Magic informationally complete POVMs with permutations

Eigenstates of permutation gates are either stabilizer states (for gates in the Pauli group) or magic states, thus allowing universal quantum computation (Planat, Rukhsan-Ul-Haq 2017 Adv. Math. Phys. 2017, 5287862 (doi:10.1155/2017/5287862)). We show in this paper that a subset of such magic states, when acting on the generalized Pauli group, define (asymmetric) informationally complete POVMs. Such informationally complete POVMs, investigated in dimensions 2–12, exhibit simple finite geometries in their projector products and, for dimensions 4 and 8 and 9, relate to two-qubit, three-qubit and two-qutrit contextuality.


Introduction
Sometimes a field of knowledge gets enriched just by looking at it from a different perspective. Here we are interested in informationally complete (IC) measurements on an unknown density matrix ρ with the perspective of universal quantum computing. In the former subfield, one knows how to build group covariant symmetric measurements (SIC-POVMs) that follow from the action of the generalized Pauli group P d on a well-chosen 'fiducial' state [1][2][3]. In the latter subfield, the group P d needs to be extended by a well-chosen 'magic' state of the corresponding dimension to allow universal quantum computation [4,5]. Bravyi & Kitaev [4] introduced the principle of 'magic state distillation': universal quantum computation may be realized thanks to the stabilizer formalism (Clifford group unitaries, preparations and measurements) and the ability to prepare an ancilla in an appropriate single qubit mixed state. Following [6,§IIC], in this paper, a non-stabilizer pure state will be called a magic state. When is such a 'magic' state 'fiducial' for an IC-POVM? To address this question, we restrict our choice to eigenstates of permutation gates not living in P d (the stabilizer subgroup of unitaries) as in the recent paper [7]. We recover the Hesse SIC for d = 3 and discover asymmetric IC-POVMs for d > 3. In this paper, we first recall in the following subsections a few necessary concepts for our purpose: POVM concepts and the generalized Pauli group. In §2, we apply the methodology to the derivation of IC-POVMs in dimensions 2-12, then we establish the link to some finite geometries and to two-qubit, three-qubit and two-qutrit contextuality. Section 3 summarizes the results.
A POVM is a collection of positive semi-definite operators {E 1 , . . . , E m } that sum to the identity. In the measurement of a state ρ, the ith outcome is obtained with a probability given by the Born rule p(i) = tr(ρE i ). For a minimal IC-POVM, one needs d 2 one-dimensional projectors Π i = |ψ i ψ i |, with Π i = dE i , such that the rank of the Gram matrix with elements tr(Π i Π j ) is precisely d 2 .
A SIC-POVM obeys the remarkable relation [1] that allows the recovery of the density matrix as [8] ρ = 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 unknown classical signal, this looks similar to the frequency spectrum).
In this paper, we discover new IC-POVMs (i.e. whose rank of the Gram matrix is d 2 ) and with Hermitian angles | ψ i | ψ j | i =j ∈ A = {a 1 , . . . , a l }, a discrete set of values of small cardinality l. A SIC is equiangular with |A| = 1 and a 1 = 1/ √ d + 1. The states encountered below are considered to live in a cyclotomic field F = Q[exp(2iπ/n)], with n = GCD(d, r), the greatest common divisor of d and r, for some r. The Hermitian angle is defined as means the field norm [9, p. 162] of the pair (ψ i , ψ j ) in F and deg is the degree of the extension F over the rational field Q. For the IC-POVMs under consideration below, in dimensions d = 3, 4, 5, 6 and 7, one has to choose n = 3, 12, 20, 6 and 21, respectively, in order to be able to compute the action of the Pauli group. Calculations are performed with Magma.

The generalized Pauli group
Later, we construct IC-POVMs using the covariance with respect to the generalized Pauli group. Let d be a prime number, the qudit Pauli group is generated by the shift and clock operators as follows: with ω = exp(2iπ/d) a dth root of unity. In dimension d = 2, X and Z are the Pauli spin matrices σ x and σ z . A general Pauli (also called Heisenberg-Weyl) operator is of the form

Permutation gates, magic states and informationally complete measurements
In the approach of magic states through permutation groups, dimension 2 is trivial as the symmetric group S 2 only contains the identity I = (1, 2) and the shift gate X = (2, 1) ≡ ( 0 1 1 0 ), that live in the ordinary Pauli group P 2 . No magic state may be derived from two-dimensional permutation groups.
The situation changes as soon as d ≥ 3 with a wealth of magic states [7] having a potential usefulness for our purpose of defining IC-POVMs. From now on we focus on magic groups generated by two magic permutation gates.

