arXiv subjects

Dariusz Chruściński

Publications and source records attributed to Dariusz Chruściński.

At least 19 recordsLinked to original sources

arXiv2026

Revivals of Bell nonlocality require Schrödinger and Heisenberg non-Markovianity

Bell nonlocality is a key resource in quantum information, demonstrating the nonclassicality of quantum theory. Noise, however, is in general detrimental to nonlocality, and can cause the loss of the ability to violate any Bell inequality. Memory effects, on the other hand, can restore this quantumness and, as recently shown, they can be differently characterized in the Schrödinger and in the Heisenberg picture. Here, we show that if memory effects allow for revivals in time of nonlocality, then the dynamics must be non-Markovian in both pictures. We showcase our findings through a device-independent quantum key distribution task, for which Bell nonlocality is necessary.

quant-ph↗

arXiv2026

Schrödinger and Heisenberg non-Markovianity in quantum information tasks

Quantum non-Markovianity has been widely studied and connected to the existence of memory effects in the dynamics of open systems. Surprisingly, working in the Schrödinger or in the Heisenberg picture can provide inequivalent description non-Markovianity: a process can appear to be memoryless in one picture, while displaying memory effects in the other. Here, we investigate which kind of memory is relevant for different quantum information tasks. Some of them, such as sending information via a noisy channel, require memory in both pictures in order to exhibit revivals in the task performance. For others, only one type of memory is sufficient. We also provide necessary conditions for non-Markovianity in both pictures by only considering the dynamics in one picture, showing for instance that the previously considered witness of Schrödinger non-Markovianity in terms of the volume of accessible states does indeed witness non-Markovianity in both pictures at the same time.

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arXiv2026

Divisibility of Non-Invertible Quantum Dynamics in the Heisenberg Picture

We study divisibility of finite-dimensional quantum dynamics in the Heisenberg picture when the dynamical maps need not be invertible. Heisenberg divisible dynamics determines unique reduced propagators on the accessible operator systems, and Heisenberg CP-divisibility is shown to be equivalent to an operator-norm monotonicity. Although each reduced propagator admits a unital completely positive extension, independently chosen extensions need not satisfy a composition law. Generalized inverses restore this law algebraically, but do not, in general, preserve complete positivity. We characterize the resulting obstruction and give a sufficient construction based on unital completely positive projections onto the accessible operator systems. It is shown that for qubit dynamics Heisenberg CP-divisibility always guarantees a completely positive generalized-inverse propagator. Finally, we analyze time-local generators at singular times and identify the associated gauge freedom on dynamically inaccessible directions.

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arXiv2026

Fifty Years of the GKLS Master Equation: Foundations and Early Developments

The year 2026 marks the fiftieth anniversary of the two seminal papers in which Gorini, Kossakowski, and Sudarshan, and independently Lindblad, characterized the generators of completely positive quantum dynamical semigroups. We revisit the conceptual and historical route leading to the Gorini-Kossakowski-Lindblad-Sudarshan (GKLS) master equation and examine several of its early developments and consequences. Our aim is to show how several initially separate mathematical and physical ideas converged in the GKLS structure theorem, imme- diately opening the way to manifold consequences for the description of dissipative quantum phenomena and how that theorem, fifty years later, remains both a foundational result and an active framework for open quantum dynamics.

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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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arXiv2026

Diagonal Unitary Covariant Superchannels

We present a complete characterization of diagonal unitary covariant (DU-covariant) superchannels, i.e. higher-order transformations transforming quantum channels into themselves. Necessary and sufficient conditions for complete positivity and trace preservation are derived and the canonical decomposition describing DU-covariant superchannels is provided. The presented framework unifies and extends known families of covariant quantum channels and enables explicit analysis of their action on physically relevant examples, including amplitude-damping, bit-flip, and Pauli channels. Our results provide a practical toolbox for symmetry-restricted higher-order quantum processes and offer a setting for exploring open problems such as the celebrated positive partial transpose (PPT)-squared conjecture.

