Observing a Quantum Measurement

Abstract

With the example of a Stern–Gerlach measurement on a spin-1/2 atom, we show that a superposition of both paths may be observed compatibly with properties attributed to state collapse—for example, the singleness (or mutual exclusivity) of outcomes. This is done by inserting a quantum two-state system (an ancilla) in each path, capable of responding to the passage of the atom, and thus acting as a virtual detector. We then consider real measurements on the compound system of atomic spin and two ancillae. Nondestructive measurements of a set of compatible joint observables can be performed, one for a superposition and others for collapse properties. A novel perspective is given as to why, within unitary quantum theory, ordinary measurements are blind to such superpositions. Implications for the theory of measurement are discussed.

Appendices

Appendix A: Density Matrix of Spin-Detectors System

Here we write out the density matrix of the spin/detectors system, and we show how the trace over the unobserved states of the detectors’ internal degrees of freedom yields the appropriate reduced density matrix, which expresses the blindness of the apparatus to superpositions of output states.

The initial mixed state of the spin/detectors system, assuming probabilities $p_{\mu _\uparrow }$ and $p_{\mu _\downarrow }$ for the microstates, $| 0,\mu _\uparrow \rangle _{D_{\uparrow }}$ and $| 0,\mu _\downarrow \rangle _{D_{\downarrow }}$ of the two detectors, is

$$\begin{aligned} \rho (t_1)= & {} \sum _{\mu _\uparrow ,\mu _\downarrow } p_{\mu _\uparrow } p_{\mu _\downarrow } \bigg ( \alpha | 1 \rangle _s +\beta | 0 \rangle _s \bigg ) | 0,\mu _\uparrow \rangle _{D_{\uparrow }} | 0,\mu _\downarrow \rangle _{D_{\downarrow }}\nonumber \\&\times \bigg ( \alpha ^* \langle 1 |_s +\beta ^* \langle 0 |_s \bigg ) \langle 0,\mu _\uparrow |_{D_{\uparrow }} \langle 0,\mu _\downarrow |_{D_{\downarrow }}. \end{aligned}$$

(12)

After the atom passes through the detectors and the paths are recombined at $t_4$, this becomes

$$\begin{aligned} \rho (t_4) = \sum _{\mu _\uparrow ,\mu _\downarrow }&p_{\mu _\uparrow } p_{\mu _\downarrow } \bigg ( \alpha | 1 \rangle _s | 1,\mu _\uparrow ' \rangle _{D_{\uparrow }} | 0,\mu _\downarrow \rangle _{D_{\downarrow }} + \beta | 0 \rangle _s | 0,\mu _\uparrow \rangle _{D_{\uparrow }} | 1,\mu _\downarrow ' \rangle _{D_{\downarrow }} \bigg ) \nonumber \\&\cdot \bigg ( \alpha ^* \langle 1 |_s \langle 1,\mu _\uparrow ' |_{D_{\uparrow }} \langle 0,\mu _\downarrow |_{D_{\downarrow }} + \beta ^* \langle 0 |_s \langle 0,\mu _\uparrow |_{D_{\uparrow }} \langle 1,\mu _\downarrow ' |_{D_{\downarrow }} \bigg ). \end{aligned}$$

(13)

Since we only read the detectors’ outputs and do not monitor the microscopic degrees of freedom (considered as the “environment” E), we trace over the latter to define the reduced density matrix describing the state of the spin and the detector displays, which is called the spin/pointer system [43]). To be more precise, each detector consists of its own pointer (with readout states 0 and 1), and its own environment (with associated states $\mu$ and $\mu '$, respectively). The trace consists of independent traces over the environments within each detector, i.e., $\rho ^r(t_4) \equiv Tr_{E_\uparrow ,E_\downarrow } \rho (t_4)$. To evaluate each of these, it is convenient to sum over $\mu$ (i.e., $\sum _\mu \langle \mu | ... | \mu \rangle$ in those terms where the pointer state 0 appears, and over $\mu '$ where it does not. The latter choice is legitimate because the two sets are related unitarily. It is straightforward then to show that

