Features of Sequential Weak Measurements

Transcription

1 Features of Sequential Weak Measurements Lajos Diósi Wigner Centre, Budapest 22 June 2016, Waterloo Acknowledgements go to: EU COST Action MP1209 Thermodynamics in the quantum regime Perimeter Institute Lajos Diósi (Wigner Centre, Budapest) Features of Sequential Weak Measurements 22 June 2016, Waterloo 1 / 11

2 1 WM vs post-selection 2 SWMs without post-selection 3 SWM of canonical variables 4 SWM of spin- 1 observables 2 5 Testing SWM in Time-Continuous Measurement 6 SWM with post-selection 7 Re-selection paradox 8 Example: spin References Lajos Diósi (Wigner Centre, Budapest) Features of Sequential Weak Measurements 22 June 2016, Waterloo 2 / 11

3 WM vs post-selection WM vs post-selection In unsharp (imprecise) measurement on ˆρ, post-measurement state preserves some well-defined features of ˆρ. 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 ˆρ 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 Diósi (Wigner Centre, Budapest) Features of Sequential Weak Measurements 22 June 2016, Waterloo 3 / 11

4 SWMs without post-selection SWMs without post-selection Â: measured; A: outcome; M: statistical mean; Â : q-expectation. MA = Â single WM MAB = 1 {Â, ˆB} double WM: order doesn t matter 2 MABC = 1 8 { Â, {ˆB, Ĉ}} triple WM: ˆB, Ĉ are interchangeable (Bednorz & Belzig 2010) Generally: MA 1 A 2... A n = 1 2 n { Â 1, {Â2, {..., {Ân 1, Ân}... }}} 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: [Â k, Â l ] = c-number (k, l = 1, 2,..., n). Then step-wise symmetrization symmetrization S: MA 1 A 2... A n = SÂ1Â2... Ân 1Ân Lajos Diósi (Wigner Centre, Budapest) Features of Sequential Weak Measurements 22 June 2016, Waterloo 4 / 11

5 SWM of canonical variables SWM of canonical variables  k = u k ˆq + v kˆp (k = 1, 2,..., n) where [ˆq, ˆp] = i Step-wise symmetrization symmetrization S = Weyl ordering! Weyl-ordered correlation functions of ˆq, ˆp = = correlation functions (moments) of Wigner function W (q, p). MA 1 A 2... A n = W (q, p)a 1 A 2... A n dqdp A 1 A 2... A n W (for n = 2: Bednorz & Belzig 2010) Direct tomography through Wigner function moments: Example: SWM of ˆq, ˆq, ˆp, ˆp (in any order) yields q W = Mq 1 = Mq 2 ; p W = Mp 1 = Mp 2 q 2 W = Mq 1 q 2 ; p 2 W = Mp 1 p 2, qp W = Mq 1 p 1 = Mq 1 p 2 = Mq 2 p 1 = Mq 2 p 2 q 2 p W = Mq 1 q 2 p 1 = Mq 1 q 2 p 2 ; p 2 q W = Mp 1 p 2 q 1 = Mp 1 p 2 q 2 q 2 p 2 W = Mq 1 q 2 p 1 p 2 Lajos Diósi (Wigner Centre, Budapest) Features of Sequential Weak Measurements 22 June 2016, Waterloo 5 / 11

6 SWM of spin- 1 2 observables SWM of spin- 1 2 observable SQM of  1 = ˆσ 1,  2 = ˆσ 2,...,  n = ˆσ n ; (ˆσ k = ˆ σ e k, e k =1) Outcomes A 1 =σ 1, A 2 =σ 2,..., A n =σ n Surprize: Mσ 1 σ 2... σ n = 1 2 n {ˆσ1, {ˆσ 2, {..., {ˆσ n 1, ˆσ n }... }}} ( ) is valid no matter the measurements are weak or strong (ideal). R.h.s. for SSM (with ˆP ± = 1 (1 ± ˆσ): 2 tr ( σ σ (n) n=±1 n ˆP σ n... σ2=±1 σ 2 ˆP (2) σ 2 ( σ 1 =±1 σ 1 (1) ˆP σ 1 ˆρˆP σ (1) )ˆP(2) ) 1 σ 2...ˆP σ (n) n Key identity σ=± σ ˆP σ ÔˆP σ = 1 {ˆσ,Ô}, using it n-times yields ( )! 2 Evaluating r.h.s. yields { ( e1 e Mσ 1 σ 2... σ n = 2 )( e 3 e 4 )... ( e n 1 e n ) n even ˆσ 1 ( e 2 e 3 )... ( e n 1 e n ) n odd Correlations are kinematically constrained: n even correlations are independent of ˆρ n odd correlations depend on ˆρ but via ˆσ 1 Lajos Diósi (Wigner Centre, Budapest) Features of Sequential Weak Measurements 22 June 2016, Waterloo 6 / 11

