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106 results on '"Ion Nechita"'

1. Algebra of Nonlocal Boxes and the Collapse of Communication Complexity

Author

Pierre Botteron, Anne Broadbent, Reda Chhaibi, Ion Nechita, and Clément Pellegrini

Subjects
Physics, QC1-999
Abstract

Communication complexity quantifies how difficult it is for two distant computers to evaluate a function $f(X,Y)$, where the strings $X$ and $Y$ are distributed to the first and second computer respectively, under the constraint of exchanging as few bits as possible. Surprisingly, some nonlocal boxes, which are resources shared by the two computers, are so powerful that they allow to $collapse$ communication complexity, in the sense that any Boolean function f can be correctly estimated with the exchange of only one bit of communication. The Popescu-Rohrlich (PR) box is an example of such a collapsing resource, but a comprehensive description of the set of collapsing nonlocal boxes remains elusive. In this work, we carry out an algebraic study of the structure of wirings connecting nonlocal boxes, thus defining the notion of the "product of boxes" $P\boxtimes Q$, and we show related associativity and commutativity results. This gives rise to the notion of the "orbit of a box", unveiling surprising geometrical properties about the alignment and parallelism of distilled boxes. The power of this new framework is that it allows us to prove previously-reported numerical observations concerning the best way to wire consecutive boxes, and to numerically and analytically recover recently-identified noisy $\texttt{PR}$ boxes that collapse communication complexity for different types of noise models.

Published
2024
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2. A refinement of Reznick's Positivstellensatz with applications to quantum information theory

Author

Alexander Müller-Hermes, Ion Nechita, and David Reeb

Subjects
Physics, QC1-999
Abstract

In his solution of Hilbert's 17th problem Artin showed that any positive definite polynomial in several variables can be written as the quotient of two sums of squares. Later Reznick showed that the denominator in Artin's result can always be chosen as an $N$-th power of the squared norm of the variables and gave explicit bounds on $N$. By using concepts from quantum information theory (such as partial traces, optimal cloning maps, and an identity due to Chiribella) we give simpler proofs and minor improvements of both real and complex versions of this result. Moreover, we discuss constructions of Hilbert identities using Gaussian integrals and we review an elementary method to construct complex spherical designs. Finally, we apply our results to give improved bounds for exponential quantum de Finetti theorems in the real and in the complex setting.

Published
2023
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3. Measurement Incompatibility versus Bell Nonlocality: An Approach via Tensor Norms

Author

Faedi Loulidi and Ion Nechita

Subjects
Physics, QC1-999, Computer software, QA76.75-76.765
Abstract

Measurement incompatibility and quantum nonlocality are two key features of quantum theory. Violations of Bell inequalities require quantum entanglement and incompatibility of the measurements used by the two parties involved in the protocol. We analyze the converse question: for which Bell inequalities is the incompatibility of measurements enough to ensure a quantum violation? We relate the two questions by comparing two tensor norms on the space of dichotomic quantum measurements, one characterizing measurement compatibility and the other characterizing violations of a given Bell inequality. We provide sufficient conditions for the equivalence of the two notions in terms of the matrix describing the correlation Bell inequality. We show that the Clauser-Horne-Shimony-Holt inequality and its variants are the only ones that satisfy it.

Published
2022
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4. Order preserving maps on quantum measurements

Author

Teiko Heinosaari, Maria Anastasia Jivulescu, and Ion Nechita

Subjects
Physics, QC1-999
Abstract

We study the partially ordered set of equivalence classes of quantum measurements endowed with the post-processing partial order. The post-processing order is fundamental as it enables to compare measurements by their intrinsic noise and it gives grounds to define the important concept of quantum incompatibility. Our approach is based on mapping this set into a simpler partially ordered set using an order preserving map and investigating the resulting image. The aim is to ignore unnecessary details while keeping the essential structure, thereby simplifying e.g. detection of incompatibility. One possible choice is the map based on Fisher information introduced by Huangjun Zhu, known to be an order morphism taking values in the cone of positive semidefinite matrices. We explore the properties of that construction and improve Zhu's incompatibility criterion by adding a constraint depending on the number of measurement outcomes. We generalize this type of construction to other ordered vector spaces and we show that this map is optimal among all quadratic maps.

Published
2022
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5. A Fisher Information-Based Incompatibility Criterion for Quantum Channels

Author

Qing-Hua Zhang and Ion Nechita

Subjects
incompatibility criterion, quantum channels, quantum measurements, Fisher information, Science, Astrophysics, QB460-466, Physics, QC1-999
Abstract

We introduce a new incompatibility criterion for quantum channels based on the notion of (quantum) Fisher information. Our construction is based on a similar criterion for quantum measurements put forward by H. Zhu. We then study the power of the incompatibility criterion in different scenarios. First, we prove the first analytical conditions for the incompatibility of two Schur channels. Then, we study the incompatibility structure of a tuple of depolarizing channels, comparing the newly introduced criterion with the known results from asymmetric quantum cloning.

Published
2022
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6. Maximal violation of steering inequalities and the matrix cube

Author

Andreas Bluhm and Ion Nechita

Subjects
Physics, QC1-999
Abstract

In this work, we characterize the amount of steerability present in quantum theory by connecting the maximal violation of a steering inequality to an inclusion problem of free spectrahedra. In particular, we show that the maximal violation of an arbitrary unbiased dichotomic steering inequality is given by the inclusion constants of the matrix cube, which is a well-studied object in convex optimization theory. This allows us to find new upper bounds on the maximal violation of steering inequalities and to show that previously obtained violations are optimal. In order to do this, we prove lower bounds on the inclusion constants of the complex matrix cube, which might be of independent interest. Finally, we show that the inclusion constants of the matrix cube and the matrix diamond are the same. This allows us to derive new bounds on the amount of incompatibility available in dichotomic quantum measurements in fixed dimension.

Published
2022
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7. Diagonal unitary and orthogonal symmetries in quantum theory

Author

Satvik Singh and Ion Nechita

Subjects
Physics, QC1-999
Abstract

We analyze bipartite matrices and linear maps between matrix algebras, which are respectively, invariant and covariant, under the diagonal unitary and orthogonal groups' actions. By presenting an expansive list of examples from the literature, which includes notable entries like the Diagonal Symmetric states and the Choi-type maps, we show that this class of matrices (and maps) encompasses a wide variety of scenarios, thereby unifying their study. We examine their linear algebraic structure and investigate different notions of positivity through their convex conic manifestations. In particular, we generalize the well-known cone of completely positive matrices to that of triplewise completely positive matrices and connect it to the separability of the relevant invariant states (or the entanglement breaking property of the corresponding quantum channels). For linear maps, we provide explicit characterizations of the stated covariance in terms of their Kraus, Stinespring, and Choi representations, and systematically analyze the usual properties of positivity, decomposability, complete positivity, and the like. We also describe the invariant subspaces of these maps and use their structure to provide necessary and sufficient conditions for separability of the associated invariant bipartite states.

Published
2021
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8. On the Separability of Unitarily Invariant Random Quantum States: The Unbalanced Regime

Author

Ion Nechita

Subjects
Physics, QC1-999
Abstract

We study entanglement-related properties of random quantum states which are unitarily invariant, in the sense that their distribution is left unchanged by conjugation with arbitrary unitary operators. In the large matrix size limit, the distribution of these random quantum states is characterized by their limiting spectrum, a compactly supported probability distribution. We prove several results characterizing entanglement and the PPT property of random bipartite unitarily invariant quantum states in terms of the limiting spectral distribution, in the unbalanced asymptotical regime where one of the two subsystems is fixed, while the other one grows in size.

