Science Research
Volume 3, Issue 5, October 2015, Pages: 240-247

Quantum State Evolution in C2 and G3+

Alexander Soiguine

Soiguine Supercomputing, Aliso Viejo, CA, USA

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To cite this article:

Alexander Soiguine. Quantum State Evolution in C2 and G3+.Science Research.Vol.3, No. 5, 2015, pp. 240-247. doi: 10.11648/

Abstract: Quantum mechanical qubit states as elements of two dimensional complex Hilbert space can be generalized to elements of even subalgebra of geometric algebra over three dimensional Euclidian space. The construction critically depends on generalization of formal, unspecified, complex plane to arbitrary variable, but explicitly defined, planes in 3D, and of usual Hopf fibration to special maps of the geometric algebra elements to the unit sphere in 3D generated by arbitrary unit value bivectors. Analysis of the structure of the map of the even subalgebra to the Hilbert space demonstrates that quantum state evolution in the latter gives only restricted information compared to that in geometric algebra.

Keywords: Qubits, Geometric Algebra, Clifford Translation, Berry Phase, Topological Quantum Computing

1. Introduction

Qubits, unit value elements of the Hilbert space  of two dimensional complex vectors, can be generalized to unit value elements [1] of even subalgebra of geometric algebra over Euclidian space . I called such generalized qubits -qubits [1].

Some minimal information about  and  is necessary. Algebraically,  is linear space with canonical basis , where  is unit value scalar, are orthonormal basis vectors in ,  are oriented, mutually orthogonal unit value areas (bivectors) spanned by  and  as edges, with orientation defined by rotation  to  by angle ; and  is unit value oriented volume spanned by ordered edges ,  and

Subalgebra  is spanned by scalar  and basis bivectors: . Variables and  in  are scalars, is a unit size oriented area, bivector, lying in an arbitrary given plane. Bivector  is linear combination of basis bivectors , see Fig.1a.

Fig. 1. Variable complex plane in different bivector bases.

It was explained in [2], [3] that elements only differ from what is traditionally called "complex numbers" by the fact that  is an arbitrary, though explicitly defined, plane in . Putting it simply,  are "complex numbers" depending on embedded into. Traditional "imaginary unit"  is actually associated with some unspecified  – everything is going on in one tacit fixed plane, not in 3D world. Usually  is considered just as a "number" with additional algebraic property . That may be a source of ambiguities, as it happens in quantum mechanics.

I will denote unit value oriented volume  by . It has the property . Actually, there always are two options to create oriented unit volume, depending on the order of basis vectors in the product. They correspond to the two types of handedness – left and right screw handedness. In Fig.1a the variant of right screw handedness is shown. It is geometrically obvious that  (bivector flips, changes orientation to the opposite, in swapping the edge vectors). Then changing of handedness, for example by , gives .

Any basis vector is conjugate (conjugation means multiplication by ), or dual, to a basis bivector and inverse. For example  . By multiplying these equations from left or right by  we get . Such duality holds for arbitrary, not just basis, vectors and bivectors. Any vector  is conjugate to some bivector:  and for any bivector  there exists  such that . Good to remember that in  vector and bivector are in handedness opposite to  while in  they are in the same handedness as .One can also think about  as a right (left) single thread screw helix of the height one. In this way  is left (right) screw helix.

All above remains true (see Fig.1b) if we replace basis bivectors  by an arbitrary triple of unit bivectors  satisfying the same multiplication rules which are valid for (see Fig.2, where right screw handedness is assumed):


Fig. 2. Oriented basis bivectors their right screw oriented volume.

2. Parameterization of Unit Value Elements in G3+ and C2 by Points of S3

Let’s take a -qubit:

, ,

I will use notation  for such elements. They can be considered as points on unit radius sphere in the sense that any point  parameterizes a -qubit.

A pure qubit state in terms of conventional quantum mechanics is two dimensional unit value vector with complex components:

We can also think about such pure qubit states as points on  because:


Explicit relations, based on their parameterization by  points, between elements  and elements  can be established through the following construction2.

Let basis bivector  is chosen as defining the complex plane , then we have (see multiplication rules (1.1)):

Hence we get the map:


In the similar way, for two other selections of complex plane we get:



So we have three different maps  defined by explicitly declared complex planes  satisfying (1.1):


There exists infinite number of options to select the triple . It means that to recover a-qubit  in 3D associated with  it is necessary, firstly, to define which bivector  in 3D should be taken as defining "complex" plane and then to choose another bivector , orthogonal to . The third bivector , orthogonal to both  and  is then defined by the first two by orientation (handedness, right screw in the used case ): .

The conclusion is that to each single element (2.1) there corresponds infinite number of elements  depending on choosing of a triple of orthonormal bivectors  in 3D satisfying (1.1), and associating one of them with complex plane. This allows constructing map, fibration  restricted to unit value elements.

3. Fiber Bundle

Take general definition of fiber bundle as a set  where  is bundle (or total) space;  - the base space; - standard fiber;  - Lie group which acts effectively on ;  - bundle projection: , such that each space , fiber at , is homeomorphic to standard fiber .

