Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

QUANTUM COMPUTING LAND with CRISTALLOGRAPHIC GROUPS

Michel R. P. Planat

Published Aug 1, 2009• 0 citations
OpenAlex

Abstract

All the distinctive features of quantum mechanics versus its classical counterpart crucially depends on the non-commutativity of observables. For instance, the existence of disjoint sets of mutually commuting operators implies quantum complementarity, in the sense that precise measurements on one set implies that possible outcomes of measurements on the other set are equally probable. The maximal number of such mutually disjoint sets (also called mutually unbiased bases) and their structure relates on concepts of finite projective geometry 1 and group theory, that we developed in the past five years 2. Among such mutually commuting sets, those sharing a base of entangled states are at the origin of paradoxes named Kochen-Specker or Bell's theorems, that recognize the contradiction between the algebraic structure of eigenvalues/ measurements and the corresponding expressions for the eigenstates 3. We investigated the zoology and geometry of all mutually unbiased bases for spaces of prime dimension 4 or composite dimension 5. Inhabitants in quantum computing world are qubits and tensor products of them, their houses are the mutually commuting sets, finite geometries are the villages and towns. I discovered a large city with 696 729 600 apartments that is, in mathematics, isomorphic to the largest crystallographic group 6, named W(E8). Groups of three inhabitants/states in the villages are in general highly connected/entangled, their contract is of the type GHZ (true tripartite union), W (three bipartite unions), CPT (a compromise), chain state (a pair can get a divorce) and so on. The population of the Pauli group Pn is 4n+1, meaning that at most n people are connected (by a tensor product). The whole unitary world is infinite, but people usually travel in a finite part of it named the Clifford group Cn. The Clifford group divides into males/females and thus into two dipolar groups denoted C±n . Transsexual people do exist in Cn, e.g. for n = 2, |P2| = 64, |C2| = 92160 and C+ 2 ∩ C−2 = 8 (as described in 7.). The group C+ 3 is the largest maximal subgroup of W(E8) and indeed the aforementioned contracts are work contracts. Following Mermin's great intuition about quantum paradoxes, I have discovered two real twoqubit matrices controling every action in Cn, the first is a braiding matrix R (of a topological character) and the second is a CPT matrix S (related to charge conjugation C, parity P and time reversal T ). The dipolar group of a male type C+ n is labeled by S and the dipolar group C−n of female type is labeled by R. The geometry of the octahedron is associated to the group R, S generated by R and S, as already anticipated in the XIXth century 8 and thoroughly developed in the context of self-dual codes 9. By the way, the CPT invariance is expressed, up to an isomorphism, by the single qubit Pauli group P1 10 and is the kernel of the threequbit representation of W(E8), the full Dirac group of gamma matrices is the two-qubit Pauli group P2 and is generated in W(E8) by matrices sustaining quantum states both of the GHZ and CPT type. Further type of three-qubit entangled states are carried by larger subgroups such as W(F4), W(H4), W(E6) and W(E7) yet to be investigated in detail, in relation to the quaternionic and octonionic representations. All these topics, and related ones, will be introduced in my series of lectures at ZIF, on August 12 to 14, 2009. 1Levay P, Saniga M and Vrana P 2008 Phys. Rev. D 78 124022. 2Planat M and Saniga M 2008 Quant. Inf. Comp. 8 127. 3Mermin N D 1993 Rev. Mod. Phys. 65 803. 4Planat M and Rosu H 2005 Eur. Phys. J. D 36 133. 5Planat M, Baboin A C and Saniga M 2008 Int. J. Theor. Phys. 47 1127. 6Planat M 2009 Preprint 0904.3691 (quant-ph). 7Planat M and Sol´e P 2008 J. Phys. A: Math. Theor. 42 8Klein F 1956 Lectures on the icosahedron and the solution of equations of the fifth degree (Dover: New York). 9Nebe G, Rains E M and Sloane N J A Self-dual codes and invariant theory (Berlin: Springer). 10Socolovsky M 2004 Int. J. Theor. Phys. 43 1941.

