arXiv subjects

Péter Lévay

Publications and source records attributed to Péter Lévay.

At least 19 recordsLinked to original sources

arXiv2026

The Finite Geometry of Breaking Quantum Secrets

Using a finite geometric framework for studying the pentagon and heptagon codes we show that the concepts of quantum secret sharing and contextuality can be studied in a nice and unified manner. The basic idea is a careful study of the respective $2+3$ and $3+4$ tensorial factorizations of the elements of the stabilizer groups of these codes. It is demonstrated in detail how finite geometric structures entailing a specific three-qubit (resp. four-qubit) embedding of binary symplectic polar spaces of rank two (resp. three), corresponding to these factorizations, govern issues of contextuality and entanglement needed for a geometric understanding of quantum secret sharing. Using these results for the $(3,5)$ and $(4,7)$ threshold schemes explicit secret breaking protocols are derived. Our results hint at a novel geometric way of looking at contextual configurations.

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arXiv2025

A holographic connection between strings and causal diamonds

In this paper we explore ideas of holography and strings living in the $d+1$ dimensional Anti-de Sitter space $AdS_{d+1}$ in a unified framework borrowed from twistor theory. In our treatise of correspondences between geometric structures of the bulk $AdS_{d+1}$, its boundary and the moduli space of boundary causal diamonds aka the kinematic space ${\mathbb K}$, we adopt a perspective offered by projective geometry. From this viewpoint certain lines in the $d+1$ dimensional real projective space, defined by two light-like vectors in ${\mathbb R}^{d,2}$ play an important role. In these projective geometric elaborations objects like Ryu-Takayanagi surfaces, spacelike geodesics with horospheres providing regularizators for them and the metric on ${\mathbb K}$ all find a natural place. Then we establish a correspondence between classical strings in $AdS_{d+1}$ and causal diamonds of its asymptotic boundary. At each point on the worldsheet, the tangent vectors $\partial_\pm X$ are projected onto boundary coordinates that identify the past and future tips of a causal diamond. Under this projection, the string equations of motion translate into a dynamics of boundary causal diamonds. A procedure for lifting up a causal diamond to get a proper string world sheet is also developed. In this context we identify an emerging $SO(1,1)\times SO(1,d-1)$ gauge structure incorporated into a Grassmannian $σ$-model targeted in ${\mathbb K}$. The $d=2$ case is worked out in detail. Surprisingly in this case $AdS_3$ with its strings seems to be a natural object which is living inside projective twistor space. On the other hand ${\mathbb K}$ (comprising two copies of two dimensional de Sitter spaces) is a one which is living inside the Klein quadric, as a real section of a complexified space time.

hep-th↗

arXiv2025

A thermofield-double model of Uhlmann anholonomy

A simple parametrized family of quantum systems consisting of two entangled subsystems, dubbed left and right ones, both of them featuring N qubits is considered in the thermofield double formalism. We assume that the system evolves in a purely geometric manner based on the parallel transport condition due to Uhlmann. We explore the different interpretations of this evolution relative to observers either coupled to the left or to the right subsystems. The Uhlmann condition breaks the symmetry between left and right by regarding one of the two possible sets of local unitary operations as gauge degrees of freedom. Then gauging the right side we show that the geometric evolution on the left manifests itself via certain local operations reminiscent of non-unitary filtering measurements. On the other hand on the right the basic evolutionary steps are organized into a sequence of unitary operations of a holonomic quantum computation. We calculate the Uhlmann connection governing the transport for our model which turns out to be related to higher dimensional instantons. Then we evaluate the anholonomy of the connection for geodesic triangles with geodesic segments defined with respect to the Bures metric. By analysing the explicit form of the local filtering measurements showing up on the left side we realize that they are also optimal measurements for distinguishing two given mixed states in the statistical sense. We also point out that by conducting an interference experiment on the right side one can observe the physical effects of the anholonomic quantum computation. We demonstrate this by calculating explicit examples for phase shifts and visibility patterns arising in such interference experiments. Finally a sequence of geodesic triangles producing the iSWAP gate via anholonomy needed for computational universality is presented.

