dorsal/arxiv
View SchemaThe Mutually Unbiased Bases Revisited
| Authors | M. Combescure |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0605090 |
| URL | https://arxiv.org/abs/quant-ph/0605090 |
Abstract
The study of Mutually Unbiased Bases continues to be developed vigorously, and presents several challenges in the Quantum Information Theory. Two orthonormal bases in $\mathbb C^d, B {and} B'$ are said mutually unbiased if $\forall b\in B, b'\in B'$ the scalar product $b\cdot b'$ has modulus $d^{-1/2}$. In particular this property has been introduced in order to allow an optimization of the measurement-driven quantum evolution process of any state $\psi \in \mathbb C^d$ when measured in the mutually unbiased bases $B\_{j} {of} \mathbb C^d$. At present it is an open problem to find the maximal umber of mutually Unbiased Bases when $d$ 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 $d$. 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 $d$-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 $d$ even or odd number, and to recover the known result of existence of $d+1$ MUB's for $d$ 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.
{
"annotation_id": "296be9f5-4883-4d3a-aff6-664428612f54",
"date_created": "2026-03-02T18:02:27.657000Z",
"date_modified": "2026-03-02T18:02:27.657000Z",
"file_hash": "080e1145e19af9d91882f95a69755c975ab4fadb5da9025799832ffec19fa786",
"private": false,
"record": {
"abstract": "The study of Mutually Unbiased Bases continues to be developed vigorously,\nand presents several challenges in the Quantum Information Theory. Two\northonormal bases in $\\mathbb C^d, B {and} B\u0027$ are said mutually unbiased if\n$\\forall b\\in B, b\u0027\\in B\u0027$ the scalar product $b\\cdot b\u0027$ has modulus\n$d^{-1/2}$. In particular this property has been introduced in order to allow\nan optimization of the measurement-driven quantum evolution process of any\nstate $\\psi \\in \\mathbb C^d$ when measured in the mutually unbiased bases\n$B\\_{j} {of} \\mathbb C^d$. At present it is an open problem to find the maximal\number of mutually Unbiased Bases when $d$ is not a power of a prime number.\n\\noindent In this article, we revisit the problem of finding Mutually Unbiased\nBases (MUB\u0027s) in any dimension $d$. The method is very elementary, using the\nsimple unitary matrices introduced by Schwinger in 1960, together with their\ndiagonalizations. The Vandermonde matrix based on the $d$-th roots of unity\nplays a major role. This allows us to show the existence of a set of 3 MUB\u0027s in\nany dimension, to give conditions for existence of more than 3 MUB\u0027s for $d$\neven or odd number, and to recover the known result of existence of $d+1$ MUB\u0027s\nfor $d$ a prime number. Furthermore the construction of these MUB\u0027s is very\nexplicit. As a by-product, we recover results about Gauss Sums, known in number\ntheory, but which have apparently not been previously derived from MUB\nproperties.",
"arxiv_id": "quant-ph/0605090",
"authors": [
"M. Combescure"
],
"categories": [
"quant-ph",
"math-ph",
"math.MP"
],
"title": "The Mutually Unbiased Bases Revisited",
"url": "https://arxiv.org/abs/quant-ph/0605090"
},
"schema_id": "dorsal/arxiv",
"source": {
"execution_id": "c8f6e5b5-0365-4feb-a037-b1afaea71aa3",
"id": "arXiv Dataset IDs",
"type": "Model",
"variant": "snapshot-2026-03-01",
"version": "0.1.0"
},
"user_id": 1000002
}