Related papers

ON MUTUALLY UNBIASED BASES

Thomas Durt, Berthold‐Georg Englert, Ingemar Bengtsson +1 more

Mutually unbiased bases for quantum degrees of freedom are central to all theoretical investigations and practical exploitations of complementary properties. Much is known about mutually unbiased bases, but there are also a fair number of important questions that have not been answered in full as yet. In particular, one can find maximal sets of N + 1 mutually unbiased bases in Hilbert spaces of prime-power dimension N = p M , with p prime and M a positive integer, and there is a continuum of mutually unbiased bases for a continuous degree of freedom, such as motion along a line. But not a single example of a maximal set is known if the dimension is another composite number (N = 6, 10, 12,…). In this review, we present a unified approach in which the basis states are labeled by numbers 0, 1, 2, …, N - 1 that are both elements of a Galois field and ordinary integers. This dual nature permits a compact systematic construction of maximal sets of mutually unbiased bases when they are known to exist but throws no light on the open existence problem in other cases. We show how to use the thus constructed mutually unbiased bases in quantum-informatics applications, including dense coding, teleportation, entanglement swapping, covariant cloning, and state tomography, all of which rely on an explicit set of maximally entangled states (generalizations of the familiar two–q-bit Bell states) that are related to the mutually unbiased bases. There is a link to the mathematics of finite affine planes. We also exploit the one-to-one correspondence between unbiased bases and the complex Hadamard matrices that turn the bases into each other. The ultimate hope, not yet fulfilled, is that open questions about mutually unbiased bases can be related to open questions about Hadamard matrices or affine planes, in particular the notorious existence problem for dimensions that are not a power of a prime. The Hadamard-matrix approach is instrumental in the very recent advance, surveyed here, of our understanding of the N = 6 situation. All evidence indicates that a maximal set of seven mutually unbiased bases does not exist — one can find no more than three pairwise unbiased bases — although there is currently no clear-cut demonstration of the case.

arXiv:1004.3348•International Journal of Quantum Information•Jun 1, 2010•727 citations

Entanglement patterns in mutually unbiased basis sets

Jay Lawrence

A few simply stated rules govern the entanglement patterns that can occur in mutually unbiased basis sets (MUBs) and constrain the combinations of such patterns that can coexist in full complements of MUBs. We consider Hilbert spaces of prime power dimensions ($D={p}^{N}$), as realized by systems of $N$ prime-state particles, where full complements of $D+1$ MUBs are known to exist, and we assume only that MUBs are eigenbases of generalized Pauli operators, without using any particular construction. The general rules include the following: (1) In any MUB, a given particle appears either in a pure state or totally entangled and (2) in any full MUB complement, each particle is pure in ($p+1$) bases (not necessarily the same ones) and totally entangled in the remaining (${p}^{N}\ensuremath{-}p$). It follows that the maximum number of product bases is $p+1$ and, when this number is realized, all remaining (${p}^{N}\ensuremath{-}p$) bases in the complement are characterized by the total entanglement of every particle. This ``standard distribution'' is inescapable for two-particle systems (of any $p$), where only product and generalized Bell bases are admissible MUB types. This and the following results generalize previous results for qubits [Phys. Rev. A 65. 032320 (2002); Phys. Rev. A 72, 062310 (2005)] and qutrits [Phys. Rev. A 70, 012302 (2004)], drawing particularly upon [Phys. Rev. A 72, 062310 (2005)]. With three particles there are three MUB types, and these may be combined in ($p+2$) different ways to form full complements. With $N=4$, there are 6 MUB types for $p=2$, but new MUB types become possible with larger $p$, and these are essential to realizing full complements. With this example, we argue that new MUB types that show new entanglement patterns should enter with every step in $N$ and, also, when $N$ is a prime plus 1, at a critical $p$ value, $p=N\ensuremath{-}1$. Such MUBs should play critical roles in filling complements.

arXiv:1103.3818•Physical Review A•Aug 25, 2011•22 citations

Three-qubit entangled embedding of CPT and Dirac groups within E8 Weyl group

Michel R. P. Planat

In quantum information context, the groups generated by Pauli spin matrices, and Dirac gamma matrices, are known as the single qubit Pauli group P, and two-qubit Pauli group P2, respectively. It has been found [M. Socolovsky, Int. J. Theor. Phys. 43, 1941 (2004)] that the CPT group of the Dirac equation is isomorphic to P. One introduces a two-qubit entangling orthogonal matrix S basically related to the CPT symmetry. With the aid of the two-qubit swap gate, the S matrix allows the generation of the three-qubit real Clifford group and, with the aid of the Toffoli gate, the Weyl group W(E8) is generated (M. Planat, Preprint 0904.3691). In this paper, one derives three-qubit entangling groups ˜ P and ˜ P2, isomorphic to the CPT group P and to the Dirac group P2, that are embedded into W(E8). One discovers a new class of pure theequbit quantum states with no-vanishing concurrence and three-tangle that we name CPT states. States of the GHZ and CPT families encode the new representation of the Dirac group and its CPT subgroup.

