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arXiv:1205.4182 · Physical Review A

Equivalence between sharing quantum and classical secrets and error correction

Anne Marin, Damian Markham

Published Oct 25, 2013• 32 citations
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Abstract

We present a general scheme for sharing quantum secrets, and an extension to sharing classical secrets, which contain all known quantum-secret-sharing schemes. In this framework, we show the equivalence of existence of both schemes, that is, the existence of a scheme sharing a quantum secret implies that the extended classical-secret-sharing scheme works, and vice versa. As a consequence of this, we find schemes sharing classical secrets for arbitrary access structures. We then clarify the relationship to quantum error correction and observe several restrictions thereby imposed, which for example indicates that for pure-state threshold schemes the share size $q$ must scale with the number of players $n$ as $q\ensuremath{\ge}\sqrt{n}$. These results also provide a way of searching for quantum-error-correcting codes.

DOI
10.1103/physreva.88.042332

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We present a general scheme for sharing quantum secrets, and an extension to sharing classical secrets, which contain all known quantum-secret-sharing schemes. In this framework, we show the equivalence of existence of both schemes, that is, the existence of a scheme sharing a quantum secret implies that the extended classical-secret-sharing scheme works, and vice versa. As a consequence of this, we find schemes sharing classical secrets for arbitrary access structures. We then clarify the relationship to quantum error correction and observe several restrictions thereby imposed, which for example indicates that for pure-state threshold schemes the share size qqq must scale with the number of players nnn as q\ensuremath≥nq\ensuremath{\ge}\sqrt{n}q\ensuremath≥n​. These results also provide a way of searching for quantum-error-correcting codes.
Secret sharing is ubiquitous throughout cryptography. All possible access structures are classically feasible, and in the case of threshold access structures, the protocols are even very efficient. However, when moving to the quantum setting, the no-cloning theorem shows that many access structures are impossible. In fact, no-cloning exactly characterizes feasibility: for thresholds, quantum secret sharing is possible if and only if the threshold ttt is strictly more than n/2n/2n/2. In this work, we propose a variant of quantum secret sharing (QSS) where two or more identical copies of the input state are provided, but the output is only required to recover one copy. This notion circumvents the simple one-copy no-cloning obstruction, though the natural kkk-copy generalization still gives much milder obstructions. For thresholds using kkk copies, no-cloning implies that QSS is impossible whenever t≤n/(k+1)t\leq n/(k+1)t≤n/(k+1). It is tempting to hypothesize that no-cloning continues to exactly characterize the many-copy case. However, we show that this is not the case. We give positive results showing that multiple copies allow for going slightly beyond the single-copy obstruction: for thresholds, we construct QSS whenever t>(n−k+1)/2t>(n-k+1)/2t>(n−k+1)/2. On the other hand, we give a novel obstruction we call the Clique Path obstruction, which applies to arbitrary access structures. For thresholds, it shows that QSS is impossible whenever t≤(n−1)/kt\leq (n-1)/kt≤(n−1)/k. Our upper and lower bounds exactly match for k=2k=2k=2. Both our results leverage connections to the kkk-colorability of certain graphs derived from the access structure. We leave closing the gap for k≥3k\geq 3k≥3 copies as a fascinating direction for future work. Secret sharing is closely related to error correction, which can also be considered in the many-copy setting. Our results imply similar obstructions for quantum error correction for erasure channels.
Quantum secret-sharing and quantum error-correction schemes rely on multipartite decoding protocols, yet the nonlocal operations involved are challenging and sometimes infeasible. Here we construct a quantum secret-sharing protocol with a reduced number of quantum communication channels between the players. We introduce a scheme based on embedding a classical linear code into a quantum error-correcting code and then mapping the latter to a quantum secret-sharing protocol. In contrast to the Calderbank-Shor-Steane construction, we do not impose any restriction on the classical code; our protocol works with any arbitrary linear code. Our work paves the way towards the more general problem of simplifying the decoding of quantum error-correcting codes.
