1 Introduction
Quantum contextuality (see, e.g., Budroni et al. (Reference Budroni, Cabello, Gühne, Kleinmann and Larsson2022) for a recent comprehensive review) is an important property of quantum mechanics saying that measurements of quantum observables cannot be regarded as revealing certain preexisting values; slightly rephrased, the result of a measurement depends on the context to which it is associated with. One of the simplest proofs of contextuality of quantum mechanics is the so-called Mermin–Peres square (Mermin, Reference Mermin1993; Peres, Reference Peres1990), which demonstrates that already in the Hilbert space of two qubits it is not possible to consistently assign certain preexisting values to all observables. Another well-known (observable-based) contextuality proof is furnished by a Mermin pentagram (Mermin, Reference Mermin1993) living in the three-qubit Hilbert space. It is interesting to realize that both the Mermin–Peres square and the Mermin pentagram are related to distinguished subgeometries of the symplectic polar spaces
$\mathcal{W}(2N-1,2)$
associated with corresponding generalized N-qubit Pauli groups: the former being isomorphic to a geometric hyperplane of
$\mathcal{W}(3,2)$
(Saniga et al., Reference Saniga, Planat, Pracna and Havlicek2007) and the latter living in a doily-related pentad of Fano planes of
$\mathcal{W}(5,2)$
(Lévay and Szabó, Reference Lévay and Szabó2017; Saniga, Reference Saniga2021). These two and several other (e.g., Planat et al. (Reference Planat, Saniga and Holweck2013); Saniga and Lévay (Reference Saniga and Lévay2012)) geometrical observations have recently prompted us (de Boutray et al., Reference de Boutray, Holweck, Giorgetti, Masson and Saniga2022) to have a more systematic look at such geometrically backed quantum contextual configurations (i.e., sets of contexts) of
$\mathcal{W}(2N-1,2)$
for some small values of N. This paper can be regarded as an organic continuation of Boutray et al. (2022) with substantially improved algorithms and a more efficient code to address contextuality issues not only for all the types of geometric hyperplanes but also for a great number of subspaces of varying dimensions found in multi-qubit symplectic spaces of rank up to 7.
The paper is organized as follows. Section 2 summarizes the most essential properties of binary symplectic polar spaces and their relation with the generalized N-qubit Pauli groups and then recalls the definition of the degree of contextuality for a quantum configuration. Our contributions begin with Section 3 that presents algorithms and programs that we created to generate quantum configurations and reveal their contextuality or evaluate their degree of contextuality. In particular, we describe in detail two new algorithms to generate totally isotropic subspaces and to compute the degree of contextuality using a SAT solver. The main computational outcomes of our work are gathered in Section 4. Section 5 then establishes or discusses some general facts based on our findings. Here, we notably prove that there are no negative subspaces of dimension 3 and higher, introduce a new doily-based contextual configuration – the so-called two-spread – and compute its contextuality degree and, last but not least, we show that the contextuality degree of the configuration comprising all 315 three-qubit lines is 63, which is much lower than the previously known value. Section 6 concludes the paper with some open questions and an outline of prospective tasks.
2. Background
In this section, we collect all the necessary concepts and introduce the symbols and notation to the extent to make the paper as self-contained as possible so that the reader should be able to follow the main line of our reasoning without the urgent need to consult relevant references (like (Saniga et al., Reference Saniga, de Boutray, Holweck and Giorgetti2021, Section 2) and (Muller et al., Reference Muller, Saniga, Giorgetti, De Boutray and Holweck2022, Introduction)). For all integers
$0 \leq k \leq 2N$
, the 2N-dimensional vector space
$\mathbb{F}_2^{2N}$
over the 2-element field
$\mathbb{F}_2=(\{0,1\},+,\times)$
has vector subspaces of dimension k. Among them, the ones that are totally isotropic, without their 0, form the symplectic space
$\mathcal{W}(2N-1,2)$
, whose name is shortened as
$W_N$
. A subspace is totally isotropic if any two vectors x and y in it are mutually orthogonal (
$(\langle{x}|{y}\rangle = 0)$
), for the symplectic form
$\langle{.}|{.}\rangle$
defined by:
\begin{equation}\langle{x}|{y}\rangle = x_1y_{N+1} + x_{N+1}y_1 + x_2y_{N+2} + x_{N+2}y_2 + \dots + x_{N} y_{2N} + x_{2N} y_{N}.\end{equation}
In other words, a (totally isotropic) subspace of
$W_N$
of (projective) dimension
$1 \leq k \leq N-1$
is a totally isotropic vector subspace of
$\mathbb{F}_2^{2N}$
of dimension
$k+1$
without its zero vector. A point of
$W_N$
is the unique element of a (totally isotropic) subspace of
$W_N$
of (projective) dimension 0. A line of
$W_N$
is a (totally isotropic) subspace of
$W_N$
of (projective) dimension 1. A generator of
$W_N$
is a (totally isotropic) subspace of
$W_N$
of (projective) dimension
$N-1$
.
It is known that for a symplectic polar space
$\mathcal{W}(2N-1,q)$
embedded in a projective space PG
$(2N-1,q)$
, the number of its k-dimensional subspaces is given by (see, e.g., (De Boeck et al., Reference De Boeck, Rodgers, Storme and Švob2019, Lemma 2.10)):
\begin{equation} \begin{bmatrix}{N}\\ {k+1}\end{bmatrix}_q \prod_{i=1}^{k+1} (q^{N+1-i} +1), \end{equation}
where
\begin{equation}\begin{bmatrix}{n}\\ {k}\end{bmatrix}_q = \prod_{i=1}^{k} \frac{q^{n-k+i} -1}{q^i -1} = \frac{(q^n-1) \dots (q^{n-k+1} -1)}{(q^k-1) \dots (q-1)},\end{equation}
is the Gaussian (binomial) coefficient.
Let
\begin{equation*}X = \left(\begin{array}{rr}0 & 1 \\1 & 0 \\\end{array}\right),~~Y = \left(\begin{array}{rr}0 & -\textrm{i} \\\textrm{i} & 0 \\\end{array}\right)~~{\rm and}~~Z = \left(\begin{array}{rr}1 & 0 \\0 & -1 \\\end{array}\right),\end{equation*}
be the Pauli matrices, I the identity matrix, “
$\otimes$
” the tensor product of matrices, and
$I^{\otimes N}\equiv I_{(1)}\otimes I_{(2)}\otimes\ldots\otimes I_{(N)}$
. From here on, all tensor products of observables
$G_1 \otimes G_2\otimes \cdots \otimes G_N$
are called N-qubit observables and denoted
$G_1 G_2 \cdots G_N$
, by omitting the symbol
$\otimes$
for the tensor product. Let “
$.$
” denote the matrix product and
$M^2$
denote
$M.M$
. It is easy to check that
$X^2 = Y^2 = Z^2 = I$
,
$X.Y = \textrm{i}Z = -Y.X$
,
$Y.Z = \textrm{i}X = -Z.Y$
, and
$Z.X = \textrm{i}Y = -X.Z$
. The N-qubit observables and the multiplicative factors
$\pm 1$
and
$\pm \textrm{i}$
, called phase, form the (generalized) (N-qubit) Pauli group
$\mathcal{P}^{\otimes N}= (\{1,-1,\textrm{i},-\textrm{i}\}\times\{I,X,Y,Z\}^{\otimes N},.)$
.
The N-qubit observable
$G_1 G_2 \cdots G_N$
, with
$G_j \in \{I, X, Y, Z \}$
for
$j \in \{1, 2, \ldots, N \}$
, is bijectively represented by the bitvector
$(g_1, g_2, \ldots, g_{2N})$
such that
$G_j \leftrightarrow (g_j, g_{j+N})$
for
$j \in \{1, 2,\dots,N\}$
, with the assumption that
$I\leftrightarrow(0,0)$
,
$X\leftrightarrow (0,1)$
,
$Y\leftrightarrow(1,1)$
and
$Z\leftrightarrow (1,0)$
.
