Pointers for Quantum Measurement Theory

Abstract

In the iconic measurements of atomic spin-1/2 or photon polarization, one employs two separate noninteracting detectors. Each detector is binary, registering the presence or absence of the atom or the photon. For measurements on a d-state particle, we recast the standard von Neumann measurement formalism by replacing the familiar pointer variable with an array of such detectors, one for each of the d possible outcomes. We show that the unitary dynamics of the pre-measurement process restricts the detector outputs to the subspace of single outcomes, so that the pointer emerges from the apparatus. We propose a physical extension of this apparatus which replaces each detector with an ancilla qubit coupled to a readout device. This explicitly separates the pointer into distinct quantum and (effectively) classical parts, and delays the quantum to classical transition. As a result, one not only recovers the collapse scenario of an ordinary apparatus, but one can also observe a superposition of the quantum pointer states.

Appendix: Recovery of Simultaneous Fourier Transforms

Here we show that the simultaneous Fourier transform description of the entangled state (6) [based on Eqs. (10) and 11)] is recovered from the general prescription given in Sec. IV. The Fourier transformed basis, $| k \rangle _{pX} = {{\mathcal {F}}}^{\dag } | k \rangle _p$, corresponds to

$$\begin{aligned} O_p \rightarrow X_p = {{\mathcal {F}}}^{\dag } Z_p {{\mathcal {F}}}, \end{aligned}$$

(26)

so that the general unitary transformations defined in Eqs. 17 and 18 are specialized to

$$\begin{aligned} U_p = {{\mathcal {F}}}^{\dag } \hspace{42.67912pt}\hbox {and} \hspace{42.67912pt}V_s = U_s^* = {{\mathcal {F}}}. \end{aligned}$$

(27)

since complex conjugation amounts to inversion of the Fourier transformation. And since this inversion amounts to a sign change of labels, the entangled state (6) is rewritten as (15). The prescription of Sec. V tells us that this state is an eigenstate of ${\tilde{O}}_s^{\dag } O_p$, and that

$$\begin{aligned} {\tilde{O}}_s = V_s Z_s V_s^{\dag } = {{\mathcal {F}}} Z_s {{\mathcal {F}}}^{\dag } = X^{\dag } \end{aligned}$$

(28)

so that

$$\begin{aligned} {\tilde{O}}_s^{\dag } O_p = X_s X_p, \end{aligned}$$

(29)

as is needed for consistency with Eq. 14. This shows that the Fourier transformed representation of Sec. III is recovered by the general prescription of Sec. IV.