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KARŞILIKLI TARAFSIZ BAZLARIN SONLU CİSİMLER ÜZERİNDEKİ BAZI İNŞALARI

Elmas, Gökhan

Published Feb 27, 2024• 0 citations
OpenAlex

Abstract

Mutually unbiased bases, as a mathematical concept, has important implications in quantum information theory where the information is encoded as linear combinations of vectors in Hilbert spaces instead of as arrays of digits. Offering a designation on the preparation and measurement of the quantum states, mutually unbiased bases provide mathematics based security to many quantum computation protocols including the famous quantum key distribution protocol named BB84. The construction of mutually unbiased bases, however, is not straight forward and it requires an extensive mathematical approach based on the properties of the finite fields.

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Mutually unbiased bases, as a mathematical concept, has important implications in quantum information theory where the information is encoded as linear combinations of vectors in Hilbert spaces instead of as arrays of digits. Offering a designation on the preparation and measurement of the quantum states, mutually unbiased bases provide mathematics based security to many quantum computation protocols including the famous quantum key distribution protocol named BB84. The construction of mutually unbiased bases, however, is not straight forward and it requires an extensive mathematical approach based on the properties of the finite fields.
The basic methods of constructing the sets of mutually unbiased bases in the Hilbert space of an arbitrary finite dimension are reviewed and an emerging link between them is outlined. It is shown that these methods employ a wide range of important mathematical concepts like, e.g., Fourier transforms, Galois fields and rings, finite and related projective geometries, and entanglement, to mention a few. Some applications of the theory to quantum information tasks are also mentioned. KEY WORDS: mutually unbiased bases; d-dimensional Hilbert space; Galois fields and rings; maximally entangled states.
Finding all the mutually unbiased bases in various dimensions is a problem of fundamental interest in quantum information theory and pure mathematics. The general problem formulated in finite-dimensional Hilbert spaces is open. In the categorical approach to quantum mechanics one can find examples of categories which behave ``like'' the category of finite-dimensional Hilbert spaces in various ways but are subtly different. One such category is the category of sets and relations, Rel\mathbf{Rel}Rel. One can formulate the concept of mutually unbiased bases here as well. In this note we classify all the mutually unbiased bases in this category by relating it to a standard question in combinatorics.
One of the essential features of quantum mechanics is that most pairs of observables cannot be measured simultaneously. This phenomenon manifests itself most strongly when observables are related to mutually unbiased bases. In this paper, we shed some light on the connection between mutually unbiased bases and another essential feature of quantum mechanics, quantum entanglement. It is shown that a complete set of mutually unbiased bases of a bipartite system contains a fixed amount of entanglement, independent of the choice of the set. This has implications for entanglement distribution among the states of a complete set. In prime-squared dimensions we present an explicit experiment-friendly construction of a complete set with a particularly simple entanglement distribution. Finally, we describe the basic properties of mutually unbiased bases composed of product states only. The constructions are illustrated with explicit examples in low dimensions. We believe that the properties of entanglement in mutually unbiased bases may be one of the ingredients to be taken into account to settle the question of the existence of complete sets. We also expect that they will be relevant to applications of bases in the experimental realization of quantum protocols in higher-dimensional Hilbert spaces.
The application of quantum mechanics contributes to the field of cryptography with very important advantage as it offers a mechanism for detecting the eavesdropper. The pioneering work of quantum key distribution uses mutually unbiased bases (MUBs) to prepare and measure qubits (or qudits). Weak mutually unbiased bases (WMUBs) have weaker properties than MUBs properties, however, unlike MUBs, a complete set of WMUBs can be constructed for systems with composite dimensions. In this paper, we study the use of weak mutually unbiased bases (WMUBs) in quantum key distribution for composite dimensional finite systems. We prove that the security analysis of using a complete set of WMUBs to prepare and measure the quantum states in the generalized BB84 protocol, gives better results than using the maximum number of MUBs that can be constructed, when they are analyzed against the intercept and resend attack.