Optics Communications 281 (2008) 4946–4950

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Optics Communications

journal homepage: www.elsevier .com/locate/optcom

Quantum state sharing using linear optical elements q

Yan Xia *, Jie Song, He-Shan SongSchool of Physics and Optoelectronic Technology, Dalian University of Technology, Dalian 116024, China

a r t i c l e i n f o

Article history:Received 22 April 2008Accepted 16 June 2008

PACS:03.67.Lx42.65.Lm

Keywords:Quantum state sharingOptical elementsLinear optical

0030-4018/$ - see front matter � 2008 Elsevier B.V. Adoi:10.1016/j.optcom.2008.06.028

q The project supported by National Natural SciencGrant No. 10575017.

* Corresponding author. Tel.: +86 411 84706201.E-mail addresses: xia-208@163.com (Y. Xia), hsson

a b s t r a c t

Motivated by protocols [G. Gordon, G. Rigolin, Phys. Rev. A 73 (2006) 062316] and [N.B. An, G. Mahler,Phys. Lett. A 365 (2007) 70], we propose a linear optical protocol for quantum state sharing of polariza-tion entangled state in terms optical elements. Our protocol can realize a near-complete quantum statesharing of polarization entangled state with arbitrary coefficients, and it is possible to achieve unity fidel-ity transfer of the state if the parties collaborate. This protocol can also be generalized to the multi-partysystem.

� 2008 Elsevier B.V. All rights reserved.

No-cloning theorem forbids a perfect copy of an arbitrary un-known quantum state. How to interchange different resourceshas ever been a question in quantum computation and quantuminformation. Using the theory of quantum mechanics in the fieldof information in the recent years has produced many interestingdevelopments, such as quantum teleportation [1,2], quantum cryp-tography [3,4], quantum secret sharing [5], and so on [6–8]. Quan-tum secret sharing (QSS), an important branch of quantumcommunication, is the generalization of classical secret sharing[9] into a quantum scenario and has attracted much attention[10–17]. Different from secret sharing of classical information viaquantum channels, Cleve et al. [18] proposed a secret sharing ofquantum state protocol, which was called quantum state sharing(QSTS) by Lance et al. [19]. We term this quantum state sharingto differentiate from the quantum secret sharing of classical infor-mation. In (k; n) threshold quantum state sharing [18], the boss(Alice) node encodes a secret state into an n-party entangled stateand distributes it to n players nodes. Any k players (the accessstructure) can collaborate to retrieve the quantum state, whereasthe remaining n � k players (the adversary structure), even whenconspiring, acquire nothing. This scheme provides quantum infor-mation networks with a secure framework for distributed quan-tum computation and quantum communication. The importantfeature of such a scheme is that at the end of the protocol the infor-

ll rights reserved.

e Foundation of China under

g@dlut.edu.cn (H.S. Song).

mation contained in the transferred state is completely available toonly one of the parties and Alice is free to choose whether Bob orCharlie will be the receiver.

Later, quantum state sharing has received much attention boththeoretically and experimentally [18–22] in recent years due to itsimportant applications in quantum communications. For example,in 2004, Lance et al. [19] demonstrated a multipartite protocol tosecurely distribute and reconstruct a quantum state. In their proto-col, a secret quantum state is encoded into a tripartite entangledstate and distributed to three players. These quantum state sharingprotocols to date are based on maximally bipartite or multipartiteentangled states. But in a realistic situation, however, decoherenceand noise degrade the channel and we do not have a maximallyentangled state anymore. To overcome this flaw, there are threeapproaches [23] to quantum state sharing that the sender andthe receivers can take: (a) one way out of this problem is to employquantum distillation protocol [24], which allows us to obtain amaximally entangled state from a large ensemble of partiallystates. But quantum distillation only achieves a maximally entan-gled state asymptotically. Thus, for finite runs of the distillationprotocol we always obtain an almost maximally entangled state.The probability for concentration is less than 1, but the fidelity ofthe teleported state using the concentrated pair as the quantumchannel is 1; (b) the sender may perform the standard Bell-statemeasurements directly with the nonmaximally entangled statesand let the receiver apply an auxiliary unitary transformation withthe aid of an auxiliary qubit to obtain the desired state probabilis-tically [2]. The success probability is less than 1, but the receiverknows whether he(she) obtained the desired state or not by

