arXiv subjects

Michel R. P. Planat

Publications and source records attributed to Michel R. P. Planat.

9 recordsLinked to original sources

arXiv2013

It from qubit: how to draw quantum contextuality

Wheeler's {\it observer-participancy} and the related {\it it from bit} credo refer to quantum non-locality and contextuality. The mystery of these concepts slightly starts unveiling if one encodes the (in)compatibilities between qubit observables in the relevant finite geometries. The main objective of this treatise is to outline another conceptual step forward by employing Grothendieck's {\it dessins d'enfants} to reveal the topological and (non)algebraic machinery underlying the measurement acts and their information content.

quant-ph↗

arXiv2012

Quantum States Arising from the Pauli Groups, Symmetries and Paradoxes

We investigate multiple qubit Pauli groups and the quantum states/rays arising from their maximal bases. Remarkably, the real rays are carried by a Barnes-Wall lattice $BW_n$ ($n=2^m$). We focus on the smallest subsets of rays allowing a state proof of the Bell-Kochen-Specker theorem (BKS). BKS theorem rules out realistic 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 BKS-proofs $v-l$ involving $v$ rays and $l$ $2n$-dimensional bases of $n$-qubits. Specifically, we look at the classes of parity proofs 18-9 with two qubits (A. Cabello, 1996), 36-11 with three qubits (M. Kernaghan & A. Peres, 1995) and related classes. One finds characteristic signatures of the distances among the bases, that carry various symmetries in their graphs.

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arXiv2010

About the Dedekind psi function in Pauli graphs

We study the commutation structure within the Pauli groups built on all decompositions of a given Hilbert space dimension $q$, containing a square, into its factors. The simplest illustrative examples are the quartit ($q=4$) and two-qubit ($q=2^2$) systems. It is shown how the sum of divisor function $σ(q)$ and the Dedekind psi function $ψ(q)=q \prod_{p|q} (1+1/p)$ enter into the theory for counting the number of maximal commuting sets of the qudit system. In the case of a multiple qudit system (with $q=p^m$ and $p$ a prime), the arithmetical functions $σ(p^{2n-1})$ and $ψ(p^{2n-1})$ count the cardinality of the symplectic polar space $W_{2n-1}(p)$ that endows the commutation structure and its punctured counterpart, respectively. Symmetry properties of the Pauli graphs attached to these structures are investigated in detail and several illustrative examples are provided.

quant-ph↗

arXiv2007

Multi-Line Geometry of Qubit-Qutrit and Higher-Order Pauli Operators

The commutation relations of the generalized Pauli operators of a qubit-qutrit system are discussed in the newly established graph-theoretic and finite-geometrical settings. The dual of the Pauli graph of this system is found to be isomorphic to the projective line over the product ring Z2xZ3. A "peculiar" feature in comparison with two-qubits is that two distinct points/operators can be joined by more than one line. The multi-line property is shown to be also present in the graphs/geometries characterizing two-qutrit and three-qubit Pauli operators' space and surmised to be exhibited by any other higher-level quantum system.

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arXiv2007

Pauli graph and finite projective lines/geometries

The commutation relations between the generalized Pauli operators of N-qudits (i. e., N p-level quantum systems), and the structure of their maximal sets of commuting bases, follow a nice graph theoretical/geometrical pattern. One may identify vertices/points with the operators so that edges/lines join commuting pairs of them to form the so-called Pauli graph P_{p^N} . As per two-qubits (p = 2, N = 2) all basic properties and partitionings of this graph are embodied in the geometry of the symplectic generalized quadrangle of order two, W(2). The structure of the two-qutrit (p = 3, N = 2) graph is more involved; here it turns out more convenient to deal with its dual in order to see all the parallels with the two-qubit case and its surmised relation with the geometry of generalized quadrangle Q(4, 3), the dual of W(3). Finally, the generalized adjacency graph for multiple (N > 3) qubits/qutrits is shown to follow from symplectic polar spaces of order two/three. The relevance of these mathematical concepts to mutually unbiased bases and to quantum entanglement is also highlighted in some detail.

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arXiv2006

A Survey of Finite Algebraic Geometrical Structures Underlying Mutually Unbiased Quantum Measurements

The basic methods of constructing the sets of mutually unbiased bases in the Hilbert space of an arbitrary finite dimension are discussed and an emerging link between them is outlined. It is shown that these methods employ a wide range of important mathematical concepts like, e.g., Fourier transforms, Galois fields and rings, finite and related projective geometries, and entanglement, to mention a few. Some applications of the theory to quantum information tasks are also mentioned.

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arXiv2006

Quantum Entanglement and Projective Ring Geometry

The paper explores the basic geometrical properties of the observables characterizing two-qubit systems by employing a novel projective ring geometric approach. After introducing the basic facts about quantum complementarity and maximal quantum entanglement in such systems, we demonstrate that the 15$\times$15 multiplication table of the associated four-dimensional matrices exhibits a so-far-unnoticed geometrical structure that can be regarded as three pencils of lines in the projective plane of order two. In one of the pencils, which we call the kernel, the observables on two lines share a base of Bell states. In the complement of the kernel, the eight vertices/observables are joined by twelve lines which form the edges of a cube. A substantial part of the paper is devoted to showing that the nature of this geometry has much to do with the structure of the projective lines defined over the rings that are the direct product of $n$ copies of the Galois field GF(2), with $n$ = 2, 3 and 4.

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arXiv2005

Huyghens, Bohr, Riemann and Galois: Phase-Locking

Several mathematical views of phase-locking are developed. The classical Huyghens approach is generalized to include all harmonic and subharmonic resonances and is found to be connected to 1/f noise and prime number theory. Two types of quantum phase-locking operators are defined, one acting on the rational numbers, the other on the elements of a Galois field. In both cases we analyse in detail the phase properties and find them related respectively to the Riemann zeta function and to incomplete Gauss sums.

math-ph↗

arXiv2005

Abstract algebra, projective geometry and time encoding of quantum information

Algebraic geometrical concepts are playing an increasing role in quantum applications such as coding, cryptography, tomography and computing. We point out here the prominent role played by Galois fields viewed as cyclotomic extensions of the integers modulo a prime characteristic $p$. They can be used to generate efficient cyclic encoding, for transmitting secrete quantum keys, for quantum state recovery and for error correction in quantum computing. Finite projective planes and their generalization are the geometric counterpart to cyclotomic concepts, their coordinatization involves Galois fields, and they have been used repetitively for enciphering and coding. Finally the characters over Galois fields are fundamental for generating complete sets of mutually unbiased bases, a generic concept of quantum information processing and quantum entanglement. Gauss sums over Galois fields ensure minimum uncertainty under such protocols. Some Galois rings which are cyclotomic extensions of the integers modulo 4 are also becoming fashionable for their role in time encoding and mutual unbiasedness.

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