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1
Geometric algebra, qubits, geometric evolution, and all that
Alexander M. SOIGUINE
Copyright © 2015
Abstract: The approach initialized in [1], [2] is used for description and analysis of qubits, geometric phase parameters – things
critical in the area of topological quantum computing [3], [4]. The used tool, Geometric (Clifford) Algebra [5], [6], is the most
convenient formalism for that case. Generalizations of formal “complex plane” to an arbitrary variable plane in 3D, and of usual
Hopf fibration to the map generated by an arbitrary unit value element
0
g
3
G
are resulting in more profound description
of qubits compared to quantum mechanical Hilbert space formalism.
1. Introduction
The working environment will be even subalgebra
3
G
of elements
S
I
1
,
and
are (real
2
)
scalars,
S
I
is a unit size oriented area, bivector, in an arbitrary given plane
3
ES
. It was explained in
[7], [8] that elements
S
I
only differ from what is traditionally called “complex numbers” by the
fact that
3
ES
is an arbitrary plane. Traditional “imaginary unit”
i
is geometrically some
S
I
when it is
not necessary to specify the containing plane – everything is going on in one fixed plane, not in 3D
world.
2. Rotations with elements of
3
G
and generalized Hopf fibration
Let’s take a unit value element of
3
G
:
Sso ,,
ii beeeeeeeebeebeeb
,
213132321213132321
,
1
22
and
1
2
3
2
2
2
1 bbb
. If
C
is a bivector,
213132321 eeCeeCeeCC
, then the map
rot
CSCsoSso ,,,, ~
,
S
ISso
~
,,
, is rotation of
C
. This rotation generates a
map
rot
CCSsoSS ,,:
23
. If, for example,
32eeC
then:
)()( 32
SS IeeI
32213132321 eeeebeebeeb
213132321 eebeebeeb
)()( 21313232132213132321 eeeeeeeeeeeeee
32
2
3
2
2
2
1
2)( ee
13213 )(2 ee
21231 )(2 ee
, (2.1)
which is classical Hopf fibration
23 SS
.
1
This sum bears the sense “something and something”. It is not a sum of similar elements giving another element
of the same kind, see [1], [9].
2
Scalars should always be real. “Complex” scalars are not scalars.
2
Also (2.1) shows that “imaginary” number
i
in this Hopf fibration written in traditional terms should
geometrically be unit value oriented area
32ee
orthogonal to the basis vector
1
e
.
Suppose bivector is expanded not in
211332 ,, eeeeee
but in any basis of bivectors
321 ,, BBB
, where
1
B
is arbitrary unit bivector in 3D, and
2
B
,
3
B
are two unit bivectors orthogonal to
1
B
and to each
other. Their mutual orientation should satisfy the same multiplication rules as
211332 ,, eeeeee
do:
132231321 ,, BBBBBBBBB
. The Hopf fibration then can be extended to an arbitrary
generating element from
3
G
:
3322110
,, BCBCBCC
Sso
0
C
131233212
2
3
2
2
2
1
2
122 BCCC
21323
2
3
2
1
2
2
2
22131 22 BCCC
3
2
2
2
1
2
3
2
332122311 22 BCCC
(2.2)
3. Quantum mechanical qubits and elements of
3
G
A pure qubit state in terms of conventional quantum mechanics is two dimensional unit value vector
with “complex” value components:
2,1,,1
~~
,
1
0
0
121
2211
2
2
2
121
2
1
kizzzzzzzzzzz
z
zkkk
Let’s find explicit relations between elements
as two-component “complex” vectors and elements
Sso ,,
belonging to
3
G
. Take an arbitrary
),,( Sso
. If bivector
32ee
is chosen as the one
defining “complex plane”
S
, we have:
21322332121321322321 )()()(),,( eeeeeeeeeeeeeeSso
21
3,2
2
3,2
1eezz
So we get correspondence:
32
23
1
3,2
2
3,2
1,,, eei
i
i
z
z
Sso
Actually, the number of such maps is infinite because “complex” plane can be an arbitrary plane in 3D.
