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International Journal of Quantum Information

Two new constructions of approximately mutually unbiased bases

Gang Wang, Min-Yao Niu, Fang‐Wei Fu

Published Jun 1, 2018• 4 citations
Publisher

Abstract

Two orthonormal bases [Formula: see text] and [Formula: see text] of a [Formula: see text]-dimensional complex inner-product space [Formula: see text] are called mutually unbiased bases (MUBs) if and only if [Formula: see text] holds for all [Formula: see text] and [Formula: see text]. The size of any set containing pairwise MUBs of [Formula: see text] cannot exceed [Formula: see text]. If [Formula: see text] is a power of a prime, then extremal sets containing [Formula: see text] MUBs are known to exist, which are called the complete MUBs of [Formula: see text]. We have not known whether there exist complete MUBs when [Formula: see text] is not a power of a prime so far. Therefore, many researchers focus their attention on approximately mutually unbiased bases (AMUB). In this paper, two new constructions of AMUB of [Formula: see text] are provided based on the mixed character sum of two special kinds of functions over finite fields.

DOI
10.1142/s0219749918500387

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Analogous to the notion of mutually unbiased bases for Hilbert spaces, we consider mutually unbiased unitary bases (MUUBs) for the space of operators, [Formula: see text], acting on such Hilbert spaces. The notion of MUUB reflects the equiprobable guesses of unitary operators in one basis of [Formula: see text] when estimating a unitary operator in another. Though, for prime dimension [Formula: see text], the maximal number of MUUBs is known to be [Formula: see text], there is no known recipe for constructing them, assuming they exist. However, one can always construct a minimum of three MUUBs, and the maximal number is approached for very large values of [Formula: see text]. MUUBs can also exist for some [Formula: see text]-dimensional subspace of [Formula: see text] with the maximal number being [Formula: see text].