The non-zero bitvectors are the points of
$W_N$
, and two observables commute if and only if their encodings are orthogonal. Therefore, we sometimes write that two points of
$W_N$
“commute.”
The perpset of the point
$p\in W_N$
is the point-line geometry whose points are all points q which commute with p (
$\langle{p}{q}\rangle=0$
) and whose lines are all the lines of
$W_N$
which contain p (and thus contain only points in the perpset).
A quantum configuration is a pair (O,C) where O is a finite set of observables and C is a finite set of subsets of O, called contexts, such that (i) each observable
$M \in O$
satisfies
$M^2=I^{\otimes N}$
(so, its eigenvalues are in
$\{-1,1\}$
); (ii) any two observables M and M’ in the same context commute, that is,
$M.M'=M'.M$
; (iii) the product of all observables in each context is either
$I^{\otimes N}$
(positive context) or
$-I^{\otimes N}$
(negative context). This sign is encoded by the context valuation
$e : C \rightarrow \{-1,1\}$
of (O,C), defined by
$e(c) = 1$
if the context c is positive, and
$e(c) = -1$
if it is negative.
The points and lines of
$W_2$
form a noticeable quantum configuration called the two-qubit doily. A N-qubit doily is a quantum configuration of
$W_N$
isomorphic to the two-qubit doily. A spread of generators is a set of pairwise disjoint generators that partition the set of points of
$W_N$
. A two-spread is a quantum configuration obtained by removing from a multi-qubit doily one spread of its lines, while keeping all its points.
2.1 Contextuality degree
Since the contextuality degree is the central subject of the present paper, this section recalls its definition and its connection with the phenomenon of quantum contextuality, both introduced in a former work (de Boutray et al., Reference de Boutray, Holweck, Giorgetti, Masson and Saniga2022).
Let (O,C) be a quantum configuration with
$p = |O|$
observables
$\{M_1,\ldots,M_p\}$
and
$l = |C|$
contexts
$\{c_1,\ldots,c_l\}$
. Its incidence matrix
$A \in \mathbb{F}_2^{l \times p}$
is defined by
$A_{i,j}= 1$
if the i-th context
$c_i$
contains the j-th observable
$M_j$
. Otherwise,
$A_{i,j} = 0$
. Its valuation vector
$E \in \mathbb{F}_2^{l}$
is defined by
$E_i = 0$
if
$e(c_i) = 1$
and
$E_i = 1$
if
$e(c_i) = -1$
, where e is the context valuation of (O,C).
With these notations, the quantum configuration (O,C) is contextual iff the linear system
\begin{equation}A x = E,\end{equation}
has no solution in
$\mathbb{F}_2^{p}$
. To be non-contextual means there is a solution
$x\in \mathbb{F}_2^l$
that satisfies (4). Such a solution x can be thought of as a set of predefined values for the observables of O that satisfies all constraints imposed by the contexts of the configuration. Note that the predefined values defined by x do not depend on the contexts. When such a solution x exists, one says that there exists a Non-Contextual Hidden Variables (NCHV) model that reproduces the outcomes of the configuration predicted by quantum mechanics (Budroni et al., Reference Budroni, Cabello, Gühne, Kleinmann and Larsson2022; de Boutray et al., Reference de Boutray, Holweck, Giorgetti, Masson and Saniga2022).
In this setting, one can measure how much contextual a given quantum configuration is. Let us denote by
$\textrm{Im}(A)$
the image of the matrix A as a linear map
$A:\mathbb{F}_2^{p}\to \mathbb{F}_2^l$
. Then, if (O,C) is contextual, necessarily
$E \notin \textrm{Im}(A)$
. A natural measure is the degree of contextuality (de Boutray et al., Reference de Boutray, Holweck, Giorgetti, Masson and Saniga2022), defined as follows:
Definition 1. (Contextuality degree). Let (O,C) be a contextual configuration with the valuation vector
$E\in\mathbb{F}_2^l$
. Let us denote by
$d_H$
the Hamming distance on the vector space
$\mathbb{F}_2^l$
. Then one defines the degree d of contextuality of (O,C) by:
\begin{equation} d=d_H(E,\textrm{Im}(A)). \end{equation}
The notion of degree of contextuality measures in some sense how far a given configuration is from being satisfied by an NCHV model. The Hamming distance tells us what is the minimal number of constraints on the contexts that one should change to make the configuration valid by a deterministic function. In other words,
$l-d$
measures the maximum number of constraints of the contextual configuration that can be satisfied by an NCHV model.
The degree of contextuality has also a concrete application as it can be used to calculate the upper bound for contextual inequalities. Let us consider a contextual configuration (O,C) and let us denote by
$C^+$
the subset of positive contexts,
$C^-$
the subset of negative ones and by
$\langle c\rangle$
the expectation for an experiment corresponding to the context c. The following inequality was established by Cabello (Reference Cabello2010):
\begin{equation} \sum_{c\in C^+} \langle c\rangle-\sum_{c \in C^-} \langle c\rangle \leq b. \end{equation}
Under the assumption of quantum mechanics, the upper bound b is the number of contexts of (O,C), that is,
$b=l$
. However, this upper bound is lower for contextual configurations under the hypothesis of an NCHV model. Indeed, as shown in Cabello (Reference Cabello2010), this bound is
$b=2s-l$
, where s is the maximum number of constraints of the configuration that can be satisfied by an NCHV model. It connects the notion of degree of contextuality with the upper bound b:
\begin{equation}b=l-2d.\end{equation}
3. Methodology
This section details the algorithms and programs that we propose to generate quantum configurations and evaluate their degree of contextuality. All our programs are written in C language because it is faster and more accessible than the Magma language of the previous implementation (de Boutray et al., Reference de Boutray, Holweck, Giorgetti, Masson and Saniga2022). The algorithm for generating perpsets and hyperbolic and elliptic quadrics is presented in de Boutray et al. (Reference de Boutray, Holweck, Giorgetti, Masson and Saniga2022), but we now implement it in C language. This algorithm consists of first generating all the lines and then selecting among them the ones belonging to a given perpset or quadric.
For the generation of totally isotropic subspaces, including lines and generators, we propose a more efficient algorithm than in de Boutray et al. (Reference de Boutray, Holweck, Giorgetti, Masson and Saniga2022), detailed in Section 3.1. The approach we propose to check contextuality (Section 3.3) and compute the contextuality degree (Section 3.2) is to use a SAT solver.
3.1 Generation of totally isotropic subspaces
Algorithm 1 presents the recursive function TotallyIsotropicSubspaces(N,k) that generates the configuration of the points of
$W_N$
whose contexts are all the totally isotropic subspaces of dimension k of
$W_N$
, for
$N \geq 2$
and
$1 \leq k \leq N-1$
. Thus, the function call TotallyIsotropicSubspaces
$(N,N-1)$
builds the generators of
$W_N$
, and the function call TotallyIsotropicSubspaces(N,1) builds the configuration of
$W_N$
whose contexts are the lines of
$W_N$
.
The parameters
$S = \{\}$
and
$l = 0$
with a default value are only useful for the recursive calls and do not have to be specified when calling the function. The default values mean that calling TotallyIsotropicSubspaces(N,k) automatically initializes S with the empty set and l with 0. The subsequent recursive calls will then assign different values to S and l.
Algorithm 1 Totally isotropic subspaces building algorithm.
