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Let $mathscr{H}$ be a finite-dimensional complex Hilbert space and $mathscr{D}$ the set of density matrices on $mathscr{H}$, i.e., the positive operators with trace 1. Our goal in this note is to identify a probability measure $u$ on $mathscr{D}$ that can be regarded as the uniform distribution over $mathscr{D}$. We propose a measure on $mathscr{D}$, argue that it can be so regarded, discuss its properties, and compute the joint distribution of the eigenvalues of a random density matrix distributed according to this measure.
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We demonstrate a general procedure to construct entanglement witnesses for any entangled state. This procedure is based on the trace inequality and a general form of entanglement witnesses, which is in the form $W=rho-c_{rho} I$, where $rho$ is a density matrix, $c_{rho}$ is a non-negative number related to $rho$, and $I$ is the identity matrix. The general form of entanglement witnesses is deduced from Choi-Jamio{l}kowski isomorphism, that can be reinterpreted as that all quantum states can be obtained by a maximally quantum entangled state pass through certain completely positive maps. Furthermore, we provide the necessary and sufficient condition of the entanglement witness $W=rho-c_{rho}I$ in operation, as well as in theory.
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