Session D15: Focus Session: Foundations of Quantum Theory I

LeRoy Apker Award Talk: Testing Hidden-Variable Theorems with Single-Photon Entangled States

An ensemble of single photons created in a hyperentangled Bell state were used to test a broad class of Hidden-Variable Theorems (HVTs). Specifically, the class of HVTs based on the joint assumption of Realism and Non-Contextuality (NC) -- the premise that values associated with one observable are independent of which commuting observables may be measured simultaneously -- known as NCHVTs, and first examined by Bell as well as Kochen and Specker, were addressed using these single-photon states entangled in polarization and direction of momentum. A Clauser-Horne-Shimony-Holt (CHSH) Inequality was applied, with the factorization condition that is usually satisfied by a Non-Locality assumption being instead satisfied by the assumption of NC, due to the inherently local nature of detection events for single particles. The basis rotations and projections necessary for testing the CHSH Inequality were accomplished using interferometers and standard polarization optical elements. A violation of the CHSH Inequality was observed, ruling out either Realism or Non-Contextuality -- or possibly both. The tenability of Contextual HVTs remains, trivially, as the predictions of such a theory can map one-to-one to the predictions of Quantum Mechanics.   

Weak values and the Leggett-Garg inequality in solid-state qubits

The seminal paper of Aharonov, Albert, and Vaidman introduces the concept of a weak value as a statistical average over realizations of a weak measurement, where the system is both pre- and post-selected. By taking restricted averages, weak values can exceed the range of eigenvalues associated with the observable in question. We discuss how to implement a weak values measurement with solid-state qubits. In parallel activity, Leggett and Garg have devised a test of quantum mechanics for a single system using different ensembles of (projective) measurements at different times and correlation functions of those outcomes. The original motivation was to test if there was a size scale where quantum mechanics would break down. Introduced as a ``Bell-inequality in time'', the assumptions of macrorealism that could be verified by a non-invasive detector imply that their correlation function obeys a Leggett-Garg inequality that quantum mechanics would violate, formally similar to the inequality of Bell. We demonstrate that the proper notion of a classical weak value also demands these assumptions, and that furthermore a weak value can be non-classical if and only if a Leggett-Garg inequality can also be violated. We will discuss generalized weak values, where post-selection occurs on a range of weak measurement results. Our analysis is presented in terms of kicked quantum nondemolition measurements on a quantum double-dot charge qubit.   

Negative probabilities and measurement disturbance

Feynman once said that ``the only difference between a probabilistic classical world and the equations of the quantum world is that somehow or other it appears as if the probabilities would have to go negative''. But what is the essential reason for probabilities going negative? Here it is demonstrated that negative probabilities are a direct consequence of measurement disturbance. The Margenau-Hill distribution, a quasiprobability taking negative values, is expressed in terms of the joint probability obtained in the successive measurement of two projectors. The quasiprobability takes negative values only if the measurement of the first projector disturbs the measurement of the second projector. The uniqueness of this quasiprobability follows by imposing a symmetry principle on measurement disturbance. The quasiprobability is made informationally complete by a complex extension. Conditions for informational completeness are derived. This quasiprobability also can be observed directly as a statistical average of pointer displacements in weak measurements.   

The meaning of negative weak values

A number of approaches to quantum mechanics incorporate negative values for quantities that were classically positive, such as the Wigner-Moyal density approach or the Feynman negative probability approach, etc. In the re-formulation of quantum mechanics using weak values and weak measurements, we encounter a new situation where weak values of projection operators turn out to be negative. We emphasize the differences between these negative weak values and the negative values encountered in the other formalisms: in the previous formalisms, the mathematical entity whose average yielded the negative values are not density operators. While they do yield the correct average of a function, they also have non-physical aspects, i.e. mathematical artifacts, when the densities become negative. The reason is that if we attempt to actually measure such ``negative" properties, then the result does not correspond to a physical observable in Hilbert Space. E.g. if we did attempt to cheat quantum mechanics by projecting onto p and x as densities simultaneously in Wigner-Moyal, then we obtain the parity operator, the most non-local result. On the other hand, in the case of weak values, we obtain a new situation: when we use a bonafide measuring device to measure these properties ideally, then the very same measuring device will yield the predicted negative weak values when the measurement interaction is simply weakened.   

