© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

Noninvasive measurement of dynamic correlations in spin systems

Philipp Uhrich

Published Dec 1, 2017• 0 citations

Abstract

ENGLISH ABSTRACT : Dynamic correlations hψ| O1(t1)O2(t2)|ψi of quantum observables are useful quantities for the study of quantum dynamics. Attempts at measuring these correlations are however complicated, due to the measurement backaction (wave function collapse) incurred during measurements at the early time t1. We propose a noninvasive measurement protocol, based on a weak ancilla target coupling, which reduces this backaction at t1. We show that both real and imaginary parts of the desired correlation can be extracted through appropriate choices of the initial ancilla state and of the ancilla target coupling Hamiltonian. The protocol is applicable to arbitrary (pseudo)spin systems with arbitrary (non)equilibrium initial states. Errors arising in experimental implementations are analysed, and we show that deviations from the desired correlation can be minimised through an optimal choice of the ancilla target coupling time. Implementation in linear ion trap experiments is discussed. We derive the positive-operator-valued measure which describes the noninvasive measurement at t1. For dynamic correlations of single-site spin-1/2 observables, this operator formalism shows that measurement backaction is of no concern. Real parts can be obtained with projective measurements of the target at t1 and t2. Imaginary parts are obtained by performing a local rotation of the target at t1, followed by a projective measurement at t2. These ancilla-free protocols are theoretically simpler than the noninvasive measurement protocol, but remain experimentally challenging. Rotations and projections performed at t1 may be subject to noise, which propagates into the measured correlation. We use Lieb-Robinson theory to bound the size of the resulting error terms. An analysis of the spatio-temporal behaviour of these errors provides guidance for experimental implementation of the ancilla-free measurement protocols.

Related papers

Noninvasive measurement of dynamic correlation functions

Philipp Uhrich, Salvatore Castrignano, Hermann Uys +1 more

The measurement of dynamic correlation functions of quantum systems is complicated by measurement backaction. To facilitate such measurements we introduce a protocol, based on weak ancilla-system couplings, that is applicable to arbitrary (pseudo)spin systems and arbitrary equilibrium or nonequilibrium initial states. Different choices of the coupling operator give access to the real and imaginary parts of the dynamic correlation function. This protocol reduces disturbances due to the early-time measurements to a minimum, and we quantify the deviation of the measured correlation functions from the theoretical, unitarily evolved ones. Implementations of the protocol in trapped ions and other experimental platforms are discussed. For spin-$1/2$ models and single-site observables we prove that measurement backaction can be avoided altogether, allowing for the use of ancilla-free protocols.

arXiv:1611.08123•Physical Review A•Aug 21, 2017•25 citations

Ancilla-assisted probing of temporal quantum correlations of large spins

Michael Kästner

When measuring quantum spins at two or more different times, the later measurements are affected by measurement backaction occurring due to the earlier measurements. This makes the measurement of temporal quantum correlation functions challenging. In this paper, I propose a measurement protocol that mitigates the effect of measurement backaction by exploiting spin selection rules. I show that, under suitable conditions, the effect of measurement backaction on two-time quantum correlations becomes negligible when probing a system consisting of spins with large spin quantum numbers $l\ensuremath{\gg}s$ by coupling it to a spin-$s$ ancilla degree of freedom. A potential application of such a measurement protocol is the probing of an array of Bose-Einstein condensates by light.

arXiv:2006.07028•Physical Review A•Aug 13, 2020•2 citations

Ancilla-free measurement of out-of-time-ordered correlation functions: General measurement protocol and Rydberg atom implementation

Michael Kästner, Philip Osterholz, Christian Groß

We introduce a protocol that gives access to out-of-time-ordered correlation functions in many-body quantum systems. Unlike other such protocols, our proposal, which can be applied to arbitrary initial states, neither requires ancilla degrees of freedom to the quantum system of interest, nor has the need for randomized measurements. Nontrivial experimental capabilities required to implement the protocol are single-site measurements, single-site rotations, and backwards time evolution. To exemplify the implementation of the protocol, we put forward a strategy for Hamiltonian sign inversion $H\ensuremath{\rightarrow}\ensuremath{-}H$ in arrays of Rydberg-dressed atoms. In this way, a complete and practical toolbox is obtained for the measurement of out-of-time-ordered correlations in equilibrium and nonequilibrium situations.

arXiv:2403.08670•Physical Review A•Jul 8, 2024•6 citations

Correlations of weak quantum measurements in a non-Markovian detection scheme

Johannes Bülte, Adam Bednorz, Wolfgang Belzig

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.

International Conference on Noise and Fluctuations•Jun 1, 2015•0 citations

Symmetrized and non-symmetrized noise from weak measurements in mesoscopic circuits

Wolfgang Belzig, Johannes Bülte, Adam Bednorz +1 more

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.