We can think about the map  as a fiber bundle. In the current case, the fiber bundle will have  as total space and  as base space. I will denote them as  and  respectively. The projection  depends on which particular  is taken from an arbitrary triple  satisfying (1.1) as associated complex plane of complex vectors of  and explicitly given by (2.2), so we should write .

By some reasons that will be explained a bit later I will use complex plane associated with , so by (2.2) the projection is:


Then for any  the fiber in  is comprised of all elements with an arbitrary triple of orthonormal bivectors  in 3D satisfying (1.1). That particularly means that standard fiber is equivalent to the group of rotations of the triple  as a whole (see Fig.3):

Fig. 3. Two sections of a fiber: The left one received from the right by 90 degree counterclockwise rotation around vertical axis.

All such rotations in  are also identified by elements of  since for any bivector  the result of its rotation is [3] (see, for example [4], [5]):


So, standard fiber is identified as  and composition of rotations is:

All that means that the fibration  is principal fiber bundle with standard fiber  and the group acting on it is the group of (right) multiplications by elements of .

Now the explanation why  was taken as complex plane. As was shown in [1], [6] the variant of classical Hopf fibration


can be received as one of the cases of generalized Hopf fibrations:

The basis bivector  gives:


and for two other basis bivectors:



In the literature mostly the third or the first variants are called Hopf fibrations. For my considerations it does not matter which variant to choose. I take the case (3.4) with as complex plane:

4. Tangent Spaces

We need to temporarily get back to the case of  - geometric algebra on a plane [2].

Let an orthonormal basis  is taken. It generates  basis  where  is usual geometric product of two vectors: first member is scalar product of the vectors and second member is oriented area (bivector) swept by rotating  to  by the angle which is less than .

The  basis vectors satisfy particularly the properties:

,  (clockwise rotation),  (counterclockwise rotation), , . I am using notation  for unit bivector .

For further convenience, let’s construct a matrix basis isomorphic to . Commonly used agreement will be that scalars are identified with scalar matrices, for example: .

For the  case the second order matrices will suffice to get necessary isomorphism. Let’s take  and [4] as corresponding to geometric basis vectors  and . They satisfy the properties mentioned above, for example

, , , etc.

The operation representing scalar product gives:


as it should be. If we take two arbitrary vectors expanded in basis :

then their product is:

The first member is usual scalar product and the second one is bivector of the value equal to the area of parallelogram with the sides  and .

Important thing to keep in mind is that multiplication of any vector by unit bivector  rotates the vector by  (see Fig.4):

(counterclockwise rotation)

(clockwise rotation)

This remains valid for any unit bivector of the same orientation as . We can conclude that such multiplications give basis vectors of the tangent spaces to the original vectors.

Fig. 4. Vector is rotated by 90 degrees when multiplied by basis bivector.

This is identical to considering even elements  corresponding to vectors and their multiplication by : . Elements  and  are orthogonal:  (index 0 means scalar part).

General element of  in matrix basis , isomorphic to geometrical basis , is:

This is arbitrary real valued matrix of the second order. Inversely, any matrix of second order can be uniquely mapped to the element of  in the basis :

To upgrade to , we need basis matrices of a higher order because a second order non-zero matrix orthogonal in the sense of scalar product (4.1) to both  and  does not exist.

It is easy to verify that the three, playing the role of three-dimensional orthonormal basis, necessary matrices can be taken as 4th-order block matrices:


where  - scalar matrix corresponding to scalar , . Easy to see that  and  are received from  and  by replacing scalars by corresponding scalar matrices.

By multiplications in full  form one can easily proof that block-wise multiplications are correct, the basis  is orthonormal, anticommutative, and ,  and  satisfy requirements (1.1).

Taking , multiplications give:


, .

Then, since , we easily get:

, ,

that is exactly (1.1) with basis bivectors ,  and .

Following all that we can write general element  of algebra  expanded in formal matrix basis  as:


This is arbitrary "complex" valued matrix where "imaginary unit" is matrix , corresponding to bivector expanded in the first two basis vectors , , that geometrically is bivector . Inversely, any "complex" valued matrix can be written as  element in matrix basis  isomorphic to geometrical basis :

If  belongs to even subalgebra  then, by identifying , ,  we have from (4.2) the correspondence:


In the same way as in the  case, multiplications of  by basis bivectors  give basis bivectors of tangent spaces to original bivectors:

These  elements are orthogonal to  and to each other, and are the tangent space basis elements at points .

Projections of  onto  are:

, ,

These elements of  are orthogonal, in the sense of Euclidean scalar product in : , to each other and to the projection  of the original state in . They are the tangent space basis elements in at points .