The strict total order
$<$
on points used in this algorithm is the lexicographic order, when points are considered as bitvectors (of even length). The expression
$\textit{max}(S)$
represents the point p in S such that
$\forall q \in S, p \geq q$
. For the case of the empty set, it is assumed that
$p > \textit{max}(\{\})$
holds for all p in
$W_N$
. Let us denote by
$S+p$
the set
$\{q + p \mid q \in S\}$
for any point p and any set of points S.
The following definition and facts justify some expected properties of the algorithm.
The effective size t(p) of a point p is the size of the associated vector when we remove all the 0’s from the beginning. For instance,
$t((0,0,0,0,0,1,0,1)) = t((1,0,1)) = 3$
.
Let B be a basis of the subspace S (of dimension
$l-1$
) at the beginning of the function. When valid is true on Line 11, the variable C stores the subspace
$S \cup \{p\} \cup (S+p)$
, of dimension l. One of its bases is
$B \cup \{p\}$
. So, the algorithm is correct.
Conversely, let C be any totally isotropic subspace of dimension l. Let t be the maximal effective size of the elements of C. Let S be the subset of the elements of C whose effective size is strictly less than t. Let m be the minimal point of C whose effective size is t. Then, C will be built on Line 11 by the function call TotallyIsotropicSubspaces
$(N,k,S,l-1)$
, when the value of p on Line 4 is m. This remark justifies that the algorithm is complete and does not generate duplicates of the same subspace.
3.2 Computing the contextuality degree with a SAT solver
This section shows how the problem of finding the contextuality degree of a configuration can be reduced to a 3-SAT problem and thus take advantage of the numerous optimizations implemented in the current SAT solvers. More precisely, we limit the effort of reduction to 3-SAT by going through an intermediary translation into the rich input language of the tool bc2cnf (Junttila and Niemelä, Reference Junttila and Niemelä2000), which creates itself a file in DIMACS format (suffix .cnf), the default format for SAT solvers.
The linear system
$Ax = E$
is a set of equations in
$\mathbb{F}_2$
(see the example for a Mermin–Peres square in Figure 1a, where
$+$
is the “exclusive or” symbol for exclusive disjunction). Each equation corresponds to a context of the configuration, with the expected value on the right-hand side. Since there are only zeros and ones, we rephrase the problem of knowing whether a configuration is contextual as finding a set of Boolean variables
$v_i$
(
$1\leq i \leq 9$
in Figure 1a) such that the exclusive disjunction of the variables is 0 for each positive context and 1 for each negative context. We can then use any SAT solver to compute a degree of contextuality, by using a feature of bc2cnf which allows us to specify bounds for the number of constraints that have to be satisfied.
Figure 1. (a) Contextuality linear system for a Mermin–Peres square and (b) the corresponding bc2cnf file to decide whether 5 to 6 out of the 6 constraints can be solved (i.e., the contextuality degree is at most 1).
A bc2cnf input (see an example in Figure 1b) mainly consists of constraints. The exclusive disjunction is represented by ^, the true and false values, respectively, by T and F, and the equality of expressions by ==. After the prefix BC1.1 specifying the version, the ASSIGN clause encapsulates the constraints under the form [low,high](formula) where low and high, respectively, are the lower and upper bounds for the number of constraints to satisfy. After some experimentations, we found that kissat_gb (Chowdhury et al., Reference Chowdhury, Müller and You2021) was the fastest SAT solver for these kinds of systems.
Algorithm 2 Algorithm for calculating the degree of contextuality of a configuration C.
The Algorithm 2 computes the contextuality degree of any configuration. The variable C is the set of the constraints of a configuration, the integer i is the current maximal known number of satisfiable constraints, and sol stores a solution for the set of constraints. The
$sat\_sol$
function called on Line 4 is such that
$sat\_sol(i,j,C)$
retrieves a solution satisfying at least i and at most j of the constraints in the set C. The
$n\_match(sol,l)$
returns the number of constraints of sol that are satisfied by l.
Let
$|C^+|$
denote the number of positive constraints. We know that we can at least satisfy this number of constraints, by assigning the value 0 to all variables in C. For every iteration of the loop, we know that at least i constraints can be satisfied, so on Line 4 we ask the solver if at least one more constraint than i can be satisfied. If yes, we retrieve a solution sol of the SAT solver and then count the number of satisfied constraints, which can be more than
$i+1$
. This number is then used as the new limit i until no solution can be found, which means that
$|C|-i$
is the minimal number of unsatisfiable constraints in the system, that is, the contextuality degree of the corresponding configuration.
3.3 Checking contextuality with a SAT solver
When computing the contextuality degree is too time-consuming, we simply check contextuality. Checking contextuality of a quantum configuration is showing that a linear system
$A x = E$
has no solution in
$\mathbb{F}_2^{p}$
, for a matrix A of size
$l\times p$
with
$l\leq p$
. The complexity of linear system resolution (e.g., by Gaussian elimination) is polynomial, and it is exponential for SAT solving. So, we also implemented the contextuality test (i.e., whether a configuration is contextual or not) of subspaces by linear system resolution, with Gaussian elimination optimized for sparse matrices, as implemented in the function lu_solve of the SpaSM tool (Bouillaguet, Reference Bouillaguet2017), to compare its efficiency in practice with SAT solving with kissat_gb, when computing times are neither negligible nor too long for this comparison.
For subspaces of dimension
$k=2$
and
$N=5$
qubits, the computation time is 3 s for kissat_gb and 40 s for SpaSM. For
$k=3$
, it is 45 s for kissat_gb and 150 s for SpaSM. This suggests that SAT solving is faster than Gaussian elimination to check non-contextual configurations. However, for contextual configurations, SpaSM appears to be faster. For
$N=7$
, SpaSM validates the contextuality of an hyperbolic quadric in 21 s, against 32 s for kissat_gb, and the contextuality of an elliptic quadric in 14 s, against 31 s for kissat_gb.
4. Computational Results
The results provided by the C program are summarized in Table 1, where N is the number of qubits, # contexts is the number of contexts (i.e., the number of rows of A and E), # neg. contexts is the number of contexts with negative sign among them, # obs. is the number of observables (i.e., the number of columns of A), d is the contextuality degree, and D(N) is the number of N-qubits doilies, given in Muller et al. (Reference Muller, Saniga, Giorgetti, De Boutray and Holweck2022, Section 3).
Table 1. Known and new results for the contextuality degree of various configurations
1 These numbers are found at least for one of the configurations.
Results are grouped by family of configurations (first column), and # config
$^{\circ}$
is the number of configurations in this family. The numbers of subspaces computed by the C program for any N and k match those given by (2). The column C/NC indicates whether all the configurations in this family are Contextual or Not Contextual. Since the generators of
$W_2$
are its lines, their properties given in the first block for lines are not repeated in the block for generators.
Table 1 significantly extends Table 2 in de Boutray et al. (Reference de Boutray, Holweck, Giorgetti, Masson and Saniga2022), in which the bounds for the contextuality degree d for lines with
$N = 3, 4, 5$
qubits should be, according to Cabello (Reference Cabello2010), the numbers 90, 1908, and 35400 of negative contexts, instead of the numbers 135, 1539, and 16155 of the corresponding bound b, with the notations of Section 2.1.
Table 2. Illustrative example for the proofs of Proposition 3 and Lemma 4
For a quicker orientation, the new results and improved bounds, detailed in the rest of this section, are shown in the bold font style in Table 1.
4.1 Results for subspaces
For lines with 3 qubit observables, the C program found a valuation with 63 invalid constraints, and no valuation with 62 invalid constraints or less, thus justifying that the contextuality degree is 63. This exact value is lower that the previously known bound, which was the number 90 of negative lines. For four and five qubits, the contextuality checks succeed, but the time to compute the contextuality degree is too high when the variable i in Algorithm 2 is initialized with the number
$|C^+|$
of positive contexts (Line 2). When initializing it with the total number
$|C|$
of contexts, we obtained at best the bounds 2268 and 40391 for the contextuality degree for 4 and 5 qubits. These bounds are not interesting, since they are higher than the number of negative contexts.
