We show that qubit and chirotope concepts are closely related. In fact, we
prove that the qubit concept leads to a generalization of the chirotope
concept, which we call qubitope. Moreover, we argue that a possible qubitope
theory may suggest interesting applications of oriented matroid theory in at
least three physical contexts, in which qubits make their appearance, namely
string theory, black holes and quantum information.Comment: 11 pages, Late

Björner

Borsten

Bossard

Castellani

Cayley

Duff

Goto

J.A. Nieto

Kallosh

Kung

Levay

Luque

Lévay

Nieto

Nishino

Schild

Whitney

Physics Letters B

English

arXiv

AbstractWe show that qubit and chirotope concepts are closely related. In fact, we prove that the qubit concept leads to a generalization of the chirotope concept, which we call qubitope. Moreover, we argue that a possible qubitope theory may suggest interesting applications of oriented matroid theory in at least three physical contexts, in which qubits make their appearance, namely string theory, black holes and quantum information

Nieto, J.A.

Elsevier - Publisher Connector 

Qubits and chirotopes 

Physics Letters B 692 (2010) 43–46Contents lists available at ScienceDirect
Physics Letters B
www.elsevier.com/locate/physletb
Qubits and chirotopes
J.A. Nieto a,b,c,∗
a Mathematical, Computational & Modeling Science Center, Arizona State University, PO Box 871904, Tempe, AZ 85287, USA
b Facultad de Ciencias Físico-Matemáticas de la Universidad Autónoma de Sinaloa, 80010, Culiacán Sinaloa, Mexico
c Departamento de Investigación en Física de la Universidad de Sonora, 83000, Hermosillo Sonora, Mexico
a r t i c l e i n f o a b s t r a c t
Article history:
Received 6 May 2010
Received in revised form 23 June 2010
Accepted 28 June 2010
Available online 13 July 2010
Editor: M. Cveticˇ
Keywords:
Qubits
Oriented matroid theory
2+ 2 dimensions
We show that qubit and chirotope concepts are closely related. In fact, we prove that the qubit concept
leads to a generalization of the chirotope concept, which we call qubitope. Moreover, we argue that
a possible qubitope theory may suggest interesting applications of oriented matroid theory in at least
three physical contexts, in which qubits make their appearance, namely string theory, black holes and
quantum information.
© 2010 Elsevier B.V.Open access under CC BY license. Recently, in a number of remarkable developments [1–8], a
relation between, apparently two different scenarios, black holes
and quantum information, has been established. The key concept
for this link has been the so-called quantum bit notion, or qubit,
which is the smallest unit of quantum information. In appropri-
ate qubit basis, the components of a pure state |ψ〉 can be written
in terms hypermatrix aa1a2...aN which in turn leads to a density
matrix ρ . It turns out that ρ can be deﬁned in terms of the hy-
perdeterminant associated with aa1a2...aN , a quantity introduced for
the ﬁrst time by Cayley in 1845 [9]. Surprisingly, in some cases the
quantity aa1a2...aN can also be related to the entropy of STU black
holes via also its hyperdeterminant (see Ref. [4] for details).
On the other hand, it is known that the chirotope concept plays
a fundamental role in oriented matroid theory [10]. In fact, the
emergence of this concept can be traced back to the origin of
matroids [11] which can be understood as a generalization of ma-
trices. From a modern perspective, however, one may introduce the
mathematical notion of chirotope, or oriented matroid, by consid-
ering a generalization of the Grassmann–Plücker relations of ordi-
nary determinants [12].
Thus, we have two generalizations of the matrix notion, namely
hypermatrix and matroid. Since a qubit is related with hyperdeter-
minants of hypermatrices and a chirotope is connected with a gen-
eralization of ordinary determinants via the Grassmann–Plücker
relations one may wonder whether these two qubit-chirotope con-
* Address for correspondence: Facultad de Ciencias Físico-Matemáticas de la Uni-
