Pre- and Post-Selection Paradoxes, Measurement-Disturbance and Contextuality in Quantum Mechanics M. S. Leifer (joint work...
Background and Motivation <ul><li>Pre- and Post-Selection was introduced by Aharonov, Bergmann and Lebowitz (ABL) in 1964....
Background and Motivation <ul><li>Suggests possibilities for interpretation of QM. </li></ul><ul><ul><li>Reality of both s...
Background and Motivation <ul><li>The counterfactual interpretation has several difficulties </li></ul><ul><ul><li>We will...
Why should we care about PPS systems? <ul><li>Despite the controversy over interpretation, the TSVF agrees with QM for act...
Outline <ul><li>Operational Notation </li></ul><ul><li>Pre- and Post-Selection in Quantum Theory </li></ul><ul><ul><li>Qua...
1) Operational Notation <ul><ul><li>M -  A measurement. </li></ul></ul><ul><ul><li>X M 2 {1, 2, ..., n} -  The outcome of ...
2)  Pre- and Post-Selection in Quantum Theory <ul><li>2.1)  Quantum Measurement Formalism </li></ul><ul><li>States: Densit...
2.2)  The ABL Rule <ul><li>Assumptions </li></ul><ul><ul><li>Density matrix is   I / d  prior to pre-selection. </li></ul>...
2.2)  The ABL Rule <ul><li>Special cases: </li></ul><ul><ul><li>Intermediate L üders rule </li></ul></ul><ul><ul><li>Rank-...
2.3)  The “Three-Box Paradox” | 1  i | 2  i | 3  i Pre-selection:   |  pre   i  = | 1  i  + | 2  i  + | 3  i Post-selecti...
3)  Hidden Variable Theories <ul><li>3.1)  The “Partitioned-Box Paradox” </li></ul><ul><ul><li>Left verification </li></ul...
3.1)  The “Partitioned Box Paradox” <ul><li>Pre-selection:  Successful Front Verification. </li></ul><ul><li>Post-selectio...
3.2)  General Formalism of HVTs <ul><li>Set of ontic states     2    </li></ul><ul><li>States correspond to probability...
3.2)  General Formalism of HVTs <ul><li>Update rule </li></ul><ul><li>3.3)  Pre- and Post-Selection in HVTs </li></ul><ul>...
3.4)  Noncontextuality <ul><li>If there is an outcome  j  of   M  and an outcome  k  of   N  that have the same probabilit...
3.5) Noncontextuality, PPS and Disturbance <ul><li>We can also try to apply these constraints to ABL probabilities. </li><...
4)  A Surprising Theorem <ul><li>Whenever a set of projectors have ABL probabilities that give rise to a logical PPS parad...
4.2)  Failure of the Product Rule <ul><li>For observables  B  =   j    j   P j   in an MNHVT we can define values by </l...
4.2)  Failure of the Product Rule <ul><li>Pre-select:  |   pre   i  = 2 -1/2  ( | 0  i   ­  | 0  i  + | 1  i   ­  | 1  i ...
4.2)  Failure of the Product Rule Q - P - R + Projectors onto Eigenvalues R +  =  P +   Q -  +  P -   Q + R +  =  P +   Q ...
5)  Conclusions and Open Questions <ul><li>We have shown that: </li></ul><ul><ul><li>The existence of Logical PPS paradoxe...
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Pre- and Post-Selection Paradoxes, Measurement-Disturbance and Contextuality in Quantum Mechanics

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Talk from circa 2004 on pre- and post-selection paradoxes in quantum theory and their relationship to contextuality. Based on the results of Phys. Rev. Lett. 95, 200405 (2005) (http://arxiv.org/abs/quant-ph/0412178) and Proceedings of QS 2004, Int. J. Theor. Phys. 44:1977-1987 (2005) (http://arxiv.org/abs/quant-ph/0412179). This talk was recorded and can be viewed online at http://pirsa.org/04110008/

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Pre- and Post-Selection Paradoxes, Measurement-Disturbance and Contextuality in Quantum Mechanics

  1. 1. Pre- and Post-Selection Paradoxes, Measurement-Disturbance and Contextuality in Quantum Mechanics M. S. Leifer (joint work with R. W. Spekkens) 10th November 2004 Coming soon to an arXiv/quant-ph near you!
