Evidence of Uncollapsed Quantum Amplitudes After Consecutive Measurements

Abstract

Two of the most common interpretations of quantum measurement disagree about the fate of quantum amplitudes after measurement, yet this disagreement has not previously led to experimentally distinguishable predictions. In the standard collapse picture, commonly linked to the Copenhagen interpretation of quantum mechanics, measurements eliminate unrealized amplitudes without leaving a memory. In contrast, in the unitary theory, the measurement device registers one of the possible outcomes while remaining part of an entangled state that continues to harbor the unrealized amplitudes. This persistence arises naturally under unitary evolution, since a measurement device that is part of an entangled system cannot serve as a faithful probe of the joint quantum state. Using single-photon measurements of a tunable quantum state, we experimentally show that these two theories make different predictions when three or more consecutive measurements are performed on the same quantum system. Analysis of the joint density matrix of the three measurements reveals coherence among them and supports the unitary theory of quantum measurement. When decoherence is explicitly introduced, the joint density matrix of the quantum system of interest and the apparatus becomes consistent with what a collapse theory would predict. This work clarifies the dynamics of consecutive quantum measurements and offers new insights into the interpretation of quantum measurements.

Figures (10)

• Figure 1: (a) Consecutive measurements of a quantum state $\rho_Q$, showing rotation of the measurement basis by angles $\theta_1$ and $\theta_2$ using polarizers. (b) Experimental realization of the scheme using half-waveplate (HWP), calcite crystals and followed by photon detection. $\rho_Q$ denotes the initial quantum state, $\rho_1$ the density matrix of the first readout $M_1$, $\rho_{12}$ the joint state of the first two readouts after $M_2$, and $\rho_{123}$ the combined state of all three readouts following $M_3$.
• Figure 2: Information-theoretic Venn diagrams showing conditional and shared entropies between the quantum system $Q$ and the measurement systems after one, two, and three consecutive measurements. The left column corresponds to the standard collapse theory and the right column to the unitary theory. Three systems $M_1$, $M_2$, and $M_3$ measure a fully mixed quantum system ($\phi=\pi/4$) with relative angles $\theta_1 = \theta_2 = \pi/4$. In the unitary case, the third measurement creates entanglement between $Q$ and the collective measurement system, signaled by negative conditional quantum entropies.
• Figure 3: (a) Conceptual view of three consecutive measurements implementing a sequence of non-commuting measurements ($\pmb{\pi}_H\pmb{\pi}_D\pmb{\pi}_H$) using calcites as beam displacer and half-waveplates. (b) The final geometry for the eight beam paths. The dashed blue lines indicate coherence according to the unitary theory.
• Figure 4: State preparation, sequential measurements, and quantum state reconstruction. Photons are generated by the SPDC process. A 4$f$-system is built to ensure all the optics are within the Rayleigh length. A combination of half-wave plates (HWP) and quarter-wave plates (QWP) prepares a mixed state, while the HWP$_S$ spins faster than the collection time of the camera. Next, a series of measurements is implemented using calcite displacement crystals (named $\delta_x$, $\delta_y$, and $\delta_X$). Lastly, another 4$f$-system ($f_1 = 1000$ mm and $f_2 = 400$ mm) is used to image the eight path modes in an electron-multiplying CCD (EMCCD) camera. Optical Fourier transform (OFT) along the OFT-axis is performed by rotating a 250 mm cylindrical lens. The final coherence readout is obtained by Discrete Fourier Transform (DFT).
• Figure 5: (a) Reconstructed joint density matrix [Eq. (\ref{['rho_123_tilde']})] for a completely mixed input state ($\phi=0$). Each square represents the absolute value of a matrix element after the third measurement, averaged over multiple experimental runs; one standard deviation is indicated numerically within each square. The color scale (right) encodes the mean amplitude. The upper-left and lower-right $4\times4$ blocks correspond to the populated subspaces, while the off-diagonal $4\times4$ blocks are zero in both theories. (b) Predicted absolute values of the joint density matrix for $\phi=0$ under the collapse model, the unitary model, and the experiment (right), shown side-by-side as Manhattan plots. Experimental amplitudes correspond to those reported numerically in panel (a). (c) Reconstructed joint density matrix for a diagonal (pure) input state, $\phi=45^{\circ}$, displayed as in (a). (d) Predicted absolute values of the joint density matrix for $\phi=45^{\circ}$ under the collapse, unitary, and experimental results, shown as in (b). Additional data for intermediate values of $\phi$ are presented in Appendix \ref{['extramat']}. In all cases, including those shown here, the experimental results agree with the unitary theory and disagree with the standard collapse theory.