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Journal of Physics A Mathematical and Theoretical

Bi-orthogonal mutually unbiased bases for N-qubit systems

Juan J Díaz, Isabel Sainz, A. B. Klimov

Published Jan 18, 2017• 0 citations
Publisher

Abstract

Abstract We have developed a general method for constructing a set of non-orthogonal bases with equal separations between all different basis’ states for N-qubit systems. Using these bases we derive an explicit expression for the optimal tomography in non-orthogonal bases and discuss the amount of non-classical resources required for the bases preparation and the precision of the reconstructed state. The special two-qubit case is analysed separately.

DOI
10.1088/1751-8121/aa532c

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Abstract We have developed a general method for constructing a set of non-orthogonal bases with equal separations between all different basis’ states for N-qubit systems. Using these bases we derive an explicit expression for the optimal tomography in non-orthogonal bases and discuss the amount of non-classical resources required for the bases preparation and the precision of the reconstructed state. The special two-qubit case is analysed separately.
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.
Based on mutually unbiased measurements, an optimal tomographic scheme for the multiqutrit states is presented explicitly. Because the reconstruction process of states based on mutually unbiased states is free of information waste, we refer to our scheme as the optimal scheme. By optimal we mean that the number of the required conditional operations reaches the minimum in this tomographic scheme for the states of qutrit systems. Special attention will be paid to how those different mutually unbiased measurements are realized; that is, how to decompose each transformation that connects each mutually unbiased basis with the standard computational basis. It is found that all those transformations can be decomposed into several basic implementable single- and two-qutrit unitary operations. For the three-qutrit system, there exist five different mutually unbiased-bases structures with different entanglement properties, so we introduce the concept of physical complexity to minimize the number of nonlocal operations needed over the five different structures. This scheme is helpful for experimental scientists to realize the most economical reconstruction of quantum states in qutrit systems.
Mutually unbiased bases are an important tool in many applications of quantum information theory. We present a new algorithm for finding the mutually unbiased bases for two-qubit systems. We derive a system of four equations in the Galois field GF(4) and show that the solutions of this system are sufficient for obtaining the most general set of mutually unbiased bases. Further, our algorithm is applied to an example and we show that there are three possible solutions of the system of four equations, each solution leading to a different set of mutually unbiased bases.
Two orthonormal bases in the complex space of dimension d, are said to be mutually unbiased if the square of the magnitude of the inner product of any vector from one basis with any vector in other basis is equal to the reciprocal of the dimension d. Mutually unbiased bases are used for optimal state determination of mixed quantum states. It is known that in any dimension d, the number of mutually unbiased bases is at most d+1. Ivanovic found a complete set of mutually unbiased bases for prime dimensions. His construction was generalized by Wootters and Fields for prime power dimensions. There is a strong connection between maximally commuting bases of orthogonal unitary matrices and mutually unbiased bases. Based on this connection, there exits a constructive proof of the existence of a complete set of mutually unbiased bases for prime power dimensions. This thesis is an exploration on construction of mutually unbiased bases.