Displacement gain dependent fidelity in quantum teleportation using entangled two-mode squeezed light

Opt Quant ElectronDOI 10.1007/s11082-013-9843-5

Displacement gain dependent fidelity in quantumteleportation using entangled two-mode squeezed light

Dilip Kumar Giri · Ravindra Pratap Singh ·Abir Bandyopadhyay

Received: 17 June 2013 / Accepted: 16 November 2013© Springer Science+Business Media New York 2013

Abstract We study a scheme for quantum teleportation of a single-mode squeezed coherentstate using entangled two-mode squeezed vacuum. We establish the analytic expression ofdisplacement gain dependent fidelity in terms of the squeezing coherent parameter r andquantum channel parameter p. The dependence of the optimum displacement gain for tele-porting a squeezed coherent state upon the EPR entanglement is discussed. It shows that thefidelity of teleportation can be improved by tuning the displacement gain. We find that thefidelity increases with the increase of EPR parameter, while it decreases with the increaseof the squeezing coherent parameter of the signal. We get infinite squeezing as a resourceis required for an ideal and perfect teleportation of unknown input states. We show thatthe nonclassical properties of an unknown state to be teleported can be preserved in theteleportation.

Keywords Quantum teleportation · Entangled states · Fidelity · Displacement gain ·Squeezed states

PACS: 03.67.Hk, 03.67.Lx, 03.65Ud

1 Introduction

Quantum entanglement is one of unique characteristic in quantum mechanics. It is consideredas the fundamental resource of quantum information processing such as quantum teleportation

Sindri College is a Constituent Unit of Vinoba Bhave University, Hazaribag.

D. K. Giri (B)Department of Physics, Sindri College, Sindri, P.O. Sindri, Dhanbad 828 122, Indiae-mail: [email protected]

D. K. Giri · R. P. Singh · A. BandyopadhyayTheoretical Physics Division, Quantum Optics and Quantum Information Group,Physical Research Laboratory, Navrangpura, Ahmedabad 380 009, India

A. BandyopadhyayHooghly Engineering and Technology College, Hooghly 712103, India

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D. K. Giri et al.

(Bennett et al. 1993; Vaidman 1994; Braunstein and Kimble 1998; Ralph and Lam 1998;Furusawa et al. 1998; Bouwmeester et al. 1997), quantum computation (Lloyd and Braunstein1999), quantum cryptography (Ralph 1999), quantum cloning (Cerf et al. 2000) and etc. In thisquantum teleportation, first proposed by Bennett et al. (1993) for systems of discrete variablesand later on proposed by others Vaidman (1994); Braunstein and Kimble (1998); Ralph andLam (1998); Furusawa et al. (1998); Bouwmeester et al. (1997) for continuous variables (CV)using Einstein–Podolsky–Rosen (EPR) entangled state (Einstein et al. 1935) and becomeindispensable for theoretical and experimental quantum communication and informationprocessing (Furusawa et al. 1998; Bouwmeester et al. 1997; Milburn and Braunstein 1999;Boschi et al. 1998; Braunstein and Pati 2003). It is a technique that can transmit an unknownquantum state from a sender to a receiver at a distant location via a secret quantum channeland a public classical one. Furthermore, distributed entangled states make it possible to sendan unknown state for a long distance. The success of the communication of information canbe characterized by the teleportation fidelity. When the output state is exactly the same as theinput state, the fidelity is equal to unity. Horodecki et al. (1999) proved the relation betweenthe optimal fidelity of teleportation and the maximal singlet fraction of the quantum channel.The quantum teleportation network has been studied by Loock and Braunstein (2000) usingmultipartite entanglement for CV. Takei et al. (2005) has established an operational methodof evaluation for quantum teleportation of a squeezed state using fidelity and discussed theclassical limit of the state. Anno et al. (2007) has investigated continuous variable quantumteleportation using non-Gaussian states of the radiation field as entangled resources andshowed that optimized squeezed Bell-like resources yield a remarkable improvement inthe fidelity of teleportation both for coherent and nonclassical input states. The optimalfidelity of teleportation of coherent states and entanglement has been studied by Mari andVitali (2008). Owari et al. (2008) has established the fidelity benchmarks for the quantumstorage and teleportation of squeezed states of continuous variable systems. Recently, Wagnerand Clemens (2010) has discussed the fidelity of quantum teleportation based on spatiallyand temporally resolved spontaneous emission. More recently, the dependence of fidelityin teleportation on the squeezing parameter of single-mode squeezed coherent states hasbeen discussed by Jing-Tao et al. (2011) and Irfan et al. (2011) has proposed a scheme forteleportation of bipartite entangled state of the two cavity modes.

