Approximate Quantum Error-Correcting Codes and Secret Sharing Schemes

It is a standard result in the theory of quantum error-correcting codes that no code of length n can fix more than n/4 arbitrary errors, regardless of the dimension of the coding and encoded Hilbert spaces. However, this bound only applies to codes which recover the message exactly. Naively, one might expect that correcting errors to very high fidelity would only allow small violations of this bound. This intuition is incorrect: in this paper we describe quantum error-correcting codes capable of correcting up to (n-1)/2 arbitrary errors with fidelity exponentially close to 1, at the price of increasing the size of the registers (i.e., the coding alphabet). This demonstrates a sharp distinction between exact and approximate quantum error correction. The codes have the property that any $t$ components reveal no information about the message, and so they can also be viewed as error-tolerant secret sharing schemes. The construction has several interesting implications for cryptography and quantum information theory. First, it suggests that secret sharing is a better classical analogue to quantum error correction than is classical error correction. Second, it highlights an error in a purported proof that verifiable quantum secret sharing (VQSS) is impossible when the number of cheaters t is n/4. More generally, the construction illustrates a difference between exact and approximate requirements in quantum cryptography and (yet again) the delicacy of security proofs and impossibility results in the quantum model.Comment: 14 pages, no figure

Communicating over adversarial quantum channels using quantum list codes

We study quantum communication in the presence of adversarial noise. In this setting, communicating with perfect fidelity requires using a quantum code of bounded minimum distance, for which the best known rates are given by the quantum Gilbert-Varshamov (QGV) bound. By asking only for arbitrarily high fidelity and allowing the sender and reciever to use a secret key with length logarithmic in the number of qubits sent, we achieve a dramatic improvement over the QGV rates. In fact, we find protocols that achieve arbitrarily high fidelity at noise levels for which perfect fidelity is impossible. To achieve such communication rates, we introduce fully quantum list codes, which may be of independent interest.Comment: 6 pages. Discussion expanded and more details provided in proofs. Far less unclear than previous versio

Quantum cryptography: key distribution and beyond

Uniquely among the sciences, quantum cryptography has driven both foundational research as well as practical real-life applications. We review the progress of quantum cryptography in the last decade, covering quantum key distribution and other applications.Comment: It's a review on quantum cryptography and it is not restricted to QK

Pauli Manipulation Detection codes and Applications to Quantum Communication over Adversarial Channels

We introduce and explicitly construct a quantum code we coin a "Pauli Manipulation Detection" code (or PMD), which detects every Pauli error with high probability. We apply them to construct the first near-optimal codes for two tasks in quantum communication over adversarial channels. Our main application is an approximate quantum code over qubits which can efficiently correct from a number of (worst-case) erasure errors approaching the quantum Singleton bound. Our construction is based on the composition of a PMD code with a stabilizer code which is list-decodable from erasures. Our second application is a quantum authentication code for "qubit-wise" channels, which does not require a secret key. Remarkably, this gives an example of a task in quantum communication which is provably impossible classically. Our construction is based on a combination of PMD codes, stabilizer codes, and classical non-malleable codes (Dziembowski et al., 2009), and achieves "minimal redundancy" (rate $1-o(1)$)

Approximate reconstructability of quantum states and noisy quantum secret sharing schemes

We introduce and analyse approximate quantum secret sharing in a formal cryptographic setting, wherein a dealer encodes and distributes a quantum secret to players such that authorized structures (sets of subsets of players) can approximately reconstruct the quantum secret and omnipotent adversarial agents controlling non-authorized subsets of players are approximately denied the quantum secret. In particular, viewing the map encoding the quantum secret to shares for players in an authorized structure as a quantum channel, we show that approximate reconstructability of the quantum secret by these players is possible if and only if the information leakage, given in terms of a certain entanglement-assisted capacity of the complementary quantum channel to the players outside the structure and the environment, is small.Comment: 6 pages, 1 figur