For planes (
$k = 2$
), the contextuality degree has been computed up to five qubits and is always 0. Table 1 also shows the number of negative planes for six and seven qubits that we were able to generate. However, we could not compute their contextuality degree, because the input files for bc2cnf and SpaSM are too large.
The totally isotropic subspaces of dimension k of
$W_N$
computed for
$(k,N) = (3,4)$
, (3,5), (3,6), (4,5), (4,6), (5,6), and (6,7) appear to be positive. These results suggest that this positivity property might hold for all
$N > k \geq 3$
, with the consequence that the configuration whose contexts are all these subspaces for given k and N is non-contextual. After further examination, this property indeed holds. We state it as Proposition 3 and prove it in Section 5.2.
4.2 Results for perpsets, quadrics, doilies, Mermin–Peres squares, and two-spreads
For quadrics and perpsets, the C program gives the same contextuality answers as in Boutray et al. (2022). Moreover, it additionally checks that all perpsets with 6 and 7 qubits are non-contextual.
For N qubits, there seems to be N possible numbers of negative lines in a perpset (de Boutray et al., Reference de Boutray, Holweck, Giorgetti, Masson and Saniga2022). These numbers seem to be four times the number of negative lines in a perpset of
$N-1$
qubits, plus an additional number that seems to match the OEIS A006516 sequence (https://oeis.org/A006516).
The case of three-qubit quadrics is particularly interesting in the sense that, in addition to reconfirming their exact degree of contextuality found earlier, we can also explicitly see the geometric patterns formed by corresponding invalid (or unsatisfiable) constraints. Thus, for each of 28 elliptic quadrics living in
$\mathcal{W}(5,2)$
, we found that 9 invalid constraints form a set of 9 pairwise disjoint lines partitioning the set of points of the quadric, that is, it is a spread of lines. Similarly, for each of 36 hyperbolic quadrics living in
$\mathcal{W}(5,2)$
, the geometry formed by the 21 unsatisfiable contexts is isomorphic to that depicted in Figure 2. This geometry is underpinned by the Heawood graph (Heawood, Reference Heawood1890). This remarkable bipartite graph has 14 vertices and 21 edges, being isomorphic to the point-line incidence (or Levi) graph of the Fano plane; in Figure 2, the seven big empty circles correspond to the seven points of the Fano plane, while the seven big filled circles correspond to the seven lines, with an edge joining an empty vertex to a filled vertex iff the corresponding point lies on the corresponding line. On a hyperbolic quadric, both seven-sets can be labeled by the points/observables of two distinct disjoint Fano planes lying on the quadric in such a way that any two observables joined by an edge commute and thus define the line/context whose the third observable (represented by a smaller gray-shaded circle) is the product of the two. Note that, as in the case of elliptic quadrics, the 21 lines entail all the 35 points/observables of the quadric.
Figure 2. An illustration of the point-line configuration formed by 21 invalid constraints of the particular three-qubit hyperbolic quadric that consists solely of symmetric observables.
Concerning Mermin–Peres squares, the table just restates a well-known result proved in Holweck and Saniga (Reference Holweck and Saniga2017) that each grid is contextual once embedded in an arbitrary multi-qubit symplectic polar space of order 2. Similarly, all multi-qubit doilies have already been shown to be contextual for any N, their degree of contextuality being always 3 (Muller et al., Reference Muller, Saniga, Giorgetti, De Boutray and Holweck2022, Proposition 1). Contextuality of two-spreads is an important novelty of this work, their degree of contextuality being proved (see Section 5.5) to be 1 irrespectively of the value of the rank of the ambient symplectic space.
The C program took on average 10 min to compute the contextuality degree 21 of 1 hyperbolic quadric of 3 qubits, with the default SAT solver minisat. Almost the entirety of the time is taken checking the unsatisfiability of the last system. This is the longest computation time, compared to perpsets and elliptics. For this reason, we checked the top three SAT solvers of 2021 (https://github.com/simewu/top-SAT-solvers-2021) and found out that kissat_gb was the fastest for our problem. Indeed, it took an average of 10 s to find the contextuality degree of an hyperbolic of three qubits. Thus, the time to compute the contextuality degree of all the hyperbolics for three qubits was reduced down to 5 min and to a few seconds for elliptics and perpsets.
The results are the same as the ones in de Boutray et al. (Reference de Boutray, Holweck, Giorgetti, Masson and Saniga2022). However, as with the Magma implementation, it does not seem possible to compute in the same way the contextuality degree of configurations with more than three qubits. From our computations up to now, it comes that for four qubits the contextuality degree of elliptics is at most 351 for one of those with 360 negative lines, at most 363 for one of those with 384 negative lines. It is at most 500 for one of the hyperbolics with 532 or 604 negative lines and at most 517 for the single one with 612 negative lines. For five qubits, when starting from the number of positive contexts, the times to compute the contextuality degree of hyperbolics and elliptics are too high. When starting from the total number of contexts, the best bounds 10878 and 9169 obtained are not interesting, since they exceed the number of negative contexts of hyperbolics and elliptics, respectively.
5. Contextuality-Related Properties
This section presents contextuality-related properties holding for all numbers of qubits beyond a certain known threshold.
5.1 Perpsets
Proposition 2. All perpsets are non-contextual.
Proof. Remember that the perpset of the point
$p\in W_N$
is the configuration of the points q commuting with p whose contexts are the lines of
$W_N$
containing p. Consequently, any point
$q \neq p$
in the perpset of p belongs to the unique line
$\{p,q,p+q\}$
. Thus, the two variables
$v_q$
and
$v_{p+q}$
(other than the variable
$v_p$
associated with p) in the sign constraint
$v_p + v_q + v_{p+q} = e$
of the line
$\{p,q,p+q\}$
appear in no other sign constraint. Whatever the values of
$v_p$
and e, it is always possible to find a value for
$v_q$
and
$v_{p+q}$
to satisfy this constraint
$v_p+ v_q + v_{p+q} = e$
. So, the entire system of sign constraints is always solvable and all perpsets are non-contextual.
5.2 Totally isotropic subspaces of dimension
$k \geq 3$
k 3
Proposition 3. For
$N > k \geq 3$
, all totally isotropic subspaces of
$W_N$
of dimension k are positive. Consequently, the quantum configuration whose contexts are the totally isotropic subspaces of
$W_N$
of dimension k is non-contextual.
Proof. A totally isotropic subspace of
$W_N$
is positive if the phase of the product of the elements in the N-qubit Pauli group that are bijectively associated with its points is 1. Thus, after introducing some notations, we reformulate and formalize the positivity property with the matrix product in this (non-commutative) group.
Let
$M = \{I,X,Y,Z\}$
. Let
$|\_|$
denote the norm on the Pauli group
$\mathcal{P}^{\otimes N} = \{1,-1,\textrm{i},-\textrm{i}\} \times M^{\otimes N}$
, defined as phase removal by
$|p\,a| = a$
for any phase
$p \in \{1,-1,\textrm{i},-\textrm{i}\}$
and any
$a \in M^{\otimes N}$
. Let us denote by
$|.|$
the binary operation on
$M^{\otimes N}$
such that, for all
$a, b \in M^{\otimes N}$
,
$a~|.|~b = |a.b|$
. (Here, the dot “
$.$
” denotes the matrix product in
$\mathcal{P}^{\otimes N} $
.). For
$\iota$
in
$\mathbb{F}_2$
(whose elements are the binary digits 0 and 1) and a in
$M^{\otimes N}$
, let
$a^{\iota}$
be the multiplication by a scalar defined by
$a^{0} = I^{\otimes N}$
and
$a^{1} = a$
. With this multiplication by a scalar and the product
$|.|$
, it is easy to check that
$M^{\otimes N}$
is a vector space (of dimension 2N) over
$\mathbb{F}_2$
.
