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Łukasz Rudnicki

Publications and source records attributed to Łukasz Rudnicki.

At least 19 recordsLinked to original sources
arXiv2026

Superluminal Quantum Reference Frames

While particles cannot travel faster than the speed of light, nor can information, this assumption has over the years been frequently questioned. Most recently, it has been argued [New J. Phys. 22, 033038 (2020)] that in a world with superluminal observers local determinism is impossible, linking the two pillars of physics-quantum theory and relativity-suggesting that the latter serves as the foundation for the former. Motivated by this approach, in this work, we extend the framework of quantum reference frames to incorporate superluminal Lorentz transformations. We apply this conceptual result to examine an apparent paradox where particles acquire negative energies after undergoing a superluminal Lorentz boost and propose a resolution within our framework. We also discuss Bell experiments under superluminal quantum reference frame transformations, showing that involved probabilities remain conserved.

quant-ph↗
arXiv2025

Quantum thermodynamics as a gauge theory

Thermodynamics is based on a coarse-grained approach, from which its fundamental variables emerge, effectively erasing the complicate details of the microscopic dynamics within a macroscopic system. The strength of Thermodynamics lies in the universality provided by this paradigm. In contrast, quantum mechanics focuses on describing the dynamics of microscopic systems, aiming to make predictions about experiments we perform, a goal shared by all fundamental physical theories, which are often framed as gauge theories in modern physics. Recently, a gauge theory for quantum thermodynamics was introduced, defining gauge invariant work and heat, and exploring their connections to quantum phenomena. In this work, we extend that theory in two significant ways. First, we incorporate energy spectrum degeneracies, which were previously overlooked. Additionally, we define gauge-invariant entropy, exploring its properties and connections to other physical and informational quantities. This results in a complete framework for quantum thermodynamics grounded in the principle of gauge invariance. To demonstrate some implications of this theory, we apply it to well-known critical systems.

quant-ph↗
arXiv2025

Universal geometries underpinning linear second order ordinary differential equations

A deep relationship [arXiv:2503.17816v1] between real linear second order ordinary differential equations $u''\left(x\right)+h\left(x\right)u\left(x\right)=0$, with differentiable $h(x)$, and two dimensional hyperbolic geometry is generalized in a multitude of ways. First, I present an equivalent relationship in which the hyperbolic geometry is replaced by a two dimensional (anti-)de Sitter geometry. I show that this equation everywhere admits a pair of linearly independent solutions locally expressed in terms of an arbitrary non-vertical geodesic curve in this geometry. I also show that every solution of a corresponding Ricatti equation $ Θ'\left(x\right)+Θ^2\left(x\right)+h(x)=0$ obtained through $u'\left(x\right)=Θ\left(x\right)u\left(x\right)$ itself is a geodesic curve in the two dimensional (anti-)de Sitter geometry. Next, after promoting $h(x)$ to a holomorphic function $h(z)$, I express two linearly independent solutions of $u''\left(z\right)+h\left(z\right)u\left(z\right)=0$ in virtually the same way as for the real scenario and hyperbolic geometry. In this case, the curves used to build the solutions are geodesic in a two dimensional complex Riemannian geometry of a sphere. Analogous results for the complex Ricatti equation follow. This geometric interpretation is independent of the function $h(z)$, while the holomorphic metric assumes the same functional form as the hyperbolic metric discovered in [arXiv:2503.17816v1]. Finally, I show that the equation in question is in an equivalent relationship with four dimensional pseudo Riemannian Kähler-Norden geometry. The added value of working with real geometry turns out to be that certain two dimensional submanifold of the Kähler-Norden manifold render the hyperbolic and the (anti-)de Sitter scenario, both relevant for the real equation.

math.CA↗
arXiv2025

Almost all real linear second order ordinary differential equations are solved by geodesic curves in two dimensional Riemannian hyperbolic geometry

