Density matrix is one important tool in quantum mechanics,and it has very broad applications.However there are different definitions about the density matrix,and they describe quite different systems.There has been some misunderstanding about the density matrix in the community,and these misunderstandings hinder the right application of the density matrix.In this article,we discuss the different definitions of density matrix. We suggest to use the full density matrix,compressed density matrix and the reduced density matrix to describe the state of a complete quantum system,the state of an averaged particle in an ensemble and the state of part of a composite system.We stress that contrary to the wide accepted understanding that ensembles with the same compressed density matrix are physically indistinguishable, they are distinguishable through the so-called ensemble measurement.As an application,we suggest that the present conclusion that the present-day nuclear magnetic resonance quantum computation does not have quantum entanglement is groundless.
We compare different strategies aimed to prepare an ensemble with a given density matrix
rho. Preparing the ensemble of eigenstates of
rho with appropriate probabilities can be treated as `generous' strategy: it provides maximal accessible information about the state. Another extremity is the so-called `Scrooge' ensemble, which is mostly stingy to share the information. We introduce `lazy' ensembles which require minimal efforts to prepare the density matrix by selecting pure states with respect to completely random choice. We consider two parties, Alice and Bob, playing a kind of game. Bob wishes to guess which pure state is prepared by Alice. His null hypothesis, based on the lack of any information about Alice's intention, is that Alice prepares any pure state with equal probability. Then, the average quantum state measured by Bob turns out to be
rho, and he has to make a new hypothesis about Alice's intention solely based on the information that the observed density matrix is
rho. The arising `lazy' ensemble is shown to be the alternative hypothesis which minimizes the Type I error.
We compare different strategies aimed to prepare an ensemble with a given density matrix ρ. Preparing the ensemble of eigenstates of ρ with appropriate probabilities can be treated as 'generous' strategy: it provides maximal accessible information about the state. Another extremity is the so-called 'Scrooge' ensemble, which is mostly stingy in sharing the information. We introduce 'lazy' ensembles which require minimal effort to prepare the density matrix by selecting pure states with respect to completely random choice. We consider two parties, Alice and Bob, playing a kind of game. Bob wishes to guess which pure state is prepared by Alice. His null hypothesis, based on the lack of any information about Alice's intention, is that Alice prepares any pure state with equal probability. Then, the average quantum state measured by Bob turns out to be ρ, and he has to make a new hypothesis about Alice's intention solely based on the information that the observed density matrix is ρ. The arising 'lazy' ensemble is shown to be the alternative hypothesis which minimizes type I error.