arXiv subjects

Michel Planat

Publications and source records attributed to Michel Planat.

At least 37 records · Page 2Linked to original sources

arXiv2016

Geometric contextuality from the Maclachlan-Martin Kleinian groups

There are contextual sets of multiple qubits whose commutation is parametrized thanks to the coset geometry $\mathcal{G}$ of a subgroup $H$ of the two-generator free group $G=\left\langle x,y\right\rangle$. One defines geometric contextuality from the discrepancy between the commutativity of cosets on $\mathcal{G}$ and that of quantum observables.It is shown in this paper that Kleinian subgroups $K=\left\langle f,g\right\rangle$ that are non-compact, arithmetic, and generated by two elliptic isometries $f$ and $g$ (the Martin-Maclachlan classification), are appropriate contextuality filters. Standard contextual geometries such as some thin generalized polygons (starting with Mermin's $3 \times 3$ grid) belong to this frame. The Bianchi groups $PSL(2,O\_d)$, $d \in \{1,3\}$ defined over the imaginary quadratic field $O\_d=\mathbb{Q}(\sqrt{-d})$ play a special role.

quant-ph↗

arXiv2016

Zoology of Atlas-groups: dessins d'enfants, finite geometries and quantum commutation

Every finite simple group P can be generated by two of its elements. Pairs of generators for P are available in the Atlas of finite group representations as (not neccessarily minimal) permutation representations P. It is unusual but significant to recognize that a P is a Grothendieck's dessin d'enfant D and that most standard graphs and finite geometries G-such as near polygons and their generalizations-are stabilized by a D. In our paper, tripods P -- D -- G of rank larger than two, corresponding to simple groups, are organized into classes, e.g. symplectic, unitary, sporadic, etc (as in the Atlas). An exhaustive search and characterization of non-trivial point-line configurations defined from small index representations of simple groups is performed, with the goal to recognize their quantum physical significance. All the defined geometries G' s have a contextuality parameter close to its maximal value 1.

quant-ph↗

arXiv2015

Geometry of contextuality from Grothendieck's coset space

The geometry of cosets in the subgroups H of the two-generator free group G =\textless{} a, b \textgreater{} nicely fits, via Grothendieck's dessins d'enfants, the geometry of commutation for quantum observables. Dessins stabilize point-line incidence geometries that reflect the commutation of (generalized) Pauli operators [Information 5, 209 (2014); 1310.4267 and 1404.6986 (quant-ph)]. Now we find that the non-existence of a dessin for which the commutator (a, b) = a^ (--1) b^( --1) ab precisely corresponds to the commutator of quantum observables [A, B] = AB -- BA on all lines of the geometry is a signature of quantum contextuality. This occurs first at index |G : H| = 9 in Mermin's square and at index 10 in Mermin's pentagram, as expected. Commuting sets of n-qubit observables with n \textgreater{} 3 are found to be contextual as well as most generalized polygons. A geometrical contextuality measure is introduced.

quant-ph↗

arXiv2015

Quantum contextual finite geometries from dessins d'enfants

We point out an explicit connection between graphs drawn on compact Riemann surfaces defined over the field $\bar{\mathbb{Q}}$ of algebraic numbers --- so-called Grothendieck's {\it dessins d'enfants} --- and a wealth of distinguished point-line configurations. These include simplices, cross-polytopes, several notable projective configurations, a number of multipartite graphs and some 'exotic' geometries. Among them, remarkably, we find not only those underlying Mermin's magic square and magic pentagram, but also those related to the geometry of two- and three-qubit Pauli groups. Of particular interest is the occurrence of all the three types of slim generalized quadrangles, namely GQ(2,1), GQ(2,2) and GQ(2,4), and a couple of closely related graphs, namely the Schläfli and Clebsch ones. These findings seem to indicate that {\it dessins d'enfants} may provide us with a new powerful tool for gaining deeper insight into the nature of finite-dimensional Hilbert spaces and their associated groups, with a special emphasis on contextuality.

quant-ph↗

arXiv2015

A moonshine dialogue in mathematical physics

Phys and Math are two colleagues at the University of Sa{\c c}enbon (Crefan Kingdom), dialoguing about the remarkable efficiency of mathematics for physics. They talk about the notches on the Ishango bone, the various uses of psi in maths and physics, they arrive at dessins d'enfants, moonshine concepts, Rademacher sums and their significance in the quantum world. You should not miss their eccentric proposal of relating Bell's theorem to the Baby Monster group. Their hyperbolic polygons show a considerable singularity/cusp structure that our modern age of computers is able to capture. Henri Poincar{é} would have been happy to see it.