$$\begin{aligned}&\rho ^r(t_4) = |\alpha |^2 | 1 \rangle _s | 1 \rangle _{D_{\uparrow }} | 0 \rangle _{D_{\downarrow }} \langle 1 |_s \langle 1 |_{D_{\uparrow }} \langle 0 |_{D_{\downarrow }} \nonumber \\&+ \alpha \beta ^* | 1 \rangle _s | 1 \rangle _{D_{\uparrow }} | 0 \rangle _{D_{\downarrow }} \langle 0 |_s \langle 0 |_{D_{\uparrow }} \langle 1 |_{D_{\downarrow }} \sum _{\mu _\uparrow ,\mu _\downarrow } p_{\mu _\uparrow } p_{\mu _\downarrow } \langle \mu _\uparrow | \mu _\uparrow ' \rangle _{D_{\uparrow }} \langle \mu _\downarrow '| \mu _\downarrow \rangle _{D_{\downarrow }} \nonumber \\&+ \alpha ^* \beta | 0 \rangle _s | 0 \rangle _{D_{\uparrow }} | 1 \rangle _{D_{\downarrow }} \langle 1 |_s \langle 1 |_{D_{\uparrow }} \langle 0 |_{D_{\downarrow }} \sum _{\mu _\uparrow ,\mu _\downarrow } p_{\mu _\uparrow } p_{\mu _\downarrow } \langle \mu _\uparrow '| \mu _\uparrow \rangle _{D_{\uparrow }} \langle \mu _\downarrow | \mu _\downarrow ' \rangle _{D_{\downarrow }} \nonumber \\&+ |\beta |^2 | 0 \rangle _s | 0 \rangle _{D_{\uparrow }} | 1 \rangle _{D_{\downarrow }} \langle 0 |_s \langle 0 |_{D_{\uparrow }} \langle 1 |_{D_{\downarrow }} \end{aligned}$$

(14)

The environmental sums in the second and third terms essentially vanish (they are undetectably small) because the inner product factors, none greater than unity in magnitude, have random phases, in contrast with analogous factors $\big (\langle \mu | \mu \rangle \langle \mu '| \mu ' \rangle = 1\big )$ which appeared in the first and fourth terms and summed to unity. Thus $\rho ^r$ is diagonal in the spin-pointer basis, which consists of $| 1 \rangle _s | 1 \rangle _{D_{\uparrow }} | 0 \rangle _{D_{\downarrow }}$ and $| 0 \rangle _s | 0 \rangle _{D_{\uparrow }} | 1 \rangle _{D_{\downarrow }}$. The surviving singleness and projection correlations result from the entanglement generated between (12), (13) by the passage of the atom. In fact it should be noted that, except for the remaining summations in the off-diagonal terms, (14) is equivalent in form to the density matrix of the spin-ancilla system (see (4) in Sect. 2). Thus, the environmental factors in (14) neatly summarize how the superposition of outcomes becomes undetectable with real detectors.

Appendix B: Detectors that Absorb

In the original Stern–Gerlach experiment [42], silver atoms were directed at a glass plate and formed two separated deposits, with segments of the glass acting as the two detectors. Imagining ideally a pair of absorbing single-atom detectors, their state at time $t_3$ could still be written as in (10), but the 1 states now represent the absorbed atom as well as excitations created by the absorption event. Natural evolution produces these states from the 0 states of the detectors multiplied by the corresponding path occupation states $| 1 \rangle _{Pk}$ of the atom: $| 1,\mu ' \rangle _{Dk} = U(t_3,t_2) | 0,\mu \rangle _{Dk} | 1 \rangle _{Pk}$. So the $X_{Dk}$ operator analogous to (11) is

$$\begin{aligned} X_{Dk} = P_k(1) U_k (t_3,t_2) | 1 \rangle _{Pk} P_k(0) \langle 1 |_{Pk} + | 1 \rangle _{Pk} P_k(0) \langle 1 |_{Pk} U_k^{-1}(t_3,t_2) P_k(1), \end{aligned}$$

(15)

and the subsequent blindness argument is unchanged.

The decoherence approach of Appendix A is similarly adapted: Since the $| \mu ' \rangle$ states include the absorbed atom, the inner product factors in (14) are replaced by $\langle 1 |_{Pk} \langle \mu | \mu ' \rangle$ or its complex conjugate. The set {$\mu '$} is not complete because it refers to more particles than $\{\mu \}$, but it includes all states generated unitarily from {$\mu$} and the incident atom.