arXiv:2001.09091v1 [math.GT] 22 Jan 2020QUANTUM COMPUTATION AND MEASUREMENTS
FROM AN EXOTIC SPACE-TIME R4
MICHEL PLANAT†, RAYMOND ASCHHEIM‡,
MARCELO M. AMARAL‡ AND KLEE IRWIN‡
Abstract. The authors previously found a model of universal quan-
tum computation by making use of the coset structure of subgroups of
a free group G with relations. A valid subgroup H of index d in G
leads to a ‘magic’ state |ψ〉 in d-dimensional Hilbert space that encodes
a minimal informationally complete quantum measurement (or MIC),
possibly carrying a finite ‘contextual’ geometry. In the present work,
we choose G as the fundamental group π1(V ) of an exotic 4-manifold
V , more precisely a ‘small exotic’ (space-time) R4 (that is homeomor-
phic and isometric, but not diffeomorphic to the Euclidean R4). Our
selected example, due to to S. Akbulut and R. E. Gompf, has two re-
markable properties: (a) it shows the occurence of standard contextual
geometries such as the Fano plane (at index 7), Mermin’s pentagram
(at index 10), the two-qubit commutation picture GQ(2, 2) (at index
15) as well as the combinatorial Grassmannian Gr(2, 8) (at index 28) ,
(b) it allows the interpretation of MICs measurements as arising from
such exotic (space-time) R4’s. Our new picture relating a topological
quantum computing and exotic space-time is also intended to become
an approach of ‘quantum gravity’.
PACS: 03.65.Fd, 03.67.Lx, 03.65.Aa, 03.65.Ta, 02.20.-a, 02.10.Kn, 02.40.Pc, 02.40.-k,
03.65.Wj
MSC codes: 81P68, 57R65, 57M05, 32Q55, 57M25, 14H30
Keywords: Topological quantum computing, 4-manifolds, Akbulut cork, exotic R4,
fundamental group, finite geometry, Cayley-Dickson algebras
1. Introduction
Concepts in quantum computing and the mathematics of 3-manifolds
could be combined in our previous papers. In the present work, 4-manifolds
are our playground and the four dimensions are interpreted as a space-time.
This offers a preliminary connection to the field called ‘quantum gravity’
with unconventional methods. In four dimensions, the topological and the
smooth structure of a manifold are apart. There exist infinitely many 4-
manifolds that are homeomorphic but non diffeomorphic to each other [1]-
[4]. They can be seen as distinct copies of space-time not identifiable to the
ordinary Euclidean space-time.
A cornerstone of our approach is an ‘exotic’ 4-manifold called an Akbulut
cork W that is contractible, compact and smooth, but not diffeomorphic
to the 4-ball [4].
In our approach, we do not need the full toolkit of 4-
manifolds since we are focusing on W and its neighboors only. All what
we need is to understand the handlebody decomposition of a 4-manifold,
the fundamental group π1(∂W ) of the 3-dimensional boundary ∂W of W ,
and related fundamental groups. Following the methodology of our previous
work, the subgroup structure of such π1’s corresponds to the Hilbert spaces
1
2MICHEL PLANAT†, RAYMONDASCHHEIM‡, MARCELOM. AMARAL‡ANDKLEE IRWIN‡
of interest. One gets a many-manifold view of space-time -reminiscent of
the many-worlds- where the many-manifolds can be considered as many-
quantum generalized measurements, the latter being POVM’s (positive op-
erator valued measures).
In quantum information theory, the two-qubit configuration and its prop-
erties: quantum entanglement and quantum contextuality have been dis-
cussed at length as prototypes of peculiarities or resources in the quantum
world. We are fortunate that the finite geometrical structures sustaining
two-qubit commutation, the projective plane or Fano plane PG(2, 2) over
the two-element field F2, the generalized quadrangle of order two GQ(2, 2),
its embedding 3-dimensional projective space PG(3, 2), and other structures
relevant to quantum contextuality, are ingredients in our quantum comput-
ing model based on an Akbulut cork. Even more appealing, is the recovery
of the Grassmannian configuration Gr(2, 8) on 28 vertices and 56 lines that
connects to Cayley-Dickson algebras up to level 8.
Our description will be as follows.
In Sec. 2, one sums up the peculiarities of small exotic 4-manifolds of
type R4. This includes the handlebody structure of an arbitrary 4-manifold
in Sec. 2.1, the concept of an Akbulut cork in Sec. 2.2 and examples of
small exotic 4-manifolds that are homeomorphic but not diffeomorphic to
the Euclidean space R4 in Sec. 2.3.
