Probabilistic resumable quantum teleportation in high dimensions
aa r X i v : . [ qu a n t - ph ] F e b Probabilistic resumable quantum teleportation in high dimensions
Xiang Chen, Jin-Hua Zhang,
2, 3 and Fu-Lin Zhang ∗ Department of Physics, School of Science, Tianjin University, Tianjin 300072, China School of Mathematical Sciences, Capital Normal University, Beijing 100048, China Department of Physics, Xinzhou Teacher’s University, Xinzhou 034000, China (Dated: February 8, 2021)Teleportation is a quantum information processes without classical counterparts, in which thesender can disembodied transfer unknown quantum states to the receiver. In probabilistic telepor-tation through a partial entangled quantum channel, the transmission is exact (with fidelity 1),but may fail in a probability and simultaneously destroy the state to be teleported. We proposea scheme for nondestructive probabilistic teleportation of high-dimensional quantum states. Withthe aid of an ancilla in the hands of sender, the initial quantum information can be recovered whenteleportation fails. The ancilla acts as a quantum apparatus to measure the sender’s subsystem,and erasing the information it records can resumes the initial state.
I. INTRODUCTION
Quantum teleportation is one of the processes uniqueto the quantum information, which serves as an impor-tant example for the most intriguing uses of entangle-ment [1–4]. It has been widely studied both theoreticallyand experimentally since it was put forwards [5–24]. Onereason it is gaining so much traction is that, the originalscheme and its various extensions play key roles in severalcontexts in quantum communication, including quantumrepeaters, quantum networks and cryptographic confer-ences [25–29].In the original and also the simplest version of telepor-tation [2], Alice (the sender) can transfer an unknownstate from a qubit to Bob (the receiver), with fidelity 1and successful probability 1, without physical transmis-sion of the qubit itself. The key ingredient in the protocolis a two-qubit Bell state shared by them as a quantumchannel. Alice makes a joint measurement on the twoqubits in her hands, projecting them onto one of the fourBell states with equal probability. Through a classicalchannel, Alice sends Bob two classical bits to inform heroutcome. According to the classical information, Bobcan perform appropriate unitary operations on his qubitto rebiuld Alice’s initial state, and thereby accomplishsthe teleportation.A variant of the process called probabilistic teleporta-tion was proposed based upon the consideration of that aquantum channel may be prepared in a partially entan-gled pure state in practice [5–9]. Such teleportation isexact (with fidelity 1), but may sometimes fail as thethe price to pay for the fidelity. In principle, Alice’sjoint measurement is destructive, and it is generally as-sumed that the information ecoded in the state to be tele-ported is lost when the teleportation fails. In their recentwork [9], Roa and Groiseau presented a nondestructivescheme for probabilistic teleportation by introducing an ∗ Corresponding author: fl[email protected] ancillary qubit. This avoids losing the initial informationand offers the chance to repeat the teleportion process.The nondestructive scheme has been extended in manybranches, including bidirectional teleportation [30], tele-portation of an entangled state [31] and multihop sce-nario [32, 33].In this work, we present a general protocol for nonde-structive probabilistic teleportation of high-dimensionalquantum states, in which Alice can resume her initialstate when the teleportation fails. The motivation forthis work is not only to extend the study of Roa andGroiseau to high-dimensional systems, but also based onthe following considerations. In theory, high-dimensionalchannels, especially the partially entangled ones, havemore rich entanglement properties. Our protocol pro-vides a sample to study the cooperative relationshipamong the enanglement, joint measurement and classi-cal information in a quantum information process. Inpractice, to teleport high-dimensional states is requiredin the task to completely rebuild the quantum states ofa a real particle remotely. And, recent experiment pro-gresses [16, 17] show the possibility of implementationthe our present protocol in optical systems. We make aremark here that, although Fu et al. [32, 33] give a ver-sion of nondestructive probabilistic teleportation in highdimension using an auxiliary particle in Bob’s hand, wemaintain the approach in [9] with an ancilla belongingto Alice. In this case, only the sender, Alice, is requiredthe ability of bipartite operations, such as generation andmeasurement of entangled states. In addtion, the ancillacan be regarded as a quantum apparatus to measure Al-ice’s system. The initial state is recovered by erasing theinformation it records.