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arXiv2024

Exploiting Finite Geometries for Better Quantum Advantages in Mermin-Like Games

Quantum games embody non-intuitive consequences of quantum phenomena, such as entanglement and contextuality. The Mermin-Peres game is a simple example, demonstrating how two players can utilise shared quantum information to win a no - communication game with certainty, where classical players cannot. In this paper we look at the geometric structure behind such classical strategies, and borrow ideas from the geometry of symplectic polar spaces to maximise this quantum advantage. We introduce a new game called the Eloily game with a quantum-classical success gap of $0.2\overline{6}$, larger than that of the Mermin-Peres and doily games. We simulate this game in the IBM Quantum Experience and obtain a success rate of $1$, beating the classical bound of $0.7\overline{3}$ demonstrating the efficiency of the quantum strategy.

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arXiv2023

Segmented strings and holography

In this paper we establish a connection between segmented strings propagating in $AdS_{d+1}$ and $CFT_d$ subsystems in Minkowski spacetime characterized by quantum information theoretic quantities calculated for the vacuum state. We show that the area of the world sheet of a string segment on the AdS side can be connected to fidelity susceptibility (the real part of the quantum geometric tensor) on the CFT side. This quantity has another interpretation as the computational complexity for infinitesimally separated states corresponding to causal diamonds that are displaced in a spacelike manner according to the metric of kinematic space. These displaced causal diamonds encode information for a unique reconstruction of the string world sheet segments in a holographic manner. Dually the bulk segments are representing causally ordered sets of consecutive boundary events in boosted inertial frames or in noninertial ones proceeding with constant acceleration. For the special case of $AdS_3$ one can also see the segmented stringy area in units of $4GL$ ($G$ is Newton's constant and $L$ is the AdS length) as the conditional mutual information $I(A,C\vert B)$ calculated for a trapezoid configuration arising from boosted spacelike intervals $A$,$B$ and $C$. In this special case the variation of the discretized Nambu-Goto action leads to an equation for entanglement entropies in the boundary theory of the form of a Toda equation. For arbitrary $d$ the string world sheet patches are living in the modular slices of the entanglement wedge. They seem to provide some sort of tomography of the entanglement wedge where the patches are linked together by the interpolation ansatz, i.e. the discretized version of the equations of motion for the Nambu-Goto action.

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arXiv2023

Fluctuations, uncertainty relations, and the geometry of quantum state manifolds

The complete quantum metric of a parametrized quantum system has a real part (usually known as the Provost-Vallee metric) and a symplectic imaginary part (known as the Berry curvature). In this paper, we first investigate the relation between the Riemann curvature tensor of the space described by the metric, and the Berry curvature, by explicit parallel transport of a vector in Hilbert space. Subsequently, we write a generating function from which the complex metric, as well as higher order geometric tensors (affine connection, Riemann curvature tensor) can be obtained in terms of gauge invariant cumulants. The generating function explicitly relates the quantities which characterize the geometry of the parameter space to quantum fluctuations. We also show that for a mixed quantum-classical system both real and imaginary parts of the quantum metric contribute to the dynamics, if the mass tensor is Hermitian. A many operator generalization of the uncertainty principle results from taking the determinant of the complex quantum metric. We also calculate the quantum metric for a number of Lie group coherent states, including several representations of the $SU(1,1)$ group. In our examples non-trivial complex geometry results for generalized coherent states. A pair of oscillator states corresponding to the $SU(1,1)$ group gives a double series for its spectrum. The two minimal uncertainty coherent states show trivial geometry, but, again, for generalized coherent states non-trivial geometry results.

