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International Conference on Quantum Information and Quantum Computing
Indian Institute of Science
Sub-Planck Structure and Weak
Measurement
INTRODUCTION
 Measurement plays a central role in quantum mechanics.
Precision of measurement is restricted not only by technology
but also by the theory itself.
 Heisenberg-limited metrology and weak measurements are the
two notions that find use in enhancing sensitivity of
measurements.
The first can increase precision whereas the second can be used
to amplify small signals.
 Both of these essentially rely on quantum mechanical
interference, but in two opposite regimes.
Sub-Planck and Weak Value
 The cat states and their generalization owe this precision to
sub-Planck structure in the phase space.
 Weak value of a dichotomic observable can also be inferred
from the cat state if the two coherent states comprising the
cat state are identified with the post-measurement meter
state in measuring that observable.
 We link these two notions of sub-Planck structure and weak
measurement in terms of the cat state.
 Specifically, we show that the parameter regime relevant to
Heisenberg limited measurement is opposite of what is
required for weak measurement.
Sub-Planck Structure
Sub-Planck structure is a striking effect of interference in phase
space. It explains oscillating photon number distribution of
squeezed states in a phase plane.
The Wigner distribution ideally captures such non-classical
interference phenomena, as it can be negative in a phase-space
region.
Zurek first demonstrated that such oscillatory structures
resulting from quantum interference, can lead to a fundamental
area in the phase space, smaller than the Planck’s constant.
Heisenberg-limited sensitivity has been demonstrated for
superposition of two coherent states and a classical wave optics
analogue has been tested experimentally .
Wigner function for compass state showing sub-Planck
scale structures in the interference region
Heisenberg Limited Sensitivity
For quasiclassical states ( e.g., coherent states) the sensitivity of
phase estimation is restricted by the standard quantum limit,
also known as the shot-noise limit.
Sub-shot-noise sensitivities can be obtained using the sensitivity
of the quantum state to displacements, which is related to the
sub-Planck phase space structure extracted from its Wigner
function.
A number of proposals have been advanced for generating
single particle cat and generalized states, showing the above
feature .
Weak Measurement
The path-breaking idea of ‘weak measurement’(WM) in QM,
proposed by Aharonov, Albert and Vaidman (AAV) .
In this measurement scenario, the empirically measured value
(coined as ‘weak value’) of an observable can go beyond the
eigenvalue spectrum of the measured observable.
WM has several implications, one being that it provides insight
into conceptual quantum paradoxes.
It has been used in identifying a tiny spin Hall effect, detecting
very small transverse beam deflections and measuring average
quantum trajectories for photons.
Ideal Quantum Measurement
System + Apparatus
|𝜓𝑖 =
𝑐𝑗 |𝑎𝑗 〉 ⊗ 𝑄0 = 𝛷𝑖
System + apparatus
𝐻 = 𝑔′ 𝑡 𝐴 ⊗ 𝑃
𝛷𝑓 =
𝑐𝑗 𝑎𝑗 |𝑄𝑗 〉
 System is coupled to apparatus, which is prepared in a known
initial state
 Through appropriate interaction, one-to-one correlation
established between apparatus and system property that is
being measured.
 Superposition in system leads to entangled system+apparatus
state.
 Experimenter records final apparatus state – entangled state
collapses stochastically; change in apparatus state proportional
to measurement outcome (eigenvalue of A).
Weak Measurement and Weak Value
 Ideal measurements require initial apparatus state to be
sharply peaked – allow uncertainty.
ideal apparatus
𝑞|𝑄0 = 𝛿(𝑞 − 𝑞0 )
practical apparatus
𝑞|𝑄0 = N exp −𝑞 2 /4𝜎 2
 After measurement interaction, post-select a final state of the
system, |𝜓𝑓 and then observe apparatus. Then, apparatus
state is
𝑄𝑓 = 𝜓𝑓 𝑒 −𝑖
𝑑𝑡𝑔′ 𝑡 𝐴𝑃
𝜓𝑖 𝑄0 = 𝜓𝑓 |𝑒 −𝑖𝑔𝐴𝑃 𝜓𝑖 𝑄0
 The measurement interaction is weak, i.e., coupling g is small.
