Session Z30: Focus Session: Quantum Information for Quantum Foundations - Quantumness versus Classicality

Nonequilibrium quantum correlations and Leggett-Garg inequalities

Theoretical guides to test 'macroscopic realism' in condensed matter systems under quantum control are highly desirable. We report the evaluation of Leggett-Garg inequalities (LGI) in an out-of-equilibrium set up consisting in two interacting qubits coupled to independent baths at different temperatures as can occur for two dipolar coupled spins or superconducting qubits in diverse solid-state environments. We find that LGI violations persist for a longer time in a thermal nonequilibrium scenario as compared with similar results at thermodynamic equilibrium. We contrast these findings with the behavior of non-locality-dominated quantum correlation measurements, such as concurrence, between the two qubits under similar temperature gradients.   

Quantum Correlations in Large-Dimensional States of High Symmetry

Multiparty quantum systems can possess non-classical correlations more general than those characterized by entanglement. In this talk, I will discuss various proposed measures of quantum correlations and investigate how these measures behave for the so-called Werner and isotropic families of states. In particular, I will provide closed expressions for the quantum discord (QD) and the relative entropy of quantumness (REQ) in these states for arbitrary dimensions. The QD and REQ will be shown to equal one another, as well as other well-known measures of quantum correlations. For all Werner states, the classical correlations are seen to vanish in high dimensions while the amount of quantum correlations becomes independent of whether or not the the state is entangled. For isotropic states, nearly the opposite effect is observed with both the quantum and classical correlations growing without bound as the dimension increases and only as the system becomes more entangled.   

Nonnegative subtheories of qubits: stabilizer states and more

Negativity in a quasi-probability representation is typically interpreted as an indication of nonclassical behavior. However, this does not preclude bases that are nonnegative from having interesting applications---the single-qubit stabilizer states have nonnegative Wigner functions and yet play a fundamental role in many quantum information tasks. We determine what other sets of quantum states and measurements of a qubit can be nonnegative in a quasi-probability distribution, and identify nontrivial groups of unitary transformations that permute such states. These sets of states and measurements are analogous to the single-qubit stabilizer states. We show that no quasi-probability representation of a qubit can be nonnegative for more than 2 bases in any plane of the Bloch sphere. Furthermore, there is a unique set of 4 bases that can be nonnegative in an arbitrary quasi-probability representation of a qubit. We provide an exhaustive list of the sets of single-qubit bases that are nonnegative in some quasi-probability distribution and are also closed under a group of unitary transformations, revealing two families of such sets of 3 bases with quasi-probability distributions defined on a space of 8 ontic states. We extend several of these results to higher dimensions.   

Symmetric States on the Octonionic Bloch Ball

Finite-dimensional homogeneous self-dual cones arise as natural candidates for convex sets of states and effects in a variety of approaches towards understanding the foundations of quantum theory in terms of information-theoretic concepts. The positive cone of the ten-dimensional Jordan-algebraic spin factor is one particular instantiation of such a convex set in generalized frameworks for quantum theory. We consider a projection of the regular 9-simplex onto the octonionic projective line to form a highly symmetric structure of ten octonionic quantum states on the surface of the octonionic Bloch ball. A uniform subnormalization of these ten symmetric states yields a symmetric informationally complete octonionic quantum measurement. We discuss a Quantum Bayesian reformulation of octonionic quantum formalism for the description of two-dimensional physical systems. We also describe a canonical embedding of the octonionic Bloch ball into an ambient space for states in usual complex quantum theory.   

Magically, the negativity of the discrete Wigner function is useful

It is possible to represent $d$-dimensional quantum states as probability distributions over a phase space of $d^{2}$ points. However, to encompass the full quantum formalism we must allow negative representations. The well known magic state model of quantum computation gives a recipe for universal quantum computation using perfect Clifford operations and repeat preparations of a noisy ancilla state. It is an open problem to determine which ancilla states enable universal quantum computation in this model. In this talk we will show that for systems of odd dimension a necessary condition for a state to enable universal quantum computation is that it have negative representation in a particular quasi-probability representation. This representation is a natural discrete analogue to the Wigner function. This condition implies the existence of a large class of bound states for magic state distillation: states which cannot be prepared using Clifford operations but which are not useful for quantum computation. This settles in the negative the conjecture that all states not representable as a convex combination of stabilizer states enable universal quantum computation.   

Anderson localization modeled by means of numerical solutions of the Schr\"{o}dinger equation

We developed codes for simulating the Schr\"{o}dinger equation based on the finite-difference time-domain (FDTD) method. We model the 2 dimensional free electron gas system using perfectly matched layers for the open surrounding space. We study the effect of localized impurities on the time evolution of the electron wave function, thereby observing dephasing introduced by the impurities. Our numerical simulations show the decoherence due to the impurities at moderate impurity densities and Anderson localization at high impurity densities. Our results are important for the implementation of quantum computing, quantum communication, and spintronics.   