versidad Autónoma de Sinaloa, 80010, Culiacán Sinaloa, Mexico.
E-mail addresses: nieto@uas.uasnet.mx, janieto1@asu.edu.0370-2693© 2010 Elsevier B.V.
doi:10.1016/j.physletb.2010.07.010
Open access under CC BY license. cepts are related. If we achieve such a relation then one may be in
a position to bring a variety of mathematical tools from oriented
matroid theory to black-hole physics and vice versa.
Our starting point is to consider a possible scenario in which
the qubit concept makes its appearance [1], namely the (2 + 2)-
signature ﬂat target “spacetime” of the Nambu–Goto action. Let us
ﬁrst observe that the line element,
ds2 = dxμ dxν ημν, (1)
of ﬂat space with (2+ 2)-signature, with ημν = diag(−1,−1,1,1),
may also be written as
ds2 = 1
2
dxab dxcd εacεbd, (2)
where the matrix coordinates xab are given by
xab =
(
x1 + x3 x2 + x4
x2 − x4 −x1 + x3
)
, (3)
and εab is the completely antisymmetric symbol with ε12 = 1.
Similarly, it is not diﬃcult to show [1] (see also Ref. [2]) that
the world sheet metric
γab = ∂axμ∂bxνημν (4)
can also be written as
γab = 12∂ax
cd∂bx
ef εceεdf . (5)
This expression motivates to write the determinant of γab ,
44 J.A. Nieto / Physics Letters B 692 (2010) 43–46detγ = 1
2
εabεcdγacγbd, (6)
in the form
detγ = 1
2
εabεcdεegε f hεruεsva
ef
a a
gh
c a
rs
b a
uv
d = Deta, (7)
with
acda ≡ ∂axcd. (8)
One recognizes in (7) the hyperdeterminant of the hypermatrix acda .
Thus, this proves that the Nambu–Goto action [13,14]
S = 1
2
∫
d2ξ
√
detγ , (9)
for a ﬂat target “spacetime” with (2 + 2)-signature can also be
written as [1]
S = 1
2
∫
d2ξ
√
Deta. (10)
We shall now show that the hyperdeterminant (7) can be linked
to the chirotope concept. For this purpose by using (4) we ﬁrst
write (6) in the alternative Schild-type [15] form
detγ = 1
2
σμνσαβημαηνβ, (11)
where
σμν = εabaμa aνb . (12)
Here, we have used the deﬁnition
aμa = ∂axμ. (13)
It turns out that the quantity χμν = signσμν is a chirotope of an
oriented matroid (see Refs. [16–18]). In fact, since σμν satisﬁes the
identity
σμ[νσαβ] ≡ 0, (14)
one can verify that χμν satisﬁes the Grassmann–Plücker relation
χμ[νχαβ] = 0, (15)
and therefore χμν is a realizable chirotope (see Ref. [10] and ref-
erences therein). Here, the bracket [ναβ] means completely anti-
symmetric.
Since the Grassmann–Plücker relation (15) holds, the ground
set
E = {1,2,3,4} (16)
and the alternating map
χμν → {−1,0,1} (17)
determine a 2-rank realizable oriented matroid M = (E,χμν). The
collection of bases for this oriented matroid is
B = {{1,2}, {1,3}, {1,4}, {2,3}, {2,4}, {3,4}}, (18)
which can be obtained by just given values to the indices μ and
ν in χμν . Actually, the pair (E,B) determines a 2-rank uniform
nonoriented ordinary matroid.
Using the deﬁnition
σ ef rs ≡ εabaefa arsb , (19)
one can show that the hyperdeterminant (7) can also be written
asdetγ = 1
2
σ ef rsσ ghuvεegε f hεruεsv = Deta. (20)
So, we have achieved our goal of writing the hyperdeterminant (7)
in terms of a “chirotope” structure (19). Our strategy was to trans-
late the “chirotope” given in (12) to the form (19). However, by
comparing (12) and (19) one ﬁnds that there are important dif-
ferences between these two expressions which suggest a possible
generalization of the chirotope concept. From (12) we obtain the
property
σμν = −σνμ, (21)
that is, σμν is completely antisymmetric (alternative) quantity,
while in (19) we have the weaker condition
σ ef rs = −σ rsef . (22)
This means that the quantity σ ef rs , which we shall call qubitope
(qubit-chirotope), is not completely antisymmetric but only alter-
native in pair of indices. Further, while in the case of (12) the
ground set E is given by (16), the expressions (19) and (20) sug-
gest to introduce the underlying ground bitset (from bit and set)
E = {1,2} (23)
and the pre-ground set
E0 =
{
(1,1), (1,2), (2,1), (2,2)
}
. (24)
So, our task is to ﬁnd the relation between E0 and E . By comparing
(16) and (24) one sees that by establishing the labels
