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Secure bits with quantum pseudo-telepathy

Tobias Ambà ⁄ hl

Published Jan 1, 2009• 0 citations
Publisher

Abstract

Motivated by the concept of quantum pseudo-telepathy games as well as by quantum key distribution protocols such as [1], the security of which is based on the non-signaling principle, we try to answer the question wheather it is possible to generate a perfectly secure, shared bit between two parties by a single usage of quantum correlations and a performance of local operations on the input and output.We provide both parties with some prior shared entanglement and the possibility of postselection, e.g.discard of inputs and outputs, aiming to prevent any eavesdropper bound by the non-signaling condition from successfully performing an attack.After, in a first step, defining the framework and setting up the requirements we are going to see that quantum mechanics does not allow for a physical simulation of such correlations.For the case of binary output we will even see that, in order to make perfectly correlated bits perfectly secure, we would need maximal nonlocal correlations which also prohibits a quantum physical representation.

DOI
10.3929/ethz-a-005747634

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Motivated by the concept of quantum pseudo-telepathy games as well as by quantum key distribution protocols such as [1], the security of which is based on the non-signaling principle, we try to answer the question wheather it is possible to generate a perfectly secure, shared bit between two parties by a single usage of quantum correlations and a performance of local operations on the input and output.We provide both parties with some prior shared entanglement and the possibility of postselection, e.g.discard of inputs and outputs, aiming to prevent any eavesdropper bound by the non-signaling condition from successfully performing an attack.After, in a first step, defining the framework and setting up the requirements we are going to see that quantum mechanics does not allow for a physical simulation of such correlations.For the case of binary output we will even see that, in order to make perfectly correlated bits perfectly secure, we would need maximal nonlocal correlations which also prohibits a quantum physical representation.
Imagine that Alice and Bob, unable to communicate, are both given a 16-bit string such that the strings are either equal, or they differ in exactly 8 positions. Both parties are then supposed to output a 4-bit string in such a way that these short strings are equal if and only if the original longer strings given to them were equal as well. It is known that this task can be fulfilled without failure and without communication if Alice and Bob share 4 maximally entangled quantum bits. We show that, on the other hand, they CANNOT win the same game with certainty if they only share classical bits, even if it is an unlimited number. This means that for fulfilling this particular distributed task, quantum entanglement can completely replace communication. This phenomenon has been called pseudo-telepathy. The results of this paper complete the analysis of the first proposed game of this type between two players.
In this work we have introduced two party pseudo telepathy games with respective winning conditions. One cannot win these games deterministically by any kind of classical protocols. Interestingly we find out that in quantum world, these winning conditions can be achieved if the players share an entangled state. We also introduced a game which is impossible to win if the players are form classical world, (both probabilistically and deterministically) yet there exists a perfect quantum strategy by following which, one can attain the winning condition of the game.
Quantum cryptography shows that one can guarantee the secrecy of correlation on the sole basis of the laws of physics, that is, without limiting the computational power of the eavesdropper. The usual security proofs suppose that the authorized partners, Alice and Bob, have a perfect knowledge and control of their quantum systems and devices; for instance, they must be sure that the logical bits have been encoded in true qubits and not in higher dimensional systems. In this paper, we present an approach that circumvents this strong assumption. We define protocols, both for the case of bits and for generic ddd-dimensional outcomes, in which the security is guaranteed by the very structure of the Alice-Bob correlations, under the no-signaling condition. The idea is that if the correlations cannot be produced by shared randomness, then Eve has poor knowledge of Alice's and Bob's symbols. The present study assumes on the one hand that the eavesdropper Eve performs only individual attacks (this is a limitation to be removed in further work), and on the other hand that Eve can distribute any correlation compatible with the no-signaling condition (in this sense her power is greater than what quantum physics allows). Under these assumptions, we prove that the protocols defined here allow extracting secrecy from noisy correlations, when these correlations violate a Bell-type inequality by a sufficiently large amount. The region in which secrecy extraction is possible extends within the region of correlations achievable by measurements on entangled quantum states.
By carrying out measurements on entangled states, two parties can generate a secret key which is secure not only against an eavesdropper bound by the laws of quantum mechanics, but also against a hypothetical 'post-quantum' eavesdroppers limited by the no-signalling principle only. We introduce a family of quantum key distribution protocols of this type, which are more efficient than previous ones, both in terms of key rate and noise resistance. Interestingly, the best protocols involve large number of measurements. We show that in the absence of noise, these protocols can yield one secret bit per entanglement bit, implying that the key rates in the no-signalling post-quantum scenario are comparable to the key rates in usual quantum key distribution.
By carrying out measurements on entangled states, two parties can generate a secret key which is secure not only against an eavesdropper bound by the laws of quantum mechanics, but also against a hypothetical "post-quantum" eavesdroppers limited by the no-signalling principle only. We introduce a family of quantum key distribution protocols of this type, which are more efficient than previous ones, both in terms of key rate and noise resistance. Interestingly, the best protocols involve large number of measurements. We show that in the absence of noise, these protocols can yield one secret bit per entanglement bit, implying that the key rates in the no-signalling post-quantum scenario are comparable to the key rates in usual quantum key distribution.
Quantum pseudo-telepathy is an intriguing phenomenon which results from the application of quantum information theory to communication complexity. To demonstrate this phenomenon researchers in the field of quantum communication complexity devised a number of quantum non-locality games. The setting of these games is as follows: the players are separated so that no communication between them is possible and are given a certain computational task. When the players have access to a quantum resource called entanglement, they can accomplish the task: something that is impossible in a classical setting. To an observer who is unfamiliar with the laws of quantum mechanics it seems that the players employ some sort of telepathy; that is, they somehow exchange information without sharing a communication channel. This paper provides a formal framework for specifying, implementing, and analysing quantum non-locality games.