Y. Xia et al. / Optics Communications 281 (2008) 4946–4950 4947

observing the state of the auxiliary qubit; (c) Alice may perform‘‘generalized measurements” upon the nonmaximally entangledpairs that distinguish nonorthogonal ‘‘Bell-type” states conclu-sively with a certain probability less than 1. For example, Gordonand Rigolin [25] present two interesting generalized quantum statesharing protocols where the channels are not maximally entangledstates. In this protocol, by properly choosing the measurement ba-sis it is possible to achieve unity fidelity transfer of the state if theparties collaborate.

It is this third approach we wish to study in detail here. In thispaper, inspired by the protocols of Gordon and Rigolin [25] and Anand Mahler [26], we present a linear optical protocol for quantumstate sharing in terms optical elements. The realization of this pro-tocol is appealing due to the fact that quantum state of light is ro-bust against the decoherence and photons are ideal carriers fortransmitting quantum information over long distances. The pres-ent case, however, holds an interesting advantage over the previ-ous case [29–38] in that the detectors do not need to distinguishbetween one and two photons.

Let us assume the sender Alice wants to transfer to the receivers(Bob and Charlie) the polarization state

j/i0 ¼ ajHi0 þ bjVi0; ð1Þ

where we encode the polarized photon states jHi and jVi as logiczero and one states, i.e., jHi ¼ j0i and jVi ¼ j1i, and a and b are com-plex coefficients to satisfy jaj2 þ jbj2 ¼ 1. Alice shares with Bob andCharlie the state [40]

jWiABC ¼ cos gjVHHiABC � sin gjHVViABC: ð2Þ

Mode A belongs to Alice, the second mode belongs to Bob, and thelast one to Charlie. The initial state can be written as

jUi0ABC ¼ j/0i � jWiABC: ð3Þ

(S1) The protocol begins when Alice implements a correspond-ing generalized Bell-type measurement. See Fig. 1 (Alice area) for apictorial representation of generalized Bell-type measurement aswell as of the whole protocol. The linear optical scheme that distin-guishes two of the Bell-type states jU�g iA0 ¼ cos gjViAjVi0�sin gjHiAjHi0 and jW�g iA0 ¼ cos gjViAjHi0 � sin gjHiAjVi0 consist oftwo beam splitters (one is 50/50 beam splitter ðBS1Þ, the otherbeam splitter ðBS2Þ of transmission coefficient t ¼ tan g), fourpolarizing beam splitters (PBS1, PBS2, PBS3, and PBS4), and fourdetectors (DE1, DE2, DF1, and DF2). The polarizing beam splitterstransmit vertically polarized photons ðjViÞ and reflect horizontallypolarized photons ðjHiÞ. In fact, one obtains through a straightfor-ward calculation that the state jUi0ABC is transformed, via the actionof PBS1, PBS2, PBS3, PBS4, BS1, and BS2, into the state

jUiEFYBC ¼12f½singðjViEjHiF � jHiFjViEÞj0iY þ

ffiffiffiffiffiffiffiffiffiffiffiffiffifficos 2g

pffiffiffi2p ðjHiEj0iF þ j0iEjHiFÞj1iY� � ðajHHiBC þ bjVViBCÞ � ½singðjVHiEj0iF

� j0iEjVHiFÞj0iY þffiffiffiffiffiffiffiffiffiffiffiffiffifficos 2g

pffiffiffi2p ðjHiEj0iF þ j0iEjHiFÞj1iY�ðajHHiBC � bjVViBCÞ

þffiffiffi2p

2½jVViEj0iF � j0iEjVViF � jHHiEj0iF þ j0iEjHHiF�j0iY

(þ

ffiffiffiffiffiffiffiffiffiffiffiffiffifficos 2g

pffiffiffi2p ðjViEj0iF þ j0iEjViFÞjViY

)ðbjHHiBC þ ajVViBCÞ

þ 1ffiffiffi2p ½jVViEj0iF � j0iEjVViF þ jHHiEj0iF � j0iEjHHiF�j0iY�

þffiffiffiffiffiffiffiffiffiffiffiffiffifficos 2g

pffiffiffi2p ðjViEj0iF þ j0iEjViFÞjViY

)ðbjHHiBC þ ajVViBCÞ; ð4Þ

where j0iE means that there is neither a vertically polarized nor ahorizontally polarized photon of mode E, jVHiE stands for the stateof one vertically polarized photon and one horizontally polarized