(1,1) ↔ 1, (1,2) ↔ 2,
(2,1) ↔ 3, (2,2) ↔ 4, (25)
such a relation is achieved. This can be understood consider-
ing that (25) is equivalent to make the identiﬁcation of indices
{a,b} ↔ μ, . . . , etc. Observe that considering this identiﬁcation the
family of bases (18) becomes
B0 =
{{
(1,1), (1,2)
}
,
{
(1,1), (2,1)
}
,
{
(1,1), (2,2)
}
,{
(1,2), (2,1)
}
,
{
(1,2), (2,2)
}
,
{
(2,1), (2,2)
}}
. (26)
Thus, from the qubitope σ ef rs , we have discovered the underlying
structure Q = (E, E0, B0). By convenience we call this new struc-
ture Q a qubitoid. The word “qubitoid” is short for qubit-matroid.
Let us try to generalize the above scenario to higher dimen-
sions. First, we would like to extend the steps in the expressions
(1) and (2). If we consider the coordinates xabc instead of xab one
ﬁnds that the null line element
ds2 = 1
2
dxabc dxdef εadεbeεcf (27)
vanishes identically. This follows because dxabc dxdef εadεbe = scf
is a symmetric quantity, while εc f is antisymmetric. Similarly
one can verify that the hyperdeterminant of the hypermatrix
abcda ≡ ∂axbcd presents some diﬃculties due to the fact that the
analogue of (7) cannot be obtained. In fact, the quantity λab =
∂axef g∂bxhrsεehε f rεgs is antisymmetric rather than symmetric as
the metric γab and therefore in this case the steps (4) and (5)
cannot follow. Hence, from the Nambu–Goto action point of view
this case, which corresponds to (4 + 4)-signature, is not very in-
teresting, although in the Polyakov action context may still be
interesting. So, we jump to the next possibility, namely the line
element
ds2 = 1 dxabcr dxdef sεadεbeεc f εrs, (28)2
J.A. Nieto / Physics Letters B 692 (2010) 43–46 45which, one can show, implies a line element of the type (1), but
now associated with a ﬂat target (8+8)-signature “spacetime”. Ex-
plicitly, we have the relations
x1111 ↔ x1 + x9, x2222 ↔ −x1 + x9, x1112 ↔ x2 + x10,
x2221 ↔ x2 − x10, x1122 ↔ x3 + x11, x2211 ↔ −x3 + x11,
x1121 ↔ x4 + x12, x2212 ↔ x4 − x12, x1212 ↔ x5 + x13,
x2121 ↔ −x5 + x13, x1211 ↔ x6 + x14, x2122 ↔ x6 − x14,
x1221 ↔ x7 + x15, x2112 ↔ −x7 + x15, x1222 ↔ x8 + x16,
x2111 ↔ x8 − x16. (29)
In this case, the hyperdeterminant of the hypermatrix
abcdea = ∂axbcde (30)
is given by (see Eq. (32) of Ref. [19])
detγ = 1
2
εabεcdεe1 f1εe2 f2εe3 f3εe4 f4εg1h1εg2h2εg3h3εg4h4
× ae1e2e3e4a a f1 f2 f5 f4c ag1g2g3g4b ah1h2h3h4d
= Deta. (31)
Thus, by substituting (31) into (10) we ﬁnd a Nambu–Goto action
for a ﬂat target “spacetime” with (8+8)-signature written in terms
of the hyperdeterminant Deta.
The qubitoid now is determined by the underlying set
E = {1,2}, (32)
and the pre-ground set
E0 =
{
(1,1,1,1), (1,1,1,2), (1,1,2,1), (1,1,2,2),
(1,2,1,1), (1,2,1,2), (1,2,2,1), (1,2,2,2),
(2,1,1,1), (2,1,1,2), (2,1,2,1), (2,1,2,2),
(2,2,1,1), (2,2,1,2), (2,2,2,1), (2,2,2,2)
}
. (33)
It is not diﬃcult to see that by making the identiﬁcations
(1,1,1,1) ↔ 1, (1,1,1,2) ↔ 2, (1,1,2,1) ↔ 3,
(1,1,2,2) ↔ 4, (1,2,1,1) ↔ 5, (1,2,1,2) ↔ 6,
(1,2,2,1) ↔ 7, (1,2,2,2) ↔ 8, (2,1,1,1) ↔ 9,
(2,1,1,2) ↔ 10, (2,1,2,1) ↔ 11, (2,1,2,2) ↔ 12,
(2,2,1,1) ↔ 13, (2,2,1,2) ↔ 14, (2,2,2,1) ↔ 15,
(2,2,2,2) ↔ 16, (34)
one obtains a relation between the pre-ground set E0 given in (33)
and the ground set
E = {1,2, . . . ,15,16}. (35)
This can be understood by considering that (34) is equivalent to
make the identiﬁcation of indices (a,b, c,d) ↔ μ, . . . , etc. It turns
out that considering these relations one ﬁnds that the collection
of bases B contains
(
16
2
)
= 120 two-element subsets of the 16-
element set E , given in (35). This 2-element subset can be ob-
tained by considering a lexicographic order of all 120 two-subsets
of {1,2, . . . ,15,16}. For instance, the ﬁrst 35 two-subsets of B are{1,2}, {1,3}, {1,4}, {1,5}, {1,6}, {1,7},
{1,8}, {1,9}, {1,10}, {1,11}, {1,12},
{1,13}, {1,14}, {1,15}, {1,16}, {2,3},
{2,4}, {2,5}, {2,6}, {2,7}, {2,8}, {2,9},
{2,10}, {2,11}, {2,12}, {2,13}, {2,14},
{2,15}, {2,16}, {3,4}, {3,5}, {3,6},
{3,7}, {3,8}, {3,9}, . . . . (36)
