2015/12/07 by J. M. Farinholt, Farinholt, J. M., A. Ghazarians +3
Mathematics · Physics and Astronomy · #FOS: Physical sciences #Mathematical Physics (math-ph) #Quantum Physics (quant-ph) #math-ph #math.MP #quant-ph
paper · pdf · doi:10.48550/arxiv.1512.02113
18 pages, 3 figures. v2: minor change to one of the references
arxiv created 2015/12/15 · arxiv updated 2015/12/17
The concept of a weak value of a quantum observable was developed in the late 1980s by Aharonov and colleagues to characterize the value of an observable for a quantum system in the time interval between two projective measurements. Curiously, these values often lie outside the eigenspectrum of the observable, and can even be complex-valued. Nevertheless, the weak value of a quantum observable has been shown to be a valuable resource in quantum metrology, and has received recent attention in foundational aspects of quantum mechanics. This paper is driven by a desire to more fully understand the underlying mathematical structure of weak values. In order to do this, we allow an observable to be any Hermitian operator, and use the pre- and post-selected states to develop well-defined linear maps between the Hermitian operators and their corresponding weak values. We may then use the inherent Euclidean structure on Hermitian space to geometrically decompose a weak value of an observable. In the case in which the quantum systems are qubits, we provide a full geometric characterization of weak values.