Thursday, July 19, 2012

I Ching and Bloch Sphere

Bloch sphere
The Bloch or Poincare sphere seems particularly instrumental to understanding how I Ching may be related to quantum mechanics; perhaps even an ancient depiction of it.  I Ching is founded on two principles, Ch'ien and K'un, or the Creative and the Receptive, commonly known as YANG and YIN, and discussed at length in various appendices to I Ching (namely, appendices III and IV).  Appendix III, the Great Treatise, is quite clear on the distinction between hexagrams #1 and #2 and the remainder of I Ching.  

Stated clearly in several verses, Ch'ien and K'un are the "pole," "axis, or "gates" of Change, without which Change could not be seen and would cease to exist. (*provide  references*).  These passages provide support to the intuition that hexagrams #1 and #2 are intended to act as basis vectors and define the spherical system that represents the I Ching.  

Spherical I Ching
We now set aside Ch'ien and K'un for consideration of the remaining thirty-one pairs of hexagrams. 

Some effort was made earlier to find meaningful correspondence between the human body's 31 pairs of spinal nerves and the 31 (+1) hexagram pairs, but these efforts have not borne fruit.

A presentation from the Vienna Center for Parallel Computing shows us that our complementary or xian tian hexagram pairs having scalar values which sum to 65 are regarded as orthogonal and identical states (except for phase/sign) on the Bloch sphere.  The model of I Ching presented in these blogs accounts not only orthogonal hexagram pairs, but also the pairs given by Wen, Duke of Chou.  
I Ching tradition describes Duke's imprisonment at the hands of the evil tyrant of Shang, and of the pairing scheme he devised during his period of captivity.  

In general, Wen's pairs are distinct figurate inversions (read: 180°rotation) except in the eight cases where inversion does not produce an hexagram distinct from the original:  
[(1,2) (51,57) (29,30) (52,58)]. In those cases, the complementary pair is taken as the Wen pair.

The graphic of the Bloch sphere (at left) highlights its important features:
basis vectors |0 and |1, arbitrary quantum state psi (Ψ), two coordinate systemspolar: denoted by angles theta (Θ) and phi (Φ); and rectangular: denoted by orthogonal axes x,y,z.  

Selecting one of the remaining 56 'non-distinct' hexagrams to approach this question spatially, we see that complementary pairs are opposite and orthogonal points on the Bloch sphere, while Wen pairs lay on the same (or opposite latitude) to the selected hexagram and  may also be complementary to the selected hexagram.  We see in the graphic above the north and south poles (denoted by |0 and |1⟩) along with psi (Ψforming an imaginary triangle.  
Arbitrarily selecting hexagram #41, we get its Wen pair #42, and #31 as its orthogonal point.  These three hexagrams form a corresponding imaginary triangle: #41 and #42 analogous to the poles, and #31 as the surface point.  The hexagram, its Wen inverse, and its opposite form a triplet: two hexagrams share a latitude while two share a diameter.  Selecting any hexagram from the eight pairs listed above produces just a diametric pair instead of a triplet since the Wen pair is also the opposite hexagram.

If we accept that I Ching is analogous with Bloch's sphere as a working model or depiction of quanta, the question begs, what is to be made of Wen's pairs?  We understand that Wen pairs are figurate inverses, but what does this mean in terms of the Bloch sphere and how might this inform us to quanta?

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