We study macroscopic observables defined as the total value of a physical quantity over a collection of quantum systems. We show that previous results obtained for an infinite ensemble of identically prepared systems lead to incorrect conclusions for finite ensembles. In particular, exact measurement of a macroscopic observable significantly disturbs the state of any finite ensemble. However, we show how this disturbance can be made arbitrarily small when the measurements are of finite accuracy. We demonstrate a general trade-off between state disturbance and measurement coarseness as a function of the size of the ensemble. Using this trade-off, we show that the histories generated by any sequence of finite accuracy macroscopic measurements always generate a consistent family in the absence of large-scale entanglement for sufficiently large ensembles. Hence, macroscopic observables behave ``classically'' provided that their accuracy is coarser than the quantum correlation length scale of the system. The role of these observable is also discussed in the context of NMR quantum information processing and bulk ensemble quantum state tomography.
We study macroscopic observables defined as the total value of a physical quantity over a collection of quantum systems. We show that previous results obtained for an infinite ensemble of identically prepared systems lead to incorrect conclusions for finite ensembles. In particular, exact measurement of a macroscopic observable significantly disturbs the state of any finite ensemble. However, we show how this disturbance can be made arbitrarily small when the measurements are of finite accuracy. We demonstrate a general trade-off between state disturbance and measurement coarseness as a function of the size of the ensemble. Using this trade-off, we show that the histories generated by any sequence of finite accuracy macroscopic measurements always generate a consistent family in the absence of large-scale entanglement for sufficiently large ensembles. Hence, macroscopic observables behave ``classically'' provided that their accuracy is coarser than the quantum correlation length scale of the system. The role of these observable is also discussed in the context of NMR quantum information processing and bulk ensemble quantum state tomography.
Assuming a well-behaving quantum-to-classical transition, measuring large quantum systems should be highly informative with low measurement-induced disturbance, while the coupling between system and measurement apparatus is fairly simple and weak. Here, we show that this is indeed possible within the formalism of quantum mechanics. We discuss an example of estimating the collective magnetization of a spin ensemble by simultaneous measuring three orthogonal spin directions. For the task of estimating the direction of a spin-coherent state, we find that the average guessing fidelity and the system disturbance are nonmonotonic functions of the coupling strength. Strikingly, we discover an intermediate regime for the coupling strength where the guessing fidelity is quasioptimal, while the measured state is almost not disturbed.
We consider finite macroscopic systems, i.e. systems of large but finite degrees of freedom, which we believe are poorly understood as compared with small systems and infinite systems. We focus on pure states that do not have the "cluster property". Such a pure state is entangled macroscopically, and is quite anomalous in view of many-body physics because it does not approach any pure state in the infinite-size limit. However, we often encounter such anomalous states when studying finite macroscopic systems, such as quantum computers with many qubits and finite systems that will exhibit symmetry breaking in the infinite-size limit. We study stabilities of such anomalous states in general systems. In contrast to the previous works, we obtain general and universal results, by making full use of the locality of the theory. Using the general results, we discuss roles of anomalous states in quantum computers, and the mechanism of emergence of a classical world from quantum theory.
The departure of the quantum world from the classical regime is captured through the observation of nonclassical correlations manifested in the behaviors of subatomic systems. However, once the dimension of the system becomes substantially large, the quantum behavior begins to decline, and subsequently it starts following the predictions of classical physics. The macroscopic limit at which such quantum-to-classical transition occurs remains one of the long-standing questions in the foundations of quantum theory. There are evidences that the macroscopic limit to which the quantumness of a system persists depends on the degree of interaction due to the measurement processes. For instance, with a system having a considerably large Hilbert-space dimension, if the measurement is performed in such a way that the outcome of the measurement only reveals a coarse-grained version of the information about the individual level of the concerned system then the disturbance due to the measurement process can be considered to be infinitesimally small. Based on such coarse-grained measurement the dependence of Bell inequality violation on the degree of coarsening has already been investigated [H. Jeong, Y. Lim, and M. S. Kim, Phys. Rev. Lett. 112, 010402 (2014)]. In this paper, we first capture the fact that, when local realism is taken to be the defining notion of classicality, the effect of the degree of coarsening on the downfall of quantumness of a macroscopic entangled state can be compensated by testing a Bell inequality of a higher number of settings from a family of symmetric Bell inequalities if the number of settings is odd. However, in contrast, we show that such compensation cannot be seen when we witness such quantum-to-classical transition using symmetric Bell inequalities having an even number of settings. Finally, complementing the above result, we show that, when unsteerability is taken as the classicality, for both odd and even numbers of settings the degree of coarsening at which the quantum-to-classical transition occurs can be consistently pushed ahead by testing a linear steering inequality of a higher number of settings and observing its violation. We further extend our treatment for mixed macroscopic entangled states.