Quantum secret sharing(QSS) is a procedure of sharing classical information or quantum information by using quantum states. This paper presents how to use a [2 k − 1, 1, k ] quantum error-correcting code (QECC) to implement a quantum ( k , 2 k − 1) threshold scheme. It also takes advantage of classical enhancement of the [2 k − 1, 1, k ] QECC to establish a QSS scheme which can share classical information and quantum information simultaneously. Because information is encoded into QECC, these schemes can prevent intercept-resend attacks and be implemented on some noisy channels.
Quantum secret sharing (QSS) allows a dealer to distribute a secret quantum state among a set of parties in such a way that certain authorized subsets can reconstruct the secret, while unauthorized subsets obtain no information about it. Previous works on QSS for general access structures focused solely on the existence of perfectly secure schemes, and the share size of the known schemes is necessarily exponential even in cases where the access structure is computed by polynomial size monotone circuits. This stands in stark contrast to the classical setting, where polynomial-time computationally-secure secret sharing schemes have been long known for all access structures computed by polynomial-size monotone circuits under standard hardness assumptions, and one can even obtain shares which are much shorter than the secret (which is impossible with perfect security). While QSS was introduced over twenty years ago, previous works only considered information-theoretic privacy. In this work, we initiate the study of computationally-secure QSS and show that computational assumptions help significantly in building QSS schemes, just as in the classical case. We present a simple compiler and use it to obtain a large variety results: We construct polynomial-time computationally-secure QSS schemes under standard hardness assumptions for a rich class of access structures. This includes many access structures for which previous results in QSS necessarily required exponential share size. In fact, we can go even further: We construct QSS schemes for which the size of the quantum shares is significantly smaller than the size of the secret. As in the classical setting, this is impossible with perfect security. We also apply our compiler to obtain results beyond computational QSS. In the information-theoretic setting, we improve the share size of perfect QSS schemes for a large class of n-party access structures to 1.5^{n+o(n)}, improving upon best known schemes and matching the best known result for general access structures in the classical setting. Finally, among other things, we study the class of access structures which can be efficiently implemented when the quantum secret sharing scheme has access to a given number of copies of the secret, including all such functions in 𝖯 and NP.
In a recent work, Markham and Sanders proposed a framework to study quantum secret-sharing (QSS) schemes using graph states. This framework unified three classes of QSS protocols, namely, sharing classical secrets over private and public channels, and sharing quantum secrets. However, previous work on graph-state secret sharing mostly focused on threshold schemes. In this paper, we focus on general access structures. We show how to realize a large class of arbitrary access structures using the graph-state formalism. We show an equivalence between [[n,1]][[n,1]][[n,1]] binary quantum codes and graph-state secret-sharing schemes sharing one bit. We also establish a similar (but restricted) equivalence between a class of [[n,1]][[n,1]][[n,1]] Calderbank-Shor-Steane codes and graph-state QSS schemes sharing one qubit. With these results we are able to construct a large class of graph-state quantum secret-sharing schemes with arbitrary access structures.
We investigate quantum secret-sharing schemes constructed from ${[[n,k,\ensuremath{\delta}]]}_{D}$ nonbinary stabilizer quantum error-correcting codes with carrier qudits of prime dimension DDD. We provide a systematic way of determining the access structure, which completely determines the forbidden and intermediate structures. We then show that the information available to the intermediate structure can be fully described and quantified by what we call the information group, a subgroup of the Pauli group of kkk qudits, and we employ this group structure to construct a method for hiding the information from the intermediate structure via twirling of the information group and sharing of classical bits between the dealer and the players. Our scheme allows for the transformation of a ramp (intermediate) quantum secret-sharing scheme into a semiquantum perfect secret-sharing scheme with the same access structure as the ramp one but without any intermediate subsets, and is optimal in the amount of classical bits the dealer has to distribute.