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arXiv2021

Cluster algebraic description of entanglement patterns for the BTZ black hole

We study the thermal state of a two dimensional conformal field theory which is dual to the static BTZ black hole in the high temperature limit. After partitioning the boundary of the static BTZ slice into $N$ subsystems we show that there is an underlying $C_{N-1}$ cluster algebra encoding entanglement patterns of the thermal state. We also demonstrate that the polytope encapsulating such patterns in a geometric manner for a fixed $N$ is the cyclohedron ${\mathcal C}_{N-1}$. Alternatively these patterns of entanglement can be represented in the space of geodesics (kinematic space) in terms of a Zamolodchikov $Y$-system of $C_{N-1}$ type. The boundary condition for such an $Y$-system is featuring the entropy of the BTZ black hole.

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arXiv2021

The associahedron as a holographic entanglement polytope

By employing the ${\rm AdS}_3/{\rm CFT}_2$ correspondence in this note we observe an analogy between the structures found in connection with the Arkani-Hamed-Bai-He-Yan (ABHY) associahedron used for understanding scattering amplitudes, and the one used for understanding space-time emerging from patterns of entanglement. The analogy suggests the natural interpretation for the associahedron as a holographic entanglement polytope associated to the ${\rm CFT}_2$ vacuum. Our observations hint at the possibility that the factorization properties of scattering amplitudes are connected to the notion of separability of space-time as used in the theory of holographic quantum entanglement.

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arXiv2020

$X$-States From a Finite Geometric Perspective

It is found that $15$ different types of two-qubit $X$-states split naturally into two sets (of cardinality $9$ and $6$) once their entanglement properties are taken into account. We {characterize both the validity and entangled nature of the $X$-states with maximally-mixed subsystems in terms of certain parameters} and show that their properties are related to a special class of geometric hyperplanes of the symplectic polar space of order two and rank two. Finally, we introduce the concept of hyperplane-states and briefly address their non-local properties.

math-ph↗

arXiv2020

Scanning space-time with patterns of entanglement

In the ${\rm AdS}_3/{\rm CFT}_2$ setup we elucidate how gauge invariant boundary patterns of entanglement of the CFT vacuum are encoded into the bulk via the coefficient dynamics of an $A_{N-3}$, $N\geq 4$ cluster algebra. In the static case this dynamics of encoding manifests itself in kinematic space, which is a copy of de Sitter space ${\rm dS}_2$, in a particularly instructive manner. For a choice of partition of the boundary into $N$ regions the patterns of entanglement, associated with conditional mutual informations of overlapping regions, are related to triangulations of geodesic $N$-gons. Such triangulations are then mapped to causal patterns in kinematic space. For a fixed $N$ the space of all causal patterns is related to the associahedron ${\mathcal K}^{N-3}$ an object well-known from previous studies on scattering amplitudes. On this space of causal patterns cluster dynamics acts by a recursion provided by a Zamolodchikov's $Y$-system of type $(A_{N-3},A_1)$. We observe that the space of causal patterns is equipped with a partial order, and is isomorphic to the Tamari lattice. The mutation of causal patterns can be encapsulated by a walk of $N-3$ particles interacting in a peculiar manner in the past light cone of a point of ${\rm dS}_2$.

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arXiv2019

Berry curvature, horocycles and scattering states in $AdS_3/CFT_2$

By studying the space of geodesics in $ADS_3/CFT_2$ and quantizing the geodesic motion, we relate scattering data to boundary entanglement of the CFT vacuum. The basic idea is to use a family of plane waves parametrized by coordinates of the space of geodesics i.e. kinematic space. This idea enables a simple calculation of the Berry curvature living on kinematic space. As a result we recover the Crofton form with a coefficient depending on the scattering energy. In arriving at these results the space of horocycles is used. We show that this new space used in concert with kinematic space incorporates naturally the gauge degrees of freedom responsible for an analogue of Berry's Phase. Horocycles also give a new geometric look to the strong subadditivity relation in terms of lambda lengths giving rise to shear coordinates of geodesic quadrangles. A generalization for geodesic polygons then reveals an interesting connection with $A_{n}$ cluster algebras. Here the cluster variables are the lambda lengths related to the regularized entropies of the boundary via the Ryu-Takayanagi relation. An elaboration of this idea indicates that cluster algebras might provide a natural algebraic means for encoding the gauge invariant entanglement patterns of certain boundary entangled states in the geometry of bulk geodesics. Finally using the language of integral geometry we show how certain propagators connected to the bulk, boundary and kinematic spaces are related to data of elementary scattering problems. We also present some hints how these ideas might be generalized for more general holographic scenarios.

