7

For any quantum state, we have a unique density matrix $\rho$. For any $\rho$, we can do the Wigner transformation to get a unique Wigner function $P(x,p)$. For any Wigner function $P(x,p)$, we can do the Weyl transformation to get back the unique $\rho$. If the construction of the Wigner function from $\rho$ was not unique, then it would not be possible to ...


5

What is non-classicality? I'm not sure if there's a universally accepted definition, but the way that I'd define it is: if all possible outcomes of experiments on a particular quantum system can be described by a probability distribution, then the system is classical. Otherwise, it is non-classical. In alternative terminology, for a classical system, people ...


3

The authors are certainly thinking about finite frames. In this case, your statement is correct, since the number of elements in every spanning set is at least the vector space dimension. As glS already pointed out, the frames could be infinite: for countable frames, the summation is then meant as convergence in the $L^2$-sense (i.e. w.r.t. Hilbert-Schmidt ...


3

You mean something like $$W_{G}(\mathbf{r}) =\frac{2^{n}}{\pi^{n} \sqrt{\operatorname{Det} \sigma}} \mathrm{e}^{-(\mathbf{r}-\overline{\mathbf{r}})^{\top} \boldsymbol{\sigma}^{-1}(\mathbf{r}-\overline{\mathbf{r}})},$$ where $W_{G}(\mathbf{r})$ is the Wigner function corresponding to a Gaussian state, $\mathbf{\sigma}$ its covariance matrix, and $\overline{r}$...


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