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arXiv:1905.09978 · New Journal of Physics

The role of system–meter entanglement in controlling the resolution and decoherence of quantum measurements

Kartik Patekar, Holger F Hofmann

Published Sep 13, 2019• 14 citations
arXivHTMLPDF

Abstract

Abstract Measurement processes can be separated into an entangling interaction between the system and a meter and a subsequent readout of the meter state that does not involve any further interactions with the system. In the interval between these two stages, the system and the meter are in an entangled state that encodes all possible effects of the readout in the form of non-local quantum correlations between the system and the meter. Here, we show that the entanglement generated in the system–meter interaction expresses a fundamental relation between the amount of decoherence and the conditional probabilities that describe the resolution of the measurement. Specifically, the entanglement generated by the measurement interaction correlates both the target observable and the back-action effects on the system with sets of non-commuting physical properties in the meter. The choice of readout in the meter determines the trade-off between irreversible decoherence and measurement information by steering the system into a corresponding set of conditional output states. The Hilbert space algebra of entanglement ensures that the irreversible part of the decoherence is exactly equal to the Hellinger distance describing the resolution achieved in the measurement. We can thus demonstrate that the trade-off between measurement resolution and back-action is a fundamental property of the entanglement generated in measurement interactions.

DOI
10.1088/1367-2630/ab4451

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Abstract Measurement processes can be separated into an entangling interaction between the system and a meter and a subsequent readout of the meter state that does not involve any further interactions with the system. In the interval between these two stages, the system and the meter are in an entangled state that encodes all possible effects of the readout in the form of non-local quantum correlations between the system and the meter. Here, we show that the entanglement generated in the system–meter interaction expresses a fundamental relation between the amount of decoherence and the conditional probabilities that describe the resolution of the measurement. Specifically, the entanglement generated by the measurement interaction correlates both the target observable and the back-action effects on the system with sets of non-commuting physical properties in the meter. The choice of readout in the meter determines the trade-off between irreversible decoherence and measurement information by steering the system into a corresponding set of conditional output states. The Hilbert space algebra of entanglement ensures that the irreversible part of the decoherence is exactly equal to the Hellinger distance describing the resolution achieved in the measurement. We can thus demonstrate that the trade-off between measurement resolution and back-action is a fundamental property of the entanglement generated in measurement interactions.
Information about the internal properties of a system can only be obtained through interactions of the system with an external meter. However, such interactions generally result in entanglement between the system and the meter, making it difficult to trace the measurement result back to a specific value of the physical property in the system. It is therefore possible that the outcomes of quantum measurements depend in a nontrivial manner on the dynamics of the measurement interaction, possibly providing a physical explanation for the role of measurement contexts in quantum mechanics. Here we show that the effects of the measurement interaction on the meter can be described entirely in terms of the quantum coherent system dynamics associated with the back-action on the system. For sufficiently small back-action uncertainties, the physical property of the system is described by a weak value obtained from the Hamilton-Jacobi equation of the back-action dynamics. At higher measurement resolutions, the observed values are determined by quantum interferences between different amounts of back-action. Eigenvalues emerge when the quantum interferences between different back-actions correspond to a Fourier transform in the back-action parameter. We conclude that the values of physical properties obtained in quantum measurements originate from the quantum coherent properties of the back-action dynamics generated by that physical property during an interaction. Measurement outcomes represent elements of the dynamics and cannot be explained by measurement independent elements of reality.
It is not possible to obtain information about the observable properties of a quantum system without a physical interaction between the system and an external meter. This physical interaction is described by a unitary transformation of the joint quantum state of the system and the meter, which means that the information transfer from the system to the meter depends on the initial quantum coherence of the meter. In the present paper, we analyze the measurement interaction in terms of the changes of the meter state caused by the interaction with the system. The sensitivity of the meter can then be defined by evaluating the distinguishability of meter states for different values of the target observable. It is shown that the sensitivity of the meter requires quantum coherences in the generator observable that determines the magnitude of the backaction of the meter on the system. The tradeoff between measurement resolution and backaction is decided by the relation between sensitivity and quantum coherent uncertainty in the external meter. No matter how macroscopic the device used as a meter is, it must necessarily be quantum coherent in the degrees of freedom that interact with the system being measured.
Theory and experiment have not fully resolved the apparent dichotomy , which has agonized physics for the past eighty years: on the one hand, the description of microsystems by quantum mechanics and, on the other, the description of macrosystems by classical dynamics or statistical mechanics. Derivations of the time-irreversible Liouville equation for an open quantum system, based on projecting out its environment, have narrowed the gap between the quantum and classical descriptions. Yet our `classical' intuition continues to be confronted by quantum-mechanical results like the Einstein--Podolsky--Rosen paradox that challenges the classical notion of locality, or the quantum Zeno effect which suggests that the isolation of a system is not the only way to preserve its quantum state. There are two key concepts in any discussion of such issues. The first, which is responsible for the most salient nonclassical properties, is entanglement , that is partial or complete correlation or, more generally, inseparability of the elements comprising a quantum ensemble . Even after their interaction has ceased, this inseparability, originating from their past interaction, can affect the state of one element when another element is subject to a nonunitary action, such as its measurement, tracing- out, or