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arXiv:math/0703333

Generation of mutually unbiased bases as powers of a unitary matrix in 2-power dimensions

Rod Gow

Published Mar 12, 2007• 38 citations
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Abstract

Let q be a power of 2. We show by representation theory that there exists a q x q unitary matrix of multiplicative order q+1 whose powers generate q+1 pairwise mutually unbiased base in C^q. When q is a power of an odd prime, there is a q x q unitary matrix of multiplicative order q+1 whose first (q+1)/2 powers generate (q+1)/2 pairwise mutually unbiased bases. We also show how the existence of these matrices implies the existence of a special type of orthogonal decomposition with respect to the Killing form of the special linear and symplectic Lie algebras.

DOI
10.48550/arxiv.math/0703333

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Let q=2^r, where r is a positive integer. We show that there is a q^2 x q^2 real orthogonal matrix D, say, of multiplicative order q+1 whose powers generate q+1 real pairwise mutually unbiased bases in R^{q^2}. This means that the scaled powers qD, qD^2, ..., qD^q are q different real Hadamard matrices. We also show that each diagonal entry of these q Hadamard matrices equals -1.

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We establish a connection between the problem of constructing maximal collections of mutually unbiased bases (MUBs) and an open problem in the theory of Lie algebras. More precisely, we show that a collection of $\mu$ MUBs in $\K^n$ gives rise to a collection of $\mu$ Cartan subalgebras of the special linear Lie algebra $sl_n(\K)$ that are pairwise orthogonal with respect to the Killing form, where $\K=\R$ or $\K=\C$. In particular, a complete collection of MUBs in $\C^n$ gives rise to a so-called orthogonal decomposition (OD) of $sl_n(\C)$. The converse holds if the Cartan subalgebras in the OD are also $\dag$-closed, i.e., closed under the adjoint operation. In this case, the Cartan subalgebras have unitary bases, and the above correspondence becomes equivalent to a result of \cite{bbrv02} relating collections of MUBs to collections of maximal commuting classes of unitary error bases, i.e., orthogonal unitary matrices. This connection implies that for $n\le 5$ an essentially unique complete collection of MUBs exists. We define \emph{monomial MUBs}, a class of which all known MUB constructions are members, and use the above connection to show that for $n=6$ there are at most three monomial MUBs.

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Real mutually unbiased bases and representations of groups of odd order by real scaled Hadamard matrices of 2-power size

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We prove the following two results relating real mutually unbiased bases and representations of finite groups of odd order. Let $q$ be a power of 2 and $r$ a positive integer. Then we can find a $q^{2r}\times q^{2r}$ real orthogonal matrix $D$, say, of multiplicative order $q^{2r-1}+1$, whose $q^{2r-1}+1$ powers $D$, \dots, $D^{q^{2r-1}+1}=I$ define $q^{2r-1}+1$ mutually unbiased bases in $\mathbb{R}^{q^{2r}}$. Thus the scaled matrices $q^rD$, \dots, $q^rD^{q^{2r-1}}$ are $q^{2r-1}$ different Hadamard matrices. When we take $q=2$, we achieve the maximum number of real mutually unbiased bases in dimension $2^{2r}$ using the elements of a cyclic group. We also prove the following. Let $G$ be an arbitrary finite group of odd order $2k+1$, where $k\geq 3$. Then $G$ has a real representation $R$, say, of degree $2^{2^{k-1}}$ such that the elements $R(σ)$, $σ\in G$, define $|G|$ mutually unbiased bases in $\mathbb{R}^{d}$, where $d= 2^{2^{k-1}}$. In addition, a group of order 5 defines five real mutually unbiased bases in $\mathbb{R}^{16}$ and a group of order 3 defines three real mutually unbiased bases in $\mathbb{R}^{4}$. Thus, an arbitrary group of odd order has a faithful representation by real scaled Hadamard matrices of 2-power size.

