Quantum coherence and negative quasi probabilities in a contextual three-path interferometer

Basic quantum effects are often illustrated using single particle interferences in two-path interferometers. A wider range of non-classical phenomena can be illustrated using three-path interferometers, but the increased complexity of quantum statistics in a three-dimensional Hilbert space makes it difficult to identify a representative set of observable properties that could be used to characterize specific phenomena. Here, I propose a characterization of pure states based on a five-stage interferometer recently introduced to demonstrate the relation between different measurement contexts (Optica Quantum 1, 63 (2023)). It is shown that the orthogonality relations between the states representing the different measurement contexts can be used to classify pure states within the three-dimensional Hilbert space according to the non-classical correlations between different contexts expressed by negative Kirkwood-Dirac distributions.

💡 Research Summary

The paper presents a comprehensive study of quantum coherence and non‑classical correlations in a five‑stage three‑path interferometer, extending the familiar two‑path interference paradigm to a three‑dimensional Hilbert space. The interferometer implements five distinct measurement contexts, each defined by a pair of orthogonal path states. In total ten physical path states are identified: five “outer” paths (|1⟩, |2⟩, |S₁⟩, |f⟩, |S₂⟩) and five “inner” paths (|3⟩, |D₁⟩, |P₁⟩, |P₂⟩, |D₂⟩). By expressing each state as a real‑valued three‑component unit vector, the authors map the full set of states onto the surface of a unit sphere. Orthogonality between two states corresponds to a great circle on this sphere; outer states lie at the intersection of four great circles, inner states at the intersection of two. This geometric representation makes the relationships among the five contexts visually transparent.

Beyond the physical path states, the analysis introduces another ten “non‑path” states that arise from orthogonality constraints between inner states belonging to different contexts. These states form the vertices of a pentagon inscribed in a pentagram whose points are the outer path states. The pentagon vertices correspond to the well‑known Hardy‑type contextuality paradox: despite certain inner paths having zero detection probability, the associated non‑path state can still have a non‑zero probability (e.g., |N_f⟩ has P(f)=1/9 even though P(D₁)=P(D₂)=0). These five states—|N_f⟩, |N₁⟩, |N_{S2}⟩, |N_{S1}⟩, |N₂⟩—violate a Clifton‑type non‑contextual inequality, thereby providing a concrete experimental signature of contextuality.

To quantify the non‑classicality of each state pair, the paper employs the Kirkwood‑Dirac (KD) quasi‑probability distribution. For any two non‑orthogonal states |a⟩ and |b⟩, the KD value K(a,b)=⟨a|b⟩⟨b|ρ|a⟩ (with ρ the system state) serves as a joint statistical weight. Positive KD values admit a classical interpretation, whereas negative values signal genuine quantum interference and contextuality. The authors compute all ten×ten KD values for the ten path states and the ten non‑path states. They find that negative KD values appear precisely along the lines that are orthogonal to one of the two states, i.e., the great circles identified earlier. In particular, the strongest negative KD contributions involve pairs of outer paths from different contexts and pairs of outer–inner paths that share a context.

The distribution of negative KD values maps directly onto the violation of the non‑contextual inequality. The five |N_i⟩ states each belong simultaneously to three of the six major classes defined on the sphere, reflecting their highly contextual nature. Conversely, the five |θ_k⟩ states sit at the intersection of four sub‑classes and exhibit comparatively milder negativity, yet still lie outside any purely classical region. By examining the sign pattern of KD values across the sphere, the authors partition the surface into six primary classes and a total of thirty‑one sub‑classes. Each class is characterized by a distinct pattern of positive/negative KD values and by the fidelity of the state with respect to the five measurement contexts.

A particularly interesting result is the construction of an orthogonal basis that simultaneously achieves high fidelity in all five contexts. By selecting appropriate states from different sub‑classes on the “quasi‑classical” belt of the sphere, the authors demonstrate that one can engineer a set of mutually orthogonal states that are each close to classical path statistics yet retain the essential contextual correlations. This provides a practical recipe for designing multi‑context quantum circuits or measurement schemes that exploit contextuality without sacrificing operational stability.

In summary, the paper delivers a unified geometric‑algebraic framework for analyzing three‑path interferometry. It shows how orthogonality relations define a spherical network of states, how Kirkwood‑Dirac quasi‑probabilities capture the emergence of negative joint probabilities, and how these negatives pinpoint the exact loci of contextuality. The resulting classification of states into six major classes and thirty‑one sub‑classes offers a detailed map of the quantum‑classical boundary in a three‑dimensional Hilbert space. These insights are directly applicable to the design of higher‑dimensional quantum information protocols, quantum simulators, and foundational tests of contextuality, marking a significant step toward intuitive yet rigorous understanding of quantum effects in larger systems.