Quantum Measurements with Post-selection

5/22/2018 Quantum Measurements with Post-selection

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Quantum MeasurementsWeak Value Amplification

State Reconstruction with Post-selected MeasurementsSummary

Quantum Measurements with Post-selection

Anirban Ch Narayan Chowdhury

IISER Pune

Supervisor

Prasanta K. Panigrahi

IISER Kolkata

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Introduction

A quantum system is initialized in a state| and later found to be in somestate b of an observable B. Can we measure another observable A in between

these two occurrences?

Aharonov, Albert and Vaidman postulated a weak measurement of A givensuch pre- and post-selection.

Weak measurement outcomes can be different from eigenvalues and exhibitother strange properties.

Used to analyse conceptual paradoxes in quantum mechanics.

Experimental technique for signal detection and amplification. Also used in quantum tomography, control of quantum systems.

Aharonov, Albert, and Vaidman, PRL 60, 13511354 (1988)

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Introduction








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Introduction








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Outline

1 Quantum MeasurementsVon Neumann Measurement Model

Measurement with Post-selection

Weak Measurement

Examples: Measurements of Spin and Polarization

Are Weak Measurements Meaningful?2 Weak Value Amplification

Large meter Shifts and Amplification

Experimental Implementations

Amplification Limits

Comparison of Shifts and SNR

Conclusion3 State Reconstruction with Post-selected Measurements

Essential Idea

State reconstruction of spin- 12

particle4 Summary

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V Ne Me s e e t M del


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Von Neumann Measurement ModelMeasurement with Post-selectionWeak MeasurementExamples: Measurements of Spin and PolarizationAre Weak Measurements Meaningful?

The von Neumann Measurement Model

A model of projective quantum measurements.

Observable A of system Smeasured through change in meter Q ofmeasuring device D.

Interaction given by H= k(t)A P, where [ Q, P] =i andt0k(t)dt= g.

System Device System+Device

|in = i ci|ai |in = |q U=eigAP i ci|ai|q+gai

Entangled state - each eigenstate of A corresponds to a unique shiftq=gai in Q; invert to obtain measurement outcome.

Observation causesstate vector collapsegiving distribution of shifts.

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Von Neumann Measurement Model


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The von Neumann Measurement Model

A model of projective quantum measurements.

Observable A of system Smeasured through change in meter Q ofmeasuring device D.

Interaction given by H= k(t)A P, where [ Q, P] =i andt0k(t)dt= g.


|in = i ci|ai |in = |q U=eigAP i ci|ai|q+gai

Entangled state - each eigenstate of A corresponds to a unique shiftq=gai in Q; invert to obtain measurement outcome.

Observation causesstate vector collapsegiving distribution of shifts.

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Von Neumann Measurement Model


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Device



.

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Q t Me s e e tsVon Neumann Measurement Model


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Measurement with Post-selectionWeak MeasurementExamples: Measurements of Spin and PolarizationAre Weak Measurements Meaningful?







Device



.

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Quantum MeasurementsVon Neumann Measurement Model


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42 . Weak


(q) exp (qg Aw)2

42

.

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Quantum Measurements Von Neumann Measurement ModelM i h P l i


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42 . Weak


(q) exp (qg Aw)2

42

.

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Quantum Measurements Von Neumann Measurement ModelMeasurement with Post selection


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QWeak Value Amplification



Weak Measurement

Initial system state superposition of eigenstates of A. But weak measurement gives a single outcome probability

distribution centered around Aw.

Approximations valid in the following limit:g | Aw|, g

fi| An|infi| A|in1/(n1) n 2

Properties of weak values

Weak value can be different from eigenvalues, and can even lie outside therange of eigenvalues.

Weak measurement outcomes have large associated uncertainty. Weak values can be complex, in which case:q=g Aw, p= g22 Aw

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Quantum Measurements Von Neumann Measurement ModelMeasurement with Post selection


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Weak Value AmplificationState Reconstruction with Post-selected Measurements

Summary


Weak Measurement








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Quantum Measurements Von Neumann Measurement ModelMeasurement with Post-selection


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Summary

Measurement with Post selectionWeak MeasurementExamples: Measurements of Spin and PolarizationAre Weak Measurements Meaningful?

Weak Measurement








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Quantum MeasurementsW k V l A lifi i

Von Neumann Measurement ModelMeasurement with Post-selection


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Summary

easu e e t t ost se ect oWeak MeasurementExamples: Measurements of Spin and PolarizationAre Weak Measurements Meaningful?