I show that a real linear second order ordinary differential equation $u''\left(x\right)+h\left(x\right)u\left(x\right)=0$, with differentiable $h(x)$, locally admits two linearly independent solutions which exist on an open interval around any $x_0\in\mathbb{R}$: \[ u_\mathtt{top}(x)=\exp\left[\int_{x_0}^{x}\!\!dξ\,Φ\left(ξ\right)\frac{Φ'\left(ξ\right)-\sqrt{\left[h\left(ξ\right)-Φ^{2}\left(ξ\right)\right]^{2}+\left[Φ'\left(ξ\right)\right]^{2}}}{h\left(ξ\right)-Φ^{2}\left(ξ\right)}\right], \] \[ u_\mathtt{bot}(x)=\exp\left[\int_{x_0}^{x}\!\!dξ\,Φ\left(ξ\right)\frac{Φ'\left(ξ\right)+\sqrt{\left[h\left(ξ\right)-Φ^{2}\left(ξ\right)\right]^{2}+\left[Φ'\left(ξ\right)\right]^{2}}}{h\left(ξ\right)-Φ^{2}\left(ξ\right)}\right], \] where $Φ(x)$ is any geodesic curve in a two dimensional hyperbolic geometry of a Riemannian manifold $\mathbb{M}_h$, which is non-vertical at $x_0$. I define $\mathbb{M}_h$ to be an upper half plane $\{\left(x,\varPhi\right)\in\mathbb{R}^2\,|\,\varPhi>0\}$, with points in which $\varPhi^2=h(x)$ being removed, equipped with metric $g_h=\left[\left(h(x)-\varPhi^2\right)^2dx^2+d\varPhi^2\right]/\varPhi^2$. A non-trivial character of the presented result stems from the fact that $g_h$ is solely defined in terms of the function $h(x)$. I also show that a local diffeomorphism between $\mathbb{M}_h$ and Poincaré upper half plane $\mathbb{H}$ is induced by any pair of linearly independent solutions of $u''\!\left(x\right)+h\left(x\right)u\left(x\right)=0$. If this pair is selected to be $u_\mathtt{top}(x)$ and $u_\mathtt{bot}(x)$, the associate geodesic curve $Φ(x)$ is mapped to a vertical geodesic curve on $\mathbb{H}$. Thus, I establish a link between linear second order ordinary differential equations and two dimensional hyperbolic geometry.

math.CA↗
arXiv2024

Quantum uncertainty of optical coherence

Light is known to exhibit quantum uncertainty in terms of its amplitude, phase, and polarization. However, quantum uncertainty related to coherence, which is also a fundamental physical property of light, has not been considered to date. Here, we formulate and explore the concept of quantum optical coherence uncertainty. We focus on the first-order coherence of the simplest possible light field, a purely monochromatic plane wave, which is classically completely stable. Starting from a scalar treatment, we show that the field displays zero coherence uncertainty only for a number state. We then proceed to the vectorial regime and establish that any state leads to coherence fluctuations, governed by a set of uncertainty relations depending on the polarization state and space-time points. Our work thus provides fundamental insights into the quantum character of optical coherence, with potential benefits in applications using highly sensitive interferometric and polarimetric techniques.

quant-ph↗
arXiv2024

Multipartite entanglement

In this contribution we present a concise introduction to quantum entanglement in multipartite systems. After a brief comparison between bipartite systems and the simplest non-trivial multipartite scenario involving three parties, we review mathematically rigorous definitions of separability and entanglement between several subsystems, as well as their transformations and measures.

quant-ph↗
arXiv2024

Formal relation between Pegg-Barnett and Paul quantum phase frameworks

The problem of defining a hermitian quantum phase operator is nearly as old as quantum mechanics itself. Throughout the years, a number of solutions was proposed, ranging from abstract operator formalisms to phase-space methods. In this work, we make an explicit connection between two of the most prominent approaches, by proving that the probability distribution of phase in the Paul formalism follows exactly from the Pegg-Barnett formalism by combining the latter with the quantum limited amplifier channel. Our findings suggest that the Paul framework may be viewed as a semi-classical limit of the Pegg-Barnett approach.

quant-ph↗
arXiv2024

Suppression of noise in separation estimation of optical sources with spatial-mode demultiplexing

Spatial mode demultiplexing was proved to be a successful tool for estimation of the separation between incoherent sources, allowing for sensitivity much below the Rayleigh limit. However, with the presence of measurement's noise, superresolution brought by this technique deteriorates rapidly. On a formal ground, this can be seen in terms of, so called, Rayleigh curse known from direct imaging, which while being absent for ideal spatial mode demultiplexing, goes back in a noisy scenario. In this article, we develop a formal procedure to suppress the destructive effect of the noise, proposing a procedure effectively working as an error correction. For noise models given by a random unitary channel generated by a polynomial of creation and annihilation operators, we demonstrate that perfect noise decoupling can be reached by repeating the mode demultiplexers and intervening them by a group of rotations, in the limit of a large number of repetitions and small noise strength. For a special case of displacement noise, our solution is simplified: by using the demultiplexer twice, and interlacing it by a parity operator, given that the noise configuration is frozen between the first and the second step, a perfect decoupling can be achieved. This allows for a recovery of superresolution for a special class of noise generated by displacement operators. Furthermore, for a strong noise correlation between these two steps, our protocol provides an improved measurement resolution.