Weak Measurement and Weak Value
𝑄𝑓 ≈ 𝜓𝑓 𝐼 − 𝑖𝑔𝐴𝑃 + ⋯ 𝜓𝑖 𝑄0 ≈ 𝜓𝑓 𝜓𝑖 1 − 𝑖𝑔𝐴𝑤 𝑃 |𝑄0
𝑞|𝑄𝑓 ≈ 𝜓𝑓 𝜓𝑖 𝑒 −
Weak Value
𝐴𝑤 =
𝑞−𝐴𝑤 2 /4𝜎 2
𝜓𝑓 𝐴 𝜓𝑖
𝜓𝑓 𝜓𝑖
 Final pointer approximately a gaussian peaked at weak value
 Weak value can be outside range of eigenvalues, depending
on pre- and post-selection
 The approximations used are valid in limited cases; we require
𝜎 ≪ 1/𝑔𝐴𝑤 and 𝜎 ≪ 𝑚𝑖𝑛
𝜓𝑓 𝐴 𝜓𝑖
𝑛=2,3…
1/(𝑛−1)
𝜓𝑓 𝐴𝑛 𝜓𝑖
 Single gaussian arising from interference effects due to large
variance in initial apparatus state. Duck, Stevenson, Sudarshan, PRD 40 6 (1989)
𝑔
Weak Measurement and Weak Value
 In contrast to strong measurement, the formalism of weak
measurement allows to extract information of a quantum system in
the limit of vanishingly small disturbance to its state.
 In strong measurement, while measuring an observable, the
pointer indicates the eigenvalues of the given observable.
 In weak measurement the pointer may indicate a value beyond
the eigenvalues range.
 The weak value can be complex. Real part gives pointer shift in
position space whereas imaginary part gives shift in momentum
space.
Weak Measurement with Stern-Gerlach
Post-selected spin state: |𝜒𝑓 〉
Initial state: |Ψ〉 = 𝜓0 𝑥 |𝜒𝑖 〉
𝐻 = 𝑔𝛿 𝑡 − 𝜏 𝑥 𝐴
Weak
Strong
𝜓0 𝑥 =
1
2𝜋𝜎 2 − 4
𝑥2
− 2
𝑒 4𝜎
After first SG the state is |Ψ ′ 〉 = 𝑒 −𝑖𝑔𝑥 𝐴 /ℏ 𝜓0 𝑥 |𝜒𝑖 〉
If the interaction is weak, 𝑒 −𝑖𝑔𝑥 𝐴 /ℏ = 1 − 𝑖𝑔𝑥 𝐴 /ℏ + O(2)- . . . . .
If particles are post-selected in the spin state |𝜒𝑓 〉, the device state is
𝜓𝑓 𝑥 = 𝜒𝑓 𝛹 ′ = 〈𝜒𝑓 |𝜒𝑖 〉[1 − 𝑖𝑔𝑥 𝐴𝑤 /ℏ]𝜓0 𝑥
= 〈𝜒𝑓 |𝜒𝑖 〉 𝑒 −𝑖𝑔𝑥 𝐴𝑤 /ℏ 𝜓0 𝑥
The final pointer state in momentum space: 𝜙𝑓 𝑝𝑥 =
〈𝜒𝑓 |𝜒𝑖 〉
2𝜎 2
𝜋ℏ2
1
4
𝑒
−
𝜎2 (𝑝𝑥 −𝑔𝐴𝑤 )^2
ℏ2
Aharonov, Albert, and Vaidman, PRL, 60, 1351 (1988)
Weak Measurement with Stern-Gerlach
The final pointer state in momentum space :
