Geometric phase in the G3 + quantum state evolution - arXiv

Geometric phase in the G3+ quantum state evolution Alexander SOIGUINE1 1 SOiGUINE

Supercomputing, Aliso Viejo, CA 92656, USA

Email address: [email protected]

Key words: Qubits, Geometric algebra, Clifford translation, Geometric phase, Topological quantum computing

Abstract: When quantum mechanical qubits as elements of two dimensional complex Hilbert space are generalized to elements of even subalgebra of geometric algebra over three dimensional Euclidian space, geometrically formal complex plane becomes explicitly defined as an arbitrary, variable plane in 3D [1]. The result is that the quantum state definition and evolution receive more detailed description, including clear calculations of geometric phase, with important consequences for topological quantum computing.

1. Introduction Qubits, unit value elements of the Hilbert space

z 

 1

C2

of two dimensional complex vectors:

 0

z1  z2 ~ z2  1 , zk  z1k  iz k2 , k  1,2    1   z1    z2   , z12  z22  z1~ z 0 1      2 can be generalized to unit value elements   I S  of even subalgebra G3 of geometric algebra G3 over Euclidian space E 3 , g-qubits [1]:

  I S      b1B1  b2 B2  b3 B3     1B1  2 B2  3 B3 , i  bi ,  i  bi ,  2   2  1 ,

b12  b22  b32  1 ,  ,  , bi scalars, Bi - unit value bivectors satisfying B1B2   B3 , B1B3  B2 , B2 B3   B1 , Bi2  1

(1.1)

Multiplication rules (1.1) assume the right screw space orientation that can be seen through the order of vectors, dual to the bivectors, used to create the space oriented unit volume I 3 (see Fig.1.1).

1

Fig.1.1. Right screw unit value oriented volume

There always are two options to create oriented unit volume, depending on the order of vectors in the product. They correspond to the two types of the three dimensional space handedness – left and right screw handedness. One can also think about I 3 as a right (left) single thread screw helix of the height one (see the above picture). In this way  I 3 is left (right) screw helix. Mappings between g-qubits and qubits are not one-to-one and are defined by:

so( ,  , S )    1B1   2 B2  3 B3     1B1  ( 3   2 B1 ) B3       2 B2  ( 1  3 B2 ) B1      3 B3  (  2  1B3 ) B2

   i1  , i  B1   3  i 2     i 2  , i  B2     i  1 3       i3  , i  B3    2  i1 

(1.2)

i ( z1 , z2 ) where that actually defines principal fiber bundle so ,  , S  

B

so( ,  , S )  g  G

 3

them as

i



i

: g  1 is total space and {z  ( z1, z2 )  C 2 ; z  1} is base space. I will denote 2

G3 |S 3 and C 2 |S respectively. The projection  : G3 |S  C 2 |S depends on which particular 3

3

3

Bi is taken from an arbitrarily selected triple B1, B2 , B3 in 3D satisfying (1.1). Bivector Bi defines  2 i C 2 |S 3 . For any complex plane for the complex vectors of C , so we should write  : G3 |S 3  B

 x  iy1     C 2 |S 3 the fiber in G3 |S 3 consists of all elements Fz  x1  y2 B1  x2 B2  y1B3 , if B3 is z   1  x2  iy2  optionally chosen as complex plane. That particularly means that standard fiber is equivalent to the group of rotations of the triple y2 B1 , x2 B2 , y1B3  as a whole. All such rotations in G3 are also 



identified by elements of G3 |S 3 since for any bivector B the result of its rotation is 1 (see, for example

1

It is convenient to write elements so ,  , S  as exponents: so ,  , S   e

2

I S

.

[2], [3]): so( ,  , S ) Bso( ,  , S ) , where so( ,  , S ) ~    1 B1   2 B2   3 B3 . So, standard fiber is ~

identified as

G3 |S 3 and the composition of rotations is: e

 I S2

e

 I S1

Be

I S1

e



I S2

 Be

I  I  ~

 e S1 e S2

I S1 I S2

e

Multiplications of so( , 1 ,  2 ,  3 )    1 B1   2 B2   3 B3 by basis bivectors Bi give basis bivectors of tangent spaces to original bivectors [4]:

