  The low-temperature regime of charge-qubit decoherence due to its Coulomb
interaction with electrons tunneling through Luttinger liquid quantum-point
contact (QPC) is investigated. The study is focused on quantum detector
properties of Luttinger liquid QPC. It is shown, that in low-temperature limit
the respective perturbative decoherence- and acquisition of information
timescales both tend to diverge, thus, shadowing a true picture of
low-temperature quantum detection for such quantum systems. Here I prove two
general mathematical statements (S-theorem and S-lemma) about exact
re-exponentiation of Keldysh-contour ordered T-exponent for arbitrary Luttinger
liquid tunnel Hamiltonian. As the result, decoherence- and acquisition of
information time-scales as well as QPC quantum detector efficiency rate are
calculated exactly and are shown to have a dramatic dependence on repulsive
interaction between electrons in 1D leads of QPC. Discovered abrupt decrease of
QPC quantum detector efficiency $ Q $ with the increase of $ g $ in the close
vicinity of value $ g_{cr}(T) $ represents a fingerprint of
interaction-dependent instability of all the quantum detection procedure for
any Luttinger liquid QPC quantum detector at definite low enough temperatures $
T_{cr}(g) $. The reasons behind these effects are discussed. Also, it is shown
that such the low-temperature detection instability effect is able to explain a
large unclear mismatch between expected and observed decoherence timescales in
two well-known experiments (J.Gorman, D.G.Hasko, D.A.Williams, Phys.Rev.Lett.,
95, 090502 (2005); K.D.Petersson, J.R.Petta, H.Lu, A.C.Gossard, Phys.Rev.Lett.,
105, 246804 (2010);) on charge-qubit quantum dynamics.