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arXiv2026

Universal bound on the Lyapunov spectrum of quantum master equations

The spectral properties of positive maps are pivotal for understanding the dynamics of quantum systems interacting with their environment. Furthermore, central problems in quantum information such as the characterization of entanglement can be reformulated in terms of spectral properties of positive maps. The present work aims to contribute to a better understanding of the spectrum of positive maps. Specifically, our main result is a new proof of a universal bound on the $d^{2}-1$ generically non vanishing decay rates $Γ_{i}$ of time-autonomous quantum master equations on a $d$-dimensional Hilbert space: $$Γ_{\mathrm{max}}\,\leq\,\varkappa_{d}\,\sum_{i=1}^{d^{2}-1}Γ_{i}$$ The prefactor $\varkappa_{d}$ %, which we explicitly determine, depends only on the dimension $d$ and varies depending on the sub-class of positive maps to which the semigroup solution of the master equation belongs. We provide a brief but self-consistent survey of these concepts. We obtain our main result by resorting to the theory of Lyapunov exponents, a central concept in the study of dynamical systems, control theory, and out-of-equilibrium statistical mechanics. We thus show that progress in understanding positive maps in quantum mechanics may require ideas at the crossroads between different disciplines. For this reason, we adopt a notation and presentation style aimed at reaching readers with diverse backgrounds.

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arXiv2026

Detecting Entanglement by State Preparation and Local Measurements

Entanglement witnesses (EWs) are a collection of observables that can characterize separable states and, experimentally, estimating EWs can verify entangled states. In this work, we show that a fixed measurement setting on a multipartite entangled state, which we introduce as a network state for the purpose, can estimate EWs. Namely, entangled states can be fully verified in a measurement-based manner, in which experimenters do not necessarily change measurement settings. We present a fixed measurement setting and network states for estimating decomposable EWs, equivalent to the partial transpose criteria. We also consider non-decomposable EWs that detect bound entangled states beyond the partial transpose criteria. The results can be extended to multipartite states such as graph states, a resource for measurement-based quantum computing, and readily applied to distributed settings such as quantum metrology or sensor networks where multipartite entangled states are resourceful.

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arXiv2026

Open quantum dynamics without Complete Positivity: a criticism

The requirement of complete positivity is very often regarded as a fundamental consistency condition for the description of open quantum dynamics. We critically examine this requirement and discuss both its physical motivations and its limitations. We analyze proposals based on restricting the domain of non-completely positive maps to subsets of compatible initial states. Using isotropic states as a concrete example, we show that such domain restrictions become increasingly severe with growing system dimension, revealing an intrinsic weakness of the compatibility-based approach.

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arXiv2026

Beyond Robertson-Schrödinger: A General Uncertainty Relation Unveiling Hidden Noncommutative Trade-offs

We report a universal improvement to the standard Robertson--Schrödinger uncertainty relation. Our result shows that the Robertson--Schrödinger lower bound can be supplemented by a new noncommutativity-induced term. This term represents a previously overlooked quantum contribution and becomes more pronounced as the state becomes more mixed. Moreover, it is expressed as the expectation value of a positive observable, namely the squared modulus of the commutator, and therefore preserves the direct, experimentally accessible character of the Robertson--Schrödinger relation. For two-level quantum systems, our relation becomes an \emph{exact equality} for \emph{any} state and \emph{any} pair of observables, thereby ensuring the tightness of the bound in the strongest possible sense. The relation also yields, as a corollary, a complete proof of a general uncertainty bound that had previously been supported only by numerical evidence.

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arXiv2026

Practical and Efficient Verification of Entanglement with Incomplete Measurement Settings

In this work, we present a practical and efficient framework for verifying entangled states when only a tomographically incomplete measurement setting is available-specifically, when access to observables is severely limited. We show how the experimental estimation of a small number of observables can be directly exploited to construct a large family of entanglement witnesses, enabling the efficient identification of entangled states. Moreover, we introduce an optimization approach, formulated as a semidefinite program, that systematically searches for those witnesses best suited to reveal entanglement under the given measurement constraints. We demonstrate the practicality of the approach in a proof-of-principle experiment with photon-polarization qubits, where entanglement is certified using only a fraction of the full measurement data. These results reveal the maximal usefulness of incomplete measurement settings for entanglement verification in realistic scenarios.

