Neutrinos belong to a class of objects allowing for ’complementary’ studies. The most common approaches of particle physics community become often successfully augmented by results concerning quantum informational properties studied from seemingly different perspective.
Our work belongs to a part of investigations indicating possibility of utilizing quantum informational perspective for better understanding the nature of neutrino. We considered geometric quantum speed limit and we showed its potential applicability as a hallmark for the
CP-violation. First, we presented how QSL as a function of time becomes modified by interaction of neutrino with normal matter formed by matter’s electrons or neutrons and generating the charge-current correction
\(V_C\) in the Hamiltonian Eq. (
8). Removal of artificial discontinuities in the QSL-time characteristics is the main effect of this interaction. Infinite peaks in
\(\nu _\mathrm{QSL}\) typical for an ideal decoherence-free system become shifted and lowered by
\(V_C\) and, for sufficiently large values of
\(V_C\) lose their direct connection to the position of extremal values of
\(P_0\) Eq. (
7). These results after inclusion of decoherence effect become supplemented by damping of oscillations of
\(\nu _\mathrm{QSL}\) which, for sufficiently high decoherence (given by an amplitude
\(\kappa \) of Lindblad dissipators Eq. (
11). Although we used purely phenomenological model of decoherence guided solely by the requirement of complete positivity, the results are generic for Markovian open quantum systems.
Geometric quantum speed limit Eq. (
2) quantifies informational (statistical) content of a time-local (calculated at a given time instant
t) rate of change of neutrino state. There are three factors affecting this rate which are studied in this paper: interaction between neutrino and normal matter (for
\(V_C\ne 0\)), Markovian decoherence (for
\(\kappa \ne 0\) in Eq. (
11) and potential violation of the
CP symmetry (for
\(\delta \ne 0\) in Eq. (
5)). From a mathematical perspective, the changes in the geometric speed limit Eq. (
2) reported here are due to a collective effect of collaboration of these three factors. In particular, the effect of
\(\delta \) in
\(\nu _\mathrm{QSL}\) starts to be visible provided that one allows for the presence of decoherence in a dynamical model of neutrino oscillation. It is due to an effective cancellation of the
CP-violating phase both for the operator norm and the Bures distance
l(
t) (related to fidelity and for pure states given by their overlap) in the numerator and denominator of Eq. (
2), respectively, for conservative model of neutrino oscillation with
\(\kappa =0\). The numerator of QSL Eq. (
2) is given by maximal singular value which, to be calculated, demands multiplication of an operator and its conjugate [
28] resulting in the above-mentioned cancellation of phase. In the presence of decoherence resulting in an information loss due to non-unitary dynamics Eq. (
11), the effect of
\(\delta \) is present indicating limiting applicability of purely Hamiltonian models of a description of at least certain properties of neutrino oscillation in time. The QSL quantifying instantaneous rate of changes of neutrino state under non-unitary evolution is a quantity which allows to exhibit otherwise ’invisible’ effects related to characteristic time scale of neutrino dynamics. The phenomenon of neutrino oscillations in time and for decoherence-free Hamiltonian description is related to the Mandelstam–Tamm time-energy uncertainty relation [
16]. Despite of the presence of certain controversial issues [
3,
15], it allows to set a characteristic time interval required for a significant change of the flavour neutrino state, i.e. an intrinsic time scale of the inter-flavour passage [
16]. Let us emphasize that in particular the Mössbauer neutrinos produced in two-body decays of nuclei embedded in a crystal lattice [
2] can be utilized for justification and verification of this property [
17]. Working beyond conservative dynamics approximation, as we do here, one can utilize QSL as a natural generalization of a (local in time) bound for a rate of neutrino oscillation. Moreover, the geometric quantum speed limit Eq. (
2) is directly associated with a quantum speed limit time [
28]
$$\begin{aligned} \tau _\mathrm{QSL}^{-1}(t)= & {} \frac{1}{t}\int _0^t \nu _\mathrm{QSL}(t') \mathrm{d}t' \end{aligned}$$
(14)
which thereafter can serve as a generalization of the characteristic time interval for a significant change of the flavour neutrino state in a very general, possibly non-Hamiltonian, systems. The quantum speed limit time Eq. (
14) is directly related to geometric quantum speed limit QSL Eq. (
2) and, by extension, it reflects the results reported in this work. The quantum speed limit time
\(\tau _\mathrm{QSL}\) is presented in Fig.
4 for the same set of parameters as previously
\(\nu _\mathrm{QSL}\). Let notice that although the results are qualitative only, one recognizes characteristic features of geometric speed limit Eq. (
2) also present in the speed limit time Eq. (
14). Following the interpretation given in Refs. [
16,
17] of the time-energy uncertainty and a related time scale as being characteristic for duration of a neutrino inter-flavour passage, one observes both an interaction with normal matter of an amplitude
\(V_C\) and Markovian decoherence quantified by
\(\kappa \) in the model Eq. (
11) significantly modifying the quantum speed limit time Eq. (
14). In particular, the effect of
\(V_C\) (as presented in the upper panel of Fig.
4 influences both a magnitude of
\(\tau _\mathrm{QSL}\) and its periodicity as it was for geometric speed limit Eq. (
2) in Fig.
1. Similar modification yet applied to an amplitude only is shared by geometric speed limit and its corresponding speed limit time under Markovian decoherence cf. Fig.
2 and the central panel of Fig.
4, respectively. The effect of
CP violation, seemingly less ’spectacular’, is also present (bottom panel of Fig.
4 indicating an enhancement of the quantum speed limit time due to an increasing value of the
CP-violating phase
\(\delta \). We strongly emphasize that all the neutrino experiments such as the recent one reported in Ref. [
54]) belong to the most sophisticated experiments performed so far and our predictions concerning QSL and the corresponding time are nothing but qualitative signatures of certain properties of neutrino oscillation.