Upper bound on singlet fraction of mixed entangled two qubitstates

Transcription

1 Upper bound on singlet raction o mixed entangled two qubitstates Satyabrata Adhikari Indian Institute o Technology Jodhpur, Rajasthan Collaborator: Dr. Atul Kumar

2 Deinitions A pure state ψ AB H H A B is separable i ψ AB = e A B where denotes the basis o e A respectively. and B H A and H B ρ A mixed state is entangled iit cannot be represented as ρ = p i ei i i i i, e,, p = 1, p i i i 0

3 Introduction Quantum entanglement has been used as an eicient resource or several quantum communication protocols. In general, i a state is maximally entangled then the optimal success o a communication protocol is a certainty. In an open system it is practically not possible to keep the state with cent percent purity. We have to deal with mixed entangled resources or quantum inormation processing.

4 Relation between teleportation and singlet raction A mixed two-qubit entangled state useul or teleportation i the singlet raction is greater than ½. Singlet raction: F( σ) = max ψ ψ ME ME σ ψme Teleportation idelity: T ( σ ) = F ( σ ) M. Horodecki, P. Horodecki, and R. Horodecki, Phys. Rev. A 60, 1888 (1999).

5 Result o Badziaget.al. Badzaigand Horodeckihave shown that there exist mixed states with idelity smaller than 1/, or which local trace preserving protocols exist that transorm this state into a state with idelity larger than 1/ without the help o classical communication. Is it possible to show that any entangled state is useul or teleportation? P. Badzia g, M. Horodecki, P. Horodecki, and R. Horodecki, Phys. Rev. A 6, (000).

6 Result o Verstraete and Verschelde Proved that the optimal trace preserving protocol or maximizing the singlet raction o a given state always belongs to a class o one-way communication (1-LOCC). Shown that any entangled two-qubit mixed state can be used as a resource or quantum teleportation using certain trace preserving local operations and classical communications. F. Verstraete and H. Verschelde, Phys. Rev. Lett. 90, (003)

7 A ilter is constructed in such a way so that the cost unction deined below is maximal. Cost unction (K): Result o Verstraete and Verschelde (1 p ) K = p ( ) AB AB F σ +, ( A I) σ( A I) + where σ =, p [( ) ( )] AB = Tr A I σ A I p AB The optimal ilter and singlet raction F can be ound by solving the convex semideinite program. + F. Verstraete and H. Verschelde, Phys. Rev. Lett. 90, (003)

8 Result o Verstraete and Verschelde Convex optimization problem: 1 F ( σ ) = Max [ Tr ( X σ Γ )] Subject to 0 X I + ψ ψ 4 I I X 4 4 X = ( A I ) ( A I ), and A represents the ilter. F. Verstraete and H. Verschelde, Phys. Rev. Lett. 90, (003)

9 Result Verstraete and Verschelde have shown that using trace preserving optimal local operations, the maximal achievable singlet raction F or the amily o states σ is F 1 1 ( σ ) = [1 + ]; 4(1 ) 3 3 F ( σ ) = ; σ = ψ ψ + (1 ) 01 01, ψ = ( ) F. Verstraete, and H. Verschelde, Phys. Rev. Lett. 90, (003)

10 Motivation

11 Theorem For any real n nmatrix C and a positive semideinite operator B, the ollowing inequality holds λ1 ( C ) Tr ( B ) Tr ( CB ) λn ( C ) Tr ( B ) C + C where C =, λn ( C ) is the nth eigenvalue o the matrix C and λ λ λ... λ. 1 3 T n Y. Fang, K. A. Loparo, and X. Feng, IEEE Transactions on Automatic Control 39, 489 (1994).

12 Upper bound on singlet raction I4 Γ For C = X and B = σ, we have I4 Γ I4 Γ I4 Γ λ1 ( X ) F = Tr[( X ) σ ] λ4( X ) + ψ ψ where X = ( A I ) ( A I ), and A represents the ilter. The upper bound depends on the state parameter and hence, must have a maximum achievable value or every particular value o the state parameter. This value would be provided by Dembo's bound.

