The impossibility of measuring non-commuting quantum mechanical observables is one of the most fascinating consequences of the quantum mechanical postulates. Hence, to date the investigation of quantum measurement and projection is a fundamentally interesting topic. We propose to test the concept of weak measurement of non-commuting observables in mesoscopic transport experiments, using a quasiprobabilistic description. We derive an inequality for current correlators, which is satisfied by every classical probability but violated by high-frequency fourth-order cumulants in the quantum regime for experimentally feasible parameters. We further address the creation and detection of entanglement in solid-state electronics, which is of fundamental importance for quantum information processing. We propose a general test of entanglement based on the violation of a classically satisfied inequality for continuous variables by 4th or higher order quantum correlation functions. Our scheme provides a way to prove the existence of entanglement in a mesoscopic transport setup by measuring higher order cumulants without requiring the additional assumption of single charge detection.
The impossibility of measuring non-commuting quantum mechanical observables is one of the most fascinating consequences of the quantum mechanical postulates. Hence, to date the investigation of quantum measurement and projection is a fundamentally interesting topic. We propose to test the concept of weak measurement of non-commuting observables in mesoscopic transport experiments, using a quasiprobabilistic description. We derive an inequality for current correlators, which is satisfied by every classical probability but violated by high-frequency fourth-order cumulants in the quantum regime for experimentally feasible parameters. We further address the creation and detection of entanglement in solid-state electronics, which is of fundamental importance for quantum information processing. We propose a general test of entanglement based on the violation of a classically satisfied inequality for continuous variables by 4th or higher order quantum correlation functions. Our scheme provides a way to prove the existence of entanglement in a mesoscopic transport setup by measuring higher order cumulants without requiring the additional assumption of single charge detection.
The impossibility of measuring noncommuting quantum mechanical observables is one of the most fascinating consequences of the quantum mechanical postulates. Hence, to date the investigation of quantum measurement and projection is a fundamentally interesting topic. We propose to test the concept of weak measurement of noncommuting observables in mesoscopic transport experiments, using a quasiprobabilistic description. We derive an inequality for current correlators, which is satisfied by every classical probability but violated by high-frequency fourth-order cumulants in the quantum regime for experimentally feasible parameters.
Quantum transport in mesoscopic conductors is essentially governed by the laws of quantum mechanics. One of the major open questions of quantum mechanics is what happens if noncommuting observables are measured simultaneously. Since current operators at different times do not commute, the high-frequency correlation functions of the current are realization of this fundamental quantum question. We formulate this problem in the context of measurements of finite-frequency current cumulants in a general quantum point contact, which are the subject to ongoing experimental effort. To this end, we present two models of detectors that correspond to a weak time-resolved measurement of the electronic current in a mesoscopic junction. In both cases, the backaction of the detector leads to observable corrections to the current correlations functions involving the so-called noise susceptibilities. As a result, we propose a reinterpretation of environmental corrections to the finite-frequency cumulants as inevitable effect resulting from basic quantum mechanical principles. Finally we make concrete predictions for the temperature, voltage, and frequency dependence of the third cumulant, which could be verified directly using current experimental techniques.
Generalized quantum measurement schemes are described by positive operator-valued measures going beyond the projection postulate, which predicts the instantaneous collapse of the systems wave function. This allows to take the noninvasive limit and investigate the correlations of such weak measurements which enables the observation of non-commuting observables within the same system. We propose a scheme in which the detector is coupled to the measured system for a finite time, as it is the case in many real setups. This leads to non-Markovian effects appearing by memory functions which are related to symmetric and antisymmetric correlators of the detector variables [12]. We investigate these functions addressing the role of equilibrium and non-equilibrium detectors and how they differ from and could realize the standard Markovian measurement respectively. The latter scheme leads to the symmetrized operator order (aka Keldysh ordering), which is widely used in quantum measurement discussions. We show that the non-Markovian measurement scheme yields information beyond the standard approach, allowing e.g. for the proof of the non classical nature of a system (quasiprobability statistics) by second-order correlation functions [13].We further propose setups in mesoscopic electronic circuits to realise those concepts. One possibility is to use two double quantum dots coupled to a common quantum system. The detectors cross correlations are read out and by tuning the dot parameters, it is possible to explore the non-Markovian nature of the measurement setup.