In dimension 3
The symmetric group S 3 contains the permutation matrices I, X and X 2 of the Pauli group, where X = Taking the eigensystem of the latter matrices, it is not difficult check that there exist two types of qutrit magic states of the form (0, 1, ±1) ≡ (|1 ± |2 )/ √ 2. Then, taking the action of the nine qutrit Pauli matrices, one arrives at the well-known Hesse SIC [10][11][12].
The Hesse configuration shown in figure 1a is a configuration [9 4 , 12 3 ] with 9 points and 12 lines, 4 lines incident on every point and 3 points on a line. It can also be seen as the three-dimensional affine plane. The reason it occurs in the context of the three-dimensional SIC is as follows. The SIC relations are tr(Π i Π j ) i =j = 1/4 and, if one takes all projectors satisfying the triple product relation tr(Π i Π j Π k ) i =j =k = ±1/8, the corresponding triples (i, j, k) define the Hesse configuration. For the Hesse SIC built from the magic state (0, 1, −1), one only needs the plus sign to recover the Hesse geometry, but for the Hesse SIC built from the magic state (0, 1, 1) both signs are needed (see also [11]).
Observe that the configuration in figure 1a is labelled in terms of qutrit operators acting on the magic states instead of the projector themselves.

In dimension 4
In dimension 4 and higher, the strategy is to restrict to permutation groups whose two generators are magic gates, gates showing one entry of 1 on their main diagonals. From now on, we call such a group a   Taking the action of the two-qubit Pauli group on the latter type of state, the corresponding pure projectors sum to four times the identity (to form a POVM) and are independent, with the pairwise distinct products satisfying the dichotomic relation tr(Π i Π j ) i =j = | ψ i | ψ j | 2 i =j ∈ {1/3, 1/3 2 }. Thus, the 16 projectors Π i build an asymmetric informationally complete measurement not discovered so far.
The organization of triple products of projectors whose trace is ±1/27 and simultaneously equal plus or minus the identity matrix I is shown in figure 1b. Instead of labelling coordinates as projectors one may label them with the two-qubit operators acting on the magic state. As a result, the two-qubit (3 × 3)-grid identifies to the standard Mermin square that is known to allow an operator proof of the Kochen-Specker theorem [13,14].

In dimension 5
Still restricting to permutation groups generated by two magic gates (magic groups), the smallest group is isomorphic to the semidirect product Z 5 Z 4 of cyclic groups Z 4 and Z 5 [7, §3.4]. One finds magic states of type (0, 1, 1, 1, 1), (0, 1, −1, −1, 1) and (0, 1, i, −i, −1). The latter two types allow one to construct IC-POVMs such that the pairwise distinct products satisfy | ψ i | ψ j | 2 = 1/4 2 , that is the POVM is equiangular with respect to the field norm defined in the introduction. The first type of magic state is dichotomic with values of the products 1/4 2 and (3/4) 2 . The trace of pairwise products of (distinct) projectors is not constant. For example, with the state (0, 1, −1, −1, 1), one gets a field norm equiangular IC-POVM in which the trace is trivalued: it is either 1/16 or (7 ± 3

In dimension 6
With the alternating group A 6 generated by two magic gates, one finds an IC-POVM associated to a magic state such as (0, 1, ω 6 − 1, 0, −ω 6 , 0) with tr(Π i Π j ) i =j = | ψ i | ψ j | 2 i =j = 1/3 or 1/3 2 . Taking the trace of 4-tuple products of projectors whose value is 1/9 and simultaneously equal ±I, one gets two copies of a geometry looking like a Borromean ring as shown in figure 2b.