physics.pop-ph↗

arXiv2014

Improving riemann prime counting

Prime number theorem asserts that (at large $x$) the prime counting function $π(x)$ is approximately the logarithmic integral $\mbox{li}(x)$. In the intermediate range, Riemann prime counting function $\mbox{Ri}^{(N)}(x)=\sum_{n=1}^N \frac{μ(n)}{n}\mbox{Li}(x^{1/n})$ deviates from $π(x)$ by the asymptotically vanishing sum $\sum_ρ\mbox{Ri}(x^ρ)$ depending on the critical zeros $ρ$ of the Riemann zeta function $ζ(s)$. We find a fit $π(x)\approx \mbox{Ri}^{(3)}[ψ(x)]$ [with three to four new exact digits compared to $\mbox{li}(x)$] by making use of the Von Mangoldt explicit formula for the Chebyshev function $ψ(x)$. Another equivalent fit makes use of the Gram formula with the variable $ψ(x)$. Doing so, we evaluate $π(x)$ in the range $x=10^i$, $i=[1\cdots 50]$ with the help of the first $2\times 10^6$ Riemann zeros $ρ$. A few remarks related to Riemann hypothesis (RH) are given in this context.

math.NT↗

arXiv2014

Drawing quantum contextuality with 'dessins d'enfants'

In the standard formulation of quantum mechanics, there exists an inherent feedback of the measurement setting on the elementary object under scrutiny. Thus one cannot assume that an 'element of reality' prexists to the measurement and, it is even more intriguing that unperformed/counterfactual observables enter the game. This is called quantum contextuality. Simple finite projective geometries are a good way to picture the commutation relations of quantum observables entering the context, at least for systems with two or three parties. In the essay, it is further discovered a mathematical mechanism for 'drawing' the contexts. The so-called 'dessins d'enfants' of the celebrated mathematician Alexandre Grothendieck feature group, graph, topological, geometric and algebraic properties of the quantum contexts that would otherwise have been 'hidden' in the apparent randomness of measurement outcomes.

quant-ph↗

arXiv2014

Veldkamp-Space Aspects of a Sequence of Nested Binary Segre Varieties

Let $S_{(N)} \equiv PG(1,\,2) \times PG(1,\,2) \times \cdots \times PG(1,\,2)$ be a Segre variety that is $N$-fold direct product of projective lines of size three. Given two geometric hyperplanes $H'$ and $H''$ of $S_{(N)}$, let us call the triple $\{H', H'', \overline{H' ΔH''}\}$ the Veldkamp line of $S_{(N)}$. We shall demonstrate, for the sequence $2 \leq N \leq 4$, that the properties of geometric hyperplanes of $S_{(N)}$ are fully encoded in the properties of Veldkamp {\it lines} of $S_{(N-1)}$. Using this property, a complete classification of all types of geometric hyperplanes of $S_{(4)}$ is provided. Employing the fact that, for $2 \leq N \leq 4$, the (ordinary part of) Veldkamp space of $S_{(N)}$ is $PG(2^N-1,2)$, we shall further describe which types of geometric hyperplanes of $S_{(N)}$ lie on a certain hyperbolic quadric $\mathcal{Q}_0^+(2^N-1,2) \subset PG(2^N-1,2)$ that contains the $S_{(N)}$ and is invariant under its stabilizer group; in the $N=4$ case we shall also single out those of them that correspond, via the Lagrangian Grassmannian of type $LG(4,8)$, to the set of 2295 maximal subspaces of the symplectic polar space $\mathcal{W}(7,2)$.

math.CO↗

arXiv2013

Singularity of type $D_4$ arising from four qubit systems

An intriguing correspondence between four-qubit systems and simple singularity of type $D_4$ is established. We first consider an algebraic variety $X$ of separable states within the projective Hilbert space $\mathbb{P}(\mathcal{H})=\mathbb{P}^{15}$. Then, cutting $X$ with a specific hyperplane $H$, we prove that the $X$-hypersurface, defined from the section $X\cap H\subset X$, has an isolated singularity of type $D_4$; it is also shown that this is the "worst-possible" isolated singularity one can obtain by this construction. Moreover, it is demonstrated that this correspondence admits a dual version by proving that the equation of the dual variety of $X$, which is nothing but the Cayley hyperdeterminant of type $2\times 2\times 2\times 2$, can be expressed in terms of the SLOCC invariant polynomials as the discriminant of the miniversal deformation of the $D_4$-singularity.

math-ph↗

arXiv2013

Grassmannian Connection Between Three- and Four-Qubit Observables, Mermin's Contextuality and Black Holes