  2. 2. Background and Motivation <ul><li>Pre- and Post-Selection was introduced by Aharonov, Bergmann and Lebowitz (ABL) in 1964. </li></ul><ul><li>Two-State-Vector Formalism (TSVF) (see Aharonov & Vaidman, 2001) </li></ul><ul><li>Foundational motivation - Time symmetry in the quantum measurement process. </li></ul>Pre-selection Measurement Post-selection t pre t t post time Measurement
  3. 3. Background and Motivation <ul><li>Suggests possibilities for interpretation of QM. </li></ul><ul><ul><li>Reality of both states - Aharonov (2001): </li></ul></ul><ul><ul><ul><li>“ Even at present, before the ‘future’ measurements, the backward evolving quantum state [...] exists! It exists in the same way as the quantum state evolving from the past exists. An element of arbitrariness: ‘Why this particular outcome and not some other?’ might discourage, but the alternative [...] – the collapse of the quantum wave – is clearly worse than that.” </li></ul></ul></ul><ul><ul><li>Counterfactual interpretation - Mohrhoff (2000): </li></ul></ul><ul><ul><ul><li>“ ABL probabilities are based on a complete set of facts, and are therefore objective, only if none of the measurements to the possible results of which they are assigned is actually performed (that is, only if between the ‘preparation’ or preselection and the ‘retroparation’ or postselection no measurement is performed).” </li></ul></ul></ul>Measurement
  4. 4. Background and Motivation <ul><li>The counterfactual interpretation has several difficulties </li></ul><ul><ul><li>We will see that “Logical PPS paradoxes” would imply that reality is contextual. </li></ul></ul><ul><ul><li>Further, a similar interpretation of “classical” probability would also imply that reality is contextual. </li></ul></ul><ul><ul><ul><li>Reason - It does not take the possibility that measurements might disturb the system into account. </li></ul></ul></ul><ul><ul><li>There is no compelling reason to treat the ABL probabilities counterfactually. </li></ul></ul><ul><li>See Kastner Phil. Sci. 70, 145 (2003) for refs. </li></ul><ul><li>Main results: </li></ul><ul><ul><li>Logical PPS paradoxes can be explained in noncontextual hidden variable theories, provided measurements disturb the values of the hidden variables. </li></ul></ul><ul><ul><li>However, for every logical PPS paradox there is a related proof of contextuality, which can be constructed from the same set of measurements. </li></ul></ul>
  5. 5. Why should we care about PPS systems? <ul><li>Despite the controversy over interpretation, the TSVF agrees with QM for actual measurements. </li></ul><ul><li>TSVF has yielded results of interest to quantum information theorists: </li></ul><ul><ul><li>The Mean King’s Problem (Vaidman et. al., 1987) </li></ul></ul><ul><ul><li>Cryptography protocols (Bub, 2000, Botero & Reznik, 1999) </li></ul></ul><ul><li>The formalism has been extended to more general types of pre- and post-selection (Aharovov & Reznik, 1995). </li></ul><ul><ul><li>e.g. correlated pre- and post-selections of the type used in standard quantum cryptography (BB84, B92, etc.) </li></ul></ul><ul><ul><li>There is an analog of entanglement, which might be regarded as a resource for qinfo. </li></ul></ul><ul><li>It seems likely that many PPS effects could be exploited in quantum information protocols. </li></ul><ul><ul><li>To find out if this is the case, we need to know whether these effects can be simulated “classically” and, if so, how efficiently? This means that we should re-examine the foundational questions with this in mind. </li></ul></ul>
