Universally valid reformulation of the Heisenberg uncertainty principle on noise and disturbance in measurement

Abstract

The Heisenberg uncertainty principle [1] states that the product of the noise in a position measurement and the momentum disturbance caused by that measurement should be no less than the limit set by Planck's constant,h/2, as demonstrated by Heisenberg's thought experiment using a γ-ray microscope [2]. Here I show that this common assumption is false: a universally valid trade-off relation between the noise and the disturbance has an additional correlation term, which is redundant when the intervention brought by the measurement is independent of the measured object, but which allows the noise-disturbance product much below Planck's constant when the intervention is dependent. A model of measuring interaction with dependent intervention shows that Heisenberg's lower bound for the noise-disturbance product is violated even by a nearly nondisturbing, precise position measuring instrument. An experimental implementation is also proposed to realize the above model in the context of optical quadrature measurement with currently available linear optical devices.

Quantum Errors and Disturbances: Response to Busch, Lahti and Werner

“…Therefore, inequality (10) is violated for every physical state. The fact that inequalities (6) and (10) are not generally valid was noted by us [27] and subsequently by Ozawa [28,29,[39][40][41]; in the case of (10) also by Hall [42]. We, Ozawa and Hall responded to these facts by trying to find alternative inequalities that are generally valid.…”

“…Let us now turn to the approach of Ozawa [28,29,40,41] and Hall [42]. In our approach, we replaced the state-dependent definitions ∆ρ e x, ∆ρ d p with the quantities ∆ e x, ∆ d p and ∆ R e x, ∆ R d p and proved inequalities applying to those.…”

“…BLW's criticisms are largely directed against the state-dependent version proposed by Ozawa [28,29]. However, we had previously proposed a state-independent version [27].…”

“…We focus on Busch, Lahti and Werner's (BLW's) criticisms [2,[23][24][25][26] of the operator approach [27][28][29][30][31] to the description of quantum errors and disturbances. Their criticisms raise some issues that are highly relevant to the above discussion and that need to be settled if we hope to make progress.…”

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

“…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]). In addition, the weak value and weak measurements have been successfully applied to quantum paradoxes such as the three-box problem [57] and Hardy's paradox [58][59][60].…”