arXiv subjects

Karol Życzkowski

Publications and source records attributed to Karol Życzkowski.

At least 19 recordsLinked to original sources

arXiv2026

Product Weyl--Heisenberg covariant mutually unbiased bases and extremal non-stabilizerness

Discrete structures in product Hilbert spaces are investigated. For monopartite systems of size $d$ one relies on the Weyl--Heisenberg group $WH(d)$, while in the case of composite Hilbert spaces with the local dimensions $d_i$ we identify designs covariant with respect to the product group, $\bigotimes_i WH(d_i)$. In analogy with magic -- a quantity attaining its maximum for states fiducial with respect to $WH(d)$ -- we introduce a similar quantifier of {\sl product magic}, defined with respect to the product group. The maximum of this quantity over all equimodular vectors yields fiducial states that generate $d = \prod_i d_i$ \textit{a priori} isoentangled mutually unbiased bases (MUBs), which, when supplemented by the identity, form their complete set. Such fiducial states are explicitly constructed in all prime-power dimensions $d = p^n$ with $p\ge 3$. The result for $p\ge 5$ extends the construction of Klappenecker and R{ö}tteler, whereas for $p=3$ it is mathematically distinct and is based on Galois rings. The global maximum of the quantifier of product magic for $d=2^3$ yields fiducial states corresponding to the symmetric informationally complete (SIC) generalized measurement of Hoggar. Our approach feeds into a unifying perspective in which highly symmetric quantum designs emerge from fiducial states with extremal properties via structured group-orbit constructions.

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arXiv2026

Statistical mechanics of multipartite entanglement in hypergraph states

We investigate multipartite entanglement in a particular family of pure $n$-qubit hypergraph states through a statistical-mechanics framework, where the average bipartite purity maps onto an effective Hamiltonian of $2^n$ classical binary spins. In this correspondence, each hypergraph state uniquely corresponds to a classical spin configuration, while temperature serves as a control parameter that continuously interpolates between a uniform ensemble of random hypergraph states at high temperature and maximally multipartite entangled states (MMES) at zero temperature. Remarkably, the exponential of the zero-temperature entropy directly gives the number of MMES within the set of hypergraph states. For small system sizes ($n \leq 5$), we perform an exact enumeration, fully characterizing the energy landscape and associated thermodynamic observables, and validating known MMES counts. For larger systems ($n = 6$ and $7$), where exact methods become computationally infeasible, we employ simulated annealing and parallel tempering algorithms to efficiently sample the exponentially large state space. Our analysis yields quantitative predictions of the number of MMES and reveals how entanglement is statistically distributed across the sets of hypergraph states. These results establish hypergraph states as an ideal platform for investigating multipartite entanglement through thermodynamic methods, offering both computational advances and physical insights into the structure of quantum entanglement in restricted families of quantum states.

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arXiv2026

Reachable sets under Kolmogorov dynamics in the probability simplex

The finite-time reachability problem - whether two probability vectors can be transformed in a time t through a generator from a given set - is a key question in Markovian dynamics. We address this problem for normalized Kolmogorov generators with tools from differential geometry. The shortest time T connecting an ordered pair of states defines an (asymmetric) quasi-distance in the probability simplex, which distinguishes between outgoing and incoming reachable sets, and captures the inherent irreversibility of Markov dynamics. The Finsler metric corresponding to normalized generators with bounded diagonal entries is characterized, and time-independent evolutions that saturate speed limits are presented. Analytical boundaries of both reachable sets are derived and the ratio of their corresponding volumes assess the bounds obtained. The asymmetry with respect to the uniform probability vector is shown to be related to the Kullback-Leibler relative entropy.

math-ph↗

arXiv2026

Entangling power and fidelity diagnostic for bipartite quantum channels

We study two distinct diagnostics of bipartite quantum channels: the average fidelity between a desired state and noisy output across different classes of input states, and generation of entanglement from pure product input states, quantified by well-defined entangling power. These diagnostics are generally independent, but together they provide a more complete description of bipartite channels. We show that the fidelity averaged over any local-unitary orbit with fixed Schmidt coefficients for equal local dimensions is completely determined by the average input-output fidelity and its restriction to product inputs. We also introduce concurrence- and negativity-based entangling power for two-qubit channels, prove their convexity and monotonicity under local postprocessing, and show that, unlike the previously proposed linear-entropy quantity, they vanish for all separable channels. Representative separable and nonseparable channels are investigated. Finally, we generalize our definitions of entangling power to nonproduct inputs, and provide an analytical lower bound for the concurrence-based entanglement variation, showing its effectiveness with specific examples.

