The concept of a \emph{weak value} of a quantum observable was developed in the late 1980s by Aharonov and colleagues to characterize the value of an observable for a quantum system in the time interval between two projective measurements. Curiously, these values often lie outside the eigenspectrum of the observable, and can even be complex-valued. Nevertheless, the weak value of a quantum observable has been shown to be a valuable resource in quantum metrology, and has received recent attention in foundational aspects of quantum mechanics. This paper is driven by a desire to more fully understand the underlying mathematical structure of weak values. In order to do this, we allow an observable to be \emph{any} Hermitian operator, and use the pre- and post-selected states to develop well-defined linear maps between the Hermitian operators and their corresponding weak values. We may then use the inherent Euclidean structure on Hermitian space to geometrically decompose a weak value of an observable. In the case in which the quantum systems are qubits, we provide a full geometric characterization of weak values.
The concept of a \emph{weak value} of a quantum observable was developed in the late 1980s by Aharonov and colleagues to characterize the value of an observable for a quantum system in the time interval between two projective measurements. Curiously, these values often lie outside the eigenspectrum of the observable, and can even be complex-valued. Nevertheless, the weak value of a quantum observable has been shown to be a valuable resource in quantum metrology, and has received recent attention in foundational aspects of quantum mechanics. This paper is driven by a desire to more fully understand the underlying mathematical structure of weak values. In order to do this, we allow an observable to be \emph{any} Hermitian operator, and use the pre- and post-selected states to develop well-defined linear maps between the Hermitian operators and their corresponding weak values. We may then use the inherent Euclidean structure on Hermitian space to geometrically decompose a weak value of an observable. In the case in which the quantum systems are qubits, we provide a full geometric characterization of weak values.
A description of quantum systems at the time interval between two successive measurements is presented. Two wave functions, the first pre-selected by the initial measurement and the second post-selected by the final measurement describe quantum systems at a single time. It is shown how this approach leads to a new concept: a weak value of an observable. Weak values represent novel characteristics of quantum systems between two measurements. They are outcomes of a standard measuring procedure that fulfills certain requirements of “weakness”. We call it weak measurement . Physical meaning and underlying mathematical structure of weak measurements are explored.
Abstract Observations in quantum weak measurements are determined by complex numbers called weak values. We present a geometrical interpretation of the argument of weak values of general Hermitian observables in N-dimensional quantum systems in terms of geometric phases. We formulate an arbitrary weak value in terms of three real vectors on the unit sphere in N 2 − 1 dimensions, S N 2 − 2 . These vectors are linked to the initial and final states, and to the weakly measured observable, respectively. We express pure states in the complex projective space of N − 1 dimensions, C P N − 1 , which has a non-trivial representation as a 2N − 2 dimensional submanifold of S N 2 − 2 (a generalization of the Bloch sphere for qudits). The argument of the weak value of a projector on a pure state of an N-level quantum system describes a geometric phase associated to the symplectic area of the geodesic triangle spanned by the vectors representing the pre-selected state, the projector and the post-selected state in C P N − 1 . We then proceed to show that the argument of the weak value of a general observable is equivalent to the argument of an effective Bargmann invariant. Hence, we extend the geometrical interpretation of projector weak values to weak values of general observables. In particular, we consider the generators of SU(N) given by the generalized Gell–Mann matrices. Finally, we study in detail the case of the argument of weak values of general observables in two-level systems and we illustrate weak measurements in larger dimensional systems by considering projectors on degenerate subspaces, as well as Hermitian quantum gates. To conclude, we discuss the interpretation and usefulness of geometric phases in weak values in connection to weak measurements.