Frame representations of quantum mechanics and the necessity of negativity in quasi-probability representations

Several finite dimensional quasi-probability representations of quantum states have been proposed to study various problems in quantum information theory and quantum foundations. These representations are often defined only on restricted dimensions and their physical significance in contexts such as drawing quantum-classical comparisons is limited by the non-uniqueness of the particular representation. Here we show how the mathematical theory of frames provides a unified formalism which accommodates all known quasi-probability representations of finite dimensional quantum systems. Moreover, we show that any quasi-probability representation satisfying two reasonable properties is equivalent to a frame representation and then prove that any such representation of quantum mechanics must exhibit either negativity or a deformed probability calculus.   

Towards Testing Quantum Mechanics with Micro-Optomechanical Systems

We review our work in micro-optomechanical systems. Motivation for work on these systems is based in proposals to test quantum mechanics in new regimes. Although extremely challenging, creating a quantum superposition of a mirco-mechanical oscillator coupled to an optical cavity seems experimentally feasible with current technology. Additionally, the optomechanical systems used for this type of research have other applications, such as optical cooling, as recently demonstrated by several independent groups. Finally we will briefly discuss the direction of our research in the near future, including the use of conventional cryogenics to cool the resonator and the prospects for several related types of devices.   

Decoherence and the Uncertainty Principle

The uncertainty principle is normally understood as representing a limit on the fundamental accuracy of simultaneous measurements of incompatible observables. In the context of consistent/decoherent histories formulations of quantum theory, we show that it may also be understood as quantifying the degree of course-graining necessary in order for histories of a quantum system to decohere. This follows as a consequence of a new inequality bounding the interference between histories in a consistent histories formulation of quantum theory.   

Noncontextuality and the Kochen-Specker Theorem

The question of noncontextuality in a simple, two-qubit system is considered. It is shown that quantum theory is consistent with a noncontextual hidden variable interpretation, contrary to the conclusions of the Kochen-Specker theorem. The key to the proof is the recognition of a subtle but fundamentally important assumption regarding the dependence of the hidden variable probability distribution on the particular set of mutually commensurate observables chosen for measurement. Recent experiments to test noncontextuality will be discussed and their results reconciled with a hidden variable interpretation.   

Path detection and interference tradeoff in the double-slit experiment

We study how the acquisition of ``which-path'' information leads to a loss of contrast in the double-slit interference setup. We show that neither the conventional uncertainty principle nor the recently-derived universal uncertainty principle place any restriction on the minimum root-mean-square momentum disturbance, once it is recognized that the which-path determination does not require one to measure $x$, but only an appropriate two-valued function of $x$. We then develop a description of the problem in terms of only two-valued variables, and consider a completely general measurement model, which allows us to distinguish between the measurement error and what we call the ``preparational error.'' We show that error-free which-path measurements are possible in this model that do not destroy the fringe visibility. On the other hand, we also show that there is a general tradeoff relation between preparational error and fringe visibility, which, for measurements obeying the ``non- destruction'' condition, reduces to Englert's inequality.   

Exact mappings between quantum relativistic and quantum optical models

We develop a novel quantum optical perspective into a couple of quantum relativistic systems: i) First we show how the two-dimensional extension of the harmonic oscillator, known as the Dirac oscillator, can be exactly mapped onto a chiral Anti-Jaynes-Cummings model of quantum optics. This equivalence allows us to predict a series of novel relativistic phenomena, such as spin-orbit {\it Zitterbewegung}. Furthermore, we also make a realistic experimental proposal, at reach with current technology, for studying the equivalence of both models using a single trapped ion [1]. ii) Second, we show that a relativistic version of Schr\"{o}dinger cat states, here called {\it Dirac cat states}, can be built in relativistic Landau levels when an external magnetic field couples to a relativistic spin $1/2$ charged particle. Under suitable initial conditions, the associated Dirac equation produces unitarily Dirac cat states involving the orbital quanta of the particle in a well defined mesoscopic regime. These states have a purely relativistic origin and cease to exist in the non-relativistic limit [2]. \newline [1] A. Bermudez et. al, Phys. Rev. A.\textbf{76}, 041801(R) (2007). \newline [2] A. Bermudez et. al, Phys. Rev. Lett.\textbf{99}, 123602 (2007).