Multipletime states

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Multiple-time states measurements
Jeff Tollaksen Chapman University
FQXi 2nd International Conference Ponta Delgada,
Azores, July 7-12, 2009
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The time reversed description of a quantum system
Backward Evolving Quantum State
The Quantum State Evolving Backward
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The two-state vector description of a quantum
system
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The two-state vector description of a quantum
system
Measurements performed on a pre- post-selected
system described by the two-state vector
Strong measurement The Aharonov-Bergmann-Lebowitz
(ABL) formula
Weak measurement The Aharonov-Albert-Vaidman
effect
Weak value
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Multiple-time states measurements Aharonov,
Popescu, Tollaksen, Vaidman, Phys Rev A 79,
052110 (May 1, 2009)
The remarkable thing about two time states is
that, similar to ordinary quantum states, we can
form superpositions.
e.g. a 2 time-state
Multiple-time measurements are measurements
consisting of multiple measurement stages, but
which cannot be decomposed into separate
measurements, 1 for each time
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PRL 58, 1385 (1987)
protection
Confirmed experimentallySchulz, et al  PRL 90
,177901 (2003)
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Multiple-time states Aharonov, Popescu,
Tollaksen, Vaidman, Phys Rev A 79, 052110 (May 1,
2009)

• Whenever we consider multiple instants of time,
  the most general object is any combination of
  bras kets, e.g.

• four-time state w/ well-defined past future
  (determined by the initial preparation final
  post-selection) two measurement periods
  (t1lttltt2) and (t3lttltt4). The multi-time state is
  a vector in the Hilbert space
• expanded in basis states

• Another 4-time state both the future and the
  past are uncertain

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Multiple-time measurements operators Aharonov,
Popescu, Tollaksen, Vaidman, Phys Rev A 79,
052110 (May 1, 2009)

• This is an observable that gives the value zero
  in the case when the x-component of the spin is
  the same at the two times, but doesn't offer any
  information about the actual value of the
  x-component
• could yield three possible values 2, 0 and -2.
  To each of these values we associate a multi-time
  projector

• These are entangled states measurements are
  collections of bras kets very similar to
  states
• no simple description in the standard quantum
  formalism

• kinematic dynamical descriptions are united

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Multiple-time states measurements Aharonov,
Popescu, Tollaksen, Vaidman, Phys Rev A 79,
052110 (May 1, 2009)

• The operator is not there in order to evolve the
  state, but is part of the state itself
• true force of the formalism w/ multi-time
• To obtain the state of the system given the
  outcome k of the POVM we insert the multi-time
  Krauss operator into the original multi-time
  state i.e. we use POVM not just to prepare a
  state
• but to test it
• multi-time state is covariant

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QM Generalization Each moment of time a new
universe

• Consider a spin 1/2 particle with constant time
  evolution

• Ques can we prepare a set of N of these
  particles such that if we perform at some given
  time t0 measurements on these N particles we'll
  get the same information as we'd have obtained by
  making measurements at t1, t2...tN on the
  original single particle? E.g.

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QM Generalization Each moment of time a new
universe

• For QM, this doesnt work, due to multi-time
  correlations

• Measuring sx (t4) - s x (t2) for the single
  spin-1/2 particle on the left will not re-produce
  multi-time correlations.

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QM Generalization Each moment of time a new
universe

• New ability to obtain a post-selected state of
  one particle that is completely correlated to the
  pre-selected state of a second particle

• stack N particles on top of another along the
  time axis

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Conclusions

• introduced new approach to quantum mechanics
  multi-time states describing experimental
  situations consisting of multiple preparation and
  measurement stages
• Put states operators on equal footing,
  complementary
• implications for the problem of the flow of time

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Time-Symmetric formulation of QM (TSQM)
To be useful and interesting, any re-formulation
of QM should meet several criteria, for example

• TSQM is consistent with all the predictions made
  by standard QM,
• TSQM brings out features in QM that were missed
  before (e.g. weak values, QRW)
• TSQM lead to simplifications in calculations and
  stimulated discoveries in other fields
• TSQM suggests generalizations of QM