thermalization. The second key concept is decoherence of open quantum systems, which is the consequence of their entanglement with their environment, a `meter' or a thermal `reservoir', followed by the tracing-out of the latter. Despite new insights into entanglement and decoherence, there are still no complete, unequivocal answers to the fundamental questions of the transition from quantal to classical behaviour: how do irreversibility and classicality emerge from unitarity as systems and their environments become increasingly complex ? At what stage does system--meter entanglement give rise to a classical readout of the meter? Is there an upper limit on the size or complexity of systems displaying entanglement? Major developments have opened new vistas into controlled entanglement , which is the resource of quantum information processing. However, these developments have mainly focused on ensembles of simple two- and three-level systems that are thoroughly isolated from their environments. Treatments of coherence in quantum computing have mostly assumed that only a single or a few the elements of the quantum ensemble may simultaneously undergo an uncontrolled intervention---a quantum error. Decoherence-control protocols for more general types of errors are still incomplete. In order to resolve the outstanding issues of the quantum--classical transition, and study the control of entanglement and decoherence without the foregoing restrictions, we must venture into the domain of Quantum Complex Systems (QUACS), either consisting of a large number of inseparable elements or having many coupled degrees of freedom. Our conviction is that fundamental understanding and manipulation of entanglement within QUACS or their entanglement with the environment or a meter, call for the creation of a new conceptual framework or paradigm, that would encompass phenomena common to cold atoms in laser fields, large molecules, Josephson junctions, quantum gases and solids, with the view of employing these systems for quantum information processing and computing. Progress within this paradigm should allow us to answer the questions: does entanglement play an essential role in the evolution of large collections of complex systems? What are the size and complexity limits of systems and ensembles still controllable by an external intervention? What are the most appropriate decoherence protection schemes and control algorithms? This special issue addresses the challenge of understanding in depth and manipulating the basic quantum properties of optical, atomic, molecular and condensed-matter QUACS, and large ensembles thereof. In view of the interdisciplinary character of this issue, we find it expedient to look at the presented articles not only according to the physical objects they describe, but also from a unifying standpoint. Several unifying themes may be discerned: memory effects in dephasing and relaxation; protection of quantum information by its distribution throughout the ensemble; control of decoherence and entanglement by time-dependent interventions or measurements; manifestations of entanglement in scattering off statistical ensembles; measurment-induced dynamics. The concerted discussion of the topics outlined above should help us advance the new paradigm that addresses our abilities to diagnose and manipulate the entangled states of complex quantum objects and their robustness against decoherence. These abilities are required for quantum information (QI) applications in matter--wave interferometry in molecular, semiconducting or superconducting systems. On the fundamental level, this book may help establish the notion of dynamical information exchange between quantum systems and chart in detail the route from unitarity to classicality. Further developments within the outlined paradigm should yield novel, advantageous QI processing schemes and high-sensitivity interferometers, owing to decoherence suppression and effective control of many degrees of freedom. In the long run, these strategies may prove to be the first step towards the (hitherto unattempted) use of entanglement in nanotechnology, metrology and chemistry, with potentially remarkable novel applications.
Measurements map the value of a target observable onto a meter shift, resulting in a meter readout that combines the initial statistics of the meter state with the quantum statistics of the target observable. Even in the limit of weak measurement interactions, some information about the fluctuations of the target observable can be extracted from the change in the readout fluctuations caused by the measurement interaction. Here, we apply the Heisenberg picture to analyze the changes in the meter readout statistics caused by sufficiently weak measurement interactions, including the effects of nonlinearities in the meter response. When additional information is obtained in a subsequent measurement of the system, the meter fluctuations are modified based on the postselected statistics of the target observable. In addition, our analysis reveals a direct modification of the meter fluctuations due to the dependence of the postselection probability on the dynamics induced by the meter in the measurement interaction. We point out that the quantum formalism makes it difficult to distinguish this dynamic term from the physical fluctuations of the target observable and stress the importance of distinguishing between genuine conditional fluctuations of the target observable and the dynamic pseudovariance associated with the measurement back-action.
Abstract Our aim is to make a step toward clarification of foundations for the notion of entanglement (both physical and mathematical) by representing it in the conditional probability framework. In Schrödinger’s words, this is entanglement of knowledge which can be extracted via conditional measurements. In particular, quantum probabilities are interpreted as conditional ones (as, e.g., by Ballentine). We restrict considerations to perfect conditional correlations (PCC) induced by measurements (‘EPR entanglement’). Such entanglement is coupled to the pairs of observables with the projection type state update as the back action of measurement. In this way, we determine a special class of entangled states. One of our aims is to decouple the notion of entanglement from the compound systems. The rigid association of entanglement with the state of a few body systems stimulated its linking with quantum nonlocality (‘spooky action at a distance’). However, already by Schrödinger entanglement was presented as knotting of knowledge (about statistics) for one observable A with knowledge about another observable B.
The dynamics of quantum measurements defines a precise relation between the information gained about one physical property of a system and the observable changes in another physical property of the same system. Here, we express this relation in terms of the Hilbert space superpositions of the corresponding eigenstates and show how the probability of an observable physical change can be obtained from the Bayesian update of the probabilities associated with the information obtained in the measurement. Our analysis demonstrates that the superposition principle provides the tightest possible expression of the trade-off between information and back action in a quantum measurement.