A scheme for performing strong and weak sequential

Authors: Aharon Brodutch Eliahu Cohen

Publish Date: 2016/08/23

Volume: 4, Issue: 1, Pages: 13-27

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Abstract

Quantum systems usually travel a multitude of different paths when evolving through time from an initial to a final state In general the possible paths will depend on the future and past boundary conditions as well as the system’s dynamics We present a gedanken experiment where a single system apparently follows mutually exclusive paths simultaneously each with probability one depending on which measurement was performed This experiment involves the measurement of observables that do not correspond to Hermitian operators Our main result is a scheme for measuring these operators The scheme is based on the erasure protocol Brodutch and Cohen Phys Rev Lett 116070404 2016 and allows a wide range of sequential measurements at both the weak and strong limits At the weak limit the back action of the measurement cannot be used to account for the surprising behaviour and the resulting weak values provide a consistent yet strange account of the system’s pastThe joint measurement of sequential observables in quantum mechanics is not canonically defined A sequential measurement of two observables at different times can lead to incompatible results that depend on how the experiment is carried out 1 2 3 When the observables are compatible ie they commute the results of a measurement can depend on the context of the measurement leading to the well known contextuality and non locality theorems 4 5 6 7 8 9 When the observables are incompatible the results of measurements can be correlated in a nontrivial manner that leads to Leggett–Garg inequalities 10 11 12 13 14 15 16 17 18 for standard invasive measurements and to other predictions when the measurements are less invasive 2 19Questions regarding sequential measurements are even more significant in systems with both past and future boundary conditions The observable quantities that depend on past preselected and future postselected boundary conditions are not limited to Hermitian operators or POVM elements but rather depend on the implementation of the measurement and how it affects the measured system ie the observable transition amplitudes depend on Kraus operators The twostate vector formalism TSVF 20 21 provides a useful platform for asking questions about pre and postselected systemsAharonov Bergman and Lebowitz ABL 20 derived a formula for calculating conditional probabilities for the outcomes of projective measurements performed on pre and postselected ensembles Each set of probabilities predicted by the ABL formula is limited to a particular measurement strategy This may lead to strange ‘counterfactual’ predictions regarding measurements which cannot be performed simultaneously 22 Aharonov Albert and Vaidman AAV introduced a new way to extract information from the system with negligible change of the measured state 23 24 25 The result of this weak measurement is a complex number called a weak value Unlike standard observables whose measurement result corresponds to a classical outcome weak values can exceed the spectrum of the weakly measured operator and form an effective ‘weak potential’ When we weakly couple a particle to an operator A describing a pre and postselected system the weak value A w will enter the interaction Hamiltonian 26 Despite the ‘weak’ method used to obtain it each ‘weak value’ is related to a particular ‘strong’ measurement strategy and can be used to understand some counterfactual probabilities obtained using the ABL formula This results from the following theorem 21 if the measured weak value of a dichotomic operator ie operator having only two eigenvalues like the projections used in our gedanken experiment is equal to one of its eigenvalues then this value could have been found with certainty upon being strongly measured see Property 4 belowWhile strong projective measurements change the probability of postselection weak measurements introduce negligible changes of the measured state and thus hardly affect subsequent interactions with the system For this reason it seems that weak values of sequential observables can shed light on and test experimentally fundamental questions such as Leggett–Garg inequalities 10 11 12 13 14 15 16 17 which are based on multipletimes observables Mitchison et al have previously used weak values of sequential observables to explain strange predictions regarding measurements in a double interferometer 27 Their theoretical proposal has been recently further analyzed 3 28 29 and demonstrated experimentally 30 A different approach to analyze consecutive measurements is described in 31 Here we extend the results of Mitchison et al and relate them to the original work on sequential observables 1 Our main result is twofold a systematic analysis of the quantum system’s path in terms of weak and strong sequential measurements as well as an erasurebased scheme for implementing them Under the right pre and postselection the system apparently traverses mutually exclusive paths each with probability 1Analysis in terms of weak values requires the introduction of a new measurement scheme first described in 32 Using the proposed scheme also helps in providing a detailed answer to the questions ‘what is the path of a particle’ 33 and ‘where have the particles been’ 34 35 36 It was suggested that inbetween two strong measurements the particles have been where they left a ‘weak trace’ ie a nonzero weak value This approach towards the past of a quantum particle has attracted much attention and generated some controversy 37 38 39 40 41 42 43The erasure scheme for the sequential operator BA To make a variable strength measurement of the operator BA we first use an ancilla to couple strongly to A at time t 1 and then use the meter to couple to both ancilla and the system to measure the joint observable The measurement is completed by erasing the result recorded in the ancilla a probabilistic procedure The strength of the measurement can be selected by changing the strength of the interaction at t 2

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