Weak value

In quantum mechanics (and computation), a weak value is a quantity related to a shift of a measuring device's pointer when usually there is pre- and postselection. It should not be confused with a weak measurement, which is often defined in conjunction. The weak value was first defined by Yakir Aharonov, David Albert, and Lev Vaidman in 1988, published in Physical Review Letters and is related to the two-state vector formalism. The first experimental realization came from researchers at Rice University in 1991. The physical interpretation and significance of weak values remains a subject of ongoing discussion in the quantum foundations and metrology literature.

In quantum mechanics (and computation), a weak value is a quantity related to a shift of a measuring device's pointer when usually there is pre- and postselection. It should not be confused with a weak measurement, which is often defined in conjunction. The weak value was first defined by Yakir Aharonov, David Albert, and Lev Vaidman in 1988, published in Physical Review Letters and is related to the two-state vector formalism. The first experimental realization came from researchers at Rice University in 1991. The physical interpretation and significance of weak values remains a subject of ongoing discussion in the quantum foundations and metrology literature. ==Definition== The weak value of the observable A is defined as: A_w = \frac{\langle\psi_f|A|\psi_i\rangle}{\langle\psi_f|\psi_i\rangle}, where |\psi_i\rangle is the initial or preselection state and |\psi_f\rangle is the final or postselection state. The nth order weak value, A^n_w is defined using the nth power of the operator in this expression. Weak values arise in small perturbations of quantum measurements. Representing a small perturbation with the operator \exp(-i\epsilon \hat{A}) , the probability of detecting a system in a final state given the initial state is P_\epsilon = |{\langle\psi_f|\exp(-i\epsilon \hat{A})|\psi_i\rangle}|^2, For small perturbations, \epsilon is small and the exponential can be expanded in a Taylor series P_\epsilon = |{\langle\psi_f|1-i\epsilon \hat{A} +\dots|\psi_i\rangle}|^2, The first term is the unperturbed probability of detection, P =|{\langle\psi_f|\psi_i\rangle}|^2 , and the first order correction involves the first order weak value: \frac{P_\epsilon}{P} \approx 1 + 2 \epsilon A_w. In general the weak value quantity is a complex number. In the weak interaction regime, the ratio P_\epsilon/P is close to one and \epsilon Im A_w is significantly larger than higher order terms. For example, two Stern-Gerlach analyzers can be arranged along the y axis, with the field of the first one along the z axis set at low magnetic field and second on along the x axis with sufficient field to separate the spin 1/2 particle beams. Going into the second analyzer is the initial state |\psi_i\rangle= \frac{1}{\sqrt{2}}\begin{pmatrix}\cos\frac{\alpha}{2}+\sin\frac{\alpha}{2} \\ \cos\frac{\alpha}{2}-\sin\frac{\alpha}{2}\end{pmatrix} and the final state will be |\psi_f\rangle=\frac{1}{\sqrt{2}}\begin{pmatrix}1 \\ 1 \end{pmatrix}. The perturbing action of the first analyzer is described with Pauli z-axis operator as A= \sigma_z giving the weak value A_w = (\sigma_z)_w = \tan\frac{\alpha}{2}. ==Derivation== The derivation below follows the presentation given References. A quantum system is measured by coupling it to an ancillary quantum system that acts as the measuring device. The joint Hilbert space is H_{\rm system} \otimes H_{\rm ancilla} . The observable to be measured on the system is A . The system and ancilla interact through the Hamiltonian H = \gamma A \otimes p, where the coupling constant is integrated over an interaction time \gamma = \int_{t_i}^{t_f} g(t) dt \ll 1 and [q, p] =i is the canonical commutator. The Hamiltonian generates the unitary U= \exp[-i \gamma A\otimes p]. Let the initial state of the ancilla be a Gaussian wavepacket in position space, |\Phi\rangle = \frac{1}{(2\pi \sigma^2)^{1/4}}\int dq' \exp[-q'^2/4\sigma^2]|q'\rangle. Its position wavefunction is \Phi(q) =\langle q|\Phi\rangle = \frac{1}{(2\pi \sigma^2)^{1/4}} \exp[-q^2/4\sigma^2], where \sigma characterizes the initial uncertainty in the pointer position of the measuring device and |q\rangle denotes an eigenstate of the position operator q . The system begins in the state |\psi_i\rangle . Thus, the combined initial state of the system and ancilla is |\Psi\rangle , jointly describing the initial state of the system and ancilla, is given then by: |\Psi\rangle =|\psi_i\rangle \otimes |\Phi\rangle. Next