Triangle inequalities for classical and quantum probabilities

Paweł Kurzyński, Dagomir Kaszlikowski

In this paper we study an application of an information distance between two measurements to the problem of non-contextuality and local realism. We postulate the triangle principle which states that any information distance is well defined on any pair of measurements, even if the two measurements cannot be jointly performed. As a consequence, the triangle inequality for this distance is obeyed for any three measurements. This simple principle is valid in any classical realistic theory, however it does not hold in quantum theory. It allows us to re-derive in an astonishingly simple way a large class of non-contextuality and local realistic inequalities via multiple applications of the triangle inequality. We also show that this principle can be applied to derive monogamy relations. The triangle principle is different than the assumption of non-contextuality and local realism, which is defined as a lack of existence of a joined probability distribution giving rise to all measurable data. Therefore, we show that one can design Bell-Kochen-Specker tests using a different principle.

Eftichios Bitsakis

Nils A. Törnqvist

It is observed that Bell's inequalities can be written as triangle inequalities for cross-sections, while quantum mechanics implies triangle inequalities for the square roots of the same cross-sections. A useful diagram displays instructively the domains separated by Bell's inequalities, on the one hand, and by quantum mechanics, on the other hand, and shows how the former lacks quantum-mechanical interference terms.

Enrico G. Beltrametti

Stephen Bruce Sontz

Yue Dai, Wen‐Long You, Yuli Dong +1 more

We provide detailed proofs of triangle inequalities in coherence measures and entanglement concurrence. If a rank-2 state $\ensuremath{\varrho}$ can be expressed as a convex combination of two pure states, i.e., $\ensuremath{\varrho}={p}_{1}|{\ensuremath{\psi}}_{1}\ensuremath{\rangle}\ensuremath{\langle}{\ensuremath{\psi}}_{1}|+{p}_{2}|{\ensuremath{\psi}}_{2}\ensuremath{\rangle}\ensuremath{\langle}{\ensuremath{\psi}}_{2}|$, a triangle inequality can be established as $|E(|{\mathrm{\ensuremath{\Psi}}}_{1}\ensuremath{\rangle})\ensuremath{-}E(|{\mathrm{\ensuremath{\Psi}}}_{2}\ensuremath{\rangle})|\ensuremath{\le}E(\ensuremath{\varrho})\ensuremath{\le}E(|{\mathrm{\ensuremath{\Psi}}}_{1}\ensuremath{\rangle})+E(|{\mathrm{\ensuremath{\Psi}}}_{2}\ensuremath{\rangle})$, where $|{\mathrm{\ensuremath{\Psi}}}_{1}\ensuremath{\rangle}=\sqrt{{p}_{1}}|{\ensuremath{\psi}}_{1}\ensuremath{\rangle}$ and $|{\mathrm{\ensuremath{\Psi}}}_{2}\ensuremath{\rangle}=\sqrt{{p}_{2}}|{\ensuremath{\psi}}_{2}\ensuremath{\rangle}$; $E$ can be considered either coherence measures or entanglement concurrence. This inequality displays mathematical beauty for its similarity to the triangle inequality in plane geometry. An illustrative example is given after the proof.