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Starting Point• Electrons must have an intrinsic state• This state differs with orientation in 3d space • states along different orientations are dependent
Describing State Prob of being in the UP state p p 1-p 1-p Prob of being in the DN statep changes withthe orientation
Transformations p q • Tzx must be a Stochastic Tzx = Transformation 1-p 1-q – Non-negative entriesTransforms state – Each column sums to 1 along Z axis tostate along X axis
Stochastic Transformations • Can two stochastic matrices multiply toTxz Tzx = I yield an identity matrix? – All matrix entries are non- negative Transforms state along – So NO, unless each matrix is I! Z axis to state along X axis and then transform back Stochastic Transformations ruled out
Revisiting the State Description Can we allow for negative values a here? a2 +b2 =1 b Points on a unit How do we circle translate these to probabilities?
Transformations a • Tzx must be preserve a’ Tzx b = Euclidean length b’ – (Tzx)T Tzx = ITransforms state Cosθ -Sinθ along Z axis tostate along X axis Sinθ Cosθ For any θ
Z Explanations I X θ -X 1 0 1/√2 -Z 0 1 1/√2 Cos(θ/2) -Sin(θ/2) 1TZZ Sin(θ/2) Cos(θ/2) 0 Initial state along Z Initial state along Z TZXθ
Z X Explanations II -X 1 0 -Z 1/√2 0 1 1/√2 1/√2 -1/√2 1TZZ 1/√2 1/√2 1 1/√2 1/√2 0 -1/√2 1/√2 0 Initial state along Z TXZ = Inverse TZX of TZX Initial state along Z Initial state along X
Z X Bringing in the Y Dimension-Y Y Initial state along 1 1 TZXTYZ = TYX Y transformed to state along X 0 0-X -Z 1/√2 -1/√2 a c 1 +/- 1/√2 = 1/√2 1/√2 b d 0 +/- 1/√2 All UPs along Y All UPs along Y translate to equal translate to equal +/- 1/√2 UPs and DNs UPs and DNs along X along X +/- 1/√2 NOT POSSIBLE!!
Revisiting the State Description Yet Again Can we allow for complex valuesa here? Complexb conjugate |a|2 +|b|2 =a a + b b = 1 How do we translate these to probabilities?
Revisiting Transformations Conjugate Transpose a • Tzx must be preserve |a|2 a’ TYX b = b’ +|b|2 – (Tzx)† Tzx = ITransforms state eiεCosθ -ei(ψ – φ+ ε) Sinθ along Z axis tostate along X axis ei(φ+ ε) Sinθ ei(ψ+ ε) Cosθ For any θ,ψ,ε
Z X Bringing in the Y Dimension-Y Y Initial state along 1 1 TZXTYZ = TYX Y transformed to state along X 0 0-X -Z 1/√2 -1/√2 1/√2 -e-iφ 1/√2 1 eiφ’’ 1/√2 = 1/√2 1/√2 eiφ 1/√2 1/√2 0 eiφ’ 1/√2 All UPs along Y All UPs along Y translate to equal translate to equal 1/√2 UPs and DNs UPs and DNs along X along X eiφ 1/√2 Φ=π/2, Φ’’=-π/4, Φ’=π/4!!
The Final Transformations 1/√2 -1/√2 1/√2 i/√2 TZX 1/√2 1/√2 TYZ i/√2 1/√2 1/√2 -1/√2 1/√2 i/√2 TYX=TZXTYZ= 1/√2 1/√2 i/√2 1/√2 e-iπ/4/√2 -e-iπ/4/√2 eiπ/4/√2 eiπ/4/√2Can you write the transformation from Z to a general direction in 3D space?
Summary• State vector v has complex entries and satisfies – |v|2 = v†v = Σ |vi|2 = 1 • vi’s are called Amplitudes• Transformations T satisfy T†T = I – T’s are called Unitary Transformations• When we measure a system in state v – We get i with Probability |vi|2
Contrast with Classical States• Take 2 bits, so state vector [p1 p2 p3 p4] corresponding to 00, 01, 10, 11 resp.• Suppose you replace the first bit by an AND of the 2 bits with prob p and by an OR with prob 1-p? – Show this can be written as a stochastic transformation.
Our Two Worlds Σ vi = 1 , 0<=vi<=1 T is stochastic (non-neg, col sums 1)Classical MeasurementQuantum |v|2 = v†v = Σ |vi|2 = 1 T is Unitary T†T = I