Measuring Incompatible Observables by Exploiting Sequential Weak Values

Abstract

One of the most intriguing aspects of quantum mechanics is the impossibility of measuring at the same time observables corresponding to noncommuting operators, because of quantum uncertainty. This impossibility can be partially relaxed when considering joint or sequential weak value evaluation. Indeed, weak value measurements have been a real breakthrough in the quantum measurement framework that is of the utmost interest from both a fundamental and an applicative point of view. In this Letter, we show how we realized for the first time a sequential weak value evaluation of two incompatible observables using a genuine single-photon experiment. These (sometimes anomalous) sequential weak values revealed the single-operator weak values, as well as the local correlation between them.

Anomalous weak values and the violation of a multiple-measurement Leggett-Garg inequality

“…In this respect, it is not surprising that the measurement procedure is insensitive to the imaginary part of the weak values [64], as discussed in Ref. [56]. We tested the inequality for different choices of the initial state α, of the orientation γ of the weak measurement, and of the final postselection δ: the four combinations have been identified to deliver a violation (indicated by the blue arrow in each plot reported in Fig.…”

“…The use of a single-mode fiber (SMF) then prepares the transverse profile F(x,y) in a Gaussian shape of width σ , which ensures that the two directions can be used as distinct pointers for the weak measurements [56]. ensures that the back-action on the incoming state is negligible; i.e., the measurement is carried out in the weak regime [56].…”

“…We perform a LGT on the polarization of single photons, estimating noncommuting observables via "weak averages" [56], and draw an explicit link to the emergence of anomalous values. Our experiment confirms the intimate connection between the observation of anomalies in the postselected statistics of quantum measurement as a genuine manifestation of the quantum character of the observed object.…”

A framework for measuring weak values without weak interactions and its diagrammatic representation

“…In addition, weak values are used for the precise measurement of the magnitudes of weak system-probe interactions [11][12][13][14][15] because they can exceed the eigenvalues of the observables. Furthermore, weak values are utilized for the direct measurement of complex functions such as wavefunctions and pseudo-probability distributions of the initial state of the system [16][17][18][19][20][21][22][23][24][25].…”

“…However, this is not the only possible method for their determination, and several alternative weak value measurement techniques that do not involve weak system-probe interactions have been recently developed, including those using strong system-probe interactions [26][27][28][29], modular values [30], quantum control interactions [31,32], an enlarged Hilbert space [33], coupling-deformed pointer observables [34], and a combination of several strong measurements of the system [35]. It should be noted here that most of them [26][27][28][29][30][31][32][33] and some weak measurement experiments [2][3][4][5][6][7][8][9][10][16][17][18][19][20][21][22][23][24] use qubit systems as the probes. They all have the same measurement procedure of the qubit probes to obtain complex weak values-measuring expectation values of two operators, such as Pauli-X and Pauli-Y.…”