Since its introduction 25 years ago, the quantum weak value has gradually transitioned from a theoretical curiosity to a practical laboratory tool. While its utility is apparent in the recent explosion of weak value experiments, its interpretation has historically been a subject of confusion. Here, a pragmatic introduction to the weak value in terms of measurable quantities is presented, along with an explanation of how it can be determined in the laboratory. Further, its application to three distinct experimental techniques is reviewed. First, as a large interaction parameter it can amplify small signals above technical background noise. Second, as a measurable complex value it enables novel techniques for direct quantum state and geometric phase determination. Third, as a conditioned average of generalized observable eigenvalues it provides a measurable window into nonclassical features of quantum mechanics. In this selective review, a single experimental configuration is used to discuss and clarify each of these applications.

A central feature of quantum mechanics is that a measurement is intrinsically probabilistic. As a result, continuously monitoring a quantum system will randomly perturb its natural unitary evolution. The ability to control a quantum system in the presence of these fluctuations is of increasing importance in quantum information processing and finds application in fields ranging from nuclear magnetic resonance 1 to chemical synthesis 2 . A detailed understanding of this stochastic evolution is essential for the development of optimized control methods. Here we reconstruct the individual quantum trajectories 3-5 of a superconducting circuit that evolves in competition between continuous weak measurement and driven unitary evolution. By tracking individual trajectories that evolve between an arbitrary choice of initial and final states we can deduce the most probable path through quantum state space. These pre-and post-selected quantum trajectories also reveal the optimal detector signal in the form of a smooth time-continuous function that connects the desired boundary conditions. Our investigation reveals the rich interplay between measurement dynamics, typically associated with wave function collapse, and unitary evolution of the quantum state as described by the Schrödinger equation. These results and the underlying theory 6 , based on a principle of least action, reveal the optimal route from initial to final states, and may enable new quantum control methods for state steering and information processing.Our experiment focuses on the dynamics of two quantum levels of a superconducting circuit (a qubit), which can be continuously measured and excited by microwave pulses. To access individual quantum trajectories, we make use of the fact that fully projective measurement (or wavefunction collapse) happens over an average timescale τ controlled by the interaction strength between the system and the detector. By recording the measurement signal in time steps much shorter than τ with high fidelity, we realize a continuous sequence of weak measurements and track the qubit state as it evolves in a single experimental iteration. Individual weak measurements have been recently employed in atomic physicsCavity input Transmon qubit experiments that probe wave function collapse 7 and perform state stabilization 8 . In the domain of superconducting circuits, weak measurements 9 have only recently been realized due to the challenge associated with high fidelity detection of near single-photon level microwave signals. Advances in superconducting parametric amplifiers have enabled continuous feedback control 10-12 , the observation of individual quantum trajectories 13,14 , the determination of weak values 15,16 , and entanglement of qubits 17,18 .Our experiment consists of a superconducting transmon circuit 19 dispersively coupled to a waveguide cavity 20 (Fig. 1a). Considering only the two lowest levels of the transmon as a qubit, our system is described by arXiv:1403.4992v1 [quant-ph] 19 Mar 2014 2 the Hamiltonian Hwhere ...

We generalize the derivation of Leggett-Garg inequalities to systematically treat a larger class of experimental situations by allowing multi-particle correlations, invasive detection, and ambiguous detector results. Furthermore, we show how many such inequalities may be tested simultaneously with a single setup. As a proof of principle, we violate several such two-particle inequalities with data obtained from a polarization-entangled biphoton state and a semi-weak polarization measurement based on Fresnel reflection. We also point out a non-trivial connection between specific two-party Leggett-Garg inequality violations and convex sums of strange weak values.

