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arXiv:1907.02726

Cyclic Mutually Unbiased Bases and Quantum Public-Key Encryption

Ulrich Seyfarth

Published Jul 5, 2019• 4 citations
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Abstract

The thesis is mainly about the construction and implementation of cyclic mutually unbiased bases, dealing with different entanglement structures by discussing the related group structures. A recursive construction for Fermat number dimensions is given and related to Wiedemann's conjecture for systems with more than 2048 qubits. The second part of the thesis analyses a quantum public-key encryption scheme.

DOI
10.48550/arxiv.1907.02726

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The thesis is mainly about the construction and implementation of cyclic mutually unbiased bases, dealing with different entanglement structures by discussing the related group structures. A recursive construction for Fermat number dimensions is given and related to Wiedemann's conjecture for systems with more than 2048 qubits. The second part of the thesis analyses a quantum public-key encryption scheme.
For a system of NNN qubits, living in a Hilbert space of dimension d=2Nd={2}^{N}d=2N, it is known that there exists d+1d+1d+1 mutually unbiased bases. Different construction algorithms exist, and it is remarkable that different methods lead to sets of bases with different properties as far as separability is concerned. Here we derive four sets of nine bases for three qubits, and show how they are unitarily related. We also briefly discuss the four-qubit case, give the entanglement structure of 16 sets of bases, and show some of them and their interrelations, as examples. The extension of the method to the general case of NNN qubits is outlined.
The number of measurements necessary to perform the quantum state reconstruction of a system of qubits grows exponentially with the number of constituents, creating a major obstacle for the design of scalable tomographic schemes. We work out a simple and efficient method based on cyclic generation of mutually unbiased bases. The basic generator requires only Hadamard and controlled-phase gates, which are available in most practical realizations of these systems. We show how complete sets of mutually unbiased bases with different entanglement structures can be realized for three and four qubits. We also analyze the quantum circuits implementing the various entanglement classes.
We construct two mutually unbiased bases by maximally entangled states (MUMEBsss) in C2⊗C3\mathbb{C}^{2}\otimes \mathbb{C}^{3}C2⊗C3. This is the first example of MUMEBsss in Cd⊗Cd′\mathbb{C}^{d}\otimes \mathbb{C}^{d'}Cd⊗Cd′ when d∤d′d\nmid d'd∤d′, namely d′d'd′ is not divisible by ddd. We show that they cannot be extended to four MUBs in C6\mathbb{C}^6C6. We propose a recursive construction of mutually unbiased bases formed by special entangled states with a fixed Schmidt number kkk (MUSEBkkks). It shows that min⁡{t1,t2}\min \{t_{1},t_{2}\}min{t1​,t2​} MUSEBk1k2k_{1}k_{2}k1​k2​s in Cpd⊗Cqd′\mathbb{C}^{pd}\otimes \mathbb{C}^{qd'}Cpd⊗Cqd′ can be constructed from t1t_{1}t1​ MUSEBk1k_{1}k1​s in Cd⊗Cd′\mathbb{C}^{d}\otimes \mathbb{C}^{d'}Cd⊗Cd′ and t2t_{2}t2​ MUSEBk2k_{2}k2​s in Cp⊗Cq\mathbb{C}^{p}\otimes \mathbb{C}^{q}Cp⊗Cq for any d,d′,p,qd,d',p,qd,d′,p,q. Further, we show that three MUMEBsss exist in Cd⊗Cd′\mathbb{C}^{d}\otimes \mathbb{C}^{d'}Cd⊗Cd′ for any d,d′d,d'd,d′ with d∣d′d\mid d'd∣d′, and two MUMEBsss exist in Cd⊗Cd′\mathbb{C}^{d}\otimes \mathbb{C}^{d'}Cd⊗Cd′ for infinitely many d,d′d,d'd,d′ with d∤d′d\nmid d'd∤d′.
Abstract Based on maximally entangled states, we explore the constructions of mutually unbiased bases in bipartite quantum systems. We present a new way to construct mutually unbiased bases by difference matrices in the theory of combinatorial designs. In particular, we establish q mutually unbiased bases with q − 1 maximally entangled bases and one product basis in C q ⊗ C q for arbitrary prime power q . In addition, we construct maximally entangled bases for dimension of composite numbers of non-prime power, such as five maximally entangled bases in C 12 ⊗ C 12 and C 21 ⊗ C 21 , which improve the known lower bounds for d = 3 m , with (3, m ) = 1 in C d ⊗ C d . Furthermore, we construct p + 1 mutually unbiased bases with p maximally entangled bases and one product basis in C p ⊗ C p 2 for arbitrary prime number p .
For the complete estimation of arbitrary unknown quantum states by measurements, the use of mutually unbiased bases has been well established in theory and experiment for the past 20 years. However, most constructions of these bases make heavy use of abstract algebra and the mathematical theory of finite rings and fields, and no simple and generally accessible construction is available. This is particularly true in the case of a system composed of several qubits, which is arguably the most important case in quantum information science and quantum computation. In this paper, we close this gap by providing a simple and straightforward method for the construction of mutually unbiased bases in the case of a qubit register. We show that our construction is also accessible to experiments, since only Hadamard and controlled-phase gates are needed, which are available in most practical realizations of a quantum computer. Moreover, our scheme possesses the optimal scaling possible, i.e., the number of gates scales only linearly in the number of qubits.