photon of mode E, and jVViE stands for the state of two verticallypolarized photons of mode E. It is clear from Eq. (4) that the tele-portation succeeds if detectors DE1 and DF2 (or DE2 and DF1) registera photon each or if detectors DE1 and DE2 (or DF1 and DF2) register aphoton each corresponding to the first square bracket (secondsquare bracket) of Eq. (4). All other measurement results corre-spond to inconclusive events. We show in Fig. 2 the success prob-ability as a function of j for this case is nearly 1/2 when g ¼ p=4.The present case, however, holds an interesting advantage over theprevious case in that there is no need for a detector to detect pho-tons of mode Y.

Alice measures modes 0 and A with detectors DE1, DE2, DF1, andDF2. We observe that neither Bob nor Charlie can recover the statein Eq. (1) in its exact form by performing any general operationsthemselves without communicating between themselves. Thoughthey have the amplitude information, that is not sufficient sincethe phase information is not available. In this case they must agreeto cooperate among themselves. Only by this way, one of them, notboth, can recover the desired state for the no-cloning theorem.

(S2) For concreteness, let us assume Alice assigns Charlie to re-cover the quantum state in Eq. (1) (The case when Bob is selected issimilar because of the Bob–Charlie symmetry in this problem andthe selection should not be learn beforehand by neither Bob norCharlie). Therefore, after measuring her two modes Alice tellsBob to let his mode B pass through a half-wave plate (HWP) [39](see Fig. 1 Bob area), whose action is given by transformationjHi ! 1ffiffi

2p ðjHi þ jViÞ and jVi ! 1ffiffi

2p ðjHi � jViÞ. After passing through

the HWP, the mode B is sent through another PBS (50/50 beamsplitter) and the outmodes 1 and 2 are detected by two conven-tional photon detectors D1 and D2. Alice and Bob then tell Charlietheir measurement outcomes. Depending on Alice’s and Bob’s re-sults Charlie implements a corresponding specific unitary opera-tion in terms optical elements to change the polarization state toEq. (1). If Charlie does not learn from Bob the outcome of thedetecting result his mode will be left in a mixed state for withoutthat information Charlie cannot know the phase of Alice’s mode.

For example, if Alice’s measurement result is that DE1 and DF2

register a photon each, see Table 1, so Bob and Charlie shareajHHiBC þ bjVViBC. Bob sent mode B pass through a half-wave plate(HWP). After passing through the HWP, one obtains the quantumstate of the system as

jWiBC ¼ jHiaðajHiC þ bjViCÞ þ jViaðajHiC � bjViCÞ: ð5Þ

Finally the mode a is sent through another PBS and the statebecomes

jWim ¼ jHi1ðajHiC þ bjViCÞ þ jVi2ðajHiC � bjViCÞ: ð6Þ

The outmodes 1 and 2 are detected by two conventional photondetectors D1 and D2. The conventional photon detector can only dis-tinguish between the presence and absence of photons, and no

Source D1

D2

PBSHWP

DE1

DE2PBS1

PBS2

PBS4

PBS3

BS1

BS2

DF2

DF1

Alice

Bob

P1

Charlie

Y

0

a1

2

Fig. 1. Linear optical scheme for a quantum state sharing. The source station generates a nonmaximally entangled state Eq. (2). DE1, DE2, DF1, DF1, D1, and D2 are detectors,PBS1, PBS2, PBS3, PBS1, and PBS are polarizing beam splitters, BS1 is a 50/50 beam splitter, and BS2 is a beam splitter of transmission coefficient t ¼ tan g. HWP is a half-waveplate. P1 is a p=2-phase shifter.