Quench dynamics in the Anderson impurity model

We study the non-equilibrium behavior of an interacting quantum dot following a quench, where it is suddenly attached to a lead. The system is modeled by a single level Anderson impurity model with infinite on-site repulsion attached via tunneling to non-interacting leads. We use the open system Bethe Ansatz solution of the Anderson model and develop a formal framework to implement Yudson's contour integral formalism in the presence of a Fermi sea. This framework allows the calculation of the full time evolution of the multi-particle wave function and various observables of the system.   

Consistent quantum prediction and decoherence in quantum cosmology

A complete ``consistent histories'' framework for certain symmetry-reduced models of quantum gravity is given, within which probabilities may be consistently extracted from quantum amplitudes. The decoherence functional for both a standard ``Wheeler-DeWitt'' quantization and a loop quantization of a flat Friedmann-Robertson-Walker cosmological model is constructed, from which consistent quantum predictions may be made in mathematically precise models of quantum cosmologies. Consistent (decoherent) families of histories are exhibited, with an emphasis on the crucial role played by the decoherence of histories in arriving at self-consistent quantum predictions for these closed quantum systems. By way of example, the problem of resolution of the classical ``big bang'' singularity is compared and contrasted in these two models. Special attention is given to consistent quantum predictions in these theories which are \emph{certain i.e.\ }predictions for which the problem of interpretation of probabilities for a closed quantum system is not present.   

Wavefunction Collapse via a Nonlocal Relativistic Variational Principle

We propose a relativistically covariant variational principle (VP) capable of describing wavefunction collapse. This produces a nonlinear, nonlocal, time-reversal-invariant theory; the hidden variable is the phase of the wavefunction. The VP is $\delta (A_1 + A_2) = 0$, in which $A_1$ and $A_2$ are positive definite integrals over all spacetime of functions of $\psi(t,\vec x)$. $A_1$ is quadratic in deviations of the wavefunction from compliance with the standard quantum mechanical (SQM) wave equation. $A_2$ takes a minimum value when the wavefunction is a state of minimal uncertainty, penalizing certain kinds of superpositions and thus driving collapse. A multiplier sets the relative size of the terms so that (1) $A_1$ dominates in isolated microscopic systems, so they evolve according to the SQM wave equation; and (2) macroscopic superpositions cause $A_2$ to dominate, driving the system to collapse. Since any macroscopic measurement apparatus is entangled with the system being measured, process (2) explains the empirical observation that measurement collapses the wavefunction. We show that $A_2$ enforces the Born rule, under suitable assumptions and approximations. As an example, the theory predicts the results of the two-slit experiment, including the delayed-choice variant.   

New proofs of the Kochen-Specker theorem for a system of three qubits

In 1995 Kernaghan and Peres gave a transparent state-independent ``parity proof'' of the Kochen-Specker theorem by using a system of three qubits. They did this by using the observables of the 3-qubit system to construct a set of 40 rays in a real 8-dimensional space that formed 25 bases, and then picked out a subset of the bases that gave a parity proof. They showed that there are 320 different (but unitarily equivalent) versions of their proof in this 40-ray set. We extend their result in a number of ways. Firstly, we show that this 40-ray set contains five new types of parity proofs in addition to the one found by Kernaghan and Peres, and that the total number of versions of all six types of proofs under the symmetries of the system is 2$^{11}$ = 2048. Secondly, we point out the existence of a large number of state-independent KS proofs in the 3-qubit system that are structurally different from the Kernaghan-Peres proof, and we explore their features. The geometry of mutually unbiased bases (MUBs) in the 3-qubit system, which played a crucial role in the discovery of these new proofs, will be discussed.   

New Algorithms for Generating Arbitrary Kochen-Specker Sets

The Kochen-Specker KS) sets (constructive proofs of quantum contextuality) have recently obtained a special significance as building blocks of quantum information protocols since quantum contextuality was revealed as property complementary to nonlocality and entanglement. [A. Cabello, {\it Phys.\ Rev.\ Lett.\/} {\bf 104}, 220401 (2010).] Thus, generating arbitrary KS sets becomes as needed as generating Bell states and this has been enabled by recent findings of a vast amount ($>10^{20}$) of new KS sets---we call them a ``KS sea.'' [N.D.\ Megill, K.\ Fresl, M.\ Waegell, P.K.\ Aravind, and M. Pavi{\v c}i{\'c}, {\it Phys.\ Lett.\ A}, {\bf 375} 3419 (2011); M.\ Waegell and P.K.\ Aravind, {it J. Phys.\ A} [to appear] (2011).] Here we present our newest algorithms and computer programs which enable us to obtain any desired KS set from the KS sea in a very short time without ever making a complete data base of KS sets---which would be an impossible task anyhow. This was made possible with the help of our representation of the KS sea as well as individual KS sets by means of MMP hypergraphs.   