Published
2018
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9. Eigenvalue and Entropy Statistics for Products of Conjugate Random Quantum Channels

Author

Benoît Collins and Ion Nechita

Subjects
random quantum channels, weingarten calculus, minimum output entropy, von neumann entropy, additivity problem, Science, Astrophysics, QB460-466, Physics, QC1-999
Abstract

Using the graphical calculus and integration techniques introduced by the authors, we study the statistical properties of outputs of products of random quantum channels for entangled inputs. In particular, we revisit and generalize models of relevance for the recent counterexamples to the minimum output entropy additivity problems. Our main result is a classification of regimes for which the von Neumann entropy is lower on average than the elementary bounds that can be obtained with linear algebra techniques.

Published
2010
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18. Multipartite Entanglement Detection Via Projective Tensor Norms

Author

Maria Anastasia Jivulescu, Cécilia Lancien, Ion Nechita, Politehnica University of Timisoara (UPT), Institut de Mathématiques de Toulouse UMR5219 (IMT), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), Laboratoire de Physique Théorique (LPT), Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)-Fédération de recherche « Matière et interactions » (FeRMI), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), Institut de Recherche sur les Systèmes Atomiques et Moléculaires Complexes (IRSAMC), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), and Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)

Subjects
Quantum Physics, Nuclear and High Energy Physics, FOS: Physical sciences, Statistical and Nonlinear Physics, Mathematical Physics (math-ph), [MATH.MATH-FA]Mathematics [math]/Functional Analysis [math.FA], Functional Analysis (math.FA), Mathematics - Functional Analysis, [PHYS.QPHY]Physics [physics]/Quantum Physics [quant-ph], [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], FOS: Mathematics, Quantum Physics (quant-ph), Mathematical Physics
Abstract

International audience; We introduce and study a class of entanglement criteria based on the idea of applying local contractions to an input multipartite state, and then computing the projective tensor norm of the output. More precisely, we apply to a mixed quantum state a tensor product of contractions from the Schatten class $S_1$ to the Euclidean space $\ell_2$, which we call entanglement testers. We analyze the performance of this type of criteria on bipartite and multipartite systems, for general pure and mixed quantum states, as well as on some important classes of symmetric quantum states. We also show that previously studied entanglement criteria, such as the realignment and the SIC POVM criteria, can be viewed inside this framework. This allows us to answer in the positive two conjectures of Shang, Asadian, Zhu, and G\"uhne by deriving systematic relations between the performance of these two criteria.

Published
2022
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23. Diagonal unitary and orthogonal symmetries in quantum theory: II. Evolution operators

Author

Satvik Singh, Ion Nechita, Department of Applied Mathematics and Theoretical Physics (DAMTP), University of Cambridge [UK] (CAM), Information et Chaos Quantiques (LPT), Laboratoire de Physique Théorique (LPT), Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)-Fédération de recherche « Matière et interactions » (FeRMI), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-Université Toulouse III - Paul Sabatier (UT3), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS), ANR-20-CE47-0014,ESQuisses,Évolutions Stochastiques Quantiques(2020), Singh, S [0000-0002-2971-4256], Nechita, I [0000-0003-3016-7795], and Apollo - University of Cambridge Repository

Subjects
Statistics and Probability, Quantum Physics, entangling power, dual unitary operators, local diagonal orthogonal invariant (LDOI) unitary operators, General Physics and Astronomy, FOS: Physical sciences, Statistical and Nonlinear Physics, Mathematical Physics (math-ph), operator Schmidt rank, [PHYS.QPHY]Physics [physics]/Quantum Physics [quant-ph], Modeling and Simulation, Quantum Physics (quant-ph), Mathematical Physics
Abstract

We study bipartite unitary operators which stay invariant under the local actions of diagonal unitary and orthogonal groups. We investigate structural properties of these operators, arguing that the diagonal symmetry makes them suitable for analytical study. As a first application, we construct large new families of dual unitary gates in arbitrary finite dimensions, which are important toy models for entanglement spreading in quantum circuits. We then analyze the non-local nature of these invariant operators, both in discrete (operator Schmidt rank) and continuous (entangling power) settings. Our scrutiny reveals that these operators can be used to simulate any bipartite unitary gate via stochastic local operations and classical communication. Furthermore, we establish a one-to-one connection between the set of local diagonal unitary invariant dual unitary operators with maximum entangling power and the set of complex Hadamard matrices. Finally, we discuss distinguishability of unitary operators in the setting of the stated diagonal symmetry.

Published
2022
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24. Measurement incompatibility vs. Bell non-locality: an approach via tensor norms

Author

Faedi Loulidi, Ion Nechita, Information et Chaos Quantiques (LPT), Laboratoire de Physique Théorique (LPT), Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)-Fédération de recherche « Matière et interactions » (FeRMI), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-Université Toulouse III - Paul Sabatier (UT3), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS), ANR-20-CE47-0014,ESQuisses,Évolutions Stochastiques Quantiques(2020), and ANR-20-CE40-0008,STARS,Espace de trafics et asymptotiques de spectres aléatoires(2020)

Subjects
Quantum Physics, General Computer Science, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], Applied Mathematics, FOS: Physical sciences, General Physics and Astronomy, High Energy Physics::Experiment, Mathematical Physics (math-ph), Electrical and Electronic Engineering, Quantum Physics (quant-ph), Mathematical Physics, Electronic, Optical and Magnetic Materials
Abstract

Measurement incompatibility and quantum non-locality are two key features of quantum theory. Violations of Bell inequalities require quantum entanglement and incompatibility of the measurements used by the two parties involved in the protocol. We analyze the converse question: for which Bell inequalities is the incompatibility of measurements enough to ensure a quantum violation? We relate the two questions by comparing two tensor norms on the space of dichotomic quantum measurements: one characterizing measurement compatibility and the second one characterizing violations of a given Bell inequality. We provide sufficient conditions for the equivalence of the two notions in terms of the matrix describing the correlation Bell inequality. We show that the CHSH inequality and its variants are the only ones satisfying it., new applications and results about the optimality of the CHSH game

Published
2022
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25. The PPT$^2$ Conjecture Holds for All Choi-Type Maps

Author

Satvik Singh, Ion Nechita, Laboratoire de Physique Théorique (LPT), Institut de Recherche sur les Systèmes Atomiques et Moléculaires Complexes (IRSAMC), Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)-Université Toulouse III - Paul Sabatier (UT3), and Université de Toulouse (UT)-Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Computer Science::Machine Learning, Nuclear and High Energy Physics, Quantum Physics, [PHYS.QPHY]Physics [physics]/Quantum Physics [quant-ph], covariance, Computer Science::Mathematical Software, FOS: Physical sciences, unitarity, Statistical and Nonlinear Physics, algebra, Quantum Physics (quant-ph), Computer Science::Digital Libraries, Mathematical Physics
Abstract

We prove that the PPT$$^2$$ 2 conjecture holds for linear maps between matrix algebras which are covariant under the action of the diagonal unitary group. Many salient examples, like the Choi-type maps, depolarizing maps, dephasing maps, amplitude damping maps, and mixtures thereof, lie in this class. Our proof relies on a generalization of the matrix-theoretic notion of factor width for pairwise completely positive matrices, and a complete characterization in the case of factor width two.

