Geometric information in eight dimensions vs. quantum information
aa r X i v : . [ qu a n t - ph ] J a n Geometric information in eight dimensions vs. quantuminformation
Victor I. Tarkhanov a and Michael M. Nesterov ba St. Petersburg State Polytechnic University, St. Petersburg, Russia b St. Petersburg Institute for Informatics and Automation, Russian Academy of Sciences, St.Petersburg, Russia
ABSTRACT
Complementary idempotent paravectors and their ordered compositions, are used to represent multivector basiselements of geometric Clifford algebra G , as the states of a geometric byte in a given frame of reference. Twolayers of information, available in real numbers, are distinguished. The first layer is a continuous one. It isused to identify spatial orientations of similar geometric objects in the same computational basis. The secondlayer is a binary one. It is used to manipulate with 8D structure elements inside the computational basis itself.An oriented unit cube representation, rather than a matrix one, is used to visualize an inner structure of basismultivectors. Both layers of information are used to describe unitary operations — reflections and rotations —in Euclidian and Hilbert spaces. The results are compared with ones for quantum gates. Some consequences forquantum and classical information technologies are discussed. Keywords:
Clifford algebra, geometric algebra, complementary paravectors, multivector, information, informa-tion layer, information image, unit cube representation, geometric information, geometric qubit.
1. INTRODUCTION
The idea to use geometric Clifford algebras in information processing technologies is not new. It was used, forexample, by T. F. Havel et al. to replace quantum mechanics in quantum NMR computing. Along with obviousadvantages of the method, it was mentioned, that geometric algebra had a lot of extra degrees of freedom. Specialcorrelators were invented to eliminate them by reducing all imaginary units to a single one.More radical idea was introduced by D. Hestenes, who suggested to use geometric algebras as a unifiedlanguage to combine all existing theories as in mathematics and in physics. On this way he had got a success, but all the efforts were directed on those parts of geometric algebras which were in common with one or moreof these theories. Again those parts and features of geometric algebras, which were beyond the existing theorieswere laid aside.It was silently admitted, that geometric algebras could be used as a universal tool to describe all existingtheories, but they would hardly bring anything new into them. Is it really so?Geometric algebra is a synthesis of two calcules: algebra and geometry. It’s main idea is to deal withgeometric objects in terms of algebra in a given vector space. But there are not only vectors and scalars amongits elements. There are objects of other grades and clusters of mixed grades as well. Some of them are stableones, like fullerenes in physics. Not all of them are suitable for measurements, especially those with rather specialor extraordinary algebraic and geometric properties. But they are sound algebraic elements which could be usedto describe physics involved into information technologies.We concentrate here on using 4D and 8D clusters of geometric algebra G , — complementary idempotentparavectors and their ordered products — to show the way to get some new ideas in classical and quantuminformation technologies. Further author information: (Send correspondence to V.I.T.)V.I.T.: E-mail: [email protected].: E-mail: [email protected] . 3D EUCLIDEAN VECTOR SPACE IN EIGHT DIMENSIONS
We begin with geometric algebra G , , which is spanned on vectors of 3D Euclidean space and has an 8Dmultivector basis : { e , e , e , e , e , e , e , e } , (1)where e ik = e i e k , i, k = 1 , ,
3, and e = e e e . There is a unit scalar (grade 0), e ; three unit vectors (grade1), e , e , e ; three unit bivectors (grade 2), e , e , e , and a unit trivector (grade 3), e , among them.Inside this basis we combine a unit scalar e and each of three basis vectors, e i , i = 1 , ,