We invoke some ideas from finite geometry to map bijectively 135 heptads of mutually commuting three-qubit observables into 135 symmetric four-qubit ones. After labeling the elements of the former set in terms of a seven-dimensional Clifford algebra, we present the bijective map and most pronounced actions of the associated symplectic group on both sets in explicit forms. This formalism is then employed to shed novel light on recently-discovered structural and cardinality properties of an aggregate of three-qubit Mermin's 'magic' pentagrams. Moreover, some intriguing connections with the so-called black-hole--qubit correspondence are also pointed out.

math-ph↗

arXiv2012

Distinguished three-qubit 'magicity' via automorphisms of the split Cayley hexagon

Disregarding the identity, the remaining 63 elements of the generalized three-qubit Pauli group are found to contain 12096 distinct copies of Mermin's magic pentagram. Remarkably, 12096 is also the number of automorphisms of the smallest split Cayley hexagon. We give a few solid arguments showing that this may not be a mere coincidence. These arguments are mainly tied to the structure of certain types of geometric hyperplanes of the hexagon. It is further demonstrated that also an (18_{2}, 12_{3})-type of magic configurations, recently proposed by Waegell and Aravind (J. Phys. A: Math. Theor. 45 (2012) 405301), seems to be intricately linked with automorphisms of the hexagon. Finally, the entanglement properties exhibited by edges of both pentagrams and these particular Waegell-Aravind configurations are addressed.

math-ph↗

arXiv2012

'Magic' Configurations of Three-Qubit Observables and Geometric Hyperplanes of the Smallest Split Cayley Hexagon

Recently Waegell and Aravind [J. Phys. A: Math. Theor. 45 (2012), 405301, 13 pages] have given a number of distinct sets of three-qubit observables, each furnishing a proof of the Kochen-Specker theorem. Here it is demonstrated that two of these sets/configurations, namely the $18_{2} - 12_{3}$ and $2_{4}14_{2} - 4_{3}6_{4}$ ones, can uniquely be extended into geometric hyperplanes of the split Cayley hexagon of order two, namely into those of types ${\cal V}_{22}(37; 0, 12, 15, 10)$ and ${\cal V}_{4}(49; 0, 0, 21, 28)$ in the classification of Frohardt and Johnson [Comm. Algebra 22 (1994), 773-797]. Moreover, employing an automorphism of order seven of the hexagon, six more replicas of either of the two configurations are obtained.

math-ph↗

arXiv2012

On small proofs of Bell-Kochen-Specker theorem for two, three and four qubits

The Bell-Kochen-Specker theorem (BKS) theorem rules out realistic {\it non-contextual} theories by resorting to impossible assignments of rays among a selected set of maximal orthogonal bases. We investigate the geometrical structure of small $v-l$ BKS-proofs involving $v$ real rays and $l$ $2n$-dimensional bases of $n$-qubits ($1< n < 5$). Specifically, we look at the parity proof 18-9 with two qubits (A. Cabello, 1996), the parity proof 36-11 with three qubits (M. Kernaghan & A. Peres, 1995 \cite{Kernaghan1965}) and a newly discovered non-parity proof 80-21 with four qubits (that improves work of P. K Aravind's group in 2008). The rays in question arise as real eigenstates shared by some maximal commuting sets (bases) of operators in the $n$-qubit Pauli group. One finds characteristic signatures of the distances between the bases, which carry various symmetries in their graphs.

quant-ph↗

arXiv2012

Finite Geometry Behind the Harvey-Chryssanthacopoulos Four-Qubit Magic Rectangle

A "magic rectangle" of eleven observables of four qubits, employed by Harvey and Chryssanthacopoulos (2008) to prove the Bell-Kochen-Specker theorem in a 16-dimensional Hilbert space, is given a neat finite-geometrical reinterpretation in terms of the structure of the symplectic polar space $W(7, 2)$ of the real four-qubit Pauli group. Each of the four sets of observables of cardinality five represents an elliptic quadric in the three-dimensional projective space of order two (PG$(3, 2)$) it spans, whereas the remaining set of cardinality four corresponds to an affine plane of order two. The four ambient PG$(3, 2)$s of the quadrics intersect pairwise in a line, the resulting six lines meeting in a point. Projecting the whole configuration from this distinguished point (observable) one gets another, complementary "magic rectangle" of the same qualitative structure.

quant-ph↗

arXiv2012

Five-Qubit Contextuality, Noise-Like Distribution of Distances Between Maximal Bases and Finite Geometry

Employing five commuting sets of five-qubit observables, we propose specific 160-661 and 160-21 state proofs of the Bell-Kochen-Specker theorem that are also proofs of Bell's theorem. A histogram of the 'Hilbert-Schmidt' distances between the corresponding maximal bases shows in both cases a noise-like behaviour. The five commuting sets are also ascribed a finite-geometrical meaning in terms of the structure of symplectic polar space W(9,2).

