I will provide an answer but from a different perspective, and hopefully convince you that there is information in a density matrix which has no classical counterpart. Furthermore this can hence be considered a quantum component, and it can be shown that this information is stored as the eigenvectors of $\rho$.

I will give an example of how this manifests. The Fisher Information $I(\theta)$ is a statistic from classical probability theory which characterises how quickly one can learn about a parameter $\theta$ which characterises a probability distribution $p(\theta)$.

Specifically the variance of an unbiased classical estimator $\hat{\theta}$ respects the Cramer Rao bound
$$\mathrm{var}(\hat{\theta})\geq \frac{1}{I(\theta)}$$

The additivity of information means that if you sample the distribution $n$ times, collecting measurements each time the expected error $\Delta \theta_c = \sqrt{\mathrm{var}(\hat{\theta})}$ of any estimator goes like
$$\Delta \theta_c \propto \frac1{\sqrt{n}}$$

This is recognised in the scaling of the standard deviation $\sigma$ in things like central limit theorem.

We can define a quantum analogue, to the fisher information $J(\theta)$ which satisfies an analogus bound, known as the Quantum Cramer Rao bound.

However it is found that by permitting entanglement between classically independent sampling events, the bound is much better. And after having collected a dataset of $n$ measurments, the best possible quantum estimator is bound only by the error
$$\Delta \theta_q \propto \frac1{n}$$.

This shows that a general quantum state $\rho$ can definately support statistics which a classical probability distribution cannot.

The quantum Fisher information of a density matrix which depends on a parameter $\theta$
$$\rho(\theta) = \sum_i p_i(\theta) |\psi_i(\theta)\rangle\langle\psi_i(\theta)|$$
can be seen to seperate into several contributions, one of which is the classical Fisher information of the spectrum $p_i(\theta)$, another of which is a Fubini-Study like term which accounts for the information stored in the basis $|\psi_i(\theta)\rangle$. The possibility of (super-classical) quantum scaling depends entirely on the existence of this quantum term.

Alternatively stated, in terms of the behaviour of the Fisher information statistic and its quantum analogues, a density matrix $\rho$ supports non classical behaviour only if the basis set $|\psi_i(\theta)\rangle$ contains information relevant to the measurment, and in this sense, information stored in this way may be considered non-classical.

# Useful stuff

If you are interested in some of the topics discussed here see this good review for an explanation.
http://arxiv.org/pdf/1102.2318v1.pdf

This for an accessible but mathematical explanation of the QFI.
http://arxiv.org/pdf/0804.2981.pdf

This post imported from StackExchange Physics at 2014-04-11 15:21 (UCT), posted by SE-user ComptonScattering