Take arbitrary mutually orthogonal unit bivectors
321 ,, BBB
satisfying multiplication rules as above. If
S
ISsoS ,,
3
is expanded in basis
321 ,, BBB
:
3
iiS bBBBBbBbBbI
,
332211332211
,
then, for example, taking
1
B
as “complex” plane we get:
3123113331211332211 BBBBBBBBBB
,
and the correspondence is:
1
23
1
2
1,,, 1
1Bi
i
i
z
z
Sso B
B
- arbitrary bivector in 3D (3.1)
So, for any
Sso ,,
we have the following mappings:
231233
123122
312311
332211332211
)(
)(
)(
)(),,(
BBB
BBB
BBB
BBBBbBbBbSso
forgottenBBi
i
i
forgottenBBi
i
i
forgottenBBi
i
i
23
12
3
12
31
2
31
23
1
;,
;,
;,
(3.2)
All that means that to recover, or establish, which
Sso ,,
in 3D is associated with
2
1
z
z
it is
necessary, firstly, to define which plane in 3D should be taken as “complex” plane and then to choose
another plane, orthogonal to the first one. The third orthogonal plane is then defined by the first two up
to orientation that formally means
1
321 BBB
.
4. New definitions
I shall give new definitions of states, observables, measurements and observable values in the case of
3D objects identifiable as
3
G
elements:
Definition 4.1.
States of a qubit and observables of a qubit are unit value elements of
3
G
. The notations will
be used in the following mainly as:
4
State -
S
ISso ,,
ii bBBBBbBbBb
,
332211332211
,
1
22
,
1
2
3
2
2
2
1 bbb
Observable -
1, 2
3
2
2
2
1
2
03322110 CCCCBCBCBCCC
Definition 4.2.
Measurement of observable
C
, measured in state
S
I
, is a generalized Hopf fibration
generated by
C
:
SS
C
S
GICIIGG :
33 3
,
which is a new element from
3
G
explicitly given in basis components by (2.2).
Definition 4.3.
Value of observable
C
, measured in state
S
I
, is a map of the measurement of the
observable to a set of measurement values.
5. Qubit states in
3
G
corresponding to quantum mechanical basis states
Quantum mechanical pure qubit state is
10 21 zz
, linear combination of two basis states
0
and
1
. In
3
G
terms these two states are, as follows from (3.1), the classes of equivalence:
- State
0
corresponds to any one of the following sets of
0
,, Sso
elements:
1
i
,
1
2
1
2
, if the “complex plane” is selected as
1
Bi
, or:
2
i
,
1
2
2
2
, if the “complex plane” is selected as
2
Bi
, or: (5.1)
3
i
,
1
2
3
2
, if the “complex plane” is selected as
3
Bi
.
- State
1
corresponds to any one the following sets of
1
,, Sso
elements:
323 )( Bi
,
1
2
2
2
3
, if the “complex plane” is selected as
1
Bi
, or:
131 )( Bi
,
1
2
3
2
1
, if the “complex plane” is selected as
2
Bi
, or: (5.2)
212 )( Bi
,
1
2
1
2
2
, if the “complex plane” is selected as
3
Bi
.
For any of the states
3,2,1, iBii
, corresponding to
0
, the value of observable
i
B
is:
iiiiiiii BBBBB )()()( 22
(5.3)
5
For any of the states
3,2,1,
3mod)1(3mod)1(3mod)2(3mod)2( iBB iiii
, corresponding to
1
(with the
agreement 3mod3=3, since index 0 does not here exist), the value of observable
i
B
is:
iiii
iiiiiiiii
BB
BBBBB
)(
)()(
23mod)1(
23mod)2(
3mod)1(3mod)1(3mod)2(3mod)2(3mod)1(3mod)1(3mod)2(3mod)2(
(5.4)
This is the actual meaning of quantum mechanical basis states:
Value of observable
i
B
is, for any pure qubit state from the set of all
0
,, Sso
, the bivector
i
B
itself:
i
BBSso i
0
,,
Value of observable
i
B
is, for any pure qubit state from the set of all
1
,, Sso
, flipped bivector
i
B
:
i
BBSso i
1
,,
Lets’ take arbitrary bivector observable
332211 BCBCBCC
. Without losing of generality we can
think that “complex” plane is defined by bivector
1
B
. Then generalized Hopf fibrations, measurements
of the observable
C
, for the states
0
,, Sso
and
1
,, Sso
, are correspondingly:
312
2
1
2
3213
2
1
2
2111111 22 BCCBCCBCBCB
33223211 2cos2sin2sin2cos BCCBCCBC
(through parameterization
cos
,
sin
1
)
and:
3
2
3
2
233222323
2
3
2
221133223322 22 BCCBCCBCBBCBB
33223211 )2cos2sin()2sin2cos( BCCBCCBC
(through parameterization
cos
2
,
sin
3
).