T1  so( , 1 ,  2 ,  3 ) B1    1 B1   2 B2   3 B3 B1  1  B1   3 B2   2 B3 T2  so( , 1, 2 , 3 ) B2    1B1  2 B2  3B3 B2  2  3B1  B2  1B3 T3  so( , 1, 2 , 3 ) B3    1B1  2 B2  3 B3 B3  3  2 B1  1B2  B3 

These G3 |S 3 elements are orthogonal to   1 B1   2 B2   3 B3 and to each other, and are the tangent space basis elements at points   1 B1   2 B2   3 B3 . Projections of Ti onto C 2 |S 3 are:

    i    1  i 2     2  i1    ,  (T2 )    ,  (T3 )   3  1  i 2     3  i     i 3 

 (T1 )  

These elements of C 2 |S 3 are mutually orthogonal, in the sense of Euclidean scalar product in C 2 :

   i 3   of the z, w  Re~ z1w1  ~ z2 w2  , and orthogonal to the projection  ( so( , 1 ,  2 ,  3 )     2  i1  original state in

 z     i 

3 G3 |S 3 . They are the tangent space basis elements in C 2 |S at points  1      z2    2  i1  3

.

2. Clifford translations Let’s take Clifford translation in C 2 |S 3 , Cl z   ei z , and lift it to

G3 |S 3 using :

FCl ( z )  r1 cos(   1 )  r2 sin(   2 ) B1  r2 cos(   2 ) B2  r1 sin(   1 ) B3 

r1 cos(   1 )  r1 sin(   1 ) B3  r2 cos(   2 )  r2 sin(   2 ) B3 B2  r1e B3 e B31  r2 e B3 e B3 2 B2  e B3 Fz 3

Translational velocity is





  B3 FCl ( z )  e Fz  B3 FCl ( z ) (2.1)   and is orthogonal to FCl ( z ) :

FCl ( z ) , B3 FCl ( z )

0



~ ~  FCl ( z ) FCl ( z ) B3



~

0

 B3 0  0 (index 0 means scalar part of element)

Two other components of the tangent space, orthogonal to FCl ( z ) and B3 FCl ( z ) at any point of the orbit, are B1 FCl ( z ) and B2 FCl ( z ) 2. Their velocities, while moving along Clifford orbit are:









  B1FCl ( z )  B1e B3 Fz  B1B3 FCl ( z )  B2 FCl ( z )  

(2.2)

(derivative of B1 FCl ( z ) is orthogonal to B1 FCl ( z ) and looking in the direction B2 FCl ( z ) )









  B2 FCl ( z )  B2e B3 Fz  B2 B3 FCl ( z )   B1FCl ( z )  

(2.3)

(derivative of B2 FCl ( z ) is orthogonal to B2 FCl ( z ) and looking in the direction  B1 FCl ( z ) ) These two equations explicitly show that the two tangents, orthogonal to Clifford translation velocity,





rotate in moving plane B1FCl ( z ) , B2 FCl ( z ) with the same, by value, rotational velocity as translation velocity is (see Fig 2.1).

Fig.2.1. Tangents rotate in their plane with the same, by value, speed as translational velocity

2

Clearly, the three Bi FCl ( z ) are identical to earlier considered tangents

4

Ti

If a fiber, g-qubit, makes full circle in Clifford translation: Fz  FCl ( z )  e

B3

Fz , 0    2 , both

B1FCl ( z ) and B2 FCl ( z ) also make full rotation in their common plane by 2 . This is special case of the 3

g-qubit geometric phase, incrementing in the S sphere big circle closed curve quantum state path. The demonstrated rotation of tangents in the plane orthogonal to the orbit of g-qubit Clifford translation – that is what actually is not intuitively obvious and is more important than all widely accepted “mysteries” of quantum mechanics. This rotation phenomena has nothing to do with the size of physical system. This is topological property of the space of dimension 4, not 3, our imagination cannot easily deal with. At the same time, we should remember that g-qubits, states, are operators acting on observables. Though observables are elements of the same space as states (see next section), action of a state on 3 G3 :   I S    I S  C   I S   , and the result of this action observable is G3 



G



C

changes differently compared to the state modification subjected to Clifford translation.