In Sec. 3, one first gives an account of the model of quantum compu-
tation developed by the authors in their previous papers. The connection
to a manifold M is through the fundamental group π1(M). The subgroup
structure of π1(M) is explored when M is the boundary ∂W of an Akbulut
cork (in Sec. 3.1), the manifoldW̄ in Akbulut h-cobordism (in Sec. 3.2)
and the middle level Q between diffeomorphic connnected sums involving
the exotic R4’s (in Sec. 3.3).
Sec. 4 is a discussion of the obtained results and their potential interest
for connecting quantum information and the structure of space-time.
Calculations in this paper are performed on the softwares Magma [5] and
SnapPy [6].
2. Excerpts on the theory of 4-manifolds and exotic R4’s
2.1. Handlebody of a 4-manifold. Let us introduce some excerpts of
the theory of 4-manifolds needed for our paper [1, 2, 3]. It concerns the
decomposition of a 4-manifold into one- and two-dimensional handles as
shown in Fig. 1 [1, Fig. 1.1 and Fig. 1.2]. Let Bn and Sn be the n-
dimensional ball and the n-dimensional sphere, respectively. An observer
is placed at the boundary ∂B4 = S3 of the 0-handle B4 and watch the
attaching regions of the 1- and 2-handles. The attaching region of 1-handle
is a pair of balls B3 (the yellow balls), and the attaching region of 2-handles
is a framed knot (the red knotted circle) or a knot going over the 1-handle
(shown in blue). Notice that the 2-handles are attached after the 1-handles.
For closed 4-manifolds, there is no need of visualizing a 3-handle since it can
be directly attached to the 0-handle. The 1-handle can also be figured out
as a dotted circle S1 × B3 obtained by squeezing together the two three-
dimensional balls B3 so that they become flat and close together [2, p. 169]
QUANTUM COMPUTATION AND MEASUREMENTS FROM AN EXOTIC SPACE-TIME R43
Figure 1. (a) Handlebody of a 4-manifold with the struc-
ture of 1- and 2-handles over the 0-handle B4 , (b) the struc-
ture of a 1-handle as a dotted circle S1 ×B3.
as shown in Fig. 1b. For the attaching region of a 2- and a 3-handle one
needs to enrich our knowledge by introducing the concept of an Akbulut
cork to be described later on. The surgering of a 2-handle to a 1-handle
is illustrated in Fig. 2a (see also [2, Fig. 5.33]). The 0-framed 2-handle
(left) and the ‘dotted’ 1-handle (right) are diffeomorphic at their boundary
∂. The boundary of a 2- and a 3-handle is intimately related to the Akbulut
cork shown in Fig 2b as described at the subsection 2.3.
2.2. Akbulut cork. A Mazur manifold is a contractible, compact, smooth
4-manifold (with boundary) not diffeomorphic to the standard 4-ball B4 [1].
Its boundary is a homology 3-sphere. If one restricts to Mazur manifolds
that have a handle decomposition into a single 0-handle, a single 1-handle
and a single 2-handle then the manifold has to be of the form of the dotted
circle S1 ×B3 (as in Fig. 2a) (right) union a 2-handle.
Recall that, given p, q, r (with p ≤ q ≤ r), the Brieskorn 3-manifold
Σ(p, q, r) is the intersection in the complex 3-space C3 of the 5-dimensional
sphere S5 with the surface of equation zp1 +z
q
2+z
r
3 = 0. The smallest known
Mazur manifold is the Akbulut cork W [4, 7] pictured in Fig 2b and its
boundary is the Brieskorn homology sphere Σ(2, 5, 7).
According to [8], there exists an involution f : ∂W → ∂W that surgers
the dotted 1-handle S1 ×B3 to the 0-framed 2-handle S2 ×B2 and back, in
the interior ofW . Akbulut cork is shown in Fig. 2b. The Akbulut cork has a
simple definition in terms of the framings ±1 of (−3, 3,−3) pretzel knot also
called K = 946 [8, Fig. 3]. It has been shown that ∂W = Σ(2, 5, 7) = K(1, 1)
and W = K(−1, 1).
4MICHEL PLANAT†, RAYMONDASCHHEIM‡, MARCELOM. AMARAL‡ANDKLEE IRWIN‡
Figure 2. (a) A 0-framed 2-handle S2 × B2 (left) and a
dotted 1-handle S1 × B3 (right) are diffeomorphic at their
boundary ∂ = S2 × S1 , (b) Two equivalent pictures of the
Akbulut cork W .