  6. 6. Outline <ul><li>Operational Notation </li></ul><ul><li>Pre- and Post-Selection in Quantum Theory </li></ul><ul><ul><li>Quantum Measurement Formalism </li></ul></ul><ul><ul><li>The ABL Rule </li></ul></ul><ul><ul><li>The “Three-Box Paradox” </li></ul></ul><ul><li>Hidden Variable Theories </li></ul><ul><ul><li>The “Partitioned-Box Paradox” </li></ul></ul><ul><ul><li>General Formalism of HVTs </li></ul></ul><ul><ul><li>Pre- and Post-Selection in HVTs </li></ul></ul><ul><ul><li>Noncontextuality </li></ul></ul><ul><ul><li>Noncontextuality, PPS and Disturbance </li></ul></ul><ul><li>A Surprising Theorem </li></ul><ul><ul><li>Example: “Three-box” and Clifton </li></ul></ul><ul><ul><li>Example: “Failure of the Product Rule” </li></ul></ul><ul><li>Conclusions and Open Questions </li></ul>
  7. 7. 1) Operational Notation <ul><ul><li>M - A measurement. </li></ul></ul><ul><ul><li>X M 2 {1, 2, ..., n} - The outcome of M. </li></ul></ul><ul><ul><li>pre/post - Refers to pre/post-selection. </li></ul></ul><ul><ul><li>A pre /A post - The occurrence of successful pre/post-selection. </li></ul></ul><ul><li>Goal: Compute p ( X M = j | A pre , A post , M) </li></ul>Pre-selection M Post-selection A pre A post X M
  8. 8. 2) Pre- and Post-Selection in Quantum Theory <ul><li>2.1) Quantum Measurement Formalism </li></ul><ul><li>States: Density operators,  , on a  d -dimensional H. S. </li></ul><ul><li>Sharp quantum measurements </li></ul><ul><ul><li>Statistical aspect of M given by a PVM { P M,j } </li></ul></ul><ul><ul><li>P M ,j 2 = P M , j  j P M, j = I </li></ul></ul><ul><ul><ul><li>Born rule: p  ( X M = j ) = Tr( P M ,j  ) </li></ul></ul></ul><ul><ul><li>Transformation aspect given by CP-maps { E M,j } </li></ul></ul><ul><ul><li>E M ,j # ( I ) = P M ,j where Tr( E # ( A ) B ) = Tr( A E ( B )) </li></ul></ul><ul><ul><li>On obtaining X M = j : </li></ul></ul><ul><ul><li> ! E M ,j (  ) / Tr( E M, j (  )) </li></ul></ul><ul><ul><li>L üders Rule (projection postulate) </li></ul></ul><ul><ul><li>E M ,j (  ) = P M, j  P M ,j </li></ul></ul>
  9. 9. 2.2) The ABL Rule <ul><li>Assumptions </li></ul><ul><ul><li>Density matrix is I / d prior to pre-selection. </li></ul></ul><ul><ul><li>Pre-selection measurement obeys L üders Rule. </li></ul></ul><ul><li>Notation </li></ul><ul><ul><li>A pre (A post ) corresponds to a projector  pre (  post ). </li></ul></ul><ul><ul><li>The measurement M corresponds to a PVM { P M, j } </li></ul></ul><ul><ul><li>and CP-maps { E M, j }. </li></ul></ul>
  10. 10. 2.2) The ABL Rule <ul><li>Special cases: </li></ul><ul><ul><li>Intermediate L üders rule </li></ul></ul><ul><ul><li>Rank-1 pre- and post-selection projectors </li></ul></ul><ul><ul><ul><li> pre = |  pre i h  pre |  post = |  post i h  post | </li></ul></ul></ul>
  11. 11. 2.3) The “Three-Box Paradox” | 1 i | 2 i | 3 i Pre-selection: |  pre i = | 1 i + | 2 i + | 3 i Post-selection: |  post i = | 1 i + | 2 i - | 3 i Intermediate measurements: M: { P M,1 = | 1 ih 1 |, P M,2 = | 2 i h 2 | + | 3 ih 3 | } N: { P N,1 = | 2 ih 2 |, P N,2 = | 1 ih 1 | + | 3 ih 3 | } p ( X M = 1 | A pre , A post , M) = p ( X N = 1 | A pre , A post , N) = 1 Reason: h  post | P M,2 |  pre i = h  post | P N,2 |  pre i = 0 Counterfactual interpretation: David Blaine is in both boxes at the same time!