Published
2022
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26. Incompatibility in General Probabilistic Theories, Generalized Spectrahedra, and Tensor Norms

Author

Andreas Bluhm, Anna Jenčová, Ion Nechita, Centre for the Mathematics of Quantum Theory (QMATH), Department of Mathematical Sciences [Copenhagen], Faculty of Science [Copenhagen], University of Copenhagen = Københavns Universitet (UCPH)-University of Copenhagen = Københavns Universitet (UCPH)-Faculty of Science [Copenhagen], University of Copenhagen = Københavns Universitet (UCPH)-University of Copenhagen = Københavns Universitet (UCPH), Slovak Academy of Sciences (SAS), Information et Chaos Quantiques (LPT), Laboratoire de Physique Théorique (LPT), Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)-Fédération de recherche « Matière et interactions » (FeRMI), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-Université Toulouse III - Paul Sabatier (UT3), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS), and ANR-20-CE47-0014,ESQuisses,Évolutions Stochastiques Quantiques(2020)

Subjects
Quantum Physics, [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph], FOS: Physical sciences, Statistical and Nonlinear Physics, Mathematical Physics (math-ph), space, Functional Analysis (math.FA), Mathematics - Functional Analysis, [PHYS.QPHY]Physics [physics]/Quantum Physics [quant-ph], FOS: Mathematics, Quantum Physics (quant-ph), vector, qubit, Mathematical Physics
Abstract

In this work, we investigate measurement incompatibility in general probabilistic theories (GPTs). We show several equivalent characterizations of compatible measurements. The first is in terms of the positivity of associated maps. The second relates compatibility to the inclusion of certain generalized spectrahedra. For this, we extend the theory of free spectrahedra to ordered vector spaces. The third characterization connects the compatibility of dichotomic measurements to the ratio of tensor crossnorms of Banach spaces. We use these characterizations to study the amount of incompatibility present in different GPTs, i.e. their compatibility regions. For centrally symmetric GPTs, we show that the compatibility degree is given as the ratio of the injective and the projective norm of the tensor product of associated Banach spaces. This allows us to completely characterize the compatibility regions of several GPTs, and to obtain optimal universal bounds on the compatibility degree in terms of the 1-summing constants of the associated Banach spaces. Moreover, we find new bounds on the maximal incompatibility present in more than three qubit measurements., 60 pages, 4 figures. Changed the notation of the norm characterizing compatibility from rho-norm to c-norm

Published
2022
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27. Sinkhorn algorithm for quantum permutation groups

Author

Ion Nechita, Simon Schmidt, Moritz Weber, Laboratoire de Physique Théorique (LPT), Institut de Recherche sur les Systèmes Atomiques et Moléculaires Complexes (IRSAMC), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées, Saarland University [Saarbrücken], Universität des Saarlandes [Saarbrücken], Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)-Fédération de recherche « Matière et interactions » (FeRMI), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), and Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)

Subjects
General Mathematics, [MATH.MATH-OA]Mathematics [math]/Operator Algebras [math.OA], [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph], MathematicsofComputing_NUMERICALANALYSIS, 0102 computer and information sciences, Permutation matrix, [MATH.MATH-FA]Mathematics [math]/Functional Analysis [math.FA], 01 natural sciences, Square matrix, [PHYS.QPHY]Physics [physics]/Quantum Physics [quant-ph], [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], Encoding (memory), Mathematics - Quantum Algebra, ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION, [MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO], FOS: Mathematics, Quantum Algebra (math.QA), 0101 mathematics, Quantum, Mathematics, 010102 general mathematics, Permutation group, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], 010201 computation theory & mathematics, Homogeneous space, [MATH.MATH-QA]Mathematics [math]/Quantum Algebra [math.QA], Algorithm
Abstract

We introduce a Sinkhorn-type algorithm for producing quantum permutation matrices encoding symmetries of graphs. Our algorithm generates square matrices whose entries are orthogonal projections onto one-dimensional subspaces satisfying a set of linear relations. We use it for experiments on the representation theory of the quantum permutation group and quantum subgroups of it. We apply it to the question whether a given finite graph (without multiple edges) has quantum symmetries in the sense of Banica. In order to do so, we run our Sinkhorn algorithm and check whether or not the resulting projections commute. We discuss the produced data and some questions for future research arising from it., Comment: Matlab/Octave code available in Ancillary files

Published
2021
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28. Enumerating meandric systems with large number of loops

Author

Motohisa Fukuda and Ion Nechita

Subjects
Statistics and Probability, Discrete mathematics, Algebra and Number Theory, Degree (graph theory), 010102 general mathematics, Generating function, FOS: Physical sciences, Statistical and Nonlinear Physics, Mathematical Physics (math-ph), 0102 computer and information sciences, Rational function, Free probability, 01 natural sciences, Meander (mathematics), Integer, 010201 computation theory & mathematics, FOS: Mathematics, Mathematics - Combinatorics, Discrete Mathematics and Combinatorics, Combinatorics (math.CO), Geometry and Topology, 0101 mathematics, Mathematical Physics, Mathematics
Abstract

We investigate meandric systems with a large number of loops using tools inspired by free probability. For any fixed integer $r$, we express the generating function of meandric systems on $2n$ points with $n-r$ loops in terms of a finite (the size depends on $r$) subclass of irreducible meandric systems, via the moment-cumulant formula from free probability theory. We show that the generating function, after an appropriate change of variable, is a rational function, and we bound its degree. Exact expressions for the generating functions are obtained for $r \leq 6$, as well as the asymptotic behavior of the meandric numbers for general $r$., Minor revision. 22 pages. To view supplementary material, please download and extract the file listed under "Other formats"

Published
2019
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29. Generating random quantum channels

Author

Ion Nechita, Karol Życzkowski, Ryszard Kukulski, Zbigniew Puchała, Łukasz Pawela, Laboratoire de Physique Théorique (LPT), Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)-Fédération de recherche « Matière et interactions » (FeRMI), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS), Centrum Fizyki Teoretycznej, Polska Akademia Nauk = Polish Academy of Sciences (PAN), Institut de Recherche sur les Systèmes Atomiques et Moléculaires Complexes (IRSAMC), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), and Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)

Subjects
[MATH.MATH-OA]Mathematics [math]/Operator Algebras [math.OA], [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph], Measure (physics), FOS: Physical sciences, Quantum channel, [MATH.MATH-FA]Mathematics [math]/Functional Analysis [math.FA], 01 natural sciences, [PHYS.QPHY]Physics [physics]/Quantum Physics [quant-ph], Quantum state, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], 0103 physical sciences, [MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO], Statistical physics, 0101 mathematics, Quantum, Mathematical Physics, ComputingMilieux_MISCELLANEOUS, Physics, Quantum Physics, Lebesgue measure, Unitarity, 010102 general mathematics, Statistical and Nonlinear Physics, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], [MATH.MATH-QA]Mathematics [math]/Quantum Algebra [math.QA], Spectral gap, 010307 mathematical physics, Quantum Physics (quant-ph), Coherence (physics)
Abstract

