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arXiv:1503.02114

Nonseparability and simultaneous readability

Riuji Mochizuki

Published Mar 7, 2015• 1 citation
arXivHTMLPDF

Abstract

In this study, we investigate quantum nonseparability between an observed system and a measuring apparatus, or multiple measuring apparatuses. We show that the physical meaning of the outcome of the measuring apparatus obtained by weak measurement with a post-selection differs critically from that without any post-selection. In this study, the nonseparability plays the essential role, which is shown to be the same in a simultaneous conventional von Neumann-type measurement of multiple observables. From this viewpoint, we suggest a new concept, known as {\it simultaneous readability}, which is the possibility that multiple measuring apparatuses will give the proper information of the observed system simultaneously. Next, we show that different components of the spin of an electron are not simultaneously measurable even if it has EPR correlation with another electron.

DOI
10.48550/arxiv.1503.02114

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In this study, we investigate quantum nonseparability between an observed system and a measuring apparatus, or multiple measuring apparatuses. We show that the physical meaning of the outcome of the measuring apparatus obtained by weak measurement with a post-selection differs critically from that without any post-selection. In this study, the nonseparability plays the essential role, which is shown to be the same in a simultaneous conventional von Neumann-type measurement of multiple observables. From this viewpoint, we suggest a new concept, known as {\it simultaneous readability}, which is the possibility that multiple measuring apparatuses will give the proper information of the observed system simultaneously. Next, we show that different components of the spin of an electron are not simultaneously measurable even if it has EPR correlation with another electron.
The possibility of performing simultaneous measurements in quantum mechanics is investigated in the context of the Curie-Weiss model for a projective measurement. Concretely, we consider a spin--12\frac{1}{2}21​ system simultaneously interacting with two magnets, which act as measuring apparatuses of two different spin components. We work out the dynamics of this process and determine the final state of the measuring apparatuses, from which we can find the probabilities of the four possible outcomes of the measurements. The measurement is found to be nonideal, as (i) the joint statistics do not coincide with the one obtained by separately measuring each spin component, and (ii) the density matrix of the spin does not collapse in either of the measured observables. However, we give an operational interpretation of the process as a generalized quantum measurement, and show that it is fully informative: The expected value of the measured spin components can be found with arbitrary precision for sufficiently many runs of the experiment.
In an EPR setup, simultaneous measurement of two non-commuting observables is considered at station A and, possibly after a time delay, at station B . At each station one apparatus measures the z -component of a spin, while a second one measures the x -component of the same spin simultaneously. The dynamics of the measurements is solved explicitly. It is shown that no violations of Bell inequalities occur in Bell-like correlation experiments, and that, in fact, experiments of this type are subject to even stronger constraints. This is caused by the mutual influence of the two apparatuses coupled to the same spin, an effect which occurs at both stations.
The process of measurement plays an important role in quantum theory. Measurements can be direct, if the physical system S interacts directly with the measurement apparatus M (as we have assumed until now), or indirect. In the latter case, the physical system first interacts with an ancillary system B , which may have a space of states that is very different from that of S , for instance much larger; after this interaction has finished, M is used to perform a measurement on B , without any direct interaction with A . Because S is then “protected” from any direct interaction with the measurement apparatus, the state of S is not necessarily strongly modified, and may even be only weakly affected. In both cases, the process implies entanglement between several physical systems. In this chapter, we study how this entanglement is created and used for measurements as well as the notion of weak and continuous measurements. These questions play an important role in several of the interpretations of quantum mechanics that we discuss in Chapter 10. Direct measurements The Von Neumann model of quantum measurement [4] provides a general frame for describing the process in terms of correlations appearing (or disappearing) in the state vector associated with the whole system S + M . In this model, the two systems S and M are initially described by a product state ∣ψ 0 〈 and interact during the time of measurement, so that they become entangled; they then reach a final state ∣ψ′〈 and do not interact anymore.
In is shown that when two observers carry out a simultaneous measurement on a pair of spin-? particles in a "singlet" state a possibility exists for an outcome that lacks physical meaning. More specifically, despite the fact that the two commuting operators formally possess simultaneous eigenvectors it is not possible for physical reasons these eigenvectors to exist simultaneously. It is pointed out that the possibility for such non-physical outcome is observed only in the case of "singlet" state which puts into question the physical meaning of such state, and following from it exotic notions such as "non-locality". This hints that probably QM in its present form should be reconsidered with regard to its limitations.
Weak measurement is a potentially powerful tool for studying post-selected quantum systems. The original approach to weak measurement requires a different von Neumann interaction between the quantum system and measurement pointer for weak values of every different observable. When studying weak values of properties of more than one particle, which is necessary for studying entanglement, these interactions can range from difficult to impossible to implement experimentally. For example, the measurement of properties on a pair of optical photons requires a strong Kerr-like nonlinearity at the two-photon level. In a recent work, we described a new method for extracting weak values of operators that can be expressed as products of two local, single-particle operators by studying the correlations in experimentally feasible local, single-particle measurements. In the present work, that theoretical approach is generalized so that weak values of products of N local, single-particle operators can be extracted from the correlations in N single-particle measurements.
The fact that not all quantum observables are jointly measurable is one of the major differences between quantum and classical theory. In the former, non-commuting observables can only be simultaneously measured with limited precision. We report on an experimental implementation of such a simultaneous measurement of two non-commuting observables based on the framework of weak von Neumann measurements. We use a photonic setup where the polarisation degree of freedom acts as the system and the two components of the transversal position correspond to the pointers of the measurement apparatuses. In addition, the theory shows that these weak measurements demonstrate a counter-intuitive non-monotonicity: weaker measurements can potentially reveal more information than stronger ones.