II. TELEPORTATION OF A QUTRIT
Let us start with the teleportation of a qutrit (a three-level quantum system) . Suppose Alice wishes to teleportto Bob an arbitrary qutrit state as | φ i = α | i + α | i + α | i , (1)with | α | + | α | + | α | = 1. The participants share atwo-qutrit entangled state as the quantum channal, | Φ i = b | i + b | i + b | i , (2)with | b | + | b | + | b | = 1. Without loss of generality,we assume the Schmidt coefficients are non-negative realnumbers and b ≤ b ≤ b . Here, the three qutrits areidentified by the subscripts 1 , | Ψ i = | φ i ⊗ | Φ i = 13 (cid:2) | ψ i ( α | i + α | i + α | i ) + | ψ i ( α | i + α ω | i + α ω | i ) + | ψ i ( α | i + α ω | i + α ω | i ) + | ψ i ( α | i + α | i + α | i ) + | ψ i ( α | i + α ω | i + α ω | i ) (3)+ | ψ i ( α | i + α ω | i + α ω | i ) + | ψ i ( α | i + α | i + α | i ) + | ψ i ( α | i + α ω | i + α ω | i ) + | ψ i ( α | i + α ω | i + α ω | i ) (cid:3) , where ω = e i π is the triple root, and | ψ nm i are ninelinearly independent two-qutrit states, equivalent to thechannel (2) under local unitary transformations, as fol-lows, | ψ i = ( b | i + b | i + b | i ) , | ψ i = ( b | i + b ω | i + b ω | i ) , | ψ i = ( b | i + b ω | i + b ω | i ) , | ψ i = ( b | i + b | i + b | i ) , | ψ i = ( b | i + b ω | i + b ω | i ) , (4) | ψ i = ( b | i + b ω | i + b ω | i ) , | ψ i = ( b | i + b | i + b | i ) , | ψ i = ( b | i + b ω | i + b ω | i ) , | ψ i = ( b | i + b ω | i + b ω | i ) . The vector of qutrit 3 in each term (each line) of the to-tal state (3) is equivalent to the initial state (1) undera unitary operator. However, the corresponding statesof qutrits 1 and 2 | ψ nm i are not orthogonal to eachother. To teleportate the initial state exactly, an optionis that, Alice performs an unambiguous quantum statediscrimination process [6, 7, 12, 13], to distinguish thenine states with no error but with a probability of fail-ures. As in the standard teleportation through a maximalentangled quantum channel [16, 17], Bob can rebuild theinitial state on his qutrit by performing appropriate uni-tary operations according to Alice’s outcome. In anotherscheme, Alice still performs a joint measurement in nine FIG. 1: (Color online) Alice’s sequential operations on hersubsystems: (1) The GCNOT gate C factorizes the statesof 1 and 2 to be discriminated; (2) The apparatus 0 measures1 by using C † ; (3) Alice performs a joint transformation D on 1 and 2 to discriminate the states of 2; (4) The GCNOTgate C erases the information of 1 recorded in 0; (5) Alicemeasures 0 to divide the procedure into success and failureparts. maximally entangled basis, while Bob needs an extract-ing quantum state process [5]. Here, we follow the formerscheme, which only requires Alice’s ability to manipulatetwo or more particles.For the sake of brevity, before putting forward our fi-nal total state in this section, we only show the states inthe hands of Alice. We show the sequential operationsof Alice in Fig. 1. To discriminate the nine two-qutritstates (4), Alice factorizes the states by applying a gener-alized controlled-NOT (GCNOT) gate onto her bipartitesystem and obtain C | ψ nm i = | m i ( b | i + b ω n | i + b ω n | i ) , (5)with m, n = 0 , ,
