Path summation and von Neumann–like quantum measurements

We demonstrate how a general von Neumann–like measurement can be analyzed in terms of histories (paths) constructed for the measured variable $A$. The Schrödinger state of a system in a Hilbert space of arbitrary dimensionality is decomposed into a set of substates, each of which corresponds to a particular detailed history of the system. The coherence between the substates may then be destroyed by meter(s) to a degree determined by the nature and the accuracy of the measurement(s) which may be of von Neumann, finite-time, or continuous type. The cases of a particle described by Feynman paths in the coordinate space and a qubit in a two-dimensional Hilbert space are studied in some detail.

Figure 1

(a) A schematic diagram of a path $\phi \left(t^{\prime }\right)$ which contributes to $\mid \mathrm{\Phi }(t\mid \left[\varphi \right])⟩$ at $t<T$ (solid). Also shown by a dashed line is the tube which contains the paths, contributing to $\mid \mathrm{\Psi }(t\mid \left[\phi \right])⟩$, obtained by Gaussian coarse graining with the width $\sigma$. (b) An eigenpath which contributes to $\mid \mathrm{\Phi }(T\mid \left[\phi \right])⟩$ for a two-level system $(a_1=1,a_2=2)$.

Figure 2

A system starts in a state ${\mathrm{\Psi }}_0⟩$ which, without measurements, would evolve into $\mathrm{\Psi }\left(t\right)⟩$. $\mathrm{\Psi }\left(t\right)⟩$ can be decomposed into a fine set of substates $\mid \mathrm{\Psi }\left[\phi \right]⟩$, each labeled by a particular history $\phi \left(t^{\prime }\right)$. A meter decomposes $\mathrm{\Psi }\left(t\right)⟩$ into less informative substates, labeled by the value $f$ of functional $F\left[\phi \right]$, which are obtained by summing $\mid \mathrm{\Psi }\left[\phi \right]⟩$ subject to the restriction $F\left[\phi \right]=f$. For the two meters shown, the substates in the shaded area are such that $F_1\left[\phi \right]=f_1$ and $F_2\left[\phi \right]=f_2$, and add up coherently to the state $\mid \mathrm{\Phi }(f_1\bullet f_2)⟩$, whose norm determines the probability to register both $f_1$ and $f_2$.