Features of Sequential Weak Measurements Lajos Di´ osi Wigner Centre, Budapest 22 June 2016, Waterloo Acknowledgements go to: EU COST Action MP1209 ‘Thermodynamics in the quantum regime ’ Perimeter Institute Lajos Di´ osi (Wigner Centre, Budapest) Features of Sequential Weak Measurements 22 June 2016, Waterloo 1 / 11
WM vs post-selection 1 SWMs without post-selection 2 SWM of canonical variables 3 SWM of spin- 1 2 observables 4 Testing SWM in Time-Continuous Measurement 5 SWM with post-selection 6 Re-selection paradox 7 Example: spin- 1 8 2 References 9 Lajos Di´ osi (Wigner Centre, Budapest) Features of Sequential Weak Measurements 22 June 2016, Waterloo 2 / 11
WM vs post-selection WM vs post-selection In unsharp (imprecise) measurement on ˆ ρ , post-measurement state preserves some well-defined features of ˆ ρ . Imprecision a of measurement can be compensated by larger ensemble statistics. Weak measurement (WM): asymptotic limit of zero precision a → ∞ (and infinite statistics): pre-measurement state ˆ ρ invariably survives the measurement (non-invasiveness). WM was used by AAV as non-invasive quantum measurement between pre- and post-selected states, resp. Non-invasiveness of WM is remarkable both with and without post-selection, can be maintained for a succession of WMs on a single quantum system. General features of such sequential WMs (SWMs): our topics. Lajos Di´ osi (Wigner Centre, Budapest) Features of Sequential Weak Measurements 22 June 2016, Waterloo 3 / 11
SWMs without post-selection SWMs without post-selection A : measured; A : outcome; M : statistical mean; � ˆ ˆ A � : q-expectation. M A = � ˆ A � — single WM 2 �{ ˆ A , ˆ M AB = 1 B }� — double WM: order doesn’t matter { ˆ A , { ˆ B , ˆ — triple WM: ˆ B , ˆ M ABC = 1 � � C }} C are interchangeable 8 (Bednorz & Belzig 2010) Generally: { ˆ A 1 , { ˆ A 2 , { . . . , { ˆ A n − 1 , ˆ 1 � � M A 1 A 2 . . . A n = A n } . . . }}} 2 n Correlation of SWM outcomes = = Step-wise symmetrized quantum correlation of operators Ordering in SWM matters but the last two ones are interchangeable. Sufficient condition of full interchangeability: [ˆ A k , ˆ A l ] = c-number ( k , l = 1 , 2 , . . . , n ). Then step-wise symmetrization ⇒ symmetrization S : S ˆ A 1 ˆ A 2 . . . ˆ A n − 1 ˆ � � M A 1 A 2 . . . A n = A n Lajos Di´ osi (Wigner Centre, Budapest) Features of Sequential Weak Measurements 22 June 2016, Waterloo 4 / 11
SWM of canonical variables SWM of canonical variables ˆ A k = u k ˆ q + v k ˆ p ( k = 1 , 2 , . . . , n ) where [ˆ q , ˆ p ] = i Step-wise symmetrization ⇒ symmetrization S = Weyl ordering! Weyl-ordered correlation functions of ˆ q , ˆ p = = correlation functions (moments) of Wigner function W ( q , p ). � M A 1 A 2 . . . A n = W ( q , p ) A 1 A 2 . . . A n d q d p ≡ � A 1 A 2 . . . A n � W (for n = 2: Bednorz & Belzig 2010) Direct tomography