the system and ancilla interact via the unitary U |\Psi\rangle . After this one performs a projective measurement of the projectors \{ |\psi_f\rangle\langle \psi_f |, I- |\psi_f\rangle\langle \psi_f |\} on the system. Postselecting (or condition) on getting the outcome |\psi_f\rangle\langle \psi_f | , then the (unnormalized) final state of the meter is \begin{align} |\Phi_f \rangle &= \langle \psi_f |U |\psi_i\rangle \otimes |\Phi\rangle\\ &\approx \langle \psi_f |(I\otimes I -i \gamma A\otimes p ) |\psi_i\rangle \otimes|\Phi\rangle \quad \text{(I)}\\ &= \langle \psi_f|\psi_i\rangle (1 -i \gamma A_w p ) |\Phi\rangle\\ &\approx \langle \psi_f|\psi_i\rangle \exp(-i \gamma A_w p) |\Phi\rangle. \quad \text{(II)} \end{align} Here it looks like the ancilla state will be shifted by \gamma A_w due to the momentum operator in the exponential. There is a way to obtain weak values without postselection.. To arrive at this conclusion, the first order series expansion of U on line (I) is used, and one requires that \begin{align} \frac {\sigma} \left|\frac{\langle \psi_f |A^n |\psi_i \rangle}{ \langle \psi_f| A |\psi_i \rangle }\right|^{1/(n-1)} \ll 1, \quad (n = 2, 3, \dots) \end{align} On line (II) the approximation that e^{-x}\approx 1-x for small x was used. This final approximation is only valid when |\gamma A_w|/\sigma \ll 1. As p is the generator of translations, the ancilla's wavefunction is now given by \Phi_f(q) = \Phi(q-\gamma A_w). This is the original wavefunction, shifted by an amount \gamma A_w . By Busch's theorem the system and meter wavefunctions are necessarily disturbed by the measurement. Although the protocol used to access the weak value is minimally disturbing in a specific sense , it still induces disturbance. ==Applications== Weak values have been proposed as potentially useful for quantum metrology and for clarifying aspects of quantum foundations. The sections below briefly outline these applications. ===Quantum metrology=== At the end of the original weak value paper the authors suggested weak values could be used in quantum metrology: In modern language, when the weak value A_w lies outside the eigenvalue range of the observable A , the effect is known as weak value amplification. In this regime, the shift of the measuring device's pointer can appear much larger than expected, for example a component of spin may seem 100 times greater than its largest eigenvalue. This amplification effect has been viewed as potentially beneficial for metrological applications where small physical signals need to be detected with high sensitivity. This weak value amplification subsequently demonstrated experimentally. . ===Quantum Tomography=== Weak values have also been explored in the context of quantum state tomography. Direct state tomography uses weak measurements and post-selection, motivated by weak-value protocols, to reconstruct the quantum state. It also provides an operational interpretation of wavefunction amplitudes. Weak-measurement tomography aims to improve upon standard tomography by exploiting the minimal disturbance from weak measurements, allowing the same system to be reused for additional measurements. ===Quantum foundations=== Weak values are used as indicators of nonclassicality, as tools for explaining quantum paradoxes, and as links between different interpretations of quantum mechanics. Anomalous weak values, those lying outside the eigenvalue range of an observable, are considered indicators of nonclassicality. They serve as proofs of quantum contextuality, showing that measurement outcomes cannot be reproduced by any noncontextual hidden-variable model. . Weak values have been used to create and explain some of the paradoxes in the foundations of quantum theory, for example, the Quantum Cheshire cat. They have also been used in experimental studies of Hardy's paradox, where joint weak measurements of entangled pairs of photons reproduced the paradoxical predictions. Weak values have been proposed as a way to define a particle's velocity at a given position, referred to as the 'naively observable velocity.' Experiments in 2010 reported photon trajectories in a double-slit interferometer that qualitatively matched earlier predictions for photons in the de Broglie-Bohm interpretation. Subsequent analyses have argued that weak velocity measurements do not provide new evidence for or against de Broglie-Bohm theory and cannot directly reveal the form of particle trajectories, even under deterministic assumptions ==References== ==Further reading==

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