Quantum secret-sharing protocol based on Grover’s algorithm Li-Yi Hsu Physics Division, National Center of Theoretical Sciences, Hsinchu, Taiwan, Republic of China ~Received 11 February 2003; published 18 August 2003! A marked state can be found with certainty in the two-qubit case of Grover’s algorithm. This property is included in the proposed quantum secret-sharing protocol. In the proposed scheme, the sender prepares some initial state in private and then performs a phase shift of the marked state as the sender’s bit. Then, the sender sends these two qubits to each of the two receivers. Only when the sender broadcasts the initially prepared state and then the receivers perform the corresponding inversion operation about the average, is the sender’s bit faithfully revealed. Moreover, the sender can detect deception using cheat-detecting states. The proposed quantum secret-sharing protocol is shown to be secure. DOI: 10.1103/PhysRevA.68.022306 PACS number~s!: 03.67.Hk Secret sharing addresses the following problem: Alice in Taipei wants a confidential action to take place in Seattle. She wants two agents, Bob and Charlie, to carry it out for her. However, she knows the following: ~1! one of the two agents—and at most one—may be dishonest, ~2! as long as the two agents work together, the honest agent will prevent the dishonest one from sabotaging the action. Consequently, she cannot entrust the two agents with the faithful message. Instead, she encrypts her message in two pieces, neither of which contains any information individually. These two agents can determine Alice’s message only when they com- bine their encrypted messages. Recently, some research has focused on quantum secret sharing because of its potential application in quantum information theory. Hillery et al. originally considered quantum secret sharing via three- particle and four-particle Greenberger-Horne-Zeilinger ~GHZ! states @1#. Karlsson et al. considered quantum secret sharing using two-particle entanglement @2#. Cleve and co- workers investigated quantum ( k , n ) threshold scheme @3,4#. Furthermore, they considered the connection between quan- tum secret sharing and quantum error-correction code @3#. Recently, Karimipour et al. explored quantum secret sharing using the entanglement swapping of d-level generalized Bell states @5#. Very recently, Cabello discussed N-party quantum secret sharing @6#. This study proposes a one-to-two-party quantum secret- sharing protocol based on Grover’s algorithm @8#. The basic idea that underlies the proposed protocol markedly differs from the ideas that underlie the protocols mentioned above. In this protocol, Alice prepares different Bell states. In addi- tion, the sender and the honest agent do not analyze a portion of the sequence of measurement outcomes to discover pos- sible cheating @1#. Under error-free conditions, the sender can find cheating immediately by observing a public mes- sage of receivers’ discussion result. However, in some of the above protocols, Alice can encrypt only random bits owing to the intrinsic randomness in the quantum measurement or the entanglement swapping @1,5#. In the proposed protocol, Alice can encode the wanted bits with the help of the Grov- er’s algorithm. Moreover, since even the three-qubit Grov- er’s algorithm has been experimentally realized @7#, our quantum secret-sharing protocol based on Grover’s algo- rithm becomes highly practical for experimental realization. Finally, through Grover’s algorithm, a (2,2) threshold scheme is established. The original two-qubit Grover’s algorithm is briefly re- viewed in Ref. @8#. Suppose we want to find a marked state u w & , where w can be 00, 01, 10, or 11. The initial state u S 1 & 5@ (1/A 2)( u 0 & 1u 1 & )] ^ 2 is prepared. Then, two unitary transformations U w and 2U S 1 are transformed in that order, where U x 51 22 u x &^ x u , yielding 2U S 1 U w u S 1 & 5u w & ; ~1! that is, the marked state can be found with certainty. In the proposed protocol, the two-qubit Grover’s algorithm with some other initial prepared state u S i & , is performed. Each qubit in u S i & can be in one of the following four states: (1/A 2)( u 0 & 1u 1 & ), (1/A 2)( u 0 & 2u 1 & ), (1/A 2)( u 0 & 1i u 1 & ), and (1/A 2)( u 0 & 2i u 1 & ), denoted as u 1& , u 2& , u 1i & , and u 2i & in Table I, respectively. For simplicity, U w u S i & is rep- resented as u S i & w . Interestingly, 2U S j u S j & w 5a u w & ~2! holds, where a is some phase term and w can be 00, 01, 10, or 11. In other words, in Grover’s algorithm, even though some other u S i & is prepared, the marked state can still be found with certainty. In the following discussion, S is the set of these 16 initial preparations, as shown in Table I. This paper divides u S j & w into two classes: the message states and the cheat-detecting states. Alice encodes the clas- sical bits 0 and 1 as the marked states u 01& and u 10& , respec- tively. That is, Alice encrypts her secret bit in the state u S j & w , where w is either 01 or 10. She tries to detect any possible eavesdropping using the state u S j & w , where w is either 00 or 11. The procedure of the secret-sharing protocol is as fol- lows: ~1! Alice randomly prepares some initial state u S i & PS, for i 51, . . . ,16, and then performs U w on u S i & . ~2! Alice sends each of the receivers, Bob and Charlie, one of the qubits. Bob and Charlie are assumed to receive the first and the second qubits, respectively. ~3! Alice has to confirm that each agent has actually received the qubit via classical communication. ~4! Alice announces her initial state u S i & in public. ~5! Only when Bob and Charlie combine their qubits and perform 2U S i on these two qubits can they both deter- mine the marked state u w & with certainty. ~6! Bob and PHYSICAL REVIEW A 68, 022306 ~2003! 1050-2947/2003/68~2!/022306~4!/$20.00 ©2003 The American Physical Society 68 022306-1