Anomalous Weak Values Without Post-Selection Alastair A. Abbott1, Ralph Silva2,3, Julian Wechs1, Nicolas Brunner2, and Cyril Branciard1 1Univ. Grenoble Alpes, CNRS, Grenoble INP, Institut Néel, 38000 Grenoble, France 2Département de Physique Appliquée, Université de Genève, 1211 Genève, Switzerland 3Institute for Theoretical Physics, ETH Zurich, 8093 Zurich, Switzerland September 2, 2019 A weak measurement performed on a pre- packet centred at a position x = 0 with spread (i.e., and post-selected quantum system can result standard deviation) σ. Assuming that we are in the in an average value that lies outside of the ob- weak measurement regime, with the coupling constant servable’s spectrum. This effect, usually re- γ and interaction time ∆t such that g := γ∆t is small ferred to as an “anomalous weak value”, is gen- enough compared to the spread of the pointer, the erally believed to be possible only when a non- global state after the coupling is given by trivial post-selection is performed, i.e., when ˆ only a particular subset of the data is consid- e−iH∆t |ψi |ϕ(0)i ≈ (1 − igAˆpˆ) |ψi |ϕ(0)i (1) ered. Here we show, however, that this is not the case in general: in scenarios in which sev- (where tensor products are implicit, and taking ~ = eral weak measurements are sequentially per- 1). For simplicity we will henceforth choose units so formed, an anomalous weak value can be ob- that g = 1; the strength of the measurement will then tained without post-selection, i.e., without dis- be controlled solely by the pointer spread σ, and the validity of the weak regime will depend only on this carding any data. We discuss several questions 1 that this raises about the subtle relation be- being sufficiently large. Next, the system is post- tween weak values and pointer positions for selected onto the state |φi (e.g. via a strong projective sequential weak measurements. Finally, we measurement). The final state of the pointer is then consider some implications of our results for (up to normalisation) the problem of distinguishing different causal φ hφ| (1 − iAˆpˆ) |ψi |ϕ(0)i = hφ|ψi (1 − iA pˆ) |ϕ(0)i structures. ψ −iAφ pˆ ≈ hφ|ψi e ψ |ϕ(0)i , (2) 1 Introduction where hφ|Aˆ|ψi Aφ := (3) All quantum measurements are subjected to a funda- ψ hφ|ψi mental trade-off between information gain and distur- is the so-called weak value of the observable Aˆ bance of the measured system. In particular, one can given the pre-selection in the state |ψi and post- perform weak measurements that provide little infor- selection in the state |φi [1]. The mean position mation but only weakly perturb the system. A partic- of the pointer is thus displaced—via the displace- ularly interesting situation arises when weak measure- −iAφ pˆ ment operator e ψ , which generates the (possi- ments are combined with post-selection [1]. This can φ φ −iAψ pˆ be conveniently described within the von Neumann bly unnormalised) state |ϕ(Aψ)i = e |ϕ(0)i; see model of quantum measurements, where the quantum Appendix—to system to be measured is coupled via a joint unitary hϕ(Aφ )| xˆ |ϕ(Aφ )i operation to another quantum system, the “pointer”, ψ ψ φ hxˆi ≈ φ φ = Re(Aψ). (4) arXiv:1805.09364v3 [quant-ph] 12 Sep 2019 which represents the measurement device. The mea- hϕ(Aψ)|ϕ(Aψ)i surement is then completed by performing a strong measurement of the pointer. Notably, the real part of the weak value can be- More formally, consider a system initially prepared come very large when the pre- and post-selected (or pre-selected) in a pure state |ψi, and an observable 1 ˆ A characteristic of the weak regime is that an individ- A to be weakly measured on it. The system-pointer ual measurement of the pointer position following the interac- interaction is generated via a Hamiltonian of the form tion with the system yields little information, since the pointer Hˆ = γAˆ⊗pˆ, where pˆ denotes the momentum operator spread is much larger than the range of mean pointer positions one obtains (which scales with g). As only the ratio between acting on the pointer. The latter is initially in a state these quantities is important, it is thus sufficient for our anal- |ϕ(0)i, which we shall take here to be a Gaussian wave ysis to take g = 1. In the Appendix we present more precise statements of the conditions for the weak regime to be satisfied A.A.A. and R.S. contributed equally to this work. in the situations we consider throughout this paper. Accepted in Quantum 2019-09-02, click title to verify. Published under CC-BY 4.0. 