  12. 12. 3) Hidden Variable Theories <ul><li>3.1) The “Partitioned-Box Paradox” </li></ul><ul><ul><li>Left verification </li></ul></ul><ul><ul><li>Similarly for Right Front and Back verification. </li></ul></ul>F B L R or ? ? ? ?
  13. 13. 3.1) The “Partitioned Box Paradox” <ul><li>Pre-selection: Successful Front Verification. </li></ul><ul><li>Post-selection: Successful Back Verification </li></ul><ul><li>Possible Intermediate measurements: </li></ul><ul><ul><li>Left Verification or Right Verification </li></ul></ul><ul><ul><li>LV case </li></ul></ul><ul><ul><li>Counterfactual interpretation: The ball is both on the left and the right. </li></ul></ul><ul><ul><li>Correct interpretation: Different measurements disturb the ball in different ways. </li></ul></ul>? ? ? ? ? ?
  14. 14. 3.2) General Formalism of HVTs <ul><li>Set of ontic states  2  </li></ul><ul><li>States correspond to probability distributions  (  ). </li></ul><ul><li>Assumption: Outcome determinism for sharp measurements. </li></ul><ul><li>Measurement M corresponds to: </li></ul><ul><ul><li>Statistical aspect: indicator functions {  M, j } </li></ul></ul><ul><ul><li> M, j (  ) = 0 or 1  j  M, j (  ) = 1 </li></ul></ul><ul><ul><li>p  ( X M = j ) = s   M ,j (  )  (  ) d  </li></ul></ul><ul><ul><li>Transformation aspect: Stochastic transformations {D M, j } </li></ul></ul><ul><ul><li>D M, j (  ,  ) is prob. of transition from   to  </li></ul></ul><ul><ul><li>Define transition matrices  M, j (  ,  ) = D M, j (  ,  )  M, j (  ) </li></ul></ul><ul><ul><li>s   M, j (  ,  ) d  =  M ,j (  ) </li></ul></ul>
  15. 15. 3.2) General Formalism of HVTs <ul><li>Update rule </li></ul><ul><li>3.3) Pre- and Post-Selection in HVTs </li></ul><ul><li>Pre-selection prepares  pre. </li></ul><ul><li>Post-selection corresponds to  post. </li></ul><ul><li>Intermediate measurement M: {  M, j }, {  M, j } </li></ul><ul><li>By Bayes' Theorem: </li></ul>
  16. 16. 3.4) Noncontextuality <ul><li>If there is an outcome j of M and an outcome k of N that have the same probability for all preparations then </li></ul><ul><li> M, j =  N, k </li></ul><ul><li>In QM equivalence  both outcomes associated to the same projector P. </li></ul><ul><li> M, j =  N, k =  P </li></ul><ul><li> projectors associated to unique properties. </li></ul><ul><li> for any distribution  can assign probabilities to projectors via p(P) = s   P (  )  (  )d  </li></ul><ul><li>Algebraic constraints </li></ul>
  17. 17. 3.5) Noncontextuality, PPS and Disturbance <ul><li>We can also try to apply these constraints to ABL probabilities. </li></ul><ul><li>Definition: We have a Logical PPS paradox whenever the ABL rule assigns probability 1 or 0 to the outcomes of measurements, such that associating the same probabilities to the projectors corresponding to those outcomes would violate the algebraic constraints. </li></ul><ul><ul><li>The Three-Box paradox is an example of this. </li></ul></ul><ul><li>You could explain Logical PPS paradoxes via contextuality. However, this is not required as our “partitioned box” example shows. </li></ul><ul><li>Reproducing ABL predictions does place nontrivial constraints on the transitions  M, j . </li></ul><ul><ul><li>They must be nontrivial  M, j (  ,  ) .   (  ,  )  M, j (  ). </li></ul></ul><ul><ul><li>Outcomes associated with the same projector, but different CP-maps, must be associated with different transitions. </li></ul></ul>
