Bounds on mixedness and entanglement of quantum teleportation resources

Bounds on mixedness and entanglement of quantum teleportation resources

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Physics Letters A ••• (••••) •••–•••

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Bounds on mixedness and entanglement of quantum teleportation resources

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Department of Physics, Pondicherry University, Puducherry 605 014, India

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K.G. Paulson, S.V.M. Satyanarayana

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Article history: Received 18 July 2016 Received in revised form 3 February 2017 Accepted 6 February 2017 Available online xxxx Communicated by P.R. Holland Keywords: Quantum entanglement Quantum teleportation Concurrence Singlet fraction Von Neumann entropy Linear entropy

For a standard teleportation protocol, rank dependent upper bounds on Von Neumann entropy and linear entropy beyond which the states of respective ranks cease to be useful for quantum teleportation are derived analytically for a general d × d bipartite systems. For two qubit mixed states, we obtain rank dependent lower bound on concurrence below which the state is useless for quantum teleportation. For two qubit system, we construct mixed states of different ranks that exhibit theoretical rank dependent upper bounds on the measures of mixedness and lower bounds on the concurrence. © 2017 Published by Elsevier B.V.

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1. Introduction

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Entangled states are used as resource for quantum teleportation [1–4]. The success of quantum teleportation is quantified by teleportation fidelity. The strength of entanglement is proportional to the success of teleportation for pure entangled resources. Maximally entangled pure states give maximum teleportation fidelity of unity. On the other hand, for mixed states, success of teleportation depends, in addition to strength of entanglement, on mixedness of the state [5,6]. It is known that entanglement itself depends on mixedness of the state [8,7]. It was shown in [9] that, for mixed entangled states, there is an upper bound on mixedness above which the state is useless for quantum teleportation. Upper bounds on measures of mixedness, namely, Von Neumann entropy and linear entropy were obtained for a general bipartite d × d state. Mixed states can be classified according to their rank r, where the rank r varies between 2 and d2 for a bipartite d × d system. In the present work, we obtain rank dependent upper bounds on measures of mixedness, above which the states of respective ranks become useless for quantum teleportation in a standard teleportation protocol. In our previous work [5], we observed that mixedness and entanglement of a mixed state resource independently influence teleportation. For a state with a fixed value of mixedness, the state being entangled is only necessary for it to be a resource for quan-

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2. Upper bounds on measures of mixedness We consider two measures of mixedness of a state ρ , namely, von Neumann entropy defined as S (ρ ) = − T r (ρ ln ρ ) [12] and 2

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tum teleportation, but not sufficient. States with low mixedness and high entanglement turn out to be ideal resources for quantum teleportation. We argued based on the numerical work on a class of maximally entangled mixed states that there exists a rank dependent lower bound on a measure of entanglement such as concurrence, below which the states are useless for quantum teleportation. In this letter, we derive rank dependent lower bounds on concurrence for a bipartite 2 × 2 systems. It is know that Werner state, defined as a probabilistic mixture of maximally entangled pure state and maximally mixed separable state, exhibits highest mixedness for a given teleportation fidelity among all the two qubit mixed entangled states known in the literature. A pair of Werner states [10] is used to teleport unknown entangled quantum states. Werner state [11] is a rank 4 state. We observe that Werner state exhibits theoretical upper and lower bounds on measures of mixedness and entanglement respectively for rank 4 states. We construct mixed states of lower ranks that exhibit the respective rank dependent bounds obtained on measures of mixedness and entanglement for a given value of teleportation fidelity.

E-mail address: [email protected] (S.V.M. Satyanarayana). http://dx.doi.org/10.1016/j.physleta.2017.02.010 0375-9601/© 2017 Published by Elsevier B.V.

linear entropy given as S L = d2d−1 [1 − T r (ρ 2 )]. Further, the maximum achievable teleportation fidelity of a bipartite d × d system

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K.G. Paulson, S.V.M. Satyanarayana / Physics Letters A ••• (••••) •••–•••

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f d+1

in the standard teleportation [6] scheme is F = d+1 , where f is the singlet fraction of ρ given by f (ρ ) = maxψ ψ|ρ |ψ. Here the maximum is over all maximally entangled states. And maximum 2 fidelity achieved classically is F cl = d+ . This shows that the state 1 is useful for quantum teleportation if f >

1 d

(F (ρ ) > F cl ). Along the

lines of successful studies where for two qubit system, f > 12 is a sufficient condition for entanglement distillability of mixed states and their super activation [13] and is a condition for purification of a state using quantum privacy amplification procedure [14], we employ f > d1 as a condition failing which the state is useless for quantum teleportation in standard teleportation protocol. Let ρr denote a bipartite mixed state of d × d system whose rank is r, 2 ≤ r ≤ rmax = d2 . Firstly, we prove a theorem that gives rank dependent upper bounds on von Neumann entropy, above which the state is useless for quantum teleportation.

