Uncertainty relations for time averaged weak values

Abstract

Time averaging of weak values using the quantum transition path time probability distribution enables us to establish a general uncertainty principle for the weak values of two not necessarily Hermitian operators. This new principle is a weak value analog of the Heisenberg-Robertson strong value uncertainty principle. It leads to the conclusion that it is possible to determine with high accuracy the simultaneous mean weak values of non-commuting operators by judicious choice of the pre- and post-selected states. Generally, when the time fluctuations of the two weak values are proportional to each other there is no uncertainty limitation on their variances and, in principle, their means can be determined with arbitrary precision even though their corresponding operators do not commute. To exemplify these properties we consider specific weak value uncertainty relations for the time-energy, coordinate-momentum and coordinate-kinetic energy pairs. In addition we analyze spin operators and the Stern-Gerlach experiment in weak and strong inhomogeneous magnetic fields. This classic case leads to anomalous spin values when the field is weak. However, anomalous spin values are also associated with large variances implying that their measurement demands increased signal averaging. These examples establish the importance of considering the time dependence of weak values in time of flight scattering experiments.

arXiv comment32 pages, 6 Figures

References in corpus (13)

• Direct Measurement of the Quantum Wavefunction
• Colloquium: Understanding Quantum Weak Values: Basics and Applications
• Universally valid reformulation of the Heisenberg uncertainty principle on noise and disturbance in measurement
• Nonperturbative theory of weak pre- and post-selected measurements
• Violation of Heisenberg's Measurement-Disturbance Relationship by Weak Measurements
• Grounding Bohmian Mechanics in Weak Values and Bayesianism
• Measuring measurement--disturbance relationships with weak values
• Experimental demonstration of direct path state characterization by strongly measuring weak values in a matter-wave interferometer
• Determination of weak values of quantum operators using only strong measurements
• The Time-Energy Uncertainty Relation
• General complex Spin Weak Values obtained in Matter-Wave Interferometer Experiments
• Weak Measurement and its Experimental Realisation
• Time averaging of weak values - consequences for time-energy and coordinate-momentum uncertainty

Cited by in corpus (4)

• Geometric Event-Based Relativistic Quantum Mechanics
• The time-energy uncertainty relation for quantum events
• Theoretical investigation of the relations between quantum decoherence and weak-to-strong measurement transition
• PT and Anti-PT Symmetry Arising in Time Dependent Weak Value Measurements