Quantum State Evolution in C^{2} and G_{3}^{+}
Alexander Soiguine
Soiguine Supercomputing, Aliso Viejo, CA, USA
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To cite this article:
Alexander Soiguine. Quantum State Evolution in C^{2} and G_{3}^{+}.Science Research.Vol.3, No. 5, 2015, pp. 240-247. doi: 10.11648/j.sr.20150305.11
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.
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):
(1.1)
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:
(2.1)
Explicit relations, based on their parameterization by points, between elements and elements can be established through the following construction^{2}.
Let basis bivector is chosen as defining the complex plane , then we have (see multiplication rules (1.1)):
Hence we get the map:
where
In the similar way, for two other selections of complex plane we get:
,
and
So we have three different maps defined by explicitly declared complex planes satisfying (1.1):
(2.2)
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:
(3.1)
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):
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]):
where
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:
(3.2)
and for two other basis bivectors:
(3.3)
(3.4)
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:
(4.1)
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.
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 4^{th}-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:
(4.2)
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:
(4.3)
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 :
, , , ,
, , ,
Then:
(5.1)
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:
(5.2)
where
(5.3)
Using (5.2) we get from (5.3) the fiber at :
(5.4)
where
(5.5)
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:
(6.1)
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
and
.
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.
References
Footnotes
[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