arXiv subjects

Karol Życzkowski

Publications and source records attributed to Karol Życzkowski.

At least 37 records · Page 2Linked to original sources

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.

quant-ph↗

arXiv2025

The max-type quasimetrics on probability simplices

Quasimetric spaces form a natural framework to study distance problems with an inherent directional asymmetry. We introduce a simple novel class of quasimetrics on probability simplices, inspired by the Chebyshev distance. It is shown that such quasimetrics have expedient geometric properties -- they induce the Euclidean topology and a Finslerian infinitesimal structure, with which the probability simplices become geodesic spaces. Moreover, we prove that the broad family of the proposed quasimetrics are monotone under bistochastic maps.

math.MG↗

arXiv2025

Matrix logistic map: fractal spectral distributions and transfer of chaos

The standard logistic map, $x'=ax(1-x)$, serves as a paradigmatic model to demonstrate how apparently simple non-linear equations lead to complex and chaotic dynamics. In this work we introduce and investigate its matrix analogue defined for an arbitrary matrix $X$ of a given order $N$. We show that for an arbitrary initial ensemble of hermitian random matrices with a continuous level density supported on the interval $[0,1]$, the asymptotic level density converges to the invariant measure of the logistic map. Depending on the parameter $a$ the constructed measure may be either singular, fractal or described by a continuous density. In a broader class of the map multiplication by a scalar logistic parameter $a$ is replaced by transforming $aX(\mathbb{I}-X)$ into $BX(\mathbb{I}-X)B^{\dagger}$, where $A=BB^{\dagger}$ is a fixed positive matrix of order $N$. This approach generalizes the known model of coupled logistic maps, and allows us to study the transition to chaos in complex networks and multidimensional systems. In particular, associating the matrix $B$ with a given graph we demonstrate the gradual transfer of chaos between subsystems corresponding to vertices of a graph and coupled according to its edges.

nlin.CD↗

arXiv2025

Certifying nonlocal properties of noisy quantum operations

Certifying quantum properties from the probability distributions they induce is an important task for several purposes. While this framework has been largely explored and used for quantum states, its extrapolation to the level of channels started recently in a variety of approaches. In particular, little is known about to what extent noise can spoil certification methods for channels. In this work we provide a unified methodology to certify nonlocal properties of quantum channels from the correlations obtained in prepare-and-measurement protocols: our approach gathers fully and semi-device-independent existing methods for this purpose, and extends them to new certification criteria. In addition, the effect of different models of dephasing noise is analysed. Some noise models are shown to generate nonlocality and entanglement in special cases. In the extreme case of complete dephasing, the measurement protocols discussed yield particularly simple tests to certify nonlocality, which can be obtained from known criteria by fixing the dephasing basis. These are based on the relations between bipartite quantum channels and their classical analogues: bipartite stochastic matrices defining conditional distributions.

quant-ph↗

arXiv2025

Fidelity decay and error accumulation in random quantum circuits

We present a comprehensive analysis of fidelity decay and error accumulation in faulty quantum circuit models. Our work devises an analytical bound for the average fidelity between desired and faulty output states, accounting for errors that may arise during the implementation of two-qubit gates and multi-qubit permutations. It is shown that fidelity decays exponentially with both circuit depth and the number of qubits raised to an architecture-dependent power, and determine the decay rates as a function of the two types of errors. Furthermore, we establish a robust linear relationship between fidelity and the heavy output frequency used in Quantum Volume tests to benchmark quantum processors, under the considered errors protocol. These findings pave the way for predicting the behavior of fidelity in the presence of specific errors and offer insights into the best strategies for increasing Quantum Volume.

