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Weak values maximizing the output of weak measurements

Antonio Di Lorenzo

Published Jul 17, 2013• 1 citation
OpenAlex

Abstract

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.

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In a weak measurement, the average output ⟨p⟩\langle p\rangle⟨p⟩ of a probe that measures an observable \HatA\Hat{A}\HatA of a quantum system undergoing both a preparation in a state ρi\rho_iρi​ and a postselection in a state EfE_fEf​ is, to a good approximation, a function of the weak value Aw=\Tr[Ef\HatAρi]/\Tr[Efρi]A_w=\Tr[E_f\Hat{A}\rho_i]/\Tr[E_f\rho_i]Aw​=\Tr[Ef​\HatAρi​]/\Tr[Ef​ρi​], a complex number. For a fixed coupling, when the overlap \Tr[Efρi]\Tr[E_f\rho_i]\Tr[Ef​ρi​] is very small, AwA_wAw​ diverges, but ⟨p⟩\langle p\rangle⟨p⟩ 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 AwA_wAw​, and provide the solutions. The results are independent of the dimensionality of the system.
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.
The weak measurement proposed by Aharonov and his colleagues extracts information about a physical quantity of the system by the postselection as shifts of the argument of the probe wave function. The more the postselected state is orthogonal to the preselected state, the larger the shift determined by the weak value becomes. Recently, the signal amplification by the weak measurement has been extensively studied. In the present work, we explicitly obtain the optimal probe wave function and the amplification factor for a given weak value, which is calculated from the experimental setup. It is shown that the amplification factor has no upper bound, in contrast to the Gaussian probe wave function, and that the amplified signal is sharp.
Weak measurement is a novel quantum measurement scheme, which is usually characterized by the weak value formalism. To guarantee the validity of the weak value formalism, the fidelity between the preselection and the postselection should not be too small generally. In this work, we study the weak measurement on a qubit system with exactly or asymptotically orthogonal pre- and postselections. We shall establish a general rigorous framework for the weak measurement beyond the weak value formalism, and obtain the average output of a weak measurement when the pre- and postselections are exactly orthogonal. We shall also study the asymptotic behavior of a weak measurement in the limiting process that the pre- and postselections tend to be orthogonal.
Since weak measurements are known to produce measurement values that can be much larger than the maximal eigenvalues of the measured observable, it is an interesting question how this enhancement of the measurement signal relates to the sensitivity of quantum measurements as investigated in the field of quantum metrology. In this presentation, it is pointed out that the estimation of a small interaction parameter using weak measurements actually corresponds to standard quantum metrology, where the logarithmic derivatives of the final measurement probabilities are proportional to the corresponding weak values. The analysis of the general weak measurement formalism then shows that extreme weak values do not improve the sensitivity. Instead, all final measurements with real weak values have the same sensitivity as a final measurement of the eigenvalues. This result supports the view that real weak values can be interpreted as precise, zero uncertainty estimates of a quantum observable, despite their deviation from the eigenvalues of the corresponding operator.
A measurement consists in coupling a system to a probe and reading the output of the probe to gather information about the system. The weaker the coupling, the smaller the back-action on the system, but also the less information conveyed. If the system undergoes a second measurement, the statistics of the first output can be conditioned on the value of the second one. This procedure is known as postselection. A postselected weak measurement of an observable can give a large average output of the probe when the postselected state is nearly orthogonal to the initial state of the system. This large value is an interference effect in the readout of the probe, which is initially in a coherent superposition of readout states (also known as pointer states). Usually, the weak interaction between system and probe is considered instantaneous, so that the dynamics of the probe can be neglected. However, for a weak measurement in solid-state devices, an interaction of finite duration is likely needed. Here, we show how this finite duration generates a contribution of the dynamical phase to the readout statistics. Furthermore, we derive interpolation formulas that are able to describe the statistics of the weak measurement for the whole range of pre- and postselected states. Phase-space averages appear in the expansion, suggesting an interpretation in terms of non-positive probabilities. Decoherence in the probe is also accounted for and it is pointed out the existence of a regime of intermediate coupling strength in which coherent oscillations can be observed in the probability of the readout.
Postselected weak measurement has been widely used in experiments to observe weak effects in various physical systems. However, it is still unclear how large the amplification ability of a weak measurement can be and what determines the limit of this ability, which is fundamental to understanding and applying weak measurements. The limitation of the conventional weak-value formalism for this problem is the divergence of weak values when the pre- and postselections are nearly orthogonal. In this paper, we study this problem by a variational approach for a general Hamiltonian ${H}_{\mathrm{int}}=gA\ensuremath{\bigotimes}\ensuremath{\Omega}\ensuremath{\delta}(t\ensuremath{-}{t}_{0}),\phantom{\rule{0.16em}{0ex}}g\ensuremath{\ll}1$. We derive a general asymptotic solution and show that the amplification limit is essentially independent of ggg and is determined by only the initial state of the detector and the number of distinct eigenvalues of AAA or $\ensuremath{\Omega}$. An example of spin-12\frac{1}{2}21​ particles with a pair of Stern-Gerlach devices is given to illustrate the results. The limiting case of continuous-variable systems is also investigated to demonstrate the influence of system dimension on the amplification limit.