Typical weak and superweak values

Oded Zilberberg, Alessandro Romito, Yuval Gefen

Michael Berry, Pragya Shukla

Weak values, resulting from the action of an operator on a preselected state when measured after postselection by a different state, can lie outside the spectrum of eigenvalues of the operator: they can be ‘superweak’. This phenomenon can be quantified by averaging over an ensemble of the two states, and calculating the probability distribution of the weak values. If there are many eigenvalues, distributed within a finite range, this distribution takes a simple universal generalized lorentzian form, and the ‘superweak probablility’, of weak values outside the spectrum, can be as large as 1–1/ = 0.293.... By contrast, the familiar expectation values always lie within the spectral range, and their distribution, although approximately gaussian for many eigenvalues, is not universal.

Lev Vaidman A

Pragya Shukla

Weak value is a physical property of a quantum system which manifests itself through a weak measurement using different preand post-selected ensembles of the system. The weak values of an operator may differ significantly from its eigenvalues and can even lie outside the spectrum if it is bound: they can be ‘superweak’. The latter, originating due to a coherent superposition of waves, may appear as a ‘supershift’ on the measuring device. This property has potential application in the amplification and detection of extremely weak signals.

Michael Berry, Mark R. Dennis, B McRoberts +1 more

The simplest weak measurement is of a component of spin 1/2. For this observable, the probability distributions of the real and imaginary parts of the weak value, and their joint probability distribution, are calculated exactly for pre- and postselected states uniformly distributed over the surface of the Poincaré-Bloch sphere. The superweak probability, that the real part of the weak value lies outside the spectral range, is 1/3. This case, with just two eigenvalues, complements our previous calculation ( Berry and Shukla 2010 J. Phys. A: Math. Theor. 43 354024 ) of the universal form of the weak value probability distribution for an operator with many eigenvalues.

Lars M. Johansen