Abstract
A weak value of an observable is studied for a quantum system which is placed under the influence of an environment, where a quantum system irreversibly evolves from a pre-selected state to a post-selected state. A general expression for a weak value influenced by an environment is provided. For a Markovian environment, the weak value is calculated in terms of the predictive and retrodictive density matrices, or by means of the quantum regression theorem. For a non-Markovian environment, a weak value is examined by making use of exactly solvable models. It is found that although the anomalous property is significantly suppressed by a Markovian environment, it can survive a non-Markovian environment.
Highlights
One of the most characteristic features of quantum mechanics lies in a measurement process which provides some information about an observable of a quantum system to be measured [1]
In a usual measurement process, the measured system is not referred after the interaction with the measuring device, though it is prepared in an initial state before the interaction
We have considered the weak value of an observable of a system interacting with an environment and we have provided the general expression of the weak value influenced by an environment
Summary
One of the most characteristic features of quantum mechanics lies in a measurement process which provides some information about an observable of a quantum system to be measured [1]. What we can obtain by quantum measurement is the eigenvalue and average value of the observable. It is interesting to investigate the decoherence of weak values during the irreversible time evolution of a system from pre-selected state to a post-selected state. The post-selection of the system that is essential for weak values makes it very difficult to investigate the irreversible time evolution by the usual method when an environment is non-Markovian.
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