Transition Probabilities

Fritz, T.

It is well-known that a quantum measurement can enhance the transition probability between two quantum states. Such a measurement operates after preparation of the initial state and before postselecting for the final state. Here we analyze this kind of scenario in detail and determine which probability distributions on a finite number of outcomes can occur for an intermediate measurement with postselection, for given values of the following two quantities: (i) the transition probability without measurement, (ii) the transition probability with measurement. This is done for both the cases of projective measurements and of generalized measurements. Among other constraints, this quantifies a trade-off between high randomness in a projective measurement and high measurement-modified transition probability. An intermediate projective measurement can enhance a transition probability such that the failure probability decreases by a factor of up to 2, but not by more.

Pieter E. Vermaas

Tobias Fritz

It is well-known that a quantum measurement can enhance the transition probability between two quantum states. Such a measurement operates after preparation of the initial state and before postselecting for the final state. Here we analyze this kind of scenario in detail and determine which probability distributions on a finite number of outcomes can occur for an intermediate measurement with postselection, for given values of the following two quantities: (i) the transition probability without measurement and (ii) the transition probability with measurement. This is done for both the cases of projective measurements and of generalized measurements. Among other constraints, this quantifies a trade-off between high randomness in a projective measurement and high measurement-modified transition probability. An intermediate projective measurement can enhance a transition probability such that the failure probability decreases by a factor of up to 2, but not by more.

Vittorio Cantoni

Enrico G. Beltrametti, Gianni Cassinelli, Peter Carruthers

Transition Probabilities between Pure States In physicists' nomenclature the expression “transition probability” generally refers to some dynamical instability, more specifically to a nonzero probability for the system to make a transition from an initial to a final state. Our use of the term is not directly related to dynamical instabilities; rather we follow von Neumann's terminology, and the transition probability between two states is meant to represent, intuitively, a measure of their overlapping. To visualize this notion in an explicit example consider the states of linear polarization of a photon beam, let a be the polarization state filtered by a Nicol prism N 1 , and let β be the state filtered by a Nicol prism N 2 rotated by an angle ϑ with respect to N 1 ; then the transition probability between a and β is the probability for a photon in state α to pass N 2 , which is empirically known to be cos 2 ϑ (Malus law), and equals the probability for a photon in state β to pass N 1 . In the Hilbert-space formulation of quantum mechanics the idea of transition probability we have in mind is simply the modulus squared of the scalar product (see Sections 2.5 and 9.3).