3, into clusters oflikeness and distinction: positive and negative idempotent paravectors, P i = ( e + e i ) and N i = ( e − e i ),respectively. They are sound 4D objects, hermitian and invariant to multiplication by themselves, which are normed to themselves , not to single scalars. Each pair of them is orthogonal (linear independent), P i N i = N i P i = 0, andcomplementary in two senses: a scalar one, P i + N i = e , and a vector one, P i − N i = e i .These complementarities are used to define geometric bits P + N = e ; P − N = e ; { + , −} ; P + N = e ; P − N = e ; { + , −} ; (2) P + N = e ; P − N = e ; { + , −} . There is a common “ground” scalar state, e , for all of them, and different “excited” vector states depending onspatial properties of unit vectors, used in their inner structure compositions.These geometric bits are combined into a geometric byte with a sort of binomial formula, A = ( P ± N )( P ± N )( P ± N ) , to get basis multivectors, e , e , e , e , e , e , e , e , as the states of the byte : e = ( P + N )( P + N )( P + N ); { + , + , + } e = ( P − N )( P + N )( P + N ); {− , + , + } e = ( P + N )( P − N )( P + N ); { + , − , + } e = ( P + N )( P + N )( P − N ); { + , + , −} (3) e = ( P − N )( P − N )( P + N ); {− , − , + } e = ( P + N )( P − N )( P − N ); { + , − , −} e = ( P − N )( P + N )( P − N ); {− , + , −} e = ( P − N )( P − N )( P − N ); {− , − , −} . Opening the brackets brings us to a set of binary coded superpositions of the same 8D structure elementsfor all basis multivectors, which have no analogs in quantum or classical theories. Each of them is a scaledsuperposition of all basis multivectors and vise versa: A = P P P = 18 ( e + e + e + e + e + e + e + e ); B = N P P = 18 ( e − e + e + e − e − e + e − e ); C = P N P = 18 ( e + e − e + e − e + e − e − e ); D = P P N = 18 ( e + e + e − e + e − e − e − e ); (4) D = N N P = 18 ( e − e − e + e + e − e − e + e ); = N P N = 18 ( e − e + e − e − e + e − e + e ); B = P N N = 18 ( e + e − e − e − e − e + e + e ); A = N N N = 18 ( e − e − e − e + e + e + e − e ) . They are associated with the oriented octants of a unit cube and are labeled with a letter in each vertex ∗ , Fig.1.They are the same oriented octants as in a Cartesian frame of reference, but scaled and shifted from the origin AD CBCB AD
Figure 1.
A unit cube representation. to compose a unit cube form.In these notations complementary basis paravectors are the opposite sides of the unit cube: P = A + B + C + D = ; N = A + B + C + D = ; P = A + B + C + D = ; N = A + B + C + D = , (5) P = A + B + C + D = ; N = A + B + C + D = . The inner structure of basis multivectors in these notations is coded as e = ( A + A ) + ( B + B ) + ( C + C ) + ( D + D ) = ; e = ( A − A ) − ( B − B ) + ( C − C ) + ( D − D ) = ; e = ( A − A ) + ( B − B ) − ( C − C ) + ( D − D ) = ; e = ( A − A ) + ( B − B ) + ( C − C ) − ( D − D ) = ; e = ( A + A ) − ( B + B ) − ( C + C ) + ( D + D ) = ; (6) e = ( A + A ) + ( B + B ) − ( C + C ) − ( D + D ) = ; ∗ An overline means a conjugate operation of space reversion, in which all vectors change their signs. = ( A + A ) − ( B + B ) + ( C + C ) − ( D + D ) = ; e = ( A − A ) − ( B − B ) − ( C − C ) − ( D − D ) = . Here green circles correspond to structure elements taken with the sign “plus”, and blue circles — to ones, takenwith the sign “minus”. So there are sets of binary coefficients +1 and −
3. TWO LAYERS OF INFORMATION
There are two distinct layers of information, available in real numbers, inside the 8D algebraic space of G , geometric algebra.The first one is used to compare and identify spatial orientations of similar geometric objects. Its informationcarriers are real coefficients (coordinates) of basis Clifford numbers, showing their role in a particular spatialdecomposition of the object. In this case basis Clifford numbers are the same, and all information about theobject is contained in their real coefficients. These may be any real numbers.When an object is rotated, some of its coordinates are changed in a harmonic or a continuous manner. Beingmeasured in time, their successive changes of values can be treated as flows of analog or digital information.The second layer is used to identify basis multivectors and their clusters algebraically and to manipulate withtheir structure elements (4). In a unit cube representation its information carriers are real binary numbers, +1and − In a given Cartesian frame of reference { e , e , e } any unit vector a is described as a = a e + a e + a e . (7)Here unit vectors e , e and e are fixed, and all information about a particular space orientation of vector a is contained in the set of direction cosines ( a , a , a ). This is the first layer of information. As there are onlythree items in the sum and each item is a vector itself, we treat equation (7) as a 3D representation (image) ofvector a .In geometric algebra G , basis unit vectors e , e , e are 8D Clifford numbers. So a can be rewritten in theform a = a + a + a . (8)Only three of eight basis multivectors are used here for the decomposition. The inner structure of these Cliffordnumbers, coded in terms of structure elements (4) helps us to identify them as unit vectors in 8D space. This isthe second layer of information. It does not change when vector a rotates. The information about the orientationof vector a is contained in the same set of direction cosines ( a , a , a ).fter combining structure elements (4) with their coefficients along the main diagonals of the cube thisexpression can be rewritten as a = ( a + a + a ) + ( − a + a + a ) ++ ( a − a + a ) + ( a + a − a ) = (9)= ( a + a + a )( A − A ) + ( − a + a + a )( B − B ) ++ ( a − a + a )( C − C ) + ( a + a − a )( D − D ) . Now there are four items in the sum, so it is a 4D image of vector a . It resembles a sort of a Radon transform.We have changed 3D unit vectors onto some 4D clusters, that is we have increased the dimensionality of basisClifford numbers. Now they are the inhabitants of a Minkowski space with { + , + , + , −} metrics: ( A − A ) = ( e + e + e + e );( B − B ) = ( − e + e + e − e );( C − C ) = ( e − e + e − e );( D − D ) = ( e + e − e − e ) . (10)They are not perpendicular to each other. They are rather orthogonal in the sense of linear independence in 4Dspace.We have changed information of the second layer. What has happened with information of the first layer?It has changed its image, but has preserved all its ingredients — the direction cosines, a , a , a . Now they arerearranged into some superpositions, but they are again just real numbers. Hence the information of the firstlayer is the same. Quaternions are 4D even Clifford numbers containing a scalar and a bivector inside. Unitary quaternions areused to describe rotations. In matrix representation a unitary contravariant quaternion usually has a form Q = αP + β ( e P ) − β ∗ ( e N ) + α ∗ N = (cid:20) α − β ∗ β α ∗ (cid:21) . (11)Here P , e P , e N and N are paravector names (labels) of matrix elements in geometric algebra G , . Complexnumbers α , β , − β ∗ and α ∗ , meeting a condition αα ∗ + ββ ∗ = 1, are called Cayley–Klein parameters.For a simple positive (anticlockwise) rotation around a unit axis c = c e + c e + c e through an angle ϑ the quaternion (11) has a form Q = exp ( − ic ϑ ϑ − ic sin ϑ α = cos ϑ − ic sin ϑ ; β = − i ( c + ic ) sin ϑ . (13)They are complex numbers, complementary in the sense, that, in any given frame of reference, the pare of themcontains all information about the angle ϑ and the axis ( c , c , c ) for the described simple rotation.In terms of structure elements (4), one can write (11) as Q ( a ) = α + β − β ∗ + α ∗ , (14)or α = ρ − iν and β = − i ( µ + iλ ) eq. (14) can be rewritten with only real coefficients inside the firstinformation layer. There are two ways to do it.The first way is a rather traditional one: Q ( a ) = ρe − νe − µe − λe = ρ − ν − µ − λ , (15)where real numbers ρ , ν , µ and λ , meeting a condition ρ + ν + µ + λ = 1, are called Euler–Rodriguesparameters. It is built in the computational basis, { e , e , e , e } , which is a basis for a Minkowski space with { + , − , − , −} metrics.The second way is a less obvious one. It is obtained by rearranging structure elements (4) with their signs(information of the second layer) and their real coefficients (information of the first layer) into another compu-tational basis: Q ( a ) = ( ρ − ν − µ − λ ) + ( ρ + ν + µ − λ ) ++ ( ρ + ν − µ + λ ) + ( ρ − ν + µ + λ ) == ( ρ − ν − µ − λ )( A + A ) + ( ρ + ν + µ − λ )( B + B ) + (16)+ ( ρ + ν − µ + λ )( C + C ) + ( ρ − ν + µ + λ )( D + D ) . The new basis, { ( A + A ) , ( B + B ) , ( C + C ) , ( D + D ) } , consists of four pairs of structure elements (4) lying againalong the main diagonals of the unit cube. But now they are combined with plus signs. They are inhabitants ofthe same 4D Minfowski space with { + , − , − , −} metrics: ( A + A ) = ( e + e + e + e );( B + B ) = ( e − e − e + e );( C + C ) = ( e − e − e + e );( D + D ) = ( e + e − e − e ) . (17)And again some rearrangements of structure elements image inside the second information layer changes theimage of the object inside the first information layer. The new image has the same information, because newcoefficients inside the first layer contain all ingredients of the previous image.