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arXiv2026

Interpolating between positive, Schwarz, and completely positive evolution for d-level systems

We study a class of quantum dynamical maps for d-level systems that interpolate between positive, Schwarz, and completely positive evolutions. Our approach is based on a geometric analysis of the parameter space, which reveals the structure of regions corresponding to different positivity classes and their boundaries. We show that dynamical trajectories naturally move across these regions, providing a clear geometric interpretation of transitions between Markovian and non-Markovian regimes. It is shown that within presented class the evolution becomes eventually entanglement breaking. This analysis highlights the role of divisibility and eternally non-Markovian evolution.

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arXiv2026

Stochastic unravelings for Heisenberg picture and trace-nonpreserving dynamics

Stochastic unravelings allow to efficiently simulate open system dynamics, yet their application has traditionally been restricted to master equations that preserve both Hermiticity and trace. In this work, we introduce a general framework that extends piecewise-deterministic unravelings to arbitrary trace-nonpreserving master equations, requiring only positivity and Hermiticity of the dynamics. Our approach includes, as special cases, unravelings of arbitrary dynamics in the Heisenberg picture, evolutions interpolating between fully Lindblad and non-Hermitian Hamiltonian generators, and equations employed in the derivation of full counting statistics, for which we show it can be used to obtain the moments of the associated probability distribution. The framework is suitable for both trace-decreasing and trace-increasing processes through stochastic disappearance and replication of the stochastic realizations, and it is compatible with different unraveling schemes and with reverse jumps in the non-Markovian regime. Thereby, our approach provides a powerful and versatile simulation method that significantly broadens the applicability of stochastic techniques for open system dynamics.

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arXiv2026

Tomiyama-type maps with a diagonal perturbation

We investigate a two-parameter family of linear maps on matrix algebras, constructed as diagonal perturbations of classical Tomiyama maps. Employing the Choi matrix method alongside block-positivity techniques, we derive explicit necessary and sufficient conditions for positivity, complete positivity, and k-positivity across arbitrary dimensions. These conditions provide a transparent geometric characterization of the positivity regions within the parameter space.

math.RA↗

arXiv2026

Constructing $k$-Kadison-Schwarz maps

We study k Kadison Schwarz (k KS) mappings on matrix algebras and derive explicit conditions ensuring the k KS property for two classes of maps parameterized by a single k-positive map.

math.FA↗

arXiv2026

Divisibility of dynamical maps: Schrödinger vs. Heisenberg picture

Divisibility of dynamical maps is a central notion in the study of quantum non-Markovianity, providing a natural framework to characterize memory effects via time-local master equations. In this work, we generalize the notion of divisibility of quantum dynamical maps from the Schrödinger to the Heisenberg picture. While the two pictures are equivalent at the level of physical predictions, we show that the divisibility properties of the corresponding dual maps are, in general, not equivalent. This inequivalence originates from the distinction between left and right generators of time-local master equations, which interchange roles under duality. We demonstrate that Schrödinger and Heisenberg divisibility are distinct concepts by constructing explicit dynamics divisible only in one picture. Furthermore, we introduce a quantifier for the violation of Heisenberg P-divisibility, analogous to the trace-distance-based measure of non-Markovianity, and provide it with an operational interpretation in terms of the guessing probability between effects. Our results show that Heisenberg divisibility is an independent witness of memory effects and highlight the need to consider both pictures when characterizing non-Markovian quantum dynamics.

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arXiv2025

Asymptotic relaxation in quantum Markovian dynamics

We investigate the long-time behavior of quantum Markovian dynamics generated by time-dependent Gorini-Kossakowski-Lindblad-Sudarshan (GKLS) master equations. We introduce a notion of weak relaxation and derive sufficient conditions guaranteeing asymptotic independence from the initial state. Our results provide a quantitative extension of the Spohn-Frigerio theorem to time-dependent generators, yielding explicit contraction bounds in terms of the instantaneous steady state and time-integrated dissipation rates. For a class of microscopically derived master equations, we further obtain a graph-theoretic characterization of the aforementioned conditions that directly links the structure of the jump operators to the relaxation properties. The general theory is illustrated by applications to driven finite-level systems, including a detailed three-level example, and is extended to a non-Markovian setting by means of time-local master equations that become of GKLS form at long times. These findings pave the way for the development of a more general theory of relaxation beyond the Markovian case.

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