A totally isotropic subspace of
$W_N$
of (projective) dimension k is bijectively associated with a vector subspace S of (vectorial) dimension
$k+1$
of
$M^{\otimes N}$
, without its neutral element
$I^{\otimes N}$
. Since multiplying with this neutral element does not change the value of the product of all the elements of S, the totally isotropic subspace of
$W_N$
is positive if and only if the matrix product (with “
$.$
”) of all the observables in S equals
$I^{\otimes N}$
.
We illustrate the notations and proof steps with the explicit example in Table 2, for
$k=3$
and
$N=4$
.
For
$m \geq 0$
, the natural number
$i = i_{m} 2^{m} + \ldots i_0 2^0$
, with
$i_{m}$
, …,
$i_0 \in \mathbb{F}_2$
, is identified with the tuple
$(i_{m},\ldots,i_0)$
of its binary digits. For
$m=3$
, these 16 numbers
$i \in [0..15]$
and their binary encoding
$i_{3}\cdots i_0$
are, respectively, listed in the first and second column of Table 2. For any tuple
$b = (b_l,\ldots,b_u)$
of N-qubit observables and any tuple
$i = (i_{l},\ldots,i_u)$
of binary digits with the same length as b, let
$b^i$
denote the matrix product
$b_l^{i_l}.\cdots.b_u^{i_u}$
in this order. (Despite the multiplicative notation, it is a linear combination of the elements of b, with 0 or 1 coefficients.)
Let
$b = (b_k,\ldots,b_0)$
be a basis of S. Its
$k+1$
elements are independent and mutually commuting N-qubit observables, chosen without phase. In Table 2, the basis b is (ZIXX,XZXY,IXXX,IIYZ). The value of each
$b^i$
is displayed in the third column. The basis vectors
$b_0$
,
$b_1$
,
$b_2$
, and
$b_3$
, respectively, appear in the rows of this column indexed by
$i= 1, 2, 4$
and 8. With the former conventions and notations, the elements of S are the norms of all the linear combinations
$b^i$
for
$0 \leq i \leq 2^{k+1}-1$
. So, formally, the positivity property to prove is
\begin{align}\prod_{0 \leq i < 2^{k+1}} \left|b^i\right| & = I^{\otimes N}, \end{align}
where
$\prod_{\ell \leq i \leq u} a_i$
denotes the generalized matrix product
$a_{\ell} . \cdots . a_u$
of all the elements in the finite sequence
$(a_i)_{\ell \leq i \leq u}$
of N-qubit observables, in this order.
For
$1 \leq j \leq N$
, let
$a_j$
denote the j-th qubit of the phase-free N-qubit observable
$a \in M^{\otimes N}$
. For
$0 \leq m \leq k$
, the j-th qubit of the m-th basis vector
$b_m$
is denoted
$b_{m,j}$
. As a direct consequence, for all s,t in
$\mathcal{P}^{\otimes N}$
,
\begin{align}\left|s.t\right|_j = \left||s|_j.|t|_j\right|.\end{align}
The matrix product of all the elements of S (left-hand side of (8)) is the tensor product for all
$j \in [1..N]$
of the matrix products of their j-th qubits. Formally,
\begin{align*}\prod_{0 \leq i < 2^{k+1}} \left|b^i\right|& = \bigotimes\limits_{1 \leq j \leq N} \, \prod_{0 \leq i < 2^{k+1}} \left|b^i\right|_j.\end{align*}
So, we prove that it is
$I^{\otimes N}$
as a consequence of the (stronger) fact that all the products of their j-th qubits equal I (Lemma 4), when the order of the qubit products is chosen to be the lexicographic order on the tuples
$(i_{m-1},\ldots,i_0)$
, corresponding to the usual order
$<$
on the natural numbers they encode. Since S comes from a totally isotropic space, all its pairs of elements mutually commute, so the order of the product of all its elements (left-hand side of (8)) can be chosen to be this lexicographic order.
In Table 2, the j-th qubit
$\left|b^i\right|_j$
of the elements of S are shown in Columns 4, 6, 8, and 10, for j = 3, 2, 1, and 0. The matrix product of all elements in each of these columns is expected to be I. The remaining contents of Table 2 are described in the proof of Lemma 4, postponed in Appendix 7 to lighten the present proof, which ends here.
Lemma 4. For
$3 \leq k \leq N-1$
, let S be a vector subspace of
$M^{\otimes N}$
, of dimension
$k+1$
, generated by a basis
$(b_k,\ldots,b_0)$
of
$k+1$
independent and mutually commuting vectors without phase. Let
$L = (\left|b^i\right|)_{0\leq i < 2^{k+1}}$
be the sequence of all the elements of S in lexicographic order. Then, for all
$1 \leq j \leq N$
, the matrix product
$\prod_{0 \leq i < 2^{k+1}} \left|b^i\right|_j$
of all the elements of the sequence of j-th qubits in L, in the same order as in L, always equals I.
5.3 Geometrical constraints and a related corollary on the sign(s) of three-dimensional subspaces
In addition to the algebraic proof of Proposition 3 presented in Section 5.2, this section provides a geometric interpretation of the positivity of every PG(3,2) of a multi-qubit
$W_N$
, for
$N \geq 4$
.
Proposition 3. Let P be a PG(3,2) of a multi-qubit
$W_N$
with
$N \geq 4$
. Then, the following four propositions are equivalent:
(a) P is negative.
(b) Each of the 56 spreads of lines of P contains an odd number of negative lines.
(c) Through each of the 35 lines of P there pass an odd number of negative planes.
(d) Through each of the 15 points of P there pass an odd number of negative lines.
Proof. As the observables of any subspace of
$W_N$
mutually commute, we can take any product of observables in P in any order.
(a)
$\Leftrightarrow$
(b) A spread of lines of P is a set of five pairwise disjoint lines that partition the set of points of P. Pick up a spread of lines, take first the product of the three observables on each line of the spread, and then multiply the five products obtained to get the sign of P. This space will be negative (i.e., the product of all the 15 observables will be equal to
$-I^{\otimes N}$
) iff the number of negative lines in the spread selected is odd. Clearly, this property must hold irrespectively of which spread of lines we select.
(a)
$\Leftrightarrow$
(c) Let us assume that the 15 points/observables of P are labeled simply as
$\{a$
, b, c, d, e, f, g, h, i, j, k, l, m, n,
$o\}$
, that a,b,c lie on a line and that the three planes through this line are
$F_1 = \{a,b,c;\ d,e,f,g\}$
,
$F_2 = \{a,b,c;\ h,i,j,k\}$
, and
$F_3 =\{a,b,c;\ l,m,n,o\}$
. Let us consider the following product of the products of all the points in each of these three planes:
\begin{align}(a.b.c.d.e.f.g).(a.b.c.h.i.j.k).(a.b.c.l.m.n.o).\end{align}
By commutation relations and
$a^2 = b^2 = c^2 = I^{\otimes N}$
, this product (10) equals the product
\begin{align}a.b.c.d.e.f.g.h.i.j.k.l.m.n.o,\end{align}
of all the points of P. So, P is negative iff one or all the three planes through the line in question are negative.
Again, as the sign of the space is an invariant this reasoning must hold for the planes through any line of P.
