Using technical noise to increase the signal-to-noise ratio of measurements via imaginary weak values

Abstract

The advantages of weak measurements, and especially measurements of imaginary weak values, for precision enhancement, are discussed. A situation is considered in which the initial state of the measurement device varies randomly on each run, and is shown to be in fact beneficial when imaginary weak values are used. The result is supported by numerical calculation and also provides an explanation for the reduction of technical noise in some recent experimental results. A connection to quantum metrology formalism is made.In 1988 Aharonov, Albert and Vaidman (AAV) [1] discovered that the measured value of an observable can be 100 times bigger than its biggest eigenvalue, provided the measurement interaction is weak and a postselection is employed. They showed that a system which is coupled weakly to another, pre-and postselected system, described by the two-state vector Φ| |Ψ , via an observable C, is effectively coupled to the weak value of the observable [2]The replacement of the interaction operator with its weak value, which is a complex number [3], is known as the AAV effect and the procedure in which the weak value is measured is referred to as a weak measurement. In this letter we will analyze the process of weak measurement as a method for precision measurements. Furthermore, we will present a concrete model for technical noise affecting the preparation of the measurement device (meter), and show that in the presence of such a noise the precision is enhanced.We start with an overview of known results regarding the precision achievable by weak measurements. Consider a physical interaction:where C is an observable on a system, P is an operator on a meter and g(t) is a coupling function satisfying g(t)dt = k. Our concern is estimating the size of k, or in some cases simply observing the interaction. A straight forward approach is to put the system in an eigenstate of C having some eigenvalue c, and the meter in a Gaussian state:where Q is a variable conjugate to P , and ∆ is its quantum uncertainty. An estimate of k can be obtained from the shift in Q due to the interaction, Q = kc , and its precision is determined by the standard deviation∆. In the case kc ∆, little information is acquired from a single measurement, but by repeating the procedure N times and averaging the results, the precision is enhanced. Strictly speaking, the amount of information gathered, regarding k, is measured by the Fisher information [15], but for our purposes we can use the more intuitive concept of signal to noise ratio (S/N ) [17], which in this case isSince our interest is in the regime where kc ∆, which is a condition for the AAV effect [18], we will, for now, assume that the AAV effect occurs and later examine its validity in more detail. Thus, we will consider the system to be initially in a state |Ψ and take into account the meter results only when the system was found in a state |Φ , after the interaction, which implies a replacement C → C w in (2) [19]. The shift in Q is given by Q Φ = kReC w [1], andwhere N Φ ∼ N | Φ|Ψ | ...

Technical Advantages for Weak-Value Amplification: When Less Is More

“…An important step was made in this question when Feizpour, Xingxing, and Steinberg [10] were able to consider a more general kind of technical noise and show that so long as it has a long correlation time, WVA also helps suppress it in the signal-to-noise ratio (SNR). In other closely related work, Kedem [17], Brunner and Simon [18], and Nishizawa et al [19] also showed an increased performance of the SNR in the presence of technical noise.…”

Seeing opportunity in every difficulty: protecting information with weak value techniques

“…An informationtheoretical advantage has been conjectured several times [15,19,28,39]: and a recent study by Harris, Boyd and Lundeen [25] gives these conjectures a theoretical underpinning. We refer the reader to [25] for a full analysis, but the rough argument is as follows: for a deterministic strategy, there will be some maximum photon flux for 2 )r , vertical grey bars) shown as a guide for the eye.…”

“…Since weak values may channel information into the s θ variable (with θ not necessarily zero, which is impossible in standard measurements), this can be useful in the case where a more stable measurement of s θ (than of x) is available [28]. See Fig.…”

“…In this scenario, the black arrow shrinks while the red ellipse remains constant, meaning that the 'amplification factor' |A w |/λ * becomes very large, while the size of the shift itself must actually remain small. This has previously been described as the requirement that the signal-to-noise ratio per data point must be low [28].…”

Weak-value amplification: state of play

“…protocols with di erent physics, there may be scope for altering the apparatus in a fashion which tailors it to better t the available hardware. In a time-domain experiment, for example, a frequency analyser is sometimes preferred to a stopwatch [10,13]. However, unless there is a severe mismatch between the quality of detection in the two variables, imaginary weak values will not provide a signi cant advantage [12].…”