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Zhongguo kexue. Wulixue Lixue Tianwenxue

Construction of orthogonal extraordinary supersquares and mutually unbiased bases

Xiaoya Cheng, Shang Yun

Published Mar 6, 2018• 0 citations
Publisher

Abstract

This article studies the connection between mutually orthogonal extraordinary supersquares and mutually unbiased bases. Firstly, we generalize extraordinary supersquares defined by Ghiu et al. to arbitrary order $d$.For order $d=p^a$ being a single prime power, we construct $d+1$ mutually orthogonal extraordinary supersquares; for arbitrary order $d$, we construct $p_1^{a_1}+1$ mutually orthogonal extraordinary supersquares, where $p_1^{a_1}$ is the minimal prime power dividing $d$.In addition, this article sets up the connection between mutually orthogonal extraordinary supersquares and mutually unbiased bases for arbitrary $d$. Then we provide a new proof for the known lower bound of mutually unbiased bases.

DOI
10.1360/sspma2018-00022

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This article studies the connection between mutually orthogonal extraordinary supersquares and mutually unbiased bases. Firstly, we generalize extraordinary supersquares defined by Ghiu et al. to arbitrary order ddd.For order d=pad=p^ad=pa being a single prime power, we construct d+1d+1d+1 mutually orthogonal extraordinary supersquares; for arbitrary order ddd, we construct p1a1+1p_1^{a_1}+1p1a1​​+1 mutually orthogonal extraordinary supersquares, where p1a1p_1^{a_1}p1a1​​ is the minimal prime power dividing ddd.In addition, this article sets up the connection between mutually orthogonal extraordinary supersquares and mutually unbiased bases for arbitrary ddd. Then we provide a new proof for the known lower bound of mutually unbiased bases.
In order to describe the right setting to handle Zauner's conjecture on mutually unbiased bases (MUBs) (saying that in Cd\mathbb{C}^dCd, a set of MUBs of the theoretical maximal size d+1d + 1d+1 exists only if ddd is a prime power), we pose some fundamental questions which naturally arise. Some of these questions have important consequences for the construction theory of (new) sets of maximal MUBs.
The study of Mutually Unbiased Bases continues to be developed vigorously, and presents several challenges in the Quantum Information Theory. Two orthonormal bases in Cd,BandB′\mathbb C^d, B {and} B'Cd,BandB′ are said mutually unbiased if ∀b∈B,b′∈B′\forall b\in B, b'\in B'∀b∈B,b′∈B′ the scalar product b⋅b′b\cdot b'b⋅b′ has modulus d−1/2d^{-1/2}d−1/2. In particular this property has been introduced in order to allow an optimization of the measurement-driven quantum evolution process of any state ψ∈Cdψ\in \mathbb C^dψ∈Cd when measured in the mutually unbiased bases B_jofCdB\_{j} {of} \mathbb C^dB_jofCd. At present it is an open problem to find the maximal umber of mutually Unbiased Bases when ddd is not a power of a prime number. \noindent In this article, we revisit the problem of finding Mutually Unbiased Bases (MUB's) in any dimension ddd. The method is very elementary, using the simple unitary matrices introduced by Schwinger in 1960, together with their diagonalizations. The Vandermonde matrix based on the ddd-th roots of unity plays a major role. This allows us to show the existence of a set of 3 MUB's in any dimension, to give conditions for existence of more than 3 MUB's for ddd even or odd number, and to recover the known result of existence of d+1d+1d+1 MUB's for ddd a prime number. Furthermore the construction of these MUB's is very explicit. As a by-product, we recover results about Gauss Sums, known in number theory, but which have apparently not been previously derived from MUB properties.
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.