Soundness and completeness of quantum root-mean-square errors

Abstract

Defining and measuring the error of a measurement is one of the most fundamental activities in experimental science. However, quantum theory shows a peculiar difficulty in extending the classical notion of root-mean-square (rms) error to quantum measurements. A straightforward generalization based on the noise-operator was used to reformulate Heisenberg's uncertainty relation on the accuracy of simultaneous measurements to be universally valid and made the conventional formulation testable to observe its violation. Recently, its reliability was examined based on an anomaly that the error vanishes for some inaccurate measurements, in which the meter does not commute with the measured observable. Here, we propose an improved definition for a quantum generalization of the classical rms error, which is state-dependent, operationally definable, and perfectly characterizes accurate measurements. Moreover, it is shown that the new notion maintains the previously obtained universally valid uncertainty relations and their experimental confirmations without changing their forms and interpretations, in contrast to a prevailing view that a state-dependent formulation for measurement uncertainty relation is not tenable.

Uncertainty Relation for Errors Focusing on General POVM Measurements with an Example of Two-State Quantum Systems

“…As is presented in later publication, the framework [ 47 ] is found to entail Ozawa’s relation [ 9 ], as well as its recent modification [ 12 ]. More specifically, an enhancement of the relation ( 8 ) to accommodate joint measurability is available, which is found to be tighter than Ozawa’s relations.…”

“…A popular model for the description of quantum measurement has been the indirect measurement scheme, which explicitly considers an external quantum system of an ancillary meter device in addition to the original quantum system of interest, thereby allowing for a physically intuitive representation of an otherwise obscure measurement process. The Arthurs–Kelly–Goodman relations [ 6 , 7 ] and the more recent Ozawa relations [ 8 , 9 ], along with their refinement [ 10 ] and modifications [ 11 , 12 ], are among the most notable formulations that are founded on this model, whereby their formulation of error (and disturbance) defined for measurements associated with positive-operator valued measures (POVMs) over the real field admits an intelligible representation. Apart form these, uncertainty relations have also been framed on the foundation of estimation theory [ 13 , 14 ] in addition to having been formulated from a measure-theoretic viewpoint [ 15 , 16 , 17 ].…”

Variational Circuit Compiler for Quantum Error Correction

“…The minimization can be accomplished by any optimization algorithm, such as simple gradient descent or the imaginary time evolution [45,46] as we presently review.…”

“…Different from the conventional variational quantum eigensolver cases where the energy spectra are unknown, here the energy spectra are known and can be altered. Next, we consider a set of parameterized ansatz circuits and make use of the variational imaginary-time-evolution method [45,46] to find the ground state, thus discovering the encoding circuit. The ansatz circuit can be tailored to meet specific requirements of any hardware system and any optimization target.…”

Tunneling through an eternal traversable wormhole

“…Later, several generalizations of the SYK model have been introduced to study different physics [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22], including tunneling spectroscopy [7][8][9][10] for generalizations with U(1) symmetry [11][12][13]. Moreover, the quantum simulation of the SYK model [23] has been performed in NMR systems [24] and there are several other proposals for realizing the SYK model in different experimental systems [25][26][27][28].…”