Quantum key distribution without the wavefunction Gerd Niestegge Zillertalstrasse 39, 81373 M¨unchen, Germany [email protected] Abstract A well-known feature of quantum mechanics is the secure exchange of secret bit strings which can then be used as keys to encrypt messages transmitted over any classical communication channel. It is demonstrated that this quantum key distribution allows a much more general and ab- stract access than commonly thought. The results include some gener- alizations for the Hilbert space version of quantum key distribution, but base upon a general non-classical extension of conditional probability. A special state-independent conditional probability is identified as origin of the superior security of quantum key distribution and may have more profound implications for the foundations and interpretation of quantum mechanics, quantum information theory, and the philosophical question what actually constitutes physical reality. Key Words: Quantum key distribution, No-cloning theorem, Founda- tions of quantum theory, Quantum logic PACS: 03.67.Dd, 03.65.Ta 1. Introduction In the past thirty years, quantum cryptography has been a large field of exten- sive theoretical and experimental research. Several quantum key distribution protocols have been elaborated, their technical feasibility has been discussed, experimental set-ups and some practical implementations have been realized. Quantum key distribution may be the first commercial application of quantum information theory. A vast amount of papers on these topics are available by now (Each of the two pioneering works [7] and [18] has more than ten thousand and almost eight thousand quotations, respectively; for a review, see [20] or [37]). The present paper, however, focuses on the theoretical foundations and concentrates on the two best-known quantum key distribution protocols. Quantum mechanics renders possible the secure exchange of secret bit strings which can then be used as keys to encrypt messages transmitted over a classical communication channel. The quantum no-cloning theorem [16, 43, 45] ensures that an eavesdropper cannot steal the key without being detected. These results follow from the usual quantum mechanical Hilbert space formalism. After it could recently be shown in Ref.[32] that the quantum no-cloning theorem can be extended to a much more general and abstract framework - the quantum 1 logics which possess a conditional probability calculus -, a similar extension of quantum key distribution shall be presented in this paper. A quantum logic E is a purely algebraic structure. It is quite common to use an orthomodular partially ordered set or lattice [5, 9, 25, 34]. Its elements can be interpreted in different ways: as events, propositions or properties of the quantum system under consideration. This is why they are called propositions by some authors and events, preferably in probabilistic approaches, by others. States are then defined in the same way as the classical probability measures, and conditional probabilities are postulated to behave like the classical ones on compatible subsets of E. Note that a subset is called compatible if it is contained in another subset of E forming a Boolean algebra (i.e., it is contained in a classical subsystem of E) [11]. Some quantum logics entail unique conditional probabilities, many others don't. The classical Boolean algebras and the Hilbert space quantum logic (consisting of the closed subspaces or, equivalently, the selfadjoint projection operators) do and, in the latter case, conditionalization becomes identical with the state transition of the L¨uders- von Neumann (i.e., projective) quantum measurement process [30]. Therefore, the quantum logics which possess unique conditional probabilities can be regarded as a generalized model of projective quantum measurement. Examples that are neither classical Boolean algebras nor Hilbert space quantum logics (nor sublogics of them) can be constructed using the exceptional real Jordan algebras [30, 31]. In this framework, a very special type of conditional probability emerges in the non-classical case [30, 31]. It describes the probability for the transition from a past event e to a future event f, independently of any underlying state, and results solely from the algebraic structure of the quantum logic E. This probability exists only for certain pairs e and f in E. It exists for all f 2 E, if e is minimal in E (i.e., e is an atom). The states resulting in this way are called atomic. They represent a generalization of the pure states in the usual quantum mechanical Hilbert space setting. After the early pioneering work by Birkhoff and von Neumann in 1936 [10], quantum logics have been studied extensively between 1960 and 1995 [5, 9, 25, 26, 33, 34, 39, 40, 41]. Various forms of conditional probability have also been considered [5, 