3. Measurement of observables in basis states Let’s consider the case when, for an arbitrary g-qubit   I S  , the plane I S is taken as playing the role

( z1, z2 ) , of complex plane. Then, due to (1.2), the C 2 |S 3 element, given by projection so ,  , S   S

   i  , i  S .  0 

is 



Let’s recall the definitions of states, observables and measurements, appropriate for the case of the G3 formalism of the two state systems [1]. States and observables are elements of G3 : Definition 3.1 (state, unit value element of G3 , defines operation acting on observable in a measurement ):

so ,  , S     I S      b1 B1  b2 B2  b3 B3     1 B1   2 B2   3 B3 ,  i  bi ,  2   2  1 , b12  b22  b32  1 Definition 3.2 (observable, element of G3 ):

C  C0  C1B1  C2 B2  C3 B3

5

Definition 3.3 (measurement): Measurement of observable C , measured in state   I S  , is generalized Hopf fibration generated by the observable: G3 C   I S  C  I S   G3  G3 :   I S   

Explicit formulas can be found in [1]. Due to definition (3.1) the state   I S  corresponds to the “stable state” (see [5]), 0 in familiar terms of quantum mechanical notations

  z1 0  z2 1 , if I S is selected as complex plane. The

state   I S  is “stable” in the sense that the measurement of any observable with the bivector part parallel to I S does not change the observable (I omit scalar part which does not change in rotations):

  I S  I S   I S    I S     I S     2I S       2I S



2

  I S  I S 2



(3.1)

The g-qubit state corresponding to 1 is  2 I S2  3 I S3 ,  22  32  1 , where I S 2 is any unit bivector orthogonal to I S and I S3  I S2 I S (if we keep right screw space orientation with multiplication rules (1.1).) Measurement of any observable with the bivector part parallel to I S in this state gives:

  I

2 S2

 

 





  3 I S3 I S  2 I S2   3 I S3    2I S3   3I S2  2 I S2   3 I S3 

  22I S   2  3   2  3   32I S   22   32 I S  I S

(3.2)

The last formula means that measurement of any observable with the bivector part parallel to I S , in the state corresponding to 1 , flips bivector part of the observable. Formulas (3.1), (3.2) retrieve the actual sense of the two basis states. Consider the results of measurements in states   I S  and  2 I S2  3 I S3 of an arbitrary observable

C  C0  C1I S  C2 I S2  C3 I S3 :









(  I S )C (  I S )  C0  C1I S  ( 2   2 )C2  2C3 I S2  ( 2   2 )C3  2C2 I S3 

C0  C1I S  C2 cos 2  C3 sin 2 I S2  C2 sin 2  C3 cos 2 I S3

(3.3)

(through parameterization   cos  ,   sin  )

  I

2 S2





 











 3 I S3 C 2 I S2  3 I S3  C0  C1I S  C2 22  32  2C3 2 3 I S2  2C2 2 3  C3 22  32 I S3 

C0  C1I S  (C2 cos 2  C3 sin 2 ) I S2  (C2 sin 2  C3 cos 2 ) I S3

6

(3.4)

(through parameterization  2  cos  ,  3  sin  ). Formulas (3.3), (3.4) mean the following: Measurement of observable C  C0  C1I S  C2 I S2  C3 I S3 in pure qubit state   I S  has bivector part with the I S component equal to unchanged value C1 . The I S 2 and I S3 measurement components are equal to I S 2 and I S3 components of C rotated by angle 2 defined by   cos  and 1  sin  , where plane of rotation is I S . Measurement of observable C  C0  C1I S  C2 I S2  C3 I S3 in pure qubit state  2 I S2  3 I S3 has bivector part with the I S component equal to flipped value  C1 (flipping in I S plane). The I S 2 and I S3 measurement components are equal to I S 2 and I S3 components of C rotated by angle 2 defined by

 2  cos ,  3  sin  , where plane of rotation is I S . The absolute value of angle of rotation is the same as for  3  I S  2 but the rotation direction is opposite to the case of  3  I S  2 . The above two results are geometrically pretty clear. The two states  3  I S  2 and

2 I S  3 I S  3  I S 2 I S only differ by additional factor I S in 3  I S  2 I S . That means that 2

3

3

3

3

measurements of an observable C , if it is pure bivector, in states  3  I S  2 and  2 I S2  3 I S3 are equivalent up to additional “wrapper” I S3 :



3



~ ~ ~  I S  2 I S3 C 3  I S 2 I S3  I S3 3  I S  2  C 3  I S  2 I S3

That simply means that the measurement in state corresponding to 1 is received from the

3  I S 2 ~ C3  I S 2  measurement, measurement in state corresponding to 0

, just by mirroring

the result relative to the plane I S3 (see Fig.3.1).