4. PROJECTIONS
Usually it is a good idea to simplify description of multidimensional geometric objects and their behavior byprojecting them onto spaces of less dimensions. There are two ways to project them in our 8D algebraic space:Cartesian and Hilbert ones.The first way is typical for a 3D vector space, and is widely used in engineering. Its extension onto 8Dmultivector space of geometric algebra G , is trivial except for the loss of commutativity in multiplication. Theresult is the reduction of the overall 8D space to three mutually orthogonal 4D subspaces. These projections areused to get orthogonal decompositions inside the first information layer. The second information layer remainsunchanged: all basis elements are in multivector form (6).The second way is a new one. It is based on specific projective properties of 4D clusters — idempotentparavectors, — which are in the focus of this article. In these projections information inside the first layer isinvariant, whereas all orthogonal decompositions are made inside the second informamtion layer. Let us considerit in more details. .1. Complementary paravectors for a unit vector As it was stated above, for any given unit vector a and the only unit scalar e one can built the only clusterof likeness, P ( a ) = ( e + a ), and the only cluster of distinction, N ( a ) = ( e − a ). They are complementaryidempotent paravectors. The first one, P ( a ), is called positive, because its Cartesian projection onto vector a isthe same P ( a ) with “plus” sign: aP ( a ) = P ( a ) a = P ( a ) . (18)The second paravector, N ( a ), is called negative, because its Cartesian projection onto vector a is the same N ( a )up to the sign. The sign is “minus”: aN ( a ) = N ( a ) a = − N ( a ) . (19)These properties can be transferred onto all other elements of the algebra through one-sided multiplications.This is the way to project them, into so called spinor ideals of Hilbert spaces, and, hence, this is the reason fora coined name.There are two orthogonal Hilbert spaces — positive one, and negative one — for each unit vector a in a givenframe of reference. They have no common points, so they are orthogonal rather in the sense of parallelism, thanin the sense of perpendicularity. And there are two spinor ideals — a contravariant one, and a covariant one, —intersecting through a body of two-sided spinors, in each of them. Let us put for simplicity a = e . Then P ( a ) = P , and N ( a ) = N . This is the only case when complementaryidempotent paravectors are associated with single matrix elements. We shall use it to compare our resultswith their analogs in quantum information theory. In the same frame of reference all the other vectorsare decomposed into complementary idempotent paravectors, which are spread over all four matrix elementssimultaneously, as in density martixes, and lose the illusion of 1D objects.To project basis multivectors (6) into an ideal of positive contravariant spinors it is enough to multiply themfrom the right side by P : e P = e P = 12 ( e + e ) = P = (cid:20) (cid:21) = ; e P = e P = 12 ( e + e ) = ( e P ) = (cid:20) (cid:21) = ; e P = e P = 12 ( e + e ) = i ( e P ) = (cid:20) i
00 0 (cid:21) = ; (20) e P = e P = 12 ( e + e ) = iP = (cid:20) i (cid:21) = . There is a two-fold degeneracy in these projections. From a single projection one cannot say for sure, whichof two multivectors was projected to get a particular image, a particular spinor shadow in the positive Hilbertspace.In matrix representation this spinor ideal has a basis consisting of two matrix elements in the left column, { P , ( e P ) } . Eight basis multivectors are projected onto them with real or imaginary coefficients.In the unit cube representation there are four basis paravector clusters with real coefficients and with thesame two-fold degeneracy in projections. Only bottom side structure elements are used here. So it is a projectionof basis multivectors (6) to the bottom side of the cube.o project the same basis multivectors (6) into an ideal of negative contravariant spinors, it is enough tomultiply them from the right side by N : e N = − e N = 12 ( e − e ) = N = (cid:20) (cid:21) = ; e N = − e N = 12 ( e − e ) = ( e N ) = (cid:20) (cid:21) = ; − e N = e N = −
12 ( e − e ) = i ( e N ) = (cid:20) i (cid:21) = ; (21) − e N = e N = −