2. Here, we define the GCNOT gate C ij acting on qutrits i and j as C ij = | i i h | ⊗ j + | i i h | ⊗ V j + | i i h | ⊗ V † j , (6)where j is the identity of j and V j = | i j h | + | i j h | + | i j h | . It shifts the target j clockwise or anticlockwisewhen the control qutrit i is a | i or | i .To distinct the nine states in (5), Alice can performa von Neumann measurement on qutrit 1 followed bya unambiguous quantum state discrimination on qutrit2. However, these operations destroy the initial state ofqutrit 1 even though the discrimination fails. Here, fol-lowing the protocol of Roa and Groiseau [9], we introducean extra auxiliary qutrit 0 which act as a quantum appa-ratus to measure qutrit 1. The key point is that, when weerase the information of qutrit 1 recorded in the ancilla,the initial state of qutrit 1 recoveries when discrimina-tion fails. Alice applies the inverse of the GCNOT gateon the ancilla initial in | i and qutrit 1, and obtain thenine states in her hands as C † | i | m i | τ n i = | m i | m i | τ n i , (7)where | τ n i = b | i + b ω n | i + b ω n | i and m, n =0 , ,
2. Then, qutrit 1 can servers as an ancilla in theunambiguous quantum state discrimination of qutrit 2.Let us define two unitary transformations on qutrit 2 tobe U = b b + s − (cid:18) b b (cid:19) V , W = b b + s − (cid:18) b b (cid:19) V , (8)and a controlled-unitary operation D = | i h | ⊗ + | i h | ⊗ U + | i h | ⊗ W . (9)Alice performs it on qutrits 1 and 2 and obtains D | m i | m i | τ n i = | m i (cid:20) √ b | m i | κ n i + | m ⊕ i ( q b − b ω n | i + q b − b ω n | i ) (cid:21) , where ⊕ denotes modulo 3 addition, and | κ n i = √ ( | i + ω n | i + ω n | i ). In the above form, thenine states | m i | m i | κ n i corresponding to successfuldiscrimination are orthogonal to each other, and orthog-onal to the states with qutrits 0 and 1 in | m i | m ⊕ i .The latter nine are for the failure of discrimination asthey are not linearly independent.The final step of Alice’s unitary operations is to erasethe information measured by qutrit 0. Applying the GC-NOT gate C , Alice can obtain C | m i | m i = | i | m i , C | m i | m ⊕ i = | i | n ⊕ i . (10)The information recorded on the apparatus qutrit 0 canbe erased partially, as it rerurns | i in the terms corre-sponding to successful discrimination but becomes | i for the case of failure. This divides the total four-qutritstate into two parts as | ∆ i = b √ | i (cid:20) | i ( | κ i | φ i + | κ i | φ i + | κ i | φ i ) + | i ( | κ i | φ i + | κ i | φ i + | κ i | φ i ) + | i ( | κ i | φ i + | κ i | φ i + | κ i | φ i ) (cid:21) + | i (cid:20)q b − b | φ i | i | i + q b − b | φ i | i | i (cid:21) , where | φ nm i are in the forms for the states of qutrit 3multiplied to | ψ nm i in (3).Alice performs a von Neumann measurements on qutrit0 in the standarad basis. It projects qutrit 0 to | i in aprobability 3 | b | . Then, she measures the nine orthog-onal direct product states | i | m i , and informs Bob toperform appropriate unitary operations on his qutrit 3to rebiuld the initial state, and thereby the teleportationsucceeds. On the contrary, the qutrit 0 could be pro- jected to | i in a probability 1 − | b | , which meansthe failure of the teleportation. Alice can recover thestate | φ i in qutrit 1, by performing a joint unitary onher two qutrits, or a von Neumann measurements on 2followed by a local unitary on 1. III. GENERAL PROTOCOL
Now we turn to the general protocol for teleporting aquNit (a N -level quantum system) through a partiallyentangled two-quNit quantum channal. Since it is di-rectly to extended the above scheme to N -level systems,we present the following results in compact general for-mulae in this part. Let an initial quNit state to be | φ i = N − P i =0 α i | i i , (11)where α i =0 , , ··· ,N − are complex numbers satisfying thenormalization condition P N − i =0 | α i | = 1. Alice and Bobshare a two-quNit entangled state as the quantum chan-nal, which can be generally written in the form of theSchmidt disposition as | Φ i = N − P j =0 b j | j i | j i , (12)with the real coefficient 0 ≤ b ≤ b ≤ . . . ≤ b N − and P N − j =0 b j = 1. The total