•Jun 15, 2009•0 citations

Mutually Unbiased Maximally Entangled Bases in Tripartite Quantum Systems

Liang Bao Tang, Fan Wu, Zhiwen Mo +1 more

International Journal of Theoretical Physics•Aug 12, 2022•0 citations

Algèbre et géométrie discrètes appliquées au groupe de Pauli et aux bases décorrélées en théorie de l’information quantique

Olivier Albouy

For d not a power of a prime, the maximal number of mutually unbiased bases (MUBs) in a d-dimensional Hilbert space is still unknown. In this thesis, we begin by an original building of MUBs by means of Gauss sums, in relation with a family of irreducible representations of the Lie algebra su(2).Then, we sytematically study the possibility of building such bases by means of Pauli operators. 1) The study of the projective line on Zdm shows that, in order to obtain maximal sets of MUBs, tensorial products of these operators are in order. 2) Lagrangian submodules of Zd2n, of which we give a complete classification, account for maximally commuting sets of Pauli operators. This classification enables to know which of these sets are likely to yield unbiased bases. They correspond to Lagrangian half-modules that can be interpreted as the isotropic points of the projective line (P(Mat(n, Zd)²),ω). Hence, we establish an isomorphism between the unbiased bases thus obtained and distant Lagrangian half-modules, which precises by the way the correspondance between Gauss sums and MUBs. 3) Corollaries on the Clifford group and the finite phase space are then developed.Finally, we present some tools inspired by the previous study. We deal with the cross-ratio on the Bloch sphere and projective geometry in higher dimension, Pauli operators with continuous exponents and we compare von Neumann entropy with a determinantal measure of entanglement

•Jun 12, 2009•2 citations

The N-qudit fabric: Pauli graph and finite geometries

Michel R. P. Planat

The commutation relations between the generalized Pauli operators of N-qudits (i. e., N p-level quantum systems), and the structure of their bases/maximal sets of commuting operators, follow a nice graph theoretical/geometrical pattern. One may identify vertices of a graph with the operators so that edges join commuting pairs of them to form the so-called Pauli graph PpN . One may identify points of a geometry with the operators so that lines correspond to the maximal commuting sets of them. 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 to see the 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. These mathematical concepts relate to mutually unbiased bases and to quantum entanglement. Michel Planat† and Metod Saniga‡ The N-qudit fabric: Pauli graph and finite geometries

