY OF TELEPORTATION AND ENTANGLEMENT

3.1. QUANTUM FIDELITY OF TELEPORTATION

To quantify the similarity of the teleported state with the received state it is

used the fidelity of teleportation. The fidelity between two states 1 and 2 is given

by:

2

F (1, 2) = Tr( 21 2) .

(8)

Fidelity takes values between 0, when the two states are orthogonal, and 1, when they

are identical. The state that we aim to teleport is a coherent state with the covariance

matrix [36]:

in =

1 2

0

0

1 2

.

(9)

As a resource we use a two-mode Gaussian state with the covariance matrix

given by Eq. (7), which can also be expressed as:

(t) =

AC CT B

,

(10)

where A =

A11 A12 A21 A22

,B=

the received state is given by

B11 B12 B21 B22

, and C =

C11 C12 C21 C22

. Then,

out =

1 2

+

A11

+

B11

-

2C11

C21 + B12 - A12 - C12

C21 + B12 - A12 - C12

1 2

+

A22

+

B22

+

2C22

.

(11)

The fidelity of teleportation has the expression [18, 37]

1

F=

,

(12)

P Q - R2

where

P

=

out11

+

1 2

,

Q

=

out22

+

1 2

,

and

R

=

out21

=

out12 ,

and

can

also

be

written in the following form:

1

F=

.

(13)

det

out

+

1 2

tr

out

+

1 4

3.2. ENTANGLEMENT

For a state to be teleported, the resource states need to be entangled. As the

measure of entanglement of the two-mode Gaussian states we use logarithmic nega-

tivity, defined as [38]

1

E(t) = max{0, - 2 log2[4f (t)]},

(14)

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Quantum fidelity of two-mode Gaussian states in a thermal reservoir

Article no. 108

where

1 f (t) = (det A + det B) - det C

2

1

1

2

2

- (det A + det B) - det C - det (t) . (15)

2

4. RESULTS AND DISCUSSIONS

4.1. RESONANT CASE

First we consider that the two modes are resonant. In this simple case the expression of the asymptotic covariance matrix is given by [38]

2L2 - q2

()

=

coth(

1 2kT

)

4(L2 - q2)

q2 -Lq

-Lq

q2 2L2 + (2 - 2)q2

-Lq q(L - q2)

-Lq

-Lq 2L2 - q2

q2

-Lq

q(L - q2) q2

,

2L2 + (2 - 2)q2

(16) where 1 = 2 = 1 and L = 2 + 1 (in the following we take m = 1, k = 1, = 0.01).

As an initial state of the two modes we take a squeezed thermal state, with the

covariance matrix given by

a 0 c 0

(0)

=

0 c

a 0

0 b

-c 0

,

(17)

0 -c 0 b

where

a

=

n1

cosh2

r

+

n2

sinh2

r

+

1 2

cosh

2r,

b

=

n1

sinh2

r

+

n2

cosh2

r

+

1 2

cosh

2r,

(18)

1 c = 2 (n1 + n2 + 1) sinh 2r.

Here, n1 and n2 denote the average thermal photon numbers, corresponding to the two modes, and r is the squeezing parameter. The initial Gaussian state is entangled

if r > re, where

cosh2

re

=

(n1 + 1)(n2 + n1 + n2 + 1

1) .

(19)

For a state to be successfully teleported, the two bosonic modes have to be en-

tangled both initially and at the moment of measurement. The evolution of fidelity

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