2.3. Exotic manifold R4. An exotic R4 is a differentiable manifold that is
homeomorphic but not diffeomorphic to the Euclidean space R4. An exotic
R4 is called small if it can be smoothly embedded as an open subset of the
standard R4 and is called large otherwise. Here we are concerned with an
example of a small exotic R4. Let us quote Theorem 1 of [4].
There is a smooth contractible 4-manifold V with ∂V = ∂W , such that V
is homeomorphic but not diffeomorphic to W relative to the boundary.
Sketch of proof [4]:
Let α be a loop in ∂W as in Fig. 3a. α is not slice in W (does not bound
an imbedded smooth B2 in W ) but φ(α) is slice. Then φ does not extend
to a self-diffeomorphism φ : W →W .
It is time to recall that a cobordism between two oriented m-manifolds
M and N is any oriented (m + 1)-manifold W0 such that the boundary
is ∂W0 =M̄ ∪ N , where M appears with the reverse orientation. The
cobordism M × [0, 1] is called the trivial cobordism. Next, a cobordism W0
QUANTUM COMPUTATION AND MEASUREMENTS FROM AN EXOTIC SPACE-TIME R45
Figure 3. (a) The loop α is not slice on the Akbulut cork,
(b) the non-trivial h-cobordism between small exotic mani-
folds V and W , (c) the mediating 4-manifoldW̄ .
between M and N is called an h-cobordism if W0 is homotopically like the
trivial cobordism. The h-cobordism due to S. Smale in 1960, states that if
Mm and Nm are compact simply-connected oriented M -manifolds that are
h-cobordant through the simply-connected (m + 1)-manifold Wm+1
0
, then
M and N are diffeomorphic [3, p. 29]. But this theorem fails in dimension
4. If M and N are cobordant 4-manifolds, then N can be obtained from M
by cutting out a compact contractible submanifold W and gluing it back in
by using an involution of ∂W . The 4-manifold W is a ‘fake’ version of the
4-ball B4 called an Akbulut cork [3, Fig. 2.23].
The h-cobordism under question in our example may be described by at-
taching an algebraic cancelling pair of 2- and 3-handles to the interior of Ak-
bulut corkW as pictured in Fig. 3b (see [4, p. 343]). The 4-manifoldW̄ me-
diating V and W is as shown in Fig. 3c [alias the 0-surgery L7a6(0, 1)(0, 1)]
(see [4, p. 355]).
Following [7], the result is relative since V
itself is diffeomorphic to W
but such a diffeomorphism cannot extend to the identity map ∂V → ∂W
on the boundary. In [7], two exotic manifolds Q1 and Q2 are built that are
homeomorphic but not diffeomorphic to each other in their interior.
By the way, the exotic R4 manifolds Q1 and Q2 are related by a diffeo-
morphism Q1#S
2 × S2 ≈ Q ≈ Q2#S
2 × S2 (where # is the connected sum
6MICHEL PLANAT†, RAYMONDASCHHEIM‡, MARCELOM. AMARAL‡ANDKLEE IRWIN‡
between two manifolds) and Q is called the middle level between such con-
nected sums. This is shown in Fig. 4 for the two R4 manifolds Q1 and Q2
[7],[9, Fig. 2].
Figure 4. Exotic R4 manifolds Q1 shown in (a) and Q2
shown in (b). The connected sumsQ1#S
2×S2 and Q2#S
2×
S2 are diffeomorphic with middle level Q shown in (c).
3. Finite geometry of small exotic R4’s and quantum computing
From now, we focus on the relationship between the aforementioned small
exotic R4’s and a model of quantum computing and (generalized) quantum
measurements based on the so-called magic states [10]-[13]. This connection
is established by computing the finite index subgroup structure of the fun-
damental group for the boundary ∂W of Akbulut cork W , forW̄ in Fig. 3
and for Q in Fig. 4 and by making use of earlier results of the authors.
Our model of quantum computing is based on the concept of a magic state
- a state that has to be added to the eigenstates of the d-dimensional Pauli
group- in order to allow universal quantum computation. This was started
by Bravyi & Kitaev in 2005 [14] for qubits (d = 2). A subset of magic states
consists of states associated to minimal informationally complete measure-
ments, that we called MIC states [10], see the appendix 1 for a definition.
We require that magic states should be MIC states as well. For getting the
candidate MIC states, one uses the fact that a permutation may be realized
as a permutation matrix/gate and that mutually commuting matrices share
eigenstates. They are either of the stabilizer type (as elements of the Pauli
group) or of the magic type. One keeps magic states that are MIC states in
order to preserve a complete information during the computation and mea-
surements. The detailed process of quantum computation is not revealed at
this stage and will depend on the physical support for the states.
