Description of weak measurements and weak values in the phase space representation of quantum mechanics

L. Rosales-Zárate, Bogdan Opanchuk, P. D. Drummond +1 more

Quantum simulations of Bell inequality violations are numerically obtained using probabilistic phase-space methods, namely, the positive-$P$ representation. In this approach the moments of quantum observables are evaluated as moments of variables that have values outside the normal eigenvalue range. There is thus a parallel with quantum weak measurements and weak values. Nevertheless, the representation is exactly equivalent to quantum mechanics. A number of states violating Bell inequalities are sampled, demonstrating that these quantum paradoxes can be treated with probabilistic methods. We treat quantum dynamics by simulating the time evolution of the Bell state formed via parametric down-conversion and discuss multimode generalizations.

M. Shioden

Allen Lobo, Carlos Alberto de Lima Ribeiro

Agus Budiyono

It is shown that a novel ``epistemically restricted phase-space distribution,'' directly derivable from the wave function, can be operationally determined by a weak measurement of momentum followed by postselection on position. The epistemically restricted phase-space representation thereby reveals deep links between a specific epistemic restriction manifesting transparently quantum uncertainty in classical phase space, weak momentum value, and complex wave function.

Werner Stulpe

Angela Karanjai, Eric G. Cavalcanti, Stephen D. Bartlett +1 more

Weak measurement of a quantum system followed by postselection based on a subsequent strong measurement gives rise to a quantity called the weak value : a complex number for which the interpretation has long been debated. We analyse the procedure of weak measurement and postselection, and the interpretation of the associated weak value, using a theory of classical mechanics supplemented by an epistemic restriction that is known to be operationally equivalent to a subtheory of quantum mechanics. Both the real and imaginary components of the weak value appear as phase space displacements in the postselected expectation values of the measurement device's position and momentum distributions, and we recover the same displacements as in the quantum case by studying the corresponding evolution in our theory of classical mechanics with an epistemic restriction. By using this epistemically restricted theory, we gain insight into the appearance of the weak value as a result of the statistical effects of post selection, and this provides us with an operational interpretation of the weak value, both its real and imaginary parts. We find that the imaginary part of the weak value is a measure of how much postselection biases the mean phase space distribution for a given amount of measurement disturbance. All such biases proportional to the imaginary part of the weak value vanish in the limit where disturbance due to measurement goes to zero. Our analysis also offers intuitive insight into how measurement disturbance can be minimized and the limits of weak measurement.