Quantum secret sharing (QSS) is a quantum cryptographic scheme for secure communication that extends beyond simple quantum key distribution. It modifies the classical secret sharing (CSS) scheme by using quantum information and the no-cloning theorem to attain the ultimate security for communications. The method of secret sharing consists of a sender who wishes to share a secret with a number of receiver parties in such a way that the secret is fully revealed only if a large enough portion of the receivers work together. However, if not enough receivers work together to reveal the secret, the secret remains completely unknown. The classical scheme was independently proposed by Adi Shamir and George Blakley in 1979. In 1998, Mark Hillery, Vladimír Bužek, and André Berthiaume extended the theory to make use of quantum states for establishing a secure key that could be used to transmit the secret via classical data. In the years following, more work was done to extend the theory to transmitting quantum information as the secret, rather than just using quantum states for establishing the cryptographic key. QSS has been proposed for being used in quantum money as well as for joint checking accounts, quantum networking, and distributed quantum computing, among other applications.
Protocol
The simplest case: GHZ states This example follows the original scheme laid out by Hillery et al. in 1998 which makes use of Greenberger–Horne–Zeilinger (GHZ) states. A similar scheme was developed shortly thereafter which used two-particle entangled states instead of three-particle states. In both cases, the protocol is essentially an extension of quantum key distribution to two receivers instead of just one. Following the typical language, let the sender be denoted as Alice and two receivers as Bob and Charlie. Alice's objective is to send each receiver a "share" of her secret key (really just a quantum state) in such a way that:
Neither Bob's nor Charlie's share contains any information about Alice's original message, and therefore neither can extract the secret on their own. The secret can only be extracted if Bob and Charlie work together, in which case the secret is fully revealed. The presence of either an outside eavesdropper or a dishonest receiver (either Bob or Charlie) can be detected without the secret being revealed. Alice initiates the protocol by sharing with each of Bob and Charlie one particle from a GHZ triplet in the (standard) Z-basis, holding onto the third particle herself:
| Ψ ⟩ G H Z = | 000 ⟩ + | 111 ⟩ 2 , {\displaystyle |\mathrm {\Psi } \rangle _{\rm {GHZ}}={\frac {|000\rangle +|111\rangle }{\sqrt {2}}},}
where | 0 ⟩ {\displaystyle |\mathrm {0} \rangle } and | 1 ⟩ {\displaystyle |\mathrm {1} \rangle } are orthogonal modes in an arbitrary Hilbert space. After each participant measures their particle in the X- or Y-basis (chosen at random), they share (via a classical, public channel) which basis they used to make the measurement, but not the result itself. Upon combining their measurement results, Bob and Charlie can deduce what Alice measured 50% of the time. Repeating this process many times, and using a small fraction to verify that no malicious actors are present, the three participants can establish a joint key for communicating securely. Consider the following for a clear example of how this will work. Let us define the x and y eigenstates in the following, standard way:
| + x ⟩ = | 0 ⟩ + | 1 ⟩ 2 , | − x ⟩ = | 0 ⟩ − | 1 ⟩ 2 {\displaystyle |\mathrm {+x} \rangle ={\frac {|0\rangle +|1\rangle }{\sqrt {2}}},|\mathrm {-x} \rangle ={\frac {|0\rangle -|1\rangle }{\sqrt {2}}}}
| + y ⟩ = | 0 ⟩ + i | 1 ⟩ 2 , | − y ⟩ = | 0 ⟩ − i | 1 ⟩ 2 {\displaystyle |\mathrm {+y} \rangle ={\frac {|0\rangle +i|1\rangle }{\sqrt {2}}},|\mathrm {-y} \rangle ={\frac {|0\rangle -i|1\rangle }{\sqrt {2}}}} . The GHZ state can then be rewritten as
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