quant-ph↗

arXiv2025

Cyclic measurements and simplified quantum state tomography

Tomographic reconstruction of quantum states plays a fundamental role in benchmarking quantum systems and accessing information encoded in quantum-mechanical systems. Among the informationally complete sets of quantum measurements, the tight ones provide a linear reconstruction formula and minimize the propagation of statistical errors. However, implementing tight measurements in the lab is challenging due to the high number of required measurement projections, involving a series of experimental setup preparations. In this work, we introduce the notion of cyclic tight measurements, which allow us to perform full quantum state tomography while considering only repeated application of a single unitary-based quantum device during the measurement stage. This type of measurement significantly simplifies the complexity of the experimental setup required to retrieve the quantum state of a physical system. Additionally, we design a feasible setup preparation procedure that produces well-approximated cyclic tight measurements in every finite dimension.

quant-ph↗

arXiv2025

Spectral delineation of Markov Generators: Classical vs Quantum

The celebrated theorem of Perron and Frobenius implies that spectra of classical Markov operators, represented by stochastic matrices, are restricted to the unit disk. This property holds also for spectra of quantum stochastic maps (quantum channels), which describe quantum Markovian evolution in discrete time. Moreover, the spectra of stochastic $N \times N$ matrices are additionally restricted to a subset of the unit disk, called Karpeleviuc region, the shape of which depends on $N$. We address the question of whether the spectra of generators, which induce Markovian evolution in continuous time, can be bound in a similar way. We propose a rescaling that allows us to answer this question affirmatively. The eigenvalues of the rescaled classical generators are confined to the modified Karpeleviuc regions, whereas the eigenvalues of the rescaled quantum generators fill the entire unit disk.

cond-mat.stat-mech↗

arXiv2025

Quantum Circuits for High-Dimensional Absolutely Maximally Entangled States

Absolutely maximally entangled (AME) states of multipartite quantum systems exhibit maximal entanglement across all possible bipartitions. These states lead to teleportation protocols that surpass standard teleportation schemes, determine quantum error correction codes and can be used to test performance of current term quantum processors. Several AME states can be constructed from graph states using minimal quantum resources. However, there exist other constructions that depart from the stabilizer formalism. In this work, we present explicit quantum circuits to generate exemplary non-stabilizer AME states of four subsystems with four, six, and eight levels each and analyze their capabilities to perform quantum information tasks.

quant-ph↗

arXiv2025

Quantum Pushforward Designs

Designs, structures connected to averaging with respect to a given measure using finite sets of points, have proven themselves as invaluable tools across the field of quantum information, finding their uses in state and process tomography, key distribution and others. In this work, we introduce a new concept of pushforward designs, which allows us to obtain new structures from already existing ones by mapping them between the spaces, with specific examples including simplex designs and mixed state designs from complex projective designs. Based on the general concept, we put forward a structure called channel $[t,k]$-design, allowing for averaging over space of quantum channels for systems in contact with an environment of dimension $k$. Based on this notion, we introduce the concept of effective environment dimensionality $k^*$, which we estimate for the IBM Kyoto quantum computer to be below $2.2$ for times up to $350μ\text{s}$.

quant-ph↗

arXiv2024

Discrete dynamics in the set of quantum measurements

A quantum measurement, often referred to as positive operator-valued measurement (POVM), is a set of positive operators $P_j=P_j^†\geq 0$ summing to identity, $\sum_jP_j=\mathbb{1}$. This can be seen as a generalization of a probability distribution of positive real numbers summing to unity, whose evolution is given by a stochastic matrix. We describe discrete transformations in the set of quantum measurements by {\em blockwise stochastic matrices}, composed of positive blocks that sum columnwise to identity, using the notion of {\em sequential product} of matrices. We show that such transformations correspond to a sequence of quantum measurements. Imposing additionally the dual condition that the sum of blocks in each row is equal to identity, we arrive at blockwise bistochastic matrices (also called {\em quantum magic squares}). Analyzing their dynamical properties, we formulate our main result: a quantum analog of the Ostrowski description of the classical Birkhoff polytope, which introduces the notion of majorization between quantum measurements. Our framework provides a dynamical characterization of the set of blockwise bistochastic matrices and establishes a resource theory in this set.