A further step in our quantum computing model was to introduce a 3-
dimensional manifold M3 whose fundamental group G = π1(M
3) would be
QUANTUM COMPUTATION AND MEASUREMENTS FROM AN EXOTIC SPACE-TIME R47
the source of MIC states [11, 12]. Recall that G is a free group with relations
and that a d-dimensional MIC state may be obtained from the permutation
group that organizes the cosets of an appropriate subgroup of index d ofG. It
was considered by us quite remarkable that two group geometrical axioms
very often govern the MIC states of interest [13], viz (i) the normal (or
conjugate) closure {g−1hg|g ∈ G and h ∈ H} of the appropriate subgroup
H of G equals G itself and (ii) there is no geometry (a triple of cosets do
not produce equal pairwise stabilizer subgroups). See [13, Sec. 1.1] for our
method of building a finite geometry from coset classes and Appendix 2 for
the special case of the Fano plane. But these rules had to be modified by
allowing either the simultaneous falsification of (i) and (ii) or by tolerating
a few exceptions. The latter item means a case of geometric contextuality,
the parallel to quantum contextuality [15].
In the present paper, we choose G as the fundamental group π1(M
4) of
a 4-manifold M4 that is the boundary ∂W of Akbulut cork W , or governs
the Akbulut h-cobordism. More precisely, one takes the manifold M4 asW̄
in Fig. 3 and Q in Fig. 4. Manifolds Q1 and Q2 are the small exotic R
4’s
of Ref. [7, Fig. 1 and 2]. There are homeomorphic but not diffeomorphic to
each other in their interiors. This choice has two important consequences.
First of all, one observes that axiom (i) is always satisfied and that (ii)
most often is true or geometric contextuality occurs with corresponding finite
geometries of great interest such as the Fano plane PG(2, 2) (at index 7),
the Mermin’s pentagram (at index 10), the finite projective space PG(3, 2)
or its subgeometry GQ(2, 2) -known to control 2-qubit commutation- [13,
Fig. 1] (at index 15), the Grassmannian Gr(2, 8) -containing Cayley-Dickson
algebras- (at index 28 ) and a few maximally multipartite graphs.
Second, this new frame provides a physical interpretation of quantum
computation and measurements as follows. Let us imagine that R4 is our
familiar space-time. Thus the ‘fake’ 4-ball W -the Akbulut cork- allows the
existence of smoothly embedded open subsets of space-time -the exotic R4
manifolds such as Q1 and Q2- that we interpret in this model as 4-manifolds
associated to quantum measurements.
3.1. The boundary ∂W of Akbulut cork. As announced earlier ∂W =
K(1, 1) ≡ Σ(2, 5, 7) is a Brieskorn sphere with fundamental group
π1(Σ(2, 5, 7)) =
〈
a, b|aBab2aBab3, a4bAb
〉
, where A = a−1, B = b−1.
The cardinality structure of subgroups of this fundamental group is found
to be the sequence
ηd[π1(Σ(2, 5, 7))] = [0, 0, 0, 0, 0, 0, 2, 1, 0, 3, 0, 0, 0,12,145, 178, 47, 0, 0,4, · · · ] .
All the subgroups H of the above list satisfy axiom (i).
Up to index 28, exceptions to axiom (ii) can be found at index d = 14, 16,
20 featuring the geometry of multipartite graphs K
(d/2)
2
with d/2 parties,
at index d = 15 and finally at index 28. Here and below the bold notation
features the existence of such exceptions.
8MICHEL PLANAT†, RAYMONDASCHHEIM‡, MARCELOM. AMARAL‡ANDKLEE IRWIN‡
Figure 5. (a) A picture of the smallest finite projective
space PG(3, 2). It is found at Frans Marcelis website [16].
The coset coordinates shown on the Fano plane with red bul-
lets of PG(3, 2) correspond the case of Table 2. (b) A picture
of the generalized quadrangle of order two GQ(2, 2) embed-
ded in PG(3, 2).
It may also be found at Frans Marcelis
website.
Apart from these exceptions, the permutation group organizing the cosets
is an alternating group Ad. The coset graph is the complete graph Kd on
d vertices. One cannot find a triple of cosets with strictly equal pairwise
stabilizer subgroups of Ad (no geometry), thus (ii) is satisfied.
At index 15, when (ii) is not satisfied, the permutation group organizing
the cosets is isomorphic to A7. The stabilized geometry is the finite projec-
tive space PG(3, 2) (with 15 points, 15 planes and 35 lines) as illustrated in
Fig. 5a 1. The geometry is contextual in the sense that all lines not going
through the identity element do not show mutually commuting cosets.