The sequence follows until the last term {15,16}. By using (34)
one ﬁnds that the ﬁrst terms of B0 look like
B0 =
{{
(1,1,1,1), (1,1,1,2)
}
,
{
(1,1,1,1), (1,1,2,1)
}
,{
(1,1,1,1), (1,1,2,2)
}
,
{
(1,1,1,1), (1,2,1,1)
}
,{
(1,1,1,1), (1,2,1,2)
}
, . . .
}
. (37)
Thus, associated with the quantity abcdea we have again a qubitoid
structure of the form Q = (E, E0, B0) which corresponds to a
ﬂat target “spacetime” of (8 + 8)-dimensions. The corresponding
qubitope is given by
σ a1a2a3a4a5a6a7a8 = 1
2
εbcaa1a2a3a4b a
a5a6a7a8
c . (38)
It is worth mentioning that while in (2+ 2)-dimensions the quan-
tity Deta is invariant under SL(2, R)3 in the case of (8+ 8)-dimen-
sions Deta must be invariant under SL(2, R)5.
The method, of course, can be extended to ( 2
2n+2
2 + 2
2n+2
2 )-
signature, n = 0,1,2, . . . , etc. For the cases of ( 22n+12 + 2
2n+1
2 )-
signature the corresponding line element vanishes identically.
It remains to explore whether the present qubitoid and qubito-
pe formalism will allow us a deeper understanding of other two
scenarios, namely black holes and quantum information. In the
ﬁrst case, as in (2+ 2)-dimensions, one may think in a black hole
with 2
2n+2
2 -electric and
22n+2
2 -magnetic charges with entropy
S = π√Deta. (39)
While in the second case, one may introduce pure states |ψ〉 as-
sociated with the 2
2n+2
2 -qubitoid system. For instance in the case
of (8 + 8)-dimensions the pure states |ψ〉 must be given by (see
Refs. [19] and [20])
|ψ〉 =
∑
a1a2a3a4a5
aa2a3a4a5a1 |a1a2a3a4a5〉. (40)
It is worth mentioning that the complete classiﬁcation of N-
qubit systems is a diﬃcult, or perhaps an impossible, task. In
Ref. [19] an interesting development for characterizing a subclass
of N-qubit entanglement has been considered. An attractive aspect
of this construction is that the N-qubit entanglement can be un-
derstood in geometric terms. The idea is based on the bipartite
partitions of the Hilbert space in the form C2
N = C L ⊗ Cl , with
L = 2N−n and l = 2n . Such a partition allows a geometric inter-
pretation in terms of the complex Grassmannian variety Gr(L, l) of
l-planes in C L via the Plücker embedding. In this case, the Plücker
coordinates of the Grassmannians are natural invariants of the the-
ory. In this scenario the 5-qubit given in (40) admits a geomet-
ric interpretation in terms of the complex Grassmannian Gr(8,4).
Considering such an interpretation it has been proved that the ex-
pression (31) is a hyperdeterminant associated with the Plücker
coordinates of the Grassmannian Gr(8,4) (see Eq. (32) of Ref. [19]
and Ref. [20] for 5-qubit discussion).
46 J.A. Nieto / Physics Letters B 692 (2010) 43–46Furthermore, it is interesting that the line element (28) also
appears on several physical contexts. In particular, extremal black
hole solutions in the STU model of D = 4, N = 2 supergravity ad-
mit a description in terms of 4-qubit systems [21,22] (for a 4-qubit
entanglement see [23] and references therein). In this case, the line
element corresponds to the moduli space M4 = [U (1)\SL(2; R)]3
rather than to the “spacetime”. Upon dimensional reduction M4
becomes M3 = [SO(4)]2\SO(4,4) or M∗3 = [SO(2,2)]2\SO(4,4) de-
pending whether the truncation is along a space-like or time-like
direction, respectively. Among other things, the relevance of this
construction in our approach is that the signature of the metric
M∗3 is also of the type (8 + 8) (see Refs. [21] and [22] for de-
tails).
It is remarkable that the Nambu–Goto action in a ﬂat target
“spacetime” with ( 2
2n+2
2 + 2
2n+2
2 )-signature emerges as the under-
lying motivation for studying the new mathematical structures of
qubitoids Q = (E, E0, B0) and the corresponding qubitopes.
Acknowledgements
I would like to thank M.C. Marín and A. León for helpful com-
ments and the Mathematical, Computational & Modeling Science
Center of the Arizona State University where part of this work was
developed.References
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