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Quantum Operations: technical or fundamental challenge?

Bogdan Mielnik

Published Aug 14, 2016• 5 citations
OpenAlex

Abstract

A class of unitary operations generated by idealized, semiclassical fields is studied. The operations implemented by sharp potential kicks are revisited and the possibility of performing them by softly varying external fields is examined. The possibility of us-ing the ion traps as ’operation factories ’ transforming quantum states is discussed. The non-perturbative algorithms indicate that the results of abstract δ-pulses of oscillator potentials can become real. Some of them, if empirically achieved, could be essential to examine certain atypical quantum ideas. In particular, simple dynamical manipula-tions might contribute to the Aharonov-Bohm criticism of the time-energy uncertainty principle, some other, to verify the existence of fundamental precision limits of the position measurements or the reality of ‘non-commutative geometries’. 1

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Quantum operations: technical or fundamental challenge?

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A class of unitary operations generated by idealized, semiclassical fields is studied. The operations implemented by sharp potential kicks are revisited and the possibility of performing them by softly varying external fields is examined. The possibility of using the ion traps as ‘operation factories’ transforming quantum states is discussed. The non-perturbative algorithms indicate that the results of abstract δ-pulses of oscillator potentials can become real. Some of them, if empirically achieved, could be essential to examine certain atypical quantum ideas. In particular, simple dynamical manipulations might contribute to the Aharonov–Bohm criticism of the time–energy uncertainty principle, while some others may verify the existence of fundamental precision limits of the position measurements or the reality of ‘non-commutative geometries’.

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Starting with the quantum Liouville equation, we write the density operator as the product of elements respectively in the left and right ideals of an operator algebra and find that the Schrodinger picture may be expressed through two representation independent algebraic forms in terms of the density and phase operators. These forms are respectively the continuity equation, which involves the commutator of the Hamiltonian with the density operator, and an equation for the time development of the phase operator that involves the anti-commutator of the Hamiltonian with this density operator. We show that this latter equation plays two important roles: (i) it expresses the conservation of energy in a system where energy is well defined and (ii) it provides a simple way to evaluate the gauge changes that occur in the Aharonov-Bohm, the Aharonov-Casher, and Berry phase effects. Both these operator (i.e. purely algebraic) equations also allow us to re-examine the Bohm interpretation, showing that it is in fact possible to construct Bohm interpretations in representations other than the x-representation. We discuss the meaning of the Bohm interpretation in the light of these new results in terms of non-commutative structures and this enables us to clarify its relation to standard quantum mechanics.

arXiv:quant-ph/0005026•May 4, 2000•17 citations
A class of unitary operations generated by idealized, semiclassical fields is studied. The operations implemented by sharp potential kicks are revisited and the possibility of performing them by softly varying external fields is examined. The possibility of us-ing the ion traps as ’operation factories ’ transforming quantum states is discussed. The non-perturbative algorithms indicate that the results of abstract δ-pulses of oscillator potentials can become real. Some of them, if empirically achieved, could be essential to examine certain atypical quantum ideas. In particular, simple dynamical manipula-tions might contribute to the Aharonov-Bohm criticism of the time-energy uncertainty principle, some other, to verify the existence of fundamental precision limits of the position measurements or the reality of ‘non-commutative geometries’. 1
A class of unitary operations generated by idealized, semiclassical fields is studied. The operations implemented by sharp potential kicks are revisited and the possibility of performing them by softly varying external fields is examined. The possibility of using the ion traps as ‘operation factories’ transforming quantum states is discussed. The non-perturbative algorithms indicate that the results of abstract δ-pulses of oscillator potentials can become real. Some of them, if empirically achieved, could be essential to examine certain atypical quantum ideas. In particular, simple dynamical manipulations might contribute to the Aharonov–Bohm criticism of the time–energy uncertainty principle, while some others may verify the existence of fundamental precision limits of the position measurements or the reality of ‘non-commutative geometries’.
The unitary evolution operators generated by periodically varying elastic potentials including the Mathieu case are studied. The evolution operations in the stability areas of the Strutt diagram admit effective (Floquet) Hamiltonians generalizing the orthodox oscillators. The points on the separatrices represent two exceptional types of unitary operations, imitating (in a soft way) the results of the sudden δ-like kicks of the elastic potential, or the distorted free evolution (in accelerated, slowed down or even negative time). In all domains the dynamical phenomena illustrate a non-trivial relation between the micro- and macroscopic motion scales.
The dynamical manipulation of physical systems is necessary to explain the structure of physical theories. However, the manipulation techniques in quantum theories progress slowly, giving still no certainty that all theoretically described states and all unitary transformations can be indeed achieved. Below, we report some elements of idealized control of the evolution processes in time dependent quadratic potentials, with attention focused on the abstract δ-pulses of the elastic forces. Some of them, if empirically approximated, could become essential for a sequence of difficult interpretation problems.
An operational description of quantum phenomena concerns developing models that describe experimentally observed behaviour. $\textit{Higher-order quantum operations}\unicode{x2014}$quantum operations that transform quantum operations$\unicode{x2014}$are fundamental to modern quantum theory, extending beyond basic state preparations, evolutions, and measurements described by the Born rule. These operations naturally emerge in quantum circuit architectures, correlated open dynamics, and investigations of quantum causality, to name but a few fields of application. This Review Article provides both a pedagogical introduction to the framework of higher-order quantum operations and a comprehensive survey of current literature, illustrated through physical examples. We conclude by identifying open problems and future research directions in this rapidly evolving field.
Starting with the quantum Liouville equation, we write the density operator as the product of elements respectively in the left and right ideals of an operator algebra and find that the Schrodinger picture may be expressed through two representation independent algebraic forms in terms of the density and phase operators. These forms are respectively the continuity equation, which involves the commutator of the Hamiltonian with the density operator, and an equation for the time development of the phase operator that involves the anti-commutator of the Hamiltonian with this density operator. We show that this latter equation plays two important roles: (i) it expresses the conservation of energy in a system where energy is well defined and (ii) it provides a simple way to evaluate the gauge changes that occur in the Aharonov-Bohm, the Aharonov-Casher, and Berry phase effects. Both these operator (i.e. purely algebraic) equations also allow us to re-examine the Bohm interpretation, showing that it is in fact possible to construct Bohm interpretations in representations other than the x-representation. We discuss the meaning of the Bohm interpretation in the light of these new results in terms of non-commutative structures and this enables us to clarify its relation to standard quantum mechanics.