Several techniques of generating random quantum channels, which act on the set of $d$-dimensional quantum states, are investigated. We present three approaches to the problem of sampling of quantum channels and show under which conditions they become mathematically equivalent, and lead to the uniform, Lebesgue measure on the convex set of quantum operations. We compare their advantages and computational complexity and demonstrate which of them is particularly suitable for numerical investigations. Additional results focus on the spectral gap and other spectral properties of random quantum channels and their invariant states. We compute mean values of several quantities characterizing a given quantum channel, including its unitarity, the average output purity and the $2$-norm coherence of a channel, averaged over the entire set of the quantum channels with respect to the uniform measure. An ensemble of classical stochastic matrices obtained due to super-decoherence of random quantum stochastic maps is analyzed and their spectral properties are studied using the Bloch representation of a classical probability vector., 29 pages, 7 figures

Published
2021
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30. The compatibility dimension of quantum measurements

Author

Ion Nechita, Faedi Loulidi, Laboratoire de Physique Théorique (LPT), Institut de Recherche sur les Systèmes Atomiques et Moléculaires Complexes (IRSAMC), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées, Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)-Fédération de recherche « Matière et interactions » (FeRMI), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), and Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)

Subjects
[MATH.MATH-OA]Mathematics [math]/Operator Algebras [math.OA], [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph], FOS: Physical sciences, [MATH.MATH-FA]Mathematics [math]/Functional Analysis [math.FA], 01 natural sciences, Theoretical physics, Dimension (vector space), [PHYS.QPHY]Physics [physics]/Quantum Physics [quant-ph], Quantum state, Quantum error correction, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], 0103 physical sciences, [MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO], Orthonormal basis, 0101 mathematics, Quantum, Mathematical Physics, Mathematics, Quantum Physics, 010102 general mathematics, Statistical and Nonlinear Physics, 16. Peace & justice, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], [MATH.MATH-QA]Mathematics [math]/Quantum Algebra [math.QA], Schrödinger picture, 010307 mathematical physics, Quantum cloning, Quantum Physics (quant-ph), Mutually unbiased bases
Abstract

We introduce the notion of compatibility dimension for a set of quantum measurements: it is the largest dimension of a Hilbert space on which the given measurements are compatible. In the Schr\"odinger picture, this notion corresponds to testing compatibility with ensembles of quantum states supported on a subspace, using the incompatibility witnesses of Carmeli, Heinosaari, and Toigo. We provide several bounds for the compatibility dimension, using approximate quantum cloning or algebraic techniques inspired by quantum error correction. We analyze in detail the case of two orthonormal bases, and, in particular, that of mutually unbiased bases., Comment: several minor changes and typos fixed

Published
2021
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32. Entanglement criteria for the bosonic and fermionic induced ensembles

Author

Stephane Dartois, Ion Nechita, Adrian Tanasa, School of Mathematics and Statistics [Melbourne], Faculty of Science [Melbourne], University of Melbourne-University of Melbourne, Information et Chaos Quantiques (LPT), Laboratoire de Physique Théorique (LPT), Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)-Fédération de recherche « Matière et interactions » (FeRMI), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-Université Toulouse III - Paul Sabatier (UT3), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS), Laboratoire Bordelais de Recherche en Informatique (LaBRI), Université de Bordeaux (UB)-École Nationale Supérieure d'Électronique, Informatique et Radiocommunications de Bordeaux (ENSEIRB)-Centre National de la Recherche Scientifique (CNRS), ANR-20-CE47-0014,ESQuisses,Évolutions Stochastiques Quantiques(2020), ANR-18-CE47-0010,QuDATA,Algorithmes quantiques pour données massives(2018), ANR-20-CE48-0018,3DMaps,Combinatoire de cartes 3D(2020), Université de Bordeaux (UB)-Centre National de la Recherche Scientifique (CNRS)-École Nationale Supérieure d'Électronique, Informatique et Radiocommunications de Bordeaux (ENSEIRB), Institut de Recherche sur les Systèmes Atomiques et Moléculaires Complexes (IRSAMC), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), and Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)

Subjects
Condensed Matter::Quantum Gases, Quantum Physics, random, [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph], Probability (math.PR), matrix model, Hilbert space, FOS: Physical sciences, Statistical and Nonlinear Physics, Mathematical Physics (math-ph), Theoretical Computer Science, Electronic, Optical and Magnetic Materials, moment, [PHYS.QPHY]Physics [physics]/Quantum Physics [quant-ph], Modeling and Simulation, Signal Processing, FOS: Mathematics, Mathematics - Combinatorics, Combinatorics (math.CO), Electrical and Electronic Engineering, entanglement, Quantum Physics (quant-ph), density matrix, Mathematical Physics, Mathematics - Probability
Abstract

We introduce the bosonic and fermionic ensembles of density matrices and study their entanglement. In the fermionic case, we show that random bipartite fermionic density matrices have non-positive partial transposition, hence they are typically entangled. The similar analysis in the bosonic case is more delicate, due to a large positive outlier eigenvalue. We compute the asymptotic ratio between the size of the environment and the size of the system Hilbert space for which random bipartite bosonic density matrices fail the PPT criterion, being thus entangled. We also relate moment computations for tensor-symmetric random matrices to evaluations of the circuit-counting and interlace graph polynomials for directed graphs., Comment: 37 pages, numerous figures and diagrammatics equations

Published
2021
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33. Maximal violation of steering inequalities and the matrix cube

Author

Andreas Bluhm, Ion Nechita, Centre for the Mathematics of Quantum Theory (QMATH), Department of Mathematical Sciences [Copenhagen], Faculty of Science [Copenhagen], University of Copenhagen = Københavns Universitet (UCPH)-University of Copenhagen = Københavns Universitet (UCPH)-Faculty of Science [Copenhagen], University of Copenhagen = Københavns Universitet (UCPH)-University of Copenhagen = Københavns Universitet (UCPH), Information et Chaos Quantiques (LPT), Laboratoire de Physique Théorique (LPT), Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)-Fédération de recherche « Matière et interactions » (FeRMI), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-Université Toulouse III - Paul Sabatier (UT3), and Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Quantum Physics, [PHYS.QPHY]Physics [physics]/Quantum Physics [quant-ph], Physics and Astronomy (miscellaneous), FOS: Physical sciences, Mathematical Physics (math-ph), Quantum Physics (quant-ph), Mathematical Physics, ComputingMilieux_MISCELLANEOUS, Atomic and Molecular Physics, and Optics
Abstract

In this work, we characterize the amount of steerability present in quantum theory by connecting the maximal violation of a steering inequality to an inclusion problem of free spectrahedra. In particular, we show that the maximal violation of an arbitrary unbiased dichotomic steering inequality is given by the inclusion constants of the matrix cube, which is a well-studied object in convex optimization theory. This allows us to find new upper bounds on the maximal violation of steering inequalities and to show that previously obtained violations are optimal. In order to do this, we prove lower bounds on the inclusion constants of the complex matrix cube, which might be of independent interest. Finally, we show that the inclusion constants of the matrix cube and the matrix diamond are the same. This allows us to derive new bounds on the amount of incompatibility available in dichotomic quantum measurements in fixed dimension., Comment: Final version: 30 pages, 2 figures

Published
2021
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35. SudoQ -- a quantum variant of the popular game