At index 28, when (ii) is not satisfied, there are two cases. In the first
case, the group P is of order 28 8! and the geometry is the multipartite graph
K
(7)
4
. In the second case, the permutation group is P = A8 and the geometry
is the configuration [286, 563] on 28 points and 56 lines of size 3. In [18], it
was shown that the geometry in question corresponds to the combinatorial
Grassmannian of type Gr(2, 8), alias the configuration obtained from the
points off the hyperbolic quadric Q+(5, 2) in the complex projective space
PG(5, 2).
Interestingly, Gr(2, 8) can be nested by gradual removal of a
so-called ‘Conwell heptad’ and be identified to the tail of the sequence of
1A different enlightning of the projective space PG(3, 2) appears in [17] with the con-
nection to the Kirkman schoolgirl problem.
QUANTUM COMPUTATION AND MEASUREMENTS FROM AN EXOTIC SPACE-TIME R49
Cayley-Dickson algebras [18, 22, Table 4]. Details for this geometry are
given in Appendix 3.
With the FreudenthalTits magic square, there exists a connection of such
a symmetry to octooctonions and the Lie group E8 [19, 20] that may be
useful for particle physics. Further, a Cayley-Dickson sequence is proposed
to connect quantum information and quantum gravity in [21].
The [286, 563] configuration. Below are given some hints about the con-
figuration that is stabilized at the index 28 subgroup H of the fundamental
group π1(∂W ) whose permutation group P organizing the cosets is isomor-
phic to A8. Recall that ∂W is the boundary of Akbulut cork W . The
28-letter permutation group P has two generators as follows
P = 〈28|g1, g2〉 with g1 = (2, 4, 8, 6, 3)(5, 10, 15, 13, 9)(11, 12, 18, 25, 17)
(14, 20, 19, 24, 21)(16, 22, 26, 28, 23), g2 = (1, 2, 5, 11, 6, 7, 3)(4, 8, 12, 19, 22, 14, 9)
(10, 16, 24, 27, 21, 26, 17)(13, 20, 18, 25, 28, 23, 15).
Using the method described in Appendix 2, one derives the configuration
[286, 563] on 28 points and 56 lines. As shown in [Table 4][18], the configu-
ration is isomorphic to the combinatorial Grassmannian Gr(2, 8) and nested
by a sequence of binomial configurations isomorphic to Gr(2, i), i ≤ 8, as-
sociated with Cayley-Dickson algebras. This statement is checked by listing
the 56 lines on the 28 points of the configuration as follows
{1, 7, 27},→Gr(2,3)
{1, 15, 23}, {15, 17, 27}, {7, 17, 23},→Gr(2,4)
{1, 5, 26}, {5, 18, 27}, {5, 15, 24}, {23, 24, 26}, {17, 18, 24}, {7, 18, 26},→Gr(2,5)
{12, 14, 17}, {1, 9, 22}, {5, 8, 9}, {9, 14, 15}, {7, 12, 22}, {8, 12, 18},
{8, 14, 24}, {8, 22, 26}, {14, 22, 23}, {9, 12, 27},→Gr(2,6)
{3, 10, 15}, {3, 6, 24}, {3, 17, 25}, {3, 23, 28}, {1, 10, 28}, {3, 14, 19}, {7, 25, 28}, {6, 8, 19},
{19, 22, 28}, {5, 6, 10}, {12, 19, 25}, {10, 25, 27}, {9, 10, 19}, {6, 18, 25}, {6, 26, 28},→Gr(2,7)
{4, 11, 12}, {11, 21, 25}, {6, 20, 21}, {2, 3, 21}, {2, 4, 14}, {7, 11, 16}, {2, 16, 23}, {1, 13, 16},
{2, 11, 17}, {4, 19, 21}, {16, 20, 26}, {2, 13, 15}, {11, 13, 27}, {16, 21, 28}, {2, 20, 24},
{5, 13, 20}, {11, 18, 20}, {4, 9, 13}, {4, 8, 20}, {4, 16, 22}, {10, 13, 21}→ Gr(2,8).
More precisely, the distinguished configuration [215, 353] isomorphic to
Gr(2, 7) in the list above is stabilized thanks to the subgroup of P iso-
morphic to A7. The distinguished Cayley-Salmon configuration [153, 203]
isomorphic to Gr(2, 6) in the list is obtained thanks to one of the two sub-
groups of P isomorphic to A6. The upper stages of the list correspond to
a Desargues configuration [103, 103], to a Pasch configuration [62, 43] and
to a single line[31, 13] and are isomorphic to the Grassmannians Gr(2, 5),
Gr(2, 4) and Gr(2, 3), respectively. The Cayley-Salmon configuration con-
figuration is shown on Fig. 6, see also [22, Fig. 12]. For the embedding of
Cayley-Salmon configuration into [215, 353] configuration, see [22, Fig. 18].
