Time in Quantum Measurement

Based on a model of quantum measurement we derive an estimate for the external measurement-time. Some interesting consequences will be analyzed. Keywords: Quantum Entropy, Energy, Measurement, Time MSC 2000 codes: 94A17, 81P15 Entropy 2006, 8, 182-187 183 1 Introduction Given an apparatus or memory A, a system S and the environment E , the measurement process M can be represented by the following sequence: M := A S ! trE (A S)! A S , 2 I: (1) The …rst step is the entanglement between the system S and the apparatus A. The second step, d; is the decoherence of the entangled AS system by the perturbation through the environment E and the creation of classical alternatives. The third step, p, is the projection to one of the eigenstates and the record of a de…nite outcome. At this stage other states of the memory are being erased. In the course of this process the entropy of the AS system is going to ‡uctuate which re‡ects the expansion and compression of the abstract phase space spanned by the eigenvectors of a corresponding observable: There is no algorithm (e.g di¤erential equation) known to describe M exactly in time. The ‡uctuation of entropy alone makes it clear that the process M cannot be described in …nite time by a Hamiltonian ‡ow and the projection postulate does not refer to time at all. If we describe the decohered AS system by its density matrix AS , there holds for the entropy S AS = tr AS log AS 0: If the environment is supposed to be in thermal equilibrium of temperature T; the generalized Landauer principle [10] ; [13] tells us that during the measurement process an amount of energy whose average over di¤erent measurement outcomes E satis…es