quant-ph↗

arXiv2024

Bulk-boundary correspondence from hyper-invariant tensor networks

We introduce a tensor network designed to faithfully simulate the AdS/CFT correspondence, akin to the multi-scale entanglement renormalization ansatz (MERA), following hyper-invariant tensor network. The proposed construction integrates bulk indices within the network architecture to uphold the key features of the HaPPY code, including complementary recovery. This framework accurately reproduces the boundary conformal field theory's (CFT) two- and three-point correlation functions, while considering the image of any bulk operator. Furthermore, we provide an explicit methodology for calculating the correlation functions in an efficient manner. Our findings highlight the physical aspects of the relation between bulk and boundary within the tensor network models, contributing to the understanding and simulation of holographic principles in quantum information.

quant-ph↗

arXiv2024

Shared entanglement for three-party causal order guessing game

In a variant of communication tasks, players cooperate in choosing their local strategies to compute a given task later, working separately. Utilizing quantum bits for communication and sharing entanglement between parties is a recognized method to enhance performance in these situations. In this work, we introduce the game for which three parties, Alice, Bob and Charlie, would like to discover the hidden order in which they make the moves. We show the advantage of quantum strategies that use shared entanglement and local operations over classical setups for discriminating operations' composition order. The role of quantum resources improving the probability of successful discrimination is also investigated.

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

Channel nonlocality under decoherence

The implementation of realistic quantum devices requires a solid understanding of the nonlocal resources present in quantum channels, and the effects of decoherence on them. Here we quantify nonlocality of bipartite quantum channels and identify its component resisting the effects of dephasing noise. Despite its classical nature, we demonstrate that the latter plays a relevant role in performing quantum protocols, such as state transformations and quantum coding for noisy communication. In the converse direction, we show that simulating certain stochastic processes with quantum channels undergoing decoherence has a communication advantage with respect to their classical simulation.

quant-ph↗

arXiv2024

Multi-Unitary Complex Hadamard Matrices

We analyze the set of real and complex Hadamard matrices with additional symmetry constrains. In particular, we link the problem of existence of maximally entangled multipartite states of $2k$ subsystems with $d$ levels each to the set of complex Hadamard matrices of order $N=d^k$. To this end, we investigate possible subsets of such matrices which are, dual, strongly dual ($H=H^{\rm R}$ or $H=H^{\rmΓ}$), two-unitary ($H^R$ and $H^Γ$ are unitary), or $k$-unitary. Here $X^{\rm R}$ denotes reshuffling of a matrix $X$ describing a bipartite system, and $X^{\rm Γ}$ its partial transpose. Such matrices find several applications in quantum many-body theory, tensor networks and classification of multipartite quantum entanglement and imply a broad class of analytically solvable quantum models in $1+1$ dimensions.

quant-ph↗

arXiv2024

Two-Unitary Complex Hadamard Matrices of Order $36$

A family of two-unitary complex Hadamard matrices (CHM) stemming from a particular matrix, of size $36$ is constructed. Every matrix in this orbit remains unitary after operations of partial transpose and reshuffling which makes it a distinguished subset of CHM. It provides a novel solution to the quantum version of the Euler problem, in which each field of the Graeco-Latin square of size six contains a symmetric superposition of all $36$ officers with phases being multiples of sixth root of unity. This simplifies previously known solutions as all amplitudes of the superposition are equal and the set of phases consists of $6$ elements only. Multidimensional parameterization allows for more flexibility in a potential experimental realization.

quant-ph↗

arXiv2024

Distribution of Fidelity in Quantum State Transfer Protocols

Quantum state transfer protocols are a major toolkit in many quantum information processing tasks, from quantum key distribution to quantum computation. To assess performance of a such a protocol, one often relies on the average fidelity between the input and the output states. Going beyond this scheme, we analyze the entire probability distribution of fidelity, providing a general framework to derive it for the transfer of single- and two-qubit states. Starting from the delta-like shape of the fidelity distribution, characteristic to perfect transfer, we analyze its broadening and deformation due to realistic features of the process, including non-perfect read-out timing. Different models of quantum transfer, sharing the same value of the average fidelity, display different distributions of fidelity, providing thus additional information on the protocol, including the minimum fidelity.

quant-ph↗