12 ( e − e ) = iN = (cid:20) i (cid:21) = . These are Hilbert projections of basis multivectors (6) to the upper side of the cube.In matrix representation this spinor ideal has a basis consisting of two matrix elements in the right column, { N , ( e N ) } . Again eight basis multivectors are projected onto them with real or imaginary coefficients. Nowthey are all different. But again there is a two-fold degeneracy in these projections, because we don’t know ifthe prototype was positive, or negative.From a unit cube representation one can see, that positive and negative Hilbert projections are complementaryspinor images (shadows) of the same object. To reconstruct the prototype unambiguously, it is enough to combinethem in a strict sum. In a given frame of reference sometimes it is convenient to express a unit vector a as a rotated (geometricallyexcited) state of a basis vector e : a = Q ( a ) e e Q ( a ) (22)Using e = P − N , one can change it into a difference of two complementary paravectors: a = Q ( a ) P e Q ( a ) − Q ( a ) N e Q ( a ) = P ( a ) − N ( a ) , (23)where P ( a ) = ( e + a ) and N ( a ) = ( e − a ). Each paravector is an ordered composition of contravariant andcovariant conjugated spinors. So (23) can be written in the form a = Q ( a ) P [ Q ( a ) P ] ∼ − Q ( a ) N [ Q ( a ) N ] ∼ = ψ + ( a ) e ψ + ( a ) − ψ − ( a ) e ψ − ( a ) . (24)In quantum mechanics positive spinors have special Dirac’s notations — bra- and ket- vectors: | a i = ψ + ( a ) = Q ( a ) P = αP + β ( e P ) = (cid:20) α β (cid:21) ; (25) h a | = e ψ + ( a ) = P e Q ( a ) = α ∗ P + β ∗ ( P e ) = (cid:20) α ∗ β ∗ (cid:21) . (26)The same Dirac’s notations are the blinders for negative spinors: ψ − ( a ) = Q ( a ) N = α ∗ N − β ∗ ( e N ) = (cid:20) − β ∗ α ∗ (cid:21) ; (27) e ψ − ( a ) = N e Q ( a ) = αN − β ( N e ) = (cid:20) − β α (cid:21) . (28)hey are never used in quantum mechanics. One of the reasons is their incompatibility with its probabilisticinterpretation. So not to lose information we prefer to avoid bra- and ket- notations and probability terms ingeometric algebra.There is a plenty of ways to get a from e in a simple rotation. They differ from each other in rotation angleand axis values, which are coded in Cayley–Klein (or Euler–Rodrigues) parameters inside the first informationlayer. Usually the shortest angle of rotation is chosen. In this case as e and a are in the same plane of rotation,the axis of rotation is in e plane, α is a real number, 4D Hilbert space is reduced to a 3D one, and thecorrespondence between spatial orientation of vector a in Euclidean space and its positive contravariant spinorimage in Hilbert space can be depictured in the same drawing. A simple example of such 2D drawing ispresented in Fig.2. Here a unit vector OA is rotated anti-clockwise around e axis, pinned to its center O’ , OAB O’ CDE αα ββ − i ( e P ) ( e P ) Figure 2.
A unit vector a in Cartesian frame of reference and its contravariant spinor image αP − iβ ( e P )in a positive Hilbert paravector space for a unit vector e . through an 130 ◦ angle into a CB position. The small circle O’ with unit diameter corresponds to Euclidianspace. It is rolled over the inner side of the big circle O with a unit radius, corresponding to a positive Hilbertspace of contravariant spinors for the selected direction e . When it is rolled through the arc of 65 ◦ , CO coincideswith OE , OB coincides with ED , CB coincides with OD . In this position Cayley–Klein parameters α and β havethe sense of directional cosines for Hilbert frame of reference. They are real in this example. Note, that in spinorrepresentation, 1/2 is a scaling factor for the angle of rotation, not for the length of the vector.In this 2D example the initial unit Euclidian vector OA (small circle) coincides with its spinor image — theHilbert vector OA (big circle), and the rotated Euclidian vector CB (small circle) has quite another spinor image— the Hilbert vector OD (big circle). Note, that in general, vectors of Euclidian space are not vectors in Hilbertspace and vice-versa. This example is of interest, because this very spinor image is used as a qubit in quantuminformation theory. There are attempts to insert it in a Bloch or in a Poincare sphere, but these are just 3Danalogs of the small Euclidian circle used in this example. Spinors and qubits live in 4D Hilbert spaces, whichin general cannot be visualized in 3D Euclidian space.In geometric algebra it is clear, that spinor images of a unit vector a cannot be normed to a single realnumber. In one order (an inner product) the composition of contravariant and covariant shadows gives the thenitial paravector for positive spinors: h a | a i = P Q ( a ) e Q ( a ) P = P = (cid:20) (cid:21) , (29)and for negative spinors: N Q ( a ) e Q ( a ) N = N = (cid:20) (cid:21) . (30)Their difference, P − N = e , gives the initial state of the rotated vector as e . The condition αα ∗ + ββ ∗ = 1is an intrinsic part of definition for Cayley–Klein parameters in geometric algebra.In the reversed order (an outer product) it gives the final paravector for positive spinors: | a ih a | = Q ( a ) P P e Q ( a ) = P ( a ) = (cid:20) αα ∗ αβ ∗ α ∗ β ββ ∗ (cid:21) , (31)and for negative spinors: Q ( a ) N N e Q ( a ) = N ( a ) = (cid:20) ββ ∗ − αβ ∗ − α ∗ β αα ∗ (cid:21) . (32)Their difference gives the final state a of the rotated vector in terms of Cayley–Klein parameters: a = P ( a ) − N ( a ) = (cid:20) a a − ia a + ia − a (cid:21) = (cid:20) αα ∗ − ββ ∗ αβ ∗ α ∗ β ββ ∗ − αα ∗ (cid:21) . (33)So if a physical object which we want to use in information processing can be described as a vector of a 3DEuclidian space, and its states are associated with spatial orientations of this vector, then all information, we areinterested in, is contained in Cayley–Klein or Euler–Rodrigues parameters inside the first information layer. Theyhave four real values, which are usually not conserved in time, and are not available for direct measurements.There are no detectors for them in Hilbert spaces, but they can be measured in their compositions in 3D Euclidianspace as stationary or rotating vector components, like in NMR pulsed experiments.
5. UNITARY OPERATIONS
Unitary operations in geometric algebra are reflections and rotations. Their main feature is that, being applied toany set of vectors, they keep all vector lengths and all angles among them invariant. There are no deformationsfor any geometric object, except inversions. So they are good for operation over composite geometric objectsand for parallel operations over distributions of simple ones. In our 8D algebraic space they can be used forinformation processing inside both information layers.
There are three kinds of reflections in a 3D Euclidian space: reflection in a point, reflection in a line, and reflectionin a plane.Reflection in a point is a displacement of the frame origin in that point and spatial inversion of all vectorsand their odd products in the new frame origin. So all the vectors and trivectors change their signs. Reflectionin the origin of the frame of reference is marked with an overline. It is one of two main conjugation operationsin geometric algebra. Structure elements (4), joined with the unit cube main diagonals, are conjugated in thatway.Reflection in a line changes signs for all vector components, perpendicular to that line. All components,collinear with that line conserve their signs. For example, a reflection of vector (7) in the line e gives e ae = a e − a e − a e . (34)It is a key to find reflection-in-line connections among 8D structure elements (4). For example, A reflected in e is B , A reflected in e is C , A reflected in e is D , B reflected in e is C .eflection in a plane changes signs for all vector components, perpendicular to that plane. For example, areflection of vector (7) in the plane e gives e ae = − a e + a e + a e . (35)Then for 8D structure elements (4) one can get, for example, A reflected in e is B , A reflected in e is C , A reflected in e is D , B reflected in e is C , etc.So all structure elements (4) are connected with each other through unitary operations of reflections in 3DEuclidian space.Reflections in planes and lines, described by basis multivectors, change images only in the second informationlayer. Information images inside the first layer remain unchanged. Reflections in arbitrary planes and lineschange information images inside both layers. Spatial rotations are noncommutative operations, which cannot be described in single real or complex numbers.They need at least four real numbers or two complex numbers for each simple rotation. Best of all they aredescribed in terms of unitary quaternions. The main sequence of their noncommutativity is that each simplerotation cannot be decomposed into superposition of simultaneous rotations in noncollinear planes or aroundnoncollinear axes. There is no superposition principle for noncollinear components of a rotation axis. A goodexample is forced rotations with nonzero offset in pulsed NMR experiments. In geometric information processing there is a practise to decompose a single simple rotation into a sequence ofother simple rotations around noncollinear axes. It can be done in a plenty of ways. The main idea is to simplifysome description in theory or to gain some advantages in excitation of inhomogeneously broadened mesomorphicstructures in practise. The price is a significant and fast increase of duration for any such composite operationwith each new discrete rotation element inside it.Simple spatial rotations and their sequences are the main tools for geometric information processing inside thefirst information layer. Unitary quaternions, used to describe them, can be treated as universal 4D informationunits, independent from the physical objects to be rotated. In that sense they are similar to bits of information,but they live in 4D space and need more complicated logic to be operated with.