through Wigner function moments: Example: SWM of ˆ q , ˆ q , ˆ p , ˆ p (in any order) yields � q � W = M q 1 = M q 2 ; � p � W = M p 1 = M p 2 � q 2 � W = M q 1 q 2 ; � p 2 � W = M p 1 p 2 , � qp � W = M q 1 p 1 = M q 1 p 2 = M q 2 p 1 = M q 2 p 2 � q 2 p � W = M q 1 q 2 p 1 = M q 1 q 2 p 2 ; � p 2 q � W = M p 1 p 2 q 1 = M p 1 p 2 q 2 � q 2 p 2 � W = M q 1 q 2 p 1 p 2 Lajos Di´ osi (Wigner Centre, Budapest) Features of Sequential Weak Measurements 22 June 2016, Waterloo 5 / 11
SWM of spin- 1 2 observables SWM of spin- 1 2 observable σ k = ˆ ˆ σ 1 , ˆ σ 2 , . . . , ˆ σ� e k , | � SQM of A 1 = ˆ A 2 = ˆ A n = ˆ σ n ; (ˆ � e k | =1) Outcomes A 1 = σ 1 , A 2 = σ 2 , . . . , A n = σ n Surprize: 1 � � M σ 1 σ 2 . . . σ n = { ˆ σ 1 , { ˆ σ 2 , { . . . , { ˆ σ n − 1 , ˆ σ n } . . . }}} ( ∗ ) 2 n is valid no matter the measurements are weak or strong (ideal). R.h.s. for SSM (with ˆ P ± = 1 2 (1 ± ˆ σ ): �� �� � � σ n = ± 1 σ n ˆ P ( n ) σ 2 = ± 1 σ 2 ˆ P (2) σ 1 = ± 1 σ 1 ˆ P (1) ρ ˆ P (1) P (2) ˆ . . . ˆ P ( n ) tr � σ n . . . σ 1 ˆ σ 2 σ 1 σ 2 σ n σ = ± σ ˆ P σ ˆ O ˆ σ, ˆ P σ = 1 Key identity � 2 { ˆ O } , using it n -times yields ( ∗ )! Evaluating r.h.s. yields � ( � e 1 � e 2 )( � e 3 � e 4 ) . . . ( � e n − 1 � e n ) n even M σ 1 σ 2 . . . σ n = � ˆ σ 1 � ( � e 2 � e 3 ) . . . ( � e n − 1 � e n ) n odd Correlations are kinematically constrained: n even — correlations are independent of ˆ ρ n odd — correlations depend on ˆ ρ but via � ˆ σ 1 � Lajos Di´ osi (Wigner Centre, Budapest) Features of Sequential Weak Measurements 22 June 2016, Waterloo 6 / 11
Testing SWM in Time-Continuous Measurement Testing SWM in Time-Continuous Measurement TCM is standard theory. TCMs are standard in lab. TCMs have WM regime! TCM of Heisenberg ˆ A t in state ˆ ρ , outcomes (signal) A t : A t � + √ α w t ; α : precision/unsharpness of TCM A t = � ˆ w t : standard white-noise TCM is invasive on the long run but it remains non-invasive as long as � t 0 � (∆ˆ A s ) 2 � ds ≪ α. That’s where SQM applies to signal’s auto-correlation: 2 �{ ˆ A t 1 , ˆ M A t 1 A t 2 = 1 A t 2 }� 2 �{ ˆ A t 1 , { ˆ A t 2 , ˆ M A t 1 A t 2 A t 3 = 1 A t 3 }}� etc. Recall r.h.s.’s must be Wigner function moments if ˆ A is harmonic, kinematically constrained if ˆ A is spin- 1 2 . Lajos Di´ osi (Wigner Centre, Budapest) Features of Sequential Weak Measurements 22 June 2016, Waterloo 7 / 11
SWM with post-selection SWM with post-selection Outcome correlations: { ˆ A 1 , { ˆ A 2 , . . . , { ˆ A n , ˆ � � Π } . . . }} M A 1 , A 2 , . . . , A n | psel = . 