4948 Y. Xia et al. / Optics Communications 281 (2008) 4946–4950

information on the exact number of photons can be obtained. Ifquantum efficiency of the detector is denoted by j, the positive-operator-valued measure for describing the conventional photondetector is given by

Yno

¼X1m¼0

ð1� jÞmjmihmj; ð7Þ

Yclick

¼ 1�Yno

¼X1m¼1

½1� ð1� jÞm�jmihmj: ð8Þ

We only consider the event that one of both detectors detects pho-tons and another does not register any photon. If the detector D1 de-tects photons and D2 does not register any photon, the state ofmode C is given by

qout ¼Tr1;2

Q1click

Q2nojWimhWj

� �Tr1;2;C

Q1click

Q2nojWimhWj

� � ; ð9Þ

0.2 0.4 0.6 0.8h

0.1

0.2

0.3

0.4

0.5

P

Fig. 2. Alice’s success detects probability P vs. g.

whereQi

j is a positive-operator-valued measure for the detectors Di

(i= 1,2). Substituting Eqs. (6)–(8) into Eq. (9), we obtain the exactform of the density operator of mode C

qout ¼ jjWiouthWj; ð10ÞjWouti ¼ ajHiC þ bjViC: ð11Þ

Otherwise, if the detector D1 does not detect any photon and D2 reg-isters one photon, the state of mode C is given by

qout ¼ jjW0iouthW0j; ð12Þ

jW0outi ¼ ajHiC � bjViC: ð13Þ

(S3) Then Bob tells his measurement results to Charlie. If detectorD1 registers one photon and detector D2 does not detect any photon,Charlie do noting and he can receiver the original state in Eq. (1); Ifdetector D2 registers one photon and detector D1 does not detectany photon, Charlie receives the state jW0outi ¼ ajHiC � bjViC, whichcan be transformed into Eq. (1) by applying a p=2-phase shifter P1

to change the sign of the polarization state jViC. The success prob-ability of obtaining Eq. (1) is j, which depends on the detector’s(D1, D2) efficiency. So Charlie can recover the original state in Eq.(1) with the help of Bob, and the total success probability isP ¼ j sin2 g. We can see from Fig. 3 that the total success probabil-

Table 1Corresponding relations among Alice’s measurement results, Bob’s measurementresult, Charlie’s state, and Charlie’s operation

Alice’s measurementresults

Bob’s measurementresult

Charlie’sstate

Charlie’soperation

DE1, DF2 or DE2, DF1 D2 ajHiC � bjViC PD1 ajHiC þ bjViC

DE1, DE2 or DF1, DF2 D2 ajHiC þ bjViCD1 ajHiC � bjViC P

00.25

0.5

0.75

1

k0

0.2

0.4

0.6

h

0

0.2

0.4P

00.25

0.5

0.75k

Fig. 3. The total success probability P vs. g and j.

Y. Xia et al. / Optics Communications 281 (2008) 4946–4950 4949

ity increases with increasing the quantum efficiency j of the detec-tor and the transmission coefficient g of the BS2.

This protocol can also be generalized to multi-party system, forexample:

Case 1: we generalize the two-party secret sharing of 1-qubitpolarization state protocol to M-qubit state case, where M P 2. Itcan be seen as M single qubit protocols implemented at once orin sequence. Suppose that Alice has an arbitrary M-qubit secretpolarization state that she would like to send to Bob and Charliein such a way that they must cooperate in order to faithfully recon-struct this state. The secret state can generally be represented as(Just like the original N-qubit state in [27])

jSiM ¼ �M

i¼1NiðxijHi þ yijViÞ; ð14Þ

where xi 6¼ xiþ1, and yi 6¼ yiþ1. Now Alice, Bob, and Charlie need shar-ing of M 3-qubit polarization state channels, which is given by MGHZ-type polarization state with different degrees of entanglement(in general mi 6¼ mj, for i 6¼ J) j/channeli ¼ �N

i¼1j/AmiBmi

Cmii. For each

M=3

00.2

0.40.6h

0

0.25

0.5

0.75

1

k0

0.0050.01

0.015P

00.2

0.40.6h

P

M=1

00.2

0.40.6h

0

0.25

0.5

0.75

1

k

0

0.2

0.4P

00.2

0.40.6h

P

Fig. 4. The total success probability

polarization entangled GHZ state, 1-qubit is with Alice, one is withBob, and the third one is with Charlie.

