arXiv subjects

Piotr Szańkowski

Publications and source records attributed to Piotr Szańkowski.

18 recordsLinked to original sources

arXiv2026

Phenomenological quantum mechanics: I. Phenomenology of quantum observables

We propose an exercise in which one attempts to deduce the formalism of quantum mechanics solely from phenomenological observations. The only assumed inputs are obtained through sequential probing of quantum systems; no presuppositions about the underlying mathematical structures are permitted. We demonstrate that it is indeed possible to derive, on this basis, a complete and fully functional formalism rooted in the structures of Hilbert spaces. However, the resulting formalism--the bi-trajectory formalism--differs significantly from the standard state-focused formulation. In Part I of the paper, we analyze the outcomes of various experiments involving sequential measurements of quantum observables. These outcomes are quantitatively described by phenomenological multi-time probability distributions, estimated from experimental data. Our first conclusion is that the theory describing these experiments must be non-classical: the measured sequences cannot be interpreted as sampling of a uni-trajectory representing the system's observable. The non-classical nature of the investigated systems manifests in a range of observed phenomena, including quantum interference, the quantum Zeno effect, and uncertainty relations between the measured observables.

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arXiv2026

Phenomenological quantum mechanics: II. Deducing the formalism from experimental observations

We propose an exercise in which one attempts to deduce the formalism of quantum mechanics solely from phenomenological observations. The only assumed inputs are the multi-time probability distributions estimated from the results of sequential measurements of quantum observables; no presuppositions about the underlying mathematical structures are permitted. In the concluding Part II of the paper, we carry out the deduction of the formalism from the phenomenological inputs described in Part I. We show that the resulting formalism exhibits an affinity with Hilbert spaces, and we derive an explicit representation in terms of those mathematical structures. Analogues of the obtained elementary building blocks -- such as projection operators -- are readily identifiable within the standard formalism. However, once these building blocks are assembled according to the blueprint of the deduced bi-trajectory formalism, it becomes evident that the new and the standard formalisms differ substantially at the conceptual level. These differences do not negate the fact that both formalisms are in perfect agreement with respect to empirically testable predictions. Rather, the emergence of a novel, non-standard formulation should be seen as a relatively rare opportunity to reassess, from a fresh perspective, some of the long-standing foundational issues in the theory. The hope is that the new approach may prove more successful in addressing problems that have resisted resolution within the established theoretical framework.

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arXiv2025

A perturbation theory for multi-time correlation functions in open quantum systems

Dynamical maps are the principal subject of the open system theory. Formally, the dynamical map of a given open quantum system is a density matrix transformation that takes any initial state and sends it to the state at a later time. Physically, it encapsulates the system's evolution due to coupling with its environment. Hence, the theory provides a flexible and accurate framework for computing expectation values of open system observables. However, expectation values -- or more generally, single-time correlation functions -- capture only the simplest aspects of a quantum system's dynamics. A complete characterization of the dynamics requires access to multi-time correlation functions as well: phenomena like detailed balance, linear and non-linear response, non-equilibrium transport in general, or even sequential measurements of system observables are all described in terms of multi-time correlations. For closed systems, such correlations are well-defined, even though knowledge of the system's state alone is insufficient to determine them fully. In contrast, the standard dynamical map formalism for open systems does not account for multi-time correlations, as it is fundamentally limited to describing state evolution. Here, we extend the scope of open quantum system theory by developing a systematic perturbation theory for computing multi-time correlation functions.

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arXiv2024

Double or nothing: a Kolmogorov extension theorem for multitime (bi)probabilities in quantum mechanics

The multitime probability distributions obtained by repeatedly probing a quantum system via the measurement of an observable generally violate Kolmogorov's consistency property. Therefore, one cannot interpret such distributions as the result of the sampling of a single trajectory. We show that, nonetheless, they do result from the sampling of one pair of trajectories. In this sense, rather than give up on trajectories, quantum mechanics requires to double down on them. To this purpose, we prove a generalization of the Kolmogorov extension theorem that applies to families of complex-valued bi-probability distributions (that is, defined on pairs of elements of the original sample spaces), and we employ this result in the quantum mechanical scenario. We also discuss the relation of our results with the quantum comb formalism.