Author

Ion Nechita, Jordi Pillet, Laboratoire de Physique Théorique (LPT), Institut de Recherche sur les Systèmes Atomiques et Moléculaires Complexes (IRSAMC), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)-Fédération de recherche « Matière et interactions » (FeRMI), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), and Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Nuclear and High Energy Physics, Computer science, [MATH.MATH-OA]Mathematics [math]/Operator Algebras [math.OA], [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph], General Physics and Astronomy, FOS: Physical sciences, [MATH.MATH-FA]Mathematics [math]/Functional Analysis [math.FA], 01 natural sciences, 010305 fluids & plasmas, Theoretical Computer Science, [PHYS.QPHY]Physics [physics]/Quantum Physics [quant-ph], [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], 0103 physical sciences, [MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO], 010306 general physics, Quantum, Commutative property, Mathematical Physics, Quantum Physics, Solution set, Statistical and Nonlinear Physics, State (functional analysis), Grid, Randomized algorithm, Algebra, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Computational Theory and Mathematics, [MATH.MATH-QA]Mathematics [math]/Quantum Algebra [math.QA], Quantum Physics (quant-ph)
Abstract

We introduce SudoQ, a quantum version of the classical game Sudoku. Allowing the entries of the grid to be (non-commutative) projections instead of integers, the solution set of SudoQ puzzles can be much larger than in the classical (commutative) setting. We introduce and analyze a randomized algorithm for computing solutions of SudoQ puzzles. Finally, we state two important conjectures relating the quantum and the classical solutions of SudoQ puzzles, corroborated by analytical and numerical evidence., Comment: Python code and examples available at https://github.com/inechita/SudoQ

Published
2020
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36. The joint distribution of the marginals of multipartite random quantum states

Author

Luca Lionni, Ion Nechita, Stephane Dartois, Laboratoire Bordelais de Recherche en Informatique (LaBRI), Université de Bordeaux (UB)-Centre National de la Recherche Scientifique (CNRS)-École Nationale Supérieure d'Électronique, Informatique et Radiocommunications de Bordeaux (ENSEIRB), Kyoto University [Kyoto], Laboratoire de Physique Théorique (LPT), Institut de Recherche sur les Systèmes Atomiques et Moléculaires Complexes (IRSAMC), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), Information et Chaos Quantiques (LPT), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Institut de Recherche sur les Systèmes Atomiques et Moléculaires Complexes (IRSAMC), Université Toulouse III - Paul Sabatier (UT3), and Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Université Toulouse III - Paul Sabatier (UT3)

Subjects
Statistics and Probability, Wishart distribution, Pure mathematics, FOS: Physical sciences, 01 natural sciences, Set (abstract data type), symbols.namesake, Quantum state, Joint probability distribution, 0103 physical sciences, FOS: Mathematics, Discrete Mathematics and Combinatorics, Mathematics - Combinatorics, 0101 mathematics, Cumulant, Mathematical Physics, Mathematics, Quantum Physics, Algebra and Number Theory, 010102 general mathematics, Probability (math.PR), Hilbert space, Mathematical Physics (math-ph), [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Multipartite, Tensor product, symbols, 010307 mathematical physics, Combinatorics (math.CO), Statistics, Probability and Uncertainty, Quantum Physics (quant-ph), Mathematics - Probability
Abstract

We study the joint distribution of the set of all marginals of a random Wishart matrix acting on a tensor product Hilbert space. We compute the limiting free mixed cumulants of the marginals, and we show that in the balanced asymptotical regime, the marginals are asymptotically free. We connect the matrix integrals relevant to the study of operators on tensor product spaces with the corresponding classes of combinatorial maps, for which we develop the combinatorial machinery necessary for the asymptotic study. Finally, we present some applications to the theory of random quantum states in quantum information theory., Comment: 53 pages, 25 figures, v2 reviewed for publication

Published
2020
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37. A graphical calculus for integration over random diagonal unitary matrices

Author

Satvik Singh, Ion Nechita, Laboratoire de Physique Théorique (LPT), Institut de Recherche sur les Systèmes Atomiques et Moléculaires Complexes (IRSAMC), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées, Indian Institute of Science Education and Research Mohali (IISER Mohali), Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)-Fédération de recherche « Matière et interactions » (FeRMI), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), and Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Diagonal, [MATH.MATH-OA]Mathematics [math]/Operator Algebras [math.OA], [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph], FOS: Physical sciences, [MATH.MATH-FA]Mathematics [math]/Functional Analysis [math.FA], 01 natural sciences, Unitary state, Matrix (mathematics), [PHYS.QPHY]Physics [physics]/Quantum Physics [quant-ph], [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], 0103 physical sciences, [MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO], Calculus, FOS: Mathematics, Discrete Mathematics and Combinatorics, Mathematics - Combinatorics, Invariant (mathematics), 010306 general physics, Mathematical Physics, Mathematics, Numerical Analysis, Quantum Physics, Algebra and Number Theory, 010308 nuclear & particles physics, Probability (math.PR), Unitary matrix, Mathematical Physics (math-ph), [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Order structure, Bipartite graph, [MATH.MATH-QA]Mathematics [math]/Quantum Algebra [math.QA], Geometry and Topology, Combinatorics (math.CO), Partially ordered set, Quantum Physics (quant-ph), Mathematics - Probability
Abstract

We provide a graphical calculus for computing averages of tensor network diagrams with respect to the distribution of random vectors containing independent uniform complex phases. Our method exploits the order structure of the partially ordered set of uniform block permutations. A similar calculus is developed for random vectors consisting of independent uniform signs, based on the combinatorics of the partially ordered set of even partitions. We employ our method to extend some of the results by Johnston and MacLean on the family of local diagonal unitary invariant matrices. Furthermore, our graphical approach applies just as well to the real (orthogonal) case, where we introduce the notion of triplewise complete positivity to study the condition for separability of the relevant bipartite matrices. Finally, we analyze the twirling of linear maps between matrix algebras by independent diagonal unitary matrices, showcasing another application of our method., Comment: Minor modifications with updated references

Published
2020
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38. RTNI—A symbolic integrator for Haar-random tensor networks

Author

Motohisa Fukuda, Ion Nechita, Robert König, Laboratoire de Physique Théorique (LPT), Institut de Recherche sur les Systèmes Atomiques et Moléculaires Complexes (IRSAMC), Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées, Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), Université Toulouse III - Paul Sabatier (UT3), and Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Université Toulouse III - Paul Sabatier (UT3)

Subjects
High Energy Physics - Theory, Statistics and Probability, [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph], MathematicsofComputing_NUMERICALANALYSIS, toy model, FOS: Physical sciences, General Physics and Astronomy, information theory: quantum, 01 natural sciences, Unitary group, 0103 physical sciences, ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION, FOS: Mathematics, Entropy (information theory), unitarity, Tensor, 0101 mathematics, Quantum information, Mathematical Physics, Mathematics, Quantum Physics, [PHYS.HTHE]Physics [physics]/High Energy Physics - Theory [hep-th], Probability (math.PR), 010102 general mathematics, Statistical and Nonlinear Physics, Mathematical Physics (math-ph), Unitary matrix, duality: holography, Symbolic computation, [PHYS.PHYS.PHYS-GEN-PH]Physics [physics]/Physics [physics]/General Physics [physics.gen-ph], Algebra, High Energy Physics - Theory (hep-th), computer: algebra, Modeling and Simulation, Integrator, network, 010307 mathematical physics, Quantum Physics (quant-ph), entropy, Symbolic integration, Mathematics - Probability
Abstract

