Weak values and the quantum phase space

Abstract

We address the issue of how to properly treat, and in a more general setting, the concept of a weak value of a weak measurement in quantum mechanics. We show that for this purpose, one must take in account the effects of the measuring process on the entire phase space of the measuring system. By using coherent states, we go a step further than Jozsa in a recent paper, and we present an example where the result of the measurement is symmetrical in the position and momentum observables and seems to be much better suited for quantum optical implementation.Comment: 07 pages, accepted for PR

Weak measurements with a qubit meter

“…See also ref. [15] for a quantum phase space perspective on real value measurements. We note, that in the same way as anomalous real weak values occur outside the spectrum of the operator A, an imaginary part of the expectation value of a Hermitian operator is anomalous, and, indeed, an interaction Hamiltonian H int = g(t)A · p is normally not expected to change the value of the Quantum Non-Demolition (QND) meter observable p. It is of course the pre-and post-selection by different states of the quantum system that change these conventional properties.…”

Typical weak and superweak values

“…A weak = φ|Â|ψ φ|ψ ≡ A + iA . (1.2) This is usually a complex number, with the following significance [2][3][4]. If the initial state of the detector is represented by a real wavepacket (e.g.…”

The Phase Space Formulation of Time-Symmetric Quantum Mechanics

“…53) have been announced without motivations and proofs in previous work [7,8]. We also mention that Lobo and Ribeiro [14] discussed weak values in the quantum phase space using methods that are very different from the Weyl-Wigner-Moyal formalism employed here.…”