How Different Can Quantum States with the Given Fidelity Be?

We address the following question: how big can the relative energy difference between two states of a harmonic oscillator (field mode) with the fixed value of fidelity $F$ be? Exact analytical bounds are found for several popular families of quantum states: coherent, squeezed, arbitrary (mixed) Gaussian, binomial and negative binomial. Numerical bounds are calculated for various superpositions of coherent states (``Schr\"odinger cat states'') and their generalizations. The restrictions on the minimal admissible fidelity levels for quite arbitrary (unknown) states belonging to selected families appear rather strong. For example, one can find two squeezed states with $F=0.9$ but with the relative mean energy difference exceeding 100\%. To guarantee the relative energy difference less than 10\% for arbitrary coherent states, the fidelity must exceed the level $0.995$. For many other sets of states (e.g., squeezed and negative binomial) the restrictions can be much stronger.   

Physical meaning of the de Broglie waves

Commonly regarded as an abstract entity, de Broglie waves represent a real thing making the wave-particle duality an operative factor with a wealth of consequences. Waves as probabilities in Schr\"{o}dinger's quantum mechanics are due to interactive holography that builds up on top of the suggested cellular automaton model of the physical Universe [1]. This model implementing a rule of mutual synchronization produces a set of helicoidal solitons that corresponds exactly to the whole spectrum of elementary particles of matter, no more and no less. The particle-like behavior of the solitons exhibits periodic synchro activities that constitute a wave. The wavelength is inversely proportional to the momentum of these elementary particles, and they distinctly show characteristic qualities of the micro world. The condition of translation-rotation congruity foresees a very small decrease of photons speed with their frequency and polarization, while smoothness of this condition for neutrinos accounts for their already observed faster-than-light propagation. Helicoidal solitons create identical impacts from rotation resulting in the same value of spin for different particles irrespective of variations in mass and speed that are related to translational motion. For any given orientation, spin is either ``up'' or ``down'' since helicoidal solitons are flipping this way with every half-turn. Deep inelastic scattering performs sampling of the wave structure of the helicoidal solitons deceptively portraying the outcomes as isolated quarks. [1] S. Berkovich, ``A comprehensive explanation of quantum mechanics,'' http://www.cs.gwu.edu/research/technical-report/170   

No Drama Quantum Theory?

Is it possible to offer a ``no drama'' quantum theory? Something as simple (in principle) as classical electrodynamics - a theory described by a system of partial differential equations (PDE) in 3+1 dimensions, but reproducing unitary evolution of a quantum field theory in the configuration space? The following results suggest an affirmative answer: 1. The scalar field can be algebraically eliminated from scalar electrodynamics; the resulting equations describe independent evolution of the electromagnetic field (EMF). 2. After introduction of a complex 4-potential (producing the same EMF as the standard real 4-potential), the spinor field can be algebraically eliminated from spinor electrodynamics; the resulting equations describe independent evolution of EMF. 3. The resulting theories for EMF can be embedded into quantum field theories. Another fundamental result: in a general case, the Dirac equation is equivalent to a 4th order PDE for just one component, which can be made real by a gauge transform. Issues related to the Bell theorem are discussed. A. Akhmeteli, Int'l Journal of Quantum Information, Vol. 9, Suppl., 17-26 (2011) A. Akhmeteli, Journal of Mathematical Physics, Vol. 52, 082303 (2011) A. Akhmeteli, quant-ph/1108.1588   

Type 2 local realism is alive and well in the quantum world

It is often, but incorrectly said, that there is no local realism in the quantum world. This is wrong because there is a second type of local realism. Type two local realism is drastically different than how most people think of their world, including Einstein. It starts with the idea that we are immersed in a sea of invisible elementary waves, traveling in all directions, at all wavelengths. Whenever a photon or other particle moves, it follows one of these waves in the reverse direction. Although this may sound preposterous, this theory can explain several quantum experiments: Jacques (2007) Wheeler thought experiment with delayed choice, Kim (1999), Quantum eraser experiment with delayed choice, all the Bell test experiments with delayed choice, and the double slit experiment. Using the Theory of Elementary Waves (TEW), all these experiments can, surprisingly, be explained using this unconventional type of local realism, with time going forward.