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arXiv2024

Objectivity of classical quantum stochastic processes

We investigate what can be concluded about a quantum system when sequential quantum measurements of its observable -- a prominent example of the so-called quantum stochastic process -- fulfill the Kolmogorov consistency condition and thus appear to an observer as a sampling of a classical trajectory. We identify a set of physical conditions imposed on the system dynamics, that when satisfied, lead to the aforementioned trajectory interpretation of the measurement results. We then show that when another quantum system is coupled to the observable, the operator representing it can be replaced by external noise. Crucially, the realizations of this surrogate (classical) stochastic process follow the same trajectories as those measured by the observer. Therefore, it can be said that the trajectory interpretation suggested by the Kolmogorov consistent measurements also applies in contexts other than sequential measurements.

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arXiv2022

Introduction to the theory of open quantum systems

This manuscript is an edited and refined version of the lecture script for a one-semester graduate course given originally at the PhD school in the Institute of Physics of Polish Academy of Sciences in the Spring/Summer semester of 2022. The course expects from the student only a basic knowledge on graduate-level quantum mechanics. The script itself is largely self-contained and could be used as a textbook on the topic of open quantum systems. The program of this course is based on a novel approach to the description of the open system dynamics: It is showed how the environmental degrees of freedom coupled to the system can be represented by a multi-component quasi-stochastic process. Using this representation one constructs the super-quasi-cumulant (or super-qumulant) expansion for the system's dynamical map -- a parametrization that naturally lends itself for the development of a robust and practical perturbation theory. Thus, even an experienced researcher might find this manuscript of some interest.

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arXiv2021

Measuring trajectories of environmental noise

In classical mechanics, a natural way to simplify a many-body problem is to ``replace'' some of the elements of the composite system with surrogate \textit{force fields}. In the realm of quantum mechanics, however, such a description is rarely compatible with the formalism of the theory. Nevertheless, the quantum version of external field models---the so-called \textit{noise representations}---can be employed in certain circumstances. The mathematics behind these models indicate that the appearing fields typically exhibit random fluctuations, hence, the name \textit{noise field} is more apt. In principle, measuring a classical force field is a trivial task; all that is needed is a probe equipped with an accelerometer that can be moved around while taking the measure of forces that affect it. Unfortunately, an analogous method cannot work in the quantum case. As indicated by the theory, the result of any measurement performed on the quantum system affected by a noise field always appears as an average over fluctuations, therefore, it is impossible to observe the noise fluctuations by measuring the state of the probe. Here, we demonstrate that this limitation can be circumvented and that it is possible to expose the noise field fluctuations to direct observation. This allows one to sample the noise trajectories (stochastic realizations), store them, and later use them to simulate the dynamics of open quantum systems affected by the noise.

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arXiv2020

Noise representations of open system dynamics

We analyze the conditions under which the dynamics of a quantum system open to a given environment can be simulated with an external noisy field that is a surrogate for the environmental degrees of freedom. We show that such a field is either a subjective or an objective surrogate; the former is capable of simulating the dynamics only for the specific system--environment arrangement, while the latter is an universal simulator for any system interacting with the given environment. Consequently, whether the objective surrogate field exists and what are its properties is determined exclusively by the environment. Thus, we are able to formulate the sufficient criterion for the environment to facilitate its surrogate, and we identify a number of environment types that satisfy it. Finally, we discuss in what sense the objective surrogate field representation can be considered classical and we explain its relation to the formation of system--environment entanglement, and the back-action exerted by the system onto environment.