1 states are almost orthogonal, i.e. |hφ|ψi| 1. In as contextuality [17], counterfactual paradoxes [18], this case, the pointer is, on average, shifted by and the nature of the wave function [19, 20]. While φ a large amount. Whenever Re(Aψ) is not in the astonishing at first sight, anomalous weak values can ˆ ˆ ˆ in fact be intuitively understood in terms of destruc- interval [λmin(A), λmax(A)] (where λmin(max)(A) = th tive interference of the pointer state, which occurs as min(max)k λk(Aˆ) and λk denotes the k eigenvalue of an observable), i.e. whenever it is outside of the (con- a result of post-selection. With this in mind and given vex hull of the) spectrum of Aˆ, the pointer’s mean po- the rudimentary analysis above, it is rather natural to sition thus moves beyond where it could have reached attribute the origin of anomalous weak values to the under simple weak measurements on an arbitrary pre- presence of post-selection; this opinion indeed seems selected state without any post-selection. Indeed, in to be widely shared in the community. the absence of post-selection one has (now with exact Here we show, however, that this is not the case in equalities) general, and that anomalous weak values can in fact be observed in the absence of post-selection and with- hxˆi = hψ| hϕ(0)| eiAˆpˆ (1 ⊗ xˆ) e−iAˆpˆ |ψi |ϕ(0)i out discarding any outcomes. Specifically, we consider a situation in which two successive weak measure- = hψ| hϕ(0)| 1 ⊗ xˆ + Aˆ ⊗ 1 |ψi |ϕ(0)i ments are performed on a quantum system. The ex- periment thus involves two pointers, one associated to = hψ| Aˆ |ψi ∈ [λ (Aˆ), λ (Aˆ)]. (5) min max each weak measurement, which are measured jointly. Note that hxˆi, both with and without post-selection, Considering observables that are simply given by pro- can be determined experimentally by performing suf- jectors, one expects to find the mean position of each ficiently many measurements, despite the large vari- pointer between 0 (the system’s state being orthogo- ance of the pointer (indeed, to obtain a given accuracy nal to the projector) and 1 (the system’s state being the number of measurements required scales propor- in the range of the projector). Yet, we will see that tionally to σ2). the average of the product of the pointer positions can The definition (3) of a weak value can be generalised become negative, something which cannot happen if to post-selections on a given result for any general the measurements are strong or classical. This may be quantum measurement [2,3]. In particular, a triv- understood in terms of the second measurement act- ial, deterministic measurement of the identity opera- ing as an effective post-selection of the system, thus tor 1 amounts to performing no post-selection. This creating the desired interference. Importantly how- allows one to also consider a weak value with no post- ever, no data is discarded. selection, defined (see Appendix) as2 Below, after discussing in detail a simple example of this effect, we turn to the more general scenario 1 ˆ Aψ := hψ|A|ψi . (6) of arbitrary sequences of weak measurements without 1 post-selection, deriving bounds on how anomalous a With this definition, Eq. (5) gives hxˆi = Aψ = weak value can be obtained in such a scenario and 1 Re(Aψ): we recover the same relation as in how these relate to the (jointly measured) pointer po- 1 Eq. (4), although now Aψ is restricted to lie in sitions. We finish by discussing the use of anomalous [λmin(Aˆ), λmax(Aˆ)] since here it is simply equal to the weak values without post-selection to certify particu- expectation value of Aˆ. lar causal structures between measurements. The phenomenon of a weak value outside the spec- trum of Aˆ is referred to as an “anomalous weak value” [1,5,6] since it conflicts with our classical intu- 2 Illustrative example ition, which would lead us to expect hxˆi to lie within the range of the spectrum of Aˆ. This has been ob- To start with, let us consider a qubit system initially served in many experiments [7–9], and appears to be prepared in the state |0i, undergoing a sequence of two directly linked to various (a priori unrelated) areas weak von Neumann measurements of the projection such as tunnelling times [10] and fast light propaga- observables |ψjihψj| (j = 1, 2), where the states |ψji tion [11, 12]. In practice, anomalous weak values allow ⊥ and their orthogonal states |ψj i are defined as for the detection and precise estimation of very small physical effects [13–16], via a form of signal amplifica- √ 1 j 3 tion, while they have also helped provide new insights |ψji = |0i − (−1) |1i , 2 2 on foundational aspects of quantum mechanics such √ 3 1 |ψ⊥i = |0i + (−1)j |1i .