(a)
$\Leftrightarrow$
(d) Let us assume that the seven lines passing via the point/observable a are
$\{a,b,c\}$
,
$\{a,d,e\}$
,
$\{a,f,g\}$
,
$\{a,h,i\}$
,
$\{a,j,k\}$
,
$\{a,l,m\}$
, and
$\{a,n,o\}$
. Consider the following product of the products of all the points in these lines:
\begin{align} (a.b.c).(a.d.e).(a.f.g).(a.h.i).(a.j.k).(a.l.m).(a.n.o).\end{align}
After simplification, it again equals the product (11) of all the points of P. So, P is negative iff the number of negative lines passing through a is odd.
Again, this must hold irrespectively of the reference point/observable chosen.
As we have proved (see Proposition 3) that there are no negative PG(3,2)’s in
$W_N$
, for
$N \geq 4$
, Proposition 5 leads to the following geometrically slanted corollary:
Corollary 6. As a PG(3,2) of a multi-qubit
$W_N$
,
$N \geq 4$
is always positive, then: a) each of its 56 spreads of lines contains an even number of negative lines; b) through each of its 35 lines there pass an even number of negative planes; and c) through each of its 15 points there pass an even number of negative lines.
5.4 No analogs of Mermin pentagrams in the four-qubit case
Given the fact, stemming from Proposition 3, that all generators – that is, PG(3, 2)s – of the four-qubit
$\mathcal{W}(7, 2)$
are positive, there will be no analogs of (three-qubit) Mermin pentagrams in the four-qubit case. We illustrate this fact in Figure 3. We took a particular quadratic
$\mathcal{W}(5, 2)$
having 42 negative planes. In this
$\mathcal{W}(5, 2)$
, we picked up one spread of planes; this spread features four negative planes and is illustrated on the left-hand side of the figure. Now, through each plane of
$\mathcal{W}(7, 2)$
there are three PG(3, 2)s, and the totality of 27 such spaces can be split into 3 sets of 9 elements each. One set has a special footing as all its PG(3, 2)s pass through a common point – the pole of that PG
$(6, 2)$
where our
$\mathcal{W}(5, 2)$
is located in (being also the nucleus of the
$\mathcal{W}(5, 2)$
) – and it will be disregarded. The other two nine-element sets are similar in the sense that each of them consists of mutually disjoint PG(3, 2)s. The affine parts, AG(3, 2)s, of these spaces of either set, however, cover the same (
$9 \times 8 =$
) 72 points; this means that each of these 72 points will lie in two different AG(3, 2)s. In analogy with a Mermin pentagram, these (
$9 \times 2=$
) 18 AG(3, 2)s will be our contexts. They are shown on the right-hand side of Figure 3; here, the two contexts originating from the same plane of the spread are arranged in the row and/or the column bearing the number of the corresponding plane. From this construction, it is clear that irrespectively of the number of negative planes our selected spread features, there will always be an even number of negative contexts in our analog of a Mermin pentagram; indeed, each PG(3, 2) – being always positive – that passes via a negative plane will have the corresponding AG
$(3, 2)$
negative, but for each plane these AG(3, 2)s come in pairs. A similar argument can be used to show that there are no higher-rank analogs of Mermin pentagrams for any
$N > 3$
.
Figure 3.
Left: A spread of planes in the selected four-qubit
$\mathcal{W}(5,2)$
, with negative lines shown in gray; each plane featuring three concurrent negative lines is negative. Right: The associated sets of 18 contexts (9 rows and 9 columns), 2 per each plane of the spread.
5.5 Contextuality properties of two-spreads: The contextuality degree of all NN-qubit two-spreads is 1
Apart from grids (aka Mermin–Peres magic squares), a multi-qubit doily hosts another prominent, yet almost virtually unknown to the quantum informational community, class of contextual configurations, the so-called two-spreads. Given a multi-qubit doily, remove from it one spread of lines while keeping all the points; what we get is a two-spread, that is, a (sub-)configuration having 15 points and 10 lines, with 3 points per line and 2 lines per point (see, e.g., Figure 8 of Polster (Reference Polster1998)). Figure 4 is an illustration of a two-spread living in a quadratic five-qubit.
Figure 4. An example of five-qubit two-spread having nine (i.e., the maximum possible number of) negative lines (boldfaced).
Let us briefly recall that a “classical” contextual point-line configuration features (a) an even number of lines/contexts per point/observable and (b) an odd number of negative lines/contexts (Holweck and Saniga, Reference Holweck and Saniga2017). We already see that a multi-qubit two-spread meets the first condition (as there are two lines per each of its points). Now we show that it also satisfies the second condition. To this end in view, we readily infer from our computer-generated results (after carefully inspecting Figure 2 of our recent paper (Muller et al., Reference Muller, Saniga, Giorgetti, De Boutray and Holweck2022)) that that if a doily is endowed with an odd number of negative lines, each of its six spreads of lines contains an even number of them, whereas if a doily features an even number of negative lines, each of its six spreads shares an odd number of them. That is, removal of a spread leads in both cases to a configuration endowed with an odd number of negative lines, as envisaged.
The former reasoning is substantiated by our exhaustive computer-aided analysis up to
$N=5$
. In
$W_N$
, for
$2 \leq N \leq 5$
, we picked up a representative doily from each type (defined by the signature of observables) and subtype (defined by the pattern formed by negative lines) as classified in Muller et al. (Reference Muller, Saniga, Giorgetti, De Boutray and Holweck2022) and found out that all six two-spreads in it were contextual. Moreover, we also found that in each case the degree of contextuality was the same and equal to 1. In fact, by analyzing the incidence matrix of the two-spread configuration, it appears that all two-spreads are contextual and the vector E of negative lines is at Hamming distance 1 of the image of the incidence matrix. In other words, we have the following proposition:
Proposition 7. A two-spread in an N-qubit doily is contextual for each
$N \geq 2$
, and its degree of contextuality is always 1.
Proof. Consider the incidence matrix of the two-spread configuration:
The linear combinations of the rows of A span the nine-dimensional space of vectors in
$\mathbb{F}_2^{10}$
having an even number of 1s in their decomposition in the standard basis. In fact,
$\textrm{Im}(A)$
is the space of length-10 binary vectors with an even number of 1s. Thus, any vector E representing a context evaluation with an odd number of negative lines, that is, an odd number of 1s, is at Hamming distance 1 of
$\textrm{Im}(A)$
. This proves that two-spreads are always contextual with contextuality degree 1.
5.6 Contextuality degree of three-qubit lines
A very instructive result in Table 1 concerns the contextuality degree of lines of
$W_3$
. This result contradicts Lemma 3 and Figure 1 of Cabello (Reference Cabello2010). Indeed the contextuality degree corresponds to the minimal number of constraints that cannot be satisfied by any NCHV. In Cabello (Reference Cabello2010), it is supposed to be always equal to the number of negative lines when we consider, for a fixed N, all N-qubit observables and all lines. But the model obtained by the SAT solver was able to satisfy, in the three-qubit case,
$315-63=252$
constraints instead of 235 as previously obtained. Note that this new result changes the contextuality bound experimentally tested in Holweck (Reference Holweck2021). The experimental evaluation of Equation (6) on IBM quantum computers described in Holweck (Reference Holweck2021) provided a result of 236, which still violates the classical bound, now known to be
$b=315-2\times 63=189$
, instead of the erroneous value
$315-2\times 90=135$
.
Figure 5. A classically embedded three-qubit split Cayley hexagon of order 2 that accommodates all the 63 invalid constraints for a particular solution found by the SAT solver. The graphical illustration of the hexagon is a simplified reproduction of that given in Polster et al. (Reference Polster, Schroth and Van Maldeghem2001).