We provide a computer algebra package called Random Tensor Network Integrator (RTNI). It allows to compute averages of tensor networks containing multiple Haar-distributed random unitary matrices and deterministic symbolic tensors. Such tensor networks are represented as multigraphs, with vertices corresponding to tensors or random unitaries and edges corresponding to tensor contractions. Input and output spaces of random unitaries may be subdivided into arbitrary tensor factors, with dimensions treated symbolically. The algorithm implements the graphical Weingarten calculus and produces a weighted sum of tensor networks representing the average over the unitary group. We illustrate the use of this algorithmic tool on some examples from quantum information theory, including entropy calculations for random tensor network states as considered in toy models for holographic duality. Mathematica and Python implementations are supplied., Comment: Code available (for Mathematica and python) at https://github.com/MotohisaFukuda/RTNI

Published
2019
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39. Operator Schmidt ranks of bipartite unitary matrices

Author

Alexander Müller-Hermes, Ion Nechita, Information et Chaos Quantiques (LPT), Laboratoire de Physique Théorique (LPT), Institut de Recherche sur les Systèmes Atomiques et Moléculaires Complexes (IRSAMC), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Institut de Recherche sur les Systèmes Atomiques et Moléculaires Complexes (IRSAMC), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées, Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Institut de Recherche sur les Systèmes Atomiques et Moléculaires Complexes (IRSAMC), and Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)

Subjects
Numerical Analysis, Algebra and Number Theory, Rank (linear algebra), [MATH.MATH-OA]Mathematics [math]/Operator Algebras [math.OA], 010103 numerical & computational mathematics, Unitary matrix, 01 natural sciences, Unitary state, Combinatorics, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Matrix (mathematics), Operator (computer programming), Tensor product, Simple (abstract algebra), [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], 0103 physical sciences, Bipartite graph, Discrete Mathematics and Combinatorics, Geometry and Topology, 0101 mathematics, 010306 general physics, ComputingMilieux_MISCELLANEOUS, Mathematics
Abstract

The operator Schmidt rank of an operator acting on the tensor product C n ⊗ C m is the number of terms in a decomposition of the operator as a sum of simple tensors with factors forming orthogonal families in their respective matrix algebras. It has been known that for unitary operators acting on two copies of C 2 , the operator Schmidt rank can only take the values 1, 2, and 4, the value 3 being forbidden. In this paper, we settle an open question, showing that the above obstruction is the only one occurring. We do so by constructing explicit examples of bipartite unitary operators of all possible operator Schmidt ranks, for arbitrary dimensions ( n , m ) ≠ ( 2 , 2 ) .

Published
2018
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40. Joint measurability of quantum effects and the matrix diamond

Author

Ion Nechita, Andreas Bluhm, Technische Universität Munchen - Université Technique de Munich [Munich, Allemagne] (TUM), Information et Chaos Quantiques (LPT), Laboratoire de Physique Théorique (LPT), Institut de Recherche sur les Systèmes Atomiques et Moléculaires Complexes (IRSAMC), Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Institut de Recherche sur les Systèmes Atomiques et Moléculaires Complexes (IRSAMC), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Institut de Recherche sur les Systèmes Atomiques et Moléculaires Complexes (IRSAMC), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées, Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse III - Paul Sabatier (UT3), and Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse)

Subjects
[PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph], Mathematics::Optimization and Control, FOS: Physical sciences, engineering.material, 01 natural sciences, Matrix (mathematics), Quantum mechanics, 0103 physical sciences, FOS: Mathematics, 0101 mathematics, Quantum information, 010306 general physics, Operator Algebras (math.OA), Quantum, Mathematical Physics, Mathematics, Quantum Physics, Operator (physics), 010102 general mathematics, Mathematics - Operator Algebras, Diamond, Statistical and Nonlinear Physics, Mathematical Physics (math-ph), Exponential function, Connection (mathematics), [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], engineering, Relaxation (approximation), Quantum Physics (quant-ph)
Abstract

In this work, we investigate the joint measurability of quantum effects and connect it to the study of free spectrahedra. Free spectrahedra typically arise as matricial relaxations of linear matrix inequalities. An example of a free spectrahedron is the matrix diamond, which is a matricial relaxation of the $\ell_1$-ball. We find that joint measurability of binary POVMs is equivalent to the inclusion of the matrix diamond into the free spectrahedron defined by the effects under study. This connection allows us to use results about inclusion constants from free spectrahedra to quantify the degree of incompatibility of quantum measurements. In particular, we completely characterize the case in which the dimension is exponential in the number of measurements. Conversely, we use techniques from quantum information theory to obtain new results on spectrahedral inclusion for the matrix diamond., 28 pages, 4 figures. v3: minor revisions

Published
2018
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41. Almost all quantum channels are equidistant

Author

Łukasz Pawela, Karol Życzkowski, Zbigniew Puchała, Ion Nechita, Information et Chaos Quantiques (LPT), Laboratoire de Physique Théorique (LPT), Institut de Recherche sur les Systèmes Atomiques et Moléculaires Complexes (IRSAMC), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Institut de Recherche sur les Systèmes Atomiques et Moléculaires Complexes (IRSAMC), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées, Centrum Fizyki Teoretycznej, Polska Akademia Nauk = Polish Academy of Sciences (PAN), Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Institut de Recherche sur les Systèmes Atomiques et Moléculaires Complexes (IRSAMC), and Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)

Subjects
Quantum Physics, 010102 general mathematics, FOS: Physical sciences, Statistical and Nonlinear Physics, Free probability, 01 natural sciences, Unitary state, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Norm (mathematics), 0103 physical sciences, Equidistant, Statistical physics, 0101 mathematics, Quantum Physics (quant-ph), 010306 general physics, Quantum, Random matrix, Mathematical Physics, ComputingMilieux_MISCELLANEOUS, Mathematics
Abstract

In this work we analyze properties of generic quantum channels in the case of large system size. We use random matrix theory and free probability to show that the distance between two independent random channels converges to a constant value as the dimension of the system grows larger. As a measure of the distance we use the diamond norm. In the case of a flat Hilbert-Schmidt distribution on quantum channels, we obtain that the distance converges to $\frac12 + \frac{2}{\pi}$, giving also an estimate for the maximum success probability for distinguishing the channels. We also consider the problem of distinguishing two random unitary rotations., Comment: 30 pages, commets are welcome

Published
2018
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42. On the Minimum Output Entropy of Random Orthogonal Quantum Channels

Author

Ion Nechita, Motohisa Fukuda, Zentrum Mathematik [Munchen] (TUM), Technische Universität Munchen - Université Technique de Munich [Munich, Allemagne] (TUM), Information et Chaos Quantiques (LPT), Laboratoire de Physique Théorique (LPT), Institut de Recherche sur les Systèmes Atomiques et Moléculaires Complexes (IRSAMC), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Institut de Recherche sur les Systèmes Atomiques et Moléculaires Complexes (IRSAMC), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Université Toulouse III - Paul Sabatier (UT3), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Institut de Recherche sur les Systèmes Atomiques et Moléculaires Complexes (IRSAMC), and Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées

Subjects
FOS: Physical sciences, Quantum entanglement, Library and Information Sciences, 01 natural sciences, Channel capacity, [PHYS.QPHY]Physics [physics]/Quantum Physics [quant-ph], [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], 0103 physical sciences, FOS: Mathematics, Entropy (information theory), Statistical physics, Orthogonal matrix, 0101 mathematics, Quantum, Mathematical Physics, ComputingMilieux_MISCELLANEOUS, Mathematics, Quantum Physics, Bell state, Probability (math.PR), 010102 general mathematics, Mathematical Physics (math-ph), Unitary matrix, Computer Science Applications, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Tensor product, 010307 mathematical physics, Quantum Physics (quant-ph), Mathematics - Probability, Information Systems
Abstract

