Mapping qubit algebras to combinatorial designs

Abstract

This work considers combinatorial designs in terms of the algebra of quantum spins, providing insights and techniques useful to both. The number 15 that is central to a famous problem of classical design theory, with origins in a recreational puzzle nearly two centuries old, coincidentally equals the number of basic operators of two quantum spins (“qubits”). This affords a 1:1 correspondence that we exploit to use the well-known Pauli spin or Lie–Clifford algebra of those 15 operators to provide specific constructions for that recreational problem. An algorithm is set up that, working with four basic operators conveniently rendered in a two-bit binary, generates solutions to quantum states. They can be mapped into four base colors or various tonal scales, thus leading to visual or acoustic realizations of each design. The Fano plane of finite projective geometry involving seven points and lines of an equilateral triangle, and the tetrahedron (a “three-dimensional simplex” of 15 points) are key objects in this study. They simplify the handling of two-qubit operator algebra and may be useful in wide contexts in the field of quantum information. In particular, the seven-parameter sub-algebra describes also what have been called X-states which, while being a subset of all two-qubit states, still embrace a wide variety of quantum correlations including entanglement and quantum discord. Extension to n qubits and higher-dimensional “qudits” with similar mappings of them to combinatorial designs and finite projective geometries is indicated.

Appendix: Color and acoustic designs of the Fano and Kirkman Patterns

1.1 Color designs for (7, 3, 1) and (15, 3, 1)

Fig. 7

$D_c(7,3,1)|Q_{10},Q_4,Q_{11}\rangle$

Fig. 8

$K_c(15,3,1)|Q_{12},Q_{10},Q_4,Q_{11}\rangle$

As noted in the text, color has played an enormous role in human perception, art, and philosophy in many cultures. The very naming of a color poses deep questions that have been addressed in literature [44]. Using the color–flavor language hereto developed, we can cast resolved design systems into color tilings. To construct a $(v,k,\lambda )$ tiling, we assign a subset of the v colors of the design to the k tiles in each block, and the v colors can be read directly from a resolved operator design with the use of our dictionary in Table 1. Hence, in Fig. 7, we see a color design that is a representation of the $D(7,3,1)|Q_{10},Q_{4},Q_{11}\rangle$ example used throughout this paper, especially in Sections IIIB and IV. Furthermore, if a Kirkman design (3n, 3, 1) is of odd n, we can find a resolved design wherein the v varieties of the design appear once in a number of discrete sets or “days.” For such a resolved Kirkman system, we construct rows of colored tiles packed into sets of all v colors. Hence, in Fig. 8, we see a Kirkman design that is a representation of the $D(15,3,1)|Q_{12},Q_{10},Q_{4},Q_{11}\rangle$ example used throughout this paper and which gives the Kirkman arrangement in Table 4.

1.2 Constructing acoustic designs

Our 15 schoolgirls have already made their appearance in music. While most western music is based on the canonical 12-tone chromatic scale, musical scales can be constructed on any number of notes through different octave divisions. The composer Tom Johnson used such an alternative octave division in his musical composition “Kirkman’s Ladies” [45], a sonic representation of a Kirkman design system in which the schoolgirls or tones are arranged into sets of three as chords over a 15-note scale. Johnson’s musical Kirkman design spans 7 intervals or “days” of 5 musical chords of three notes or “sets of three girls” such that 15 notes that form the chords obey the criteria of the Kirkman design system. At a notional level, musical composition strongly resembles the method of design construction. A composer arranges the distinct notes of the scale into blocks that form chords and then harmonies.

Following Gabor’s construction of acoustic quanta as products of windowing functions and harmonic oscillations, we choose our wavelets, taking special care to construct wavelets that when sounded together produce distinct chords for the human ear. We first address the windowing function, which can be written $W_n(t)=W(t-nh)$, where n denotes the temporal displacement and h denotes the hop size of the windowing function. There are a number of possible choices for the windowing function, including the square window, the B-spline, the Cosine window, and Gaussian function. Each type of windowing function introduces artifacts in the frequency spectrum of a note, yet we find that, as long as the windowing function is sufficiently smooth, the structure of the window has only a small effect on the way that notes sound together in chords, this being rather a consequence of the harmonic oscillations modulating each window. Thus, we choose the Gaussian function to window our notes and we adjust the width of the Gaussian so that the onsets and decays of the chords can be distinguished.

Because the harmonic aspect of the wavelet has the most significant effect on the distinguishability of different tonal bases, we must be careful with the definition of our 15-note scale. Inspired by the chromatic scale, we might be tempted to choose 15-note equal temperament. However, we will find that equal temperament will not provide a set of notes that are readily distinguishable by the human ear. The problem arises because the notes of an equal tempered scale are not very consonant among themselves, and so will produce chords that all sound very similar. It is for this reason that western music generally works with a subset of the chromatic scale, such as the C-major scale, to create music. In order to maximize the distinguishability of the notes, we must turn to some non-traditional ideas in music theory. One such idea is the use of “combination product sets” (CPS s) for scale production, which Ervin Wilson used to construct his famous “Hexany” system [44, 46]. To construct a CPS, one first picks a number N of prime numbers, excluding two as the octave number, and a combinatoric number M. The octave divisions of the scale are then found by multiplying M prime numbers and then rescaling the product by a power of two such that the elements lie on a range [1, 2] in the octave. A ‘tonic’ is then chosen for the scale, which defines the frequency of the lowest note. All further frequencies are then given by the product of the tonic and the other scaled products. We employ this method to construct our scale, using a $\hbox {CPS}(6,2)$ for the 15-note scale.

Using 15 tonal bases as a 15-note scale, the Fano (7, 3, 1) design rendered previously in color in Figs. 4 and 7 takes the acoustical form shown in Fig. 9 and the Kirkman (15, 3, 1) design of Figs. 6 and 8 the form shown in Fig. 10. Note how the (7, 3, 1) design in Fig. 9 is a subset formed by the first vertical bars of each of the 7 day panels in the (15, 3, 1) design of Fig. 10, an aspect of the nesting of the Fano triangle’s 2-simplex of Fig. 4 within the tetrahedron’s 3-simplex of Fig. 6 discussed earlier.

Fig. 9

$D_s(7,3,1)|Q_{10},Q_4,Q_{11}\rangle$

Fig. 10

$K_s(15,3,1)|Q_{12},Q_{10},Q_4,Q_{11}\rangle$