quant-ph↗

arXiv2012

Twelve-dimensional Pauli group contextuality

The goal of the paper is to check whether the real eigenstates of the observables in the single qudit Pauli group may lead to quantum contextuality, the property that mutually compatible and independent experiments depend on each other. We find that quantum contextuality crops up at dimension twelve in various configurations with a few rays. We use the Shannon capacity for characterizing the corresponding orthogonality graphs. Some arithmetical properties underlying the qudit contextuality are outlined.

quant-ph↗

arXiv2011

Chebyshev's bias and generalized Riemann hypothesis

It is well known that $li(x)>π(x)$ (i) up to the (very large) Skewes' number $x_1 \sim 1.40 \times 10^{316}$ \cite{Bays00}. But, according to a Littlewood's theorem, there exist infinitely many $x$ that violate the inequality, due to the specific distribution of non-trivial zeros $γ$ of the Riemann zeta function $ζ(s)$, encoded by the equation $li(x)-π(x)\approx \frac{\sqrt{x}}{\log x}[1+2 \sum_γ\frac{\sin (γ\log x)}γ]$ (1). If Riemann hypothesis (RH) holds, (i) may be replaced by the equivalent statement $li[ψ(x)]>π(x)$ (ii) due to Robin \cite{Robin84}. A statement similar to (i) was found by Chebyshev that $π(x;4,3)-π(x;4,1)>0$ (iii) holds for any $x<26861$ \cite{Rubin94} (the notation $π(x;k,l)$ means the number of primes up to $x$ and congruent to $l\mod k$). The {\it Chebyshev's bias}(iii) is related to the generalized Riemann hypothesis (GRH) and occurs with a logarithmic density $\approx 0.9959$ \cite{Rubin94}. In this paper, we reformulate the Chebyshev's bias for a general modulus $q$ as the inequality $B(x;q,R)-B(x;q,N)>0$ (iv), where $B(x;k,l)=li[ϕ(k)*ψ(x;k,l)]-ϕ(k)*π(x;k,l)$ is a counting function introduced in Robin's paper \cite{Robin84} and $R$ resp. $N$) is a quadratic residue modulo $q$ (resp. a non-quadratic residue). We investigate numerically the case $q=4$ and a few prime moduli $p$. Then, we proove that (iv) is equivalent to GRH for the modulus $q$.

math.NT↗

arXiv2011

Efficient prime counting and the Chebyshev primes

The function $ε(x)=\mbox{li}(x)-π(x)$ is known to be positive up to the (very large) Skewes' number. Besides, according to Robin's work, the functions $ε_θ(x)=\mbox{li}[θ(x)]-π(x)$ and $ε_ψ(x)=\mbox{li}[ψ(x)]-π(x)$ are positive if and only if Riemann hypothesis (RH) holds (the first and the second Chebyshev function are $θ(x)=\sum_{p \le x} \log p$ and $ψ(x)=\sum_{n=1}^x Λ(n)$, respectively, $\mbox{li}(x)$ is the logarithmic integral, $μ(n)$ and $Λ(n)$ are the Möbius and the Von Mangoldt functions). Negative jumps in the above functions $ε$, $ε_θ$ and $ε_ψ$ may potentially occur only at $x+1 \in \mathcal{P}$ (the set of primes). One denotes $j_p=\mbox{li}(p)-\mbox{li}(p-1)$ and one investigates the jumps $j_p$, $j_{θ(p)}$ and $j_{ψ(p)}$. In particular, $j_p<1$, and $j_{θ(p)}>1$ for $p<10^{11}$. Besides, $j_{ψ(p)}<1$ for any odd $p \in \mathcal{\mbox{Ch}}$, an infinite set of so-called {\it Chebyshev primes } with partial list $\{109, 113, 139, 181, 197, 199, 241, 271, 281, 283, 293, 313, 317, 443, 449, 461, 463, \ldots\}$. We establish a few properties of the set $\mathcal{\mbox{Ch}}$, give accurate approximations of the jump $j_{ψ(p)}$ and relate the derivation of $\mbox{Ch}$ to the explicit Mangoldt formula for $ψ(x)$. In the context of RH, we introduce the so-called {\it Riemann primes} as champions of the function $ψ(p_n^l)-p_n^l$ (or of the function $θ(p_n^l)-p_n^l$ ). Finally, we find a {\it good} prime counting function $S_N(x)=\sum_{n=1}^N \frac{μ(n)}{n}\mbox{li}[ψ(x)^{1/n}]$, that is found to be much better than the standard Riemann prime counting function.

math.NT↗