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arXiv2018

A finite geometric toy model of space-time as an error correcting code

A finite geometric model of space-time (which we call the bulk) is shown to emerge as a set of error correcting codes. The bulk is encoding a set of messages located in a blow up of the Gibbons-Hoffman-Wootters (GHW) discrete phase space for $n$-qubits (which we call the boundary). Our error correcting code is a geometric subspace code known from network coding, and the correspondence map is the finite geometric analogue of the Plücker map well-known form twistor theory. The $n=2$ case of the bulk-boundary correspondence is precisely the twistor correspondence where the boundary is playing the role of the twistor space and the bulk is a finite geometric version of compactified Minkowski space-time. For $n\geq 3$ the bulk is identified with the finite geometric version of the Brody-Hughston quantum space-time. For special regions on both sides of the correspondence we associate certain collections of qubit observables. On the boundary side this association gives rise to the well-known GHW quantum net structure. In this picture the messages are complete sets of commuting observables associated to Lagrangian subspaces giving a partition of the boundary. Incomplete subsets of observables corresponding to subspaces of the Lagrangian ones are regarded as corrupted messages. Such a partition of the boundary is represented on the bulk side as a special collection of space-time points. For a particular message residing in the boundary, the set of possible errors is described by the fine details of the light-cone structure of its representative space-time point in the bulk. The geometric arrangement of representative space-time points, playing the role of the variety of codewords, encapsulates an algebraic algorithm for recovery from errors on the boundary side.

hep-th↗

arXiv2018

A fermionic code related to the exceptional group $E_8$

In this paper we study the Hamming-like fermionic code encoding three-qubits into sixteen Majorana modes recently introduced by Hastings. We show that although this fermionic code cannot be obtained from a single qubit stabilizer code via the usual procedure however, it can be obtained from two, qubit stabilizer ones via a glueing procedure combining both single and double occupancy embeddings of qubits into fermionic Fock space. This technique identifies the code subspace as a Cartan subspace of the largest exceptional group $E_8$. In arriving at these results we develop a general formalism for implementing this glueing procedure via the use of intertwiners between different subsectors of the Fock space realizing embedded qubits.

quant-ph↗

arXiv2017

The magic three-qubit Veldkamp line: A finite geometric underpinning for form theories of gravity and black hole entropy

We investigate the structure of the three-qubit magic Veldkamp line (MVL). This mathematical notion has recently shown up as a tool for understanding the structures of the set of Mermin pentagrams, objects that are used to rule out certain classes of hidden variable theories. Here we show that this object also provides a unifying finite geometric underpinning for understanding the structure of functionals used in form theories of gravity and black hole entropy. We clarify the representation theoretic, finite geometric and physical meaning of the different parts of our MVL. The upshot of our considerations is that the basic finite geometric objects enabling such a diversity of physical applications of the MVL are the unique generalized quadrangles with lines of size three, their one point extensions as well as their other extensions isomorphic to affine polar spaces of rank three and order two. In a previous work we have already connected generalized quadrangles to the structure of cubic Jordan algebras related to entropy fomulas of black holes and strings in five dimensions. In some respect the present paper can be regarded as a generalization of that analysis for also providing a finite geometric understanding of four-dimensional black hole entropy formulas. However, we find many more structures whose physical meaning is yet to be explored. As a familiar special case our work provides a finite geometric representation of the algebraic extension from cubic Jordan algebras to Freudenthal systems based on such algebras.