Example: Spin-12 Particles in modified Stern-Gerlach

Figure: Measurements with post-selection on spin- 12 particles

Momentum Pz as meter. Weak value:zw= tan 2 Initial:

pz| = 1(22)1/4 expp2z

42

Final:

f(P) =

cos 2 + sin

2

exp

(P s)2

+

cos 2 sin 2

exp(P+ s)2

P= pz2, s= g2

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Summary

Weak MeasurementExamples: Measurements of Spin and PolarizationAre Weak Measurements Meaningful?


4 2 2 4P

0.1

0.2

0.3

0.4

0.5

fP

s2.0, 0.4

(a) s=2. Two unequal peaks are

clearly observed, located at2.

3 2 1 1 2 3P

0.1

0.2

0.3

0.4

0.5

0.6

fP

s1.0, 0.4

(b) s=1. The two Gaussians

interfere, the peaks here are locatedat P= 1.047 and P=1.022.

Figure: Graph of f(P) against P for= 0.4 and s=2, 1.

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Summary



2 1 1 2P

0.2

0.4

0.6

0.8

fP

s0.2, 0.4

(a) Destructive interferenceproduces a probability distribution

with one large and one small peakfor s=0.2, located at P= 1.014and P=0.514 respectively.

0.10 0.05 0.05 0.10 0.15 0.20P

0.770

0.775

0.780

0.785

0.790

0.795

0.800

fPs0.01, 0.4

(b)Final probability distribution isapproximately a single Gaussian for

s=0.01, peaked around P=0.049.

Figure: Graph of f(P) against P for= 0.4 and s=0.5, 0.01.

Duck, Stevenson and Sudarshan, Phys. Rev. D, 40, 21122117 (1989) 10/28


Von Neumann Measurement ModelMeasurement with Post-selectionW k M


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Summary


Experimental Realization: Polarization of Light

Figure: Optical setup for weak measurement of polarization states of light.Polarizer P1 pre-selects initial polarization. A birefringent crystal Q performsmeasurement interaction by spatially separating x and y polarizations. P2post-selects, and intensity is recorded by photodetector D.

Ritchie, Story and Hulet, PRL 66, 9 (1991)

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Von Neumann Measurement ModelMeasurement with Post-selectionW k M t


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ea a ue p cat oState Reconstruction with Post-selected Measurements

Summary


Are Weak Measurements Meaningful?

Weak values have been denoted asactual outcomes of weakmeasurements on pre- andpost-selected ensembles.

Used to analyse long-standingquantum paradoxes such as Hardysparadox.

Larger than any eigenvalue,

negative, even complexmeasurement outcomes?

Large uncertainty, so effectively aconditional average. Figure: Hardys thought experiment

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Large meter Shifts and AmplificationExperimental ImplementationsAmplification Limits


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pState Reconstruction with Post-selected Measurements

Summary

Amplification LimitsComparison of Shifts and SNRConclusion


Projective measurements give meter shifts g A, gai. Weak measurements yield shifts, qw= g Aw, pw= g22 Aw Aw can be larger than maximum ai.

Possibility of amplifying small signals that are otherwise not detectable due tonoise.

Q2w,P2w are unaffected to 1st order in g.

In the weak measurement limit, no additional noise (measured by q, p) is

introduced.

Quality of amplification measured by signal-to-noise ratio (SNR):

R=N

qq

(projective), R=NW

qwqw

(weak), W= |fi|in|2.

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introduced.


R=N

qq

(projective), R=NW

qwqw


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S R h P l d M



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introduced.


R=N

qq

(projective), R=NW

qwqw


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St t R t ti ith P t l t d M t



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pComparison of Shifts and SNRConclusion






introduced.


R=N

qq

(projective), R=NW

qwqw


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State Reconstruction with Post selected Measurements



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pComparison of Shifts and SNRConclusion

Experimental Implementations

Hosten and Kwiat, Observation ofthe Spin Hall Effect of Light via

Weak Measurements, Science 319,787 (2008)

Dixon, Starling, Jordan, andHowell, Ultrasensitive BeamDeection Measurement via

Interferometric Weak Value

Amplication, PRL 102, 173601(2009)

Starling, Dixon, Jordan, andHowell, Precision frequencymeasurements with

interferometric weak values, PRA82, 063822 (2010)

Figure: Experimental Setup of DSJHfor measuring beam-deflection.

Which-path information in Sagnacinterferometer as system and positionof laser beam as meter. Beamdeflected by tilted mirror to left orright, depending on which-path state.