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Theorem. If Von Neumann entropy S (ρr ) of a given state ρr of rank r of √ d × d system exceeds ln rmax + (1 − √r1 ) ln √rr −1−1 ), then the state

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is not useful for quantum teleportation.

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max

max

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Proof. Consider ρr , a mixed state of rank r of d × d system (2 ≤ r ≤ rmax = d2 ). The singlet fraction f (ρr ) > d1 implies the state is useful for quantum teleportation. And it is known that Von Neumann entropy S (ρr ) of a given state ρr is less than or equal to Shannon entropy. If λ1 , λ2 , · · · , λr are eigenvalues of ρr in descending order, then the Shannon entropy is maximized for λ1 = f and other r − 1 eigenvalues are equal. Subjected to the constraint f > √r1 , max

we get the upper bound as

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S (ρr ) = 

r 

√ rmax λi ln λi = √ rmax



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− 1− √

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1





ln

rmax

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This implies,

S (ρr ) ≤ ln





1

rmax + 1 −

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(1 −

1 rmax

) ln

√ r −1 , rmax −1

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r−1

ln √ rmax − 1

(2.1)

state

S L (ρr ) =

(2.2)

r (rmax − 1) − 2 rmax

√

rmax − 1

(rmax − 1) (r − 1) 3r − 4 3(r − 1)

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where λi are the eigenvalues of the state ρ (λ1 ≥ λ2 ≥ λ3 ≥ λ4 and λ1 + λ2 + λ3 + λ4 = 1) and maximally entangled Bell states |ψ ±  = √1 (|01 ± |10). The Werner state, which is a convex sum

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of maximally entangled pure state and maximally mixed separable state, given by

W 4 = (1 − p )

I4 4

(2.6)

corresponds to a choice of eigenvalues given by λ1 =

λ2 = λ 3 = λ 4 =

1− p . 4



(2.3)

(2.4)

where r is the rank of the state, which varies from 2 to 4. In our previous work [5], based on the analysis of a class of maximally entangled mixed states of 2 × 2 bipartite system given in [15], we observed that for a given value of linear entropy, there exists a rank dependent upper bound on the teleportation fidelity and the upper bound increases with increase in the rank. This is equivalent to stating that for a given value of optimal teleportation fidelity, there exists a rank dependent upper bound on linear entropy and the upper bound increases with rank. The above result allows us to calculate the upper bounds explicitly. If the teleportation fidelity is fixed as 2/3 ( f = 1/2), the classical fidelity, the upper bound on linear entropy for states of second, third and fourth ranks are

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+ p |ψ + ψ + | 1+3p 4

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and

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Werner state is a state of rank 4. The singlet

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1+3p 4

fraction for Werner state is and linear entropy is estimated as 1 − p 2 . The value of linear entropy at which fidelity is equal to classical limit [18] can be found as 89 , which is same as the upper bound obtained above for states of rank 4. This correspondence motivated us to construct lower rank maximally entangled mixed states that exhibit theoretical bounds derived in this section. A rank 3 boundary MEMS can be constructed by using eigenvalues 1+2p 1− p 1+2p as λ1 = 3 , λ2 = λ3 = 3 and λ4 = 0. We have f ( W 3 ) = 3 8(1− p ) . 9 2

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Thus it is clear

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that f ( W 3 ) ≤ 12 for a value of linear entropy greater than 56 , which is the obtained upper bound beyond which a rank 3 state is useless for quantum teleportation. To construct a rank 2 boundary 1+ p 1− p MEMS we take λ1 = 2 , λ2 = 2 and λ3 = λ4 = 0. We get sin-

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and respectively. glet fraction and linear entropy as It clearly shows that rank 2 boundary MEMS is useless for teleportation when linear entropy is greater than 23 , coinciding with the theoretical upper bound shown above. Thus, we illustrate that the constructed lower rank boundary MEMS exhibit the theoretical upper bounds on the linear entropy for a given value of optimal teleportation fidelity. 2

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2(1− p 2 ) 3

ρr is useless for quantum teleportation

For d = 2, we have

S L (ρr ) =

ρ= λ1 |ψ − ψ − | + λ2 |0000| + λ3 |ψ + ψ + | + λ4 |1111| (2.5)

1+ p 2

max



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1− √ rmax



in a standard teleportation protocol. We also obtain an analytical expression for the rank dependent upper bound on linear entropy as a measure of mixedness as

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r−1

1

This shows that for S (ρr ) violating eq. (2.2), the singlet fraction √ cannot be greater than √r1 . Thus, we prove if S (ρr ) > ln rmax +

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rmax

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and linear entropy is equal to S L ( W 3 ) =

ln

i =1

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2/3, 5/6 and 8/9 respectively. States with linear entropy above the respective rank dependent upper bounds are useless for quantum teleportation. Our results are in agreement with the observation that there are no entangled states if the linear entropy is greater than 8/9 [16]. A class of two qubit maximally entangled mixed states (MEMS) is constructed by Ishizaka et al. [15,17] and the construction is as follows.

