Quantum Foundations

There is a common framework for the measurement problem for sensors such as radars, sonars, and optics in a common language by casting analysis of signals in the language of quantum mechanics (Rigged Hilbert Space). The use of this language can reveal a more detailed understanding of the underlying interactions of a return signal that are not usually brought out by standard signal processing design techniques. The weak measurement Ansatz first provided by the Aharonov, Albert and Vaidman paper (A2V) that introduced weak values to the world provides an explicit means to consider all interactions of a signal with an object by using what we term the Aharonov Ansatz. The Aharonov Ansatz for sensing can summarized as: 1. Any sensor measurement process, whether active or passive can be thought of as determining the mathematical operator's characteristics of a signal's interaction with a object. 2. Certain types of interaction operators can be "post-selected" for in the return signal when the broadcast signal is known for either a single or multiple operators so receiver design can be optimized. 3. In principle detectors can design can be optimized, "matched" to signal interaction for these operators (operator matched filter), so mathematical solutions to receiver (in the classical sense) design or the design of apparatus of difficult to measure quantum interactions can be improved as has been reported in the literature. 4. Matching or post-selection to a given operator, when possible, maximizes ability to detect a "signal" or the characteristics of an interaction. Finally, in this talk we note a connection between this work and a the variational functional used in perturbation theory in quantum mechanics.

In 2005 R. Spekkens presented a generalization of noncontextuality that applies to imperfect measurements (POVMs) by allowing the underlying ontological model to be indeterministic. Unlike traditional Bell-Kochen-Specker noncontextuality, ontological models of a single qubit were shown to be contextual under this definition. Recently, M. Pusey showed that, under certain conditions, exhibiting an anomalous weak value implies contextuality. We will present a single qubit prepare and measure QKD protocol that uses observation of anomalous weak values of particular observables to estimate the quantum channel error rate and certify the security of the channel. We will also argue that it is the “degree” of contextuality of the noisy qubits exiting the channel that fundamentally determine the secure key rate. A benefit of this approach is that the security does not depend on the fair sampling assumption, and so is not compromised by Eve controlling Bob’s measurement devices. Thus it retains much of the benefit of “Measurement Device Independent” QKD protocols while only using single photon preparation and measurement.

Measurements performed at variable strengths show that non-commuting physical properties are related by complex-valued statistics, where the complex phase expresses the action of transformations along orbits represented by the eigenstates. In strong measurements, the dynamics along the orbits is completely randomized, which means that the pure states prepared by such a measurement actually represent ergodic statistics where the coherence between components originates from quantum dynamics. The complex algebra of Hilbert space inner products describes the intersection of two ergodically randomized orbits, where the complex phase describes the action of propagation along the orbits. Since the same action also appears in classical descriptions of the dynamics it is possible to derive quantum states and their time evolution directly from the classical equations of motion, without the abstractions of operator algebra. A representative example of this fundamental relation between classical dynamics and quantum coherence is the multi-photon interference in two-path interferometers, where the multi-photon interference fringes can be explained by the action enclosed by two classical orbits corresponding to the input and output photon number states. This example shows how the non-classical features of quantum statistics emerge from the effects of enclosed actions on the causality relations between the initial orbit prepared by ergodic randomization and the final orbit along which the system was sampled during the measurement. Since action relations take the same form in quantum mechanics and in the classical limit, any attempt to explain quantum mechanics should start with an analysis of the dynamics. The conventional sense of reality only emerges from the consistency of causality relations,not from any abstract ``knowledge of reality''. Our concepts of particles and trajectories only have an approximate validity which breaks down in the limit of small action. Reality always requires the dynamics of interaction, and hbar is an absolute limitation of physical reality. In this presentation, I hope to clarify that this absence of a microscopic material reality can be understood quite naturally in terms of the well known physics of dynamics and interactions, removing the need for any untestable platonic assumptions about a hypothetical ``reality out there''.