We get the following:
Measurement of observable
332211 BCBCBCC
in pure qubit state
0
,, Sso
, is bivector
with the
1
B
component equal to unchanged value
1
C
. The
2
B
and
3
B
measurement components are
equal to
2
B
and
3
B
components of
C
rotated by angle
2
defined by
cos
and
sin
1
,
where plane of rotation is
1
B
.
6
Measurement of observable
332211 BCBCBCC
in pure qubit state
1
,, Sso
, is bivector
with the
1
B
component equal to flipped value
1
C
(qubit flips in
1
B
plane). The
2
B
and
3
B
measurement components are equal to
2
B
and
3
B
components of
C
rotated by angle
2
defined
by
cos
2
,
sin
3
, where plane of rotation is
1
B
. The absolute value of angle of rotation is
the same as for
0
,, Sso
but the rotation direction is opposite to the case of
0
,, Sso
.
The above results are geometrically pretty clear. The two states,
0
,, Sso
and
1
,, Sso
represented in
3
G
correspondingly by
11
B
and
3213 )( BB
, only differ by additional factor
3
B
in
1
,, Sso
. That means that measurements of bivector observable
C
in states
0
,, Sso
and
1
,, Sso
are equivalent up to additional “wrapper”
3
B
:
3
0
~
0
3
1
~
1,,,,
~
,,,, BSCsoSsoBSCsoSso
(5.5)
That simply means that the measurement on the left side is received from the
0
~
0,,,, SCsoSso
measurement just by flipping of the latter relative to the plane
3
B
.
6. Berry parameters for the
3
G
qubit states
Quantum mechanical transformations
i
e
, considered as transformations on
3
S
, are often
called Clifford translations. Clifford translation of state
),,( Sso
with explicitly defined “complex”
plane
Cl
should be written as
),,(),,( SsoeSso Cl
I
.
Let’s take transformation:
),,,(),,(),,(),,( tSsoSsoeSsoeSso tH
H
H
Ht
(6.1)
where
H
is constant value generic Hamiltonian of the system, bivector of
3
G
with the plane not
coinciding with
S
. Clearly,
),,,( tSso
satisfies Schrödinger equation:
),,,(),,,( tSsoH
H
H
tSso
dt
d
with unit bivector
H
H
in 3D replacing “imaginary unit”
i
of the conventional quantum mechanics case.
This is important thing: in terms of
3
G
the Schrödinger equation is equation for the result of
7
transformation of a
3
G
qubit state
),,(),,( SsoeSso Ht
generated by
3
G
bivector
Hamiltonian which is generic Hamiltonian of the system.
