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# Rules of a Quantum World

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### Rules of a Quantum World

1. 1. Rules of a Quantum World
2. 2. The Stern-Gerlach Experiment N ½Electron gun ½ S Ignore horizontal Beam splits into deflection as per two! Not a Fleming’s Left continuous Hand Rule spread
3. 3. Abstract Representation ½ UP Electron gun DN ½ Arrow points to the North pole
4. 4. Cascading Devices Z Z Z ½ ½Electron gun ½ -Z Only UP
5. 5. Z Cascading Devices Z -Z ½Electron gun -Z ½ ½ All DN!
6. 6. Z Cascading Devices Z X X Arrow goes into the screen ¼ -X ½Electron gun ¼ -Z ½ Half UP, Half DN!!
7. 7. Z Cascading Devices Z Xθ X θ At angle θ -X -Z Cos2 (θ/2)/2 ½Electron gun Sin2 (θ/2)/2 ½
8. 8. Z X Cascading Devices Z X Z -X -Z 1/8 ¼ ½ 1/8Electron gun ¼ ½ Down along Z reappears!
9. 9. How do we model this behaviour?
10. 10. Starting Point• Electrons must have an intrinsic state• This state differs with orientation in 3d space • states along different orientations are dependent
11. 11. 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
12. 12. 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
13. 13. 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
14. 14. 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?
15. 15. 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
16. 16. 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
17. 17. 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
18. 18. 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!!
19. 19. 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?
20. 20. 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
21. 21. 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!!
22. 22. 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?
23. 23. 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
24. 24. 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.
25. 25. 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
26. 26. What does this mean for computation?