Projective ring line encompassing two-qubits

Abstract

The projective line over the (non-commutative) ring of two-by-two matrices with coefficients in GF (2) is found to fully accommodate the algebra of 15 operators -generalized Pauli matrices -characterizing two-qubit systems. The relevant sub-configuration consists of 15 points each of which is either simultaneously distant or simultaneously neighbor to (any) two given distant points of the line. The operators can be identified with the points in such a one-to-one manner that their commutation relations are exactly reproduced by the underlying geometry of the points, with the ring geometrical notions of neighbor/distant answering, respectively, to the operational ones of commuting/non-commuting. This remarkable configuration can be viewed in two principally different ways accounting, respectively, for the basic 9+6 and 10+5 factorizations of the algebra of the observables. First, as a disjoint union of the projective line over GF (2) × GF (2) (the "Mermin" part) and two lines over GF (4) passing through the two selected points, the latter omitted. Second, as the generalized quadrangle of order two, with its ovoids and/or spreads standing for (maximum) sets of five mutually non-commuting operators and/or groups of five maximally commuting subsets of three operators each. These findings open up rather unexpected vistas for an algebraic geometrical modelling of finite-dimensional quantum systems and give their numerous applications a wholly new perspective. Projective lines defined over finite associative rings with unity/identity 1−7 have recently been recognized to be an important novel tool for getting a deeper insight into the underlying algebraic geometrical structure of finite dimensional quantum systems. 8−10 Focusing almost uniquely on the two-qubit case, i.e., the set of 15 operators/generalized four-by-four Pauli spin matrices, of particular importance turned out to be the lines defined over the direct product of the simplest Galois fields, GF (2) × GF (2) × . . . × GF (2). Here, the line defined over GF (2) × GF (2) plays a prominent role in grasping qualitatively the basic structure of so-called Mermin squares, 9,10 i. e., three-by-three arrays in certain remarkable 9 + 6 split-ups of the algebra of operators, whereas the line over GF (2) × GF (2) × GF (2) reflects some of the basic features of a specific 8 + 7 ("cubeand-kernel") factorization of the set. 10 Motivated by these partial findings, we started our quest for such a ring line that would provide us with a complete picture of the algebra of all the 15 operators/matrices. After examining a large number of lines defined over commutative rings, 6,7 we gradually realized that a proper candidate is likely to be found in the non-commutative domain and 1

The Veldkamp Space of Two-Qubits

“…As discovered in [1] (see also [3]), the fifteen generalized Pauli operators/matrices associated with the Hilbert space of two-qubits (see, e.g., [9]) can be put into a one-to-one correspondence with the fifteen points of the generalized quadrangle W (2) in such a way that their commutation algebra is completely and uniquely reproduced by the geometry of W (2) in which the concept commuting/non-commuting translates into that of collinear/non-collinear. Given this mapping, it was possible to ascribe a definitive geometrical meaning to sets of three pairwise commuting generalized Pauli operators in terms of lines of W (2) and to other three kinds of distinguished subsets of the operators having their counterparts in geometric hyperplanes of W (2) as shown in Table 2 (see [1,3] for more details).…”

“…V(Γ) is the space in which (i) a point is a geometric hyperplane of Γ and (ii) a line is the collection H 1 H 2 of all geometric hyperplanes H of Γ such that H 1 H 2 = H 1 H = H 2 H or H = H i (i = 1, 2), where H 1 and H 2 are distinct points of V(Γ). 1 If Γ = S, from the preceding paragraph we learn that the points of V(S) are, in general, of three different types.…”

“…Given this mapping, it was possible to ascribe a definitive geometrical meaning to sets of three pairwise commuting generalized Pauli operators in terms of lines of W (2) and to other three kinds of distinguished subsets of the operators having their counterparts in geometric hyperplanes of W (2) as shown in Table 2 (see [1,3] for more details). Yet, V(W (2)) puts this bijection in a different light, in which other three subsets of the Pauli operators come into play, namely those represented by the two types of a triad and by the specific pentads occurring as the core-sets of the lines of V(W (2)) ( Table 1).…”

“…Three kinds of the distinguished subsets of the generalized Pauli operators of two-qubits (PO) viewed as the geometric hyperplanes in the generalized quadrangle of order two (GQ) [1,3].…”

“…Recently, we made an important step in this respect by demonstrating that the commutation algebra of the generalized Pauli operators on the 2 N -dimensional Hilbert spaces is embodied in the geometry of the symplectic polar space of rank N and order two [1,2,3]. The case of two-qubit operator space, N = 2, was scrutinized in very detail [1,3] by explicitly demonstrating, in different ways, the correspondence between various subsets of the generalized Pauli operators/matrices and the fundamental subgeometries of the associated rank-two polar space -the (unique) generalized quadrangle of order two. In this paper we will reveal another interesting geometry hidden behind the Pauli operators of two-qubits, namely that of the Veldkamp space defined on this generalized quadrangle.…”