•Mar 30, 2007•0 citations
All the distinctive features of quantum mechanics versus its classical counterpart crucially depends on the non-commutativity of observables. For instance, the existence of disjoint sets of mutually commuting operators implies quantum complementarity, in the sense that precise measurements on one set implies that possible outcomes of measurements on the other set are equally probable. The maximal number of such mutually disjoint sets (also called mutually unbiased bases) and their structure relates on concepts of finite projective geometry 1 and group theory, that we developed in the past five years 2. Among such mutually commuting sets, those sharing a base of entangled states are at the origin of paradoxes named Kochen-Specker or Bell's theorems, that recognize the contradiction between the algebraic structure of eigenvalues/ measurements and the corresponding expressions for the eigenstates 3. We investigated the zoology and geometry of all mutually unbiased bases for spaces of prime dimension 4 or composite dimension 5. Inhabitants in quantum computing world are qubits and tensor products of them, their houses are the mutually commuting sets, finite geometries are the villages and towns. I discovered a large city with 696 729 600 apartments that is, in mathematics, isomorphic to the largest crystallographic group 6, named W(E8). Groups of three inhabitants/states in the villages are in general highly connected/entangled, their contract is of the type GHZ (true tripartite union), W (three bipartite unions), CPT (a compromise), chain state (a pair can get a divorce) and so on. The population of the Pauli group Pn is 4n+1, meaning that at most n people are connected (by a tensor product). The whole unitary world is infinite, but people usually travel in a finite part of it named the Clifford group Cn. The Clifford group divides into males/females and thus into two dipolar groups denoted C±n . Transsexual people do exist in Cn, e.g. for n = 2, |P2| = 64, |C2| = 92160 and C+ 2 ∩ C−2 = 8 (as described in 7.). The group C+ 3 is the largest maximal subgroup of W(E8) and indeed the aforementioned contracts are work contracts. Following Mermin's great intuition about quantum paradoxes, I have discovered two real twoqubit matrices controling every action in Cn, the first is a braiding matrix R (of a topological character) and the second is a CPT matrix S (related to charge conjugation C, parity P and time reversal T ). The dipolar group of a male type C+ n is labeled by S and the dipolar group C−n of female type is labeled by R. The geometry of the octahedron is associated to the group R, S generated by R and S, as already anticipated in the XIXth century 8 and thoroughly developed in the context of self-dual codes 9. By the way, the CPT invariance is expressed, up to an isomorphism, by the single qubit Pauli group P1 10 and is the kernel of the threequbit representation of W(E8), the full Dirac group of gamma matrices is the two-qubit Pauli group P2 and is generated in W(E8) by matrices sustaining quantum states both of the GHZ and CPT type. Further type of three-qubit entangled states are carried by larger subgroups such as W(F4), W(H4), W(E6) and W(E7) yet to be investigated in detail, in relation to the quaternionic and octonionic representations. All these topics, and related ones, will be introduced in my series of lectures at ZIF, on August 12 to 14, 2009. 1Levay P, Saniga M and Vrana P 2008 Phys. Rev. D 78 124022. 2Planat M and Saniga M 2008 Quant. Inf. Comp. 8 127. 3Mermin N D 1993 Rev. Mod. Phys. 65 803. 4Planat M and Rosu H 2005 Eur. Phys. J. D 36 133. 5Planat M, Baboin A C and Saniga M 2008 Int. J. Theor. Phys. 47 1127. 6Planat M 2009 Preprint 0904.3691 (quant-ph). 7Planat M and Sol´e P 2008 J. Phys. A: Math. Theor. 42 8Klein F 1956 Lectures on the icosahedron and the solution of equations of the fifth degree (Dover: New York). 9Nebe G, Rains E M and Sloane N J A Self-dual codes and invariant theory (Berlin: Springer). 10Socolovsky M 2004 Int. J. Theor. Phys. 43 1941.
Mutually unbiased bases for quantum degrees of freedom are central to all theoretical investigations and practical exploitations of complementary properties. Much is known about mutually unbiased bases, but there are also a fair number of important questions that have not been answered in full as yet. In particular, one can find maximal sets of N + 1 mutually unbiased bases in Hilbert spaces of prime-power dimension N = p M , with p prime and M a positive integer, and there is a continuum of mutually unbiased bases for a continuous degree of freedom, such as motion along a line. But not a single example of a maximal set is known if the dimension is another composite number (N = 6, 10, 12,…). In this review, we present a unified approach in which the basis states are labeled by numbers 0, 1, 2, …, N - 1 that are both elements of a Galois field and ordinary integers. This dual nature permits a compact systematic construction of maximal sets of mutually unbiased bases when they are known to exist but throws no light on the open existence problem in other cases. We show how to use the thus constructed mutually unbiased bases in quantum-informatics applications, including dense coding, teleportation, entanglement swapping, covariant cloning, and state tomography, all of which rely on an explicit set of maximally entangled states (generalizations of the familiar two–q-bit Bell states) that are related to the mutually unbiased bases. There is a link to the mathematics of finite affine planes. We also exploit the one-to-one correspondence between unbiased bases and the complex Hadamard matrices that turn the bases into each other. The ultimate hope, not yet fulfilled, is that open questions about mutually unbiased bases can be related to open questions about Hadamard matrices or affine planes, in particular the notorious existence problem for dimensions that are not a power of a prime. The Hadamard-matrix approach is instrumental in the very recent advance, surveyed here, of our understanding of the N = 6 situation. All evidence indicates that a maximal set of seven mutually unbiased bases does not exist — one can find no more than three pairwise unbiased bases — although there is currently no clear-cut demonstration of the case.