Frank Marcelis provides a parametrization of the Cayley-Salmon config-
uration in terms of 3-qubit operators [16].
10MICHEL PLANAT†, RAYMONDASCHHEIM‡, MARCELOM. AMARAL‡ANDKLEE IRWIN‡
Figure 6. The Cayley-Salmon configuration built around
the Desargues configuration (itself built around the Pasch
configuration) as in [22, Fig. 12].
Not surprisingly, geometric contextuality (in the coset coordinatization
not given here) is a common feature of all lines except for the ones going
through the identity element.
3.2. The manifoldW̄ mediating the Akbulut cobordism between
exotic manifolds V and W . The fundamental group π1(W̄ ) of the h-
cobordismW̄ is as follows
π1(W̄ ) =
〈
a, b|a3b2AB3Ab2, (ab)2aB2Ab2AB2
〉
.
The cardinality structure of subgroups of this fundamental group is
ηd[π1(W̄ )] = [0, 0, 0, 0, 1, 1, 2, 0, 0,1, 0,5, 4,9,7, 1 · · · ]
As for the previous subsection, all the subgroups H of the above list
satisfy the axiom (i). When axiom (ii) is not satisfied, the geometry is
contextual. All the cases are listed in Table 1. Column 4 and 5 give details
about the existence of a MIC at the corresponding dimension when this can
be checked, i.e. is the cardinality of P is low enough.
A picture of (contextual) Mermin pentagram stabilized at index 10 is
given in Fig. 7.
QUANTUM COMPUTATION AND MEASUREMENTS FROM AN EXOTIC SPACE-TIME R4
11
Figure 7. Mermin pentagram as the contextual geometry
occuring for the index 10 subgroup in the fundamental group
π1(W̄ ) of the h-cobordismW̄ between exotic manifolds V
and W .
3.3. The middle level Q between the diffeomorphic connected sums.
The fundamental group for the middle level Q between diffeomorphic con-
nected sums Q1#S
2 × S2 and Q2#S
2 × S2 is as follows
π1(Q) =
〈
a, b|a2bA2ba2BAB, a2BA3Ba2b3
〉
.
The cardinality structure of subgroups of this fundamental group is
ηd[π1(Q)] = [0, 0, 0, 0, 0, 2,2, 1, 0, 1, 0, 0, 0,2,2, 3 · · · ]
As for the previous subsection, all the subgroupsH of the above list satisfy
the axiom (i). When axiom (ii) is not satisfied, the geometry is contextual.
All the cases are listed in Table 2. The coset coordinates of the (contextual)
Fano plane are shown on Fig. 5a (see also Fig. 8 in appendix 2). Column
4 and 5 give details about the existence of a MIC at the corresponding
dimension when this can be checked, i.e. if the cardinality of P is low enough.
Notice that at index 15, both the the finite projective space PG(3, 2) and
the embedded generalized quadrangle of order two GQ(2, 2) (shown in Fig.
5) are stabilized. Both geometries are contextual. The latter case is pictured
in [13, Fig. 1] with a double coordinatization: the cosets or the two-qubit
operators.
12MICHEL PLANAT†, RAYMONDASCHHEIM‡, MARCELOM. AMARAL‡ANDKLEE IRWIN‡
d P
geometry
MIC fiducial
pp
5 A5
K5
(0, 1,−1,−1, 1)
1
6 A6
K6
(1, ω6 − 1, 0, 0,−ω6, 0)
2
7 A7
K7
10 A5
Mermin pentagram no
12 A12
K12
A5
K(2,2,2,2,2,2)
no
13 A13
K13
14 G1092, A14
K14
26 ⋊A5
K(2,2,2,2,2,2,2)
15 A15
K15
A5
K(3,3,3,3,3)
yes
3
A7
PG(3,2)
16 A16
K16
Table 1. Geometric structure of subgroups of the funda-
mental group π1(W̄ ) for the h-cobordismW̄ between exotic
manifolds V and W . Bold characters are for the contextual
geometries. G1092 is the simple group of order 1092. When
a MIC state can be found, details are given at column 4, the
number pp of distinct values of pairwise products in the MIC
is at column 5.