Fig.3.1. Results of measurement of C in the two stable states. C0, any measurement in a state corresponding to other measurement in a state corresponding to

1

0

, and C1,

, can be made mirrored of each other by rotating in a plane parallel to IS.

The IS components are the same, in absolute value, for all the three C, C0 and C1.

7

4. Sequence of infinitesimal Clifford translations 

Since any bivector in G3 can be generally taken as playing the role of “complex” plane, let’s take some arbitrary unit bivector I S 3 3 and make infinitesimal Clifford translation of an arbitrary g-qubit, state:

so ,  , S   e

I S3 d





so ,  , S   Cl I S3 , d so ,  , S 





Instant translational velocity tangent of it is: I S3 Cl I S3 , d so ,  , S  . We also need two bivectors (planes) to create rotational tangent components. The first one, I S1 , is any unit bivector orthogonal to

I S 3 , so defined up to arbitrary angle of rotation around normal to I S 3 . The bivector for the second tangent can be taken, in the case of the right screw space orientation, as I S2  I S1 I S3 . The translational velocity tangent will rotate by the value d in the direction opposite to





Cl I S3 , d so ,  , S  because from (2.1):

 I S FCl ( z )  I S3 I S3 FCl ( z )   FCl ( z )  3  The rotational velocity tangents will rotate by the same d value in their plane orthogonal to translational velocity tangent as normal (see Fig. 2.1, replacing Bi to I Si ). All that means that while moving in a sequence of infinitesimal Clifford translations the two rotational tangents rotate in each infinitesimal step by the same angle in their plane as translational velocity tangent rotates in plane I S 3 moving along orbit lying on S 3 . We saw above that it does not matter do we look at the angle accumulated by translational velocity in the varying plane spanned by this velocity together with instant state g-qubit, or accumulated angle of rotation of any of the two tangents of rotational speed - they are equal! Let we move along some arbitrary path on S 3 . The sphere is center symmetrical surface, so at any instant value of state on S 3 infinitesimal incrementing of translational velocity angle does not depend in what direction the displacement happens. If the path is approximated with infinitesimal pieces of geodesics then the accumulated angle between translational velocity and instant geodesic is obviously equal to the total length of the path (see Fig.4.2).

3

Index “3” here is taken to stress that this bivector will play the same role as the previously used

Hopefully, reader remembers that

B3 in Sec. 3.

S 3 is a plane in 3D bearing index “3”, and S 3 is unit 3-dimensional sphere in 4-

dimensional space.

8

Fig.4.2. Accumulating of angle while moving along path. Also equals by value to rotation angle of two vectors in plane remaining orthogonal to translational velocity

This is purely geometrical phase, separated from g-qubit exponent phase modifications caused by external factors defining the g-qubit (state) path on the S 3 . Infinite composition of infinitesimal Clifford translations along any S 3 path L with varying I S 3 gives the final state g-qubit. Take a sequence of infinitesimal Clifford transformations:

e

I S3 ( l N ) l N

...e

I S3 ( l 2 ) l2

e

I S3 ( l1 ) l1

so ,  , S 

By taking the logarithm, approaching N   and getting back to exponent we receive the final state:

e

I S 3 ( l ) dl

so ,  , S 

(4.1)

L

5. Conclusions 

Evolution of a quantum state described in terms of G3 gives more detailed information about two state 2

system compared to the C 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. The paradigm spreads from trivial phenomena like tossed coin experiment [6] to recent results on entanglement and Bell theorem [7] where the former was demonstrated as not exclusively quantum property.

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Works Cited [1] A. Soiguine, "Geometric Algebra, Qubits, Geometric Evolution, and All That," January 2015. [Online]. Available: http://arxiv.org/abs/1502.02169. [2] D. Hestenes, New Foundations of Classical Mechanics, Dordrecht/Boston/London: Kluwer Academic Publishers, 1999. [3] C. Doran and A. Lasenby, Geometric Algebra for Physicists, Cambridge: Cambridge University Press, 2010. [4] A. Soiguine, "Quantum state evolution in C2 and G3+," Science Research, vol. 3, no. 5, August 2015. [5] A. Soiguine, "What quantum "state" really is?," June 2014. [Online]. Available: http://arxiv.org/abs/1406.3751. [6] A. Soiguine, "A tossed coin as quantum mechanical object," September 2013. [Online]. Available: http://arxiv.org/abs/1309.5002. [7] 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.

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