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arXiv2026

Absolutely Maximally Entangled States of $2q$ Parties in Every Odd Prime-Power Dimension $q$

Absolutely maximally entangled (AME) states represent an extreme form of multipartite entanglement: every reduced system containing at most half of the parties is maximally mixed. These states provide perfect tensors and optimal quantum error-correcting codes, yet their existence is known only in restricted parameter regimes. For every odd prime power $q=p^e\ge3$, we construct a stabilizer $\mathrm{AME}(2q,q)$ state whose normalized one-party projection yields a stabilizer $\mathrm{AME}(2q-1,q)$ state. A closed-form $q\times q$ bordered-circulant matrix $A_q$ over $\mathbb{F}_{q^2}$ generates a Hermitian self-dual maximum distance separable (MDS) code $[2q,q,q+1]_{q^2}$, which lies outside the extended Reed-Solomon classes. In suitable bases, the amplitude tensors define normalized $q$-unitary complex Hadamard matrices of order $q^q$ with $p$th-root phases. Additional constructions yield $\mathrm{AME}(q+3,q)$ states and families at intermediate particle numbers through explicit rescalings of selected submatrices. We also provide nine explicit parent matrices and the corresponding one-party projections.

quant-ph↗

arXiv2026

Error accumulation in dissipative quantum circuits

Present-day quantum processors are open systems in which dissipation and decoherence degrade the information carried by a quantum state as it propagates through a circuit. We study this degradation in dissipative random quantum circuits, modeling each two-qubit gate as a diluted unitary that interpolates between the intended unitary operation and a random dissipative quantum channel. Using the fidelity between the ideal and noisy output states as a diagnostic, we show that sufficiently random unitary gates or dissipative Kraus operators lead to a universal decay of the average fidelity. This decay is determined only by the dissipation strength, system size, and circuit depth, while microscopic details of the gates and noise, including the Kraus rank, appear only in subleading corrections to higher moments. We derive a closed-form expression for the average fidelity in terms of three physically meaningful error parameters, dissipation strength, coherent two-qubit gate-error strength, and connectivity-error probability. Finally, we identify the cases when dissipative effects can be reproduced by an unitary-noise model and when dissipation-induced decoherence remains distinguishable from coherent noise at the level of the average fidelity.

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arXiv2026

Approximate pushforward designs and image bounds on approximations

We extend the framework of quantum pushforward designs to the approximate setting, in which moment operators agree only up to finite precision. We first formulate a general transfer theorem for maps that act linearly at the level of moments. This separates direct applications, controlled by standard Schatten-norm estimates, from refinements that use additional tensorial structure. Dephasing is shown to be contractive and therefore sends an approximate projective design to a simplex design without increasing its error. Ordinary partial-trace estimates give immediate bounds for mixed-state and channel designs. We then prove a sharp Schatten-norm bound for the partial trace restricted to the totally symmetric subspace. This replaces the full-environment norm coefficient by one governed by a symmetric-subspace dimension and yields asymptotically tighter estimates for both mixed-state and channel designs. Numerical simulations for induced mixed-state designs are consistent with the resulting hierarchy of bounds.

quant-ph↗

arXiv2026

Spin-coherent quantum designs

Coherent states bridge the gap between quantum and classical physics, but their overcomplete and nonorthogonal nature makes it difficult to identify the minimal discrete set needed to reconstruct quantum information. Finite spin-coherent tomography and discrete coherent-state operator bases are known, but here we address the more specific rank-resolved problem of preserving the canonical contravariant-symbol representation. We show that the canonical finite coherent-state formula reconstructs every operator in the rank-$S$ sector exactly if and only if the sampling points form a spherical $(2J+S)$-design. We call the associated configurations spin-coherent quantum designs. We further give a fully explicit positive-weight Gauss-Legendre construction that avoids the need for an equal-weight spherical design. Together, these results establish a unified framework for reading out physical observables from a handful of measurement samples, playing for spin systems the role that the so-called von Neumann lattice plays for canonical coherent states. Finally, we derive practical protocols for estimating moments of spin operators from these constructions, with direct applications to polarimetry, magnetometry, and quantum state tomography.

