Welch bounds

In mathematics, Welch bounds are a family of inequalities pertinent to the problem of evenly spreading a set of unit vectors in a vector space. The bounds are important tools in the design and analysis of certain methods in telecommunication engineering, particularly in coding theory. The bounds were originally published in a 1974 paper by L. R. Welch.

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Mathematical statement

If \{x_1,\ldots,x_m\} are unit vectors in \mathbb{C}^n, define c_\max = \max_{i\neq j} |\langle x_i, x_j \rangle|, where \langle\cdot,\cdot\rangle is the usual inner product on \mathbb{C}^n. Then the following inequalities hold for k=1,2,\dots:

(c_\max)^{2k} \geq \frac{1}{m-1} \left[ \frac{m}{\binom{n%2Bk-1}{k}}-1 \right]

Applicability

If m\leq n, then the vectors \{x_i\} can form an orthonormal set in \mathbb{C}^n. In this case, c_\max=0 and the bounds are vacuous. Consequently, interpretation of the bounds is only meaningful if m>n. This will be assumed throughout the remainder of this article.

Proof for k = 1

The "first Welch bound," corresponding to k=1, is by far the most commonly used in applications. Its proof proceeds in two steps, each of which depends on a more basic mathematical inequality. The first step invokes the Cauchy-Schwarz inequality and begins by considering the m\times m Gram matrix G of the vectors \{x_i\}; i.e.,

G=\left[ \begin{array}{ccc} \langle x_1, x_1 \rangle & \cdots & \langle x_1, x_m \rangle \\ \vdots & \ddots & \vdots \\ \langle x_m, x_1 \rangle & \cdots & \langle x_m, x_m \rangle \end{array}\right]

The trace of G is equal to the sum of its eigenvalues. Because the rank of G is at most n, and it is a positive semidefinite matrix, G has at most n positive eigenvalues with its remaining eigenvalues all equal to zero. Writing the non-zero eigenvalues of G as \lambda_1,\ldots,\lambda_r with r\leq n and applying the Cauchy-Schwarz inequality to the inner product of an r-vector of ones with a vector whose components are these eigenvalues yields

(\mathrm{Tr}\;G)^2 = \left( \sum_{i=1}^r \lambda_i \right)^2 \leq r \sum_{i=1}^r \lambda_i^2 \leq n \sum_{i=1}^m \lambda_i^2

The square of the Frobenius norm (Hilbert–Schmidt norm) of G satisfies

||G||^2 = \sum_{i=1}^{m} \sum_{j=1}^m |\langle x_i , x_j \rangle|^2 = \sum_{i=1}^m \lambda_i^2

Taking this together with the preceding inequality gives

\sum_{i=1}^m \sum_{j=1}^m |\langle x_i , x_j \rangle|^2\geq \frac{(\mathrm{Tr}\;G)^2}{n}

Because each x_i has unit length, the elements on the main diagonal of G are ones, and hence its trace is \mathrm{Tr}\;G = m. So,

\sum_{i=1}^{m} \sum_{j=1}^m |\langle x_i , x_j \rangle|^2 = m%2B\sum_{i\neq j} |\langle x_i , x_j \rangle|^2 \geq \frac{m^2}{n}

or

\sum_{i\neq j} |\langle x_i , x_j \rangle|^2 \geq \frac{m(m-n)}{n}

The second part of the proof uses an inequality encompassing the simple observation that the average of a set of non-negative numbers can be no greater than the largest number in the set. In mathematical notation, if a_{\ell}\geq 0 for \ell=1,\ldots, L, then

\frac{1}{L}\sum_{\ell=1}^L a_{\ell} \leq \max a_{\ell}

The previous expression has m(m-1) non-negative terms in the sum,the largest of which is c_\max^2. So,

(c_\max)^2\geq \frac{1}{m(m-1)}\sum_{i\neq j} |\langle x_i , x_j \rangle|^2\geq\frac{m-n}{n(m-1)}

or

(c_\max)^2\geq \frac{m-n}{n(m-1)}

which is precisely the inequality given by Welch in the case that k=1

Achieving Welch bound equality

In certain telecommunications applications, it is desirable to construct sets of vectors that meet the Welch bounds with equality. Several techniques have been introduced to obtain so-called Welch Bound Equality (WBE) sets of vectors for the k = 1 bound.

The proof given above shows that two separate mathematical inequalities are incorporated into the Welch bound when k=1. The Cauchy–Schwarz inequality is met with equality when the two vectors involved are collinear. In the way it is used in the above proof, this occurs when all the non-zero eigenvalues of the Gram matrix G are equal, which happens precisely when the vectors \{x_1,\ldots,x_m\} constitute a tight frame for \mathbb{C}^n.

The other inequality in the proof is satisfied with equality if and only if |\langle x_i, x_j \rangle| is the same for every choice of i\neq j. In this case, the vectors are equiangular. So this Welch bound is met with equality if and only if the set of vectors \{x_i\} is an equiangular tight frame in \mathbb{C}^n.

References