d P
geometry
MIC fiducial
pp
6 A6
K6
(1, ω6 − 1, 0, 0,−ω6, 0)
2
7 PSL(2, 7)
Fano plane
(1, 1, 0,−1, 0,−1, 0)
2
8 PSL(2, 7)
K8
no
10 A6
K10
yes
5
14 PSL(2, 7) K(2,2,2,2,2,2,2)
no
15 A6
PG(3,2), GQ(2,2) yes
4
16 A16
K16
no
SL(2, 7)
K(2,2,2,2,2,2,2,2)
no
Table 2. Geometric structure of subgroups of the fun-
damental group π1(Q) for the middle level Q of Akbu-
lut’s h-cobordism between connected sums Q1#S
2 × S2 and
Q2#S
2 × S2. Bold characters are for the contextual geome-
tries. When a MIC state can be found, details are given at
column 4, the number pp of distinct values of pairwise prod-
ucts in the MIC is at column 5. GQ(2, 2) is the generalized
quadrangle of order two embedded in PG(3,2) as shown in
Fig. 5b.
These results about the h-cobordism of R4 exotic manifolds and the rela-
tion to quantum computing, as developed in our model, are encouraging us
to think about a physical implementation. But this left open in this paper.
QUANTUM COMPUTATION AND MEASUREMENTS FROM AN EXOTIC SPACE-TIME R4
13
4. Conclusion
As recalled in Sec. 2.3, the h-cobordism theorem in four dimensions is
true topologically - there is an homotopy equivalence of the inclusion maps
of 4-dimensional manifolds M and N into the 5-dimensional cobordism -
but is false piecewise linearly and smoothly - a piecewise linear structure
or atlas cannot be smoothed in an unique way. These statements had to
await topologists Michael Friedmann and Simon Donaldson in twentieth
century to be rigorously established [1]-[3]. Handle decomposition in a way
described in Sec. 2, handle trading, handle cancellation and handle sliding
are the main techniques for establishing the failure of the 4-dimensional h-
cobordism theorem and the possible existence of infinitely many (exotic) R4
manifolds that are homeomorphic but not diffeomorphic to the Euclidean
R
4. It has been emphasized that the non-trivial h-cobordism between exotic
R4 manifolds may occur through a ‘pasting’ ingredient called an Akbulut
cork W whose 3-dimensional boundary ∂W is described explicitely as the
Brieskorn sphere Σ(2, 5, 7).
We did use of the fundamental group π1(∂W ) as a witness of symmetries
encoded into the 4-manifolds under investigation. Interpreting R4 as the
ordinary space-time, the symmetries have potential applications to space-
time physics (e.g. cosmology) and quantum mechanics (e.g. particle physics
and quantum computation). In the past, the Brieskorn sphere Σ(2, 3, 5),
alias the Poincaré dodecahedral space, was proposed as the ‘shape’ of space
because such a 3-manifold -the boundary of the 4-manifold E8- explains
some anomalies in the cosmic microwave background [23],[24].
It would
be interesting to use Σ(2, 5, 7) as an alternative model in the line of work
performed in [25],[26] and related work.
We found in Sec. 3 subgroups of index 15 and 28 of π1(∂W ) connect-
ing to the 3-dimensional projective space PG(3, 2) and to the Grassman-
nian Gr(2, 8), respectively. The former finite geometry has relevance to the
two-qubit model of quantum computing and the latter geometry connect
to Cayley-Dickson algebras already used for particle physics. In our previ-
ous work, we found that MICs (minimal informationally complete POVMs)
built from finite index subgroups of the fundamental group π1(M
3) of a
3-manifold M3 serve as models of universal quantum computing (UQC). A
MIC-UQC model based on a finite index subgroup of the fundamental group
π1(M
4) of an exotic M4 has interest for interpreting (generalized) quantum
measurements as corresponding to the non-diffeomorphic branches (to the
exotic copies) of M4 or its submanifolds.
To conclude, we are confident that an exotic 4-manifold approach of the-
oretical physics allows to establish bridges between space-time physics and
quantum physics as an alternative or as a complement to string theory and
loop quantum gravity.
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QUANTUM COMPUTATION AND MEASUREMENTS FROM AN EXOTIC SPACE-TIME R4
15
Appendix 1
A POVM is a collection of positive semi-definite operators {E1, . . . , Em}
that sum to the identity. In the measurement of a state ρ, the i-th outcome
is obtained with a probability given by the Born rule p(i) = tr(ρEi). For a
minimal and informationally complete POVM (or MIC), one needs d2 one-
dimensional projectors Πi = |ψi〉 〈ψi|, with Πi = dEi, such that the rank of
the Gram matrix with elements tr(ΠiΠj), is precisely d
2.