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arXiv2026

Absolutely maximally entangled pure states of multipartite quantum systems

Absolutely maximally entangled (AME) pure states of a system composed of $N$ parties are distinguished by the property that for any splitting at least one partial trace is maximally mixed. Due to maximal possible correlations between any two selected subsystems these states have numerous applications in various fields of quantum information processing including multi-user teleportation, quantum error correction and secret sharing. We present an updated survey of various techniques to generate such strongly entangled states, including those going beyond the standard construction of graph and stabilizer states. Our contribution includes, in particular, analysis of the degree of entanglement of reduced states obtained by partial trace of AME projectors, states obtained by a symmetric superposition of GHZ states, an orthogonal frequency square representation of the "golden" AME state and an updated summary of the number of local unitary equivalence classes.

quant-ph↗

arXiv2026

Graph restricted tensors: building blocks for holographic networks

We analyze few-body quantum states with particular correlation properties imposed by the requirement of maximal bipartite entanglement for selected partitions of the system into two complementary parts. A novel framework to treat this problem by encoding these constraints in a graph is advocated; the resulting objects are called ``graph-restricted tensors''. This framework encompasses several examples previously treated in the literature, such as 1-uniform multipartite states, quantum states related to dual unitary operators and absolutely maximally entangled states (AME) corresponding to 2-unitary matrices. Original examples of presented graph-restricted tensors are motivated by tensor network models for the holographic principle. In concrete cases we find exact analytic solutions, demonstrating thereby that there exists a vast landscape of non-stabilizer tensors useful for the lattice models of holography.

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arXiv2026

The joint numerical range of three hermitian $4\times 4$ matrices

We analyze the joint numerical range $W$ of three hermitian matrices of order four. In the generic case, this three-dimensional convex set has a smooth boundary. We analyze non-generic structures. Fifteen possible classes regarding the numbers of non-elliptic faces in the boundary of $W$ are identified and an explicit example is presented for each class. Secondly, it is shown that a nonempty intersection of three mutually distinct one-dimensional faces is a corner point. Thirdly, introducing a tensor product structure into $\mathbb C^4=\mathbb C^2\otimes\mathbb C^2$, one defines the separable joint numerical range - a subset of $W$ useful in studies of quantum entanglement. The boundary of the separable numerical range is compared with that of $W$.

math.FA↗

arXiv2026

Multipartite entanglement of random states of qubits

We investigate multipartite entanglement via the statistical properties of pure quantum states of n-qubits. By analyzing the distribution of purity among balanced bipartitions, we compare Haar-typical states, uniformly distributed on the unit sphere of states, with Hadamard states, being characterized by equal weights in the computational basis. We analyze different ensembles of Hadamard states characterized by their phase distributions. Through analytical and numerical calculations, we show that Hadamard states exhibit, on average, a higher degree of entanglement than Haar-typical states. In addition, we show that a particular class of Hadamard states, characterized by real coefficients with alternating signs, known as hypergraph states, appears especially relevant in the search for maximally multipartite entangled states, both for their structural simplicity and the increased likelihood of sampling highly entangled states. These results identify Hadamard states as a tractable yet promising class for exploring multipartite entanglement structures and advancing the characterization of maximally multipartite entangled quantum states.

quant-ph↗

arXiv2026

Benchmarking quantum devices beyond classical capabilities

Rapid development of quantum computing technology has led to a wide variety of sophisticated quantum devices. Benchmarking these systems becomes crucial for understanding their capabilities and paving the way for future advancements. The Quantum Volume (QV) test is one of the most widely used benchmarks for evaluating quantum computer performance due to its architecture independence. However, as the number of qubits in a quantum device grows, the test faces a significant limitation. It requires determining the subspace of the most probable outcomes, a task that is typically performed via classical simulation of the quantum circuit and therefore incurs an exponential computational cost. In this work, we propose modifications to the QV test, by adopting a carefully restricted circuit ensemble generated from a gate set that remains universal for quantum computation, that allows for the direct determination of the heavy-output subspace. Crucially, the modified circuits remain capable of general quantum computation. This approach overcomes the scalability barrier of the Quantum Volume test beyond classical computational limits, while still probing the key features of universal quantum computing.

