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arXiv:quant-ph/0005042

Linking Classical and Quantum Key Agreement: Is There "Bound Information"?

Nicolas Gisin, Stefan Wolf

Published May 11, 2000• 3 citations
arXivHTMLPDF

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 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.

DOI
10.48550/arxiv.quant-ph/0005042

Related papers

Linking Classical and Quantum Key Agreement: Is There "Bound Information"?

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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

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Non-additivity is one of the distinctive traits of Quantum Information Theory: the combined use of quantum objects may be more advantageous than the sum of their individual uses. Non-additivity effects have been proven, for example, for quantum channel capacities, entanglement distillation or state estimation. In this work, we consider whether non-additivity effects can be found in Classical Information Theory. We work in the secret-key agreement scenario in which two honest parties, having access to correlated classical data that are also correlated to an eavesdropper, aim at distilling a secret key. Exploiting the analogies between the entanglement and the secret-key agreement scenario, we provide some evidence that the secret-key rate may be a non-additive quantity. In particular, we show that correlations with conjectured bound information become secret-key distillable when combined. Our results constitute a new instance of the subtle relation between the entanglement and secret-key agreement scenario.

Quantum Information and Computation•Mar 1, 2013•1 citation
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
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.
It was recently pointed out that there is a close connection between information-theoretic key agreement and quantum entanglement purification. This suggests that the concept of bound entanglement (entanglement which cannot be purified) has a classical counterpart: bound information, which cannot be used to generate a secret key by any protocol. We analyse a probability distribution which results when a specific bound entangled quantum state is measured. We show strong evidence for the fact that the corresponding mutual information is indeed bound. The probable existence of such information stands in contrast to previous beliefs in classical information theory. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
We consider the extraction of shared secret key from correlations that are generated by either a classical or quantum source. In the classical setting, two honest parties (Alice and Bob) use public discussion and local randomness to distill secret key from some distribution pXYZp_{XYZ}pXYZ​ that is shared with an unwanted eavesdropper (Eve). In the quantum settings, the correlations pXYZp_{XYZ}pXYZ​ are delivered to the parties as either an \textit{incoherent} mixture of orthogonal quantum states or as \textit{coherent} superposition of such states; in both cases, Alice and Bob use public discussion and local quantum operations to distill secret key. While the power of quantum mechanics increases Alice and Bob's ability to generate shared randomness, it also equips Eve with a greater arsenal of eavesdropping attacks. Therefore, it is not obvious who gains the greatest advantage for distilling secret key when replacing a classical source with a quantum one. In this paper we first demonstrate that the classical key rate is equivalent to the quantum key rate when the correlations are generated incoherently in the quantum setting. For coherent sources, we next show that the rates are incomparable, and in fact, their difference can be arbitrarily large in either direction. However, we identify a large class of non-trivial distributions pXYZp_{XYZ}pXYZ​ that possess the following properties: (i) Eve's advantage is always greater in the quantum source than in its classical counterpart, and (ii) for the quantum entanglement shared between Alice and Bob in the coherent source, the so-called entanglement cost/squashed entanglement/relative entropy of entanglement can all be computed. With property (ii), we thus present a rare instance in which the various entropic entanglement measures of a quantum state can be explicitly calculated.
Secure key distribution between two remote parties is impossible when both are classical, unless some unproven (and arguably unrealistic) computation-complexity assumptions are made, such as the difficulty of factorizing large numbers. On the other hand, a secure key distribution is possible when both parties are quantum. What is possible when only one party (Alice) is quantum, yet the other (Bob) has only classical capabilities? Recently, a semiquantum key distribution protocol was presented [M. Boyer, D. Kenigsberg, and T. Mor, Phys. Rev. Lett. 99, 140501 (2007)], in which one of the parties (Bob) is classical, and yet, the protocol is proven to be completely robust against an eavesdropping attempt. Here we extend that result much further. We present two protocols with this constraint and prove their complete robustness against attacks: we prove that any attempt of an adversary to obtain information (and even a tiny amount of information) necessarily induces some errors that the legitimate parties could notice. One protocol presented here is identical to the one referred to above; however, its robustness is proven here in a much more general scenario. The other protocol is very different as it is based on randomization.
Non-additivity is one of the distinctive traits of Quantum Information Theory: the combined use of quantum objects may be more advantageous than the sum of their individual uses. Non-additivity effects have been proven, for example, for quantum channel capacities, entanglement distillation or state estimation. In this work, we consider whether non-additivity effects can be found in Classical Information Theory. We work in the secret-key agreement scenario in which two honest parties, having access to correlated classical data that are also correlated to an eavesdropper, aim at distilling a secret key. Exploiting the analogies between the entanglement and the secret-key agreement scenario, we provide some evidence that the secret-key rate may be a non-additive quantity. In particular, we show that correlations with conjectured bound information become secret-key distillable when combined. Our results constitute a new instance of the subtle relation between the entanglement and secret-key agreement scenario.