Abstract
A weak measurement performed on a pre- and post-selected quantum system can result in an average value that lies outside of the observable's spectrum. This effect, usually referred to as an ``anomalous weak value'', is generally believed to be possible only when a non-trivial post-selection is performed, i.e., when only a particular subset of the data is considered. Here we show, however, that this is not the case in general: in scenarios in which several weak measurements are sequentially performed, an anomalous weak value can be obtained without post-selection, i.e., without discarding any data. We discuss several questions that this raises about the subtle relation between weak values and pointer positions for sequential weak measurements. Finally, we consider some implications of our results for the problem of distinguishing different causal structures.
Highlights
All quantum measurements are subjected to a fundamental trade-off between information gain and disturbance of the measured system
A interesting situation arises when weak measurements are combined with post-selection [1]. This can be conveniently described within the von Neumann model of quantum measurements, where the quantum system to be measured is coupled via a joint unitary operation to another quantum system, the “pointer”, which represents the measurement device
Compare the situation of a sequential weak measurement of two observables Aand Bwith the alternative in which a bipartite system |ψab ∈ Ha ⊗ Hb is prepared and Aand Bare weakly measured on the two different substituent systems
Summary
All quantum measurements are subjected to a fundamental trade-off between information gain and disturbance of the measured system. A trivial, deterministic measurement of the identity operator 1 amounts to performing no post-selection This allows one to consider a weak value with no postselection, defined (see Appendix) as. While astonishing at first sight, anomalous weak values can be intuitively understood in terms of destructive interference of the pointer state, which occurs as a result of post-selection. We will see that the average of the product of the pointer positions can become negative, something which cannot happen if the measurements are strong or classical This may be understood in terms of the second measurement acting as an effective post-selection of the system, creating the desired interference. We finish by discussing the use of anomalous weak values without post-selection to certify particular causal structures between measurements
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