In this paper, we investigate the proposed teleportation scheme of radiation field via entan-gled two-mode squeezed vacuum state. The paper is organized as follows: Section 2 givesthe description of the proposed teleportation scheme. We establish the analytic expressionof displacement gain dependent fidelity in terms of the squeezing parameter r and quantumchannel parameter p in Sect. 3. The dependence of the optimum displacement gain for tele-porting a squeezed coherent state upon the EPR entanglement is also incorporated in thissection. Finally, we conclude this paper in Sect. 4.

2 Quantum teleportation of a squeezed coherent state

We concentrate our attention to study quantum teleportation fidelity with entangled two-mode squeezed vacuum state. The squeezing in two-mode quadratures may be detected byparametric down-conversion process (Ou et al. 1992) in nonlinear optical media when thehigher-order effective nonlinear susceptibilities are taken into account.

Consider two-mode squeezed vacuum state of an electromagnetic field in the proposedteleportation scheme in which squeezed coherent state as an input one to be teleported bythis scheme.

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Displacement gain dependent fidelity in quantum teleportation

Fig. 1 Schematic representation for quantum teleportation of a squeezed coherent state

Figure 1 shows a simplified schematic representation for teleportation of an unknownquantum state which is characterized by a pure input state ρin of the squeezed coherent state.Consider that the sender Alice wants to teleport the quantum state A1 to the Bob. In the firststage, Alice prepares the continuous-variables (CV) entangled EPR pairs beam A4 and A5by combining two squeezed vacuum state mode A2 with mode A3 on 50–50 beam splitterBS1, where A2 is the ‘position’ X = 1

2√

2

∑2j=1 (A j + A†

j ) and A3 is the ‘momentum’

P = 12i

√2

∑2j=1 (A j − A†

j ) quadratures in two-mode squeezed state respectively. Thusthe applied beam splitter to a position-squeezed and a momentum squeezed vacuum mode,yields

X4 = 1√2

e−p X (0)2 + 1√

2ep X (0)

3 , P4 = 1√2

ep P(0)2 + 1√

2e−p P(0)

3 (1)

X5 = 1√2

e−p X (0)2 − 1√

2ep X (0)

3 , P5 = 1√2

ep P(0)2 − 1√

2e−p P(0)

3 (2)

where

X1 =e−r X (0)1 , P1 =er P(0)

1 ,

X2 =e−p X (0)2 , p2 =ep P(0)

2 ,

X3 =ep X (0)3 , P3 =e−p P(0)

3

⎫⎪⎪⎬

⎪⎪⎭

and

⟨X (0)

1

⟩=

⟨P(0)

1

⟩= input state (coherent state),

⟨X (0)

2

⟩=

⟨P(0)

2

⟩=

⟨X (0)

3

⟩=

⟨P(0)

3

⟩=0(vacuum state)

⎫⎬

⎭

(3)

where superscript ‘(0)’ denotes initial vacuum state and r , p are the squeezed coherentparameter and squeezed channel parameter. The output modes A4 and A5 are now entangled toa finite degree in two-mode squeezed vacuum state. In the limit of infinite squeezing,p → ∞,the individual output modes become infinitely noisy, but the EPR correlations between thembecome ideal: (X5 + X4) → 0, (P5 − P4) → 0.

Now one of the EPR beams A4 is sent to Alice (the sender) and the other A5 is to Bob(the receiver). Note that an input state is unknown to both Alice and Bob in an ideal case.

