Dualities

C. Y. Li, W. Y. Wang, C. Wang +6 more

Quantum mechanical systems exhibit particle wave duality property. This duality property has been exploited for information processing. A duality quantum computer is a quantum computer on the move and passing through a multi‐slits. It offers quantum wave divider and quantum wave combiner operations in addition to those allowed in an ordinary quantum computer. It has been shown that all linear bounded operators can be realized in a duality quantum computer, and a duality quantum computer with n qubits and d‐slits can be realized in an ordinary quantum computer with n qubits and a qudit in the so‐called duality quantum computing mode. The quantum particle‐wave duality can be used in providing secure communication. In this paper, we will review duality quantum computing and duality quantum key distribution.

Sebastian De Haro, Enrico Cinti

Emilio Cobanera, Gerardo Ortíz, Zohar Nussinov

An algebraic theory of dualities based on the notion of bond algebras is developed. This framework deals with classical and quantum dualities in a unified fashion and its implementation includes discrete lattice, continuum field, and gauge theories. Dualities are revealed to be structure-preserving local mappings between model-specific bond algebras, and can be implemented as unitary (or projective-unitary) transformations. This characterization allows to search systematically for dualities or self-dualities in, finite or infinite, quantum and classical models of arbitrary complexity and number of spatial dimensions. Non-local transformations, like dual or disorder variables and Jordan-Wigner dictionaries, are algorithmically derived from these local duality mappings of bonds. An exact connection between quantum dualities and the low temperature (strong-coupling)/high temperature (weak-coupling) dualities of classical statistical mechanics (or path integrals), as well as the connection between quantum dual and classical disorder variables, is thus established. Our bond-algebraic approach goes beyond the former standard approach to classical dualities, and could help resolve the long standing problem of non-Abelian dualities. In particular, we present new dualities in any spatial dimension for the quantum Heisenberg model. Remarkably, the elimination of gauge constraints can be handled as a bond-algebraic duality (a gauge-reducing duality). In this way, we show that the 2 Higgs model is dual to the extended toric code model in any space dimension. We also show how, in specific models, the phenomenon of dimensional reduction can be accomplished by exact duality mappings. Finally, we discuss various applications including location of phase boundaries, spectral behaviour and, notably, show how bond-algebraic dualities help constrain and realize fermionization in an arbitrary number of spatial dimensions.

Sebastian De Haro, Jeremy Butterfield

Abstract This Chapter gives an in-depth discussion of four simple examples, and of how they illustrate the Schema: (i) Position-momentum duality in elementary quantum mechanics amounts to a Fourier transformation. The position and momentum models each represent the Hilbert space and its algebra of operators. (ii) Wave-particle duality, though often called a ‘duality’, is in general not a duality in the Schema’s sense. Thus the Chapter disambiguates different uses of ‘wave-particle duality’ and clarifies the relation of these uses to the conception used in the Schema. It also discusses ‘complementarity’. (iii) Electric-magnetic duality in Maxwell’s theory. Here, the interpretation of the common core is discussed in analogy with the interpretation of the Lorentz transformations. Dirac monopoles are also discussed. (iv) Kramers-Wannier duality for the Ising model. This is paradigmatic for many physical behaviours associated with dualities, and is the book’s first example of a duality for a probabilistic theory.