How to share a quantum secret

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Abstract

We investigate the concept of quantum secret sharing. In a ((k,n)) threshold scheme, a secret quantum state is divided into n shares such that any k of those shares can be used to reconstruct the secret, but any set of k-1 or fewer shares contains absolutely no information about the secret. We show that the only constraint on the existence of threshold schemes comes from the quantum "no-cloning theorem", which requires that n < 2k, and, in all such cases, we give an efficient construction of a ((k,n)) threshold scheme. We also explore similarities and differences between quantum secret sharing schemes and quantum error-correcting codes. One remarkable difference is that, while most existing quantum codes encode pure states as pure states, quantum secret sharing schemes must use mixed states in some cases. For example, if k <= n < 2k-1 then any ((k,n)) threshold scheme must distribute information that is globally in a mixed state.

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Mixed State Entanglement and Quantum Error Correction

William Wootters, John Smolin, David P. DiVincenzo … (1996)

Entanglement purification protocols (EPP) and quantum error-correcting codes (QECC) provide two ways of protecting quantum states from interaction with the environment. In an EPP, perfectly entangled pure states are extracted, with some yield D, from a mixed state M shared by two parties; with a QECC, an arbi- trary quantum state \(|\xi\rangle\) can be transmitted at some rate Q through a noisy channel \(\chi\) without degradation. We prove that an EPP involving one- way classical communication and acting on mixed state \(\hat{M}(\chi)\) (obtained by sharing halves of EPR pairs through a channel \(\chi\)) yields a QECC on \(\chi\) with rate \(Q=D\), and vice versa. We compare the amount of entanglement E(M) required to prepare a mixed state M by local actions with the amounts \(D_1(M)\) and \(D_2(M)\) that can be locally distilled from it by EPPs using one- and two-way classical communication respectively, and give an exact expression for \(E(M)\) when \(M\) is Bell-diagonal. While EPPs require classical communica- tion, QECCs do not, and we prove Q is not increased by adding one-way classical communication. However, both D and Q can be increased by adding two-way com- munication. We show that certain noisy quantum channels, for example a 50% depolarizing channel, can be used for reliable transmission of quantum states if two-way communication is available, but cannot be used if only one-way com- munication is available. We exhibit a family of codes based on universal hash- ing able toachieve an asymptotic \(Q\) (or \(D\)) of 1-S for simple noise models, where S is the error entropy. We also obtain a specific, simple 5-bit single- error-correcting quantum block code. We prove that {\em iff} a QECC results in high fidelity for the case of no error the QECC can be recast into a form where the encoder is the matrix inverse of the decoder.