  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., in [Phys. Rev. Lett. 88, 230402 (2002)],
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, even though it is possible to formally
define complementarity in any Lorentz frame, in principle, there's no
relationship between the complementarity relations of different Lorentz frames.
However, our result shows that it's possible to connect complete
complementarity relations in different Lorentz frames. 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.