13 Dembo sbound Theorem: For any n n positive semideinite operator R n with eigenvalues λ λ λ λ, Dembo sbound can be given by n n c + η c η 1 1 c + η c η n 1 n b b λn ( Rn ) + + b b Rn 1 b where R n =, is the lower bound on the minimal eigenvalue o R 1 1, T η n ( b ) c η is the upper bound on the maximal eigenvalue o R and bis an eigenvector o n 1 dimension n 1. A. Dembo, IEEE Trans. Inorm. Theory 34, 35 (1988).

14 Modiied upper bound on singlet raction Using Dembo sbound it can be easily shown that the upper bound on optimal singlet raction is I c η c η 4 Γ I ( -X ) 4 Γ 3 b, FD = + + b b X = T ( b ) c I where η is the upper bound on the maximal eigenvalue o ( 4 Γ -X ) 3 3.

15 Calculation o Dembo sbound The upper bound on singlet raction F D is F D 1 ( σ ) = ; 4(1 ) 3 3 FD ( σ ) = ; 3 where σ = ψ ψ + (1 ) 01 01, ψ = 1 ( ) I 1 F 0 0 4(1 F) 1 F X = 8(1 F) F (1 F) 4 Γ

16 Comparison o maximal singlet raction F and upper bound on singlet raction F D obtained using Dembo's bound

17 Local operations and classical communication Ournext taskis to ind a way by which we can obtain this bound experimentally i.e. would it be possible to increase the value o optimal singlet raction perorming local operations and classical communication on the iltered state i.e. can we achieve the upper bound o singlet raction given by Dembo's bound? Our results show that the bound is indeed achievable by perorming local operations and classical communication.

18 Singlet raction ater second iltering operation In order to enhance the value o optimal singlet raction F, we perorm another iltering operation on the iltered state such that the singlet raction o the output state can be given as F = p F ( σ ) + (1 p ) F ( σ ) opt where p is the success probability multiplied with the optimal singlet raction o the state coming out o the second ilter.

19 Singlet raction ater second iltering operation Deine 1 p = p AB where AB denotes the success probability o the irst ilter. Then or F( σ ) = ψ σ ψ, where σ =. p ( A I ) σ ( A I ) p AB + F opt can be re-expressed as opt AB F = (1 p ) F ( σ ) + tr[( A I) σ ( A I) ψ ψ ] +

20 Comparison between the singlet raction obtained ater the irst and second ilter For the state described by the density operator 1 σ = ψ ψ + (1 ) 01 01, ψ = ( ) 3 1 pab = + +, 8(1 ) 4(1 ) F ( σ ) = [1 + ]; 4(1 ) 3 3 A = (1 )

21 Optimal value o the probability For p to be high, pab must be minimized. The minimum value o p AB should be chosen in such a way that the value o singlet raction or the second ilter must not exceed Dembo's bound. The minimum value o would be p min AB D p AB F Tr [( A I) σ ( A I) ψ ψ ] = 1 F ( ρ ) p AB With this minimum value o, we have Fopt = FD This shows that one can achieve the maximum singlet raction equals to Dembo'sbound by using the iltering operations twice. T p AB

22 Illustration For the state 1 ρ = ψ ψ + (1 ) 01 01, ψ = ( ) The minimum success probability o the ilter is p min AB Hence the optimal singlet raction would be F = (1 )( ) opt = 4(1 )

23 Illustration In the above example, we have F opt = FD Applying the ilter twice will always result in achieving the upper bound on singlet raction or any two-qubit mixed density operator.

24 Comparison between the probabilities o success ater passing through the First Filter

25 Summary We have established a relation between Dembo'supper bound and singlet raction (and hence with teleportation idelity) o a mixed two-qubit entangled state. This relation is used to demonstrate that any two-qubit mixed entangled state can be used as a resource to achieve maximum possible teleportation idelity. we ound that the maximum idelity obtained earlier can be increased with additional local operations with certain non-zero probability.

26 THANK YOU