  18. 18. 4) A Surprising Theorem <ul><li>Whenever a set of projectors have ABL probabilities that give rise to a logical PPS paradox, and the pre- and post-selection projectors are not orthogonal, one can use fine grainings of the projectors, together with the pre- and post-selection projectors to show that a MNHVT is impossible. </li></ul><ul><li>4.1) Three box paradox and Clifton’s proof </li></ul>
  19. 19. 4.2) Failure of the Product Rule <ul><li>For observables B =  j  j P j in an MNHVT we can define values by </li></ul><ul><ul><li>v ( B ) =  k iff p ( P k ) = 1 </li></ul></ul><ul><li>Algebraic conditions imply that for [ A,B ] = 0 </li></ul><ul><ul><li>v ( B ) + v ( C ) = v ( B + C ) Sum Rule </li></ul></ul><ul><ul><li>v ( BC ) = v ( B ) v ( C ) Product Rule </li></ul></ul><ul><li>Similarly we can define ABL values </li></ul><ul><ul><li>v ABL ( B ) =  k iff p ( X B = j | A pre , A post , B ) = 1 </li></ul></ul><ul><ul><li>These can violate Sum and Product rules even in noncontextual models, but whenever they do we can find a related proof of contextuality. </li></ul></ul>
  20. 20. 4.2) Failure of the Product Rule <ul><li>Pre-select: |  pre i = 2 -1/2 ( | 0 i ­ | 0 i + | 1 i ­ | 1 i ) </li></ul><ul><li>Post-select: |  post i = | 0 i ­ | + i </li></ul><ul><li>Intermediate observables: </li></ul><ul><ul><li>X ­ I v ABL ( X ­ I ) = 1 </li></ul></ul><ul><ul><li>I ­ Z v ABL ( I ­ Z ) = 1 </li></ul></ul><ul><ul><li>X ­ Z v ABL ( X ­ Z ) = -1 </li></ul></ul>
  21. 21. 4.2) Failure of the Product Rule Q - P - R + Projectors onto Eigenvalues R + = P + Q - + P - Q + R + = P + Q + + P - Q - X ­ Z Q - Q + I ­ Z P - P + X ­ I -1 +1 Observable
  22. 22. 5) Conclusions and Open Questions <ul><li>We have shown that: </li></ul><ul><ul><li>The existence of Logical PPS paradoxes in a theory does not imply contextuality. </li></ul></ul><ul><ul><li>Nonetheless, each Logical PPS paradox is related to a proof of contextuality. </li></ul></ul><ul><li>Possible reason for discrepancy: </li></ul><ul><ul><li>HVTs have to satisfy additional constraints in order to reproduce quantum predictions. </li></ul></ul><ul><ul><li>Conjecture: Existence of Logical PPS paradoxes + some suitable analog of the uncertainty principle implies contextuality. </li></ul></ul><ul><ul><ul><li>Justification: Toy theory of Spekkens has such an analog, is noncontextual and seems to be devoid of logical PPS paradoxes. </li></ul></ul></ul><ul><li>Applications of logical PPS paradoxes in quantum information theory? </li></ul><ul><li>Use of our results in axiomatics of quantum theory? </li></ul>

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