With the C implementation of the contextuality degree algorithm 2, we managed to compute the contextuality degree of three-qubit lines. The intermediate partial solutions found had the following Hamming distances: 89, 86, 79, 72, and 63. This means that it first checked for a Hamming distance that is at most the number of negative lines, 90, and found a solution with 89 invalid constraints, and then it checked for at most 1 less invalid constraints and found 86 of them. This process was repeated until it found a configuration with 63 invalid constraints, after which the solver found no solution when asked to let at most 62 invalid constraints satisfied. Thus, the contextuality degree of the configuration of the lines of
$W_3$
is 63.
It is worth concluding this section by mentioning that in this case we carried out several runs with the SAT solver and, strikingly, in each case the set of 63 invalid constraints/lines was found to be isomorphic to a copy of the split Cayley hexagon of order 2 when embedded classically into
$W_3$
; the copy for one of these sets is illustrated in Figure 5. This finding sheds a rather unexpected new light on the contextuality role of the hexagon in the three-qubit symplectic polar space discovered in Holweck et al. (Reference Holweck, de Boutray and Saniga2022) and will be treated in more detail in a separate paper.
6. Conclusion
One of the most intriguing outcomes of our work is the fact that there are no negative subspaces of dimension 3 and higher. This has a couple of interesting implications. The first one concerns Mermin–Peres squares and Mermin pentagrams. In the former case, the contexts correspond to generators of the ambient
$\mathcal{W}(3,2)$
, while in the latter case the contexts represent affine parts of generators of
$\mathcal{W}(5,2)$
. The non-existence of negative generators for
$k \geq 3$
then simply means that there will be no higher-dimensional analogs of these two important classes of contextual configurations in polar spaces of rank 5 and higher. The second implication concerns contextuality properties of s-qubit
$\mathcal{W}(2s-1,2)'$
s living in the ambient N-qubit
$\mathcal{W}(2N-1,2)$
, for
$s < N$
. We have already verified that all multi-qubit doilies (
$s=2$
) are contextual for any N, and that their degree of contextuality is 3 (Muller et al., Reference Muller, Saniga, Giorgetti, De Boutray and Holweck2022, Proposition 1). As the next natural step, it would be vital to address the corresponding behavior of rank 3 and, especially, rank 4 polar spaces.
Next, given a multi-qubit
$\mathcal{W}(2N-1,2)$
, remove from it a spread of generators while keeping all the observables; what we get is an analog of a doily’s two-spread. It would be desirable to check, at least for a few randomly selected cases, if these higher-dimensional analogs of two-spreads of a multi-qubit doily are contextual, or not.
A final prospective task is to address the contextuality of quadrics when instead of lines we consider their planes as contexts. A particular worth-deserving object in this respect is a hyperbolic quadric of the four-qubit symplectic polar space. This quadric, sometimes called the triality quadric, possesses an unexpected threefold symmetry and, in addition, it hosts 960 ovoids, each of them being related via a particular Grassmannian mapping to a spread of planes in the three-qubit polar space.
Financial Support.
This project is supported by the PEPR integrated project EPiQ ANR-22-PETQ-0007 part of Plan France 2030, by the project TACTICQ of the EIPHI Graduate School (contract ANR-17-EURE-0002). This work also received a partial support from the Slovak VEGA grant agency, Project 2/0004/20.
Competing Interests.
The authors declare no competing interests.
Appendix A. Proof of Lemma 4
Proof. We still write
$\prod_{0 \leq i < 2^m}$
for the product in the lexicographic order or sometimes use the more compact notation
$\prod_{|i|=m}$
, where
$|i|$
denotes the length of the tuple i. Let us prove that the product in lexicographic order of the j-th qubit of the norm of all linear combinations of length
$|i|=k+1$
of the basis vectors, that is, of all elements of S, equals the identity matrix. Formally,
\begin{align}\prod_{|i|=k+1} |b^i|_j = & I \end{align}
for all
$1 \leq j \leq N$
.
If i is the k-tuple
$(i_{k},\ldots,i_{1})$
, let 2i denote the
$(k+1)$
-tuple
$(i_{k},\ldots,i_{1},0)$
and
$2i+1$
denote the
$(k+1)$
-tuple
$(i_{k},\ldots,i_{1},1)$
. For two tuples i and i’ of bits (in
$\mathbb{F}_2$
) with the same length, let
$i \oplus i'$
denote the bitwise addition (“exclusive or”) of their bits. From these notations, it comes that
$2i \oplus 2i' = 2(i \oplus i')$
.
By associating each
$\left|b^{2i}\right|_j$
with its lexicographic successor
$\left|b^{2i+1}\right|_j$
in the product, we get
\begin{align*}\prod_{|i|=k+1} \left|b^i\right|_j = & \prod_{|i|=k} \left(\left|b^{2i}\right|_j.\left|b^{2i+1}\right|_j\right)\end{align*}
for
$k \geq 1$
. Since
$b^{2i+1} = b^{2i}.b_0$
and
$b^{2i}.b^{2i} =I^{\otimes N}$
, all products
$\left(\left|b^{2i}\right|_j.\left|b^{2i+1}\right|_j\right)$
equal
$b_{0,j}$
multiplied by some complex phase that will be precised below (in Table 2, this product is shown in Columns 5, 7, 9 and 11). This is a key property for the end of the proof, because the product of all these
$b_{0,j}$
with their phase commute, which allows us to re-order them in any order, until showing that the product of all these phases is 1.
A trivial case is when
$b_{0,j} = I$
. It is illustrated by Columns 4–7 of Table 2, for
$j = 2$
and
$j = 3$
. In this case all the products
$\left(\left|b^{2i}\right|_j.\left|b^{2i+1}\right|_j\right)$
of two consecutive qubits equal I, as detailed below:
\begin{align*}\left|b^{2i}\right|_j.\left|b^{2i+1}\right|_j& = \left|b^{2i}\right|_j.\left|b^{2i}.b_0\right|_j = \left|b^{2i}\right|_j.\left|\left|b^{2i}\right|_j.b_{0,j}\right| = \left|b^{2i}\right|_j.\left|\left|b^{2i}\right|_j.I\right| = \left(\left|b^{2i}\right|_j\right)^2 = I.\end{align*}
Thus, the complete product also equals I. Apart from this trivial case, the following two complementary cases can be considered.
Case 1:
Consider first the case when the inequality
\begin{align}\left|b^{2r}\right|_j \neq b_{0,j}\end{align}
holds for all tuples of coefficients r of length
$|r|=k$
. This case is illustrated by Columns 8-9 of Table 2, for
$j=1$
and
$b_{0,1} =Y$
.
Let
$i = (i_k,\ldots,i_1)$
be any tuple of binary coefficients of length
$|i|=k$
. The inequality (14) with
$r=0$
implies that
$b_{0,j}$
is different from I. So, the set
$\{X,Y,Z\}$
of Pauli matrices is composed of
$b_{0,j}$
and the other two Pauli matrices, hereafter denoted by U and V. For instance, in Columns 8-9 of Table 2, for
$j=1$
,
$b_{0,1} =b_{0,j} = Y$
, so U can be X and V can be Z.