2 n � ˆ � Π Generic post-selection (D. 2006, Silva & al. 2014) : 0 ≤ ˆ Π ≤ 1. ρ = | i �� i | , ˆ For pure state pre/post-selection ˆ Π = | f �� f | , introduce sequential weak values: f | ˆ A n ˆ A n − 1 . . . ˆ � A 1 | i � ( A 1 , A 2 , . . . , A n ) w = � f | i � M A 1 , A 2 , . . . , A n | psel = 1 � ( A i 1 , A i 2 , . . . , A i r ) w ( A j 1 , A j 2 , . . . , A j n − r ) ⋆ w 2 n Σ for all partitions ( i 1 , i 2 , . . . , i r ) ∪ ( j 1 , j 2 , . . . , j n − r ) = (1 , 2 , . . . , n ) where i ’s and j ’s remain ordered. Degenerate partitions r = 0 , n , too, must be counted. (Mitchison, Jozsa, Popescu 2007) n = 1 reduces to AAV 1988. n = 2 contains a new paradox. Lajos Di´ osi (Wigner Centre, Budapest) Features of Sequential Weak Measurements 22 June 2016, Waterloo 8 / 11
Re-selection paradox Re-selection paradox Special post-selection: | i � = | f � , call it re-selection . For single WM, re-selection is equivalent with no-post-selection: M A = M A | rsel = � ˆ A � WMs are non-invasive, we expect re-selection and no-post-selection are equivalent. But they aren’t, already for n =2 and ˆ A 1 = ˆ A 2 = ˆ A : � i | ˆ A 2 | i � , M A 1 A 2 = 1 A 2 | i � + 1 2 � i | ˆ 2( � i | ˆ A | i � ) 2 M A 1 A 2 | rsel = Re-selection decreases M A 1 A 2 by half of (∆ A ) 2 in state | i � : M A 1 A 2 − M A 1 A 2 | rsel = 1 2(∆ A ) 2 . (1) Unexpected anomaly! Reason is finite contribution of outcomes discarded by re-selection. Lajos Di´ osi (Wigner Centre, Budapest) Features of Sequential Weak Measurements 22 June 2016, Waterloo 9 / 11
Example: spin- 1 2 Example: spin- 1 2 R disc — rate of discards; a — precision/unsharpness of measurements M . . . | rsel = M · · · − lim a →∞ ( R disc M . . . | disc ) In WM limit a → ∞ of re-selection: R disc → 0. Single WM of ˆ σ ≡ ˆ σ x , outcome σ 1 with re-selection | i � = | f � = |↑� : R disc ∼ (1 / 4 a 2 ) → 0. M σ 1 | disc =0 hence R disc M σ 1 | disc =0 anyway. SWM of ˆ σ 1 =ˆ σ 2 ≡ ˆ σ x , outcomes σ 1 , σ 2 with re-selection | i � = | f � = |↑� : R disc ∼ (1 / 2 a 2 ) → 0. M σ 1 σ 2 | disc = a 2 hence R disc M σ 1 σ 2 | disc → 1 / 2, QED. Correlation of double ˆ σ x WM in state |↑� diverges on the discarded events in re-selection. Explains why re-selection differs from no-post-selection. Novel SWM anomalies add to AAV88. Lajos Di´ osi (Wigner Centre, Budapest) Features of Sequential Weak Measurements 22 June 2016, Waterloo 10 / 11
References References Y. Aharonov, D.Z. Albert and L. Vaidman, Phys. Rev. Lett. 60 , 1351 (1988). L. Di´ osi: Quantum mechanics: weak measurements, v4, p276 in: Encyclopedia of Mathematical Physics, eds.: J.-P. Fran¸ coise, G.L. Naber, and S.T. Tsou (Elsevier, Oxford, 2006). G. Mitchison, R. Jozsa, and S. Popescu, Phys. Rev. A 76 , 062105 (2007). A. Bednorz and W. Belzig, Phys. Rev. Lett. 105 , 106803 (2010); Phys. Rev. A 83 , 052113 (2011). R. Silva, Y. Guryanova, N. Brunner, N. Linden, A. J. Short, and S. Popescu, Phys. Rev. A 89 , 012121 (2014). L. Di´ osi, Phys. Rev. A (accepted); arXiv:1511.03923 Lajos Di´ osi (Wigner Centre, Budapest) Features of Sequential Weak Measurements 22 June 2016, Waterloo 11 / 11
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