Let’s consider the Hamiltonian corresponding to constant value
H
magnetic field slowly rotating
around axis orthogonal, for example, to plane
3
B
(axis along vector
3
e
). Initial plane of the field
H
is
inclined by some angle
relative to vector
3
e
. It may be initialized as rotation of
3
B
in plane
1
B
:
2
3
2
3
11
BB eBHeBH
The inclined field rotates around
3
e
depending on angle
)(t
:
2)(
2
3
22 )( 3113
)( t
BBB
t
BeeBeeHtH
(6.2)
Calculate (6.2) in two steps. First, incline
3
B
:
2
3
2
3
11
BB eBHeBH
that gives
cossin 32 BBH
. Then rotate the inclined bivector:
2)(
2
3
22 )( 3113
)( t
BBB
t
BeeBeeHtH
2)(
32
2)( 33 cossin t
B
t
BeBBeH
cos)(cossin)(sinsin 321 BtBtBH
(6.3)
Let’s now calculate Berry curvature and connection in full
3
G
terms. Forgetting about
H
and using
notation
U
for
cos)(cossin)(sinsin 321 BtBtBH
we have:
cossincoscossinsinsin
~32321 3BBeBBBU B
,
3
sincos 32
3BBe
UB
, then
3
23
~B
GeB
U
BUA
From another derivative:
sin
3
1B
eB
U
,
cossinsin
~3
1
2
3B
GeB
U
BUA
3
I am also using easily verified
33 22 BB eBBe
8
We see that full
3
G
values
G
A
and
G
A
differ from usual quantum mechanical case by additional
bivector terms. The Berry curvature also has additional bivector term:
2cos12sin 3
1 B
GGG eBAAF
The additional bivector terms may be an indicator of a kind of torsion caused by (6.2)
g
-qubit
transformation, though it needs further elaboration.
7. Geometric evolution.
With explicitly defined variable “imaginary unit” many things become not just more informative but also
much simpler. As an example take the
3
G
variant of geometric evolution derived in usual quantum
mechanical approach through holonomy (see, for example, [4]).
Transformation (6.1) can be straightforwardly generalized to:
),,,(),,(),,( )(
)( )(
tSsoSsoeSso tH
tH tH
The “complex” plane defined by unit bivector
)( )(tH tH
then becomes variable and
)(tH
plays the role of
angle in this plane. The evolution equation for the result of transformation is:
tSsotHtH
tH tH
tSso
dt
d,,,)()(
)( )(
,,,
,
which is a generalization of Schrödinger equation with variable “imaginary unit”
)( )(
)( tH tH
ti
.
Let‘s make assumption that
)(tU
in
)()(
~
)( 0tUHtUtH
just generates rotation, so
0
)( HtH
. If
the initial value of Hamiltonian also lies on
3
S
we have:
tSsotHtHtSso
dt
d,,,)()(,,,
tSsotH
dt
d,,,)(
2
12
with the solution:
0,,,,,, )(
2
1
2
122
0SsoeetSso tHH
9
8. Conclusions.
The idea of using variable plane in 3D for playing the role of “complex plane”, in combination with
3
G
algebra, resulted in more detailed description of quantum states and observables usually formalized in
terms of two-dimensional “complex” vectors and Hermitian matrices. New additional components in
Berry parameters may be used to analyze possible anyonic states in the topological computing related
structures.
Works Cited
[1]
A. Soiguine, "A tossed coin as quantum mechanical object," September 2013. [Online]. Available:
http://arxiv.org/abs/1309.5002.
[2]
A. Soiguine, "What quantum "state" really is?," June 2014. [Online]. Available:
http://arxiv.org/abs/1406.3751.
[3]
C. Nayak, S. H. Simon, A. Stern, M. Freedman and S. Das Sarma, "Non-Abelian Anyons and
Topological Quantum Computation," Rev. Mod. Phys., vol. 80, p. 1083, 2008.
[4]
J. K. Pachos, Introduction to Topological Quantum Computation, Cambridge: Cambridge University
Press, 2012.
[5]
D. Hestenes, New Foundations of Classical Mechanics, Dordrecht/Boston/London: Kluwer Academic
Publishers, 1999.
[6]
C. Doran and A. Lasenby, Geometric Algebra for Physicists, Cambridge: Cambridge University Press,
2010.
[7]
A. Soiguine, "Complex Conjugation - Realtive to What?," in Clifford Algebras with Numeric and
Symbolic Computations, Boston, Birkhauser, 1996, pp. 285-294.
[8]
A. Soiguine, Vector Algebra in Applied Problems, Leningrad: Naval Academy, 1990 (in Russian).
[9]
J. W. Arthur, Understanding Geometric Algebra for Electromagnetic Theory, John Wiley & Sons,
2011.