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arXiv:1406.0740 · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Phase-space noncommutative formulation of Ozawa’s uncertainty principle

Catarina Bastos, Alex E. Bernardini, Orfeu Bertolami A, Nuno Costa Dias, João Nuno Prata

Published Aug 19, 2014• 36 citations
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Abstract

Ozawa's measurement-disturbance relation is generalized to a phase-space noncommutative extension of quantum mechanics. It is shown that the measurement-disturbance relations have additional terms for backaction evading quadrature amplifiers and for noiseless quadrature transducers. Several distinctive features appear as a consequence of the noncommutative extension: measurement interactions which are noiseless, and observables which are undisturbed by a measurement, or of independent intervention in ordinary quantum mechanics, may acquire noise, become disturbed by the measurement, or no longer be an independent intervention in noncommutative quantum mechanics. It is also found that there can be states which violate Ozawa's universal noise-disturbance trade-off relation, but verify its noncommutative deformation.

DOI
10.1103/physrevd.90.045023

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Ozawa's measurement-disturbance relation is generalized to a phase-space noncommutative extension of quantum mechanics. It is shown that the measurement-disturbance relations have additional terms for backaction evading quadrature amplifiers and for noiseless quadrature transducers. Several distinctive features appear as a consequence of the noncommutative extension: measurement interactions which are noiseless, and observables which are undisturbed by a measurement, or of independent intervention in ordinary quantum mechanics, may acquire noise, become disturbed by the measurement, or no longer be an independent intervention in noncommutative quantum mechanics. It is also found that there can be states which violate Ozawa's universal noise-disturbance trade-off relation, but verify its noncommutative deformation.
We revisit Ozawa's uncertainty principle (OUP) in the framework of noncommutative (NC) quantum mechanics. We derive a matrix version of OUP accommodating any NC structure in the phase space, and compute NC corrections to lowest order for two measurement interactions, namely the backaction evading quadrature amplifier and noiseless quadrature transducers. These NC corrections alter the nature of the measurement interaction, as a noiseless interaction may acquire noise, and an interaction of independent intervention may become dependent on the object system. However the most striking result is that noncommutativity may lead to a violation of the OUP itself. The NC corrections for the backaction evading quadrature amplifier reveal a new term which may potentially be amplified in such a way that the violation of the OUP becomes experimentally testable. On the other hand, the NC corrections to the noiseless quadrature transducer shows an incompatibility of this model with NC quantum mechanics. We discuss the implications of this incompatibility for NC quantum mechanics and for Ozawa's uncertainty principle.
A universal formulation of the quantum uncertainty regarding quantum indeterminacy, quantum measurement, and its inevitable observer effect is presented with additional focus on the representability of quantum observables over a given state. The operational tangibility of the framework assures that the resultant general relations admit natural operational interpretations and characterisations, and thereby perhaps most importantly, their experimental verifiability. In view of the universal formulation, Heisenberg's original philosophy of the uncertainty principle, most typically exemplified in his famous gamma-ray microscope Gedankenexperiment, is revisited; it is reformulated and restated as a refined no-go theorem, albeit perhaps in a weaker form than was originally intended. The relations entail, in essence as corollaries to their special cases, several previously known relations, including most notably the standard Kennard-Robertson relation, the Arthurs-Kelly-Goodman relations for joint measurement as well as the Ozawa and the Watanabe-Sagawa-Ueda relations for error and error-disturbance.
Heisenberg's uncertainty principle has been understood to set a limitation on measurements; however, the long-standing mathematical formulation established by Heisenberg, Kennard, and Robertson does not allow such an interpretation. Recently, a new relation was found to give a universally valid relation between noise and disturbance in general quantum measurements, and it has become clear that the new relation plays a role of the first principle to derive various quantum limits on measurement and information processing in a unified treatment. This paper examines the above development on the noise–disturbance uncertainty principle in the model-independent approach based on the measurement operator formalism, which is widely accepted to describe a class of generalized measurements in the field of quantum information. We obtain explicit formulae for the noise and disturbance of measurements given by measurement operators, and show that projective measurements do not satisfy the Heisenberg-type noise–disturbance relation that is typical in the gamma-ray microscope thought experiments. We also show that the disturbance on a Pauli operator of a projective measurement of another Pauli operator constantly equals , and examine how this measurement violates the Heisenberg-type relation but satisfies the new noise–disturbance relation.
We argue for an operational requirement that all state-dependent measures of disturbance should satisfy. Motivated by this natural criterion, we prove that in any ddd-dimensional Hilbert space and for any pair of noncommuting operators, AAA and BBB, there exists a set of at least ${2}^{d\ensuremath{-}1}$ zero-noise, zero-disturbance (ZNZD) states, for which the first observable can be measured without noise and the second will not be disturbed. Moreover, we show that it is possible to construct such ZNZD states for which the expectation value of the commutator [A,B][A,B][A,B] does not vanish. Therefore any state-dependent error-disturbance relation, based on the expectation value of the commutator as a lower bound, must violate the operational requirement. We also discuss Ozawa's state-dependent error-disturbance relation in light of our results and show that the disturbance measure used in this relation exhibits unphysical properties. We conclude that the trade-off is inevitable only between state-independent measures of error and disturbance.
Heisenberg's uncertainty principle has been understood to set a limitation on measurements; however, the long standing mathematical formulation established by Heisenberg, Kennard, and Robertson does not allow such an interpretation. Recently, a new relation was found to give a universally valid relation between noise and disturbance in general quantum measurements, and it has become clear that the new relation plays a role of the first principle to derive various quantum limits on measurement and information processing in a unified treatment. This paper examines the above development on the noise-disturbance uncertainty principle in the model independent approach based on measurement operator formalism, which is widely accepted to describe a class of generalized measurements in the field of quantum information. We obtain explicit formulas for noise and disturbance of measurements given by measurement operators, and show that projective measurements do not satisfy the Heisenberg-type noise-disturbance relation that is typical in the gamma-ray microscope thought experiments. We also show that the disturbance on a Pauli operator of a projective measurement of another Pauli operator constantly equals to the square root of 2, and examine how this measurement violates the Heisenberg-type relation but satisfies the new noise-disturbance relation.