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Alice mode (EPR beam A4) is coupled with the unknown input squeezed coherent beam at a50-50 beam splitter BS2and measures the certain classical values in terms of the real part of‘position’ quadrature X6 of A6and the imaginary part of ‘momentum’ quadrature P7 of A7

by two homodyne detectors Ax and Ap respectively and yields the quadratures of the outputmodes A6 and A7 as

X6 = 1√2(X1 + X4) and P7 = 1√

2(P1 − P4) (4)

However, all information about the observables corresponding to the imaginary part of ‘posi-tion’ quadrature X6 of A6 and the real part of ‘momentum’ quadrature P7 of A7 are lost(Braunstein and Kimble 1998) i.e.

P6 = 1√2(P1 + P4) = 0 = X7 = 1√

2(X1 − X4) (5)

The output measured results of modes A6 and A7 are transmitted to Bob through classicalchannels. After receiving the classical information from Alice, Bob reconstructs the teleported

output state by performing a phase-space displacement(√

2X6 + i√

2P7

)on the mode A5

beam. His displacement process is based on the classical information from Alice.Hence Bob displaces his mode correspondingly,

X8 = X5 + gx√

2X6,

P8 = P5 + gp√

2P7

}

(6)

where the gx and gp are normalized displacement gain for the transformation from classicalchannel to output field (Furusawa et al. 1998) and defined (Takei et al. 2005) as

andgx = 〈X8〉

/〈X1〉

gp = 〈P8〉/

〈P1〉

⎫⎬

⎭(7)

Using Eqs. (2), (4) and (6) for an arbitrary gain (Loock and Braunstein 2000), then thequadratures of the output state can be written as

X8 = gx X1 + (gx + 1)√2

e−p X (0)2 + (gx − 1)√

2ep X (0)

3 (8)

and

P8 = gp P1 − (gp − 1)√2

ep P(0)2 − (gp + 1)√

2e−p P(0)

3 (9)

Assuming that the entanglement used in the teleportation protocol is produced by twoequally amplitude squeezed beams, then the quadrature variances of the output state are

(�X8)2 = g2

x X21 + (gx + 1)2

8e−2p + (gx − 1)2

8e2p (10)

and

(�P8)2 = g2

p P21 + (gp − 1)2

8e2p + (gp + 1)2

8e−2p (11)

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3 Measurement of fidelity in teleportation

The success or failure of quantum teleportation can be characterized by the teleportationfidelity. It is defined by the overlap between the input state and the output state (Furusawa etal. 1998). To evaluate a quantitative measure of the quality of the output/teleported state, wenote that the strongest measure of fidelity of a teleported state relative to the input/originalstate is given by (Jozsa 1994), based on Uhlmann’s transition probability (Uhlmann 1975).The fidelity of a large class of single mode gaussian states was obtained in Scutaru (1998).It is an important quantity in describing the transmission of quantum information through aquantum channel.

Consider two quantum states ρin and ρout , the input-output fidelity ‘F ′ is defined (Nhaand Carmichael 2005; Olivares et al. 2006; Yukawa et al. 2008) for pure state as,

F(ρin, ρout ) =(

T r√√

ρinρout√

ρin

)2 ∼= T r(ρinρout )

=[

2√� + δ − √

δexp

{−βT (�1 + �8)

−1 β}]

(12)

where ρin and ρout indicate the input and output density operators and Γi & αi (i = 1, 8) arerespectively the covariance matrices and mean amplitudes of the modes A1 & A8 and

� = det(�1 + �8)

δ = (det �1 − 1)(det �8 − 1)

β = α8 − α1

⎫⎬

⎭(13)

3.1 Effect on fidelity with displacement gain factor

Using Eq. (3) we have covariance matrix (Tan 1999) and the mean values (Feng and Yi Zhang2009) of the original mode A1, as

�1 =(

e−2r 00 e2r

)

(14)

and

α1 =(

X1

P1

)

(15)

If gx = gp = g and introducing new parameter g ± (≡ g ± 1) and using Eqs. (10) and (11)we get the teleported state for mode A8 as

Γ8 =⎛

⎝ g2e−2r + (g+)2

2 e−2p + (g-)2

2 e2p 0

0 g2e2r + (g+)2

2 e−2p + (g-)2

2 e2p

⎞

⎠ (16)

and

α8 =(

X1

P1

)

(17)

Equation (16) is a positive definite and fulfill the uncertainty relations between canonical oper-ators and that impose a constraint on the covariance matrix, corresponding to the inequalityΓ8 + i ≥ 0, to ensure its correspondence to a physical state (Simon et al. 1987). Where

is the element of the symplectic matrix.