We consider sequences of random quantum channels defined using the Stinespring formula with Haar-distributed random orthogonal matrices. For any fixed sequence of input states, we study the asymptotic eigenvalue distribution of the outputs through tensor powers of random channels. We show that the input states achieving minimum output entropy are tensor products of maximally entangled states (Bell states) when the tensor power is even. This phenomenon is completely different from the one for random quantum channels constructed from Haar-distributed random unitary matrices, which leads us to formulate some conjectures about the regularized minimum output entropy., Comment: minor modifications

Published
2018
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43. Positive reduction from spectra

Author

Nicolae Lupa, David Reeb, Ion Nechita, Maria Anastasia Jivulescu, Department of Mathematics, Universitatea Politehnica Timisoara, Information et Chaos Quantiques (LPT), Laboratoire de Physique Théorique (LPT), Institut de Recherche sur les Systèmes Atomiques et Moléculaires Complexes (IRSAMC), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Institut de Recherche sur les Systèmes Atomiques et Moléculaires Complexes (IRSAMC), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), Zentrum Mathematik [Munchen] (TUM), Technische Universität Munchen - Université Technique de Munich [Munich, Allemagne] (TUM), Université Toulouse III - Paul Sabatier (UT3), and Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Université Toulouse III - Paul Sabatier (UT3)

Subjects
Discrete mathematics, Quantum Physics, Numerical Analysis, Algebra and Number Theory, FOS: Physical sciences, Quantum entanglement, Reduction criterion, Squashed entanglement, Multipartite entanglement, Reduction (complexity), [PHYS.QPHY]Physics [physics]/Quantum Physics [quant-ph], Qubit, Transpose, 15B48, 81P45, Discrete Mathematics and Combinatorics, Geometry and Topology, Quantum Physics (quant-ph), Mathematics, Peres–Horodecki criterion
Abstract

We study the problem of whether all bipartite quantum states having a prescribed spectrum remain positive under the reduction map applied to one subsystem. We provide necessary and sufficient conditions, in the form of a family of linear inequalities, which the spectrum has to verify. Our conditions become explicit when one of the two subsystems is a qubit, as well as for further sets of states. Finally, we introduce a family of simple entanglement criteria for spectra, closely related to the reduction and positive partial transpose criteria, which also provide new insight into the set of spectra that guarantee separability or positivity of the partial transpose., Comment: Linear Algebra and its Applications (2015)

Published
2015
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44. On bipartite unitary matrices generating subalgebra-preserving quantum operations

Author

Ion Nechita, Tristan Benoist, Information et Chaos Quantiques (LPT), Laboratoire de Physique Théorique (LPT), Institut de Recherche sur les Systèmes Atomiques et Moléculaires Complexes (IRSAMC), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Institut de Recherche sur les Systèmes Atomiques et Moléculaires Complexes (IRSAMC), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées, ANR-12-IS01-0001,RMTQIT,Techniques de matrices aléatoires en théorie quantique de l'information(2012), ANR-14-CE25-0003,StoQ,Méthodes Stochastiques en Mécanique Quantique(2014), Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Institut de Recherche sur les Systèmes Atomiques et Moléculaires Complexes (IRSAMC), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse III - Paul Sabatier (UT3), and Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse)

Subjects
Quantum t-design, [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph], FOS: Physical sciences, 01 natural sciences, SIC-POVM, POVM, [PHYS.QPHY]Physics [physics]/Quantum Physics [quant-ph], Quantum phase estimation algorithm, Quantum operation, Discrete Mathematics and Combinatorics, 0101 mathematics, Mathematical Physics, Mathematics, Quantum Physics, Numerical Analysis, Algebra and Number Theory, Mathematics::Operator Algebras, 010102 general mathematics, Mathematical Physics (math-ph), Unitary matrix, 010101 applied mathematics, Algebra, [MATH.MATH-QA]Mathematics [math]/Quantum Algebra [math.QA], Quantum algorithm, Quantum Fourier transform, Geometry and Topology, Quantum Physics (quant-ph)
Abstract

We study the structure of bipartite unitary operators which generate via the Stinespring dilation theorem, quantum operations preserving some given matrix algebra, independently of the ancilla state. We characterize completely the unitary operators preserving diagonal, block-diagonal, and tensor product algebras. Some unexpected connections with the theory of quantum Latin squares are explored, and we introduce and study a Sinkhorn-like algorithm used to randomly generate quantum Latin squares., a new reference added and minor modifications

Published
2017
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45. Asymptotically Well-Behaved Input States Do Not Violate Additivity for Conjugate Pairs of Random Quantum Channels

Author

Ion Nechita, Motohisa Fukuda, Institute for Quantum Computing [Waterloo] (IQC), University of Waterloo [Waterloo], Information et Chaos Quantiques (LPT), Laboratoire de Physique Théorique (LPT), Institut de Recherche sur les Systèmes Atomiques et Moléculaires Complexes (IRSAMC), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Institut de Recherche sur les Systèmes Atomiques et Moléculaires Complexes (IRSAMC), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées, Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Institut de Recherche sur les Systèmes Atomiques et Moléculaires Complexes (IRSAMC), and Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)

Subjects
Quantum Physics, Bell state, Complex conjugate, Probability (math.PR), [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph], Complex system, FOS: Physical sciences, 15A52, Statistical and Nonlinear Physics, Mathematical Physics (math-ph), Quantum channel, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Tensor product, [PHYS.QPHY]Physics [physics]/Quantum Physics [quant-ph], [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], Additive function, FOS: Mathematics, Entropy (information theory), Statistical physics, Quantum Physics (quant-ph), Quantum, Mathematics - Probability, Mathematical Physics, Mathematics
Abstract

International audience; It is now well-known that, with high probability, the additivity of minimum output entropy does not hold for a pair of a random quantum channel and its complex conjugate. We investigate asymptotic behavior of output states of $r$-tensor powers of such pairs, as the dimension of inputs grows. We compute the limit output states for any sequence of well-behaved inputs, which consist of a large class of input states having a nice set of parameters. Then, we show that among these input states tensor products of Bell states give asymptotically the least output entropy, giving positive mathematical evidence towards additivity of above pairs of channels.