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The Mutually Unbiased Bases Revisited

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Orthogonal pairs and mutually unbiased bases

Alexey Igorevich Bondal, Il'ya Yur'evich Zhdanovskiy

The paper is devoted to representation theoretic and algebraic geometric aspects of the theory of orthogonal decompositions of Lie algebra sl(n) into Cartan subalgebras orthogonal with respect to Killing form and the relevant theory of mutually unbiased bases. We describe the main steps of our proof for the existence of 4-dimensional family of orthogonal pairs of Cartan subalgebras in sl(6). As an application, we prove existence of a real 4-dimensional family of pairs of mutually unbiased bases in C^6, thus solving the long standing problem in Quantum Information Theory.

arXiv:1510.05317•Oct 18, 2015•0 citations
Let q be a power of 2. We show by representation theory that there exists a q x q unitary matrix of multiplicative order q+1 whose powers generate q+1 pairwise mutually unbiased base in C^q. When q is a power of an odd prime, there is a q x q unitary matrix of multiplicative order q+1 whose first (q+1)/2 powers generate (q+1)/2 pairwise mutually unbiased bases. We also show how the existence of these matrices implies the existence of a special type of orthogonal decomposition with respect to the Killing form of the special linear and symplectic Lie algebras.
We establish a connection between the problem of constructing maximal collections of mutually unbiased bases (MUBs) and an open problem in the theory of Lie algebras. More precisely, we show that a collection of μ\muμ MUBs in $\K^n$ gives rise to a collection of μ\muμ Cartan subalgebras of the special linear Lie algebra $sl_n(\K)$ that are pairwise orthogonal with respect to the Killing form, where $\K=\R$ or $\K=\C$. In particular, a complete collection of MUBs in $\C^n$ gives rise to a so-called orthogonal decomposition (OD) of $sl_n(\C)$. The converse holds if the Cartan subalgebras in the OD are also †\dag†-closed, i.e., closed under the adjoint operation. In this case, the Cartan subalgebras have unitary bases, and the above correspondence becomes equivalent to a result of \cite{bbrv02} relating collections of MUBs to collections of maximal commuting classes of unitary error bases, i.e., orthogonal unitary matrices. This connection implies that for n≤5n\le 5n≤5 an essentially unique complete collection of MUBs exists. We define \emph{monomial MUBs}, a class of which all known MUB constructions are members, and use the above connection to show that for n=6n=6n=6 there are at most three monomial MUBs.
We prove the following two results relating real mutually unbiased bases and representations of finite groups of odd order. Let qqq be a power of 2 and rrr a positive integer. Then we can find a q2r×q2rq^{2r}\times q^{2r}q2r×q2r real orthogonal matrix DDD, say, of multiplicative order q2r−1+1q^{2r-1}+1q2r−1+1, whose q2r−1+1q^{2r-1}+1q2r−1+1 powers DDD, \dots, Dq2r−1+1=ID^{q^{2r-1}+1}=IDq2r−1+1=I define q2r−1+1q^{2r-1}+1q2r−1+1 mutually unbiased bases in Rq2r\mathbb{R}^{q^{2r}}Rq2r. Thus the scaled matrices qrDq^rDqrD, \dots, qrDq2r−1q^rD^{q^{2r-1}}qrDq2r−1 are q2r−1q^{2r-1}q2r−1 different Hadamard matrices. When we take q=2q=2q=2, we achieve the maximum number of real mutually unbiased bases in dimension 22r2^{2r}22r using the elements of a cyclic group. We also prove the following. Let GGG be an arbitrary finite group of odd order 2k+12k+12k+1, where k≥3k\geq 3k≥3. Then GGG has a real representation RRR, say, of degree 22k−12^{2^{k-1}}22k−1 such that the elements R(σ)R(σ)R(σ), σ∈Gσ\in Gσ∈G, define ∣G∣|G|∣G∣ mutually unbiased bases in Rd\mathbb{R}^{d}Rd, where d=22k−1d= 2^{2^{k-1}}d=22k−1. In addition, a group of order 5 defines five real mutually unbiased bases in R16\mathbb{R}^{16}R16 and a group of order 3 defines three real mutually unbiased bases in R4\mathbb{R}^{4}R4. Thus, an arbitrary group of odd order has a faithful representation by real scaled Hadamard matrices of 2-power size.
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.
The paper is devoted to representation theoretic and algebraic geometric aspects of the theory of orthogonal decompositions of Lie algebra sl(n) into Cartan subalgebras orthogonal with respect to Killing form and the relevant theory of mutually unbiased bases. We describe the main steps of our proof for the existence of 4-dimensional family of orthogonal pairs of Cartan subalgebras in sl(6). As an application, we prove existence of a real 4-dimensional family of pairs of mutually unbiased bases in C^6, thus solving the long standing problem in Quantum Information Theory.