The formalism of weak measurement in quantum mechanics has revealed profound connections between measurement theory, quantum foundations, and signal processing. In this paper, we develop a pointer-free derivation of superoscillations, demonstrating that they are a natural and necessary consequence of the geometric structure underlying weak values. We argue that the weak value is best understood as a ratio of geometric deformation, quantifying how an observable transforms the structure of Hilbert space relative to a reference provided by the standard inner product. This deformation acts as a conceptual lens, warping the local structure of quantum states to produce oscillations far exceeding the global Fourier bandwidth. We formalize this by interpreting the weak value as a comparison between a deformed sesquilinear form and the standard one, and explore its deep connections to generalized Rayleigh quotients and the projective geometry of quantum states. This perspective unifies weak values and superoscillations as two facets of a single underlying geometric principle.
Full text: Along with the well-known concept of projective measurements, quantum mechanics allows various kinds of generalized measurement operators. Two important examples are: joint measurements on two or more quantum systems that cannot be achieved by local operations (LOCC); and weak measurements that obtain less information about a system than does a projective measurement, but with correspondingly less disturbance. Unlike the result of a strong measurement, the average value of a weak measurement of an observable (its weak value), when followed by projective postselection in a complementary basis, can lie outside the range of eigenvalues. This discrepancy is not observed in analogous classical measurements. Weak values aid the resolution of quantum paradoxes, and can simplify analysis of weakly coupled systems. We use a generalized measurement device to measure the weak value of a photon's polarization in the horizontal/vertical basis (the Stokes operator S1 = |H> weak up to 47, outside the usual range -1 {<=} S1 {<=} 1. Unlike previous observations of weak values, our measurement works by entangling two separate systems, and thus can only be described by quantum theory, not a classical wave theory. Also, we have used a two-qubit joint measurement based on a controlled-NOT gate by which certain twoqubit unentangled states can be more reliably distinguished than by using LOCC. We quantify this using a payoff function, for which the optimal LOCC measurement attains 2/3, and our experimental measurement attains 0.72 {+-} 0.02, close to the global optimum of 3/4. (author)
'Weak measurements'—involving a weak unitary interaction between a quantum system and a meter followed by a projective measurement—are investigated when the system has a non-Hermitian Hamiltonian. We show in particular how the standard definition of the 'weak value' of an observable must be modified. These studies are undertaken in the context of bound-state scattering theory, a non-Hermitian formalism for which the involved Hilbert spaces are unambiguously defined and the metric operators can be explicitly computed. Numerical examples are given for a model system. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to 'Quantum physics with non-Hermitian operators'.
A general prescription for the construction of quasi-joint-probability distributions intended to describe the "joint behaviour" of an arbitrary pair of (generally non-commuting) quantum observables, along with its application to the geometric/statistical interpretation of Aharonov's weak value and a proposal for new uncertainty relations, are presented.The introduction of such distributions is motivated by two complementary approaches: one from a bottom-up and strictly operational construction by carefully examining the mathematical framework of the conditioned measurement scheme, and the other from a top-down viewpoint by applying the results of spectral theorem for normal operators and its Fourier transforms.It is revealed that, for a pair of simultaneously measurable observables, the distribution reduces to the unique standard joint-probability distribution of the pair as expected, whereas for a non-commuting pair, there exists an inherent indefiniteness for the choice of such distributions, admitting a multitude of candidates that qualify to describe the "joint behaviour" of the pair.The L 2 structures induced by the quasi-joint-probability distributions are then found to provide statistical interpretation of the geometric structures that can be naturally introduced on the space of operators on the underlying Hilbert space.As possible applications of our findings, we focus on orthogonal projections and inner products of observables, and observe that such geometric concepts can respectively be given statistical interpretations as "conditionings" and "correlations".Incidentally, weak values A w are given a geometric/statistical interpretation as either the orthogonal projection of an observable A on the subspace generated by another observable B (or equivalently, as the conditioning of A given B with respect to the quasi-joint-probability distribution under consideration).Moreover, the Cauchy-Schwartz inequality applied on inner products of two observables (or equivalently, correlations between the pair) leads to novel inequalities interpreted as uncertainty relations for approximation/estimation, which entails the time-energy uncertainty relation as its special case.