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Two orthonormal bases [Formula: see text] and [Formula: see text] of a [Formula: see text]-dimensional complex inner-product space [Formula: see text] are called mutually unbiased bases (MUBs) if and only if [Formula: see text] holds for all [Formula: see text] and [Formula: see text]. The size of any set containing pairwise MUBs of [Formula: see text] cannot exceed [Formula: see text]. If [Formula: see text] is a power of a prime, then extremal sets containing [Formula: see text] MUBs are known to exist, which are called the complete MUBs of [Formula: see text]. We have not known whether there exist complete MUBs when [Formula: see text] is not a power of a prime so far. Therefore, many researchers focus their attention on approximately mutually unbiased bases (AMUB). In this paper, two new constructions of AMUB of [Formula: see text] are provided based on the mixed character sum of two special kinds of functions over finite fields.
Mutually unbiased bases (MUBs) are widely used in quantum information processing and play an important role in quantum cryptography, quantum state tomography and communications. It's difficult to construct MUBs and remains unknown whether complete MUBs exist for any non prime power. Therefore, researchers have proposed the solution to construct approximately mutually unbiased bases (AMUBs) by weakening the inner product conditions. This paper constructs q AMUBs of ℂq, (q+1) AMUBs of ℂq-1 and q AMUBs of ℂq-1 by using character sums over Galois rings and finite fields, where q is a power of a prime. The first construction of q AMUBs of ℂq is new which illustrates K AMUBs of ℂK can be achieved. The second and third constructions in this paper include the partial results about AMUBs constructed by W. Wang et al. in [9].
Construction of a large class of Mutually Unbiased Bases (MUBs) for non-prime power composite dimensions (d=k×sd = k\times sd=k×s) is a long standing open problem, which leads to different construction methods for the class Approximate MUBs (AMUBs) by relaxing the criterion that the absolute value of the dot product between two vectors chosen from different bases should be ≤βd\leq \fracβ{\sqrt{d}}≤d​β​. In this chapter, we consider a more general class of AMUBs (ARMUBs, considering the real ones too), compared to our earlier work in [Cryptography and Communications, 14(3): 527--549, 2022]. We note that the quality of AMUBs (ARMUBs) constructed using RBD(X,A)(X,A)(X,A) with ∣X∣=d|X|= d∣X∣=d, critically depends on the parameters, ∣s−k∣|s-k|∣s−k∣, μμμ (maximum number of elements common between any pair of blocks), and the set of block sizes. We present the construction of O(d)\mathcal{O}(\sqrt{d})O(d​) many βββ-AMUBs for composite ddd when ∣s−k∣<d|s-k|< \sqrt{d}∣s−k∣<d​, using RBDs having block sizes approximately d\sqrt{d}d​, such that ∣⟨ψil∣ψjm⟩∣≤βd|\braket{ψ^l_i|ψ^m_j}| \leq \fracβ{\sqrt{d}}∣⟨ψil​∣ψjm​⟩∣≤d​β​ where β=1+∣s−k∣2d+O(d−1)≤2β= 1 + \frac{|s-k|}{2\sqrt{d}}+ \mathcal{O}(d^{-1}) \leq 2β=1+2d​∣s−k∣​+O(d−1)≤2. Moreover, if real Hadamard matrix of order kkk or sss exists, then one can construct at least N(k)+1N(k)+1N(k)+1 (or N(s)+1N(s)+1N(s)+1) many βββ-ARMUBs for dimension ddd, with β≤2−∣s−k∣2d+O(d−1)<2β\leq 2 - \frac{|s-k|}{2\sqrt{d}}+ \mathcal{O}(d^{-1})< 2β≤2−2d​∣s−k∣​+O(d−1)<2, where N(w)N(w)N(w) is the number of MOLS(w)(w)(w). This improves and generalizes some of our previous results for ARMUBs from two points, viz., the real cases are now extended to complex ones too. The earlier efforts use some existing RBDs, whereas here we consider new instances of RBDs that provide better results. Similar to the earlier cases, the AMUBs (ARMUBs) constructed using RBDs are in general very sparse, where the sparsity (ε)(ε)(ε) is 1−O(d−12)1 - \mathcal{O}(d^{-\frac{1}{2}})1−O(d−21​).
Mutually unbiased bases (MUB) and symmetric informationally complete positive operator-valued measure (SIC-POVM) are both important objects in quantum information theory.While people do not know if there exists a complete MUB for non-prime-power dimension,several versions of approximately MUB have been considered by relaxed the inner product condition.So far there are only finite number of K such that SICPOVMs in C k have been found.As in the MUB case,several versions of approximately SIC-POVM have been considered by relaxed the inner product condition.In this paper,we use the definitions of approximate MUB and SIC-POVM given by Klappenecker et al.For prime power q,we present simple constructions of q approximately MUB (AMUB) for dimension q-1,q+1 AMUB for dimension q-1,which shows the number of orthonormal bases of an AMUB in C k can be more than K+1,and q AMUB for dimension q+1 by Gauss and Jacobi sums.We also present a construction of approximately SIC-POVM (ASIC-POVM) in dimension q-1 by Gauss sum.
Mutually Unbiased bases has various application in quantum information procession and coding theory. There can be maximum d + 1 MUBs in C^d and d/2 +1 MUBs in R^d. But , over R^d MUBs are known to be non existent when d is odd and for most of the other even d there are mostly 3 Real MUBs. In case of C^d the construction for complete set of MUBs are known for only Prime Power dimension. Thus in general large set of MUBs are not known, particularly for composite dimensions which are not of the form of prime powers. Because of this, there are many constructions of Approximate version of MUBs. In this paper we make an attempt to define certain measures to characterise the AMUBs. Our construction of measures derives its inspiration from the applications of MUBs, and based on them, we define certain quantifiable measures, which are can be computed and gives estimates of how close the Approximate MUBs are to the MUBs. We use geometric interpretation, projective design features of MUBs and applications like Optimal State determination and Entropic Uncertainty of MUBs. We show generic relationship between these measures and show that it can be evaluated for APMUBs without known the exact construction details, thereby showing that definition of APMUB is sufficient completely characterise it. We also evaluate these measure for an interesting class of AMUBs called Weak MUBs and certain AMUBs constructed using RBDs.
Mutually unbiased bases (MUB) have many applications in quantum information processing and quantum cryptography. Several complex MUB's in Cd\mathbb{C}^dCd for some dimension ddd and with larger size have been constructed. On the other hand, real MUB's with larger size are rare which lead to consider constructing approximate MUB (AMUB). In this paper we present a general and useful way to get real AMUB in R2d\mathbb{R}^{2d}R2d from any complex AMUB in Cd\mathbb{C}^dCd. From this method we present many new series of real AMUB's with parameters better than previous results.
Analogous to the notion of mutually unbiased bases for Hilbert spaces, we consider mutually unbiased unitary bases (MUUBs) for the space of operators, [Formula: see text], acting on such Hilbert spaces. The notion of MUUB reflects the equiprobable guesses of unitary operators in one basis of [Formula: see text] when estimating a unitary operator in another. Though, for prime dimension [Formula: see text], the maximal number of MUUBs is known to be [Formula: see text], there is no known recipe for constructing them, assuming they exist. However, one can always construct a minimum of three MUUBs, and the maximal number is approached for very large values of [Formula: see text]. MUUBs can also exist for some [Formula: see text]-dimensional subspace of [Formula: see text] with the maximal number being [Formula: see text].