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The low-temperature regime of charge-qubit decoherence due to its Coulombinteraction with electrons tunneling through Luttinger liquid quantum-pointcontact (QPC) is investigated. The study is focused on quantum detectorproperties of Luttinger liquid QPC. It is shown, that in low-temperature limitthe respective perturbative decoherence- and acquisition of informationtimescales both tend to diverge, thus, shadowing a true picture oflow-temperature quantum detection for such quantum systems. Here I prove twogeneral mathematical statements (S-theorem and S-lemma) about exactre-exponentiation of Keldysh-contour ordered T-exponent for arbitrary Luttingerliquid tunnel Hamiltonian. As the result, decoherence- and acquisition ofinformation time-scales as well as QPC quantum detector efficiency rate arecalculated exactly and are shown to have a dramatic dependence on repulsiveinteraction between electrons in 1D leads of QPC. Discovered abrupt decrease ofQPC quantum detector efficiency $ Q $ with the increase of $ g $ in the closevicinity of value $ g_{cr}(T) $ represents a fingerprint ofinteraction-dependent instability of all the quantum detection procedure forany Luttinger liquid QPC quantum detector at definite low enough temperatures $T_{cr}(g) $. The reasons behind these effects are discussed. Also, it is shownthat such the low-temperature detection instability effect is able to explain alarge unclear mismatch between expected and observed decoherence timescales intwo 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.

Theory of interaction-dependent instability in quantum detection by means of Luttinger liquid tunnel junction: a rigorous theorem