International Conference on Noise and Fluctuations•Jun 1, 2017•0 citations

Dynamical structure factor from weak measurements

Emine Altuntaş, R. G. Lena, Stuart Flannigan +2 more

Abstract Much of our knowledge of quantum systems is encapsulated in the expectation value of Hermitian operators, experimentally obtained by averaging projective measurements. However, dynamical properties are often described by products of operators evaluated at different times; such observables cannot be measured by individual projective measurements, which occur at a single time. For example, the dynamical structure factor (DSF) describes the propagation of density excitations, such as phonons, and is derived from the spatial density operator evaluated at different times. In equilibrium systems this can be obtained by first exciting the system at a specific wavevector and frequency, then measuring the response. Here, we describe an alternative approach using a pair of time-separated weak measurements, and analytically show that their cross-correlation function directly recovers the DSF, for all systems, even far from equilibrium. This general schema can be applied to obtain the cross-correlation function of any pair of weakly observable quantities. We provide numerical confirmation of this technique with a matrix product states simulation of the one-dimensional Bose–Hubbard model, weakly measured by phase contrast imaging. We explore the limits of the method and demonstrate its applicability to real experiments with limited imaging resolution.

arXiv:2409.07030•Quantum Science and Technology•Jun 10, 2025•2 citations

ENGLISH ABSTRACT : Dynamic correlations hψ| O1(t1)O2(t2)|ψi of quantum observables are useful quantities for the study of quantum dynamics. Attempts at measuring these correlations are however complicated, due to the measurement backaction (wave function collapse) incurred during measurements at the early time t1. We propose a noninvasive measurement protocol, based on a weak ancilla target coupling, which reduces this backaction at t1. We show that both real and imaginary parts of the desired correlation can be extracted through appropriate choices of the initial ancilla state and of the ancilla target coupling Hamiltonian. The protocol is applicable to arbitrary (pseudo)spin systems with arbitrary (non)equilibrium initial states. Errors arising in experimental implementations are analysed, and we show that deviations from the desired correlation can be minimised through an optimal choice of the ancilla target coupling time. Implementation in linear ion trap experiments is discussed. We derive the positive-operator-valued measure which describes the noninvasive measurement at t1. For dynamic correlations of single-site spin-1/2 observables, this operator formalism shows that measurement backaction is of no concern. Real parts can be obtained with projective measurements of the target at t1 and t2. Imaginary parts are obtained by performing a local rotation of the target at t1, followed by a projective measurement at t2. These ancilla-free protocols are theoretically simpler than the noninvasive measurement protocol, but remain experimentally challenging. Rotations and projections performed at t1 may be subject to noise, which propagates into the measured correlation. We use Lieb-Robinson theory to bound the size of the resulting error terms. An analysis of the spatio-temporal behaviour of these errors provides guidance for experimental implementation of the ancilla-free measurement protocols.

The measurement of dynamic correlation functions of quantum systems is complicated by measurement backaction. To facilitate such measurements we introduce a protocol, based on weak ancilla-system couplings, that is applicable to arbitrary (pseudo)spin systems and arbitrary equilibrium or nonequilibrium initial states. Different choices of the coupling operator give access to the real and imaginary parts of the dynamic correlation function. This protocol reduces disturbances due to the early-time measurements to a minimum, and we quantify the deviation of the measured correlation functions from the theoretical, unitarily evolved ones. Implementations of the protocol in trapped ions and other experimental platforms are discussed. For spin-

1/2

models and single-site observables we prove that measurement backaction can be avoided altogether, allowing for the use of ancilla-free protocols.

When measuring quantum spins at two or more different times, the later measurements are affected by measurement backaction occurring due to the earlier measurements. This makes the measurement of temporal quantum correlation functions challenging. In this paper, I propose a measurement protocol that mitigates the effect of measurement backaction by exploiting spin selection rules. I show that, under suitable conditions, the effect of measurement backaction on two-time quantum correlations becomes negligible when probing a system consisting of spins with large spin quantum numbers $l\ensuremath{\gg}s$ by coupling it to a spin-

s

ancilla degree of freedom. A potential application of such a measurement protocol is the probing of an array of Bose-Einstein condensates by light.

We introduce a protocol that gives access to out-of-time-ordered correlation functions in many-body quantum systems. Unlike other such protocols, our proposal, which can be applied to arbitrary initial states, neither requires ancilla degrees of freedom to the quantum system of interest, nor has the need for randomized measurements. Nontrivial experimental capabilities required to implement the protocol are single-site measurements, single-site rotations, and backwards time evolution. To exemplify the implementation of the protocol, we put forward a strategy for Hamiltonian sign inversion $H\ensuremath{\rightarrow}\ensuremath{-}H$ in arrays of Rydberg-dressed atoms. In this way, a complete and practical toolbox is obtained for the measurement of out-of-time-ordered correlations in equilibrium and nonequilibrium situations.

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.

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.

Abstract Much of our knowledge of quantum systems is encapsulated in the expectation value of Hermitian operators, experimentally obtained by averaging projective measurements. However, dynamical properties are often described by products of operators evaluated at different times; such observables cannot be measured by individual projective measurements, which occur at a single time. For example, the dynamical structure factor (DSF) describes the propagation of density excitations, such as phonons, and is derived from the spatial density operator evaluated at different times. In equilibrium systems this can be obtained by first exciting the system at a specific wavevector and frequency, then measuring the response. Here, we describe an alternative approach using a pair of time-separated weak measurements, and analytically show that their cross-correlation function directly recovers the DSF, for all systems, even far from equilibrium. This general schema can be applied to obtain the cross-correlation function of any pair of weakly observable quantities. We provide numerical confirmation of this technique with a matrix product states simulation of the one-dimensional Bose–Hubbard model, weakly measured by phase contrast imaging. We explore the limits of the method and demonstrate its applicability to real experiments with limited imaging resolution.