Modular variables in quantum theory

Yakir Aharonov, Aage Petersen

Augusto C. Lobo, Yakir Aharonov, Jeff Tollaksen +2 more

M. A. D. Carvalho, José Ferraz, G. F. Borges +3 more

We experimentally detect entanglement in modular position-momentum variables of down-converted photon pairs which have passed through D-slit apertures. We employ two entanglement criteria, one using variances of the modular variables, the other using the Shannon entropy.

M. A. D. Carvalho, José Ferraz, G. F. Borges +3 more

We experimentally detect entanglement in modular position and momentum variables of photon pairs which have passed through $D$-slit apertures. We first employ an entanglement criteria recently proposed in [Phys. Rev. Lett. 106, 210501 (2011)], using variances of the modular variables. We then propose an entanglement witness for modular variables based on the Shannon entropy and test it experimentally. Finally, we derive criteria for Einstein-Podolsky-Rosen-steering correlations using variances and entropy functions. In both cases, the entropic criteria are more successful at identifying quantum correlations in our data.

Vladimir Georgievich Turaev

Abdulsalam Al-Mayahi

A foundational ambiguity persists in modern physics: angular phase is frequently conflated with geometric arc length, encouraging the view that periodicity is a property of spatial motion rather than an intrinsic variable. We decouple the dimensionless phase θ from the radius-dependent arc length s and introduce an intrinsic ordering parameter—internal time τ—associated with a system’s phase structure. We show that τ provides a unifying perspective on internally generated temporal flows, linking modular flow in Tomita–Takesaki theory, the imaginary-time structure underlying equilibrium (KMS) states, and phase coherence in quantum dynamics. This reframing clarifies how evolution and stability criteria can be formulated without privileging an external clock, with implications for background-independent dynamics, information-theoretic descriptions of modular flow, and coherence-focused quantum technologies. By elevating phase from a geometric surrogate to a foundational variable, the paper offers a concise bridge between thermodynamic and quantum temporal structure and more general treatments of time in modern theory.