By Lomonaco S.J.

The hot software of the foundations of quantum mechanics to cryptography has ended in a awesome new measurement in mystery conversation. because of those new advancements, it's now attainable to build cryptographic verbal exchange platforms which discover unau- unauthorized eavesdropping may still it take place, and which offer a warrantly of no eavesdropping should still it now not happen.

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Now suppose that (φ - φ') = 0. (71) Then we can write p(+,+) in a matrix formulation as "row matrix"  p(+,+) =  cos      θ' ,sin θ' 2 2 ⋅    θ  sin 2 2 cos θ 2 (7 2) . ) This interesting structure can be repeated when (φ - φ') ≠ 0. Consider the quantity Q =   -iφ'/2   e  ∗  iφ'/2 cos θ' ,e sin θ' ⋅ 2 2 e-iφ /2  iφ/2 e cos θ   2  sin θ  2  2 (73) Working backward, we can express this as Q = |cos 2θ θ' θ θ' cos 2 + sin 2 sin 2 ei(φ - φ') |2 . (74) Now 2 |a + b| = (a + b)∗(a + b) 2 = |a| + |b|2 + a∗b + b∗a , (75) } 2 Re(a ∗b) 2 θ 2 θ' 2 θ 2 θ' ⇒ Q = cos 2 cos 2 + sin 2 sin 2 θ θ' θ θ' + 2 cos 2 cos 2 sin 2 sin 2 cos(φ - φ').

141) ^ ⋅→ ^ is a unit spatial vector (n ^2 where n σ = n1σ1 + n2σ2 + n3σ3 and n = 1). ^ = e ^ , we can write (138) and Using (140) or (141) with n 3 (139) above as _ σ1 = e-iφσ3 σ1 (142) _ σ1 = σ1eiφσ3 . 58 [There are many ways of seeing the equality of (142) and (143). Since σ1 and σ3 anticommute, we have for example σ3σ1 = σ1(-σ3) and 2 σ3σ1 = σ1(-σ3)2. ] We now wish to prove that iλσ3 e eiλ'σ3 = ei(λ + λ')σ3 . (144) We can rewrite eiλσ3 = cos λ + iσ3 sin λ, ⇒ eiλσ3 = cos λ(|+| + |-|) + i(|+| - |-|)sin λ, ⇒ eiλσ3 = |+|eiλ + |-|e-iλ (145) Using (145), we now have iλσ3 e eiλ'σ3 = (|+|eiλ + |-|e-iλ) (|+|eiλ' + |-|e-iλ') = |+|ei(λ + λ') + |-|e-i(λ + λ') = ei(λ + λ')σ3 , which proves (144) above.

116) It is also clear that |a'v a"'||a"a'| = 0, a"' ≠ a" . (117) The algebraic properties of our two types of measurement symbols are summarized in |a'||a"| = δa'a"|a'|, |a'v a"'||a"a'| = δa"' a"|a'v a'|. , the order of the operations doesn't matter). However, these new types of measurements are not reversible. b. b. |a"a'| ⇒ |a"a'||a'| ≠ |a'||a"a'|. (120) Eqn(120) follows mathematically from (119) and the fact that |a'| = |a'a'|. b. b. 51 ⇒ |a'a"||a"a'| ≠ |a"a'||a'a"|. (121) The algebra of the |a'a"|-type symbols is non-commutative.

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