Quantum Measurement, Information, and Completely Positive Maps

Axiomatic approach to measurement theory is developed. All the possible statistical properties of apparatuses measuring an observable with nondegenerate spectrum allowed in standard quantum mechanics are characterized.Comment: 9 pages, LaTeX, invited talk at 5th Int'l Conf. on Quantum Communication, Measurement and Computing (Capri, Iatly, July 3--8, 2000

Measurability and Computability

The conceptual relation between the measurability of quantum mechanical observables and the computability of numerical functions is re-examined. A new formulation is given for the notion of measurability with finite precision in order to reconcile the conflict alleged by M. A. Nielsen [Phys. Rev. Lett. 79, 2915 (1997)] that the measurability of a certain observable contradicts the Church-Turing thesis. It is argued that any function computable by a quantum algorithm is a recursive function obeying the Church-Turing thesis, whereas any observable can be measured in principle.Comment: 8 pages, RevTe

Soundness and completeness of quantum root-mean-square errors

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 for 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 imprecise 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 precise 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.Comment: 11 pages, late

Transfer principle in quantum set theory

In 1981, Takeuti introduced quantum set theory as the quantum counterpart of Boolean valued models of set theory by constructing a model of set theory based on quantum logic represented by the lattice of closed subspaces in a Hilbert space and showed that appropriate quantum counterparts of ZFC axioms hold in the model. Here, Takeuti's formulation is extended to construct a model of set theory based on the logic represented by the lattice of projections in an arbitrary von Neumann algebra. A transfer principle is established that enables us to transfer theorems of ZFC to their quantum counterparts holding in the model. The set of real numbers in the model is shown to be in one-to-one correspondence with the set of self-adjoint operators affiliated with the von Neumann algebra generated by the logic. Despite the difficulty pointed out by Takeuti that equality axioms do not generally hold in quantum set theory, it is shown that equality axioms hold for any real numbers in the model. It is also shown that any observational proposition in quantum mechanics can be represented by a corresponding statement for real numbers in the model with the truth value consistent with the standard formulation of quantum mechanics, and that the equality relation between two real numbers in the model is equivalent with the notion of perfect correlation between corresponding observables (self-adjoint operators) in quantum mechanics. The paper is concluded with some remarks on the relevance to quantum set theory of the choice of the implication connective in quantum logic.Comment: 25 pages, to appear in JS

Uncertainty principle for quantum instruments and computing

Recently, universally valid uncertainty relations have been established to set a precision limit for any instruments given a disturbance constraint in a form more general than the one originally proposed by Heisenberg. One of them leads to a quantitative generalization of the Wigner-Araki-Yanase theorem on the precision limit of measurements under conservation laws. Applying this, a rigorous lower bound is obtained for the gate error probability of physical implementations of Hadamard gates on a standard qubit of a spin 1/2 system by interactions with control fields or ancilla systems obeying the angular momentum conservation law.Comment: 13 pages, Revtex

Comment on "Proof of Heisenberg's error-disturbance principle"

Recently, Kosugi [arXiv:1504.03779v2 [quant-ph]] argued that Heisenberg's error-disturbance relation (EDR) must be interpreted as being between the resolution, the preparational error for the post-measurement observable, and the disturbance. He further claimed that Heisenberg's EDR can be proven to hold true in general, when the meter observable is modified as one of its functions. Here, some comments are given to suggest that the above claims are not supported.Comment: 1 pag

