Quantum Behavior and Negativity of Wigner Functions

Question

Let us consider a scenario where we have a dataset $\mathbf{X}$, which is a collection of vectors $\mathbf{x}_i \in \mathbb{R}^n$. We encode each component $x_j \in \mathbb{R}$ of $\mathbf{x}$ in a coherent state $|x_j\rangle$ (not a position eigenstate) via some fixed scheme. Now, consider a one-parameter unitary transformation $U(\theta(t)) : |x_j\rangle \mapsto |y_j\rangle,$ according to some dynamical parameter $\theta(t)$. One can calculate the so-called negativity of the Wigner function of a state $|\psi\rangle$ from the equation, $$ N_{|\psi\rangle} = \int_{\mathbb{R}^2} \big|W_{|\psi\rangle}(x,p)\big| \, dx \, dp - 1.$$

Let us say, we calculate an average negativity $\overline{N} \in \mathbb{R}$ associated to each vector $\mathbf{x}_i \in \mathbb{R}^n$ and plot its time evolution. What precisely can I infer from such plots? Are these average negativities a good indicator of how quantum the state after $U(\theta)$ is?

ADDENDUM:

The parameters $\theta(t)$ are chosen randomly at $t = 0$ and $\theta(t+1)$ is determined via a gradient descent on the mean square error $\left \langle \left( \langle y_j |\hat{q}| y_j\rangle - x_j \right)^2 \right \rangle$ at every time step. Here, $\hat{q}$ is a quadrature operator. The time evolution of the average negativities $\overline{N}(t)$ is computed starting from $t = 0$. Thus for every time step, we would get a different $|y_j\rangle$ according to $U(\theta(t))$ for the same $|x_j\rangle$.

I expect that in such a problem, if $U$ is not gaussian, then $\forall t > 0 : \overline{N}(t) > 0$. However, I doubt that this is a good measure for how quantum my state is, especially for small values, say, $0 < \overline{N} \leq 0.5$. I would have a similar concern for the WLN mentioned in @Alex's answer. Are there other quantities that can help quantify non-classicality besides or in addition to the WLNs or are my concerns misguided?

Answer 1

Starting from a coherent state, which possesses a positive Wigner function and thus has $N_{|\alpha\rangle} = 0$, and evolving $|\alpha\rangle$ to a final state $|\psi\rangle$, for which $N_{|\psi\rangle} \geq 0$, indicates that some non-classicality, as evidenced by the emergence of negativity, was generated during the unitary process. This suggests that there is no classical theory capable of fully explaining the statistics of your data. However, caution is warranted as the negativity is not a necessary and sufficient condition for non-classicality.

For example, in the paper "Quantum catalysis in cavity QED", the authors discuss the possibility of generating non-classical states of light using a specific mechanism. They employ two figures of merit to characterise these states: the negativity of the Wigner function and second-order coherence. While neither figures of merit is individually necessary and sufficient to define non-classicality, their combined use is particularly effective for the task at hand.

Typically, the Wigner logarithmic negativity (WLN) is used instead of the standard negativity. This is defined as

$$\mathsf{W} \left ( \rho \right) := \log \left( \int \! \mathrm{d} x \,\mathrm{d} p \, \left| W_\rho \left(x,p \right) \right| \right),$$

where $W_\rho$ is the Wigner function of the state $\rho$. This quantity is related to the negativity via

$$\mathsf{W} = \log(N+1).$$

The WLN is an additive monotone (take a look at Appendix C of arXiv.1804.06763) since the Wigner function of separable states can be factorised. The aim of this function is to quantify a measure of non-Gaussianity, which has full monotonicity under Gaussian protocols.

Finally and more towards your last question, I would look at the average of WLN after $U[\theta(t)]$. This indeed would tell us the average of non-classicality generated by this protocol.