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Fig. 2 Dependence of fidelity in teleportation on g(r = 0) with p = 0, 1, 2, 3, 4, 5


Using Eqs. (14)–(17) in Eq. (13), we obtain

Δ = 14 [e4p(g−)4 + e−4p(g+)4 + 4e2p−2r (g+)2(g−)2

+ 2(g+)2(g2+){e−2p−2r + e−2p+2r

} + (6 + 4g2 + 6g4)]δ = 0β = 0

⎫⎪⎪⎬

⎪⎪⎭

(18)

Putting the value of Eq. (18) into (12), we obtain the fidelity in terms of the squeezingparameter r , the quantum channel parameter p and the displacement gain factor g, as

F = 2√

14 [e4p(g−)4 + e−4p(g+)4 + 4e2p−2r (g+)2(g−)2

+2(g+)2(g2+){e−2p−2r + e−2p+2r

} + (6 + 4g2 + 6g4)]

(19)

Figures 2, 3, 4 and 5 show the gain dependence of the teleportation fidelity at different valuesof the squeezed channel parameter p.

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A comparative study of Figs. 2, 3, 4 and 5 show that at r = 0 i.e. at coherent state, thedisplacement gain dependent fidelity shifts to maximum and gets broader at g = 0. Thebroadening of the peak indicates that the fidelity is less sensitive to displacement gain tuningfor low values of p. Further, when increase the values of squeezing coherent parameter r , thefidelity gets lower peak and broader at g = 0. Moreover, the displacement gain dependentfidelity shifts to higher values of g and gets higher and sharper at g = 1 as the squeezedchannel parameter p increases. This observation is compatible with the result (Simon et al.1994). From the Figs. 2, 3, 4 and 5, we also observe that when higher the values of r , thefidelity decreases the sharp peak but maintain the sensitiveness at g = 1 as the p increases. Itsuggests that for large squeezing of the entanglement resource, the unity displacement gainis the best for the teleportation of unknown input states. Hence, we infer that the fidelity ofteleportation can be improved by tuning the displacement gain.

In order to achieve optimum displacement gain for ideal teleportation of a coherent-state,we derive the Eq. (19) for satisfying (d F/dg) = 0. We plot a graph between the optimizeddisplacement gain (g) versus entangled quantum channel parameter (p) in Fig. 6.

The Fig. 6 shows that the displacement gain factor increases initially, then saturates atg = 1 with the increase of quantum channel parameter p. It indicates that the normalizeddisplacement gain depends on entangled quantum channel parameter p but no matter how the

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Fig. 6 Dependence of optimized displacement gain (g) in teleportation on p with r = 0, 1, 2

large p is. These results suggest that for better fidelity in teleportation, the optimized displace-ment gain can be varied from 0 to 1 in public channel. Hence, the optimized displacementgain dependence of the fidelity is suitable for small squeezing of the entanglement resource.

Now, when the displacement gains are adjusted to unity (gx = gp = g = 1) and usingEqs. (8) and (9) then we obtain quantum output (ρout ) of mode A8 that Alice finally wantsto teleport to Bob as

X8 = X1 + √2e−p X (0)

2

P8 = P1 − √2e−p P(0)

3

}

(20)

If lim p→∞ X8 = X1; lim p→∞ P8 = P1 i.e. infinite squeezing, then the Eq. (20) corre-sponds to ideal quantum teleportation i.e. the teleported mode becomes the input mode.Using Eq. (20) we get teleported state of mode A8 as

�8 =(

e−2r + 2e−2p 00 e2r + 2e−2p

)

(21)