The development of solid-state quantum technologies requires the understanding of quantum measurements in interacting, nonisolated quantum systems.In general, a permanent coupling of detectors to a quantum system leads to memory effects that have to be taken into account in interpreting the measurement results.In this thesis we develop a microscopic framework for correlations in non-Markovian weak measurements.A special point of interest are the different orders of quantum mechanical operators that appear naturally in such a scheme.First, we analyze a generic setup of two detectors coupled to a quantum system and derive a compact formula in the weak-measurement limit that interpolates between an instantaneous (text-book type) and almost continuous -detector dynamics-dependent -measurement.A quantum memory effect that we term system-mediated detector-detector interaction is crucial to observe noncommuting observables simultaneously.A response function of the observed system variables introduces an antisymmetric operator order to the total result.We propose a mesoscopic double-dot detector setup in which the memory effect is tunable and that can be used to explore the transition to non-Markovian quantum measurements experimentally.The response and noise functions which we find to describe the system-mediated detector-detector interaction are analyzed in detail.They are compared to the phenomenological "equilibrium order" approach for non-Markovian measurements which was introduced in Ref. [1].We give an overview of the broad spectrum of results beyond "equilibrium order" in the microscopic model in a compendium of different detectors and discuss general mathematical properties of the detector functions.As the quantum measurement problem is not banned from our theory by operation in the weak measurement limit, but deferred from the system of interest to the detectors, we comment on the influence of the detector readout procedure on the results.Finally, we combine the phenomenological proposal for nonsymmetrized correlations in quantum noninvasive measurements [1] and our microscopic derivation of the systemmediated detector-detector interaction of second order to investigate higher-order correlations of non-Markovian weak quantum measurements.The general third-order correlator is introduced and the underlying physical processes are identified and discussed.We find a more complex behavior than in second-order correlations with four types of system operator orders which we denote third cumulant, noise susceptibility I and II and second-order susceptibility.We introduce a convenient diagrammatic description which we illustrate by an example of measuring the third-order correlations of the tunnel current through a barrier.v I would like to express my deep gratitude to all the people who contributed to this thesis.Special thanks to my supervisor Wolfgang Belzig for offering me the opportunity to work in his research group, for the numerous helpful discussions in which I learned a lot and which gave impetus to this work again and again, for being always patient and an excellent guidance during my studies.I also wish to express great thanks to my other collaborators, Adam Bednorz, whose ideas are an important part of the foundation of this work and who was always ready to comment new results and bring things forward, and Christoph Bruder, who was a great help in thinking through results and formulating them in an understandable manner while preparing parts of this work for publication.Many thanks to Maxim Trushin, who was willing to discuss our weak measurement formalism with the example of a graphene coherence measurement, and who contributed the initial idea of a optical detection of pseudospin superposition states as well as the description of the carrier statistics using the Liouville-von Neumann equation and very important parts of the discussion of the results.In this context I also want to acknowledge valuable discussions with Daniele Brida about the optical detection scheme and with Bertrand Reulet about third-order current correlations.Finally, I wish to thank the current and former members of the Quantum Transport Group
Bipartite quantum entangled systems can exhibit measurement correlations that violate Bell inequalities, revealing the profoundly counter-intuitive nature of the physical universe. These correlations reflect the impossibility of constructing joint probability distribution for all values of all the different properties observed in Bell inequality tests. Physically, the impossibility of measuring such distribution experimentally, as set of relative frequencies, is due to the quantum back-action of projective measurements. Weakly coupling to quantum probe, however, produces minimal back-action, and so enables weak measurement of the projector of one observable, followed by projective measurement of non-commuting observable. By this technique it is possible to empirically measure probabilities for all of the values of the observables relevant to Bell test. The marginals of this joint distribution, which we experimentally determine, reproduces all of the observable quantum statistics including violation of the Bell inequality, which we independently measure. This is possible because our distribution, like the weak values for projectors on which it is built, is not constrained to the interval [0, 1]. It was first pointed out by Feynman that, for explaining singlet-state correlations within a [local] hidden variable view of nature ... everything works fine if we permit negative probabilities. However, there are infinitely many such theories. Our method, involving weak-valued singles out unique set of probabilities, and moreover does so empirically.
Generalized quantum measurement schemes are described by positive operator-valued measures going beyond the projection postulate, which predicts the instantaneous collapse of the systems wave function. This allows to take the noninvasive limit and investigate the correlations of such weak measurements which facilitate the observation of non-commuting observables within the same system. We propose a scheme in which the detector is coupled to the measured system for a finite time, as it is the case in many real setups. This leads to non-Markovian effects appearing by memory functions which are related to symmetric and antisymmetric correlators of the detector variables. We investigate these functions addressing the role of equilibrium and non-equilibrium detectors and how they differ from and could realize the standard Markovian measurement respectively. The latter scheme leads to the symmetrized operator order (aka Keldysh ordering), which is widely used in quantum measurement discussions. We show that the non-Markovian measurement scheme yields information beyond the standard approach, allowing e.g. for the prove of the non classical nature of a system (quasiprobability statistics) by second-order correlation functions.