# Quantum Behavior and Negativity of Wigner Functions

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?

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?

• Is your Hamiltonian the oscillator one? In that case, the displaced coherent states rotate rigidly in phase space. But Kolmogorov’s third axiom is violated, so you may not call this classicality… Nov 14, 2023 at 11:34
• @CosmasZachos I am not sure I understand your comment. Which Hamiltonian are you referring to here? The one that generates the unitary? If so, then this is not the oscillator one. If you are referring to the time evolution, then I am not concerned about it. My computation of $\overline{N}(t)$ is using the same $| x_j \rangle$ at different times. Nov 15, 2023 at 19:24
• I'm speaking standard deformation quantization. If the evolution operator is not generated by the oscillator hamiltonian, but depends on the $x_i$, then, the coherent states may diffuse, in general, and generate negative points. I am emphasizing the Wigner function cannot be a probability distribution, even if it is positive semidefinite. Nov 15, 2023 at 20:11
• @CosmasZachos Indeed. However, my unitary does not explicitly depend on $x_i$, just the parameter $\theta(t)$ which is also not dependent on $x_i$ (although one could maybe define it with respect to the $x_i$ perhaps). As a result, unless I am mistaken, one should expect some negativity to arise from the action of a general unitary, so in the context of this problem I am not so sure I should be concerned with non-classicality of PSD Wigner functions. Nov 15, 2023 at 20:35
• However, I do see that even with small negativities, it is possible that this may not give a good enough measure of non-classicality in general, hence the question. Are there more figures of merit that can (and ar needed to) reinforce these negativities? Such as the second-order coherence mentioned in @Alex's answer? Nov 15, 2023 at 20:42

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.

• Thank you. This is helpful. As an addendum, I would like to know if the WLN is a sufficient quantifier by itself or does it only work in conjunction with this second order coherence? Nov 15, 2023 at 20:08
• In addition to this, are there any ballpark ranges of values for the WLN in existing literature that indicate a high or low degree of non-classicality? Nov 15, 2023 at 20:43
• The WLN isn't the only criterion to certify non-classicality, but it's a very effective measure when working with Wigner negativity. Generally, there are no specific ballpark ranges of values for the WLN that indicate high or low non-classicality. However, it's important to note that a low WLN value, or one that approaches zero, suggests a lower degree of non-classicality. Based on my experience, a WLN value significantly greater than 1 usually implies a highly non-classical state.
– Alex
Nov 17, 2023 at 10:47
• Alright. That helps. Thank you. Nov 17, 2023 at 19:17