5. Hamiltonians in C2︱S3 and Their Lifts in G3

Take self-adjoint (Hamiltonian) linear operator in  in its most general form:. Its corresponding element, see (4.3), is  and does not belong to , though the result of its action on an element from  (qubit), has preimage in . Let’s prove that. Take a qubit and a Hamiltonian in :

, , , ,

, , ,



For each single qubit  its fiber (full preimage) in  is  with an arbitrary triple of orthonormal bivectors  in 3D satisfying (1.1). In exponential form:




Using (5.2) we get from (5.3) the fiber at :




We can also explicitly write the element  which, acting (from the right) on  gives :

From (5.1) and (5.4):

where , ,  and  are defined by (5.3) and (5.5).

6. Clifford Translations

Suppose, action on  in  is multiplication by an exponent: , often called Clifford translation[5], that’s:

Then the fiber of  in  is:


In other words, Clifford translations in  are equivalent to multiplications of fibers in  by standard fiber elements with bivector part  (which is associated with formal imaginary unit ) and the same phase .

In commonly used quantum mechanics all states from the set  are considered as identical to the state , they have the same values of observables. That is not commonly true for -qubits. Only in the case when all the observables common plane, say , (unspecified in the  model[6]) is the same as the plane associated with complex plane in , the -qubit  leaves any such observable  unchanged:

Suppose angle  in  is varying. Then the tangent to Clifford orbit is


It is orthogonal to  in the sense of usual scalar product in : , and remains on :  (translational velocity of Clifford translation is one).

For any  define . The elements ,  and  are orthogonal to  and pairwise orthogonal. That means they span tangent space and the plane spanned by  is orthogonal to the Clifford orbit plane spanned by .

The vector of translational velocity rotates while moving in the orbit plane. Both  and  also rotates with the same unit value angular velocity since



To make the planes of rotations clearly identified, that is difficult to do with formal imaginary unit, let’s lift Clifford translation to  using (6.1). Translational velocity, similar to the case, is  and is orthogonal to :

Two other components of the tangent space, orthogonal to  and  at any point of the orbit, are  and [7]. Their velocities while moving along Clifford orbit are:

(derivative of  is orthogonal to  and looking in the direction )

(derivative of  is orthogonal to  and looking in the direction )

These two equations explicitly show that the two tangents, orthogonal to Clifford translation velocity, rotate in moving plane  with the same unit value rotational velocity. Interesting to notice that the triple of the translational velocity and two rotational velocities has orientation opposite to the triple of tangents: if the tangents  have right screw orientation, the speed triple  is left screw.

If a fiber, -qubit, makes full circle in Clifford translation: , , both  and  also make full rotation in their own plane by . This is special case of the -qubit Berry phase incrementing in a closed curve quantum state path.

7. Conclusions

Evolution of a quantum state described in terms of  gives more detailed information about two state system compared to the  Hilbert space model. It confirms the idea that distinctions between "quantum" and "classical" states become less deep if a more appropriate mathematical formalism is used. This paradigm spreads from trivial phenomena like tossed coin experiment [7] to recent results on entanglement and Bell theorem [8] where the former was demonstrated as not exclusively quantum property.


  1. A. Soiguine, "Geometric Algebra, Qubits, Geometric Evolution, and All That," January 2015. [Online]. Available:
  2. A. Soiguine, Vector Algebra in Applied Problems, Leningrad: Naval Academy, 1990 (in Russian).
  3. A. Soiguine, "Complex Conjugation - Realtive to What?," in Clifford Algebras with Numeric and Symbolic Computations, Boston, Birkhauser, 1996, pp. 285-294.
  4. D. Hestenes, New Foundations of Classical Mechanics, Dordrecht/Boston/London: Kluwer Academic Publishers, 1999.
  5. C. Doran and A. Lasenby, Geometric Algebra for Physicists, Cambridge: Cambridge University Press, 2010.
  6. A. Soiguine, "What quantum "state" really is?," June 2014. [Online]. Available:
  7. A. Soiguine, "A tossed coin as quantum mechanical object," September 2013. [Online]. Available:
  8. X.-F. Qian, B. Little, J. C. Howell and J. H. Eberly, "Shifting the quantum-classical boundary: theory and experiment for statistically classical optical fields," Optica, pp. 611-615, 20 July 2015.
  9. J. W. Arthur, Understanding Geometric Algebra for Electromagnetic Theory, John Wiley & Sons, 2011.
  10. D. Chruscinski and A. Jamiolkowski, Geometric Phases in Classical and Quantum Mechanics, Boston: Birkhauser, 2004.


[1] This sum bears the sense "something and something". It is not a sum of geometrically similar elements giving another element of the same type, see [7], [9].

[2] Though both  and  can be parametrized by points of , it is not correct to say that  is sphere  or -qubit is sphere .

[3] It is often convenient to write elements  as exponents: .

[4] I begin with ordinary real component matrices. That is why the order of Pauli matrices in my following definition is different from usual one. An analogue of "imaginary" unit should logically be the last one, after pure real items.

[5] It is called "translation" because does not change distance between elements.

[6] In the  model observables are also elements of , see [1], [6]

[7] Clearly, the three  are identical to earlier considered tangents

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