6. COMPARISON WITH QUANTUM INFORMATION6.1. Geometric qubit
Using additive decomposition of the unit scalar into P and N paravectors, one can decompose unitary quater-nion Q ( a ) into a direct sum of a positive contravariant spinor, Q ( a ) P , and a negative contravariant spinor, Q ( a ) N : Q ( a ) = Q ( a )( P + N ) = Q ( a ) P + Q ( a ) N . (36)Positive spinor Q ( a ) P = αP + β ( e P ) = (cid:20) α β (cid:21) = α + β (37)is a geometric analog of the object which is used in quantum mechanics as a wave function or a qubit. We shallcall it geometric qubit here. In terms of Euler–Rodrigues parameters it has a form Q ( a ) P = ρ + ν + µ + λ (38)r Q ( a ) P = ( ρ − ν − µ − λ ) + ( ρ + ν + µ − λ ) + (39)+ ( ρ + ν − µ + λ ) + ( ρ − ν + µ + λ ) . One can see that this is a 4D object. Its complementary counterpart is Q ( a ) N = − β ∗ ( e N ) + α ∗ N = (cid:20) − β ∗ α ∗ (cid:21) = − β ∗ + α ∗ (40)In terms of Euler–Rodrigues parameters it has a form Q ( a ) N = ρ + ν + µ + λ (41)or Q ( a ) N = ( ρ − ν − µ − λ ) + ( ρ + ν + µ − λ ) + (42)+ ( ρ + ν − µ + λ ) + ( ρ − ν + µ + λ ) . Positive spinor, Q ( a ) P , can be treated either as an abstract mathematical quantity — a part of a quaternion,— describing an operation of a simple rotation, or as a result of its application to the vector e or to the scalar e . As mathematical quantity it is used to design calculation algorithms. Simple rotations are composed insequences through ordered products of corresponding unitary quaternions. And there are two complementaryspinors inside the same resulting unitary quaternion, associated with the same geometric qubit.As a result of a unitary operation spinor changes its image inside the first information layer. But for a singleHilbert projection one cannot distinguish between a rotated vector e and a rotated scalar e . In geometry thereis a difference. Vector changes its orientation in Euclidian space, and hence its state or image inside the firstinformation layer. Scalar is invariant to any rotation. So there will be no changes in the first information layer,and the equation of identity will always be true. To prevent the loss of information inside the first layer oneneeds to use as positive, and negative spinor images for any qubit.In geometric algebra not only vectors, but also bivectors and their cluster combinations with all other elementsof G , algebra can be rotated. Sometimes they have or gain explicit vector components, which could interfere withcalculation results designed only for vectors. Such unpredictable gains and losses of information are inadmissiblefor large-scale calculations, especially for quantum ones. So it is a good idea to treat them geometrically as well.In quantum information processing real or complex numbers inside the first layer are treated as probabilitiesor their amplitudes and are thrown away in averaging operations. Hence all calculations seem to be performedonly inside the second information layer, over structure elements (4) of a computational basis, that is overgeometric byte structure. If it is so, then it is a pure mathematics, and where is physics? Usually as a qubit and its geometric analog is written in a form (37). Its computational basis consists of twoparavectors: P and ( e P ). Hadamard gate is used to change it to positive and negative superpositions ofboth. Due to two-fold degeneracy in Hilbert projection results, there are two ways to do it. If we assume, that = e P , then we could sum or subtract unit vectors e and e in Cartesian frame of reference, norm them to e , Hilbert project them with P , and use them as new basis elements, { √ ( e + e ) P , √ ( e − e ) P } . Thisway is used in quantum information processing. The other way is to assume, that P = e P , then to decompose e and e into their clusters of likeness anddistinction, e = P + N and e = P − N . After that one can decompose basis paravectors, P and ( e P ),into e P = P P + N P and ( e P ) = P P − N P , and to regroup them: Q ( a ) P = αP + β ( e P ) = α ( P P + N P ) + β ( P P − N P ) == ( α + β )( P P ) + ( α − β )( N P ) = ( α + β )( A + C ) + ( α − β )( B + D ) . (43)This is an ordinary regrouping operation for structure elements (4) inside the second information layer. Note,that 1/2 factor is attributed now to P and N to make them idempotent ones. This way is typical for geometricinformation processing. Both in quantum and in geometric information processing, reflections in lines, associated with basis vectors ofCartesian frame of reference, are used to implement NOT operations. In geometric algebra it is trivial to usereflection in e as NOT operation for any unit vector in (22) form. Its positive contravariant Hilbert projectionhas a geometrical qubit (37) form. Multiplication by e from