Quantum secret splitting based on non-orthogonal multi-particle states

A quantum-based method of secret splitting between two parties, comprising:

• preparing two qubits each having two possible states given by:
• |Ψ0>=cos θ0|00>+sin θ0|11>,
• |Ψ1>=cos θ0|00>−sin θ0|11>,
• where 0≦θ0≦π/4;
• sending the qubits to respective receivers A and B;
• at receiver A, performing a measurement of the qubit sent to receiver A using POVM elements |rj><rj|, for j=0, 1, with
• 
• r
• 0
• 〉
• =
• 1
• 2
• ⁢
• (
• 
• 0
• 〉
• +
• 
• 1
• 〉
• )
• ,
• ⁢
• 
• r
• 1
• 〉
• =
• 1
• 2
• ⁢
• (
• 
• 0
• 〉
• -
• 
• 1
• 〉
• )
• .
• so that receiver A obtains only the results 0 or 1 and never receives a failure result;
• at receiver B, using POVM elements
• 1
• 2
• ⁢
• (
• 1
• +
• tan
• 2
• ⁢
• θ
• 0
• )
• ⁢
• 
• s
• j
• 〉
• ⁢
• 〈
• s
• j
• 
• ,
• for
• ⁢
• ⁢
• j
• =
• 0
• ,
• 1
• ,
• with
• 
• s
• 0
• 〉
• =
• 1
• 2
• ⁢
• (
• sin
• ⁢
• ⁢
• θ
• 0
• ⁢
• 
• 0
• 〉
• -
• cos
• ⁢
• ⁢
• θ
• 0
• ⁢
• 
• 1
• 〉
• )
• ,
• ⁢
• 
• s
• 1
• 〉
• =
• 1
• 2
• ⁢
• (
• sin
• ⁢
• ⁢
• θ
• 0
• ⁢
• 
• 0
• 〉
• +
• cos
• ⁢
• ⁢
• θ
• 0
• ⁢
• 
• 1
• 〉
• )
• ,
• and
• (1−tan2 θ0)|0><0|,
• which respectively corresponding to measuring 0, 1 and a measurement failure;
• and
• sharing between A and B the outcome of their respective measurements so as to form a common bit string.

A quantum-based method of secret splitting between more than two parties, comprising:

• preparing two different quantum N-particle states of the form
• |Ψ0>=cos θ0|00 . . . 0>+sin θ0|11 . . . 1>,
• |Ψ1>=cos θ0|00 . . . 0>−sin θ0|11 . . . 1>,
• where 0≦θ0≦π/4, one of the states represents the bit value 0 and the other represents the bit value 1;
• sending a particle to each one of the receivers, say, A1, . . . , AN;
• performing at each receiver A1 through AN-1 a quantum measurement on the particle in the {r0,r1} basis, where
• 
• r
• 0
• 〉
• =
• 1
• 2
• ⁢
• (
• 
• 0
• 〉
• +
• 
• 1
• 〉
• )
• ,
• and
• 
• r
• 1
• 〉
• =
• 1
• 2
• ⁢
• (
• 
• 0
• 〉
• -
• 
• 1
• 〉
• )
• ,
• wherein receiver AN performs the unambiguous-state discrimination procedure between states |ψ0N>=cos θ0|0>+(−1)n1 sin θ0|1>ψ1N>=cos θ0|0>−(−1)n1sin θ0|1>, where n1 is the number of results of |r1> obtained by receivers A1 through AN-1.

A quantum-based secret-splitting method, comprising:

• generating at a sender a pair of entangled photons and providing the photons to two respective parties, Alice and Bob;
• at Alice, measuring a state of the photon she receives as either a 0 or a 1;
• at Bob, measuring a state of the photon he receives as either a 0, 1 or ƒ, wherein ƒ represents a failure to measure the proper photon state;
• and
• communicating between Alice, Bob and Charlie the outcome of their respective measurements so as to deduce the state of the qubits sent to Alice and Bob.