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arXiv2020

Influence of nuclear spin polarization on spin echo signal of NV center qubit

We consider the spin echo dynamics of a nitrogen-vacancy center qubit based the $S\!= \! 1$ ground state spin manifold, caused by a dynamically polarized nuclear environment. We show that the echo signal acquires then a nontrivially time-dependent phase shift. This effect should be observable for polarization $\approx \! 0.5$ of nuclei within $\sim \! 1$ nm from the qubit, and for the NV center initialized in a superposition of $m\! = \! 0$ and either $m\! =\! 1$ or $m\! =\! -1$ states. This phase shift is much smaller when the NV center is prepared in a superposition of $m\! = \! 1$ and $m\! =\! -1$ states, i.e. when the qubit couples to the spin environment in a way analogous to that of spin-$1/2$. For nuclear environment devoid of spins strongly coupled to the qubit, the phase shift is well described within Gaussian approximation, which provides an explanation for the dependence of the shift magnitude on the choice of states on which the qubit is based, and makes it clear that its presence is related to the linear response of the environment perturbed by an evolving qubit. Consequently, its observation signifies the presence environment-mediated self-interaction of the qubit, and hence, it invalidates the notion that the nuclear environment acts as a source of external noise driving the qubit. We also show how a careful comparison of the echo signal from qubits based on $m\! = \! 0,1$ and $m\! =\! \pm 1$ manifolds, can distinguish between effectively Gaussian and non-Gaussian environment.

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arXiv2019

Transition between continuous and discrete spectra in dynamical-decoupling noise spectroscopy

Here, we explore the prospects of carrying out the single qubit spectroscopy of environmental noise when the resolution of the frequency filters produced by the dynamical decoupling pulse sequences can be set high enough to reveal the discrete structure of the noise spectral density. The standard form of spectroscopy is applicable when the filter is unable to resolve the discrete spectral lines and only coarse grained approximation of the spectrum is recorded in the qubit's decoherence rate. When the discrete structure becomes accessible, the qubit probe evolves in a qualitatively different manner, and the procedure for recovering spectral density has to be redesigned.

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arXiv2019

The dynamical-decoupling-based spatiotemporal noise spectroscopy

Here we demonstrate how the standard, temporal-only, dynamical-decoupling-based noise spectroscopy method can be extended to also encompass the spatial degree of freedom. This spatiotemporal spectroscopy utilizes a system of multiple qubits arranged in a line that are undergoing pure dephasing due to environmental noise. When the qubits are driven by appropriately coordinated sequences of $π$ pulses the multi-qubit register becomes decoupled from all components of the noise, except for those characterized by frequencies and wavelengths specified by the pulse sequences. This allows for employment of the procedure for reconstruction of the two-dimensional spectral density that quantifies the power distribution among spatial and temporal harmonic components of the noise.

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arXiv2018

Decoherence-assisted detection of entanglement of two qubit states

We show that the decoherence, which in the long run destroys quantum features of a system, can be used to reveal the entanglement in a two-qubit system. To this end, we consider a criterion that formally resembles the Clauser-Horne-Shimony-Holt (CHSH) inequality. In our case the local observables are set by the coupling of each qubit to the environmental noise, controlled with the dynamical decoupling method. We demonstrate that the constructed inequality is an entanglement criterion---it can only be violated by non-separable initial two-qubit states, provided that the local noises are correlated. We also show that for a given initial state, this entanglement criterion can be repurposed as a method of discriminating between Gaussian and non-Gaussian noise generated by the environment of the qubits. The latter application is important for ongoing research on using qubits to characterize the dynamics of environment that perturbs them and causes their decoherence

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arXiv2018

Accuracy of dynamical-decoupling-based spectroscopy of Gaussian noise

The fundamental assumption of dynamical decoupling based noise spectroscopy is that the coherence decay rate of qubit (or qubits) driven with a sequence of many pulses, is well approximated by the environmental noise spectrum spanned on frequency comb defined by the sequence. Here we investigate the precise conditions under which this commonly used spectroscopic approach is quantitatively correct. To this end we focus on two representative examples of spectral densities: the long-tailed Lorentzian, and finite-ranged Gaussian---both expected to be encountered when using the qubit for nano-scale nuclear resonance imaging. We have found that, in contrast to Lorentz spectrum, for which the corrections to the standard spectroscopic formulas can easily be made negligible, the spectra with finite range are more challenging to reconstruct accurately. For Gaussian line-shape of environmental spectral density, direct application of the standard dynamical decoupling based spectroscopy leads to erroneous attribution of long-tail behavior to the reconstructed spectrum. Fortunately, artifacts such as this, can be completely avoided with the simple extension to standard reconstruction method.

