Complete complementarity relations and their Lorentz invariance

Abstract

It is well known that entanglement under Lorentz boosts is highly dependent on the boost scenario in question. For single-particle states, a spin-momentum product state can be transformed into an entangled state. However, entanglement is just one of the aspects that completely characterizes a quantum system. The other two are known as the wave-particle duality. Although the entanglement entropy does not remain invariant under Lorentz boosts, and neither do the measures of predictability and coherence, we show here that these three measures taken together, in a complete complementarity relation (CCR), are Lorentz invariant. Peres et al. (Peres et al. 2002 Phys. Rev. Lett. 88 , 230402. ( doi:10.1103/PhysRevLett.88.230402 )) realized that even though it is possible to formally define spin in any Lorentz frame, there is no relationship between the observable expectation values in different Lorentz frames. Analogously, one can, in principle, define complementary relations in any Lorentz frame, but there is no obvious transformation law relating complementary relations in different frames. However, our result shows that the CCRs have the same value in any Lorentz frame, i.e. there is a transformation law connecting the CCRs. In addition, we explore relativistic scenarios for single and two-particle states, which helps in understanding the exchange of different aspects of a quantum system under Lorentz boosts.

Quantum coherence versus interferometric visibility in a generalized Mach-Zehnder interferometer

“…( 3) has astonishing aspects that were shown recently: it is intrinsically connected to the notion of realism defined by Bilobran and Angelo [23], as discussed in Ref. [24]; it is invariant by global unitary operations, which implies that it is preserved under unitary evolution and it is Lorentz invariant [25]; it remains valid in curved spacetimes as a quanton travels along its world lines, as shown in Ref. [26].…”

Quantum simulation of the generalized-entangled quantum eraser and the related complete complementarity relations

“…( ) is an entanglement monotone, as shown in [26], that in this case measures the entanglement between B¢ with the rest of the system as a whole. Besides, as shown in [27], equation ( 8) is invariant under global unitary operations which implies that such relation remains valid under unitary evolution and, therefore, can be applied in each step of the intereferometer.…”

“…Recently, it has been shown that duality inequalities and triality equalities can be derived from the basic properties of the quantum density matrix [23][24][25]. This framework has lead to fundamental connections of complete complementarity relations (CCRs) with uncertainty relations [25], entanglement theory [26], and Lorentz invariance [27]. In [28], a CCR was applied to quantitatively understand a QE, analog to that of [1], but also considering partial entanglement of the quanton with the path marker and with an auxiliary system simulating the environment's action.…”