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Multiple-time states & measurements

Jeff Tollaksen Chapman University. Multiple-time states & measurements. FQXi 2nd International Conference Ponta Delgada, Azores, July 7-12, 2009. The time reversed description of a quantum system. Backward Evolving Quantum State. The Quantum State Evolving Backward.

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Multiple-time states & measurements

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  1. Jeff Tollaksen Chapman University Multiple-time states & measurements FQXi 2nd International Conference Ponta Delgada, Azores, July 7-12, 2009

  2. The time reversed description of a quantum system Backward Evolving Quantum State TheQuantum StateEvolving Backward

  3. The two-state vector description of a quantumsystem

  4. The two-state vector description of a quantumsystem 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

  5. 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

  6. PRL 58, 1385 (1987) protection Confirmed experimentally:Schulz, et al  PRL 90 ,177901 (2003)

  7. 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 (t1<t<t2) and (t3<t<t4). 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

  8. 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

  9. 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

  10. 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.:

  11. QM Generalization: Each moment of time a new universe • For QM, this doesn’t work, due to multi-time correlations: • Measuring σx (t4) - σx (t2) for the single spin-1/2 particle on the left will not re-produce multi-time correlations.

  12. 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:

  13. 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

  14. 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

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