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arXiv2017

Mermin pentagrams arising from Veldkamp lines for three qubits

We study the geometry of the space of Mermin pentagrams, objects that are used to rule out the existence of noncontextual hidden variable theories as alternatives to quantum theory. It is shown that this space of 12096 possible pentagrams is organized into 1008 families, with each family containing a "double six" of pentagrams. The 1008 families are connected to a special set of Veldkamp lines in the Veldkamp space of three qubits an object well-known to finite geometers but has only been introduced to physics recently. Due to the transitive action of the symplectic group on this set of Veldkamp lines it is enough to study only one "canonical" double six configuration of pentagrams. We prove that the geometry of this double-six configuration is encapsulated in the weight diagram of the 20 dimensional irreducible representation of the group $SU(6)$. As an interesting by-product of our approach we show that Mermin pentagrams and a class of Mermin squares labelled by three qubit Pauli operators are inherently related. We conjecture that by studying the representation theoretic content of other Veldkamp lines of the Veldkamp space for $N$-qubits makes it possible to find new contextual configurations in a systematic manner.

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arXiv2016

The Coupled Cluster Method and Entanglement

The Coupled Cluster (CC) and full CI expansions are studied for three fermions with six and seven modes. Surprisingly the CC expansion is tailor made to characterize the usual SLOCC entanglement classes. It means that the notion of a SLOCC transformation shows up quite naturally as a one relating the CC and CI expansions, and going from the CI expansion to the CC one is equivalent to obtaining a form for the state where the structure of the entanglement classes is transparent. In this picture entanglement is characterized by the parameters of the cluster operators describing transitions from occupied states to singles, doubles and triples of non occupied ones. Using the CC parametrization of states in the seven mode case we give a simple formula for the unique SLOCC invariant $\mathcal{J}$. Then we consider a perturbation problem featuring a state from the unique SLOCC class characterized by $\mathcal{J}\neq 0$. For this state with entanglement generated by doubles we investigate the phenomenon of changing the entanglement type due to the perturbing effect of triples. We show that there are states with real amplitudes such that their entanglement encoded into configurations of clusters of doubles is protected from errors generated by triples. Finally we put forward a proposal to use the parameters of the cluster operator describing transitions to doubles for entanglement characterization. Compared to the usual SLOCC classes this provides a coarse grained approach to fermionic entanglement.

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arXiv2015

Embedding qubits into fermionic Fock space, peculiarities of the four-qubit case

We give a fermionic Fock space description of embedded entangled qubits. Within this framework the problem of classification of pure state entanglement boils down to the problem of classifying spinors. The usual notion of separable states turns out to be just a special case of the one of pure spinors. By using the notion of single, double and mixed occupancy representation with intertwiners relating them a natural physical interpretation of embedded qubits is found. As an application of these ideas one can make a physically sound meaning of some of the direct sum structures showing up in the context of the so-called Black-Hole/Qubit Correspondence. We discuss how the usual invariants for qubits serving as measures of entanglement can be obtained from invariants for spinors in an elegant manner. In particular a detailed case study for recovering the invariants for four-qubits within a spinorial framework is presented. We also observe that reality conditions on complex spinors defining Majorana spinors for embedded qubits boil down to self conjugate states under the Wootters spin flip operation. Finally we conduct a study on the explicit structure of $Spin(16,\mathbb{C})$ invariant polynomials related to the structure of possible measures of entanglement for fermionic systems with 8 modes. Here we find an algebraically independent generating set of the generalized SLOCC invariants and calculate their restriction to the dense orbit. We point out the special role the largest exceptional group $E_8$ is playing in these considerations.

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arXiv2015

Classification of multipartite systems featuring only $|W\rangle$ and $|GHZ\rangle$ genuine entangled states

In this paper we present several multipartite quantum systems featuring the same type of genuine (tripartite) entanglement. Based on a geometric interpretation of the so-called $|W\rangle$ and $|GHZ\rangle$ states we show that the classification of all multipartite systems featuring those and only those two classes of genuine entanglement can be deduced from earlier work of algebraic geometers. This classification corresponds in fact to classification of fundamental subadjoint varieties and establish a connection between those systems, well known in Quantum Information Theory and fundamental simple Lie algebras.

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