7 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  t in state ˆρ, outcomes (signal) A t : A t = Ât + α : precision/unsharpness of TCM αw t ; w t : standard white-noise TCM is invasive on the long run but it remains non-invasive as long as t ( Âs) 2 ds α. 0 That s where SQM applies to signal s auto-correlation: MA t1 A t2 = 1 { 2 t1,  t2 } MA t1 A t2 A t3 = 1 { 2 t1, { t2,  t3 }} etc. Recall r.h.s. s must be Wigner function moments if  is harmonic, kinematically constrained if  is spin Lajos Diósi (Wigner Centre, Budapest) Features of Sequential Weak Measurements 22 June 2016, Waterloo 7 / 11

8 SWM with post-selection SWM with post-selection Outcome correlations: MA 1, A 2,..., A n psel = {  1, {Â2,..., {Ân, ˆΠ}... }} 2 n ˆΠ. Generic post-selection (D. 2006, Silva & al. 2014): 0 ˆΠ 1. For pure state pre/post-selection ˆρ = i i, ˆΠ = f f, introduce sequential weak values: f  n  n 1... (A 1, A 2,..., A n ) w = Â1 i f i MA 1, A 2,..., A n psel = 1 2 n (Ai1, A i2,..., A ir ) w (A j1, A j2,..., A jn r ) w 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 n = 2 contains a new paradox. Lajos Diósi (Wigner Centre, Budapest) Features of Sequential Weak Measurements 22 June 2016, Waterloo 8 / 11

9 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: MA = MA rsel =  WMs are non-invasive, we expect re-selection and no-post-selection are equivalent. But they aren t, already for n =2 and Â1 =Â2 =Â: MA 1 A 2 = i  2 i, MA 1 A 2 rsel = 1 2 i Â2 i ( i  i )2 Re-selection decreases MA 1 A 2 by half of ( A) 2 in state i : MA 1 A 2 MA 1 A 2 rsel = 1 2 ( A)2. (1) Unexpected anomaly! Reason is finite contribution of outcomes discarded by re-selection. Lajos Diósi (Wigner Centre, Budapest) Features of Sequential Weak Measurements 22 June 2016, Waterloo 9 / 11

10 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 ˆσ ˆσ x, outcome σ 1 with re-selection i = f = : R disc (1/4a 2 ) 0. Mσ 1 disc =0 hence R disc Mσ 1 disc =0 anyway. SWM of ˆσ 1 =ˆσ 2 ˆσ x, outcomes σ 1, σ 2 with re-selection i = f = : R disc (1/2a 2 ) 0. Mσ 1 σ 2 disc =a 2 hence R disc Mσ 1 σ 2 disc 1/2, QED. Correlation of double ˆσ 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 Diósi (Wigner Centre, Budapest) Features of Sequential Weak Measurements 22 June 2016, Waterloo 10 / 11

11 References References Y. Aharonov, D.Z. Albert and L. Vaidman, Phys. Rev. Lett. 60, 1351 (1988). L. Diósi: Quantum mechanics: weak measurements, v4, p276 in: Encyclopedia of Mathematical Physics, eds.: J.-P. Françoise, G.L. Naber, and S.T. Tsou (Elsevier, Oxford, 2006). G. Mitchison, R. Jozsa, and S. Popescu, Phys. Rev. A 76, (2007). A. Bednorz and W. Belzig, Phys. Rev. Lett. 105, (2010); Phys. Rev. A 83, (2011). R. Silva, Y. Guryanova, N. Brunner, N. Linden, A. J. Short, and S. Popescu, Phys. Rev. A 89, (2014). L. Diósi, Phys. Rev. A (accepted); arxiv: Lajos Diósi (Wigner Centre, Budapest) Features of Sequential Weak Measurements 22 June 2016, Waterloo 11 / 11