QUANTUM PROBABILITIES a novel derivation of the probability postulate

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QUANTUM PROBABILITIESa novel derivation of the
probability postulate

• Jason Doukas
• Goss Seminar
• 27/May/2005

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Standard Quantum Probability Postulate

• BORNS RULE
• When we perform a measurement M, on a state ?
  it collapses into an eigenstate of M, F, with
  probability ltF ?gt2.

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New philosophy
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Notation states(?SE)

• Can not use a vector ie ?S
• Need to allow for pure or mixed states.
• Can not use density matrix

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EnvarianceW. H. Zurek, Phys. Rev. Lett. 90,
120404 (2003)

• H S?E
• S System
• E Environment
• Environment Assisted Invariance
  UE(US ?SE)
  ?SE

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What does envariance tell us?

• Theorem
• If
• UE(US?SE) ?SE
  
• then the StateS(?SE) is unchanged by US.
• Proof
• StateS(?SE) StateS(UE(US?SE) ) by envariance
• StateS(US?SE)

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Schmidt decomposition?SE gt ?0s0gte0gt
?1s1gte1gt

• If ?0 ? ?1 then Us ei?k, Ue e-i?k
• Ue Us ?segt ?0 ei?0 e-i?0 s0gte0gt ?1 ei?1
  e-i?1 s1gte1gt
• ?segt
• Hence, state of S is independent of phases!

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What is the state of S?

• The state of S can not be a pure state
• StateS (?segt) ?sgt
• as0gt ßs1gt
• ? as0gt - ßs1gt
• ? ?sgt
• Therefore state of S is a mixed state!
• Ie 20 in s0gt and
• 80 in s1gt (or - s1gt doesnt
  matter!)

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Equal coefficient state

• ?segt (s0e0gt s1e1gt),
• Consider the swaps
• US s0gt ? s1gt
• UE e0gt ? e1gt
• For equal coefficients swaps are envariant.
• ? StateS(?se) is independent of swaps of the
  sigt states.

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Procedure

• Take an ensemble of ?segt and measure S and E for
  each

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• Experimentally Pr favourable outcomes/N
• Note Perfect correlation implies
• Pr (sigt) Pr (eigt)

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Borns Rule

• ?segt (s0e0gt s1e1gt)
• Us ?segt (s1e0gt s0e1gt)
• UE Us ?segt (s1e1gt s0e0gt)
• Pr (s1gt ?segt) Pr (e1gt ?segt)
• Pr (e1gt Us ?segt)
• Pr (s0gt Us ?segt)
• Pr (s0gt UE Us ?segt)
• Pr (s0gt ?segt)
• Pr(s0gt) Pr(s1gt)
• This is Borns rule!

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Unequal coefficients example

• ?segt (v2s0e0gt s1e1gt)
• Step 1 Remote environment
• e0gt e0e0gt
• e1gt e1e0gt
• ?segt ? v2s0e0e0gt s1e1e0gt

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• Step 2 fine-graining
• Let e0gt (e2gt e3gt) then
• ?segt ? v2s0e0e0gt s1e1e0gt
• ? s0e2e0gt s0e3e0gt s1e1e0gt
• Equal coefficient but no entanglement

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• Step 3 Control-shift operator,
• Ce eie0gt? eieigt
• Ce?segt ? s0e2gte2gt s0e3gte3gt s1e1gte1gt
• Equal coefficient form
• Pr(s0e2 gt) Pr(s0e3 gt) Pr(s1e1 gt) 1/3
• So,
• Pr(s0gt) 2/3
• Pr(s1gt) 1/3
• Thus, we have shown Borns rule.

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Conclusions

• Results completely generalise.
• The amplitude2 is shown to be equal to the
  probability.
• Many worlds
• Usually rejected because it can not explain
  probabilities.
• We might still exist in many worlds after all?