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arXiv:2008.09131 · Physical Review Letters

Weak Measurement of a Superconducting Qubit Reconciles Incompatible Operators

Jonathan Monroe, Nicole Yunger Halpern, Taeho Lee, Kater W. Murch

Published Mar 11, 2021• 36 citations
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Abstract

Traditional uncertainty relations dictate a minimal amount of noise in incompatible projective quantum measurements. However, not all measurements are projective. Weak measurements are minimally invasive methods for obtaining partial state information without projection. Recently, weak measurements were shown to obey an uncertainty relation cast in terms of entropies. We experimentally test this entropic uncertainty relation with strong and weak measurements of a superconducting transmon qubit. A weak measurement, we find, can reconcile two strong measurements' incompatibility, via backaction on the state. Mathematically, a weak value-a preselected and postselected expectation value-lowers the uncertainty bound. Hence we provide experimental support for the physical interpretation of the weak value as a determinant of a weak measurement's ability to reconcile incompatible operations.

DOI
10.1103/physrevlett.126.100403

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Traditional uncertainty relations dictate a minimal amount of noise in incompatible projective quantum measurements. However, not all measurements are projective. Weak measurements are minimally invasive methods for obtaining partial state information without projection. Recently, weak measurements were shown to obey an uncertainty relation cast in terms of entropies. We experimentally test this entropic uncertainty relation with strong and weak measurements of a superconducting transmon qubit. A weak measurement, we find, can reconcile two strong measurements' incompatibility, via backaction on the state. Mathematically, a weak value-a preselected and postselected expectation value-lowers the uncertainty bound. Hence we provide experimental support for the physical interpretation of the weak value as a determinant of a weak measurement's ability to reconcile incompatible operations.
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.
How violently do two quantum operators disagree? Different fields of physics feature different measures of incompatibility: (i) In quantum information theory, entropic uncertainty relations constrain measurement outcomes. (ii) In condensed matter and high-energy physics, the out-of-time-ordered correlator (OTOC) signals scrambling, the spread of information through many-body entanglement. We unite these measures, deriving entropic uncertainty relations for scrambling. The entropies are of distributions over weak and strong measurements' possible outcomes. Weakness causes the OTOC quasiprobability (a nonclassical generalization of a probability, in terms of which the OTOC decomposes) to govern terms in the uncertainty bound. Scrambling strengthens the bound, we show, in numerical simulations of a spin chain. Beyond scrambling, we derive entropic uncertainty relations satisfied by commonly performed weak-measurement experiments. We unveil a physical significance of common quasiprobabilities and weak values (conditioned expectation values): as governing terms in entropic uncertainty bounds.
The uncertainty principle is an important tenet and active field of research in quantum physics. Information-theoretic uncertainty relations, formulated using entropies, provide one approach to quantifying the extent to which two non-commuting observables can be jointly measured. Recent theoretical analysis predicts that general quantum measurements are necessary to saturate some such uncertainty relations and thereby overcome certain limitations of projective measurements. Here, we experimentally test a tight information-theoretic measurement uncertainty relation with neutron spin- 1 / 2 qubits. The noise associated to the measurement of an observable is defined via conditional Shannon entropies and a tradeoff relation between the noises for two arbitrary spin observables is demonstrated. The optimal bound of this tradeoff is experimentally obtained for various non-commuting spin observables. For some of these observables this lower bound can be reached with projective measurements, but we observe that, in other cases, the tradeoff is only saturated by general quantum measurements (i.e. positive-operator valued measures) as predicted theoretically. These results showcase experimentally the advantage obtainable by general quantum measurements over projective measurements when probing certain uncertainty relations.
Information-theoretic uncertainty relations formulate the joint immeasurability of two noncommuting observables in terms of information entropies.The tradeoff of the accuracy in the outcome of two successive measurements manifests in entropic noise-disturbance uncertainty relations.Recent theoretical analysis predicts that projective measurements are not optimal, with respect to the noise-disturbance tradeoffs.Therefore, the results in our previous paper [Phys. Rev. Lett. 115, 030401 (2015)] are outperformed by general quantum measurements.Here, we experimentally test a tight information-theoretic measurement uncertainty relation for three-outcome positive-operator-valued measures, using neutron spin-1 2 qubits.The obtained results violate the lower bound for projective measurements as theoretically predicted.
We realize indirect partial measurement of a transmon qubit in circuit quantum electrodynamics by interaction with an ancilla qubit and projective ancilla measurement with a dedicated readout resonator. Accurate control of the interaction and ancilla measurement basis allows tailoring the measurement strength and operator. The tradeoff between measurement strength and qubit backaction is characterized through the distortion of a qubit Rabi oscillation imposed by ancilla measurement in different bases. Combining partial and projective qubit measurements, we provide the solid-state demonstration of the correspondence between a nonclassical weak value and the violation of a Leggett-Garg inequality.
Projective measurement is a commonly used assumption in quantum mechanics. However, advances in quantum measurement techniques allow for partial measurements, which accurately estimate state information while keeping the wavefunction intact. We employ partial measurements to study two phenomena. First, we investigate an uncertainty relation—in the style of Heisenberg’s 1929 thought experiment—which includes partial measurements in addition to projective measurements. We find that a weak partial measurement can decrease the uncertainty between two incompatible (non-commuting) observables. In the second study, we investigate the foundation of irreversible dynamics resulting from partial measurements. We do so by comparing the forward and time-reversed probabilities of measurement outcomesresulting from post-selected feedback protocols with both causal and reversed-causal order. We find that the statistics of partial measurements produce entropy in accordance with generalized second laws of thermodynamics. We perform these experiments using superconducting qubits. We describe the fabrication process for these devices and detail a novel fabrication technique that allows fast, single-step lithography of Josephson-junction-based superconducting circuits. The technique simplifies processing by utilizing a direct-write photolithography system, in contrast to traditional