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Chapter 4

Hypothesis Detection of Noncommutative Random Matrices

# 4.1 Why Noncommutative Random Matrices?

The most basic building block for quantum information is the covariance matrix. We are dealing with the matrix space whose elements are covariance matrices. The sufficient and necessary conditions for a matrix to be a covariance matrix are semidefinite positive. As a result, the basic elements for us to manipulate are the SDP matrices. Naturally, convex optimization (SDP matrices are of course convex) is the new calculus under this context.

For any two elements (matrices) A and B, we need to define the basic metric to order them. If they are random matrices, we call this order the stochastic order, for example,

if B is stochastically greater than A.

More generally, A and B are two matrix-valued random variables, in contrast with the scalar random variables. Recall that every entry of A and B is a scalar random variable. The focus of the current engineering curriculum is on the scalar random variable. When we deal with “Big Data” [1] in a high-dimensional vector space, the most natural objects of mathematical operations are such (SDP) matrix-valued random variables.

The matrix operation is fundamentally different from its scalar counterpart in that the matrix multiplication is not communicative. The quantum mechanics is built upon this mathematical fact.

When we process the data, ...

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