A few simply stated rules govern the entanglement patterns that can occur in mutually unbiased basis sets (MUBs) and constrain the combinations of such patterns that can coexist in full complements of MUBs. We consider Hilbert spaces of prime power dimensions (D=pND={p}^{N}D=pN), as realized by systems of NNN prime-state particles, where full complements of D+1D+1D+1 MUBs are known to exist, and we assume only that MUBs are eigenbases of generalized Pauli operators, without using any particular construction. The general rules include the following: (1) In any MUB, a given particle appears either in a pure state or totally entangled and (2) in any full MUB complement, each particle is pure in (p+1p+1p+1) bases (not necessarily the same ones) and totally entangled in the remaining (${p}^{N}\ensuremath{-}p$). It follows that the maximum number of product bases is p+1p+1p+1 and, when this number is realized, all remaining (${p}^{N}\ensuremath{-}p$) bases in the complement are characterized by the total entanglement of every particle. This ``standard distribution'' is inescapable for two-particle systems (of any ppp), where only product and generalized Bell bases are admissible MUB types. This and the following results generalize previous results for qubits [Phys. Rev. A 65. 032320 (2002); Phys. Rev. A 72, 062310 (2005)] and qutrits [Phys. Rev. A 70, 012302 (2004)], drawing particularly upon [Phys. Rev. A 72, 062310 (2005)]. With three particles there are three MUB types, and these may be combined in (p+2p+2p+2) different ways to form full complements. With N=4N=4N=4, there are 6 MUB types for p=2p=2p=2, but new MUB types become possible with larger ppp, and these are essential to realizing full complements. With this example, we argue that new MUB types that show new entanglement patterns should enter with every step in NNN and, also, when NNN is a prime plus 1, at a critical ppp value, $p=N\ensuremath{-}1$. Such MUBs should play critical roles in filling complements.
In quantum information context, the groups generated by Pauli spin matrices, and Dirac gamma matrices, are known as the single qubit Pauli group P, and two-qubit Pauli group P2, respectively. It has been found [M. Socolovsky, Int. J. Theor. Phys. 43, 1941 (2004)] that the CPT group of the Dirac equation is isomorphic to P. One introduces a two-qubit entangling orthogonal matrix S basically related to the CPT symmetry. With the aid of the two-qubit swap gate, the S matrix allows the generation of the three-qubit real Clifford group and, with the aid of the Toffoli gate, the Weyl group W(E8) is generated (M. Planat, Preprint 0904.3691). In this paper, one derives three-qubit entangling groups ˜ P and ˜ P2, isomorphic to the CPT group P and to the Dirac group P2, that are embedded into W(E8). One discovers a new class of pure theequbit quantum states with no-vanishing concurrence and three-tangle that we name CPT states. States of the GHZ and CPT families encode the new representation of the Dirac group and its CPT subgroup.
For d not a power of a prime, the maximal number of mutually unbiased bases (MUBs) in a d-dimensional Hilbert space is still unknown. In this thesis, we begin by an original building of MUBs by means of Gauss sums, in relation with a family of irreducible representations of the Lie algebra su(2).Then, we sytematically study the possibility of building such bases by means of Pauli operators. 1) The study of the projective line on Zdm shows that, in order to obtain maximal sets of MUBs, tensorial products of these operators are in order. 2) Lagrangian submodules of Zd2n, of which we give a complete classification, account for maximally commuting sets of Pauli operators. This classification enables to know which of these sets are likely to yield unbiased bases. They correspond to Lagrangian half-modules that can be interpreted as the isotropic points of the projective line (P(Mat(n, Zd)²),ω). Hence, we establish an isomorphism between the unbiased bases thus obtained and distant Lagrangian half-modules, which precises by the way the correspondance between Gauss sums and MUBs. 3) Corollaries on the Clifford group and the finite phase space are then developed.Finally, we present some tools inspired by the previous study. We deal with the cross-ratio on the Bloch sphere and projective geometry in higher dimension, Pauli operators with continuous exponents and we compare von Neumann entropy with a determinantal measure of entanglement
The commutation relations between the generalized Pauli operators of N-qudits (i. e., N p-level quantum systems), and the structure of their bases/maximal sets of commuting operators, follow a nice graph theoretical/geometrical pattern. One may identify vertices of a graph with the operators so that edges join commuting pairs of them to form the so-called Pauli graph PpN . One may identify points of a geometry with the operators so that lines correspond to the maximal commuting sets of them. 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 to see the 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. These mathematical concepts relate to mutually unbiased bases and to quantum entanglement. Michel Planat† and Metod Saniga‡ The N-qudit fabric: Pauli graph and finite geometries