Trace-distance wave-particle inequality in dimensions four and five

Determine whether the trace-distance wave-particle inequality \(\mathcal{W}_{\rm tr}(\rho)^2+\mathcal{P}_{\rm tr}(\rho)^2\leq 1\) holds for all quantum states in four- and five-dimensional systems.

Background

The paper defines trace-distance-based waveness and particleness for a $d$-path interferometer as normalized trace distances between a quantum state and its diagonal part, and between its diagonal part and the maximally mixed state, respectively. It proves the inequality $$\mathcal{W}_{\rm tr}(\rho)^2+\mathcal{P}_{\rm tr}(\rho)^2\leq 1$$ for qubit and qutrit systems, while numerical experiments report violations in dimension $d=6$. The authors explicitly leave unresolved whether the same inequality remains valid in dimensions $d=4$ and $d=5$.