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The relationship between the concurrence as a measure of entanglement of state ρ and it’s purity is well studied, the degree of entanglement decreases as purity decreases. The maximum possible value of concurrence of a state ρ for a spectrum of eigenvalues [15,17,20] (λ1 , λ2 , λ3 , λ4 ) in descending order is given as

C max (ρ ) = max(0, λ1 − λ3 − 2 λ2 λ4 )

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3. Lower bounds on measure of entanglement for two qubit mixed states



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(3.1)

In our previous work [5], we also observed that there exist a rank dependent lower bound on concurrence for a fixed value of teleportation fidelity and the lower bound decreases with increase in the rank of the state. In the present work, we obtain the exact rank dependent lower bounds on the concurrence for a 2 × 2 bipartite states. For a given value of singlet fraction f , a two qubit state of rank r (2 ≤ r ≤ 4) with eigenvalue λ1 = f and other r − 1 eigenvalues equal leads to maximum mixedness. Since concurrence decreases with increase in mixedness, the maximum concurrence for a fixed value of mixedness or equivalently the eigenspectrum gives a lower bound on concurrence of all mixed entangled states of rank r.

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K.G. Paulson, S.V.M. Satyanarayana / Physics Letters A ••• (••••) •••–•••

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Fig. 1. Teleportation fidelity as a function of concurrence for rank dependent boundary MEMS. Rank 4 boundary MEMS is the well known Werner state.

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By substituting the values for λi , that is, corresponding to rank r, λ1 = 12 and rest of (r − 1) eigenvalues are equal, we get the lower bound on the concurrence of state ρr of rank r below which the state fails to be a source for quantum teleportation. We obtain the values of lower bound on concurrence for the failure of teleportation for rank 4, 3 and 2 states as follows, C 4 = 0, C 3 = 14 and C 2 = 12 respectively. From this we clearly show that there exists a lower bound on concurrence for a fixed value of fidelity, this lower bound decreases as rank increases for two qubit systems. The concurrence for failure of quantum teleportation for rank dependent boundary MEMS constructed in previous section coincides with the theoretical bounds derived in this work.

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4. Discussion

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Lower and upper bounds on teleportation fidelity as a function of concurrence are obtained in [19]. The upper and lower C bounds on fidelity F for a concurrence C are given by F ≤ 2+ 3

C 2C +1 and F ≥ max( 3+ , 3 ) respectively. The upper bound on fi6 delity as a function of concurrence coincides with Werner state of rank 4. Based on the properties of lower rank boundary MEMS constructed in this work, we conjecture that there exist rank dependent bounds on fidelity as a function of concurrence. The maximum amount of fidelity of a rank three state as a function of concurrence is given as F ≤ ( 5+94C ), in the same way the amount of teleportation fidelity for rank two state is given by MEMS of C rank 2 as F ≤ ( 2+ ). 3 Teleportation fidelity as a function of concurrence for rank dependent boundary MEMS is presented in Fig. 1. MEMS of different ranks exhibit respective rank dependent lower bounds on concurrence below which the state is useless for quantum teleportation. It can be seen that, in the fidelity – concurrence plane, the allowed values of teleportation fidelity for a fixed value of concurrence of 2 × 2 states are bounded below and above by curves corresponding to second and fourth rank MEMS respectively. Fidelity of second and fourth rank boundary MEMS also coincides respectively with lower and upper bounds on fidelity of 2 × 2 states obtained in [19]. The curve corresponding to boundary MEMS of rank 3, which lies between the curves of upper and lower bounds, serves as an upper bound on teleportation fidelity of rank 3 mixed states. This implies, quantum teleportation can be achieved with a resource of low value of concurrence, then it has to be a high rank state. For example, if we have to use a state with concurrence 0.1 as a quantum teleportation resource, it has to be necessarily a fourth rank state. In [21] it is shown that local environment can enhance fidelity of quantum teleportation. It is shown that for a class of