13, 17, 19, 23, 24]. However, the quantum logics which possess unique conditional probabilities and particularly the special type of the state- independent conditional probability had not received any attention before the author's work [29, 30, 31]. This special type of conditional probability is used in the present paper to study two well-known quantum key distribution protocols (including a variant of one of them). The first one was invented by Charles H. Bennett and Gilles Brassard in 1984 [7], based on earlier work in the 1970ies by Stephen Wiesner which was not published before 1983 [42]. Its variant is due to the first co- inventor [6]. The second protocol was proposed by Artur K. Ekert in 1991 [18]. These quantum key distribution protocols are generalized in a two-fold way. First, they are transferred to the quantum logics possessing the conditional 2 probability calculus; this shows that quantum key distribution does not nec- essarily require the full-blown Hilbert space formalism of quantum mechanics and works as well in a much more general and abstract model of the projective quantum measurement process. Second, system properties are used instead of states. Pure states, wavefunctions and their collapse are not needed. Before turning to quantum key distribution, the no-cloning theorem of Ref. [32] is revisited and extended by treating the cloning of system properties, since only the cloning of atomic states was considered in Ref. [32]. Moreover, the special type of conditional probability mentioned above is identified as the quantum origin of the no-cloning theorem as well as of the superior security of the key distribution processes. Other generalizations of the no-cloning theorem [1, 2] and quantum key distribution [4], using the generalized probabilistic theories, do not cover the results presented here, since they do not capture this special type of conditional probability. The paper is organized as follows. The next two sections restate some ma- terial from Refs. [30, 31, 32] as far as needed in the present paper. In section 2, the algebraic structure of a quantum logic is outlined and, in section 3, the conditional probability calculus. In section 4, the extension of the no-cloning theorem is introduced. The generalized versions of the quantum key distribu- tion protocols are presented in section 5. 2. Compatibility in orthomodular partially ordered sets In quantum mechanics, the measurable quantities of a physical system are re- presented by observables. Most simple are those observables where only the two values `true' and `false' (or `1' and `0') are possible as measurement outcome. They are elements of a mathematical structure called quantum logic, are usually called propositions, and they are called events in probabilistic approaches. The elements of the quantum logic can also be understood as potential properties of the system under consideration. If, for instance, this system is an electron, its spin along each spatial axis ~x can take one of the two values +~=2 and −~=2 and, for each ~x, `spin along ~x = +~=2' and `spin along ~x = −~=2' represent two elements of the quantum logic. In this paper, a quantum logic shall be an orthomodular partially ordered set E with the partial ordering ≤, the orthocomplementation 0, the smallest element 0 and the largest element I [5, 9, 25, 34]. This means that the following conditions are satisfied by all e; f 2 E: (A) e ≤ f implies f 0 ≤ e0. (B) (e0)0 = e. (C) e ≤ f 0 implies e _ f, the supremum of e and f, exists. (D) e _ e0 = I. (E) f ≤ e implies e = f _ (e ^ f 0). (orthomodular law) 3 Here, e ^ f denotes the infimum of e and f, which exists if and only if e0 _ f 0 exists. Two elements e; f 2 E are called orthogonal if e ≤ f 0 or, equivalently, f ≤ e0. An element e 6= 0 in E is called an atom if there is no element f in E with f ≤ e and 0 6= f 6= e. The interpretation of this mathematical terminology is as follows: two or- thogonal elements represent mutually exclusive events, propositions or system properties; e0 is the negation of e, and e_f is the disjunction of the two exclusive elements e and f. It is not assumed that E is a lattice (in a lattice, there is a smallest upper bound e _ f and largest lower bound e ^ f for any two elements e and f). If E were a lattice and satisfied the distributivity law e ^ (f _ g) = (e ^ f) _ (e ^ g) for all e; f; g 2 E, E would become a Boolean lattice or Boolean algebra. The orthomodular law is a weakening of the distributivity law. Classical probability theory uses Boolean algebras as mathematical structure for the random events, and it can be expected that those subsets of E, which are Boolean algebras, behave classically. Therefore, a subset E0 of E is called compatible if there is a Boolean algebra B with E0 ⊆ B ⊆ E. Two elements e and f in E are called compatible, if fe; fg forms a compatible subset.