Violation of the Leggett–Garg inequality with weak measurements of photons

Abstract

By weakly measuring the polarization of a photon between two strong polarization measurements, we experimentally investigate the correlation between the appearance of anomalous values in quantum weak measurements and the violation of realism and nonintrusiveness of measurements. A quantitative formulation of the latter concept is expressed in terms of a Leggett-Garg inequality for the outcomes of subsequent measurements of an individual quantum system. We experimentally violate the Leggett-Garg inequality for several measurement strengths. Furthermore, we experimentally demonstrate that there is a one-to-one correlation between achieving strange weak values and violating the LeggettGarg inequality.here has been much debate in quantum physics over the question of whether measurable quantities have definite values prior to their measurement. Key ideas addressing this question include the Bell inequality, which considers correlations between measurements on components of a composite system that are space-like separated (1, 2) and contextuality tests, which examine whether identical experiments produce results in different "classically equivalent" contexts (3, 4). A conceptually elegant extension to these ideas is the Leggett-Garg inequality (LGI) (5), which is an inequality constructed from the correlation functions of a series of three consecutive measurements on a single system. Leggett and Garg derive limits based on the joint assumptions of (i) macroscopic realism: An observable for a system will have a definite value at all times; and (ii) noninvasive measurement: It is possible to determine this value with arbitrarily small disturbance on the subsequent evolution of the system. The limits on the value of the inequality derived from these assumptions differ from the predictions of quantum mechanics. Thus the LGI tests the limits of measurement and macroscopic realism.Here we present an experimental test of a generalized LGI using weak measurements (6-9) of the polarization of single photons and measure violations by up to 14 standard deviations. Additionally, we experimentally demonstrate a one-to-one relation (10, 11) between LGI violations and strange weak-valued measurements (6-8), which also arise from the inability to assign values to physical quantities between an earlier and a later measurement.Testing the LGI requires monitoring the system without projecting it into a specific state. For a quantum system a quantum nondemolition (QND) experiment (12-14) would be one way to do this. But QND measurements are not the only way to perform a noninvasive measurement. A generalization of the QND measurement is the so-called weak measurement (6). A weak measurement is one for which it is possible to adjust the strength of the measurement and, in principle, to reduce the back action on the system to an arbitrarily small amount. In other words, a weak measurement is one for which the level of "invasivness" can be controlled.

Fundamental Features of Quantum Dynamics Studied in Matter-Wave Interferometry—Spin Weak Values and the Quantum Cheshire-Cat

“…A review focusing on significant amplification of the pointer deflection is given in [46], where in addition a non-perturbative theory of weak pre-and post-selected measurements is presented. The weak value's property of allowing information to be extracted from a quantum system with minimal disturbance is applied as a new method for estimation of quantum states [47][48][49][50] (see [51] for a recent review). This particular property of the weak value is also utilized in experimental tests of a reformulation of Heisenberg's uncertainty principle [52][53][54][55] (an overview of different viewpoints of that topic can be found in [56]).…”

Quantum State Estimation Using Weak Measurements

“…This is called the weak or unsharp measurement regime. Weak measurements have been employed in developing recipes for the violation of Bell inequalities 28 and Leggett-Garg inequalities 29 . These have also been recently used to study super-quantum discord 30 .…”

What Is a Quantum-Mechanical “Weak Value” the Value of?

“…The use of the technique for amplification purposes has been spectacular [7][8][9]. Weak measurements have also been employed to illuminate the fundamental difference between classical and quantum mechanics exhibited by violation of Leggett-Garg inequalities [3,[10][11][12][13][14], also called "Bell inequalities in time".…”

“…Note that I do not here question the weak value as a measureable entity pertaining to the system under study, nor the many ingenious ways experimentalist have exploited the weak measurement + post-selection idea ( [5][6][7][8][9][11][12][13][14]; see also [3] for a brief review). What I do question is whether a weak value can be given an 'ordinary' meaning.…”