We review and re-examine the description and separation of the spin and orbital angular momenta (AM) of an electromagnetic field in free space. While the spin and orbital AM of light are not separately meaningful physical quantities in orthodox quantum mechanics or classical field theory, these quantities are routinely measured and used for applications in optics. A meaningful quantum description of the spin and orbital AM of light was recently provided by several authors, which describes separately conserved and measurable integral values of these quantities. However, the electromagnetic field theory still lacks corresponding locally conserved spin and orbital AM currents. In this paper, we construct these missing spin and orbital AM densities and fluxes that satisfy the proper continuity equations. We show that these are physically measurable and conserved quantities. These are, however, not Lorentz-covariant, so only make sense in the single laboratory reference frame of the measurement probe. The fluxes we derive improve the canonical (nonconserved) spin and orbital AM fluxes, and include a 'spin-orbit' term that describes the spin-orbit interaction effects observed in nonparaxial optical fields. We also consider both standard and dual-symmetric versions of the electromagnetic field theory. Applying the general theory to nonparaxial optical vortex beams validates our results and allows us to discriminate between earlier approaches to the problem. Our treatment yields the complete and consistent description of the spin and orbital AM of free Maxwell fields in both quantum-mechanical and field-theory approaches.Keywords: spin and orbital angular momentum of light, electromagnetic field theory, conservation laws 0 extracted from the spin and orbital AM tensors in the Coulomb gauge (ε ijk is the Levi-Civita symbol), yield the same values S and L that appear in optical experiments with monochromatic fields [12]. Moreover, the integral values of the spin and orbital AM, ∫ V S d and ∫ V L d (volume integrals for sufficiently localized fields are assumed), are conserved, i.e., time-independent in free space [13]. This hints that the electromagnetic spin and orbital AM are separate physically meaningful quantities, and that the fundamental problems with the quantum-mechanical and field-theory approaches can and should be overcome. 2 New J. Phys. 16 (2014) 093037 K Y Bliokh et al 4 New J. Phys. 16 (2014) 093037 K Y Bliokh et al 1 1As usual in optics, the bilinear quantities calculated for monochromatic fields will be averaged over oscillations in time. 5 New J. Phys. 16 (2014) 093037 K Y Bliokh et al γ αβγ αβγ βαγ J J J 0, . (3.9) Here = α α J J 00 , and equation (3.9) also includes a continuity equation for the spin S dual :

We introduce contextual values as a generalization of the eigenvalues of an observable that takes into account both the system observable and a general measurement procedure. This technique leads to a natural definition of a general conditioned average that converges uniquely to the quantum weak value in the minimal disturbance limit. As such, we address the controversy in the literature regarding the theoretical consistency of the quantum weak value by providing a more general theoretical framework and giving several examples of how that framework relates to existing experimental and theoretical results.

Weak values arise experimentally as conditioned averages of weak (noisy) observable measurements that minimally disturb an initial quantum state, and also as dynamical variables for reduced quantum state evolution even in the absence of measurement. These averages can exceed the eigenvalue range of the observable ostensibly being estimated, which has prompted considerable debate regarding their interpretation. Classical conditioned averages of noisy signals only show such anomalies if the quantity being measured is also disturbed prior to conditioning. This fact has recently been rediscovered, along with the question whether anomalous weak values are merely classical disturbance effects. Here we carefully review the role of the weak value as both a conditioned observable estimation and a dynamical variable, and clarify why classical disturbance models will be insufficient to explain the weak value unless they can also simulate other quantum interference phenomena.

We present a stochastic path integral formalism for continuous quantum measurement that enables the analysis of rare events using action methods. By doubling the quantum state space to a canonical phase space, we can write the joint probability density function of measurement outcomes and quantum state trajectories as a phase space path integral. Extremizing this action produces the most-likely paths with boundary conditions defined by preselected and postselected states as solutions to a set of ordinary differential equations.As an application, we analyze continuous qubit measurement in detail and examine the structure of a quantum jump in the Zeno measurement regime.

Unlike the real part of the generalized weak value of an observable, which can in a restricted sense be operationally interpreted as an idealized conditioned average of that observable in the limit of zero measurement disturbance, the imaginary part of the generalized weak value does not provide information pertaining to the observable being measured. What it does provide is direct information about how the initial state would be unitarily disturbed by the observable operator. Specifically, we provide an operational interpretation for the imaginary part of the generalized weak value as the logarithmic directional derivative of the postselection probability along the unitary flow generated by the action of the observable operator. To obtain this interpretation, we revisit the standard von Neumann measurement protocol for obtaining the real and imaginary parts of the weak value and solve it exactly for arbitrary initial states and postselections using the quantum operations formalism, which allows us to understand in detail how each part of the generalized weak value arises in the linear response regime. We also provide exact treatments of qubit measurements and Gaussian detectors as illustrative special cases, and show that the measurement disturbance from a Gaussian detector is purely decohering in the Lindblad sense, which allows the shifts for a Gaussian detector to be completely understood for any coupling strength in terms of a single complex weak value that involves the decohered initial state.