𝜙𝑓 𝑝𝑥 = 〈𝜒𝑓 |𝜒𝑖 〉
2𝜎 2
𝜋ℏ2
1
4
𝜎2 (𝑝𝑥 −𝑔𝜏𝐴𝑤 )^2
−
ℏ2
𝑒
Pre-selected state : |𝜒𝑖 〉 = +𝜃 = 𝐶𝑜𝑠
𝜃
2
𝜃
2
+𝑧 + 𝑆𝑖𝑛 | − 𝑧〉
Post-selected state: 𝜒𝑓 = | + 𝑧〉
Measure A=𝜎𝑥
Weak Value: (𝜎𝑥 )_𝑤 = 𝑇𝑎𝑛
𝜃
2
𝜃
   then  x w   But, prob. of post-selection 〈𝜒𝑓 |𝜒𝑖 |2 = 𝐶𝑜𝑠 2 2 → 0
Experiments with weak measurement
Observation of the Spin Hall Effect
O. Hosten and P. Kwiat Science 319, 787(2008).
Ultrasensitive Beam Deflection
P.B. Dixon, et al., PRL,102,173601 (2009).
Violation of Leggett-Garg inequality
N. S. Willams and A. N. Jordan, PRL, 100, 026804 (2008) .
J. Dressel et al. PRL, 106, 040402(2011).
Probing Quantum paradoxes
J. S. Lundeen and A. M. Steinberg, PRL 102, 020404(2009).
K. Yokota, et al. New J. Phys. 11, 033011(2009).
Direct measurement of the quantum wavefunction
J. S. Lundeen et al. Nature, 474, 188(2011).
Cat State
The cat state is a superposition of two coherent states
where
are essentially shifted Gaussians.
The inner product is crucial:
Sub-Fourier Sensitivity of Cat State
Let the phase be shifted by 𝛿:
The overlap is given by
𝜓𝑐 ′ 𝜓𝑐
where
Given other parameters, the overlap function vanishes for
multiple values of 𝛿, allowing Heisenberg-limited metrology.
Sub-Fourier Sensitivity of Cat State
Sub-Fourier Sensitivity of Cat State
Sub-Planck structure can also be inferred from the Wigner
function
Sub-Fourier Sensitivity of Cat State
Weak measurement & cat-states
Weak value:
Suppose
,
For
, weak value is
Weak value can be arbitrarily large as 𝑎1 is made small.
We now generate the cat state as our final pointer state after a
weak measurement using a Stern-Gerlach apparatus.
Then we will explore the connection with sub-Planck.
Generating cat state with Stern-Gerlach
Initial
state
Interaction Hamiltonian
Generating cat state with Stern-Gerlach
After interaction , final state is
where the components satisfy the Pauli equation
Generating cat state with Stern-Gerlach
After post-selecting in the state
final state is
where
and
The overlap between the two components
Weak interaction requires 𝐼 ≈ 1
But sub-Planck structures require 𝐼 ≈0
Probe state after weak interaction
Taking 𝛼 = 𝑝′𝑥 𝜏/2𝑚 and 𝑝0 = 𝑝𝑥 we rewrite
𝐼 ≈ 1 ensured by small 𝛼 and 𝑝0 .
Expanding to first order in 𝑝0
Weak value is
Thus, after weak interaction, we get as probe state
whose momentum distribution is
Illustrations 𝐼 ≈ 0,
𝑎 = 𝑏 = 1/ 2
• Oscillatory overlap function
• Interference in phase space (zeros
in Wigner function)
• No weak value – strong
measurement condition; two
peaks in probability distribution
Illustrations 𝐼 ≈ 0,
𝑎=
1+𝑖
,𝑏
2
=
1−𝑖
2
• Oscillatory overlap function
• No interference in phase space
• Weak value is imaginary, so
position distribution peaks at weak
value
Illustrations 𝐼 ≈ 1,
𝑎 = 0.866, 𝑏 =-0.5
• No oscillatory overlap
• No interference in phase space
• Real weak value, momentum
distribution peaked at weak value
Conclusions
There is no sub-Planck structure in the limit where
we obtain a weak value
Sub-Planck structures are obtained in the opposite
limit that coincides with the requirement for strong
measurement
Oscillatory overlap function does not guarantee subPlanck structure
Thanks
Collaborators
Dr. Alok K. Pan, Anirban Ch. N. Chowdhury