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State Reconstruction with Post selected Measurements



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Comparison of Shifts and SNRConclusion


Weak measurement approximation (small g, large ) suggests indefinitelylarge shifts and amplification.

Upper bounds to shifts and SNR obtainable from higher order corrections/exact calculations.

Use of alternate meter distributions other than simple gaussian may achievebetter amplification.

Calculate exact expressions of shifts and SNR for A2 =Itype observables.

Compare shifts and SNR for three different pointer distributions:(i) (x) e

x242 (ii) (x) xe

x242 (HG)

(iii) (x,y) (x+ iy)ex2+y2

42 (LG)

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State Reconstruction with Post-selected Measurements

Large meter Shifts and AmplificationExperimental ImplementationsAmplification LimitsC i f Shif d SNR


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State Reconstruction with Post selected MeasurementsSummary

Comparison of Shifts and SNRConclusion


Weak measurement approximation (small g, large ) suggests indefinitelylarge shifts and amplification.

Upper bounds to shifts and SNR obtainable from higher order corrections/exact calculations.

Use of alternate meter distributions other than simple gaussian may achievebetter amplification.

Calculate exact expressions of shifts and SNR for A2 =Itype observables.

Compare shifts and SNR for three different pointer distributions:(i) (x) e

x242 (ii) (x) xe

x242 (HG)

(iii) (x,y) (x+ iy)ex2+y2

42 (LG)

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Large meter Shifts and AmplificationExperimental ImplementationsAmplification LimitsC i f Shift d SNR


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SummaryComparison of Shifts and SNRConclusion

Exact Expressions

Interaction Hamiltonian: Hint=g(t)A P

Exact expressions for gaussian distribution of meter

gPf= Awses

Z;

Xfg

= AwZ

g2P2w= s4Z

1 + | Aw|2

+ es

| Aw|2 1

(2s 1)

X2wg2

= 1

4sZ1 | Aw|2

es +

1 + | Aw|2(1 + 2s)

Z=

1

2

1 + | Aw|2 +

1 | Aw|2

es

All quantities expressed in terms ofs= g2

22.

Similar expressions for other distributions.

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Large meter Shifts and AmplificationExperimental ImplementationsAmplification LimitsComparison of Shifts and SNR


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Calculating Exact Meter Shifts

Pre-selection:|in = 12

cos2 + sin2

| +cos 2 sin 2 | .Post-selection:

|

fi= 1

2(|

+|

).

To observe shifts in coordinate, perform weak measurement ofz whichgives,zw= tan 2.To observe shifts in momentum, perform weak measurement ofx which

gives,xw= i tan 2.

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Comparison of Meter Shifts

Shifts in x

0.5 1.0 1.5 2.0 2.5 3.0

0.2

0.4

0.6

0.8

1.0

1.2

x

s0.5

0.5 1.0 1.5 2.0 2.5 3.0

0.2

0.4

0.6

0.8

1.0

x

s1.0

0.5 1.0 1.5 2.0 2.5 3.0

0.2

0.4

0.6

0.8

1.0

x

s1.5

Figure: x= X

g as a function of the pre-selection angle for different values

of the coupling parameter s= g2

22. solid, dotted and dashed lines correspond to

Gaussian, HG and LG modes respectively.

There exists an optimum pre-selection angle which maximizes the pointer shiftfor given s.

In the small s limit, gaussian gives larger shifts in x whereas in the large s limit,HG gives larger shifts.

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State Reconstruction with Post-selected MeasurementsS



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Comparison of Meter Shifts

Shifts in p

0.5 1.0 1.5 2.0 2.5 3.0

0.1

0.2

0.3

0.4

0.5

0.6

p

s0.5

0.5 1.0 1.5 2.0 2.5 3.0

0.1

0.2

0.3

0.4p

s1.0

0.5 1.0 1.5 2.0 2.5 3.0

0.05

0.10

0.15

0.20

0.25

0.30

p

s1.7

Figure: p=P

g as a function of the pre-selection angle for different values

of the coupling parameter s= g2

22. solid, dotted and dashed lines correspond to

Gaussian, HG and LG modes respectively.

In the small s limit, larger shifts in p can be achieved using alternate meterstates.

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State Reconstruction with Post-selected MeasurementsS



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Summaryp

Conclusion

Comparison of Signal-to-Noise Ratios

SNR for x:

0.5 1.0 1.5 2.0 2.5 3.0

0.5

1.0

1.5

SNR

s0.5

0.5 1.0 1.5 2.0 2.5 3.0

0.5

1.0

1.5

2.0

SNR

s1.0

0.5 1.0 1.5 2.0 2.5 3.0

1

2

3

4

SNR

s4.5

Figure: Representative plots of the SNR for x, R(x)w /

N, plotted against for

s= 0.5, 1.0, 4.5. Thin, dotted and dashed lines correspond to Gaussian, HG

and LG modes respectively.