Error-disturbance relations in mixed states

Heisenberg's uncertainty principle was originally formulated in 1927 as a quantitative relation between the "mean error" of a measurement of one observable and the disturbance thereby caused on another observable. Heisenberg derived this famous relation under an additional assumption on quantum measurements that has been abandoned in the modern theory, and its universal validity was questioned in a debate on the sensitivity limit to gravitational-wave detectors in 1980s. A universally valid form of the error-disturbance relation was shown to be derived in the modern framework of general quantum measurements in 2003. We have experienced a considerable progress in theoretical and experimental study of error-disturbance relations in the last decade. In 2013 Branciard showed a new stronger form of universally valid error-disturbance relations, one of which is proved tight for spin measurements carried out in "pure" states. Nevertheless, a recent information-theoretical study of error-disturbance relations has suggested that Branciard relations can be considerably strengthened for measurements in mixed states. Here, we show a method for strengthening Branciard relations in mixed states and derive several new universally valid and stronger error-disturbance relations in mixed states. In particular, it is proved that one of them gives an ultimate error-disturbance relation for spin measurements, which is tight in any state. The new relations will play an important role in applications to state estimation problems including quantum cryptographic scenarios.Comment: 11 pages, late

Heisenberg's original derivation of the uncertainty principle and its universally valid reformulations

Heisenberg's uncertainty principle was originally posed for the limit of the accuracy of simultaneous measurement of non-commuting observables as stating that canonically conjugate observables can be measured simultaneously only with the constraint that the product of their mean errors should be no less than a limit set by Planck's constant. However, Heisenberg with the subsequent completion by Kennard has long been credited only with a constraint for state preparation represented by the product of the standard deviations. Here, we show that Heisenberg actually proved the constraint for the accuracy of simultaneous measurement but assuming an obsolete postulate for quantum mechanics. This assumption, known as the repeatability hypothesis, formulated explicitly by von Neumann and Schr\"{o}dinger, was broadly accepted until the 1970s, but abandoned in the 1980s, when completely general quantum measurement theory was established. We also survey the author's recent proposal for a universally valid reformulation of Heisenberg's uncertainty principle under the most general assumption on quantum measurement.Comment: v3:10 peges, typos corrected. v2:10 pages, changed title, based on an invited talk at the Discussion Meeting on Quantum Measurements, Bangalore, India, 22-24 October 201

Realization of Measurement and the Standard Quantum Limit

This paper, following [M. Ozawa, Phys. Rev. Lett. 60, 385 (1988)], reports a refutation of the claim that for monitoring the position of a free mass such as gravitational-wave interferometers the sensitivity is limited by the so called standard quantum limit (SQL) due to the uncertainty principle. The latest proof of the SQL is analyzed to revleal an unsupported assumption on quantum measurements. Quantum measurement theory is introduced to give a general criterion for physically realizable measurements in quantum mechanics. A theory of approximate position measurements is developed to obtain a rigorous condition for the SQL and also to show that a precise position measurement can leave the object in an arbitrary family of states independent of the input state. This concludes that Yuen's proposal of breaking the SQL by a contractive state measurement, a measurement of the position leaving the free mass in a state with the position uncertainty decreasing in time, is physically realizable in principle. To enforce this conclusion, a model for error-free position measurement that leaves the object in a contractive state is constructed with a solvable Hamiltonian for measuring interaction. Finally, this model is shown to break the SQL with arbitrary accuracy.Comment: 24 pages, based on the author's invited address at a NATO Advanced Research Workshop on Squeezed and Non-Classical Light, held January 25-29, 1988, in Cortina d' Ampezzo, Italy. in Squeezed and Nonclassical Light (edited by P. Tombesi and R. Pike), NATO ASI Series Volume 190 (Plenum, New York, 1989), pp. 263-28

Quantum Limits of Measurements and Uncertainty Principle

In this paper, we show how the Robertson uncertainty relation gives certain intrinsic quantum limits of measurements in the most general and rigorous mathematical treatment. A general lower bound for the product of the root-mean-square measurement errors arising in joint measurements of noncommuting observables is established. We give a rigorous condition for holding of the standard quantum limit (SQL) for repeated measurements, and prove that if a measuring instrument has no larger root-mean-square preparational error than the root-mean-square measurement errors then it obeys the SQL. As shown previously, we can even construct many linear models of position measurement which circumvent this condition for the SQL.Comment: 15 pages, presented at the International Workshop on Quantum Aspects of Optical Communications, C.N.R.S. Paris, France, November 26-28, 199