state of the tripartite system isgiven by | Ψ i = | φ i | Φ i = 1 N N − ,N − X m =0 ,n =0 | ψ nm i N − P f =0 α f ⊕ m e − i πfnN | f i , where ⊕ denotes modulo N addition, and | ψ nm i are N linearly independent bipartite states | ψ nm i = N − P k =0 b k e i πknN | k ⊕ m i | k i . (13)In the above form, the states multiplied to | ψ nm i areequivalent to the state (11) on quNit 3 under unitarytransformations as N − P f =0 α f ⊕ m e − i πfnN | f i = U ( n,m )3 | φ i (14)with U ( n,m ) j = N − P f =0 e − i πfnN | f i j h f ⊕ m | , (15)and the subscript j denoting the j th subsystem.Alice’s first two operations are to disentangle subsys-tems 1 and 2 in the states | ψ nm i , and to measure 2by using an auxiliary apparatus, quNit 0 initial in | i .Here, we define the N -level GCNOT gate as C ij = N − P y =0 | y i i h y | ⊗ U (0 ,y ) j . (16)The four-partite state becomes | Ω i = C † (cid:18) | i C | Ψ i (cid:19) = 1 N N − ,N − X m =0 ,n =0 | mm i N − P j =0 b j e i πjnN | j i ! U ( n,m )3 | φ i . To unambiguously discriminate the states of particle2, Alice applies a joint joint unitary transformation on 1and 2 D = N − P y =0 | y i h y | ⊗ b b y + s − (cid:18) b b y (cid:19) U (0 , , obtains the whole system in the state | Γ i = D | Ω i = b N N − ,N − X m =0 ,n =0 | mm i N − P j =0 e i πjnN | j i ! U ( n,m )3 | φ i + N − ,N − X m =0 ,j =1 q b j − b | m i α j ⊕ m | m ⊕ N − i | jj i . When Alice erases the information in the auxiliary appa-ratus, 0, the changes of the quNit 1 are recorded as | i ,and the total state becomes | ∆ i = C | Γ i = | i b N N − ,N − X m =0 ,n =0 | m i N − P j =0 e i πjnN | j i ! U ( n,m )3 | φ i + | i N − X j =1 q b j − b U (0 ,j ⊕ | φ i | jj i . Obviously, it is divided into two parts of success andfailure, which can be collapsed by Alice’s measurements in the standard basis. When Alice outcomes | i in theprobability N b , the teleportation can be accomplishedby two local von Neumann measurement on quNits 1 and2. One the other hand, Alice can recover the initial stateby a joint unitary operation when the task of teleportaionfails in the probability 1 − N b . IV. SUMMARY
We present a scheme for nondestructive probabilisticteleportation of high-dimensional quantum states. A par-tial entangled pure state severs as the quantum channel,whose smallest coefficient determines the the successfulprobability of exactly teleporting a state. With the aidof an auxiliary particle, Alice can recover her initial stateto be teleported when teleportation fails. Compared tothe existing results [32, 33], our protocol only requiresthe sender, Alice to have the ability to perform bipartiteoperations, while the dimension of the ancilla needs to bethe same as the state to be teleported. In addition, theancilla acts as a quantum apparatus to measure Alice’ssystem. The process of resuming the initial state can beregarded as erasing information recorded in the ancilla.As the follow research, it is a fundamental problem toexplore the roles of quantum correlations in our four-party procedure, which is a fundamental problem inquantum information. In addtion, the relation betweenour protocol and the theory of extracting informationfrom a quantum system by multiple observers [34] wouldbe interesting, since the quantum correlations in the lat-ter are studied in many works [35–37]. While we focusedhere on the teleportation using quantum channels withthe same dimension as the state to be teleported, it is anatural extension to apply the present ideas in the casewith different dimension [14]. And finally, we hope theprocess can be implemented in laboratories with the helpof the techniques recently developed in optical systems[16, 17] .
