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A linear oscillator very weakly coupled with the object quantum system isproposed as a detector measuring the lifetime of irreversibility exhibited bythe system, and classically chaotic coupled kicked rotors are examined as idealexamples. The lifetime increases drastically in close correlation with theenhancement of entanglement entropy(EE) between the kicked rotors. In thetransition regime to the full entanglement, the EE of individual eigenstatesfluctuates anomalously, and the lifetime also fluctuates in correlation withthe EE. In the fully entangled regime the fluctuation disappear, but thelifetime is not yet unique but increases in proportion to the number ofsuperposed eigenstates and is proportional to the square of Hilbert spacedimension in the full superposition.

Lifetime of the arrow of time inherent in chaotic eigenstates: case of coupled kicked rotors