Now, assume that there are two tuples i and i’ of binary coefficients of length k such that
$\left|b^{2i}\right|_j = U$
and
$\left|b^{2i'}\right|_j =V$
. Then
$\left|b^{2(i\oplus i')}\right|_j = \left|b^{2i\oplus 2i'}\right|_j =\left|b^{2i}. b^{2i'}\right|_j = \left|\left|b^{2i}\right|_j.\left|b^{2i'}\right|_j\right| = \left|U.V\right| = b_{0,j}$
, which contradicts the inequality (14). Consequently, there is a unique Pauli matrix, different from
$b_{0,j}$
, hereafter assumed to be U without loss of generality, such that the j-th qubit of each vector
$\left|b^{2i}\right|$
with
$|i|=k$
is either I or U. For instance, in Columns 8-9 of Table 2, U is X. It follows that any pair
$(\left|b^{2i}\right|_j, \left|b^{2i+1}\right|_j) = (\left|b^{2i}\right|_j,\left|b^{2i}.b_0\right|_j)$
of consecutive j-th qubits whose first element comes from a linear combination without
$b_{0}$
is either
$(I,b_{0,j})$
(for
$i= 000, 011, 101$
, and 110 in Table 2) or
$(U,\left|U.b_{0,j}\right|)$
(for
$i = 001, 010, 100$
, and 111 in Table 2, gray cells in Column 9).
The sets of pairs in these two cases are in one-to-one correspondence according to the involution
$W \mapsto \left|U.W\right|$
, so their cardinality is half the total cardinality, that is,
$2^k/2=2^{k-1}$
.
In the first case,
$\left|b^{2i}\right|_j.\left|b^{2i+1}\right|_j = I.b_{0,j} =b_{0,j}$
, without phase. In the second case,
$\left|b^{2i}\right|_j.\left|b^{2i+1}\right|_{j} = U.\left|U.b_{0,j}\right| =U.\left|\pm\textrm{i}\, V\right|$
, because the product of two distinct Pauli matrices always yields the third one, which is V here, up to a phase which is either i or
$-\textrm{i}$
. Finally,
$\left|b^{2i}\right|_j.\left|b^{2i+1}\right|_j = U.V = p\, b_{0,j}$
for the same reason, with the same phase p among i and
$-\textrm{i}$
for all these products. For instance, this phase p is
$-\textrm{i}$
in the gray cells in Column 9 of Table 2.
By separating the complete product along these two cases, it comes
\begin{align*}\prod_{|i|=k+1} \left|b^i\right|_j = & \prod_{|i|=k} \left|b^{2i}\right|_j.\left|b^{2i+1}\right|_j = \left(\prod_{|i|=k,\ \left|b^{2i}\right|_j=I} \left|b^{2i}\right|_j.\left|b^{2i+1}\right|_j\right) \left(\prod_{|i|=k,\ \left|b^{2i}\right|_j=U} \left|b^{2i}\right|_j.\left|b^{2i+1}\right|_j\right)\\ = & \left(\prod_{|i|=k,\ \left|b^{2i}\right|_j=I} b_{0,j}\right)\! \left(\prod_{|i|=k,\ \left|b^{2i}\right|_j=U}p\! b_{0,j}\right) = p^{2^{k-1}}\! \left(\prod_{|i|=k}\! b_{0,j}\right) = (\pm \textrm{i})^{2^{k-1}}\, b_{0,j}^{2^k} = 1 \, I = I\end{align*}
when
$k \geq 3$
.
Case 2:
Finally, consider the complementary case when there is some tuple of coefficients
$r \neq 0$
of length
$|r|=k$
such that
$\left|b^{2r}\right|_j = b_{0,j}$
. This case is illustrated by Columns 10-11 in Table 2, with
$j=0$
and
$r = 011$
, since
$\left|b^{0110}\right|_0 = b_{0,0} = Z$
in this example (
$r = 110$
could be another valid choice).
In
$r = (r_{k-1}, \ldots, r_0) \neq 0$
, let
$0 \leq q < k$
be any non-zero coefficient (
$r_q = 1$
). For instance, since
$r = (0,1,1)$
in Table 2 for
$j=0$
, let us choose
$q = 1$
(
$q=0$
would also hold) in this example. Let
$R_0 = \{ i \in [0..2^k-1] \ | \ i_q = 0\}$
be the subset of numbers in
$[0..2^k-1]$
whose q-th bit is the opposite of the q-th bit
$r_q = 1$
in r. Then,
$R_0$
and
$([0..2^k-1] - R_0)$
partition
$[0..2^k-1]$
into two equal subsets, in bijection according to the involution
$i\mapsto r \oplus i$
. For instance, in Table 2,
$R_0$
is
$\{ 000,001, 100, 101\}$
.
The product
$\prod_{|i|=k} \left|b^{2i}\right|_j.\left|b^{2i+1}\right|_j$
is re-organized according to
$R_0$
and simplified as follows:
\begin{align*}\prod_{|i|=k+1} \left|b^i\right|_j = &\prod_{|i|=k} \left|b^{2i}\right|_j.\left|b^{2i}.b_0\right|_j = \left(\prod_{|i|=k,\, i_q=0} \left|b^{2i}\right|_j.\left|b^{2i}.b_0\right|_j\right) \left(\prod_{|i|=k,\ i_q=1} \left|b^{2i}\right|_j.\left|b^{2i}.b_0\right|_j\right)\\ = &\left(\prod_{|i|=k,\, i_q=0} \left|b^{2i}\right|_j.\left|b^{2i}.b_0\right|_j\right) \left(\prod_{|i|=k,\ i_q=0} \left|b^{2(r\oplus i)}\right|_j.\left|b^{2(r\oplus i)}.b_0\right|_j\right)\\ = &\left(\prod_{|i|=k,\,i_q = 0} \left|b^{2i}\right|_j.\left|b^{2i}.b_0\right|_j .\left|b^{2r}.b^{2i}\right|_j.\left|b^{2r}.b^{2i}.b_0\right|_j\right).\end{align*}
For instance, in Table 2, for
$i = 001$
, the subproduct
$\left|b^{2i}\right|_j.\left|b^{2i}.b_0\right|_j.\left|b^{2r}.b^{2i}\right|_j.\left|b^{2r}.b^{2i}.b_0\right|_j$
in this product is
\begin{align*}&\left|b^{0010}\right|_0.\left|b^{0010}.b_0\right|_0.\left|b^{0110}.b^{0010}\right|_0.\left|b^{0110}.b^{0010}.b_0\right|_0 =X.\left|X.Z\right|.\left|Z.X\right|.\left|Z.X.Z\right|\\& \quad =X.(Y.Y) .X = X.(I).X = I.\end{align*}
This reduction to I holds in all generality, as detailed below. From (9) and commutation relations, it comes that
\begin{align*}\left|b^{2r}.b^{2i}\right|_j & = \left|\left|b^{2r}\right|_j.\left|b^{2i}\right|_j\right| = \left|b_{0,j}.\left|b^{2i}\right|_j\right| = \left|\left|b^{2i}\right|_j.b_{0,j}\right| = \left|b^{2i}.b_0\right|_j\end{align*}
so
\begin{align*}\prod_{|i|=k+1} \left|b^i\right|_j & = \left(\prod_{|i|=k,\,i_q = 0} \left|b^{2i}\right|_j.\left(\left|b^{2i}.b_0\right|_j\right)^2.\left|b^{2r}.b^{2i}.b_0\right|_j\right) = \left(\prod_{|i|=k,\,i_q = 0} \left|b^{2i}\right|_j.\left|b^{2r}.b^{2i}.b_0\right|_j\right).\end{align*}
Similarly,
\begin{align*}\left|b^{2r}.b^{2i}.b_0\right|_j & = \left|\left|b^{2r}\right|_j.\left|b^{2i}\right|_j.b_{0,j}\right| = \left|b_{0,j}.\left|b^{2i}\right|_j.b_{0,j}\right| = \left|b_{0,j}^2.\left|b^{2i}\right|_j\right| = \left|b^{2i}\right|_j\end{align*}
so
\begin{align*}\prod_{|i|=k+1} \left|b^i\right|_j & = \prod_{|i|=k,\,i_q = 0} \left(\left|b^{2i}\right|_j\right)^2 = \prod_{|i|=k,\,i_q = 0} I = I\end{align*}
that completes the proof.