Using Eqs. (14),(15),(17) and (21) in Eq. (13), we obtain

Δ = 4(1 + 2e−2p cosh 2r + e−4p

)

δ = 0β = 0

⎫⎬

⎭(22)

Substituting the value of Eq. (22) into (12), we obtain the fidelity in terms of the squeezingparameter r and the quantum channel parameter p as

F = 1√(

1 + 2e−2p cosh 2r + e−4p) (23)

We plot a graph between the fidelity (F) and quantum channel parameter (p) with dif-ferent values of squeezing coherent parameter (r) in Fig. 7. The curve evidently shows thatthe fidelity increases and approaches to 1 with the increase of the quantum channel para-meter p. Further, when higher the values of r , the fidelity decreases and becomes zero thatmeans lowering the classical limitation of the fidelity. It indicates that the teleported stateis orthogonal to the input state (Takei et al. 2005). The results of Fig. 7 suggest that once

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Fig. 7 Dependence of fidelity in teleportation on p with r = 0, 1, 2, 3

Fig. 8 Dependence of fidelity in teleportation on r with p = 0, 1, 2, 3

the better EPR pair entangled quantum channel parameter is built then the fidelity increasesand achieve an ideal condition (F = 1) to be teleported unknown squeezed coherent state. Itinfers that the nonclassical properties of an unknown state to be teleported can be preservedin the teleportation. It also infers that for an ideal and perfect teleportation infinite squeezingas a resource is required.

Figure 8 shows the dependence of fidelity with squeezing coherent parameter havingdifferent values of p. The steady fall of the curve shows that the fidelity decreases with theincrease of the squeezing parameter r . At p = 0 (without entanglement), the fidelity F = 0.5corresponds to classical teleportation. We observe that the coherent state (r = 0) respondsoptimum fidelity (F = 1) at the highest value of p than the lower one. It confirms that thecoherent state is the best quantum signal for quantum teleportation after the quantum channelis built. We find that the fidelity depends on the squeezing coherent parameter as well as onthe entangled channel parameter.

4 Conclusions

We have established the analytical expression of displacement gain dependent fidelity inteleportation using entangled two-mode squeezed vacuum.

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We have observed that when the entangled quantum channel parameter p increases, thedisplacement gain dependent fidelity shifts to maximum and sharper at g = 1. It is concludedthat for large squeezing the unity displacement gain provides the high sensitive fidelity whichis the best for the teleportation of unknown input state. It is found that the fidelity increaseswith the increase of the quantum channel parameter p, while the fidelity decreases with theincrease of the squeezing parameter r of the squeezed coherent state. It is also observedthat normalized displacement gain depends on entangled quantum channel parameter p butno matter how the large p. Hence, the optimization of the gain dependence of the fidelityis suitable for small squeezing of the entangled resource. It is inferred that the fidelity ofteleportation can be improved by tuning the displacement gain from 0 to 1 in public channel.

We have concluded that the fidelity (F = 0.5) corresponds to classical teleportationwithout entanglement (p = 0). Further, it is shown that the coherent state (r = 0) is thebest quantum signal (F = 1) for quantum teleportation once the better quantum channelis provided. We have analyzed that the nonclassicality of fidelity decreases and becomeszero when higher the values of squeezing parameter r but once the better EPR pair entangledquantum channel parameter is built then the fidelity increases and approaches to 1 and achievean ideal condition to be teleported unknown coherent state. It is confirmed that the nonclassicalproperties of an unknown state to be teleported can be preserved in the teleportation.

These results may pave the way for designing teleporting systems with nonclassical statesto improve the quality of the teleported quantum state and may be expected for fabricationof future teleporting devices.

Acknowledgments We acknowledge to colleagues of ‘Theoretical Physics Division’ at PRL especially Mr.Shashi Prabhakar for their continuous support during the summer research project work. One of the authors(DKG) is supported by IAS-INSA-NASI, ‘Summer Research Fellowships’, India. We would like to thank thereferees for his comments and valuable suggestions.

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Displacement gain dependent fidelity in quantum teleportation using entangled two-mode squeezed light