Conceptually different from the decoherence program, we present a novel theoretical approach to macroscopic realism and classical physics within quantum theory. It focuses on the limits of observability of quantum effects of macroscopic objects, i.e., on the required precision of our measurement apparatuses such that quantum phenomena can still be observed. First, we demonstrate that for unrestricted measurement accuracy, no classical description is possible for arbitrarily large systems. Then we show for a certain time evolution that under coarse-grained measurements, not only macrorealism but even classical Newtonian laws emerge out of the Schrödinger equation and the projection postulate.
This chapter presents the measurement problem from the point of view of the thermal interpretation of quantum physics. Everything physicists measure is measured in a thermal environment, for which statistical thermodynamics is relevant. The thermal interpretation agrees with how one interprets measurements in thermodynamics, the macroscopic part of quantum physics, derived via statistical mechanics. By its very construction, the thermal interpretation naturally matches the classical properties of our quantum world: The thermal interpretation assigns states-and a realistic interpretation for them-to individual quantum systems, in a way that large quantum systems are naturally described by classical observables. Since measurement devices are large quantum systems, this is the key to solving the measurement problem. We discuss the role played by macroscopic systems and the weak law of large numbers in getting readings with small uncertainty. Since quantum physics makes many deterministic predictions, for example regarding observed spectra, but also assertions about probabilities, we distinguish deterministic and statistical measurements. To clarify the meaning of the concept of measurement, we postulate (in Section 11.1) a measurement principle that defines what it means in the thermal interpretation to measure a quantity with a specified uncertainty. This turns the measurement problem from a philosophical riddle into a scientific problem in the domain of quantum statistical mechanics, namely how the quantum dynamics correlates macroscopic readings from an instrument with properties of the state of a measured microscopic system. The essence of scientific practice is the reproducibility of measurements, discussed in Section 11.2. Unlike in traditional interpretations, single, nonreproducible observations do not count as measurements, since this would violate the reproducibility of measurements-the essence of scientific practice. As a consequence, the measurement of a Hermitian quantity A is regarded as giving an uncertain value approximating the q-expectation {A} rather than (as tradition wanted to have it) as an exact revelation of an eigenvalue of A. This difference is most conspicuous in the interpretation of single discrete microscopic events. Except in very special circumstances, these are not reproducible. Thus, they have no scientific value in themselves and do not constitute measurement results. Scientific value is, however, in ensembles of such observations, which result in approximate measurements of q-probabilities and q-expectations. Sections 11.3 and 11.4 distinguish deterministic and statistical measurements, depending on whether a single observation is reproducible, and discuss the role played by macroscopic systems and the weak law of large numbers in getting readings with small uncertainty. How to measure the q-probabilities of events, described in terms of POVMs, is discussed in Section 11.5. Section 11.6 poses the measurement problem as a problem of the statistical mechanics of measurement devices, and discuss recent work by Allahverdyan, Balian, and Nieuwenhuizen on its resolution. Section 11.7 discusses the problem how, in the thermal interpretation, discrete events can arise, although qexpectations are by their very nature continuous. This is resolved by pointing to work by Breuer and Petruccione on piecewise deterministic processes (PDP). Section 11.8 sheds additional light on the origin of discrete events by an analysis of the role dissipative bistability plays in obtaining discreteness in classical physics.
We follow Kofler and Brukner (2007 Phys. Rev. Lett. 99 180403) in studying the conditions under which a classical picture emerges from the results of not too accurate quantum measurements made on large macroscopic objects. We show that for such objects, consisting of a large number of microscopic elements obeying quantum laws, the Central Limit theorem guarantees the existence of classical values for collective variables, even if the corresponding operators do not commute. Owing to localisation of the overall wave function in any chosen representation, these values can be measured to a small relative error without significantly altering the state of the object. We study a simple model, which includes a rudimentary observer capable of detecting in the coordinate space the position of a macroscopic pointer. The pointer can be employed to measure such quantities, not directly accessible to the observer, as linear or angular momenta. A purely classical picture arises provided the measurements are made on macroscopic objects. Results of measurements, made on small quantum objects, cannot be predicted with certainty, but acquire certain objectivity when encoded in macroscopic pointers’ positions accessible to all observers. Our estimates show that the classical conditions could, in principle, be realised for systems with number of constituent parts of the order of the Avogadro constant. It is possible that the approach captures the essential features of the quantum-to-classical behaviour, although its extension to more realistic systems is likely to be required.