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arXiv:2412.13399 · Physical Review A

Experimental investigation of the uncertainty relation in pre- and postselected systems

Yue Zhang, Xinyang Che, Yaxing Wei, Rui Tian, Yian Li, Miao Zhang, Shuai Li, Bo Liu

Published Jun 2, 2025• 0 citations
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Abstract

Uncertainty principle is one of the fundamental principles of quantum mechanics. Exploring such uncertainty relations in pre- and postselected (PPS) systems, where weak measurements on post-selected states have been used as a powerful tool for exploring the foundation of quantum mechanics, has so far eluded experimental effort. In this work, we experimentally investigate the Robertson-Heisenberg-type uncertainty relation for two incompatible observables in a PPS system. Through conducting a von Neumann-type measurement, uncertainty relations between two non-commuting Pauli observables are experimentally unveiled. Such uncertainty relations in the PPS system impose limitations on the joint sharp preparation of pre- and postselected states for two incompatible observables. Our experiments would be useful for the development of amplification techniques for precision measurements.

DOI
10.1103/y287-sf69

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Uncertainty principle is one of the fundamental principles of quantum mechanics. Exploring such uncertainty relations in pre- and postselected (PPS) systems, where weak measurements on post-selected states have been used as a powerful tool for exploring the foundation of quantum mechanics, has so far eluded experimental effort. In this work, we experimentally investigate the Robertson-Heisenberg-type uncertainty relation for two incompatible observables in a PPS system. Through conducting a von Neumann-type measurement, uncertainty relations between two non-commuting Pauli observables are experimentally unveiled. Such uncertainty relations in the PPS system impose limitations on the joint sharp preparation of pre- and postselected states for two incompatible observables. Our experiments would be useful for the development of amplification techniques for precision measurements.
In this work, we derive Robertson-Heisenberg--type uncertainty relation for two incompatible observables in a pre- and postselected (PPS) system. The newly defined standard deviation and the uncertainty relation in the PPS system have physical meanings which we present here. We demonstrate two unusual properties in the PPS system using our uncertainty relation. First, for commuting observables, the lower bound of the uncertainty relation in the PPS system does not become zero even if the initially prepared state, i.e., preselection, is the eigenstate of both the observables when specific postselections are considered. This implies that for such case, two commuting observables can disturb each other's measurement results which is in fully contrast with the Robertson-Heisenberg uncertainty relation. Second, unlike the standard quantum system, the PPS system makes it feasible to prepare sharply a quantum state (preselection) for noncommuting observables (to be detailed in the main text). Some applications of uncertainty and uncertainty relation in the PPS system are provided: (i) detection of mixedness of an unknown state, (ii) stronger uncertainty relation in the standard quantum system, (iii) ``purely quantum uncertainty relation'' that is, the uncertainty relation which is not affected (i.e., neither increasing nor decreasing) under the classical mixing of quantum states, (iv) state-dependent tighter uncertainty relation in the standard quantum system, and (v) tighter upper bound for the out-of-time-order correlation function.
The Heisenberg-Robertson uncertainty relation quantitatively expresses the impossibility of jointly sharp preparation of incompatible observables. However, it does not capture the concept of incompatible observables because it can be trivial even for two incompatible observables. We experimentally demonstrate that the new stronger uncertainty relations proposed by Maccone and Pati [Phys. Rev. Lett. 113, 260401 (2014)] relating to the sum of variances are valid in a state-dependent manner and that the lower bound is guaranteed to be nontrivial when two observables are incompatible on the state of the system being measured. The behavior we find agrees with the predictions of quantum theory and obeys the new uncertainty relations even for the special states which trivialize the Heisenberg-Robertson relation. We realize a direct measurement model and perform an experimental investigation of the strengthened relations.
We prove uncertainty relations that quantitatively express the impossibility of jointly sharp preparation of pre- and post-selected quantum states for measuring incompatible observables during the weak measurement. By defining a suitable operator whose average in the pre-selected quantum state gives the weak value, we show that one can have new uncertainty relations for variances of two such operators corresponding to two non-commuting observables. These generalize the recent stronger uncertainty relations that give non-trivial lower bounds for the sum of variances of two observables which fully capture the concept of incompatible observables. Furthermore, we show that weak values for two non-commuting projection operators obey a complementarity relation. Specifically, we show that for a pre-selected state if we measure a projector corresponding to an observable AAA weakly followed by the strong measurement of another observable BBB (for the post-selection) and, for the same pre-selected state we measure a projector corresponding to an observable BBB weakly followed by the strong measurement of the observable AAA (for the post-selection), then the product of these two weak values is always less than one. This shows that even though individually they are complex and can be large, their product is always bounded.
The Heisenberg-Robertson uncertainty relation is the hallmark of quantum physics and has been widely investigated. However, it does not capture the concept of incompatible observables because it can be trivial even for incompatible observables. Recently some stronger uncertainty relations relating the sums of variances were proposed. Here we experimentally demonstrate that these stronger multiobservable uncertainty relations are valid in a state-dependent manner and that the lower bound is guaranteed to be nontrivial for multiple observables that are incompatible on the state of the system being measured. We find that the behavior of multiple high-dimensional observables agrees with the predictions of quantum theory. Our experimental results not only foster insight into a fundamental limitation of measurements with multiple observables but also contribute to the study of the precision measurement technology in quantum information processing.