quant-ph↗
arXiv2024

Superresolution in separation estimation between two dynamic incoherent sources using spatial demultiplexing

Achieving resolution in the sub-Rayleigh regime (superresolution) is one of the rapidly developing topics in quantum optics and metrology. Recently, it was shown that perfect measurement based on spatial mode demultiplexing (SPADE) in Hermite-Gauss modes allows one to reach the quantum limit of precision for estimation of separation between two weak incoherent stationary sources. Since then, different imperfections such as misalignment or crosstalk between modes have been studied to check how this result translates into more realistic experimental setups. In this paper, we consider another deviation from the perfect setup by discarding the assumption about the stationarity of the sources. This is relevant for example for astrophysical applications where planets necessarily orbit around the star. We analyze two examples of dynamics: rotations and oscillations, showing the robustness of the SPADE-based measurement against them. The analysis is based on Fisher information, which allows one to obtain the precision limit through Cramér-Rao bound. Furthermore, we formulate a measurement algorithm that allows for the reduction of one parameter for estimation (system orientation angle) in the stationary sources scenario, maintaining the measurement precision despite the lack of knowledge about this parameter.

quant-ph↗
arXiv2024

Spectral and temporal metrology with bandlimited functions and finite-time measurements

We perform an analysis supplementing the metrology toolbox in the time-frequency domain. While the relevant time-frequency-based metrological protocols can be borrowed from the spatial domain, where they have recently been well developed, their ultimate practical usefulness is shown to be restricted by limits put on the bandwidth of both the signal and measurements, as well as by the finite measurement time. As we demonstrate for the well-known problem of multiparameter estimation for two incoherent, point-like sources, the impact of these experimental limitations on the optimal protocol's efficiency can be detrimental. Nonetheless, we propose necessary operational criteria for attainability of the quantum Cramér-Rao bound under the discussed restrictions.

quant-ph↗
arXiv2024

Open dynamics of entanglement in mesoscopic bosonic systems

A key issue in Quantum Information is finding an adequate description of mesoscopic systems that is simpler than full quantum formalism yet retains crucial information about non-classical phenomena like entanglement. In particular, the study of fully bosonic systems undergoing open evolution is of great importance for the advancement of photonic quantum computing and communication. In this paper, we propose a mesoscopic description of such systems based on boson number correlations. This description allows for tracking Markovian open evolution of entanglement of both non-Gaussian and Gaussian states and their sub-Poissonian statistics. It can be viewed as a generalization of the reduced state of the field formalism [Entropy 2019, 21(7), 705], which by itself does not contain information about entanglement. As our approach adopts the structure of the description of two particles in terms of first quantization, it allows for broad intuitive usage of known tools. Using the proposed formalism, we show the robustness of entanglement against low-temperature damping for four-mode bright squeezed vacuum state and beam-splitted single photon. We also present a generalization of the Mandel Q parameter. Building upon this, we show that the entanglement of the state obtained by beam splitting of a single occupied mode is fully inherited from sub-Poissonian statistics of the input state.

quant-ph↗
arXiv2024

Gauge invariant quantum thermodynamics: consequences for the first law

Universality of classical thermodynamics rests on the central limit theorem, due to which, measurements of thermal fluctuations are unable to reveal detailed information regarding the microscopic structure of a macroscopic body. When small systems are considered and fluctuations become important, thermodynamic quantities can be understood in the context of classical stochastic mechanics. A fundamental assumption behind thermodynamics is therefore that of coarse-graning, which stems from a substantial lack of control over all degrees of freedom. However, when quantum systems are concerned, one claims a high level of control. As a consequence, information theory plays a major role in the identification of thermodynamic functions. Here, drawing from the concept of gauge symmetry, essential in all modern physical theories, we put forward a new possible, intermediate route. Working within the realm of quantum thermodynamics we explicitly construct physically motivated gauge transformations which encode a gentle variant of coarse-graining behind thermodynamics. As a consequence, we reinterpret quantum work and heat, as well as the role of quantum coherence.

