Modular variables in quantum theory

Aharonov, Yakir; Pendleton, Hugh; Petersen, Aage

doi:10.1007/bf00670008

Cited by 153 publications

(197 citation statements)

References 14 publications

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“…Unlike ordinary momentum, modular momentum becomes, on detecting (or failing to detect) the particle at a particular slit, maximally uncertain. The effect of introducing a potential at a distance from the particle (i.e., of opening a slit) is equivalent to a nonlocal rotation in the space of the modular variable (22). Denote it by θ ∈ [0, 2π).…”

Section: Nonlocal Dynamics and Wave-like Behaviormentioning

confidence: 99%

Finally making sense of the double-slit experiment

Aharonov

Cohen

Colombo

et al. 2017

Proc. Natl. Acad. Sci. U.S.A.

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Feynman stated that the double-slit experiment ". . . has in it the heart of quantum mechanics. In reality, it contains the only mystery" and that "nobody can give you a deeper explanation of this phenomenon than I have given; that is, a description of it" [Feynman R, Leighton R, Sands M (1965) The Feynman Lectures on Physics]. We rise to the challenge with an alternative to the wave function-centered interpretations: instead of a quantum wave passing through both slits, we have a localized particle with nonlocal interactions with the other slit. Key to this explanation is dynamical nonlocality, which naturally appears in the Heisenberg picture as nonlocal equations of motion. This insight led us to develop an approach to quantum mechanics which relies on preand postselection, weak measurements, deterministic, and modular variables. We consider those properties of a single particle that are deterministic to be primal. The Heisenberg picture allows us to specify the most complete enumeration of such deterministic properties in contrast to the Schrödinger wave function, which remains an ensemble property. We exercise this approach by analyzing a version of the double-slit experiment augmented with postselection, showing that only it and not the wave function approach can be accommodated within a time-symmetric interpretation, where interference appears even when the particle is localized. Although the Heisenberg and Schrödinger pictures are equivalent formulations, nevertheless, the framework presented here has led to insights, intuitions, and experiments that were missed from the old perspective.Heisenberg picture | two-state vector formalism | modular momentum | double slit experiment B eginning with de Broglie (1), the physics community embraced the idea of particle-wave duality expressed, for example, in the double-slit experiment. The wave-like nature of elementary particles was further enshrined in the Schrödinger equation, which describes the time evolution of quantum wave packets.It is often pointed out that the formal analogy between Schrödinger wave interference and classical wave interference allows us to interpret quantum phenomena in terms of the familiar classical notion of a wave. Indeed, wave-particle duality was construed by Bohr and others as the essence of the theory and, in fact, its main novelty. Even so, the foundations of quantum mechanics community have consistently raised many questions (2-5) centered on the physical meaning of the wave function.From our perspective and consistent with ideas first expressed by Born (6) and thereafter extensively developed by Ballentine (7, 8), a wave function represents an ensemble property as opposed to a property of an individual system.What then is the most thorough approach to ontological questions concerning single particles (using standard nonrelativistic quantum mechanics)? We propose an alternative interpretation for quantum mechanics relying on the Heisenberg picture, which although mathematically equivalent to the Schrödinger picture, is very dif...

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Section: Nonlocal Dynamics and Wave-like Behaviormentioning

confidence: 99%

Finally making sense of the double-slit experiment

Aharonov

Cohen

Colombo

et al. 2017

Proc. Natl. Acad. Sci. U.S.A.

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“…Within this formulation, Alice's atom has a deterministic operator with regard to its energy (i.e., when projecting on the atom's energy she finds an excited state with certainty), yet it also has a nonlocal deterministic operator sensitive to the relative phase between |e A |g B and |g A |e B . It is this operator which generalizes the singleparticle notion of modular momentum [56], accounting for the nonlocal correlations with Bob's atom. In other words, the initial (entangled) state of the system suggests that the operator σ A y σ B x is a deterministic operator, while the final strong measurements disentangle the state, giving rise to σ A z and σ B x as the set of deterministic operators.…”

Section: Discussionmentioning

confidence: 99%

Interaction-Free Effects Between Distant Atoms

et al. 2017

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A Gedanken experiment is presented where an excited and a ground-state atom are positioned such that, within the former's half-life time, they exchange a photon with 50% probability. A measurement of their energy state will therefore indicate in 50% of the cases that no photon was exchanged. Yet other measurements would reveal that, by the mere possibility of exchange, the two atoms have become entangled. Consequently, the "no exchange" result, apparently precluding entanglement, is non-locally established between the atoms by this very entanglement. This quantum-mechanical version of the ancient Liar Paradox can be realized with already existing transmission schemes, with the addition of Bell's theorem applied to the no-exchange cases. Under appropriate probabilities, the initially-excited atom, still excited, can be entangled with additional atoms time and again, or alternatively, exert multipartite nonlocal correlations in an interaction free manner. When densely repeated several times, this result also gives rise to the Quantum Zeno effect, again exerted between distant atoms without photon exchange. We discuss these experiments as variants of interactionfree-measurement, now generalized for both spatial and temporal uncertainties. We next employ weak measurements for elucidating the paradox. Interpretational issues are discussed in the conclusion, and a resolution is offered within the Two-State Vector Formalism and its new Heisenberg framework.

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“…Adopted to the present notational conventions, Aharonov's modular position and momentum operators [17], X m and P m , are introduced by the symmetric relations 1) where N x and N p are the integer operators…”

Section: Aharonov's Modular Operatorsmentioning

confidence: 99%

“…Section 6 deals with the relation of the Zak bases and operators to the modular variables introduced by Aharonov et al in Ref. [17]. A possible application for the Zak bases in quantum information theory is hinted at in section 7.…”

Section: Introductionmentioning

confidence: 99%

Periodic and discrete Zak bases

Englert

Lee

Mann

et al. 2006

J. Phys. A: Math. Gen.

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Abstract. Weyl's unitary operators for displacement in position and momentum commute with one another if the product of the elementary displacements equals Planck's constant. Then, their common eigenstates constitute the Zak basis, with each state specified by two phase parameters. Accordingly, the transformation function from the position basis to the Zak basis maps the Hilbert space on the line onto the Hilbert space on the torus. This mapping is one-to-one provided that the Zak basis states are periodic functions of their phase parameters, but then the mapping cannot be continuous on the whole torus. With the periodicity of the Zak basis enforced, the basis has a double Fourier series. The Fourier coefficients identify a discrete basis which complements the periodic Zak basis to form a pair of mutually unbiased bases. The discrete basis states are the common eigenstates of the two complementary partners to the two unitary displacement operators. These partner operators are of angular momentum type, with integer eigenvalues, and generate the fundamental rotations of the torus. Conversely, the displacement operators are the ladder operators for their partners. For each consistent phase convention for the periodic Zak basis, and thus for the line-onto-torus mapping, there is a corresponding discrete Zak basis and a corresponding pair of partner operators. Examples of particular interest are the conventions that give a continuous mapping in one phase parameter or are symmetric in both phase parameters. The latter emphasizes the Heisenberg-Weyl symmetry between position and momentum. We discuss briefly the relation between the Zak bases and Aharonov's modular operators. Finally, as an application of the Zak operators for the torus, we mention how they can be used to associate with the single degree of freedom of the line a pair of genuine qubits that are potentially entangled.

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Modular variables in quantum theory

Cited by 153 publications

References 14 publications

Finally making sense of the double-slit experiment

Finally making sense of the double-slit experiment

Interaction-Free Effects Between Distant Atoms

Periodic and discrete Zak bases

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