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arXiv2017

Localization of a magnetic moment using a two-qubit probe

A nanomagnet precessing in an external magnetic field can be treated as a source of narrow-bandwidth magnetic noise, that leaves characteristic fingerprints in decoherence of a nearby spin qubit undergoing dynamical decoupling. We show how, by measurements of two-qubit coherence, a noise sensor composed of qubit pair can be used to reconstruct the position of the nanomagnet. Such localization of noise source is possible with only two qubit probes, because the course of coherence decay under appropriately designed dynamical decoupling sequences contain information not only about noises experienced by each qubit, but also about their cross-correlations. We test the applicability of the proposed protocol on an example of two qubits coupled to the nanomagnet via dipolar interaction. We also show how, using a two-qubit sensor possessing a particular symmetry, one can localize the nanomagnet even when the sensor-magnet coupling law is unknown.

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arXiv2017

Environmental noise spectroscopy with qubits subjected to dynamical decoupling

A qubit subjected to pure dephasing due to classical Gaussian noise can be turned into a spectrometer of this noise by utilizing its readout under properly chosen dynamical decoupling (DD) sequences to reconstruct the power spectral density of the noise. We review the theory behind this DD-based noise spectroscopy technique, paying special attention to issues that arise when the environmental noise is non-Gaussian and/or it has truly quantum properties. While we focus on the theoretical basis of the method, we connect the discussed concepts with specific experiments and provide an overview of environmental noise models relevant for solid-state based qubits, including quantum-dot based spin qubits, superconducting qubits, and NV centers in diamond.

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arXiv2015

Spectroscopy of cross-correlations of environmental noises with two qubits

A single qubit driven by an appropriate sequence of control pulses can serve as a spectrometer of local noise affecting its energy splitting. We show that by driving and observing two spatially separated qubits, it is possible to reconstruct the spectrum of cross-correlations of noises acting at various locations. When the qubits are driven by the same sequence of pulses, real part of cross-correlation spectrum can be reconstructed, while applying two distinct sequence to the two qubits allows for reconstruction of imaginary part of this spectrum. The latter quantity contains information on either causal correlations between environmental dynamics at distinct locations, or on the occurrence of propagation of noisy signals through the environment. We illustrate the former case by modeling the noise spectroscopy protocol for qubits coupled to correlated two-level systems. While entanglement between the qubits is not necessary, its presence enhances the signal from which the spectroscopic information is reconstructed.

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arXiv2015

Quantum coherence and correlations in cold atom systems

Although the foundations of quantum and classical physics are much different, it is often difficult to pinpoint which features of a particular system are intrinsically "quantum". Perhapse, the most clear-cut distinction between "classical" and "quantum" can be made for systems composed of many particles when the properties of the ensemble are determined by the correlations between the constituents. The issue of grasping the nature of entanglement (i.e. quantum correlations) lies in its formal, discriminative definition: "if state is not classical then it is entangled". A possible remedy would be to approach the problem from the utilitarian point of view. The idea is that certain tasks can be performed better when entangled states are used instead of states that are only classically correlated. An example of such task is an entanglement-enhanced interferometer utilizing ensembles of ultra-cold atoms. This line of reasoning leads to the concept of classification of entanglement by the degree of usefulness and allows to relate it to certain well-defined physical properties of the state. The aim of this work is to formalize this abstraction and build upon it an intuitive picture which can help us understand what the entanglement is.

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arXiv2013

Spin decoherence due to Fluctuating Fields

The dynamics of a spin in the presence of a deterministic and a fluctuating magnetic field is solved for analytically to obtain the averaged value of the spin as a function of time for various kinds of fluctuations (noise). Specifically, analytic results are obtained for the time dependence of the expectation value of the spin, averaged over fluctuations, for Gaussian white noise and Guassian colored noise, as well as non-Gaussian telegraph noise. Fluctuations cause the decay of the average spin vector (decoherence). For noise with finite temporal correlation time, a deterministic component of the field can suppress decoherence of the spin component along the field. Hence, decoherence can be manipulated by controlling the deterministic magnetic field. A simple universal physical picture emerges which explains the mechanism of the suppression of the decay.

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