Acknowledgments
This work was supported by the NSF of China (GrantsNo. 11675119, No. 11575125, and No. 11105097). [1] M. A. Nielsen and I. L. Chuang,
Quantum Computationand Quantum Information (Cambridge University Press,Cambridge, 2000).[2] C. H. Bennett, G. Brassard, C. Cr´epeau, R. Jozsa,A. Peres, and W. K. Wootters, Phys. Rev. Lett. , 1895(1993).[3] N. Brunner, N. Gisin, and V. Scarani, New J. Phys. ,88 (2005).[4] R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, Rev. Mod. Phys. , 865 (2009).[5] W.-L. Li, C.-F. Li, and G.-C. Guo, Phys. Rev. A ,034301 (2000).[6] K. Banaszek, Phys. Rev. A , 024301 (2000).[7] L. Roa, A. Delgado, and I. Fuentes-Guridi, Phys. Rev. A , 022310 (2003).[8] F. Verstraete and H. Verschelde, Phys. Rev. Lett. ,097901 (2003).[9] L. Roa and C. Groiseau, Phys. Rev. A , 012344 (2015). [10] A. Karlsson and M. Bourennane, Phys. Rev. A , 4394(1998).[11] X.-H. Li and S. Ghose, Phys. Rev. A , 052305 (2014).[12] F.-L. Zhang, J.-L. Chen, L. C. Kwek, and V. Vedral, Sci.Rep. , 2134 (2013).[13] F.-L. Zhang and T. Wang, Europhys. Lett. , 10013(2017).[14] X. Chen, Y. Shen, and F.-L. Zhang, preprintarXiv:2101.06693 (2020).[15] Y. Huang and W. Yang, Chinese Journal of Electronics , 228 (2020).[16] Y.-H. Luo, H.-S. Zhong, M. Erhard, X.-L. Wang, L.-C.Peng, M. Krenn, X. Jiang, L. Li, N.-L. Liu, C.-Y. Lu,et al., Phys. Rev. Lett. , 070505 (2019).[17] X.-M. Hu, C. Zhang, B.-H. Liu, Y. Cai, X.-J. Ye, Y. Guo,W.-B. Xing, C.-X. Huang, Y.-F. Huang, C.-F. Li, et al.,Phys. Rev. Lett. , 230501 (2020).[18] D. Bouwmeester, J.-W. Pan, K. Mattle, M. Eibl, H. We-infurter, and A. Zeilinger, Nature , 575 (1997).[19] D. Boschi, S. Branca, F. De Martini, L. Hardy, andS. Popescu, Phys. Rev. Lett. , 1121 (1998).[20] A. Furusawa, J. L. Sorensen, S. L. Braunstein, C. A.Fuchs, H. L. Kimble, and E. S. Polzik, Science , 706(1998).[21] D. Gottesman and I. L. Chuang, Nature , 390 (1999).[22] C. Nolleke, A. Neuzner, A. Reiserer, C. Hahn, G. Rempe,and S. Ritter, Phys. Rev. Lett. , 140403 (2013).[23] W. Pfaff, B. J. Hensen, H. Bernien, S. B. van Dam, M. S.Blok, T. H. Taminiau, M. J. Tiggelman, R. N. Schouten, M. Markham, D. J. Twitchen, et al., Science , 532(2014).[24] J. M. Torres, J. Z. Bernad, and G. Alber, Phys. Rev. A , 012304 (2014).[25] N. Sangouard, C. Simon, H. de Riedmatten, andN. Gisin, Rev. Mod. Phys. , 33 (2011).[26] E. Biham, B. Huttner, and T. Mor, Phys. Rev. A ,2651 (1996).[27] S. Bose, V. Vedral, and P. L. Knight, Phys. Rev. A ,822 (1998).[28] P. Townsend, Nature , 47 (1997).[29] B. Aoun and M. Tarifi, arXiv: pp. quant–ph/0401076(2004).[30] Y.-T. Gou, H.-L. Shi, X.-H. Wang, and S.-Y. Liu, Quan-tum. Inf. Process. , 278 (2017).[31] Z.-Y. Wang, Y.-T. Gou, J.-X. Hou, L.-K. Cao, and X.-H.Wang, Entropy. , 352 (2019).[32] F. Fu and M. Jiang, J. Opt. Soc. Am. B , 233 (2020).[33] F. Fu, H. Li, S. Xue, and M. Jiang, J. Opt. Soc. Am. B , 1896 (2020).[34] J. Bergou, E. Feldman, and M. Hillery, Phys. Rev. Lett. , 100501 (2013).[35] C.-Q. Pang, F.-L. Zhang, L.-F. Xu, M.-L. Liang, andJ.-L. Chen, Phys. Rev. A , 052331 (2013).[36] J.-H. Zhang, F.-L. Zhang, and M.-L. Liang, QuantumInf. Process. , 260 (2018).[37] J.-H. Zhang, F.-L. Zhang, Z.-X. Wang, L.-M. Lai, andS.-M. Fei, Phys. Rev. A101