In a weak measurement, the average output $\langle p\rangle$ of a probe that measures an observable $\Hat{A}$ of a quantum system undergoing both a preparation in a state $\rho_i$ and a postselection in a state $E_f$ is, to a good approximation, a function of the weak value $A_w=\Tr[E_f\Hat{A}\rho_i]/\Tr[E_f\rho_i]$, a complex number. For a fixed coupling, when the overlap $\Tr[E_f\rho_i]$ is very small, $A_w$ diverges, but $\langle p\rangle$ stays finite, often tending to zero for symmetry reasons. This paper answers the question: which are the extremal values of the output as a function of the weak value? We derive equations for the optimal values of $A_w$ , and provide the solutions. The results are independent of the dimensionality of the system.In a weak measurement, the average output $\langle p\rangle$ of a probe that measures an observable $\Hat{A}$ of a quantum system undergoing both a preparation in a state $\rho_i$ and a postselection in a state $E_f$ is, to a good approximation, a function of the weak value $A_w=\Tr[E_f\Hat{A}\rho_i]/\Tr[E_f\rho_i]$, a complex number. For a fixed coupling, when the overlap $\Tr[E_f\rho_i]$ is very small, $A_w$ diverges, but $\langle p\rangle$ stays finite, often tending to zero for symmetry reasons. This paper answers the question: which are the extremal values of the output as a function of the weak value? We derive equations for the optimal values of $A_w$, and provide the solutions. The results are independent of the dimensionality of the system.

Jul 17, 20131 citation

Demystifying Weak Measurements

Ruth E. Kastner

arXiv:1702.04021Foundations of PhysicsApr 8, 201723 citations

Full counting statistics of weak-value measurement

Antonio Di Lorenzo

A weak measurement consists of coupling a system to a probe in such a way that constructive interference generates a large output. So far, only the average output of the probe and its variance were studied. Here, the characteristic function for the moments of the output is provided. The outputs considered are not limited to the eigenstates of the pointer or of its conjugate variable, so the results apply to any observable $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{o}$ of the probe. Furthermore, a family of well-behaved complex quantities, the normal weak values, is introduced, in terms of which the statistics of the weak measurement can be described. It is shown that, within a good approximation, the whole statistics of weak measurement is described by a complex parameter, the weak value, and a real one.A weak measurement consists of coupling a system to a probe in such a way that constructive interference generates a large output. So far, only the average output of the probe and its variance were studied. Here, the characteristic function for the moments of the output is provided. The outputs considered are not limited to the eigenstates of the pointer or of its conjugate variable, so the results apply to any observable $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{o}$ of the probe. Furthermore, a family of well-behaved complex quantities, the normal weak values, is introduced, in terms of which the statistics of the weak measurement can be described. It is shown that, within a good approximation, the whole statistics of weak measurement is described by a complex parameter, the weak value, and a real one.

arXiv:1109.4183Physical Review AMar 5, 201249 citations

Reconstructing weak values without weak measurements

Lars M. Johansen

arXiv:0705.0108Physics Letters AFeb 17, 200726 citations

Simultaneous path weak-measurements in neutron interferometry

Armin Danner, Ismaele Vincent Masiello, Andreas Dvorak +4 more

The statistical properties of the detection events constituting the interference fringes at the output of an interferometer are well-known. Nevertheless, there is still no unified view of what is happening to a quantum system inside an interferometer. Strong measurements of path operators destroy the interference effect. In weak measurements, an observable is weakly coupled to a pointer system and the resulting weak values quantify the observable by minimally disturbing the system. Previous which-way experiments with weak measurements could extract either the real or imaginary part of a single weak value with each ensemble. Here, we present the simultaneous full complex quantification of two path weak values with a single ensemble in a Mach-Zehnder neutron interferometer. Magnetic fields, oscillating with different frequencies, change the energy state in each interferometer path. The time-dependent phase between the energy states distinctly marks each path. The resulting beating intensity modulation at the interferometer output gives both path weak values. For the present experiment, the weak values' absolute value and phase directly describe the observed amplitude and phase of the intensity modulation.The statistical properties of the detection events constituting the interference fringes at the output of an interferometer are well-known. Nevertheless, there is still no unified view of what is happening to a quantum system inside an interferometer. Strong measurements of path operators destroy the interference effect. In weak measurements, an observable is weakly coupled to a pointer system and the resulting weak values quantify the observable by minimally disturbing the system. Previous which-way experiments with weak measurements could extract either the real or imaginary part of a single weak value with each ensemble. Here, we present the simultaneous full complex quantification of two path weak values with a single ensemble in a Mach-Zehnder neutron interferometer. Magnetic fields, oscillating with different frequencies, change the energy state in each interferometer path. The time-dependent phase between the energy states distinctly marks each path. The resulting beating intensity modulation at the interferometer output gives both path weak values. For the present experiment, the weak values' absolute value and phase directly describe the observed amplitude and phase of the intensity modulation.

Scientific ReportsOct 29, 20243 citations

In a weak measurement, the average output

\langle p\rangle

of a probe that measures an observable $\Hat{A}$ of a quantum system undergoing both a preparation in a state

\rho_i

and a postselection in a state

E_f

is, to a good approximation, a function of the weak value $A_w=\Tr[E_f\Hat{A}\rho_i]/\Tr[E_f\rho_i]$, a complex number. For a fixed coupling, when the overlap $\Tr[E_f\rho_i]$ is very small,

A_w

diverges, but

\langle p\rangle

stays finite, often tending to zero for symmetry reasons. This paper answers the question: which are the extremal values of the output as a function of the weak value? We derive equations for the optimal values of

A_w

, and provide the solutions. The results are independent of the dimensionality of the system.