arXiv:quant-ph/0412058

The fundamental problem of quantum cryptography

Marcin Pawłowski, Marek Czachor

Published Dec 7, 2004• 0 citations

Abstract

It is known that statistical predictions of quantum theory do not depend on its interpretation. In particular, an experiment cannot distinguish between the Copenhagen interpretation (involving no hidden variables) and the de Broglie-Bohm interpretation based on nonlocal hidden variables. Quantum cryptographic protocols, such as BB84 or E91, are secure and mutually equivalent as long as one works within the framework of Copenhagen interpretation. But they are inequivalent and insecure if one considers attacks allowed by the de Broglie-Bohm interpretation. The fundametal problem of quantum cryptography is therefore this: Are all the statements about security of quantum protocols based on our belief in one of the two allowed interpretations of quantum mechanics? We show that this is not the case. Ekert-type protocols can be modified in a way that makes them secure even if the de Broglie-Bohm nonlocal hidden variables exist. Bennett-Brassard-type cryptography does not seem to allow for such a correction.

DOI: 10.48550/arxiv.quant-ph/0412058

Related papers

Security in quantum cryptography

Christopher Portmann, Renato Renner

Secure communications is a vital need of the world, and the basic features of quantum mechanics seemingly offer fundamental new tools for achieving this security. But do these tools really work? This review gives due attention to the real-world problems with experimental quantum cryptography, but the heart of the review is an in-depth survey of the analyses developed to prove the security of the basic quantum techniques introduced by Wiesner, Ekert, Bennett, and Brassard. A broader set of cryptographic tasks have quantum aspects which are also discussed here.

arXiv:2102.00021Reviews of Modern PhysicsJun 29, 2022405 citations

Entangled-state cryptographic protocol that remains secure even if nonlocal hidden variables exist and can be measured with arbitrary precision

Diederik Aerts, Marek Czachor, Marcin Pawłowski

Standard quantum cryptographic protocols are not secure if one assumes that nonlocal hidden variables exist and can be measured with arbitrary precision. The security can be restored if one of the communicating parties randomly switches between two standard protocols.

arXiv:quant-ph/0501003Physical Review AMar 14, 20063 citations

The de Broglie–Bohm theory is and is not a hidden variable theory

Jean Bricmont

Abstract We will first define what is meant by “hidden variables”. Then, we will review theorems proving the impossibility of theories introducing such variables in a naive way. Next, we will outline the de Broglie-Bohm theory and show that it is not refuted by those theorems. To do that, we will explain how “measurements” work in that theory, using the examples of spin measurements and of momentum measurements. Those measurements do not reveal pre-existing properties of the quantum particle but are the result of interactions between that particle and the measuring apparatus. And this is why the de Broglie-Bohm theory is not refuted by the theorems on the impossibility of hidden variables. We will also explain the relationship between the no “hidden variables” theorems and nonlocality, with and without introducing Bell’s inequalities.

Apr 1, 20251 citation

Towards a formal definition of security for quantum protocols

Claude Crépeau, Jeroen van de Graaf

This thesis presents a formal model for quantum protocols, that is, cryptographic protocols based on quantum mechanics. It also develops a definition of security (which includes notions such as privacy and correctness) for quantum protocols. First, existing classical (non-quantum) definitions of security are studied. There are essentially two approaches: one based on Turing Machines and simulation, the other using information theory. Both approaches are explored in detail and compared, then a particular variation of the first approach is chosen. Security is defined by introducing an interface (also called a simulator) which is capable of translating an attack on protocol 1 into an attack on protocol 2. The profile of a protocol is a stochastic variable, defined as the output of the participants and the view of the adversary who performs the attack. If an interface exists such that the profile of protocol 1 is indistinguishable from the profile of protocol 2 with the help of this interface, then protocol 1 is at least as secure as protocol 2. An absolute notion of security is obtained by comparing the protocol implementation with its 'ideal' equivalent. In this approach, and in classical cryptography in general, indistinguishability of probability distributions is a crucial tool. The meaning of this notion is explored in the context of density matrices, the mathematical formalism which is used to describe the state of a quantum system. Various measures of distinguishability of quantum states relevant from a cryptographic point of view are investigated: probability of error, trace norm (related to the Kolmogorov distance), overlap (related to the Bhattacharyya coefficient) and Shannon distinguishability (defined through the mutual information). It is shown that if we require that the distinguishability between two families of quantum states vanish exponentially fast, these four distinguishability measures are equivalent. Along the way upper and lower bounds for the Shannon distinguishability are obtained. This is a very useful result, because explicit computation of the Shannon distinguishability between two quantum states is often difficult (it requires solving a transcendental equation), whereas the other distinguishability measures, in particular the trace norm, are usually much easier to compute. In the last chapter the two previous topics are combined. A formal model for quantum protocols is formulated. It uses Quantum Turing Machines to represent the participants, who communicate through a quantum channel. The different nature of quantum information as opposed to classical information is explored in detail, including the implications this has for the definition. As in the classical case, we introduce an interface to compare the security of two protocols for the same task. Subsequently some consequences of the proposed definition are investigated. In particular, strong evidence is given that one classical technique often used in proofs of security for classical protocols, called rewinding, is impossible in the quantum setting.

Jan 1, 199847 citations

The De Broglie-Bohm theory {\it is} and {\it is not} a hidden variable theory

Jean Bricmont

We will first define what is meant by ``hidden variables". Then, we will review various theorems proving the impossibility of theories introducing such variables and then show that the de Broglie-Bohm theory is not refuted by those theorems. We will also explain the relation between those theorems and nonlocality, with or without introducing Bell's inequalities.

arXiv:2307.05148Jul 11, 20230 citations

De Broglie-Bohmova interpretacija kvantne mehanike

Doris Todorović

The de Broglie–Bohm interpretation (Bohmian mechanics, hidden-variables theory, causal interpretation or pilot-wave theory) is one of the interpretations of quantum mechanics which opposes to the claim of standard interpretation on the completeness of description of the state of quantummechanical system, that description in Bohmian mechanics is completed by defining positions of the particles of the system i.e. hidden variables and their dynamics. These facts are the basis of motivation for the introduction of a new theory that can describe dubious axiomatic requirements of standard interpretation. The presented theory is deterministic and explicitly nonlocal. Originators of Bohmian mechanics were L. de Broglie (1927.) and D. Bohm (1952.) who presented mathematically more general and conceptual elaborate formulation. Along with the historical overview, a note is given about the main properties of the theory, such as the issue of nonlocality and hidden variables. The aim of this paper is to present and explain the main ideas and properties of the theory as well as to give a systematic overview of present and historical mathematical construction of the theory and most of all to discuss implications of de Broglie–Bohm theory on the ontology of quantum-mechanical picture, showcasing examples of the equivalency of the description of a free particle as a propagation of the plane wave in space or as the ensemble of the trajectories and of the spreading of the Gaussian wave packet. The main example of the obvious conceptual clashing of the interpretations is presented: the double-slit experiment, with a detailed examination of the Bohmian mechanics’ predictions using presented formalism on the two double-slit experiment. The graphs of trajectories for given examples are obtained.

Sep 11, 20190 citations