3

density matrices, interaction with environment enhances the singlet fraction above 12 , and thus making the state useful for quantum teleportation. On the other hands, if the state is distillable, then here are methods [22–27] like entanglement purification, local filtering, entanglement concentration for single and multiple number of qubits to improve the entanglement and the teleportation fidelity of quantum channels. Further, upper bounds on singlet fraction that can be achieved by local operation and classical communication have been derived [29,28]. We would like to emphasize that the bounds derived in the present work depend on the rank and mixedness of the state. Local operations involved in protocols to enhance singlet fraction decrease the mixedness and also in some cases the rank. Therefore, it can be stated that for fixed values of concurrence and linear entropy of a state of rank r, the maximum achievable teleportation fidelity is that of boundary MEMS of rank r, that we have constructed in this work, which in turn coincides with the theoretical bounds derived. We proved the existence of rank dependent upper bound on the von Neumann entropy as well as linear entropy as measures of mixedness of a general mixed state of a bipartite d × d system for failure of the state to be resource for quantum teleportation in a standard teleportation protocol. We constructed rank 3 and rank 2 boundary MEMS with rank 4 boundary MEMS being the well known Werner state. Rank dependent boundary MEMS exhibit the theoretical upper bounds obtained on von Neumann entropy and linear entropy. Further, we proved the existence of rank dependent lower bounds on the concurrence of a 2 × 2 mixed states for the failure of states as resource for quantum teleportation and showed that the lower bound on concurrence for rank dependent boundary MEMS coincides with the theoretical lower bounds. We also showed the rank dependent boundary MEMS give upper and lower bounds on fidelity as a function concurrence, which are consistent with the results in the literature.

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References

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[1] C.H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, Asher Peres, W.K. Wooters, Phys. Rev. Lett. 70 (1993) 1895. [2] Sandu Popescu, Phys. Rev. Lett. 72 (1994) 797. [3] R. Horodecki, M. Horodecki, P. Horodecki, Phys. Lett. A 222 (1996) 21. [4] Michael A. Nielsen, Issac L. Chuang, Quantum Computation and Quantum Information, Cambridge University press, Cambridge, England, 2000. [5] K.G. Paulson, S.V.M. Satyanarayana, Quantum Inf. Comput. 14 (2014) 1227. [6] M. Horodecki, P. Horodecki, R. Horodecki, Phys. Rev. A 60 (1999) 1888. [7] Gerardo Adesso, Fabrizio Illuminati, Silvio De Siena, Phys. Rev. A 68 (2003) 062318. [8] J. Batle, M. Casas, A.R. Plastino, A. Plastino, Phys. Lett. A 296 (2002) 251. [9] S. Bose, V. Vedral, Phys. Rev. A 61 (2000) 040101. [10] Jinhyoung Lee, M.S. Kim, Phys. Rev. Lett. 84 (2000) 4236. [11] Reinhard F. Werner, Phys. Rev. A 40 (1989) 4277. [12] J. von Neumann, Mathematical Foundation of Quantum Mechanics, Princeton Univeristy Press, 1995. [13] Daniel Cavalcanti, Antonio Acin, Nicolas Brunner, Tamas V’ertesi, Phys. Rev. A 87 (2013) 042104. [14] David Deutsch, Artur Ekert, Richard Jozsa, Chiara Macchiavello, Sandu Popescu, Anna Sanpera, Phys. Rev. Lett. 77 (1996) 2818. [15] Satoshi Ishizaka, Tohya Hiroshima, Phys. Rev. A 62 (2000) 022310. [16] Tzu-Chieh Wei, Kae Nemoto, Paul M. Goldbart, Paul G. Kwiat, William J. Munro, Frank Verstraete, Phys. Rev. A 67 (2003) 022110. [17] Frank Verstraete, Koenraad Audenaert, Tijl De Bie, Bart De Moor, Phys. Rev. A 64 (2001) 012316. [18] N. Gisin, Phys. Lett. A 210 (1996) 157. [19] Frank Verstraete, Henri Verschelde, Phys. Rev. A 66 (2002) 022307. [20] W.K. Wootters, Quantum Inf. Comput. 1 (2001) 27. [21] P. Badziag, M. Horodecki, P. Horodecki, R. Horodecki, Phys. Rev. A 62 (2000) 012311. [22] Charles H. Bennett, Gilles Brassard, Sandu Popescu, Benjamin Schumacher, John A. Smolin, William K. Wootters, Phys. Rev. Lett. 76 (1996) 722. [23] Charles H. Bennett, Herbert J. Bernstein, Sandu Popescu, Benjamin Schumacher, Phys. Rev. A 53 (1996) 2046.

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[24] Liang Qiua, Gang Tang, Xianqing Yang, Anmin Wang, Ann. Phys. 350 (2014) 137. [25] A. D’Arrigoa, R. Lo Franco, G. Benenti, E. Paladino, G. Falci, Ann. Phys. 350 (2014) 211. [26] Frank Verstraete, Jeroen Dehane, Bart DeMoor, Phys. Rev. A 64 (2001) 010101(R).

[27] Paul G. Kwiat, Salvador Barraza-Lopez, Andr’e Stefanov, Nicolas Gisin, Nature 409 (2001) 1014. [28] S. Adhikari, A. Kumar, Quantum Inf. Process. 15 (2016) 2797. [29] Frank Verstraete, Henri Verschelde, Phys. Rev. Lett. 90 (2003) 097901.

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