arXiv subjects

Rafał Bistroń

Publications and source records attributed to Rafał Bistroń.

13 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

Distances between pure quantum states induced by a distance matrix

Given an $n \times n$ distance matrix, we construct an infinite family of distances $d_p$ (where $p \geq 2$) on the complex projective space $\mathbb{P}(\mathbb{C}^n)$ modelling the space of pure states of an $n$-level quantum system. The construction can be seen as providing a natural way to isometrically embed any given finite metric space into the space of pure quantum states 'spanned' upon it. In order to show that the maps $d_p$ are indeed distance functions -- in particular, that they satisfy the triangle inequality -- we employ methods of analysis, multilinear algebra and convex geometry, obtaining a nontrivial auxiliary convexity result in the process. In addition, a way of extending the distances $d_p$ to mixed states is proposed for a broad class of distance matrices.

math-ph↗

arXiv2026

A Study in Thermal: Advantage framework for resource engines

Thermal engines have been one of the principal topics of thermodynamics ever since its beginning. Today, in the era of the second quantum revolution, although the thermal processes in constant temperatures are relatively well understood in the language of resource theories, a framework to describe thermal engines is still in its infancy. In this work we formalize the resource theory of engines, initially put forward in~\cite{KAMIL}, and define such quantities as engine efficiency. Then, we turn to a detailed study of thermal engines based on free operations arising from the resource theory of athermality under different restrictions: thermal operation, semilocal thermal operations and local thermal operation with classical communication. In order to provide analytic lower bounds for thermal engine operation we construct tree-states -- free states, which can be obtained from Gibbs state in simple protocol consisting solely of two-level operations. Furthermore, we derive a full description of the engine based on semilocal thermal operations by describing free states, faithful monotones and catalytic advantages.

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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

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 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

Local Thermal Operations and Classical Communication

In quantum thermodynamics, understanding the interplay between locality, thermal constraints, and communication remains an open challenge. In this manuscript, we introduce Local Thermal Operations and Classical Communication (LTOCC), a novel operational framework that unifies the distant laboratories paradigm with thermodynamic restrictions, defining the fundamental limits on transformations between spatially separated systems. We establish a hierarchy of LTOCC protocols, demonstrating inclusion relations between different levels and revealing their deep connection to semilocal thermal operations. To formalize this framework, we develop thermal tensors and bithermal tensors, extending stochastic and tristochastic tensors to thermodynamic settings and providing new mathematical tools for constrained quantum processes. Finally, we present limitations imposed by LTOCC on single- and multi-copy CHSH scenario, demonstrating no violation in former and a gap between thermal and athermal local operations in the latter with respect to their capability to detect entanglement.

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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.

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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.

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arXiv2023

Braided symmetries of $SU_{q,ϕ}(2)$ and Podleś Spheres

We present an explicit form of braided symmetries of the quantum spheres, by introducing a braided quantum Hopf algebra $\cU_{q, ϕ}$ and demonstrating that they are braided Hopf modules over this braided Hopf algebra. To obtain this result, we systematically develop braided versions of structures like pairing between Hopf algebras, left module algebra as well as their compatibility with the $*$-structure.

math.QA↗

arXiv2023

A new class of distances on complex projective spaces

The complex projective space $\mathbb{P}(\mathbb{C}^n)$ can be interpreted as the space of all quantum pure states of size $n$. A distance on this space, interesting from the perspective of quantum physics, can be induced from a classical distance defined on the $n$-point probability simplex by the `earth mover problem'. We show that this construction leads to a quantity satisfying the triangle inequality, which yields a true distance on complex projective space belonging to the family of quantum $2$-Wasserstein distances.

math-ph↗

arXiv2023

Tristochastic operations and convolutions of quantum states

The notion of convolution of two probability vectors, corresponding to a coincidence experiment can be extended for a family of binary operations determined by (tri)stochastic tensors, to describe Markov chains of a higher order. The problem of associativity, commutativity and the existence of neutral elements and inverses is analyzed for such operations. For a more general setup of multi-stochastic tensors, we present the characterization of their probability eigenvectors. Similar results are obtained for the quantum case: we analyze tristochastic channels, which induce binary operations defined in the space of quantum states. Studying coherifications of tristochastic tensors we propose a quantum analogue of the convolution of probability vectors defined for two arbitrary density matrices of the same size. Possible applications of this notion to construct schemes of error mitigation or building blocks in quantum convolutional neural networks are discussed.

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arXiv2022

Monotonicity of the quantum 2-Wasserstein distance

We study a quantum analogue of the 2-Wasserstein distance as a measure of proximity on the set $Ω_N$ of density matrices of dimension $N$. We show that such (semi-)distances do not induce Riemannian metrics on the tangent bundle of $Ω_N$ and are typically not unitary invariant. Nevertheless, we prove that for $N=2$ dimensional Hilbert space the quantum 2-Wasserstein distance (unique up to rescaling) is monotonous with respect to any single-qubit quantum operation and the solution of the quantum transport problem is essentially unique. Furthermore, for any $N \geq 3$ and the quantum cost matrix proportional to a projector we demonstrate the monotonicity under arbitrary mixed unitary channels. Finally, we provide numerical evidence which allows us to conjecture that the unitary invariant quantum 2-Wasserstein semi-distance is monotonous with respect to all CPTP maps in any dimension $N$.

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