the left changes P to ( e P ), and ( e P ) to P ,because e is a unit vector, e = e . Information inside the first information layer remains unchanged. Theseoperations are applied only to the second information layer.Note that in geometric algebra P and N are associated with “spin-up” and “spin-down” states, correspond-ingly, rather than basis paravectors P and ( e P ), which are just positive contravariant Hilbert projections forperpendicular basis vectors e and e of Cartesian frame of reference, respectively. Both of them are in the samepositive Hilbert space, defined by P paravector. Inside the first information layer each state of a geometric qubit (37) is fully described by two complementarycomplex Cayley–Klein parameters, α and β . They are related with equations (13), which are more strict andgeneral, then in eq. (1.3) of quantum information. Although they are complex numbers and can be representedin a “module–phase” form, it is impossible to compare them by their phase, because their phases are defined inquite different planes.In quantum information basis paravectors, P and ( e P ), are treated as the states, | i and | i , of a qubit,and eq. (37) — as their superposition, with probability amplitudes α and β . The latter are used as a measureof coherence between | i and | i states.In geometric algebra the picture is quite different. Basis paravectors, P and ( e P ), are not the states, butrather Hilbert projections of Cartesian frame of reference, which is always a stable and unchangeable object.They are always coherent as parts of the same frame of reference. They have nothing to do with a particularphysical or geometric object we are going to describe in that frame of reference. Cayley–Klein parameters arecoherent, but not through their phase relations. They are coherent in the sense, that they describe one of fourHilbert projections for a particular simple rotation. One can feel some decoherence only for a rotation in aspatially unstable plane or around some axis with unstable spatial orientation.Geometric algebra is not blind to phase. Each phase is defined in its plane of rotation. And there are plentyof such planes in its 8D algebraic space. It contains as operators, and operands. In geometric algebra operatorscan be expressed in terms of operands and vise-versa. But in this algebra there is no place for such monsters asphase operator S = P + iN . . DISCUSSION The described approach is of interest for perfection of information processing technologies in a number of ways.Its main idea is very close to ideas of R.B. Fuller and N. Tesla: each simplicity is a pretty organized complexity.We tried to extend calculation inside geometric Clifford algebras onto some new sound 4D paravector clusterswith some new algebraic properties. We used these clusters to define geometric bits and to organize theminto geometric byte structure. It gave us the possibility to define two layers of information: a geometric andan algebraic ones. To deal inside them with only real numbers we changed matrix representations onto anoriented unit cube ones. We described two ways to project algebraic and geometric objects onto subspaces ofless dimensions and gave some examples to apply them in practise.We tried to use mathematical similarities in geometric algebra formulas with those in quantum mechanicsto compare our approach with that of quantum information processing. We hope that geometric approach willhelp to avoid some information ambiguities in future technologies of quantum computations.The possibility to work only with real numbers is very good for ordinary classical computers. Now theyare smart enough to process eight flows of information in parallel. There are two layers of information to workwith in each flow. If information processing inside the first layer is reduced to manipulations with separatesymbols, information processing inside the second layer can be associated with fast reading technologies. Foreight independent flows of a discrete or analog information inside the first layer there is a possibility to mix themcoherently with operations inside the second layer and to separate them afterwards in constructive interferenceprocesses without any claims on their coherence or multiplexing in temporal or frequency domains. For coherentflows of information, for example ones describing coordinates of some moving (rotated) 8D geometric object,there is a unique possibility to simulate the behavior of such an object on a classical computer.This article is one of the first steps in this direction. We hope that some of our ideas could be helpful for ourcolleagues as in Clifford algebras applications and in quantum information processing area.
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