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arXiv:quant-ph/9902048 · Physical Review Letters

Quantum Cryptography on Noisy Channels: Quantum versus Classical Key-Agreement Protocols

Nicolas Gisin, Stefan Wolf

Published Nov 15, 1999• 67 citations
arXivHTMLPDF

Abstract

When the 4-state or the 6-state protocol of quantum cryptography is carried out on a noisy quantum channel, then the raw key has to be processed to reduce the information of a spy down to an arbitrarily low value, providing Alice and Bob with a secret key. In principle, quantum algorithms as well as classical algorithms can be used for this processing. A natural question is: Up to which error rate on the raw key is a secret-key agreement at all possible? Under the assumption of incoherent eavesdropping, we find that the quantum and classical limits are precisely the same: As long as Alice and Bob share some entanglement, both quantum and classical protocols provide secret keys.

DOI
10.1103/physrevlett.83.4200

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When the 4-state or the 6-state protocol of quantum cryptography is carried out on a noisy quantum channel, then the raw key has to be processed to reduce the information of a spy down to an arbitrarily low value, providing Alice and Bob with a secret key. In principle, quantum algorithms as well as classical algorithms can be used for this processing. A natural question is: Up to which error rate on the raw key is a secret-key agreement at all possible? Under the assumption of incoherent eavesdropping, we find that the quantum and classical limits are precisely the same: As long as Alice and Bob share some entanglement, both quantum and classical protocols provide secret keys.
After carrying out a protocol for quantum key agreement over a noisy quantum channel, the parties Alice and Bob must process the raw key in order to end up with identical keys about which the adversary has virtually no information. In principle, both classical and quantum protocols can be used for this processing. It is a natural question which type of protocols is more powerful. We prove for general states but under the assumption of incoherent eavesdropping that Alice and Bob share some so-called intrinsic information in their classical random variables, resulting from optimal measurements, if and only if the parties' quantum systems are entangled. In addition, we provide evidence that the potentials of classical and of quantum protocols are equal in every situation. Consequently, many techniques and results from quantum information theory directly apply to problems in classical information theory, and vice versa. For instance, it was previously believed that two parties can carry out unconditionally secure key agreement as long as they share some intrinsic information in the adversary's view. The analysis of this purely classical problem from the quantum information-theoretic viewpoint shows that this is true in the binary case, but false in general. More explicitly, bound entanglement, i.e., entanglement that cannot be purified by any quantum protocol, has a classical counterpart. This "bound intrinsic information" cannot be distilled to a secret key by any classical protocol. As another application we propose a measure for entanglement based on classical information-theoretic quantities.
Abstract. After carrying out a protocol for quantum key agreement over a noisy quantum channel, the parties Alice and Bob must process the raw key in order to end up with identical keys about which the adversary has virtually no information. In principle, both classical and quantum protocols can be used for this processing. It is a natural question which type of protocols is more powerful. We show that the limits of tolerable noise are identical for classical and quantum protocols in many cases. More specifically, we prove that a quantum state between two parties is entangled if and only if the classical random variables resulting from optimal measurements provide some mutual classical information between the parties. In addition, we present evidence which strongly suggests that the potentials of classical and of quantum protocols are equal in every situation. An important consequence, in the purely classical regime, of such a correspondence would be the existence of a classical counterpart of so-called bound entanglement, namely “bound information” that cannot be used for generating a secret key by any protocol. This stands in sharp contrast to what was previously believed. Keywords. Secret-key agreement, intrinsic information, secret-key rate, quantum privacy amplification, purification, entanglement. 1
Characterizing secret communication over noisy quantum channels is an interesting problem from both a practical and theoretical perspective. Suppose Alice and Bob wish to communicate secret information so that an eavesdropper Eve will not suspect any type of encoded communication between the two. Classical or quantum cryptography will not suffice since it is always clear secret communication is taking place. Therefore Alice and Bob must execute what is known as a quantum steganographic protocol. Assuming Eve only has partial knowledge of the channel connecting Alice and Bob, we show that for the bit-flip and depolarizing channels Alice can use Eve's lack of knowledge of the channel parameter to encode quantum information steganographically. We give an explicit encoding procedure and calculate the rate at which Alice and Bob can communicate secretly. We also show that our encoding is optimal for nondegenerate quantum codes. We calculate the rate at which a secret key must be consumed. Finally, we discuss the possibility of steganographic communication over more general quantum channels and conjecture a general formula for the steganographic rate.
Steganography is the technique of hiding secret information by embedding it in a seemingly ``innocent'' message. We present protocols for hiding quantum information by disguising it as noise in a codeword of a quantum error-correcting code. The sender (Alice) swaps quantum information into the codeword and applies a random choice of unitary operation, drawing on a secret random key she shares with the receiver (Bob). Using the key, Bob can retrieve the information, but an eavesdropper (Eve) with the power to monitor the channel, but without the secret key, cannot distinguish the message from channel noise. We consider two types of protocols: one in which the hidden quantum information is stored locally in the codeword, and another in which it is embedded in the space of error syndromes. We analyze how difficult it is for Eve to detect the presence of secret messages, and estimate rates of steganographic communication and secret key consumption for specific protocols and examples of error channels. We consider both the case where there is no actual noise in the channel (so that all errors in the codeword result from the deliberate actions of Alice), and the case where the channel is noisy and not controlled by Alice and Bob.
We have implemented the 'six-state' quantum cryptography protocol using polarization-entangled photon pairs, in which the polarization of each photon of a pair is measured in one of three randomly chosen bases. For a given amount of eavesdropping, this protocol results in a larger error rate than in four- or two-state protocols, but reduces the number of key-producing events. We have experimentally investigated several incoherent eavesdropping strategies, and verified the predicted enhancement in error rate. However, we demonstrate that for low error rates, the efficiency for secret key generation is higher when using the four-state protocol.
It has been said that quantum cryptography in general offers a secure solution to the problem of key enhancement. This means that two parties who already share a small secret key, can use quantum protocols to establish a new large secret key. This large secret key can be arbitrarily long and is unbreakable. Thus, to date, the main contribution of quantum cryptography has been believed to be quantum key enhancement. This paper shows that quantum cryptography can do significantly more. The quantum protocol described here distributes an unbreakable secret key to the two parties by relying on public information only. This is the first time that quantum cryptography is shown to be able to produce secret information using only public information. This contribution is also unique for cryptography in general, classical and quantum.