A SIC (a symmetric informationally complete POVM) obeys the remark-
able relation
|〈ψi|ψj〉|
2 = tr(ΠiΠj) =
dδij + 1
d+ 1
,
that allows the recovery of the density matrix as
ρ =
d2
∑
i=1
[
(d+ 1)p(i)−
1
d
]
Πi.
This type of quantum tomography is often known as quantum-Bayesian,
where the p(i)’s represent agent’s Bayesian degrees of belief, because the
measurement depends on the filtering of ρ by the selected SIC (for an un-
known classical signal, this looks similar to the frequency spectrum) [27].
In [10], new MICs are derived with Hermitian angles |〈ψi|ψj〉|i
6=j ∈ A =
{a1, . . . , al}, a discrete set of values of small cardinality l. A SIC is equian-
gular with |A| = 1 and a1 =
1
√
d+1
.
Appendix 2
Let us summarize how a finite geometry is build from coset classes [13, 15].
Let H be a subgroup of index d of a free group G with generators and
relations. A coset table over the subgroup H is built by means of a Coxeter-
Todd algorithm. Given the coset table, on builds a permutation group P
that is the image of G given by its action on the cosets of H.
First one asks that the d-letter group P acts faithfully and transitively
on the set Ω = {1, 2, · · · , d}. The action of P on a pair of distinct elements
of Ω is defined as (α, β)p = (αp, βp), p ∈ P , α
6= β. The number of orbits on
the product set Ω × Ω is called the rank r of P on Ω. Such a rank of P is
at least two, and it also known that two-transitive groups may be identified
to rank two permutation groups.
One selects a pair (α, β) ∈ Ω×Ω, α
6= β and one introduces the two-point
stabilizer subgroup P(α,β) = {p ∈ P |(α, β)
p = (α, β)}. There are 1 < m ≤ r
such non-isomorphic (two-point stabilizer) subgroups of P . Selecting one
of them with α
6= β, one defines a point/line incidence geometry G whose
points are the elements of the set Ω and whose lines are defined by the
subsets of Ω that share the same two-point stabilizer subgroup. Two lines
of G are distinguished by their (isomorphic) stabilizers acting on distinct
subsets of Ω. A non-trivial geometry is obtained from P as soon as the rank
of the representation P of P is r > 2, and at the same time, the number of
non isomorphic two-point stabilizers of P is m > 2. Further, G is said to be
16MICHEL PLANAT†, RAYMONDASCHHEIM‡, MARCELOM. AMARAL‡ANDKLEE IRWIN‡
Figure 8. The Fano plane as the geometry of the
subgroup of index 7 of fundamental group π1(W̄ ) for
the manifoldW̄ mediating the Akbulut cobordism be-
tween exotic manifold V and W. The cosets of π1
are organized through the permutation group P =
〈7|(1, 2, 4, 5, 6, 7, 3), (2, 5, 6)(3, 7, 4)〉. The cosets are labelled
as [1, . . . , 7] =
[
e, a, a−1, a2, ab, ab−1, a−1b
]
. The two-point
stabilizer subgroups of P for each line are distinct (act-
ing on different subsets) but isomorphic to each other and
to the Klein group Z2 × Z2. They are as follows s1 =
〈(2, 7)(4, 5), (2, 4)(5, 7)〉, s2 = 〈(1, 6)(5, 7), (1, 7)(5, 6)〉, s3 =
〈(1, 5)(2, 3), (1, 2)(3, 5)〉, s4 = 〈(3, 5)(4, 6), (3, 6)(4, 5)〉, s5 =
〈(2, 6)(3, 7), (2, 7)(3, 6)〉, s6 = 〈(1, 7)(3, 4), (1, 4)(3, 7)〉, s7 =
〈(1, 2)(4, 6), (1, 6)(2, 4)〉.
contextual (shows geometrical contextuality) if at least one of its lines/edges
is such that a set/pair of vertices is encoded by non-commuting cosets [15].
One of the simplest geometries obtained from coset classes is that of the
Fano plane shown in Fig. 8. See also Figs 5 to 7 in this paper.
† Université de Bourgogne/Franche-Comté, Institut FEMTO-ST CNRS UMR
6174, 15 B Avenue des Montboucons, F-25044 Besançon, France.
E-mail address: michel.planat@femto-st.fr
‡ Quantum Gravity Research, Los Angeles, CA 90290, USA
E-mail address: raymond@QuantumGravityResearch.org
E-mail address: Klee@quantumgravityresearch.org
E-mail address: Marcelo@quantumgravityresearch.org