Abstract
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.
Highlights
With every probability distribution μ over wave functions, i.e., over the unit sphere S(H ) in a complex Hilbert space H, there is associated a density matrix ρ= μ(dψ) |ψ ψ| . (1) S(H )In this note, in contrast, we consider a probability distribution over density matrices, and we ask whether there exists a distribution that should be regarded as the uniform distribution u over all density matrices
In contrast, we consider a probability distribution over density matrices, and we ask whether there exists a distribution that should be regarded as the uniform distribution u over all density matrices
Density matrices can arise as encoding random ψs, and as partial traces of states of larger systems; it is conceivable that even the fundamental quantum state of the universe is represented by a density matrix ρ
Summary
With every probability distribution μ over wave functions, i.e., over the unit sphere S(H ) in a complex Hilbert space H , there is associated a density matrix ρ=. For infinite-dimensional Hilbert spaces H , it does not seem that a uniform distribution exists over the density matrices on H , which is not surprising as there is no uniform distribution either over H itself or S(H ). Our reasoning yields, for any subspace H of a bigger Hilbert space K with dim H < ∞, a uniform probability distribution over the density matrices concentrated on H , regardless of whether dim K is finite or infinite.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