quant-ph↗
arXiv2024

Quantum switch as a thermodynamic resource in the context of passive states

In recent years many works have explored possible advantages of indefinite causal order, with the main focus on its controlled implementation known as quantum switch. In this paper, we tackle advantages in quantum thermodynamics, studying whether quantum switch is capable of activating a passive state: either alone or with extra resources (active control state) and/or operations (measurement of the control system). By disproving the first possibility and confirming the second one, we show that quantum switch is not a thermodynamic resource in the discussed context, though, it can facilitate work extraction given external resources. We discuss our findings by considering specific examples: a qubit system subject to rotations around the x and y axes in the Bloch sphere, as well as general unitaries from the U(2) group; and the system as a quantum harmonic oscillator with displacement operators, and with a combination of displacement and squeeze operators.

quant-ph↗
arXiv2024

Relating the Glauber-Sudarshan, Wigner and Husimi quasiprobability distributions operationally through the quantum limited amplifier and attenuator channels

The Glauber-Sudarshan, Wigner and Husimi quasiprobability distributions are indispensable tools in quantum optics. However, although mathematical relations between them are well established, not much is known about their operational connection. In this paper, we prove that a single composition of finite-strength quantum limited amplifier and attenuator channels, known for their noise-adding properties, turns the Glauber-Sudarshan distribution of any input operator into its Wigner distribution, and its Wigner distribution into its Husimi distribution. As we dissect, the considered process, which can be performed in a quantum optical laboratory with relative ease, may be interpreted as realizing a quantum-to-classical transition.

quant-ph↗
arXiv2023

Application range of crosstalk-affected spatial demultiplexing for resolving separations between unbalanced sources

Superresolution is one of the key issues at the crossroads of contemporary quantum optics and metrology. Recently, it was shown that for an idealized case of two balanced sources, spatial mode demultiplexing (SPADE) achieves resolution better than direct imaging even in the presence of measurement crosstalk [Phys. Rev. Lett. 125, 100501 (2020)]. In this work, we consider arbitrarily unbalanced sources and provide a systematic analysis of the impact of crosstalk on the resolution obtained from SPADE. As we dissect, in this generalized scenario, SPADE's effectiveness depends non-trivially on the strength of crosstalk, relative brightness and the separation between the sources. In particular, for any source imbalance, SPADE performs worse than ideal direct imaging in the asymptotic limit of vanishing source separations. Nonetheless, for realistic values of crosstalk strength, SPADE is still the superior method for several orders of magnitude of source separations.

quant-ph↗
arXiv2023

Salient signatures of entanglement in the surrounding environment

We develop a model in which presence of entanglement in a quantum system can be confirmed through coarse observations of the environment surrounding the system. This counter-intuitive effect becomes possible when interaction between the system and its environment is proportional to an observable being an entanglement witness. While presenting intuitive examples we show that: i) a cloud of an ideal gas, when subject to a linear potential coupled with the entanglement witness, accelerates in the direction dictated by the sign of the witness; ii) when the environment is a radiation field, the direction of dielectric polarization depends on the presence of entanglement; iii) quadratures of electromagnetic field in a cavity coupled with two qubits (or a four-level atom) are displaced in the same manner.

quant-ph↗
arXiv2023

Spectral stabilizability

Decoherence represents a major obstacle towards realizing reliable quantum technologies. Identifying states that can be uphold against decoherence by purely coherent means, i.e., {\it stabilizable states}, for which the dissipation-induced decay can be completely compensated by suitable control Hamiltonians, can help to optimize the exploitation of fragile quantum resources and to understand the ultimate limits of coherent control for this purpose. In this work, we develop conditions for stabilizability based on the target state's eigendecomposition, both for general density operators and for the covariance matrix parameterization of Gaussian states. Unlike previous conditions for stabilizability, these spectral conditions are both necessary and sufficient and are typically easier to use, extending their scope of applicability. To demonstrate its viability, we use the spectral approach to derive upper bounds on stabilizability for a number of exemplary open system scenarios, including stabilization of generalized GHZ and W states in the presence of local dissipation and stabilization of squeezed thermal states under collective damping.

quant-ph↗