Published
2014
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47. On symmetric decompositions of positive operators

Author

Ion Nechita, Maria Anastasia Jivulescu, Paşc Găvruţa, Department of Mathematics, Polytechnic University of Timisoara, Information et Chaos Quantiques (LPT), Laboratoire de Physique Théorique (LPT), Institut de Recherche sur les Systèmes Atomiques et Moléculaires Complexes (IRSAMC), Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Institut de Recherche sur les Systèmes Atomiques et Moléculaires Complexes (IRSAMC), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), ANR-12-IS01-0001,RMTQIT,Techniques de matrices aléatoires en théorie quantique de l'information(2012), ANR-14-CE25-0003,StoQ,Méthodes Stochastiques en Mécanique Quantique(2014), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Institut de Recherche sur les Systèmes Atomiques et Moléculaires Complexes (IRSAMC), and Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées

Subjects
Statistics and Probability, Pure mathematics, Rank (linear algebra), Equiangular polygon, FOS: Physical sciences, General Physics and Astronomy, 01 natural sciences, symbols.namesake, Identity (mathematics), [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], 0103 physical sciences, FOS: Mathematics, 0101 mathematics, Quantum information, Mathematical Physics, Mathematics, Quantum Physics, 010308 nuclear & particles physics, 010102 general mathematics, Hilbert space, Statistical and Nonlinear Physics, Mathematical Physics (math-ph), Functional Analysis (math.FA), [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Mathematics - Functional Analysis, Modeling and Simulation, symbols, Quantum Physics (quant-ph), Focus (optics), Constant (mathematics), Unit (ring theory)
Abstract

We present results concerning decompositions of positive operators acting on finite-dimensional Hilbert spaces. Our motivation is the study of a generalized version of the SIC–POVM problem, which has applications to Quantum Information Theory. We relax some of the conditions in the SIC–POVM setting (the elements sum up to the identity, resp. the elements have unit rank), and we focus on equiangular decompositions (the elements of the decomposition should have the same length, and pairs of distinct elements should have constant angles). We characterize all such decompositions, comparing our results with the case of SIC–POVMs. We also generalize some existing Welch-type inequalities.

Published
2016
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48. Flat matrix models for quantum permutation groups

Author

Ion Nechita, Teodor Banica, Analyse, Géométrie et Modélisation (AGM - UMR 8088), and CY Cergy Paris Université (CY)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Conjecture, Pauli matrices, Generalization, Applied Mathematics, Probability (math.PR), 010102 general mathematics, Mathematics - Operator Algebras, Permutation group, 01 natural sciences, Combinatorics, Matrix (mathematics), symbols.namesake, 0103 physical sciences, FOS: Mathematics, symbols, Rank (graph theory), 010307 mathematical physics, [MATH]Mathematics [math], 0101 mathematics, Operator Algebras (math.OA), Representation (mathematics), Quantum, Mathematics - Probability, Mathematics
Abstract

We study the matrix models $\pi:C(S_N^+)\to M_N(C(X))$ which are flat, in the sense that the standard generators of $C(S_N^+)$ are mapped to rank 1 projections. Our first result is a generalization of the Pauli matrix construction at $N=4$, using finite groups and 2-cocycles. Our second result is the construction of a universal representation of $C(S_N^+)$, inspired from the Sinkhorn algorithm, that we conjecture to be inner faithful., Comment: 22 pages

Published
2016
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49. Almost one bit violation for the additivity of the minimum output entropy

Author

Ion Nechita, Benoît Collins, Serban T. Belinschi, Department of Mathematics & Statistics, Queen's University, Institute of Mathematics 'Simion Stoilow', Romanian Academy of Sciences, Institut de Mathématiques de Toulouse UMR5219 (IMT), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), Department of Mathematics and Statistics [Ottawa], University of Ottawa [Ottawa], Information et Chaos Quantiques (LPT), Laboratoire de Physique Théorique (LPT), Institut de Recherche sur les Systèmes Atomiques et Moléculaires Complexes (IRSAMC), Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)-Institut de Recherche sur les Systèmes Atomiques et Moléculaires Complexes (IRSAMC), Université de Toulouse (UT)-Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Institut de Recherche sur les Systèmes Atomiques et Moléculaires Complexes (IRSAMC), and Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées

Subjects
Unit sphere, [MATH.MATH-OA]Mathematics [math]/Operator Algebras [math.OA], [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph], FOS: Physical sciences, Quantum channel, 01 natural sciences, [PHYS.QPHY]Physics [physics]/Quantum Physics [quant-ph], [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], Additive function, 0103 physical sciences, FOS: Mathematics, Entropy (information theory), 0101 mathematics, Operator Algebras (math.OA), Quantum, Mathematical Physics, Eigenvalues and eigenvectors, Mathematics, Discrete mathematics, Quantum Physics, Bell state, 010102 general mathematics, Mathematics - Operator Algebras, Statistical and Nonlinear Physics, Mathematical Physics (math-ph), 16. Peace & justice, Compact space, 010307 mathematical physics, Quantum Physics (quant-ph), 81P45, 52A22
Abstract

In a previous paper, we proved that the limit of the collection of possible eigenvalues of output states of a random quantum channel is a deterministic, compact set K_{k,t}. We also showed that the set K_{k,t} is obtained, up to an intersection, as the unit ball of the dual of a free compression norm. In this paper, we identify the maximum of l^p norms on the set K_{k,t} and prove that the maximum is attained on a vector of shape (a,b,...,b) where a > b. In particular, we compute the precise limit value of the minimum output entropy of a single random quantum channel. As a corollary, we show that for any eps > 0, it is possible to obtain a violation for the additivity of the minimum output entropy for an output dimension as low as 183, and that for appropriate choice of parameters, the violation can be as large as log 2 - eps. Conversely, our result implies that, with probability one, one does not obtain a violation of additivity using conjugate random quantum channels and the Bell state, in dimension 182 and less., v3: appendix replaced by Lemma 3.8

Published
2016
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50. Random matrix techniques in quantum information theory

Author

Ion Nechita, Benoît Collins, Institut Camille Jordan [Villeurbanne] (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS), Department of Mathematics and Statistics [Ottawa], University of Ottawa [Ottawa], Information et Chaos Quantiques (LPT), Laboratoire de Physique Théorique (LPT), Institut de Recherche sur les Systèmes Atomiques et Moléculaires Complexes (IRSAMC), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Institut de Recherche sur les Systèmes Atomiques et Moléculaires Complexes (IRSAMC), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), ANR-12-IS01-0001,RMTQIT,Techniques de matrices aléatoires en théorie quantique de l'information(2012), ANR-14-CE25-0003,StoQ,Méthodes Stochastiques en Mécanique Quantique(2014), ANR-11-BS01-0008,OSQPI,Interactions entre espaces d'opérateurs et probabilités quantiques et leurs applications à l'information quantique(2011), Université de Lyon-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Institut National des Sciences Appliquées (INSA)-Université de Lyon-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet [Saint-Étienne] (UJM)-Centre National de la Recherche Scientifique (CNRS), Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Université Toulouse III - Paul Sabatier (UT3), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Institut de Recherche sur les Systèmes Atomiques et Moléculaires Complexes (IRSAMC), and Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées

Subjects
Computer science, FOS: Physical sciences, Quantum entanglement, 01 natural sciences, Quantum state, 0103 physical sciences, FOS: Mathematics, Entropy (information theory), Statistical physics, 0101 mathematics, Quantum information, 010306 general physics, Quantum, Randomness, Mathematical Physics, Quantum Physics, Stochastic process, 010102 general mathematics, Probability (math.PR), TheoryofComputation_GENERAL, Statistical and Nonlinear Physics, Mathematical Physics (math-ph), [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Quantum Physics (quant-ph), Random matrix, Mathematics - Probability
Abstract

International audience; The purpose of this review article is to present some of the latest developments using random techniques, and in particular, random matrix techniques in quantum information theory. Our review is a blend of a rather exhaustive review, combined with more detailed examples -- coming from research projects in which the authors were involved. We focus on two main topics, random quantum states and random quantum channels. We present results related to entropic quantities, entanglement of typical states, entanglement thresholds, the output set of quantum channels, and violations of the minimum output entropy of random channels.

Published
2016
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