In dimension 8
In dimension d = 8, no IC-POVM was discovered from permutation groups. But it is time to introduce the well-known Hoggar SIC [15,16]. The Hoggar SIC follows from the action of the three-qubit Pauli group on a fiducial state such as (−1 ± i, 1, 1, 1, 1, 1, 1, 1).
It has been found that triple products are related to combinatorial designs [16]. There are 4032 (resp. 16128) triples of projectors whose products have trace equal to −1/27 (resp. 1/27) [16, (29)]. Within the 4032 triples, those whose product of projectors equal ±I (with I the identity matrix) are organized into a geometric configuration [63 3 ] whose incidence graph is of spectrum [6 1 , 3 31 , −1 27 , −3 14 ] and automorphism group G 2 (2) = U 3 (3) Z 2 of order 12 096, as in [17]. It is known that there exist two isospectral configurations of this type, one is the so-called generalized hexagon GH(2,2) (also called split Cayley hexagon) and the other one is its dual [18]. These configurations are related to the 12 096 Mermin pentagrams that build a proof of the three-qubit Kochen-Specker theorem [17,19]. From the structure of hyperplanes of our [63 3 ] configuration, one learns that we are concerned with the dual of G 2 as shown in figure 3 (see also [20, fig. 6a]).

In dimension 9
Let us consider a magic group isomorphic to Z 2 3 Z 4 generated by two magic gates. One finds a few magic states such as (1, 1, 0, 0, 0, 0, −1, 0, −1) that not only can be used to generate a dichotomic IC-POVM with distinct pairwise products | ψ i |ψ j | 2 equal to 1/4 or 1/4 2 , but also show a quite simple organization of triple products. Defining lines as triple of projectors with trace 1/8, one gets a geometric configuration of type [81 3 ] that split into nine disjoint copies of type [9 3 ]. One of the copies is shown in figure 5.
The configuration [9 3 ] labelled by the operators of figure 5 may be used to provide an operator proof of the Kochen-Specker theorem with two qutrits. The proof is in the same spirit as the one derived for two or three qubits [14] (see also [21] for two-qutrit contextuality). The vertices are projectors instead of just Hermitian operators. On one hand, every operator O can be assigned a value ν(O) which is an eigenvalue of O, that is 1, ω 3 or ω 2 3 (with ω 3 3 = 1). Taking the product of eigenvalues over all operators on a line and over all nine lines, one gets 1 since every assigned value occurs three times.
Thus the product law ν( The left-hand side equals ω 3 while the righthand side equals 1. No non-contextual hidden variable theory is able to reproduce these results. Since the lines are not defined by mutually commuting operators, it is not possible to arrive at a proof of the   two-qutrit Kochen-Specker theorem based on vectors instead of operators. In this sense, the proof of contextuality is weaker that the one obtained for two or three qubits.

Higher dimensions
The same method based on eigenstates of permutation matrices leads to IC-POVMs in dimensions higher than 9. In the next subsection, we provide details about a 12-dimensional IC-POVM covariant under the two-qubit/qutrit Pauli group because the associated triple products contain some geometrical structures as was the case in lower dimensions.

In dimension 12
One can build an IC-POVM using a magic group isomorphic to Z 2 2 (Z 2 3 Z 2 2 ). A magic state can be taken as (0, 1, ω 6 − 1, ω 6 − 1, 1, 1, ω 6 − 1, −ω 6 , −ω 6 , 0, −ω 6 , 0) or the state obtained from it by permuting   [1,2,3,4,5,6,7,8,9] , where X and Z are the qutrit shift and clock operators. The IC-POVM, as labelled, can be used to prove the Kochen-Specker theorem for two qutrits. This is related to the fact that the selected product of operators on a line is the identity matrix I except for the dotted line where it is ω 3 I (see details in the text). the entries ω 6 − 1 and −ω 6 and vice versa. The IC is obtained thanks to the action of the 2QB-QT Pauli group on this state. The distinct trace products are multivalued with eight values. There is an interesting structure of the 144 triple products whose trace is −1/27. They are organized into 12 distinct configurations of the type shown in figure 6 (with only four values of pair products occurring).

Summary and conclusion
The main contribution of our work is the construction of asymmetric IC-POVMs built thanks to the action of the Pauli group on appropriate permutation generated magic/fiducial states. A summary of the work is in table 1. It is remarkable that the same corpus of ideas may be used simultaneously for permutation groups, universal quantum computing, unambiguous quantum state recovery and also