Maximum SNR possible is larger for gaussian initial state, for all s.

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Summaryp

Conclusion

Comparison of Signal-to-Noise Ratios

SNR for p:

0.5 1.0 1.5 2.0 2.5 3.0

0.2

0.4

0.6

0.8

1.0

1.2

SNR

s0.5

0.5 1.0 1.5 2.0 2.5 3.0

0.05

0.10

0.15

0.20

0.25

SNR

s1.7

0.5 1.0 1.5 2.0 2.5 3.0

0.02

0.04

0.06

0.08

0.10

0.12

SNR

s3.5

Figure: Representative plots of the SNR for x, R(x)w /

N, plotted against for

s= 0.5, 1.0, 4.5. Thin, dotted and dashed lines correspond to Gaussian, HGand LG modes respectively.

Alternate meter distributions give larger maximum SNR in p than gaussian forsome values ofs, especially in the small s limit.

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SummaryConclusion

Enhancement in Amplification?

Using Hamiltoniang(t)A P, using modified pointer distributions give highermeter shifts and SNRs in the weak measurement limit (small s), for P

observable.

But s= g2

22 where = x (initial uncertainty in x).

p 1.

Small s implies small p.Higher SNR in p because initial uncertainty in p is small.

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Essential IdeaState reconstruction of spin-1

2 particle


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Summary

State Reconstruction using Post-selection

Post-selection so far

Perform von Neumman type interaction with A observable, followed by

projective measurement of observable B, picking out one|bi as post-selection.

Instead, measure average meter shift individually for each|bi.Reconstruct initial quantum state from these shifts.

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2 particle


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y

State Reconstruction using Post-selection

Post-selection so far

Perform von Neumman type interaction with A observable, followed by

projective measurement of observable B, picking out one|bi as post-selection.

Instead, measure average meter shift individually for each|bi.Reconstruct initial quantum state from these shifts.

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2 particle


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y

Pure State Reconstruction

Initial unknown state|in =cos| + eisin| .Von Neumann interaction ofz observable, H=g(t)zP.

Post-select on the basis:| = 12

(| |)

Approximate (1st order in g) calculations

q+

g=

cos2

1 + cossin 2

qg

= cos2

1 cossin 2

Invert to obtain:

cos2=

q+q

g(q++q)

cos= 1

sin2

q q+q +q+

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2 particle


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(| |)


q+

g=

cos2

1 + cossin 2

qg

= cos2

1 cossin 2

Invert to obtain:

cos2=

q+q

g(q++q)

cos= 1

sin2

q q+q +q+

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2 particle


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(| |)


q+

g=

cos2

1 + cossin 2

qg

= cos2

1 cossin 2

Invert to obtain:

cos2=

q+q

g(q++q)

cos= 1

sin2

q q+q +q+

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2 particle


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(| |)

Exact calculations

q+

g=

cos2

1 + cossin 2es

qg

= cos2

1 cossin 2es

Invert to obtain:

cos2=

q+q

g(q++q)

cos= 1

es sin2q q+q +q+

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2 particle


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Mixed State Reconstruction

Density matrix for spin- 12 particle

=

sin2 +pcos2 sinsin 2

sinsin 2 cos2 pcos2

Meter shifts

q+

g= (cos2)

2p c 11 c2

qg

= (cos2)2p+ c 1

1

c2

gp+= (ses sinsin 2)

2p+ c 11 c2

gp= (ses sinsin 2)2p c 1

1

c2

where c=es cossin 2

Measure qand pshifts on separate sub-ensembles to obtain full density matrix.

Wu, Sci. Rep., vol. 3, (2013)

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Summary

Review of quantum measurements

von Neumann modelpost-selectionweak measurement leading to non-eigenvalue meter shift; strange

weak valuesWeak value amplification

Larger meter shifts and SNR than projective measurementsExact expressions for A2 =Itype observablesComparison of maximum shifts and SNR for alternate meter states

State ReconstructionMeasure shifts induced by each projection in post-selection stepState reconstruction for spin- 12 particles

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Acknowledgements

Dr. Alok K. Pan

Prof. Prasanta K. Panigrahi

Prof. Dr. T.S. Mahesh

Department of Physical Sciences, IISER Kolkata

IISER Pune

THANKS

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