quant-ph↗

arXiv2026

Quantum Resource Theories beyond Convexity

A class of quantum resource theories, based on non-convex star-shape sets, presented in this work captures the key quantum properties that cannot be studied by standard convex theories. We provide operational interpretations for a resource of this class and demonstrate its advantage to improve performance of correlated quantum discrimination tasks and testing of quantum combs. Proposed techniques provide useful tools to describe quantum discord, total correlations in composite quantum systems and to estimate the degree of non-Markovianity of an analyzed quantum dynamics. Other applications include the problem of unistochasticity of a given bistochastic matrix, with relevance for quantization of classical dynamics and studies of violation of CP-symmetry in high energy physics. In all these cases, the non-linear witnesses introduced here outperform the standard linear witnesses. Importance of our findings for quantum information theory is also emphasized.

quant-ph↗

arXiv2026

Quantum Chaos and Quantum Information: Interactions and Implications

The notion of Shannon entropy is crucial for the theory of classical information. In quantum information theory, an analogous key role is played by the von Neumann entropy: quantum information processing is closely related to entropy dynamics. This reveals a direct link with the theory of quantum chaotic systems, which can be characterized by a positive entropy production. Furthermore, noise, which inevitably affects any quantum system, can be modeled by a random quantum operation or by coupling to an environment in a generic chaotic state. In this contribution, we emphasize the universality of quantum chaotic dynamics and discuss its implications for quantum information processing.

quant-ph↗

arXiv2026

Quantum convolutional channels and multiparameter families of 2-unitary matrices

Many alternative approaches to construct quantum channels with large entangling capacity were proposed in the past decade, resulting in multiple isolated gates. In this work, we put forward a novel one, inspired by convolution, which provides greater freedom of nonlocal parameters. Although quantum counterparts of convolution have been shown not to exist for pure states, several attempts with various degrees of rigorousness have been proposed for mixed states. In this work, we follow the approach based on coherifications of multi-stochastic operations and demonstrate a surprising connection to gates with high entangling power. In particular, we identify conditions necessary for the convolutional channels constructed using our method to possess maximal entangling power. Furthermore, we establish new, continuous classes of bipartite 2-unitary matrices of dimension $d^2$ for $d = 7$ and $d = 9$, with $2$ and $4$ free nonlocal parameters beyond simple phasing of matrix elements, corresponding to perfect tensors of rank $4$ or 4-partite absolutely maximally entangled states.

quant-ph↗

arXiv2026

Effective criteria for entanglement witnesses in small dimensions

We present an effective set of necessary and sufficient criteria for block-positivity of matrices of order $4$ over $\mathbb{C}$. The approach is based on Sturm sequences and quartic polynomial positivity conditions presented in recent literature. The procedure allows us to test whether a given $4\times 4$ complex matrix corresponds to an entanglement witness, and it is exact when the matrix coefficients belong to the rationals, extended by $\mathrm{i}$. The method can be generalized to $\mathcal{H}_2\otimes\mathcal{H}_d$ systems for $d>2$ to provide necessary but not sufficient criterion for block-positivity. We also outline an alternative approach to the problem relying on Gröbner bases.

quant-ph↗

arXiv2025

Random matrix perspective on probabilistic error cancellation

Probabilistic error cancellation is an attempt to reverse the effect of dissipative noise channels on quantum computers by applying unphysical channels after the execution of a quantum algorithm on noisy hardware. We investigate on general grounds the properties of such unphysical quantum channels by considering a random matrix ensemble modeling noisy quantum algorithms. We show that the complex spectra of denoiser channels inherit their structure from random Lindbladians. Additional structure imposed by the locality of noise channels of the quantum computer emerges in terms of a hierarchy of timescales.

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