The rest of the protocol is similar to two-party sharing of one-qubit polarization state case. (a) Firstly Alice sends mode (0M ,AM) pass through PBS1, PBS2, PBS3, PBS4, BS1, and BS2 in sequence.Alice performs M measurements with the detectors DE1, DE2, DF1,and DF2. (b) Alice informs Bob and Charlie of the acceptable results,and she assigns one of the agents (Charlie) to recover the quantumstate in Eq. (14). (c) Then Bob performs M operations as that in (S2)on his M modes, respectively. (B area of Fig. 1). (d) The receiver canalso recover the original polarization state in Eq. (14) with the helpof Bob’s classical information as (S3). The total success probabilityis P ¼ jðsin2 gÞM . We can see from Fig. 4 that the total successprobability decline with increasing the number of the qubit (M).

Case 2: This protocol can also be generalized to multi-party (N)sharing of multi-qubit ðMÞ state case. That is, the secret state canalso generally be represented as Eq. (14). The sender Alice, the Nagents BN need sharing of M N þ 1-qubit polarization state chan-nels, which is given by M N þ 1-qubit polarization entangledGHZ-type states like case 1. For each N þ 1-qubit polarizationentangled GHZ state, the mode 1 is with Alice. The rest of the Nmodes belong to N agents, respectively. Alice takes some opera-tions as that in (a) in case 1 and informs the N agents of her mea-surements results. If she assigns the Nth agent to recover heroriginal state. Then all the N � 1 agents are take as controller. Ifall the controller would like to help the receiver N, they performoperations as that in (S2) on their modes, respectively. After classi-cal communications, the receiver can also recover the original statein Eq. (14) as (S3). The total success probability isP ¼ jN�1ðsin2 gÞM , which denotes that P decline with increasingthe number M and N.

The security of the quantum state sharing protocol againsteavesdropping and cheating can be shown by the same methodspresented in Ref. [28]. Actually, as a matter of fact, it is Alice’s abil-ity to choose whether Bob or Charlie will receive the transferredstate which prevents cheating by one of the parties. If she thinks

M=4

00.2

0.40.6h

0

0.25

0.5

0.75

1

k

00.0010.0020.003

00.2

0.40.6h

M=2

00.2

0.40.6h

0

0.25

0.5

0.75

1

k

00.020.040.060.08

00.2

0.40.6h

P vs. g, j and M (M = 1,2,3,4).

4950 Y. Xia et al. / Optics Communications 281 (2008) 4946–4950

one of the parties is the dishonest one, she can choose the otherone to be the receiver and by comparing a subset of the states re-ceived by the latter with the states transmitted Alice can detect ifthe former is cheating.

In conclusion, we have proposed an optical protocol for thequantum state sharing by using optical elements such as beamsplitters, phase shifters, and photo detectors. Although our presentprotocol is probabilistic, the average fidelity can approach unity fora large amplitude of the polarization state. Hence, this is a near-complete protocol. Compared with Ref. [25], the realization of thisprotocol is appealing due to the fact that quantum state of light isrobust against the decoherence and photons are ideal carriers fortransmitting quantum information over long distances. The pres-ent case, however, holds an interesting advantage over the previ-ous case in that the detectors do not need to distinguish betweenone and two photons. Experimental techniques for single-photondetection have made tremendous progress [41]. What we used alsoconsists of linear optical elements, and photon detectors, whichhave been widely used to entangle photons. This is an advantageof the present protocol that may make it easier to be demonstratedexperimentally. This protocol can also be generalized to the multi-party system and the total success probability is P ¼ jN�1ðsin2 gÞM .Also, note that the sender and receivers could perform the opera-tions on the N modes, each one on their side, using either a setof beam splitters and detectors sequentially or N sets of splittersand detectors in parallel. The latter case has a lower time complex-ity than the former, but requires more resources (Higher spacecomplexity). This protocol may play an important role in the actualimplementation of linear optical scheme for quantum communica-tion. Future work will comprise investigating more extensivequantum information processing procedures.

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